research article fault feature analysis of a cracked gear...
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Research ArticleFault Feature Analysis of a Cracked Gear Coupled Rotor System
Hui Ma Rongze Song Xu Pang and Bangchun Wen
School of Mechanical Engineering and Automation Northeastern University Shenyang Liaoning 110819 China
Correspondence should be addressed to Hui Ma mahui 2007163com
Received 19 March 2014 Accepted 22 May 2014 Published 24 June 2014
Academic Editor Minping Jia
Copyright copy 2014 Hui Ma et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Considering themisalignment of gear root circle and base circle and accurate transition curve an improvedmesh stiffnessmodel forhealthy gear is proposed and it is validated by comparison with the finite element method On the basis of the improved methoda mesh stiffness model for a cracked gear pair is built Then a finite element model of a cracked gear coupled rotor system in aone-stage reduction gear box is established The effects of crack depth width initial position and crack propagation directionon gear mesh stiffness fault features in time domain and frequency domain and statistical indicators are investigated Moreoverfault features are also validated by experiment The results show that the improved mesh stiffness model is more accurate thanthe traditional mesh stiffness model When the tooth root crack appears distinct impulses are found in time domain vibrationresponses and sidebands appear in frequency domain Amplitudes of all the statistical indicators ascend gradually with the growthof crack depth and width decrease with the increasing crack initial position angle and firstly increase and then decrease with thegrowth of propagation direction angle
1 Introduction
A crack may initiate under the alternative load which willreduce the structural strength and may lead to tooth fracturewith crack propagation Tooth fracture is the most seriousfault in gear box and it may cause complete failure of the gearIf the tooth crack can be detected early and crack propagationcan be monitored a disastrous breakdown can be avoided byreplacing damaged gear The status of the tooth crack can beevaluated by the vibration response of the gear pair or gearcoupled rotor system for condition monitoring purposes
Vibration response of the gear pair is closely related to thetime-varying mesh stiffness (TVMS) of the gear pair Manyresearchers have developedmany analyticalmethods to studythe TVMS of healthy gears [1ndash9] Considering Hertzianenergy bending energy and axial compressive energy Yangand Lin [1] calculated theTVMSof a gear pair by the potentialenergy principle And this model was further refined by Tian[2] andWu [3] by taking the shear energy into considerationAssuming the gear body as a cantilever beam in the halfplane Zhou et al [4] took the effect of the fillet foundationdeflection into account and improved the method to cal-culate the TVMS Based on the corrected tooth foundationdeformation proposed by Sainsot et al [5] Chaari et al [6]
developed a model to calculate the TVMS Considering thatroot circle and base circle aremisaligned exactlyWan et al [7]developed amodifiedmethod to obtain the TVMS accuratelyby judging the number of gear teeth Including the effect ofthe gear tooth errors Chen and Shao [8] proposed a generalanalytical mesh stiffness model Considering the friction andthe possibility of contacts on both flanks Fernandez DelRincon et al [9] presented a procedure for determiningthe TVMS In addition to the analytical methods manyresearchers also applied finite element (FE)model to calculateTVMS [10ndash14]
Investigations on cracked gears have also been carriedout by many researchers A lot of researches focused on thecrack propagation and the effect of the rim thickness on thecrack propagation paths [15ndash18] In order to study the effect ofcrack propagation on the TVMS different models of crackedgears were also developed based on analytical method and FEmethod Based on the potential energy method proposed byYang and Lin [1] Tian [2] discussed the effects of a chippedtooth a cracked tooth and a broken tooth on TVMS byconsidering the shear energy and compared the vibrationacceleration responses of the gear system under different faulttypes On the basis of [2] Wu et al [19] studied the effects oftooth crack growth on the vibration response of a one-stage
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 832192 22 pageshttpdxdoiorg1011552014832192
2 Mathematical Problems in Engineering
gearbox with spur gears by computer simulation and devel-oped an analytical model for calculating the TVMS undera cracked tooth with different crack levels Chaari et al [6]presented an original analytical model of tooth crack quan-tified the gear mesh stiffness reduction due to spur gear toothcrack and verified the results obtained analytically by com-paring with the results of FE model Considering the toothroot crack propagations along both tooth width and crackdepth Chen and Shao [20] proposed an analytical model toinvestigate the effect of gear tooth crack on the TVMS Bythis analytical formulation the mesh stiffness of a spur gearpair with different crack lengths and depths can be obtainedBased on an improved potential energy method Zhou et al[4] developed a modified mathematical model for simulatinggear crack from root with linear growth path in a pinionby considering the deformation of gear body By assuminga parabolic curve instead of a straight line to deal with thetooth thickness reduction Mohammed et al [21] presented anew method to calculate the TVMS of the gear pair for apropagating crack in the tooth root and investigated theinfluence of gear mesh stiffness on the vibration-based faultdetection indicators the RMS Kurtosis and the crest factor
As the above literatures show extensive efforts have beendevoted to study the TVMS of meshing gear pairs with orwithout tooth crack However previous studies mostlyfocused on gears themselves [20ndash22] and in fact gear trans-mission systems are only one part of the whole rotor systemTo study the dynamic characteristics of the whole rotorsystem better the flexibility of shafts needs to be considered[23ndash31] Kahraman et al [23] developed a FE model of ageared rotor system on flexible bearings including rotaryinertia axial loading stiffness and damping of the gearmesh Kubur et al [25] established a dynamic FE model of ahelical gear transmission multi-parallel-shaft rotor systemand analyzed its dynamic characteristics Taking the flexuralrotary and torsional degrees of freedom into account Leeet al [26 27] established a FE model of a turbo-chillerrotor-bearing systemwith a bull-pinion speed increasing gearand analyzed the coupled natural frequencies and unbalanceresponses of this system By using an extended FE model of atest gear-shaft-bearing systemVelex andAjmi [30] comparedthe dynamic results from the formulations based on transmis-sion error with the reference solutions Ma et al [31] estab-lished a FE model of a geared rotor system and investigatedthe effects of tip relief on vibration responses of this systemOmar et al [32] presented a nine degree-of-freedom (DOF)model of a gear transmission system in which the gearboxstructure is coupled with the gear shafts Jia et al [33] pre-sented a dynamic model of three shafts and two pairs of gearsin mesh with 26 DOF including the effects of TVMS pitchand profile errors friction and a localized tooth crack on oneof the gears
In the abovementioned researches the gear tooth is gen-erally modeled as a nonuniform cantilever beam on base cir-cle when the TVMS of spur gear is solved based on the energymethod However root circle and base circle are misalignedexactly so the solution error of TVMS will appear Aiming atthis situation an improved mesh stiffness model is proposedin which the gear tooth is simplified as a cantilever beam on
the root circle In addition the accurate transition curve isalso considered in the improved method This study focuseson the computation of the TVMS of a cracked gear pairbased on the improved mesh stiffness model In addition thedynamic model of cracked gear coupled rotor system is alsoestablished in which the meshing model of the gear pairwith tooth crack is acquired by a lumpedmass model consid-ering TVMS and constant load torque [31] Finally vibrationresponses are obtained by Newmark-120573 method and the timedomain frequency domain and statistical indicators featuresare analyzed under different crack parameters
The structure of the paper is as follows After this intro-duction mesh stiffness calculation of a spur gear pair withcrack is presented in Section 2 An improved mesh stiffnessmodel for a healthy gear pair is established and verified inSections 21 and 22 respectively an improved mesh stiffnessmodel for a cracked gear pair is built in Section 23 on thebasis of Section 23 TVMS for crack propagation along thetooth width is introduced in Section 24 In Section 3 a FEmodel of a cracked gear coupled rotor system is developedVibration responses of the cracked gear coupled rotor systemare analyzed in Section 4 Effects of crack depth width initialposition and propagation direction on the system vibrationresponses are discussed in Sections 41 42 43 and 44respectively Measured vibration responses of the crackedgear coupled rotor system are performed in Section 5 Finallyconclusions are drawn in Section 6
2 Mesh Stiffness Calculation ofa Spur Gear Pair with Crack
21 An ImprovedMesh Stiffness Model for a Healthy Gear PairIn many published literatures the gear tooth is generallymodeled as a nonuniform cantilever beam on base circleHowever root circle and base circle are misaligned exactlyespecially when the number of teeth is larger or smaller than41 In this section an improved mesh stiffness model ispresented in which the misalignment effect between the rootcircle and base circle is considered and the gear tooth issimplified as a cantilever beam on the root circle In additionthe accurate transition curve is also considered in theimproved method
Before tooth profile curve equations are presented somebasic variable symbols have to be introduced 119898 is themodule 119873 is the number of teeth 120572 is the pressure angle ofthe gear pitch circle 119903 119903
119886 and 119903
119887are the radiuses of the pitch
circle addendum circle and base circle of the gear ℎlowast119886is the
addendum coefficient and 119888lowast is the tip clearance coefficientTooth profile curve can be divided into four parts addendumcurve 119860119861 involute curve 119861119862 transition curve 119862119863 anddedendum curve 119863119864 (see Figure 1) Equations of involutecurve are expressed as follows
119909 = 119903119894sin120593
119910 = 119903119894cos120593
120593 =120587
2119873minus (inv120572
119894minus inv120572)
(1)
Mathematical Problems in Engineering 3
F
Oy
x
C
A
B
D
E
Root circle
Base circle
Addendum circle
x
120573
120591
120591
Fa
Fb
xC120579c
120591c
rc
rb
y120573
y
yC
yD
x120573
Figure 1 Geometric model of the gear profile
where 119903119894= 119903
119887 cos120572
119894 120572119894(120572119888le 120572
119894le 120572
119886) is the pressure angle
of the arbitrary point at the involute curve in which 120572119888
=arccos(119903
119887119903119888) and 120572
119886= arccos(119903
119887119903119886) are the pressure angle
of the involute starting point and the addendum circlerespectively 119903
119888= radic(119903
119887tan120572 minus ℎlowast
119886119898 sin120572)2 + 1199032
119887is the radius
of the involute starting point circle inv120572119894is the involute
function inv120572119894= tan120572
119894minus 120572
119894 inv120572 = tan120572 minus 120572
Transition curve is formed in the process ofmachining byrack which is not engaged in meshing but is of greatimportance Its equations are defined as [34]
119909 = 119903 times sin (Φ) minus (1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
119910 = 119903 times cos (Φ) minus (1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
120572 le 120574 le120587
2
(2)
where Φ = (1198861 tan 120574 + 119887
1)119903 119903
120588= 119888lowast119898(1 minus sin120572) 119886
1=
(ℎlowast119886+ 119888lowast) times 119898 minus 119903
120588 and 119887
1= 1205871198984 + ℎlowast
119886119898 tan120572 + 119903
120588cos120572
Based on elastic mechanics the relationships betweenHertzian contact stiffness 119896
ℎ bending stiffness 119896
119887 shear
stiffness 119896119904 axial compressive stiffness 119896
119886 fillet foundation
stiffness 119896119891andHertzian energy119880
ℎ bending energy119880
119887 shear
energy119880119904 axial compressive energy119880
119886 and fillet foundation
energy 119880119891are obtained which are written as [1 2 20]
119880ℎ=
1198652
2119896ℎ
119880119887=
1198652
2119896119887
119880119904=
1198652
2119896119904
119880119886=
1198652
2119896119886
119880119891=
1198652
2119896119891
(3)
where 119865 is the total acting force on the contact teethThe summation of Hertzian bending shear and axial
compressive and fillet foundation energies constitutes the
total potential energy 119880 stored in a single pair of meshingteeth which can be expressed as
119880 =1198652
2119896
= 119880ℎ+ 119880
1198871+ 119880
1199041+ 119880
1198861+ 119880
1198911+ 119880
1198872+ 119880
1199042+ 119880
1198862+ 119880
1198912
=1198652
2(
1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
(4)
where 119896 represents the total effective mesh stiffness of thepair of meshing teeth and subscripts 1 and 2 denote pinionand gear respectively
According to (4) single-tooth-pair mesh stiffness can begiven as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
minus1
(5)
Consequently the total TVMSof the gear pair in themeshcycle with contact ratio between 1 and 2 can be calculated as
11989612
=
119895
sum119894=1
119896119894 (119895 = 1 2) (6)
where 11989612is the total TVMS of the gear pair in themesh cycle
and 119895 denotes thenumber of meshing tooth pair at the sametime
Hertzian contact stiffness and fillet foundation stiffness[6 20] can be found in (7) and (8)
119896ℎ=
120587119864119871
4 (1 minus V2) (7)
where 119864 119871 and V represent Youngrsquos modulus tooth widthand Poissonrsquos ratio respectively
1
119896119891
=cos2120573119864119871
119871lowast(119906119891
119878119891
)
2
+ 119872lowast (119906119891
119878119891
) + 119875lowast (1 + 119876lowasttan2120573)
(8)
where 120573 is the operating pressure angle (see Figure 1) andparameters 119906
119891 119878119891 119871lowast 119872lowast 119875lowast and 119876lowast can be found in [5]
4 Mathematical Problems in Engineering
F
y
xO
F
120573
120573
120575x
120575120575y
Figure 2 FE model of a healthy gear
Based on the beam theory axial compressive bendingand shear energy of gear tooth comprised of involute andtransitional part can be calculated by
119880119886=
1198652
2119896119886
= int119910119862
119910119863
1198652119886
21198641198601199101
d1199101+ int
119910120573
119910119862
1198652119886
21198641198601199102
d1199102
119880119887=
1198652
2119896119887
= int119910119862
119910119863
1198722
1
21198641198681199101
d1199101+ int
119910120573
119910119862
1198722
2
21198641198681199102
d1199102
119880119904=
1198652
2119896119904
= int119910119862
119910119863
121198652119887
21198661198601199101
d1199101+ int
119910120573
119910119862
121198652119887
21198661198601199102
d1199102
(9)
where 119865119887= 119865 cos120573 119865
119886= 119865 sin120573 119909
120573is the distance between
the contact point and the central line of the tooth 119910120573is the
distance between the contact point and original point in thehorizontal direction 119866 = 1198642(1 + V) is the shear modulus1199101 1199102represent the horizontal coordinates of arbitrary point
at transition curve and involute curve 119910119862 119910119863denote the
horizontal coordinates of the starting point and ending pointof the transition curve 119868
1199101 119860
1199101 1198681199102 and 119860
1199102represent
the area moment of inertia and the cross-sectional arearespectively and torques119872
1= 119865
119887(119910120573minus119910
1) minus 119865
119886119909120573and 119872
2=
119865119887(119910120573minus 119910
2) minus 119865
119886119909120573denote the bending effect of 119865
119887and 119865
119886on
the part of transition and involute curvesFor convenience angular displacement is adopted in the
calculation According to the characteristics of involute andtransition curve 119909
120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860
1199101 and 119860
1199102
can be expressed as the functions of angular 120574 120573 or 120591 (see theappendix) and then axial compressive bending and shearstiffness can be written as follows based on (9)
1
119896119886
= int120572
1205872
sin2120573119864119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
sin2120573119864119860
1199102
d1199102
d120591d120591 (10)
1
119896119887
= int120572
1205872
[cos120573 (119910120573minus 119910
1) minus 119909
120573sin120573]
2
1198641198681199101
d1199101
d120574d120574
+ int120573
120591119888
(cos120573 (119910120573minus 119910
2) minus 119909
120573sin120573)
2
1198641198681199102
d1199102
d120591d120591
(11)
1
119896119904
= int120572
1205872
12cos2120573119866119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
12cos2120573119866119860
1199102
d1199102
d120591d120591 (12)
where d1199101d120574 and d119910
2d120591 are shown in the appendix
22 Verification for Improved Mesh Stiffness Model by FEModel FE method is considered as an efficient and reliabletool for solving problems and it is widely applied to calculateTVMS [10ndash14] To reduce the computational time analysis isperformed by a 2Dmodel with only one tooth (see Figure 2)which is widely used and accepted in [6 13] The load isconsidered to be in the plane of the gear body and uniformlydistributed along the tooth width No load is applied in theaxial direction and the stress is negligible Without consid-ering the contact between two gear teeth meshing forces areapplied on the tooth profile under the plane strain assump-tion The inner ring nodes of the gear are coupled withthe master node (the geometric centre of the gear) and themaster node is restrained from all degrees of freedom Subse-quently according to [21] the deformation 120575 in the directionof action line can be acquired by deformations 120575
119909and 120575
119910
which are taken from FE model and given out as
120575 = 120575119909cos120573 + 120575
119910sin120573 (13)
The single-tooth stiffness without considering the contactbetween two teeth is calculated as
119896119905=
119865
120575 (14)
Mathematical Problems in Engineering 5
0 10 20 302
4
6
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(a)
5
4
3
20 4 8 12
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(b)
Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)
G
Q
PB
x
yo
C
D
C
Dq = q1
hqhB
q1
120595υ
(a)
υ
G
Q
B
υ
Px
yo
C
D
C
D
120595
q = q1max + q2
hqq2
hB
hBymax
q1max
(b)
Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902
1max
Considering the influence ofHertzian contact 119896ℎis intro-
duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as
119896 =1
(1119896ℎ) + (1119896
1199051) + (1119896
1199052) (15)
where subscripts 1 and 2 denote the pinion and gear respec-tively
Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM
23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The
Table 1 Parameters of spur gear pairs
Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear
Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ
119886
lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357
geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position
Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
gearbox with spur gears by computer simulation and devel-oped an analytical model for calculating the TVMS undera cracked tooth with different crack levels Chaari et al [6]presented an original analytical model of tooth crack quan-tified the gear mesh stiffness reduction due to spur gear toothcrack and verified the results obtained analytically by com-paring with the results of FE model Considering the toothroot crack propagations along both tooth width and crackdepth Chen and Shao [20] proposed an analytical model toinvestigate the effect of gear tooth crack on the TVMS Bythis analytical formulation the mesh stiffness of a spur gearpair with different crack lengths and depths can be obtainedBased on an improved potential energy method Zhou et al[4] developed a modified mathematical model for simulatinggear crack from root with linear growth path in a pinionby considering the deformation of gear body By assuminga parabolic curve instead of a straight line to deal with thetooth thickness reduction Mohammed et al [21] presented anew method to calculate the TVMS of the gear pair for apropagating crack in the tooth root and investigated theinfluence of gear mesh stiffness on the vibration-based faultdetection indicators the RMS Kurtosis and the crest factor
As the above literatures show extensive efforts have beendevoted to study the TVMS of meshing gear pairs with orwithout tooth crack However previous studies mostlyfocused on gears themselves [20ndash22] and in fact gear trans-mission systems are only one part of the whole rotor systemTo study the dynamic characteristics of the whole rotorsystem better the flexibility of shafts needs to be considered[23ndash31] Kahraman et al [23] developed a FE model of ageared rotor system on flexible bearings including rotaryinertia axial loading stiffness and damping of the gearmesh Kubur et al [25] established a dynamic FE model of ahelical gear transmission multi-parallel-shaft rotor systemand analyzed its dynamic characteristics Taking the flexuralrotary and torsional degrees of freedom into account Leeet al [26 27] established a FE model of a turbo-chillerrotor-bearing systemwith a bull-pinion speed increasing gearand analyzed the coupled natural frequencies and unbalanceresponses of this system By using an extended FE model of atest gear-shaft-bearing systemVelex andAjmi [30] comparedthe dynamic results from the formulations based on transmis-sion error with the reference solutions Ma et al [31] estab-lished a FE model of a geared rotor system and investigatedthe effects of tip relief on vibration responses of this systemOmar et al [32] presented a nine degree-of-freedom (DOF)model of a gear transmission system in which the gearboxstructure is coupled with the gear shafts Jia et al [33] pre-sented a dynamic model of three shafts and two pairs of gearsin mesh with 26 DOF including the effects of TVMS pitchand profile errors friction and a localized tooth crack on oneof the gears
In the abovementioned researches the gear tooth is gen-erally modeled as a nonuniform cantilever beam on base cir-cle when the TVMS of spur gear is solved based on the energymethod However root circle and base circle are misalignedexactly so the solution error of TVMS will appear Aiming atthis situation an improved mesh stiffness model is proposedin which the gear tooth is simplified as a cantilever beam on
the root circle In addition the accurate transition curve isalso considered in the improved method This study focuseson the computation of the TVMS of a cracked gear pairbased on the improved mesh stiffness model In addition thedynamic model of cracked gear coupled rotor system is alsoestablished in which the meshing model of the gear pairwith tooth crack is acquired by a lumpedmass model consid-ering TVMS and constant load torque [31] Finally vibrationresponses are obtained by Newmark-120573 method and the timedomain frequency domain and statistical indicators featuresare analyzed under different crack parameters
The structure of the paper is as follows After this intro-duction mesh stiffness calculation of a spur gear pair withcrack is presented in Section 2 An improved mesh stiffnessmodel for a healthy gear pair is established and verified inSections 21 and 22 respectively an improved mesh stiffnessmodel for a cracked gear pair is built in Section 23 on thebasis of Section 23 TVMS for crack propagation along thetooth width is introduced in Section 24 In Section 3 a FEmodel of a cracked gear coupled rotor system is developedVibration responses of the cracked gear coupled rotor systemare analyzed in Section 4 Effects of crack depth width initialposition and propagation direction on the system vibrationresponses are discussed in Sections 41 42 43 and 44respectively Measured vibration responses of the crackedgear coupled rotor system are performed in Section 5 Finallyconclusions are drawn in Section 6
2 Mesh Stiffness Calculation ofa Spur Gear Pair with Crack
21 An ImprovedMesh Stiffness Model for a Healthy Gear PairIn many published literatures the gear tooth is generallymodeled as a nonuniform cantilever beam on base circleHowever root circle and base circle are misaligned exactlyespecially when the number of teeth is larger or smaller than41 In this section an improved mesh stiffness model ispresented in which the misalignment effect between the rootcircle and base circle is considered and the gear tooth issimplified as a cantilever beam on the root circle In additionthe accurate transition curve is also considered in theimproved method
Before tooth profile curve equations are presented somebasic variable symbols have to be introduced 119898 is themodule 119873 is the number of teeth 120572 is the pressure angle ofthe gear pitch circle 119903 119903
119886 and 119903
119887are the radiuses of the pitch
circle addendum circle and base circle of the gear ℎlowast119886is the
addendum coefficient and 119888lowast is the tip clearance coefficientTooth profile curve can be divided into four parts addendumcurve 119860119861 involute curve 119861119862 transition curve 119862119863 anddedendum curve 119863119864 (see Figure 1) Equations of involutecurve are expressed as follows
119909 = 119903119894sin120593
119910 = 119903119894cos120593
120593 =120587
2119873minus (inv120572
119894minus inv120572)
(1)
Mathematical Problems in Engineering 3
F
Oy
x
C
A
B
D
E
Root circle
Base circle
Addendum circle
x
120573
120591
120591
Fa
Fb
xC120579c
120591c
rc
rb
y120573
y
yC
yD
x120573
Figure 1 Geometric model of the gear profile
where 119903119894= 119903
119887 cos120572
119894 120572119894(120572119888le 120572
119894le 120572
119886) is the pressure angle
of the arbitrary point at the involute curve in which 120572119888
=arccos(119903
119887119903119888) and 120572
119886= arccos(119903
119887119903119886) are the pressure angle
of the involute starting point and the addendum circlerespectively 119903
119888= radic(119903
119887tan120572 minus ℎlowast
119886119898 sin120572)2 + 1199032
119887is the radius
of the involute starting point circle inv120572119894is the involute
function inv120572119894= tan120572
119894minus 120572
119894 inv120572 = tan120572 minus 120572
Transition curve is formed in the process ofmachining byrack which is not engaged in meshing but is of greatimportance Its equations are defined as [34]
119909 = 119903 times sin (Φ) minus (1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
119910 = 119903 times cos (Φ) minus (1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
120572 le 120574 le120587
2
(2)
where Φ = (1198861 tan 120574 + 119887
1)119903 119903
120588= 119888lowast119898(1 minus sin120572) 119886
1=
(ℎlowast119886+ 119888lowast) times 119898 minus 119903
120588 and 119887
1= 1205871198984 + ℎlowast
119886119898 tan120572 + 119903
120588cos120572
Based on elastic mechanics the relationships betweenHertzian contact stiffness 119896
ℎ bending stiffness 119896
119887 shear
stiffness 119896119904 axial compressive stiffness 119896
119886 fillet foundation
stiffness 119896119891andHertzian energy119880
ℎ bending energy119880
119887 shear
energy119880119904 axial compressive energy119880
119886 and fillet foundation
energy 119880119891are obtained which are written as [1 2 20]
119880ℎ=
1198652
2119896ℎ
119880119887=
1198652
2119896119887
119880119904=
1198652
2119896119904
119880119886=
1198652
2119896119886
119880119891=
1198652
2119896119891
(3)
where 119865 is the total acting force on the contact teethThe summation of Hertzian bending shear and axial
compressive and fillet foundation energies constitutes the
total potential energy 119880 stored in a single pair of meshingteeth which can be expressed as
119880 =1198652
2119896
= 119880ℎ+ 119880
1198871+ 119880
1199041+ 119880
1198861+ 119880
1198911+ 119880
1198872+ 119880
1199042+ 119880
1198862+ 119880
1198912
=1198652
2(
1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
(4)
where 119896 represents the total effective mesh stiffness of thepair of meshing teeth and subscripts 1 and 2 denote pinionand gear respectively
According to (4) single-tooth-pair mesh stiffness can begiven as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
minus1
(5)
Consequently the total TVMSof the gear pair in themeshcycle with contact ratio between 1 and 2 can be calculated as
11989612
=
119895
sum119894=1
119896119894 (119895 = 1 2) (6)
where 11989612is the total TVMS of the gear pair in themesh cycle
and 119895 denotes thenumber of meshing tooth pair at the sametime
Hertzian contact stiffness and fillet foundation stiffness[6 20] can be found in (7) and (8)
119896ℎ=
120587119864119871
4 (1 minus V2) (7)
where 119864 119871 and V represent Youngrsquos modulus tooth widthand Poissonrsquos ratio respectively
1
119896119891
=cos2120573119864119871
119871lowast(119906119891
119878119891
)
2
+ 119872lowast (119906119891
119878119891
) + 119875lowast (1 + 119876lowasttan2120573)
(8)
where 120573 is the operating pressure angle (see Figure 1) andparameters 119906
119891 119878119891 119871lowast 119872lowast 119875lowast and 119876lowast can be found in [5]
4 Mathematical Problems in Engineering
F
y
xO
F
120573
120573
120575x
120575120575y
Figure 2 FE model of a healthy gear
Based on the beam theory axial compressive bendingand shear energy of gear tooth comprised of involute andtransitional part can be calculated by
119880119886=
1198652
2119896119886
= int119910119862
119910119863
1198652119886
21198641198601199101
d1199101+ int
119910120573
119910119862
1198652119886
21198641198601199102
d1199102
119880119887=
1198652
2119896119887
= int119910119862
119910119863
1198722
1
21198641198681199101
d1199101+ int
119910120573
119910119862
1198722
2
21198641198681199102
d1199102
119880119904=
1198652
2119896119904
= int119910119862
119910119863
121198652119887
21198661198601199101
d1199101+ int
119910120573
119910119862
121198652119887
21198661198601199102
d1199102
(9)
where 119865119887= 119865 cos120573 119865
119886= 119865 sin120573 119909
120573is the distance between
the contact point and the central line of the tooth 119910120573is the
distance between the contact point and original point in thehorizontal direction 119866 = 1198642(1 + V) is the shear modulus1199101 1199102represent the horizontal coordinates of arbitrary point
at transition curve and involute curve 119910119862 119910119863denote the
horizontal coordinates of the starting point and ending pointof the transition curve 119868
1199101 119860
1199101 1198681199102 and 119860
1199102represent
the area moment of inertia and the cross-sectional arearespectively and torques119872
1= 119865
119887(119910120573minus119910
1) minus 119865
119886119909120573and 119872
2=
119865119887(119910120573minus 119910
2) minus 119865
119886119909120573denote the bending effect of 119865
119887and 119865
119886on
the part of transition and involute curvesFor convenience angular displacement is adopted in the
calculation According to the characteristics of involute andtransition curve 119909
120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860
1199101 and 119860
1199102
can be expressed as the functions of angular 120574 120573 or 120591 (see theappendix) and then axial compressive bending and shearstiffness can be written as follows based on (9)
1
119896119886
= int120572
1205872
sin2120573119864119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
sin2120573119864119860
1199102
d1199102
d120591d120591 (10)
1
119896119887
= int120572
1205872
[cos120573 (119910120573minus 119910
1) minus 119909
120573sin120573]
2
1198641198681199101
d1199101
d120574d120574
+ int120573
120591119888
(cos120573 (119910120573minus 119910
2) minus 119909
120573sin120573)
2
1198641198681199102
d1199102
d120591d120591
(11)
1
119896119904
= int120572
1205872
12cos2120573119866119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
12cos2120573119866119860
1199102
d1199102
d120591d120591 (12)
where d1199101d120574 and d119910
2d120591 are shown in the appendix
22 Verification for Improved Mesh Stiffness Model by FEModel FE method is considered as an efficient and reliabletool for solving problems and it is widely applied to calculateTVMS [10ndash14] To reduce the computational time analysis isperformed by a 2Dmodel with only one tooth (see Figure 2)which is widely used and accepted in [6 13] The load isconsidered to be in the plane of the gear body and uniformlydistributed along the tooth width No load is applied in theaxial direction and the stress is negligible Without consid-ering the contact between two gear teeth meshing forces areapplied on the tooth profile under the plane strain assump-tion The inner ring nodes of the gear are coupled withthe master node (the geometric centre of the gear) and themaster node is restrained from all degrees of freedom Subse-quently according to [21] the deformation 120575 in the directionof action line can be acquired by deformations 120575
119909and 120575
119910
which are taken from FE model and given out as
120575 = 120575119909cos120573 + 120575
119910sin120573 (13)
The single-tooth stiffness without considering the contactbetween two teeth is calculated as
119896119905=
119865
120575 (14)
Mathematical Problems in Engineering 5
0 10 20 302
4
6
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(a)
5
4
3
20 4 8 12
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(b)
Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)
G
Q
PB
x
yo
C
D
C
Dq = q1
hqhB
q1
120595υ
(a)
υ
G
Q
B
υ
Px
yo
C
D
C
D
120595
q = q1max + q2
hqq2
hB
hBymax
q1max
(b)
Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902
1max
Considering the influence ofHertzian contact 119896ℎis intro-
duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as
119896 =1
(1119896ℎ) + (1119896
1199051) + (1119896
1199052) (15)
where subscripts 1 and 2 denote the pinion and gear respec-tively
Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM
23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The
Table 1 Parameters of spur gear pairs
Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear
Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ
119886
lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357
geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position
Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
F
Oy
x
C
A
B
D
E
Root circle
Base circle
Addendum circle
x
120573
120591
120591
Fa
Fb
xC120579c
120591c
rc
rb
y120573
y
yC
yD
x120573
Figure 1 Geometric model of the gear profile
where 119903119894= 119903
119887 cos120572
119894 120572119894(120572119888le 120572
119894le 120572
119886) is the pressure angle
of the arbitrary point at the involute curve in which 120572119888
=arccos(119903
119887119903119888) and 120572
119886= arccos(119903
119887119903119886) are the pressure angle
of the involute starting point and the addendum circlerespectively 119903
119888= radic(119903
119887tan120572 minus ℎlowast
119886119898 sin120572)2 + 1199032
119887is the radius
of the involute starting point circle inv120572119894is the involute
function inv120572119894= tan120572
119894minus 120572
119894 inv120572 = tan120572 minus 120572
Transition curve is formed in the process ofmachining byrack which is not engaged in meshing but is of greatimportance Its equations are defined as [34]
119909 = 119903 times sin (Φ) minus (1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
119910 = 119903 times cos (Φ) minus (1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
120572 le 120574 le120587
2
(2)
where Φ = (1198861 tan 120574 + 119887
1)119903 119903
120588= 119888lowast119898(1 minus sin120572) 119886
1=
(ℎlowast119886+ 119888lowast) times 119898 minus 119903
120588 and 119887
1= 1205871198984 + ℎlowast
119886119898 tan120572 + 119903
120588cos120572
Based on elastic mechanics the relationships betweenHertzian contact stiffness 119896
ℎ bending stiffness 119896
119887 shear
stiffness 119896119904 axial compressive stiffness 119896
119886 fillet foundation
stiffness 119896119891andHertzian energy119880
ℎ bending energy119880
119887 shear
energy119880119904 axial compressive energy119880
119886 and fillet foundation
energy 119880119891are obtained which are written as [1 2 20]
119880ℎ=
1198652
2119896ℎ
119880119887=
1198652
2119896119887
119880119904=
1198652
2119896119904
119880119886=
1198652
2119896119886
119880119891=
1198652
2119896119891
(3)
where 119865 is the total acting force on the contact teethThe summation of Hertzian bending shear and axial
compressive and fillet foundation energies constitutes the
total potential energy 119880 stored in a single pair of meshingteeth which can be expressed as
119880 =1198652
2119896
= 119880ℎ+ 119880
1198871+ 119880
1199041+ 119880
1198861+ 119880
1198911+ 119880
1198872+ 119880
1199042+ 119880
1198862+ 119880
1198912
=1198652
2(
1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
(4)
where 119896 represents the total effective mesh stiffness of thepair of meshing teeth and subscripts 1 and 2 denote pinionand gear respectively
According to (4) single-tooth-pair mesh stiffness can begiven as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
1198961198872
+1
1198961199042
+1
1198961198862
+1
1198961198912
)
minus1
(5)
Consequently the total TVMSof the gear pair in themeshcycle with contact ratio between 1 and 2 can be calculated as
11989612
=
119895
sum119894=1
119896119894 (119895 = 1 2) (6)
where 11989612is the total TVMS of the gear pair in themesh cycle
and 119895 denotes thenumber of meshing tooth pair at the sametime
Hertzian contact stiffness and fillet foundation stiffness[6 20] can be found in (7) and (8)
119896ℎ=
120587119864119871
4 (1 minus V2) (7)
where 119864 119871 and V represent Youngrsquos modulus tooth widthand Poissonrsquos ratio respectively
1
119896119891
=cos2120573119864119871
119871lowast(119906119891
119878119891
)
2
+ 119872lowast (119906119891
119878119891
) + 119875lowast (1 + 119876lowasttan2120573)
(8)
where 120573 is the operating pressure angle (see Figure 1) andparameters 119906
119891 119878119891 119871lowast 119872lowast 119875lowast and 119876lowast can be found in [5]
4 Mathematical Problems in Engineering
F
y
xO
F
120573
120573
120575x
120575120575y
Figure 2 FE model of a healthy gear
Based on the beam theory axial compressive bendingand shear energy of gear tooth comprised of involute andtransitional part can be calculated by
119880119886=
1198652
2119896119886
= int119910119862
119910119863
1198652119886
21198641198601199101
d1199101+ int
119910120573
119910119862
1198652119886
21198641198601199102
d1199102
119880119887=
1198652
2119896119887
= int119910119862
119910119863
1198722
1
21198641198681199101
d1199101+ int
119910120573
119910119862
1198722
2
21198641198681199102
d1199102
119880119904=
1198652
2119896119904
= int119910119862
119910119863
121198652119887
21198661198601199101
d1199101+ int
119910120573
119910119862
121198652119887
21198661198601199102
d1199102
(9)
where 119865119887= 119865 cos120573 119865
119886= 119865 sin120573 119909
120573is the distance between
the contact point and the central line of the tooth 119910120573is the
distance between the contact point and original point in thehorizontal direction 119866 = 1198642(1 + V) is the shear modulus1199101 1199102represent the horizontal coordinates of arbitrary point
at transition curve and involute curve 119910119862 119910119863denote the
horizontal coordinates of the starting point and ending pointof the transition curve 119868
1199101 119860
1199101 1198681199102 and 119860
1199102represent
the area moment of inertia and the cross-sectional arearespectively and torques119872
1= 119865
119887(119910120573minus119910
1) minus 119865
119886119909120573and 119872
2=
119865119887(119910120573minus 119910
2) minus 119865
119886119909120573denote the bending effect of 119865
119887and 119865
119886on
the part of transition and involute curvesFor convenience angular displacement is adopted in the
calculation According to the characteristics of involute andtransition curve 119909
120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860
1199101 and 119860
1199102
can be expressed as the functions of angular 120574 120573 or 120591 (see theappendix) and then axial compressive bending and shearstiffness can be written as follows based on (9)
1
119896119886
= int120572
1205872
sin2120573119864119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
sin2120573119864119860
1199102
d1199102
d120591d120591 (10)
1
119896119887
= int120572
1205872
[cos120573 (119910120573minus 119910
1) minus 119909
120573sin120573]
2
1198641198681199101
d1199101
d120574d120574
+ int120573
120591119888
(cos120573 (119910120573minus 119910
2) minus 119909
120573sin120573)
2
1198641198681199102
d1199102
d120591d120591
(11)
1
119896119904
= int120572
1205872
12cos2120573119866119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
12cos2120573119866119860
1199102
d1199102
d120591d120591 (12)
where d1199101d120574 and d119910
2d120591 are shown in the appendix
22 Verification for Improved Mesh Stiffness Model by FEModel FE method is considered as an efficient and reliabletool for solving problems and it is widely applied to calculateTVMS [10ndash14] To reduce the computational time analysis isperformed by a 2Dmodel with only one tooth (see Figure 2)which is widely used and accepted in [6 13] The load isconsidered to be in the plane of the gear body and uniformlydistributed along the tooth width No load is applied in theaxial direction and the stress is negligible Without consid-ering the contact between two gear teeth meshing forces areapplied on the tooth profile under the plane strain assump-tion The inner ring nodes of the gear are coupled withthe master node (the geometric centre of the gear) and themaster node is restrained from all degrees of freedom Subse-quently according to [21] the deformation 120575 in the directionof action line can be acquired by deformations 120575
119909and 120575
119910
which are taken from FE model and given out as
120575 = 120575119909cos120573 + 120575
119910sin120573 (13)
The single-tooth stiffness without considering the contactbetween two teeth is calculated as
119896119905=
119865
120575 (14)
Mathematical Problems in Engineering 5
0 10 20 302
4
6
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(a)
5
4
3
20 4 8 12
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(b)
Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)
G
Q
PB
x
yo
C
D
C
Dq = q1
hqhB
q1
120595υ
(a)
υ
G
Q
B
υ
Px
yo
C
D
C
D
120595
q = q1max + q2
hqq2
hB
hBymax
q1max
(b)
Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902
1max
Considering the influence ofHertzian contact 119896ℎis intro-
duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as
119896 =1
(1119896ℎ) + (1119896
1199051) + (1119896
1199052) (15)
where subscripts 1 and 2 denote the pinion and gear respec-tively
Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM
23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The
Table 1 Parameters of spur gear pairs
Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear
Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ
119886
lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357
geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position
Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
F
y
xO
F
120573
120573
120575x
120575120575y
Figure 2 FE model of a healthy gear
Based on the beam theory axial compressive bendingand shear energy of gear tooth comprised of involute andtransitional part can be calculated by
119880119886=
1198652
2119896119886
= int119910119862
119910119863
1198652119886
21198641198601199101
d1199101+ int
119910120573
119910119862
1198652119886
21198641198601199102
d1199102
119880119887=
1198652
2119896119887
= int119910119862
119910119863
1198722
1
21198641198681199101
d1199101+ int
119910120573
119910119862
1198722
2
21198641198681199102
d1199102
119880119904=
1198652
2119896119904
= int119910119862
119910119863
121198652119887
21198661198601199101
d1199101+ int
119910120573
119910119862
121198652119887
21198661198601199102
d1199102
(9)
where 119865119887= 119865 cos120573 119865
119886= 119865 sin120573 119909
120573is the distance between
the contact point and the central line of the tooth 119910120573is the
distance between the contact point and original point in thehorizontal direction 119866 = 1198642(1 + V) is the shear modulus1199101 1199102represent the horizontal coordinates of arbitrary point
at transition curve and involute curve 119910119862 119910119863denote the
horizontal coordinates of the starting point and ending pointof the transition curve 119868
1199101 119860
1199101 1198681199102 and 119860
1199102represent
the area moment of inertia and the cross-sectional arearespectively and torques119872
1= 119865
119887(119910120573minus119910
1) minus 119865
119886119909120573and 119872
2=
119865119887(119910120573minus 119910
2) minus 119865
119886119909120573denote the bending effect of 119865
119887and 119865
119886on
the part of transition and involute curvesFor convenience angular displacement is adopted in the
calculation According to the characteristics of involute andtransition curve 119909
120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 119860
1199101 and 119860
1199102
can be expressed as the functions of angular 120574 120573 or 120591 (see theappendix) and then axial compressive bending and shearstiffness can be written as follows based on (9)
1
119896119886
= int120572
1205872
sin2120573119864119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
sin2120573119864119860
1199102
d1199102
d120591d120591 (10)
1
119896119887
= int120572
1205872
[cos120573 (119910120573minus 119910
1) minus 119909
120573sin120573]
2
1198641198681199101
d1199101
d120574d120574
+ int120573
120591119888
(cos120573 (119910120573minus 119910
2) minus 119909
120573sin120573)
2
1198641198681199102
d1199102
d120591d120591
(11)
1
119896119904
= int120572
1205872
12cos2120573119866119860
1199101
d1199101
d120574d120574 + int
120573
120591119888
12cos2120573119866119860
1199102
d1199102
d120591d120591 (12)
where d1199101d120574 and d119910
2d120591 are shown in the appendix
22 Verification for Improved Mesh Stiffness Model by FEModel FE method is considered as an efficient and reliabletool for solving problems and it is widely applied to calculateTVMS [10ndash14] To reduce the computational time analysis isperformed by a 2Dmodel with only one tooth (see Figure 2)which is widely used and accepted in [6 13] The load isconsidered to be in the plane of the gear body and uniformlydistributed along the tooth width No load is applied in theaxial direction and the stress is negligible Without consid-ering the contact between two gear teeth meshing forces areapplied on the tooth profile under the plane strain assump-tion The inner ring nodes of the gear are coupled withthe master node (the geometric centre of the gear) and themaster node is restrained from all degrees of freedom Subse-quently according to [21] the deformation 120575 in the directionof action line can be acquired by deformations 120575
119909and 120575
119910
which are taken from FE model and given out as
120575 = 120575119909cos120573 + 120575
119910sin120573 (13)
The single-tooth stiffness without considering the contactbetween two teeth is calculated as
119896119905=
119865
120575 (14)
Mathematical Problems in Engineering 5
0 10 20 302
4
6
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(a)
5
4
3
20 4 8 12
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(b)
Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)
G
Q
PB
x
yo
C
D
C
Dq = q1
hqhB
q1
120595υ
(a)
υ
G
Q
B
υ
Px
yo
C
D
C
D
120595
q = q1max + q2
hqq2
hB
hBymax
q1max
(b)
Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902
1max
Considering the influence ofHertzian contact 119896ℎis intro-
duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as
119896 =1
(1119896ℎ) + (1119896
1199051) + (1119896
1199052) (15)
where subscripts 1 and 2 denote the pinion and gear respec-tively
Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM
23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The
Table 1 Parameters of spur gear pairs
Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear
Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ
119886
lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357
geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position
Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
0 10 20 302
4
6
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(a)
5
4
3
20 4 8 12
TMIM
FEM
Angular position (∘)
TVM
S (times10
8N
m)
(b)
Figure 3 TVMS based on different methods (a) gear pair 1 (b) gear pair 2 (TM stands for traditional method IM improved method andFEM finite element method)
G
Q
PB
x
yo
C
D
C
Dq = q1
hqhB
q1
120595υ
(a)
υ
G
Q
B
υ
Px
yo
C
D
C
D
120595
q = q1max + q2
hqq2
hB
hBymax
q1max
(b)
Figure 4 Schematic of crack propagation along the depth direction (a) 119902 le 1199021max (b) 119902 gt 119902
1max
Considering the influence ofHertzian contact 119896ℎis intro-
duced as is shown in (7) and the single-tooth-pair meshstiffness can be given as
119896 =1
(1119896ℎ) + (1119896
1199051) + (1119896
1199052) (15)
where subscripts 1 and 2 denote the pinion and gear respec-tively
Two gear pairs are selected to verify the validity of theimproved mesh stiffness model and their parameters arelisted in Table 1 TVMS based on traditional method (TM) inwhich the gear tooth is modeled as a nonuniform cantileverbeam on the base circle improved method (IM) in which thegear tooth is simplified as a cantilever beam on the root circleand FE method (FEM) with precise profile curve are dis-played in Figure 3 Obviously the results from IM and FEMshow good agreement while TM shows great difference fromFEMThus it can be seen that IM ismore accurate to calculatethe TVMS than TM compared with the result of FEM
23 An Improved Mesh Stiffness Model for a Cracked GearPair In order to simplify the crack model the crack path isassumed to be a straight The crack starts at the root of thedriven gear and then propagates as is shown in Figure 4The
Table 1 Parameters of spur gear pairs
Parameters Gear pair 1 Gear pair 2Pinion Gear Pinion Gear
Number of teeth N 22 22 62 62Youngrsquos modulus E (GPa) 206 206 206 206Poissonrsquos ratio v 03 03 03 03Modulem (mm) 3 3 3 3Addendum coefficient ℎ
119886
lowast 1 1 1 1Tip clearance coefficient clowast 025 025 025 025Face width L (mm) 20 20 20 20Pressure angle 120572 (∘) 20 20 20 20Hub bore radius rint (mm) 117 117 357 357
geometrical parameters of the crack (119902 120592 120595) are presentedin Figure 4 in which 119902 denotes the crack depth 120592 the crackpropagation direction and 120595 the crack initial position
Assume that the crack extends along the whole toothwidth with a uniform crack depth distribution and the toothwith this crack is still considered as a cantilevered beamThecurve of the tooth profile remains perfect and the foundationstiffness is not affected In this case based on (7) theHertzian
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
contact stiffness will still be a constant since the work surfaceof the tooth has no defect and the width of the effective worksurface will always be a constant 119871 For the axial compressivestiffness it will be considered the same with that under theperfect condition in that the crack part can still bear theaxial compressive force as if no crack exists Therefore forthe gear with a cracked tooth Hertzian fillet foundation andaxial compressive stiffness can still be calculated according to(7) (8) and (10) respectively However the bending andshear stiffnesses will change due to the influence of the crackTherefore based on (5) single-tooth-pair mesh stiffness for acracked gear pair can be calculated as
119896 = 1 times (1
119896ℎ
+1
1198961198871
+1
1198961199041
+1
1198961198861
+1
1198961198911
+1
119896119887 crack
+1
119896119904 crack
+1
1198961198862
+1
1198961198912
)
minus1
(16)
where 119896119887 crack and 119896
119904 crack denote bending and shear stiffnesseswhen a crack is introduced and subscripts 1 and 2 representpinion and gear respectively
When the tooth crack is present and 119902 le 1199021max (see
Figure 4(a)) the effective area moment of inertia and area ofthe cross section at the position of 119910 can be calculated as
119868 =
1
12(ℎ119902+ 119909)
3
119871 119910119875le 119910 le 119910
119866
1
12(2119909)3119871 =
2
31199093119871 119910 lt 119910
119875or 119910 gt 119910
119866
119860 =
(ℎ119902+ 119909) 119871 119910
119875le 119910 le 119910
119866
2119909119871 119910 lt 119910119875or 119910 gt 119910
119866
(17)
where ℎ119902is the distance from the root of the crack to the
central line of the tooth which corresponds to point119866 on thetooth profile and ℎ
119902can be calculated by
ℎ119902= 119909
119876minus 119902
1sin 120592 (18)
When the crack propagates to the central line of the tooth1199021reaches its maximum value 119902
1max = 119909119876 sin 120592 Then the
crack will change direction to 1199022 which is assumed to be
exactly symmetricwith 1199021 and in this stage 119902 = 119902
1max+1199022 (seeFigure 4(b)) In theory maximum value of 119902
2is 119902
1max how-ever the tooth is expected to suffer sudden breakage beforecrack runs through the whole tooth therefore the maximumvalue of 119902
2is smaller than 119902
1max When the crack propagates
along 1199022 the effective area moment of inertia and area of the
cross section at the position of 119910 can be calculated as
119868 =
2
31199093119871 119910 lt 119910max
1
12[119909 minus
(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]
3
119871 119910max le 119910 lt 119910119875
1
12(119909 minus ℎ
119902)3
119871 119910119875le 119910 le 119910
119866
0 119910 gt 119910119866
119860 =
2119909119871 119910 lt 119910max
[119909 minus(119910 minus 119910max) ℎ119902
119910119875minus 119910max
]119871 119910max le 119910 lt 119910119875
(119909 minus ℎ119902) 119871 119910
119875le 119910 le 119910
119866
0 119910 gt 119910119866
(19)
where 119910max denotes the leftmost position of crack propaga-tion (shown in Figure 4(b)) and ℎ
119902= 119902
2sin 120592
Based on (11) and (12) the bending and shear stiffness ofthe cracked gear pair can be calculated The schematic of thecalculation process is presented in Figure 5
24 TVMS for Crack Propagation along the ToothWidth Themesh stiffness model proposed in [20] divided the tooth intosome independent thin slices to represent the crack propa-gation along the tooth width as is shown in Figure 6 Whend119911 is small the crack depth can be assumed to be a constantthrough the width for each slice and stiffness of each slicedenoted as 119896(119911) can be calculated by (16) Then the stiffnessof the whole tooth can be obtained by integration of 119896(119911)along the tooth width
119896 = int119871
0
119896 (119911) (20)
Assume that the crack propagation is in the plane (seeA-A in Figure 6) The crack depth along tooth width can bedescribed as a function of 119911 in the coordinate system 11991010158401199001015840119911which can be written as
119902 (119911) = 119891 (119911) (21)The distribution of the crack depth is assumed to be a
parabolic function along the tooth width to investigate theinfluence of crack width on TVMS [20] When the crackwidth is less than the whole tooth width (see the solid curvein Figure 7)
119902 (119911) = 1199020radic
119911 + 119871119888minus 119871
119871119888
119911 isin [119871 minus 119871119888 119871]
119902 (119911) = 0 119911 isin [0 119871 minus 119871119888]
(22)
where 119871119888is the crack width and 119902
0is the crack depth on crack
starting surfaceWhen the crack width extends through the whole tooth
width (see the dashed curve in Figure 7)
119902 (119911) = radic 11990220minus 1199022
119890
119871119911 + 1199022
119890 (23)
where 119902119890is the crack depth on crack ending surface
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Involute
Involute
Yes
Yes
Yes
No
No
No
No
Yes
Transition curve
Involute
Involute
Yes
No
No
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
Yes
No
No
No
Yes
No
Yes
No
Yes
No
No
Yes
Crack parameters
Transition curve
Transition curve
Transition curve
(q 120592 120595)
q le q1max
yP le y le yG y lt yPyG le yB
y120573 le yG
y le yG
y lt ymax
ymax le y le yP
yP le y le yG
yG le yB
y120573 le yG
y le yG
B2 =2
3x3L
B3 =1
12[x minus
(y minus ymax)hq
yP minus ymax]3
L
B4 =1
12(x minus hq)
3L
C1 = (hq + x)L
C2 = 2xL
C3 = [x minus(y minus ymax)hq
yP minus ymax]L
C4 = (x minus hq)L
ymax le y lt yP
yltymax
Equations (14) and (15)
kb crack ks crack
G isin CD
G isin CD
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy1 = 0
Ay1 = 0
Iy2 = 0
Ay2 = 0
Iy1 = B2
Ay1 = C2
Iy1 = B3
Ay1 = C3
Iy1 = B4
Ay1 = C4
Iy2 = 0
Ay2 = 0Iy2 = B4
Ay2 = C4
Iy1 = B2
Ay1 = C2
Iy1 = B1
Ay1 = C1
Iy1 = B1
Ay1 = C1
Iy1 = B2Ay1 = C2
Iy1 = B2
Ay1 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B2
Ay2 = C2
Iy2 = B1
Ay1 = C1
B1 =1
12(hq + x)3L
Figure 5 Schematic to obtain bending and shear stiffness for a cracked tooth
3 FE Model of a Cracked GearCoupled Rotor System
In order to research on fault features of a cracked gear itis necessary to consider the dynamic characteristics of thegeared rotor system which owns realistic significance Basedon [31] a geared rotor system model will be established
Dynamic model of a spur gear pair formed by gears 1 and2 is shown in Figure 8 where the typical spur gear meshingis represented by a pair of rigid disks connected by a spring-damper set along the plane of action which is tangent to thebase circles of the gears119874
1and119874
2are their geometrical cen-
tersΩ1andΩ
2are the rotating speeds and 119903
1198871and 119903
1198872are the
radiuses of the gear base circles respectively 11989012(119905) denotes
nonloaded static transmission error which is formed solelyby the geometric deviation component Assuming that thespur gear pair studied in the paper is perfect nonloaded statictransmission error will be zero that is 119890
12(119905) = 0 119888
12(119905) rep-
resents the mesh damping which is assumed to be zero Theangle between the plane of action and the positive 119910-axis isrepresented by 120595
12defined as
12059512
= minus120572 + 120572
12Ω1 Counterclosewise
120572 + 12057212
minus 120587 Ω1 Clockwise
(24)
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
A
Az
L
z
q0
120592
y998400
x998400
o998400
dz
(a)
q(z)
k(z)
120592
dz
(b)
z
L
Lc
q(z)q0
z
f(z)
y998400
o998400
A-A
dz
(c)
Figure 6 Crack model at gear tooth root
qe
y998400
o998400
L
z
q0q(z)
Lc
Figure 7 Crack depth along tooth width
where 12057212
(0 le 12057212
le 2120587) represents the angle between theline connecting the gear centers and the positive 119909-axis of thepinion and in this work 120572
12= 0
In order to consider the influence of the rotating directionof the pinion the function 120590 is introduced as
120590 = 1 Ω
1 Counterclockwise
minus1 Ω1 Clockwise
(25)
y1
120579y1
rb1
120579z112057212
O1
k12(t)
e12(t)
c12(t)
x1
120579x1
12059512
y2
120579y2
rb2
120579z2 x2
120579x2
Gear 2 (gear)
Gear 1 (pinion)
O2
Ω2
Ω1
Figure 8 Dynamic model of a spur gear pair
Each gear owns six degrees of freedom including transla-tion in 119909 119910 and 119911 directions and rotation about these threeaxesWith these six degrees of freedom for each gear the gearpair has a total of 12 degrees of freedom that defines thecoupling between the two shafts holding the gears Without
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
considering the tooth separation backlash friction forces atthe mesh point and the position change of the pressure linethe motion equations of the gear pair are written as
11989811minus 119896
1211990112
(119905) sin12059512
minus 11988812
(119905) 12
(119905) sin12059512
= 0
1198981
1199101+ 119896
1211990112
(119905) cos12059512
+ 11988812
(119905) 12
(119905) cos12059512
= 0
11989811= 0
1198681199091
1205791199091
+ 1198681199111Ω1
1205791199101
= 0
1198681199101
1205791199101
minus 1198681199111Ω1
1205791199091
= 0
1198681199111
1205791199111
+ 120590 times 11990311988711198961211990112
(119905) + 120590 times 119903119887111988812
(119905) 12
(119905) = 1198791
11989822+ 119896
1211990112
(119905) sin12059512
+ 11988812
(119905) 12
(119905) sin12059512
= 0
1198982
1199102minus 119896
1211990112
(119905) cos12059512
minus 11988812
(119905) 12
(119905) cos12059512
= 0
11989822= 0
1198681199092
1205791199092
+ 1198681199112Ω2
1205791199102
= 0
1198681199102
1205791199102
minus 1198681199112Ω2
1205791199092
= 0
1198681199112
1205791199112
+ 120590 times 11990311988721198961211990112
(119905) + 120590 times 119903119887211988812
(119905) 12
(119905) = 1198792
(26)
where 1198981and 119898
2are the mass of gears 1 and 2 and 119868
1199091 1198681199101
1198681199111 1198681199092 1198681199102 and 119868
1199112 respectively represent the moments of
inertia about the 119909- 119910- and 119911-axis of gears 1 and 2 1198791and 119879
2
are constant load torques The term 11990112(119905) represents the
relative displacement of the gears in the direction of the actionplane and is defined as
11990112
(119905) = (minus1199091sin120595
12+ 119909
2sin120595
12+ 119910
1cos120595
12minus 119910
2cos120595
12
+120590 times 11990311988711205791199111
+ 120590 times 11990311988721205791199112) minus 119890
12(119905)
(27)
Equation (26) can be written in matrix form as follows
M12X12
+ (C12
+ G12) X
12+ K
12X12
= F12 (28)
where X12is the displacement vector of a gear pair M
12K12
C12 and G
12represent the mass matrix mesh stiffness
matrix mesh damping matrix and gyroscopic matrix of thespur gear pair respectively F
12is the exciting force vector of
the spur gear pair These matrices and vectors are definedthoroughly in [31]
Motion equations of the whole geared rotor system can bewritten in matrix form as
Mu + (C + G) u + Ku = Fu (29)
where M K C and G are the mass stiffness damping andgyroscopic matrix of the global system u and Fu denote thedisplacement and external force vectors of the global systemAll these matrixes and vectors can be obtained in [31]
In this paper the cracked gear coupled rotor systemincludes two shafts a gear pairwith a tooth crack in the driven
Shaft 2
Shaft 1 z
x
y
Bearing 1 Bearing 2
Bearing 3 Bearing 4
Gear 1 (node 8)
Gear 2 (node 22)
Figure 9 FE model of a cracked gear coupled rotor system
gear (gear 2) and four ball bearingsTheparameters of the sys-tem are listed in Table 2 and the FEmodel of the geared rotorsystem is shown in Figure 9Material parameters of the shaftsare as follows Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratioV = 03 and density 120588 = 7850 kgm3 Shafts 1 and 2 are bothdivided into 13 elements and gears 1 and 2 are located at nodes8 and 22 respectively (see Figure 9)
4 Vibration Response Analysis ofa Cracked Gear Coupled Rotor System
Based on the cracked gear coupled rotor system proposed inSection 3 the system dynamic responses can be calculated byNewmark-120573 numerical integrationmethod Firstly a crackedgear with a constant depth 3mm through the whole toothwidth is chosen as an example to analyze the influence of thecrack at the root of the gear on the systemvibration responsesThe parameters are as follows rotating speed of gear 1 Ω
1=
1000 revmin 120595 = 35∘ 120592 = 45∘ and crack depth 119902 = 3mm(1199021max = 32mm under these parameters)In this paper vibration responses at right journal of
the driving shaft in 119909 direction are selected to analyze thefault features under tooth crack condition The accelerationresponses of the healthy and cracked gears under steady stateare compared and the results are shown in Figure 10 In thefigure it is shown that three distinct impulses representingthe mating of the cracked tooth appear compared with thehealthy condition And the time interval between every twoadjacent impulses is exactly equal to the rotating period of thedriven gear (00818 s) because the influence caused by crackedtooth repeats only once in a revolution of the driven gear Inorder to clearly reflect the vibration features when the tooth
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
Table 2 Parameters of the cracked gear coupled rotor system
SegmentsShaft dimensions
Shaft 1 Shaft 2Diameters (mm) Length (mm) Diameters (mm) Length (mm)
1 30 20 30 192 35 28 35 283 42 12 42 124 35 30 35 305 32 30 32 306 30 19 30 20
Gear parametersGears 119868
119909= 119868119910(kgsdotmm2) 119868
119911(kgsdotmm2) m (kg) Pressure angle (∘) Pitch diameter (mm) Number of teeth N Tooth width (mm)
1 124799 406931 153 20 110 55 202 233884 789228 301 20 150 75 20
Bearing parametersBearings 119896
119909119909(MNm) 119896
119910119910(MNm) 119896
119911119911(MNm) 119896
120579119909120579119909(MNsdotmrad) 119896
120579119910120579119910(MNsdotmrad)
All 200 200 200 01 01
100
0
minus100
002 006 01 014 018 022
Time (s)
a(m
s2)
(a)
002 006 01 014 018 022
Time (s)
200
0
minus200a(m
s2) Δt = 00818 s
(b)
200
100
0
minus100
0032 0034 0036
Time (s)
a(m
s2)
Healthy gearCracked gear
(c)
Figure 10 Accelerations in time domain (a) healthy gear (b) cracked gear and (c) enlarged view
with crack is in mesh the enlarged views of vibration acceler-ation are displayed in Figure 10(c) in which the gray regionrepresents the double-tooth meshing area and the whiteregion denotes the single-toothmeshing area From the figureit can be seen that the vibration level increases apparentlycompared with that of the healthy gear
Amplitude spectra of the acceleration are shown inFigure 11 In the figure 119891
119890denotes the meshing frequency
(91667Hz) The figure shows that the amplitudes of 119891119890and
those of its harmonics (2119891119890 3119891
119890 etc) for the cracked gear
hardly change compared with the healthy gear but the side-bands appear under the tooth crack condition which can beobserved clearly in Figure 11(b) The interval Δ119891 betweenevery two sideband components is exactly equal to therotating frequency of the cracked gear that is 119891
2= 12Hz So
a conclusion can be drawn that the sideband components canbe used to diagnose the crack fault of gear
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
Table 3 Crack case data under different crack depths
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm)
Crackcase
Crackdepth ()
Crack size(mm) Condition
1 (A) 0 q = 0 12 22 119902 = 141 23 44 119902 = 282
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
119871119888= 119871
119902119890= 119902
0= q
1199021max= 321mm
2 2 q = 013 13 24 119902 = 154 24 46 119902 = 295
3 4 q = 026 14 26 119902 = 167 25 48 119902 = 308
4 6 119902 = 039 15 28 119902 = 180 26 50 119902 = 321
5 8 119902 = 051 16 30 119902 = 193 27 52 119902 = 334
6 10 119902 = 064 17 32 119902 = 205 28 54 119902 = 347
7 12 119902 = 077 18 34 119902 = 218 29 56 119902 = 360
8 14 119902 = 090 19 36 119902 = 231 30 58 119902 = 372
9 16 119902 = 103 20 38 119902 = 244 31 (D) 60 119902 = 385
10 18 119902 = 116 21 (C) 40 119902 = 257
11 (B) 20 119902 = 128 22 42 119902 = 270
fe
2fe3fe
4fe
5fe
6fe
7fe
8fe
9fe
10fe
11fe
12fe
13fe
14fe15fe
16fe
0 5 10 15
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
30
20
10
0
Healthy gearCracked gear
times103
(a)
04
02
04500 4600 4700
5fe
Δf = 12Hz
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Healthy gearCracked gear
(b)
Figure 11 Amplitude spectra of the acceleration (a) amplitude spectra (b) local enlarged view (119891119890 2119891
119890 16119891
119890stand for mesh frequency and
its harmonic components)
41 Effects of Crack Depth In this section the angle 120592 standsfor crack propagation direction and is set at a constant 45∘and the angle 120595 stands for crack initial position and is set at35∘ In order to analyze the influence of crack depth on thegeared rotor system 31 crack depth cases are consideredwhich are shown in Table 3 The crack depth percentage isdefined as the ratio of actual crack depth with theoreticalthrough-tooth crack size The rotating speed of gear 1 Ω
1=
1000 revmin (in the following study the same speed is cho-sen) Four cases each of which concerns a different crack sizeare selected to study the influences of different crack depthsThe selected cases are case 1 which is the healthy case case 11which represents a smaller crack case 21 which represents amoderate crack finally case 31 which is a deep crack Thesefour cases selected for stiffness and vibration responsescomparison are referred to as A B C and D respectively
(see Table 3) In order to verify the validity of the improvedmesh stiffness model for a cracked gear pair a FE model isalso established in which the crack is modeled using 2Dsingularity elements (see Figure 12)
TVMS under the four crack sizes are calculated as a func-tion of the shaft rotation angle by the improvedmesh stiffnessmodel and the FE model And the results are shown inFigure 13 There will be eight lines when four crack cases oftwomodels are considered In order to clearly show the resultdifference between the improved method (IM) and finiteelement method (FEM) under different crack cases the fourcases are shown in two figures (see Figures 13(a) and 13(b))In the figures 1 and 2 stand for IM and FEM respectively AB C and D stand for different crack cases It can be seenfrom the figure that distinct reduction of stiffness appearswhen tooth crack is introduced and the maximum stiffness
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
Table 4 Comparisons of mean TVMS under different crack depths
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3511 minus198 2B 3194 minus3881C 3327 minus712 2C 3031 minus8791D 2959 minus1739 2D 2869 minus1366
Singularity elements
F
Crack
120573
Figure 12 FE model of a tooth with crack
reduction during the mesh process for a fixed crack appearswhere the cracked tooth is just going to engage This isbecause the relative bigger flexibility of the tooth at the adden-dum circle compared with that at the dedendum circle IMshows a better agreement with the FEM under small crackdepth conditions such as cases A B andCThe result of FEMis slightly larger than that of the IM under the large crackcondition such as case D In order to clearly display the vari-ation of the stiffness with crack depths the comparisons ofthe mean stiffness 119896
119898about the four cases are shown in
Table 4 In the table reduction percentages refer to thereduction percentage of the stiffness in case B (or C D)relative to case A It can be seen that the stiffness decreaseswith the growth of crack depth and the FEM shows the samechanging trends as the IM In the analysis of vibrationresponses the TVMS acquired from IM will be used
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 14 For visual convenience only the local enlargedview is displayed It is very hard to observe the change inacceleration waveform produced by the gear tooth root crackwhen it is at early stage such as case B However with thegrowth of gear tooth crack depth the amplitudes of theimpulsive vibration signals increase The impulses causedby tooth crack are very obvious when the crack propagatesto 60 (case D) In the acceleration amplitude spectra themagnitudes of sidebands ascend with the crack propagationalong crack depth
Statistical features which are commonly used to provide ameasurement of the vibration level are widely used inmechanical fault detection [3 20 21] RMS andKurtosis indi-cators are used to explore the effect of the crack propagationalong depth [3] The RMS value is defined as
RMS = radic1
119873
119873
sum119899=1
(119909 (119899) minus 119909)2 119909 =1
119873
119873
sum119899=1
119909 (119899) (30)
And Kurtosis is calculated by
Kurtosis =1119873sum
119873
119899=1(119909 (119899) minus 119909)4
[1119873sum119873
119899=1(119909 (119899) minus 119909)2]
2 (31)
Variation of statistical indicators of vibration accelerationwith the increase of crack depth is shown in Figure 15 Andindicators are acquired from two types of signals (original sig-nal and residual signal [3]) which are shown in Figures 15(a)and 15(b) respectively For original signals in order to reflectthe vibration level compared with the healthy condition theindicators are expressed as percentages which are defined as
119903Kurtosis =Kurtosiscrack minus Kurtosishealth
Kurtosishealthtimes 100
119903RMS =RMScrack minus RMShealth
RMShealthtimes 100
(32)
where 119903 shows the relative change of cracked signal indicatorsubscripts Kurtosis and RMS denote different statisticalindicators and subscripts crack and health are cracked andhealthy gear
Relative change ratios in Figure 15(b) refer to the ratiosof RMS and Kurtosis with those under the maximum crackdepth (60 under this condition) It can be seen from thefigures that for original signals RMS and Kurtosis changelittle when the crack depth is smaller but when the crackdepth becomes higher than 40 Kurtosis exhibits an obviousincreasing trend while RMS increases slightly For residualsignals unlike the results obtained fromoriginal signals RMSshows obvious increasing pattern when the crack depthexceeds 20 Compared with RMS however Kurtosischanges less obviously SoRMSacquired from residual signalsis a robust indicator for early fault identification which issimilar to the results shown in [3]
42 Effects of Crack Width In order to investigate theinfluences of crack width on the geared rotor system 22 crackwidth cases are selectedwhich are shown inTable 5The crackwidth percentage is defined as the ratio of actual crack width119871119888to the whole tooth width (119871 = 20mm) It is noted that
120 of crack width represents the case when 119871119888
= 119871 =20mm and 119902
119890= 119902
0= 3mm Five cases corresponding to
cases 1 6 11 21 and 22 in Table 5 are referred to as A B Cand D and E TVMS under the five cases are analyzed toillustrate the influences of crack width by IM (see Figure 16)Since the FE model used in the paper is a 2D model it ispowerless to simulate the crack propagation along the toothwidth And because the crack depth is changing along the
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
4
3
20 4 8 12
TVM
S (times10
8N
m)
1A1B
2A2B
Angular position (∘)
(a)
4
3
2
10 4 8 12
TVM
S (times10
8N
m)
Angular position (∘)
1C1D
2C2D
(b)
Figure 13 TVMS under different crack depths (a) crack cases A and B and (b) crack cases C and D
AB
CD
300
200
100
0
minus100
minus200
a(m
s2)
0032 0034 0036
Time (s)
(a)
AB
CD
1
05
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 14 Acceleration responses under different crack depths (a) time domain waveform and (b) amplitude spectra
Kurtosis
25
15
5
minus50 20 40 60
Crack depth ()
r Kur
tosi
sr R
MS
()
RMS
(a)
100
50
0
Rela
tive c
hang
e rat
io (
)
Kurtosis
0 20 40 60
Crack depth ()
RMS
(b)
Figure 15 Acceleration statistical indicators under different crack depths (a) original signal and (b) residual signal
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
Table 5 Crack propagation case data under different crack widths
Crackcase
Crack width()
Crack size(mm)
Crackcase
Crack width() Crack size (mm) Crack
caseCrack width
() Crack size (mm) Condition
1 (A) 0 119902119890= 0 119871119888 = 0 9 40 119902
119890= 0 119871
119888= 8 17 80 119902
119890= 0 119871
119888= 16
Ω1 = 1000 revmin120592 = 45∘
120595 = 35∘
1199020= 3mm
2 5 119902119890= 0 119871119888 = 1 10 45 119902
119890= 0 119871
119888= 9 18 85 119902
119890= 0 119871
119888= 17
3 10 119902119890= 0 119871
119888= 2 11 (C) 50 119902
119890= 0 119871
119888= 10 19 90 119902
119890= 0 119871
119888= 18
4 15 119902119890= 0 119871
119888= 3 12 55 119902
119890= 0 119871119888 = 11 20 95 119902
119890= 0 119871
119888= 19
5 20 119902119890= 0 119871
119888= 4 13 60 119902
119890= 0 119871
119888= 12 21 (D) 100 119902
119890= 0 119871
119888= 20
6 (B) 25 119902119890= 0 119871
119888= 5 14 65 119902
119890= 0 119871
119888= 13 22 (E) 120 119902
119890= 3 119871
119888= 20
7 30 119902119890= 0 119871
119888= 6 15 70 119902
119890= 0 119871
119888= 14
8 35 119902119890= 0 119871
119888= 7 16 75 119902
119890= 0 119871
119888= 15
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
ABC
DE
Figure 16 TVMS under different crack widths by IM
tooth width even if a 3D model is adopted some difficultieswill also occur So in this section the comparison of themesh stiffness between the improved model and the FEmodel under different crack widths is not carried out Thecomparisons of the mean stiffness 119896
119898about the five cases are
shown in Table 6 It can be seen from Figure 16 and Table 6that the stiffness decreases with the growth of crack width
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 17 The amplitudesof the impulsive vibration and the magnitudes of sidebandsincrease with the growth of crack width For statisticalfeatures (see Figure 18) RMS and Kurtosis of original signalincrease slightly when the crack width is smaller When thecrack width becomes higher than 80 Kurtosis exhibits anobvious increasing trend Reversely RMS increases slightlyFor residual signals unlike the results obtained from theoriginal signals RMS shows obvious increasing pattern Inaddition there is a prompt increase for the two indicatorswhen the crack propagates along the whole tooth width andcontinues to propagate along the crack depth
Table 6 Comparisons ofmean TVMS under different crack widths
Case (IM) 119896119898(times108 Nm) Reduction percentage
A 3582 0B 3534 minus134C 3490 minus257D 3401 minus505E 3231 minus980
43 Effects of Crack Initial Position In this section 12 crackinitial position cases are considered which are shown inTable 7 Four cases marked as A (stands for healthy gear) BC and D (see Table 7) are selected to study the influencesof different crack initial positions on TVMS and vibrationresponses The mesh stiffness curves obtained by IM andFEM under the above four cases are displayed in Figure 19The comparisons of themean stiffness 119896
119898about the four cases
are shown inTable 8 It can be seen fromFigure 19 andTable 8that the stiffness ascends with the increase of the crack initialposition angle 120595 and the FEM shows the same changingtrends as IM
Vibration acceleration responses and the correspond-ing amplitude spectra under the four cases are shown inFigure 20 With the growth of initial position angle 120595 theamplitudes of the impulsive vibration signals decrease Simi-larly in the acceleration amplitude spectra themagnitudes ofsidebands also decrease with the growth of initial positionangle 120595
The variation of statistical indicators acquired from orig-inal signals and residual signals with the increase of initialposition angle 120595 is shown in Figure 21 For residual signalsthe relative change ratio of120595=65∘ crack is defined as 100Asseen from the figures for original signals Kurtosis exhibits anobvious decreasing trend while RMS decreases slightly Forresidual signals just the opposite RMS shows obviousdecreasing pattern with the growth of initial position angle120595In addition compared with original signals RMS acquiredfrom residual signals changes more obviously
44 Effects of Crack Propagation Direction In this section 19crack propagation direction cases are considered as shown inTable 9 Seven cases marked as A (stands for healthy gear)
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
200
100
0
minus100
a(m
s2)
0032 0034 0036
Time (s)
ABC
DE
(a)
4500 4600 4700
04
03
02
01
0
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
ABC
DE
(b)
Figure 17 Acceleration responses under different crack widths (a) time domain waveform and (b) amplitude spectra
3
2
1
0
0 40 80 120
Crack width ()
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
100
50
0
Relat
ive c
hang
e rat
io (
)
0 40 80 120
Crack width ()
Kurtosis
RMS
(b)
Figure 18 Acceleration statistical indicators under different crack widths (a) original signal and (b) residual signal (120 represents the case119871119888= 119871 = 20mm and 119902
119890= 119902
0= 3mm)
Table 7 Crack case data under different crack initial positions
Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Crack case Crack position 120595 (∘) Condition1 15 5 35 9 55 Ω1 = 1000 revmin
120592 = 45∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 20 6 40 10 603 (B) 25 7 (C) 45 11 (D) 65
4 30 8 50 12
Table 8 Comparisons of mean TVMS under different crack initial positions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3200 minus1066 2B 2964 minus10801C 3264 minus888 2C 2997 minus9811D 3335 minus690 2D 3031 minus879
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B
2A2B
(a)
45
35
25
150 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1C1D
2C2D
(b)
Figure 19 TVMS under different crack initial positions (a) crack cases A and B and (b) crack cases C and D
200
100
0
minus100
a(m
s2 )
0032 0034 0036
Time (s)
AB
CD
(a)
AB
CD
04
02
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
Δf = 12Hz
5fe
(b)
Figure 20 Acceleration responses under different crack initial positions (a) time domain waveform and (b) amplitude spectra45
3
15
015 25 35 45 55 65
Crack position 120595 (∘)
r Kur
tosi
sr R
MS
()
Kurtosis
RMS
(a)
180
140
100
Rela
tive c
hang
e rat
io (
)
15 25 35 45 55 65
Crack position 120595 (∘)
Kurtosis
RMS
(b)
Figure 21 Acceleration statistical indicators under different crack initial positions (a) original signal and (b) residual signal
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 17
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
1A1B1C1D
1E1F1G
(a)
4
3
2
10 4 8 12
Angular position (∘)
TVM
S (times10
8N
m)
2A2B2C2D
2E2F2G
(b)
Figure 22 TVMS under different crack propagation directions (a) IM and (b) FEM
Table 9 Crack case data under different crack propagation directions
Crack case Crack propagationdirection 120592 (∘) Crack case Crack propagation
direction 120592 (∘) Crack case Crack propagationdirection 120592 (∘) Condition
1 (B) 0 8 35 15 (E) 70
Ω1 = 1000 revmin120595 = 35∘
119871119888= 119871 = 20mm
119902119890= 119902
0= 3mm
2 5 9 40 16 753 10 10 45 17 (F) 80
4 15 11 50 18 855 20 12 55 19 (G) 90
6 25 13 (D) 60
7 (C) 30 14 65
B C D E F and G (see Table 9) are chosen to study theinfluences of crack propagation direction on TVMS andvibration responses
Based on IM and FEM the mesh stiffness curves underthe above seven cases are displayed in Figure 22 The com-parisons of the mean stiffness 119896
119898about the seven cases are
shown in Table 10 It can be seen from Figure 22 and Table 10that the stiffness firstly decreases and then increases with thegrowth of crack propagation direction angle 120592This is becausethe involute and transitional part of the gear losesmuch rigid-ity and becomes more and more flexible with the increaseof crack propagation direction angle 120592 however a crackwith a pretty large 120592 has little effect on the rigidity of thetransitional part The FEM shows the same changing trendsas the IM
Vibration acceleration responses and the correspondingamplitude spectra are shown in Figure 23 The amplitudesof the impulsive vibration and the magnitudes of sidebandsfirstly increase and then decrease with the growth of propa-gation direction angle 120592
The variation of statistical indicators acquired fromoriginal signals and residual signals with the increase of
propagation direction angle 120592 is shown in Figure 24 Forresidual signals the relative change ratio of 120592 = 90∘ crack isdefined as 100 It is seen from the figures that for originalsignals RMS and Kurtosis change a little when the crackpropagation direction angle 120592 is less than 50∘ and thenKurtosis exhibits an obvious increasing trend and reaches themaximum when 120592 is close to 80∘ and after that Kurtosisdecreases regularly RMS has the same phenomenon butchanges less obviously For residual signals just the oppositeRMS shows obvious increasing pattern with the growth ofpropagation direction angle 120592 and reaches the maximumwhen 120592 is close to 80∘ In addition compared with originalsignals RMS acquired from residual signals changes moreobviously
5 Measured Vibration Response ofa Cracked Gear Coupled Rotor System
The one-stage reduction gearbox with crack defect (seeFigure 25) is designed to validate the theoretical results andthe parameters of the rotor and the spur gear pair are listed
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
18 Mathematical Problems in Engineering
400
200
0
minus200
a(m
s2)
0032 0034 0036
Time (s)
ABCD
EFG
(a)
16
08
04500 4600 4700
Frequency (Hz)
Acce
lera
tion
ampl
itude
(ms2)
5fe
Δf = 12Hz
ABCD
EFG
(b)
4595 4605 4615
Frequency (Hz)
12
06
0
Δf = 12Hz
Acce
lera
tion
ampl
itude
(ms2)
ABCD
EFG
(c)
Figure 23 Acceleration responses under different crack propagation directions (a) time domain waveform (b) amplitude spectra and (c)enlarged view of amplitude spectra
Table 10 Comparisons of mean TVMS under different crack propagation directions
Case (IM) 119896119898(times108 Nm) Reduction percentage Case (FEM) 119896
119898(times108 Nm) Reduction percentage
1A 3582 0 2A 3323 01B 3580 minus006 2B 3183 minus4211C 3406 minus491 2C 3060 minus7911D 3027 minus1549 2D 2879 minus13361E 2905 minus1890 2E 2824 minus15021F 2820 minus2127 2F 2794 minus15921G 2861 minus2013 2G 2797 minus1583
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 19
80
50
20
minus10
r Kur
tosi
sr R
MS
()
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(a)
120
80
40
0
Rela
tive c
hang
e rat
io (
)
0 30 60 90
Crack propagation direction 120592 (∘)
RMS
Kurtosis
(b)
Figure 24 Acceleration statistical indicators under different crack propagation directions (a) original signal and (b) residual signal
in Table 2 The crack is manufactured near the root of thedriven gear and the crack depth is 3mm Vibration signalsunder tooth crack condition are measured by accelerom-eters whose locations are shown in Figure 25 The samplefrequency is 10000Hz and the rotating speed of driving shaftis 1000 revmin So rotating frequencies of the driving anddriven gear are about 167Hz and 122Hz which are denotedby 119891
1and 119891
2 respectively and meshing frequency 119891
119890is equal
to 9167Hz Time domain waveform of the vibration acceler-ation and its frequency spectrum at the right bearing of thedriving shaft in vertical direction of healthy gear systemand cracked gear system are displayed in Figures 26 and 27respectively
Comparation with Figures 26(a) and 27(a) shows theexistence of large impulse under crack state and frequency ofimpulse 1Δ119905 = 122Hz is just the rotating frequency of thecracked gearThe impulse displayed in Figure 27(a) shows thesame features as that in Figure 10(b) For the healthy gearedrotor system and cracked geared rotor system the amplitudespectra all show the meshing frequency and its harmonics(see Figures 26(b) and 27(b))Moreover there also exist inter-ference frequencies and sideband frequencies For healthygeared rotor system probably due to the mismachining tol-erance sideband frequencies around the meshing frequencyand its harmonics are detected which are related to therotating frequency of the driving gear 119891
1 for example
9328Hz asymp 119891119890+ 119891
1 1815Hz asymp 2119891
119890minus 119891
1 For cracked geared
rotor system the sideband frequency components aroundmeshing frequency and its harmonics are different fromhealthy system which are related to the rotating frequency ofthe driven gear for example 9035Hz asymp 119891
119890minus 119891
2 1844Hz asymp
2119891119890+ 119891
2
In addition owing to the influences of many factors suchas the accuracy of involute cylindrical gears variations ofbearing stiffness friction interference of current and noiseon experiment results frequency components of system arecomplicated and many interference frequency componentsappear which leads to irregular shock in time domainwaveformof healthy geared rotor system But even if there area lot of uncertain factors for cracked geared rotor system thevibration responses of the experimental signal agree with thetheoretical results
Electromotor
One-stage reduction gear box
Detent
Cracked gear
Coupling
Foundation
Acceleration sensor
Crack position
Figure 25 Test rig of a cracked gear coupled rotor system
6 Conclusions
Considering the misalignment of gear root circle and basecircle an improved mesh stiffness model for a healthy gearpair is proposed and it is validated by comparisonwith the FEresults On the basis of the improvedmethod amesh stiffnessmodel for a cracked gear pair is established Then the meshstiffness is introduced into a geared rotor system to simulatethe vibration response under different crack casesThe effectsof tooth crack parameters including crack depth widthinitial position and crack propagation direction on gearmeshstiffness vibration responses and statistical indicators areinvestigated In the end an experiment of the gearbox with acracked tooth in driven gear has been designed to validate thefailure characteristics Some conclusions can be summarizedas follows
(1) The improved mesh stiffness model in which the geartooth is modeled as a nonuniform cantilever beam onthe root circle is more accurate compared with thetraditional mesh stiffness model in which the gear
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
20 Mathematical Problems in Engineering
40
20
0
minus20
minus4011 12 13 14 15
Time (s)
a(m
s2)
(a)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
0 2500 5000
Frequency (Hz) Frequency (Hz) Frequency (Hz)
fe
2fe
3fe9160
18320
27510
1800 1820 1840 1860 1880880 900 920 940 960
8826
8994
9160
9328
9492
17990
18320
18150 18470
18650
fe minus 2f1
fe minus f1
fe
fe + f1
fe + 2f1
2fe
2fe minus 2f1
2fe minus f12fe + 1
2fe + 2f1
f1
1
08
06
04
02
0
04
03
02
01
0
04
03
02
01
0
(b)
Figure 26 Measured acceleration responses of healthy gear (a) time domain waveform and (b) amplitude spectra
60
0
minus60
Δt = 00818 s
Time (s)45 46 47 48
a(m
s2)
(a)
0 2500 5000 880 900 920 940 1800 1820 1840 1860
Frequency (Hz) Frequency (Hz) Frequency (Hz)
08
04
0
08
04
0
025
020
015
010
005
0
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
Am
plitu
de (m
s2)
27470 8913
90359158
91589279
9406
18090
18200
18440
1856018310
fe 2fe
3fe
18310
fe minus 2f2
fe minus f2
fe fe + f2
2fe
2fe minus 2f22fe + f2
2fe + 2f22fe minus f2
fe + 2f2
(b)
Figure 27 Measured acceleration responses of cracked gear (a) time domain waveform and (b) amplitude spectra
tooth is modeled as a cantilever beam on the basecircle
(2) When tooth crack is introduced distinct reduction ofstiffness can be observed and the stiffness decreaseswith the growth of crack depth and width increases
with the increasing crack initial position angle 120595 andfirstly decreases and then increases with the increas-ing crack propagation direction angle 120592
(3) When tooth root crack is introduced distinct impul-ses are found in time domain vibration responses and
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 21
the time interval between every two adjacent impulsesis exactly equal to the rotating period of the crackedgear In frequency domain the amplitudes ofmeshingfrequency and its harmonics hardly change comparedwith healthy gear but the sidebands appear under thetooth crack conditionThe interval between every twosideband components is exactly equal to the rotatingfrequency of the cracked gear The amplitudes of theimpulses and the magnitudes of sidebands increasewith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofcrack propagation direction angle 120592 In addition timeand frequency domain vibration featuresmeasured bythe experiment agree qualitatively with the analyticalresults
(4) RMS and Kurtosis indicators acquired from originalsignals and residual signals are used to analyze theeffect of different crack parameters When crackdepth width and propagation direction angle aresmaller it is hard to detect the presence of crack dueto the little change of these statistical features Ampli-tudes of all the statistical indicators ascend graduallywith the growth of crack depth and width decreasewith the increase of crack initial position angle120595 andfirstly increase and then decrease with the growth ofpropagation direction angle 120592 In addition for origi-nal signals Kurtosis changes more obviously and forresidual signals RMS shows obvious changing pat-tern But compared with original signals RMSacquired from residual signals changes more obvi-ously
It is worth mentioning that the improved mesh stiffnessmodel proposed in this paper cannot deal with this situationwhen the tooth crack propagates to the foundation of thegear In future articles emphasis will be given to the effect oftooth crack on the rigidity of the foundation In addition a 3Dmodel considering the crack propagation along tooth widthis another research focus
Appendix
119909120573 119910120573 1199091 1199101 1199092 1199102 1198681199101 1198681199102 1198601199101 and 119860
1199102can be expressed
as the function of angular 120574 120573 or 120591 which can be expressedas
119909120573= 119903119887[ (120573 + 120579
119887) cos120573 minus sin120573]
119910120573= 119903119887[ (120573 + 120579
119887) sin120573 + cos120573]
1199091= 119903 times sin (Φ) minus (
1198861
sin 120574 + 119903120588
) times cos (120574 minus Φ)
1199101= 119903 times cos (Φ) minus (
1198861
sin 120574 + 119903120588
) times sin (120574 minus Φ)
1199092= 119903119887[ (120591 + 120579
119887) cos 120591 minus sin 120591]
1199102= 119903119887[ (120591 + 120579
119887) sin 120591 + cos 120591]
1198601199101
= 21199091119871
1198601199102
= 21199092119871
1198681199101
=2
311990931119871
1198681199102
=2
311990932119871
(A1)
So d1199101d120574 and d119910
2d120591 can be calculated as follows
d1199101
d120574=
1198861sin ((119886
1 tan 120574 + 119887
1) 119903) (1 + tan2120574)
tan2120574
+1198861cos 120574
sin2120574sin(120574 minus
1198861 tan 120574 + 119887
1
119903) minus (
1198861
sin 120574+ 119903120588)
times cos(120574 minus1198861 tan 120574 + 119887
1
119903)(1 +
1198861(1 + tan2120574)119903tan2120574
)
d1199102
d120591= 119903119887(120591 + 120579
119887) cos 120591
(A2)
where 120579119887is the half tooth angle on the base circle of the gear
120579119887= 1205872119873 + inv120572
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the Program for New CenturyExcellent Talents inUniversity (Grant noNCET-11-0078) andthe Fundamental Research Funds for the Central Universities(Grant no N130403006) for providing financial support forthis work
References
[1] D C H Yang and J Y Lin ldquoHertzian damping tooth frictionand bending elasticity in gear impact dynamicsrdquo Journal ofMechanisms Transmissions and Automation in Design vol 109no 2 pp 189ndash196 1987
[2] X H Tian Dynamic simulation for system response of gearboxincluding localized gear faults [MS thesis] University ofAlberta Edmonton Canada 2004
[3] S Wu Gearbox dynamic simulation and estimation of faultgrowth [MS thesis] University of Alberta Edmonton Canada2007
[4] X Zhou Y Shao Y Lei and M Zuo ldquoTime-varying meshingstiffness calculation and vibration analysis for a 16DOFdynamic
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
22 Mathematical Problems in Engineering
model with linear crack growth in a pinionrdquo Journal of Vibrationand Acoustics Transactions of the ASME vol 134 no 1 ArticleID 011011 2012
[5] P Sainsot P Velex andODuverger ldquoContribution of gear bodyto tooth deflectionsmdasha new bidimensional analytical formulardquoJournal of Mechanical Design Transactions of the ASME vol126 no 4 pp 748ndash752 2004
[6] F Chaari T Fakhfakh and M Haddar ldquoAnalytical modellingof spur gear tooth crack and influence on gearmesh stiffnessrdquoEuropean Journal of Mechanics ASolids vol 28 no 3 pp 461ndash468 2009
[7] ZWan Y Zi H Cao Z He and SWang ldquoTime-varying meshstiffness algorithm correction and tooth crack dynamic model-ingrdquo Journal of Mechanical Engineering vol 49 no 11 pp 153ndash160 2013
[8] Z Chen and Y Shao ldquoMesh stiffness calculation of a spur gearpair with tooth profile modification and tooth root crackrdquoMechanism and Machine Theory vol 62 pp 63ndash74 2013
[9] A Fernandez Del Rincon F Viadero M Iglesias P Garcıa ADe-Juan and R Sancibrian ldquoA model for the study of meshingstiffness in spur gear transmissionsrdquo Mechanism and MachineTheory vol 61 pp 30ndash58 2013
[10] S Jia and I Howard ldquoComparison of localised spalling andcrack damage from dynamicmodelling of spur gear vibrationsrdquoMechanical Systems and Signal Processing vol 20 no 2 pp 332ndash349 2006
[11] JWang and IHoward ldquoFinite element analysis ofHighContactRatio spur gears inmeshrdquo Journal of Tribology vol 127 no 3 pp469ndash483 2005
[12] J F Li M T Xu and S Y Wang ldquoFinite element analysisof instantaneous mesh stiffness of cylindrical gears (with andwithout flexible gear body)rdquo Communications in NumericalMethods in Engineering vol 15 no 8 pp 579ndash587 1999
[13] J DWangNumerical and experimental analysis of spur gears inmesh [PhD thesis] Curtin University of Technology BentleyAustralia 2003
[14] J Wang and I Howard ldquoThe torsional stiffness of involute spurgearsrdquo Proceedings of the Institution of Mechanical Engineers CJournal ofMechanical Engineering Science vol 218 no 1 pp 131ndash142 2004
[15] D G Lewicki and R Ballarini ldquoEffect of rim thickness ongear crack propagation pathrdquo Journal of Mechanical DesignTransactions of the ASME vol 119 no 1 pp 88ndash95 1997
[16] Y Pandya and A Parey ldquoFailure path basedmodified gear meshstiffness for spur gear pair with tooth root crackrdquo EngineeringFailure Analysis vol 27 pp 286ndash296 2013
[17] Y Pandya and A Parey ldquoSimulation of crack propagation inspur gear tooth for different gear parameter and its influence onmesh stiffnessrdquoEngineering Failure Analysis vol 30 pp 124ndash1372013
[18] S Zouari M Maatar T Fakhfakh and M Haddar ldquoFollowingspur gear crack propagation in the tooth foot by finite elementmethodrdquo Journal of Failure Analysis and Prevention vol 10 no6 pp 531ndash539 2010
[19] SWuM J Zuo and A Parey ldquoSimulation of spur gear dynam-ics and estimation of fault growthrdquo Journal of Sound and Vibra-tion vol 317 no 3ndash5 pp 608ndash624 2008
[20] Z Chen and Y Shao ldquoDynamic simulation of spur gear withtooth root crack propagating along tooth width and crackdepthrdquo Engineering Failure Analysis vol 18 no 8 pp 2149ndash2164 2011
[21] O D Mohammed M Rantatalo and J Aidanpaa ldquoImprovingmesh stiffness calculation of cracked gears for the purpose ofvibration-based fault analysisrdquoEngineering Failure Analysis vol34 pp 235ndash251 2013
[22] RMa andYChen ldquoResearch on the dynamicmechanismof thegear system with local crack and spalling failurerdquo EngineeringFailure Analysis vol 26 pp 12ndash20 2012
[23] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design Transactions of the ASME vol 114 no 3pp 507ndash514 1992
[24] J S Rao T N Shiau and J R Chang ldquoTheoretical analysis oflateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998
[25] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements model and experimentrdquo Journal of Vibration andAcoustics Transactions of the ASME vol 126 no 3 pp 398ndash4062004
[26] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003
[27] A S Lee and J W Ha ldquoPrediction of maximum unbalance res-ponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005
[28] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012
[29] Y Cui Z Liu Y Wang and J Ye ldquoNonlinear dynamic of ageared rotor system with nonlinear oil film force and nonlinearmesh forcerdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 134 no 4 Article ID 041001 2012
[30] P Velex andMAjmi ldquoOn themodelling of excitations in gearedsystems by transmission errorsrdquo Journal of Sound andVibrationvol 290 no 3ndash5 pp 882ndash909 2006
[31] HMa J Yang R Z Song et al ldquoEffects of tip relief on vibrationresponses of a geared rotor systemrdquoProceedings of the Institutionof Mechanical Engineers C Journal of Mechanical EngineeringScience vol 28 pp 1132ndash1154 2014
[32] F K Omar K A F Moustafa and S Emam ldquoMathematicalmodeling of gearbox including defects with experimental verifi-cationrdquo Journal of Vibration and Control vol 18 no 9 pp 1310ndash1321 2012
[33] S Jia I Howard and J Wang ldquoThe dynamic modeling ofmultiple pairs of spur gears inmesh including friction and geo-metrical errorsrdquo International Journal of Rotating Machineryvol 9 pp 437ndash442 2003
[34] Z Guo and Y D Zhang ldquoParameterization precise modeling ofinvolutes spur gear based on APDLrdquoMachinery Manufacturingand Research vol 39 pp 33ndash36 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of