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Research ArticleOn Dynamic Mesh Force Evaluation of Spiral Bevel Gears

Xiaoyu Sun Yongqiang Zhao Ming Liu and Yanping Liu

School of Mechatronics Engineering Harbin Institute of Technology No 92 Xidazhi Street Nangang District HarbinHeilongjiang China

Correspondence should be addressed to Yongqiang Zhao zhyqhithiteducn

Received 22 June 2019 Revised 16 September 2019 Accepted 25 September 2019 Published 20 October 2019

Academic Editor Miguel Neves

Copyright copy 2019 Xiaoyu Sun et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e mesh model and mesh stiffness representation are the two main factors affecting the calculation method and the results of thedynamic mesh force Comparative studies considering the two factors are performed to explore appropriate approaches toestimate the dynamic meshing load on each contacting tooth flank of spiral bevel gears First a tooth pair mesh model is proposedto better describe the mesh characteristics of individual tooth pairs in contact e mesh parameters including the mesh vectortransmission error and mesh stiffness are compared with those of the extensively applied single-point mesh model of a gear pairDynamic results from the proposed model indicate that it can reveal a more realistic and pronounced dynamic behavior of eachengaged tooth pair Second dynamic mesh force calculations from three different approaches are compared to further investigatethe effect of mesh stiffness representations One method uses the mesh stiffness estimated by the commonly used average slopeapproach the second method applies the mesh stiffness evaluated by the local slope approach and the third approach utilizes aquasistatically defined interpolation function indexed by mesh deflection and mesh position

1 Introduction

Spiral bevel gears are commonly used in power transmissionbetween intersecting shafts but often suffer from fatiguefailures caused by excess dynamic loads e estimation ofdynamic loads carried by each pair of teeth in contact laysthe root for analyses of bending and contact stresses of gearteeth gear tooth surface wear (pitting and scoring) andlubrication performance between the mating tooth flankswhich are all effective ways to investigate the mechanism ofgear failures Consequently a reasonably accurate predictionof dynamic mesh force (DMF) is the cornerstone for failureanalysis

Two factors have a dominant influence on the calcula-tion approaches and results of DMF firstly the mesh modelwhich determines the calculation of mesh parameters in-cluding effective mesh point coordinates line of actionvectors mesh stiffness and transmission errors and secondlythe mesh stiffness representations

e determination of mesh parameters for spiral bevelgears is complicated not only because of the complex toothflank geometries but also because the mesh properties are

strongly influenced by load conditions Nowadays loadedtooth contact analysis (LTCA) based on the conventionalfinite element method [1ndash4] or finite elementcontactmechanism method [5] is widely applied to obtain the meshparameters of spiral bevel gears under the loaded conditionHowever in the absence of the sophisticated commercialfinite element software (eg ABAQUS and ANSYS) and aspecialized LTCA tool [6] the loaded mesh parameters werenot available and hence experimental or simple analyticalmodels which preclude exact mesh geometries were used inmost early dynamic studies [7 8] Until the beginning of thiscentury Cheng and Lim first attempted to integrate thetime-variant mesh parameters obtained from tooth contactanalysis (TCA) [9] and LTCA [10] of gears with exact toothflank geometry into a mesh model e model was exten-sively applied in the subsequent studies on dynamics of thehypoid and bevel gear pair or geared system [10ndash17] Inaddition to the time-dependent mesh characteristics someother special factors affecting the dynamic responses wereinvestigated such as backlash nonlinearity [10ndash13] friction[13ndash15] mesh stiffness asymmetry [12] gyroscopic effect[13] assembly errors [13] and geometric eccentricity [13] as

HindawiShock and VibrationVolume 2019 Article ID 5614574 26 pageshttpsdoiorg10115520195614574

well as other driveline components such as the bearing [16]and elastic housing [17]

In all contributions mentioned above the single-pointcoupling mesh model where the gear mesh coupling ismodeled by a single spring-damper unit located at the ef-fective mesh point and acting along the instantaneous line ofaction was applied Although the model is believed to be ableto catch the overall effect of the mesh characteristics of a gearpair it cannot reflect the mesh properties of each meshinterface when more than one tooth pair is engaged inmeshing and hence the DMF for each couple of teeth incontact cannot be predicted directly Karagiannis et al [15]utilized the load distribution factor defined and calculated inquasistatic LTCA to estimate the dynamic contact load pertooth when load sharing occurs is method may be fea-sible but the results cannot accurately reflect the real fea-tures of the contact force under the dynamic condition Toovercome the deficiency of the single-point coupling meshmodel Wang [18] proposed a multipoint coupling meshmodel in which each mesh interface is represented by therespective effective mesh point line of action and meshstiffness e mesh properties and dynamic responses fromthe model were then compared with those from a meshmodel whose parameters are obtained from pitch conedesign [19]e differences between them stem from the factthat the mesh positions and line of action vectors which aretime-variant for the multipoint coupling mesh model areconstant for the mesh model based on the pitch cone designconcept e authors also believed that the former is capableof describing the exact mesh properties of loaded gearswhile the latter is only valid for no-load and light-loadedconditions as the mesh parameters of them are obtainedunder loaded and unloaded conditions respectively So farthe multipoint coupling mesh model has not been used aswidely as the single-point coupling mesh model for dynamicanalysis probably because most studies focus on the overalleffect of mesh characteristics on dynamics of a gear pairrather than the detailed dynamic responses of each meshingtooth pair Among recent studies maybe the multipointcoupling mesh model was only applied by Wang et al [20]eir research proved that the model has the capability ofpredicting the DMF-time history for each tooth pair within afull tooth engagement cycle Even so because the model stilluses the transmission error of a gear pair to determine thetooth contact and contact deformation of each engagedtooth pair the results cannot properly reflect the real contactstate of each pair of teeth when multiple pairs of teeth are inthe zone of contact

e accuracy of DMF calculation results is to a largeextent dependent on the accuracy of mesh stiffness edeformation of an engaged gear pair or an individualcontacting tooth pair is nonlinear with the applied loadbecause of the nonlinearity of the local contact deformationwith the contact force It means that mesh stiffness shouldvary with mesh deformation to accurately describe thenonlinear force-deflection relationship However in pre-vious studies although the variation of mesh stiffness withthe mesh position was considered the change in meshstiffness with deformation at a given mesh position was

rarely taken into consideration in DMF calculation ereare generally two methods to calculate mesh stiffness thecommonly used one is called the average slope approach andthe other which is seldom applied but may be more ap-propriate for dynamic analysis is called the local slopeapproach [21] e mesh stiffness evaluated by the twoapproaches can both be applied in DMF estimation but toaccurately calculate the DMF an error-prone point whichshould be paid enough attention is that the calculationformulas of DMF are entirely different for different types ofmesh stiffness applied It is worth mentioning that not alldynamic models utilize prespecified mesh stiffness for thecalculation of DMF In researches on the dynamics ofparallel axis gears [22 23] for example the mesh stiffness isevaluated according to instantaneous mesh deflection ateach time step of the dynamic simulationis kind of modelis referred to as the coupled dynamics and contact analysismodel as the dynamic and contact behaviors are mutuallyinfluential e variation of mesh stiffness with deformationis realized in this kind of model However the model iscurrently not computationally feasible for spiral bevel gearsbecause of the more complex and time-consuming TCAcomputation

is study aims to explore appropriate methods for moreaccurate prediction of the DMF of each meshing tooth pairof spiral bevel gears through comparative analyses consid-ering the two factors mentioned above First an improvedmesh model of a tooth pair (MMTP) which represents thevariation of the mesh properties of a tooth pair throughout afull engagement cycle was proposed e model applies thetransmission error curve of a tooth pair instead of that of agear pair to estimate mesh stiffness and identify the contactfor each couple of teeth is improvement enables themodel to capture more realistic mesh characteristics of eachcontacting tooth pair than the multipoint coupling meshmodel Second we compared the dynamic responses fromthe proposed mesh model and the widely used single-pointcoupling mesh model for a generic spiral bevel gear pair onthe one hand to validate the proposed one and on the otherhand to demonstrate the superiority of the proposed modelin the evaluation of DMF per tooth pair It is shown that thetwo models predict the similar variation tendencies of DMFresponse with the mesh frequency Although some differ-ences exist in the response amplitudes particularly in ahigher mesh frequency range the resonance frequenciesestimated by the two models are almost the same Besidesthe time histories of DMF per tooth pair predicted by the twomodels are significantly different but the results from theproposed model can provide much better insight into dy-namic contact behavior of each contacting tooth pair irdwe performed a comparative study on three calculationapproaches of DMF to further investigate the effect of meshstiffness representations on DMF calculations One methoduses the mesh stiffness estimated by the average slope ap-proach another one applies the mesh stiffness evaluated bythe local slope approach and the third method utilizes aquasistatically defined interpolation function indexed by themesh deflection and the pinion roll angle Weighing theaccuracy and cost of the computation we found that the

2 Shock and Vibration

local slope approach is preferred for DMF evaluation ofspiral bevel gears

2 Mesh Model

is study applies the mesh model with prespecified meshparameters obtained through quasistatic LTCA which isperformed with the aid of the commercial finite elementanalysis software ABAQUS [24] For the development offinite element models the technique without the applicationof the CAD software proposed by Litvin et al [1] is adoptedIn this way not only are the nodes of the finite element meshlying on tooth surfaces guaranteed to be the points of theexact tooth surfaces (which are calculated through themathematical model of tooth surface geometry [25]) but alsothe inaccuracy of solid models due to the geometric ap-proximation by the spline functions of the CAD software canbe avoided e procedure of using ABAQUS for LTCA hadbeen introduced in detail by Zhou et al [4] and Hou andDeng [26] and thus will not be described here e cor-rectness of the finite element model is verified by comparingthe obtained kinematic transmission error (KTE) curve withthat obtained by traditional TCA as shown in Figure 1where the two curves are approximately consistent A pair ofmismatched face-milled Gleason spiral bevel gears is se-lected to be the example for analysis e contact pattern islocated in the center of the tooth surface and there is nocontact in the extreme part including the tip toe and heel ofthe teeth Table 1 lists the design parameters with which themachine tool setting parameters used to derive tooth ge-ometry are obtained by Gleason CAGE4Win

For the convenience of the following discussions thecoordinate systems (Figure 2) used in this study are in-troduced first e LTCA is performed in the fixed referenceframe Oxyz whose origin is at the intersection point of thepinion and gear rotation axes Other coordinate systems areused for dynamic analysis e coordinate systemsOpx0

py0pz0

p and Ogx0gy0

gz0g are the local inertial reference

frames of the pinion and gear respectively whose originsand z-axes coincide with the mass centroids and rolling axesof each body under the initial static equilibrium conditione body-fixed coordinate systems Cpx4

py4pz4

p andCgx4

gy4gz4

g for the pinion and gear are also located with theirorigins at the mass centroids and all the coordinate axes areassumed to coincide with the principal axes of the inertia ofeach body

21 MeshModel of a Gear Pair (MMGP) In the single-pointcoupling mesh model (Figure 3) the two mating gears arecoupled by a single spring-damper element whose stiffnesskm damping cm acting point and direction vary with re-spect to the angular position of the pinion θp e springstiffness km which is the function of bending shear andcontact deformations of all meshing tooth pairs in the zoneof contact and the base rotation compliance represents theeffective mesh stiffness of a gear pair e KTE ek due totooth surface modification machining error and assemblymisalignment also changes with the pinion roll angle θp

while the backlash b is often regarded as a constant forsimplicity

e synthesis of the single-point coupling mesh modelrequires reducing the distributed contact forces obtained byLTCA to an equivalent net mesh force Fm(Fmx Fmy Fmz)

(Figure 4(a)) e direction of the net mesh force is definedas the line of action vector Lm which points from the pinionto the gear and the point of action is the effective mesh pointrepresenting the mean contact position e calculationmethod of contact parameters applied in previous studies[10ndash14] can be used here while if LTCA is undertaken usingABAQUS another similar but more convenient approachcan be applied As long as the history outputs of CFN CMNand XN for the contact pair are requested [24] the effectivemesh force Fmi(Fmxi Fmyi Fmzi) and the mesh momentMmi(Mmxi Mmyi Mmzi) due to contact pressure on eachmeshing tooth pair i and the mesh force acting point vectorrmi(xmi ymi zmi) can be obtained directly at each specifiedtime step us the magnitude of equivalent net mesh forceFnet is calculated by

Fnet F2

mx + F2my + F2

mz

1113969

Fml 1113944

N

i1Fmli l x y z

(1)

e line of action vector Lm(nx ny nz) of the net meshforce is obtained by

nl Fml

Fnet

l x y z (2)

e total contact-induced moments about each co-ordinate axis are as follows

Mml 1113944

N

i1Mmli l x y z (3)

e effective mesh point position rm(xm ym zm) can becalculated from the following equation

xm 1113936

Ni1xmiFmi

1113936Ni1Fmi

zm Mmy + Fmzxm1113872 1113873

Fmx

ym Mmx + Fmyzm1113872 1113873

Fmz

(4)

In above-mentioned equations N is the number ofengaged tooth pairs at each pinion roll angle In this workthe single-point coupling mesh model is named the meshmodel of a gear pair (MMGP) as it reflects the overall meshproperties of a gear pair

22 Mesh Model of a Tooth Pair (MMTP) In the multipointcoupling mesh model (Figure 5(b)) [18] there are a variablenumber of spring-damper units coupling the pinion andgear in each mesh position e spring-damper element

Shock and Vibration 3

represents the effective mesh point line of action and meshstiffness of each mesh interface and the number of elementswhich depends on the gear rotational position and loadtorque indicates the number of tooth pairs in the zone ofcontact is model describes the variation of the meshcharacteristics of each tooth pair in the contact zone over ameshing period and thereby the mesh characteristics of theindividual tooth pair can be considered the basic unit Based

on this point of view the mesh model of a tooth pair(MMTP) (Figure 5(a)) which describes the mesh charac-teristic variation of a tooth pair over one tooth engagementcycle is proposed e MMTP facilitates the development ofmodels handlingmore complex situations such as themodelwith nonidentical mesh characteristics of the tooth pair (egone or several teeth have cracks surface wear or any otherfeatures influencing mesh properties)

e synthesis of the effective mesh point and line ofaction for the MMTP is the same as that for themultipoint coupling mesh model (Figure 4(b)) [18] How-ever as mentioned before the effective mesh force Fmi

(Fmxi Fmyi Fmzi) due to contact pressure on each meshingtooth pair i and the corresponding acting point vectorrmi(xmi ymi zmi) can be obtained directly from LTCAperformed with ABAQUS e direction of the effectivemesh force is defined as the line of action for each contactingtooth pair and the acting point vector is just the effectivemesh point position vector e calculation approach ofmesh stiffness and DMF for the MMTP is modified to makethe model describe the meshing state of each pair of teethmore accurately than the multipoint coupling mesh modelese will be elaborated in the following correspondingsections

0 02 04 06 08 1 12Pinion roll angle (rad)

ndash16ndash14ndash12ndash10


024

Ang

ular

TE

(rad

)

times10ndash5

DA B C

Contact zone 3Contact zone 1

Profileseparationof toothpair 2

Contact zone 2


KTE-tooth pair 3KTE-tooth pair 2

LTE-gear pairKTE-gear pairKTE-tooth pair 1

Figure 1 Angular transmission error curves

O

y

x

z

Og

Cg

Cp

x0g

x0p

z0p

y0p

x4p

x4g

z4p

z4g

z0g y4

py4g

Op

y0g

Figure 2 Coordinate systems of a spiral bevel gear pair

Table 1 Design parameters of the example spiral bevel gear pair

Parameters Pinion (left hand) Gear (right hand)Number of teeth 11 34Helical angle (rad) 0646Pressure angle (rad) 0349Pitch angle (rad) 0313 1257Pitch diameter (mm) 715 221Pitch cone distance (mm) 116139Width of the tooth (mm) 35Outer transverse module 65Virtual contact ratio 1897

Instantaneous line of action at angular position i

Instantaneous line of action at angular position i + 1

Gear

Pinion

km (θp)

ek (θp)

Lm (θp)

b

rpm (θp)

cm (θp) rgm (θp)

Figure 3 Single-point coupling mesh model


23 Comparison of Mesh Parameters In addition to com-paring the mesh properties embodied on the individualtooth pair and the gear pair the effect of the applied load ontheir variations was investigated To this end we synthesizedthe mesh parameters for the MMTP and the MMGP at50Nm and 300Nm torque loads e mesh vectors whichrepresent the magnitude direction and point of action ofthe equivalent mesh force at each mesh position throughoutone tooth engagement cycle for the MMTP and over onemesh cycle for the MMGP are presented in Figures 6(a) and6(b) respectively As can be seen the magnitude of theequivalent mesh force for the MMTP (ie the effective meshforce on a single pair of meshing teeth) varies obviously withthe mesh position while only the tiny change can be ob-served in the size of the equivalent net mesh force for theMMGP (ie the combined mesh force of all engaged toothpairs) e former is mainly attributed to the load sharingamong the tooth pairs in contact simultaneously whereasthe latter is only caused by small changes in the directionalrotation radius (which can be regarded as the arm of force) at

different mesh positions Figure 6 also shows that the di-rection of the equivalent mesh force changes little for boththe MMTP and the MMGP

Furthermore the line of action directional cosines andeffective mesh point coordinates of the two models over afull tooth (tooth pair 2) engagement cycle are compared inFigure 7 It shows clearly that all mesh parameters of thetooth pair are altered significantly with the gear rotationrelative to the corresponding changes in the mesh param-eters of the gear pair In addition from both Figures 6 and 7we can observe that the range of change in mesh parametersof the gear pair becomes small under the heavy-loadedcondition (300Nm) but no obvious change for the meshparameters of the tooth pair was observed A reasonableexplanation is as follows with the increase of load theextended tooth contact appears because of the deformationof gear teeth It is manifested in the advance engagement ofthe incoming tooth pair 3 and the deferred disengagement ofthe receding tooth pair 1 in Figure 7 is change increasesthe contact ratio and meanwhile decreases the amplitude of

Lmi (θp)

bi eki (θp)

kmi (θp)

cmi (θp)

(a) (b)

Figure 5 Illustrations of the (a) mesh model of a tooth pair and (b) multipoint coupling mesh model

x

yz

Mm

Effective mesh point of a gear pair

Fm (Fmx Fmy Fmz)

rm (xm ym zm)O

(a)

O

x

y zMmi

Effective mesh point of a tooth pair

Fmi (Fmxi Fmyi Fmzi)

Fmi+1 (Fmxi+1 Fmyi+1 Fmzi+1)

rmi (xmi ymi zmi)

rmi+1 (xmi+1 ymi+1 zmi+1)

(b)

Figure 4 Synthesis of the effective mesh parameters of the (a) single-point coupling mesh model and (b) multipoint coupling mesh model


40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)

Figure 6 Mesh vectors of the MMTP (a) and the MMGP (b)

Tooth pair 1 at 300NmTooth pair 2 at 300NmTooth pair 3 at 300NmGear pair at 300Nm

02 03 04 05 06 07 08 0901 111Pinion roll angle (rad)


ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued




02 03 04 05 06 07 08 09 1 1101Pinion roll angle (rad)

0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued


the variation of the mesh parameters for the gear pair Notethat it does not influence the mesh frequency of the gear pairbecause according to the compatibility condition the toothrotation due to deformation for each couple of teeth shouldbe the same to maintain tooth contact and continuity inpower transmission and hence the tooth pitch does notchange Overall the results indicate that although the time-varying mesh characteristics exhibited on each pair of teethare prominent their overall effect on the gear pair isweakened and this weakening will become more pro-nounced as the contact ratio increases

24 Remark on the Mesh Model Based on Quasistatic LTCAFrom the above analysis it is substantiated again that themesh parameters of spiral bevel gears are load-dependenterefore the mesh model based on quasistatic LTCA issuperior to the mesh model relying on the geometric re-lationship in terms of describing the mesh properties ofloaded gear pairs However as the mesh parameters areestimated at a constant applied torque the mesh model istheoretically only suitable for the dynamic analysis of a gearpair subjected to this torque load and it seems to be in-competent in the study of a gear pair subjected to a time-variant torque load Nevertheless it has been discovered thatthe mesh stiffness shows more prominent time-varyingparametric excitation effects than the mesh vector [11 13]erefore a compromising way to calculate the DMF isapplying the mesh vector predicted at the mean appliedtorque and the calculation approach using the mesh forceinterpolation function (as introduced in DMF Calculation)which takes the load effects on mesh stiffness intoconsideration

3 Mesh Stiffness Evaluation

e force-deflection curve (a typical one is illustrated inFigure 8) describes the nonlinear relationship between the

force and the deformation precisely erefore using thiscurve to predict the DMF for a known mesh deformationcould be the most accurate approach However it is by nomeans the most efficient method since the establishment ofthe curve requires a series of time-consuming LTCAs atdifferent load levels Usually the DMF for a given torqueload is predicted by using the mesh stiffness evaluated at thesame load Note that although at each rolling position themesh stiffness is a constant it varies with respect to meshpositions Hence the mesh stiffness has a spatially varyingfeature

In general there are two calculation approaches for meshstiffness namely the average slope approach and the localslope approach [21] e corresponding calculated meshstiffness is named the average secant mesh stiffness (ASMS)




ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )

Figure 7 Comparison of time-varying line of action directional cosines (andashc) and effective mesh point coordinates (dndashf) between theMMTP and the MMGP at 50Nm and 300Nm torque loads

Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)

0005 001 0015 002 00250Mesh deflection (mm)

δd

δT

kTma = Fm

T δT

kTml = dFmdδ|δ=δT

Figure 8 A typical fitted force-deflection curve and illustration ofthe local tangent mesh stiffness (LTMS) and the average secantmesh stiffness (ASMS)


and local tangent mesh stiffness (LTMS) in this paper Asshown in Figure 8 they are the slope of the secant (bluedashed line) and the slope of the tangent (red dash-dottedline) of the force-deflection curve respectively

31 Average Secant Mesh Stiffness e average secant meshstiffness has been extensively used to calculate DMF fordifferent types of gears such as spur gears [27ndash29] helicalgears [29 30] and spiral bevel or hypoid gears [9 1013ndash18 20] It is calculated through dividing the magnitudeof mesh force Fm by the mesh deflection along the line ofaction δ

kma Fm

δ (5)

e difference between the translational KTE ε0 and theloaded transmission error (LTE) ε can be attributed to themesh deflection δ

δ ε0 minus ε (6)

In practice the KTE always exists for spiral bevel gearseven if there are no machining and assembly errors becausegear tooth flank profiles are often modified to be mis-matched to avoid contact of extreme parts (top heel or toe)of the teeth and to reduce the sensitivity of transmissionperformance to installation and machining errors Becauseof the mismatch the basic mating equation of the contactingtooth flanks is satisfied only at one point of the path ofcontact It has been proved that although the KTE is alwayssmall compared to the LTE the KTE cannot be omitted inthe calculation of mesh stiffness especially for the light-loaded condition [13]

e translational KTE and LTE ε0 and ε are the pro-jections of the corresponding angular transmission errors e0and e along the line of action As y-axis of the coordinatesystem Oxyz is the gear-shaft rotation axis (Figure 2) atwhich the angular transmission error is expressed about thetranslational KTE and LTE are calculated by

ε0 e0λy

ε eλy(7)

where λy nxzm minus nzxm is the directional rotation radiusfor the angular transmission error Note that equations(5)ndash(7) are derived based on the assumption that the meshvectors for unloaded and loaded conditions are identicalHowever the results shown in Figure 7 indicate that thehypothesis deviates from the fact and thereby inevitablybrings about calculation errors

e angular KTE and LTE e0 and e can be determined as

e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)

where Δθp0 and Δθg0 (obtained by TCA) are the unloadedrotational displacements of the pinion and gear relative to

the initial conjugated contact position where the trans-mission error equals zero Δθp and Δθg (obtained by LTCA)are the corresponding loaded rotational displacements andNp and Ng are the tooth numbers of the pinion and gearrespectively

Equations (5)ndash(8) can be directly applied to calculate theASMS of a gear pair In this case Fm is the magnitude of thenet mesh force of the gear pair and δ is the total deformationof all meshing tooth pairs in the zone of contact

For the MMTP the ASMS of a single tooth pair needs tobe estimated e effective mesh force on each meshingtooth pair Fmi can be obtained directly through LTCA whilethe troublesome issue is how to obtain the mesh deflection ofa single tooth pair when multiple sets of teeth are in contactIn this work the compatibility condition is applied to ad-dress this problem It is based on the fact that the total toothrotation Δθ (which is the sum of the composite deformationof the tooth and base ΔB caused by bending moment andshearing load contact deformation ΔH and profile sepa-ration ΔS divided by the directional rotation radius aboutthe rolling axis λ) must be the same for all simultaneouslyengaged tooth pairs to maintain the tooth contact andcontinuity in power transmission [2] If three pairs of teethare in contact the compatibility condition can be expressedbyΔSiminus 1 + ΔBiminus 1 + ΔHiminus 1

λiminus 1ΔSi + ΔBi + ΔHi

λi

ΔSi+1 + ΔBi+1 + ΔHi+1

λi+1 Δθ

(9)

If the ASMS of a tooth pair is defined as the mesh forcerequired to produce a unit total deformation (which con-tains the composite deformation of the tooth and base ΔBand the contact deformation ΔH) equation (9) can be re-written in a form with the ASMS of each meshing tooth pairkmai as follows

ΔSiminus 1 + Fmiminus 1kmaiminus 11113872 1113873

λiminus 1ΔSi + Fmikmai1113872 1113873

λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)

32 Local Tangent Mesh Stiffness e local tangent meshstiffness can be approximately calculated by the centraldifferencemethod which requires the LTCA to be performedat three different torque loads one at the nominal torqueload another above it and the third below it

kml dFm

dδ δδT

1113868111386811138681113868 asymp12

FTm minus FTminus ΔT

m

δT minus δTminus ΔT +FT+ΔT

m minus FTm

δT+ΔT minus δT1113890 1113891 (11)

where ΔT is a specified small change in torque load Asδ ε0 minus ε equation (11) can be transformed to


kml asymp12


m

εTminus ΔT minus εT+

FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)

Equation (12) can be directly used to calculate the LTMSof both a gear pair and a tooth pair as the profile separationof each tooth pair ΔSi which influences the calculation of theASMS of a tooth pair does not function in the calculation ofthe LTMS of a tooth pair e reason is that the KTE ε0which contains the profile separation does not appear inequation (12) In contrast with the calculation of the ASMSfor the evaluation of the LTMS assuming themesh vectors atthe torque loads of T T + ΔT and T minus ΔT are identical isreasonable when ΔT is small

33 Example Taking the evaluation of the mesh stiffness ata torque load of 300Nm as an example we demonstrate thegeneral calculation approaches of the ASMS and LTMSFigure 9 shows the effective mesh force of three consec-utively engaged tooth pairs and the net mesh force of thegear pair over one tooth engagement cycle Note that theactual length of the period of one tooth engagement islonger than the one shown in Figure 9 since tooth pair 2comes into contact at some pinion roll angle less than 0171radians and leaves the contact at some pinion roll anglegreater than 1107 radians By assuming that at the be-ginning and end of a tooth engagement cycle the meshforce of each meshing tooth pair increases and decreaseslinearly the pinion roll angles where tooth contact beginsand ends can be approximately predicted through linearextrapolation Using the data given in Figure 9 we canpredict that tooth pair 2 engages at approximately 0159radians and disengages at nearly 1110 radians and toothpair 3 engages at about 0730 radianserefore if the pitcherror is not considered the tooth meshing is repeated every0571 radians and the actual length of one tooth engage-ment cycle is 0951 radians With the known roll angles atthe beginning of engagement and disengagement meshparameters at the two points can be predicted In this workthe LS-SVMlab toolbox [31] is utilized to estimate thefunctions of mesh parameters with respect to the rotationangle of the pinion tooth and then the parameters at thestarting positions of engagement and disengagement areevaluated by the estimated functions

Figure 1 shows the angular KTE and LTE of the examplegear pair e KTE of the gear pair is obtained by LTCAperformed at a small torque load is method is named theFEM-based TCAe torque load should be carefully chosen(3Nm in this work) to keep the teeth in contact but not tocause appreciable deformation e KTEs of tooth pairs 1 2and 3 which are necessary for the calculation of the meshdeflection of the tooth pair are acquired by traditional TCA[32] (to the authorsrsquo knowledge the complete KTE curve of atooth pair cannot be obtained by the FEM-based TCA) eappropriateness of the selected torque for FEM-based TCAcan be validated if the obtained KTE is approximatelyconsistent with KTEs acquired by traditional TCA It can beseen from Figure 1 that in contact zone 1 the KTE (absolutevalue) of tooth pair 1 is less than that of tooth pair 2 which

means in this region tooth pair 1 is in contact whereas toothpair 2 is not in contact in the no-loading condition It isbecause the mating tooth surfaces of tooth pair 2 are sep-arated from each other by the profile separation which canbe measured by the difference in KTEs between tooth pairs 1and 2 Likewise for contact zones 2 and 3 only tooth pairs 2and 3 are in contact respectively in the unloaded conditionat is why the KTEs of the gear pair in contact zones 1 2and 3 correspond to the KTEs of tooth pairs 1 2 and 3respectively

e total deformation of a gear pair is usually consideredthe difference between the KTE and the LTE of the gear pairis difference however is not always themesh deflection ofa tooth pair caused by the load it bears e schematicdiagram for illustrating the deflection of the meshing toothpair is shown in Figure 10 As previously mentioned at anyrolling position in contact zone 1 (shown in Figure 1) onlytooth pair 1 is in contact in the no-loading condition and themating tooth surfaces of tooth pair 2 are separated by theprofile separation as shown in Figure 10 With the augmentof torque load the LTE of the gear pair increases graduallybecause of the mesh deflection of tooth pair 1 If the LTE ofthe gear pair at any rolling position in contact zone 1 isgreater than the KTE of tooth pair 2 which means the profileseparation of tooth pair 2 at this position has been closedbecause of tooth rotation tooth pair 2 will be in contact andshare the whole load with tooth pair 1 erefore as illus-trated in Figure 10 part of the rotation of tooth pair 2compensates the profile separation between the matingflanks and the remaining part is the result of mesh deflectionAt any rolling position in contact zone 2 as tooth pair 2 is incontact initially the tooth rotation of tooth pair 2 is whollyattributed to its deflection In contact zone 3 however theprofile separation of tooth pair 2 must be closed beforecontact and hence the remaining part of the tooth rotationwhich excludes the angular profile separation is the rota-tional deformation of tooth pair 2 Based on above analysisit can be speculated that at the torque load of 300Nm toothpairs 1 and 2 are in contact in region AB only tooth pair 2 isin contact in region BC and tooth pairs 2 and 3 are incontact in region CD is speculation is proved to be trueby the time histories of mesh force on each engaged toothpair shown in Figure 9

Above discussions elaborate the meaning of equation(10) Namely the tooth rotations Δθ for all engaged toothpairs are the same and equal the rotation of the pinion andgear blanks which is measured by the difference between theangular KTE and LTE of the gear pair while the meshdeflection along the line of action δi for the tooth pair i dueto the load it shares should be calculated by subtracting theprofile separation ΔSi from the projection of the tooth ro-tation Δθ along the line of action Actually the mesh de-flection for a tooth pair equals the difference between theKTE of the tooth pair and the LTE of the gear pair as shownin Figure 1erefore in the MMTP the KTE of a tooth pairis applied instead of the KTE of a gear pair in the estimationof mesh stiffness and the identification of the contact foreach pair of teeth It is the main difference from the mul-tipoint coupling mesh model


Figure 11 shows the mesh deflections along the line ofaction for the gear pair and each tooth pair e toothengagement cycle is divided into regions where two meshingtooth pairs are in contact and one meshing tooth pair is incontact (denoted by DTC and STC respectively inFigures 11ndash13) In the double-tooth contact zone althoughthe rotational deformation of the gear pair equals the ro-tational deformation of the tooth pair which is initially incontact in the unloaded condition their translational de-flections along the line of action are somewhat differentisis because in the double-tooth contact zone the effectivemesh point of the gear pair is different from that of the toothpair as shown in Figure 4 and thus the corresponding linesof action of the net mesh force acting on the effective meshpoint are different as well

Figure 12 compares the ASMS calculations of the gearpair and the tooth pair It can be seen that in the double-tooth contact zone the ASMS of the gear pair experiences agradual increase and then a steady decrease e variationsignificantly differs from that of unmodified spur gearswhose mesh stiffness presents a sharp increase and decreaseat the instants when the approaching tooth pair comes intocontact and the receding tooth pair leaves the mesh re-spectively [21] Apart from that with the contact ratio

increase the variation of the mesh stiffness of the gear pairtends to become smaller is conclusion can be proved bythe comparison of the ASMS calculations between this studyand studies of Peng [13] and Wang et al [20]

An important fact shown in Figure 12 is that in thedouble-tooth contact zone the ASMS of a gear pair does notequal the sum of the ASMS of each meshing tooth pair if theKTE of a tooth pair is applied to estimate the mesh stiffnessof a tooth pair except for the position where the twomeshing tooth pairs are in contact in the unloaded condi-tion such as point A in the graph e reason is explained indetail in Appendix A In many previous studies on meshstiffness calculation of spur gears [33ndash36] the total meshstiffness of the gear pair in the multiple-tooth engagementregion is calculated by summation of the mesh stiffness of alltooth pairs in contact is estimation method is probablyinapplicable for spiral bevel or hypoid gears in terms of theASMS evaluation whereas it is proved to be feasible for theLTMS calculation

For the calculation of the LTMS expressed in equation(12) LTCA has to be performed at the torque loads of TT + ΔT and T minus ΔT to obtain the corresponding LTEs εTεT+ΔT and εTminus ΔT Since the mesh parameters are all load-dependent ΔT should be properly chosen to ensure that theeffective mesh points and the directions of the line of actionare almost the same for the three torque loads at each rollingposition However it does not mean that the smaller the ΔTis the better the result will be since the effect of numericalcalculation errors on the calculation results will becomemore pronounced as ΔT decreases In the present studyΔT 10Nm can weigh the two factors mentioned aboveFigure 13 compares the LTMS calculations of the gear pairand the tooth pair It can be seen that in the double-toothcontact zone the LTMS of the gear pair approximatelyequals the sum of the LTMS of simultaneously contactingtooth pairs e reason is mathematically clarified in Ap-pendix A Comparing calculations in Figures 12 and 13 wecan observe that for both the mesh stiffness of a gear pairand a tooth pair the LTMS is larger than the ASMS over theentire tooth engagement cycle It is consistent with the

5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584

Pinion roll angle (rad)

One tooth engagement

Tooth pair 1Tooth pair 2

Tooth pair 3Gear pair

Figure 9 Finite element calculation of the effective mesh force of the tooth pair and the net mesh force of the gear pair over one toothengagement cycle

Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear

Figure 10 Deflection of the meshing tooth pair at the meshingposition in contact zone 1


theoretical prediction from the concave force-deflectioncurve shown in Figure 8 It has been known that thenonlinear contact deformation due to the growth of the

contact area as applied load increases results in the concaveshape of the force-deflection curve Note that the differencebetween the computed results of the ASMS and LTMS is the

DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)

02 04 06 08 1 120Pinion roll angle (rad)



Figure 11 Mesh deflection along the line of action of the gear pair and the tooth pair over one tooth engagement cycle at 300Nm appliedtorque

DTC STC DTC

Tooth pair 1Tooth pair 2Tooth pair 3

Gear pair

times105

02 04 06 080 121Pinion roll angle (rad)

1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05

Figure 12 e ASMS of the gear pair and the tooth pair over one tooth engagement cycle at 300Nm applied torque




times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)

Figure 13 e LTMS of the gear pair and the tooth pair over one tooth engagement cycle at 300Nm applied torque


result of their definitions and calculation approaches and wecannot make a conclusion about which prediction is moreaccurate Both the ASMS and LTMS can be used to calculateDMF but the corresponding calculation formulas are en-tirely different

4 DMF Calculation

41 Calculation Approach Using ASMS A common repre-sentation of DMF Fm d utilizing the ASMS is [13 14 18 20]

Fmd

kma εd minus ε0( 1113857 + cm _εd minus _ε0( 1113857 εd minus ε0 gt 0( 1113857

0 minus ble εd minus ε0 le 0( 1113857

kma εd minus ε0 + b( 1113857 + cm _εd minus _ε0( 1113857 εd minus ε0 lt minus b( 1113857

⎧⎪⎪⎨

⎪⎪⎩

(13)

where kma and cm are the ASMS and proportional meshdamping and εd and ε0 are the translation form of the dy-namic transmission error and KTE along the line of actione gear backlash b is expressed on the coast side here It isnoted that here the translational KTE ε0 should be de-termined by ε0 ((NpNg)Δθp0 minus Δθg0)λ which is differentfrom the traditional definition of the KTE (equation (8))

e DMF is composed of the elastic contact force Fma

kma(εd minus ε0) and the viscous damping force Fc cm(_εd minus _ε0)e elastic contact force Fma is illustrated by the blue dashedline in Figure 8 It can be observed that as long as the dynamicmesh deflection δd is not equal to the static mesh deflectionδT caused by the nominal load the calculated contact forcewill differ from the actual one Fm e deviation can beacceptable if the dynamic mesh deflection fluctuates aroundthe nominal static mesh deflection within a small rangeHowever as for moderate and large amplitude vibrationswhere a substantial difference exists between δd and δT thevalidity of Fma may be questioned

42 Calculation Approach Using LTMS As shown in Fig-ure 8 for small mesh deflections relative to the nominalstatic deflection δT the elastic contact force Fml (demon-strated by the red dash-dotted line) evaluated by using theLTMS is closer to the true value Fm than the contact forceFma erefore in this case it could be more appropriate toapply the LTMS to calculate DMF e DMF Fm d consid-ering backlash nonlinearity can be defined as

Fmd

Fm0 + kml εd minus εl( 1113857 + cm _εd minus _εl( 1113857 Fr gt 0( 1113857

0 Fr le 0 and εd minus ε0 ge minus b( 1113857

kma εd minus ε0 + b( 1113857 + cm _εd minus _ε0( 1113857 Fr le 0 and εd minus ε0 lt minus b( 1113857

⎧⎪⎪⎨

⎪⎪⎩

(14)

where kml is the LTMS and εl is the translational LTE alongthe line of action e nominal contact force Fm0 can beacquired from LTCAe reference force Fr which is used todetermine the operational condition is given by

Fr Fm0 + kml εd minus εl( 1113857 (15)

Since the calculation approach using the LTMS is onlysuitable for small mesh deflections about the nominal static

deflection this approach is only applied to calculate theDMF for the drive side while the DMF for the coast side isstill estimated by the approach using the ASMS It has beenfound that mesh properties for the coast side have less effecton dynamic responses than those of the drive side and theirinfluences appear only in double-sided impact regions forlight-loaded cases [12] erefore if the evaluation of DMFfor the drive side is of primary interest it is reasonable toignore the mesh characteristic asymmetry In this study themesh parameters of the drive side are used to approximatelyevaluate the DMF of the coast side for simplicity

It is noteworthy that the representations of the elasticcontact force in equations (13) and (14) are fundamentallydifferent and hence it is incorrect to apply the ASMS inequation (14) and apply the LTMS in equation (13) Forexample the use of the LTMS in equation (13) will give afalse result Fme (demonstrated by the green dash-dotted linein Figure 8) which differs from the actual one Fm

dramatically

43 Calculation Approach Using Mesh Force InterpolationFunction As illustrated in Figure 8 both the calculationapproach using the ASMS and the one applying the LTMSwill lose accuracy when the dynamic mesh deflection sub-stantially differs from the nominal static mesh deflectionbecause at any mesh position the elastic contact is modeledas a linear spring and therefore the nonlinear force-de-flection relationship is ignored e effect of such simpli-fication will be examined later by comparing the DMFcalculated by the two approaches with the DMF interpolatedfrom the force-deflection function Since the mesh force alsovaries with the contact position of meshing tooth pairs theinterpolation function indexed by the nominal mesh posi-tion and the mesh deflection at this position is requiredismethod is also applied by Peng [13] and recently by Dai et al[37] to the prediction of the total DMF of hypoid gear pairsand spur gear pairs respectively Here the interpolationfunction for the mesh force of individual tooth pairs at eachmesh position over one tooth engagement cycle is developedthrough a series of quasistatic LTCAs with the appliedtorque load ranging from 10Nm to 3000Nm is range farexceeds the nominal load (50Nm and 300Nm in the presentwork) that the gear bears because we need to obtain therelationship between the mesh force and the mesh deflectionat the beginning and end of the tooth meshing cycle underthe nominal load Looking back at Figure 6(a) we canobserve that the mesh force at these two positions will bezero if the applied load is below the nominal load or be verysmall if the applied load is just over the nominal loaderefore in order to obtain sufficient data for the devel-opment of the force-deflection relationship at these twopositions the range of the applied torque load should bewide enough e force-deflection function at each meshposition over one tooth engagement cycle is graphicallyshown in Figure 14 Note that even when the load applied forLTCA is 3000Nm the data for the mesh deformation ex-ceeding 0004mm at the position of tooth engagement anddisengagement are not available Likewise because of load


sharing the data points marked in Figure 14 cannot beobtained by LTCA with the maximum applied load of3000Nm ese marked data are estimated by the LS-SVMlab toolbox [31] based on the data obtained from LTCAAlthough the predicted data may not be very precise theyhave a marginal effect on the steady-state dynamic responsesince the dynamic mesh deformation of a tooth pair in theregion of tooth engagement and disengagement is also smalland thereby the precise data from LTCA are used

e interpolated DMF Fm d is expressed as follows

Fm d

fd εd minus ε0 θt( 1113857 + cm _εd minus _ε0( 1113857 εd minus ε0 gt 0( 1113857


fc εd minus ε0 + b θt( 1113857 + cm _εd minus _ε0( 1113857 εd minus ε0 lt minus b( 1113857

⎧⎪⎪⎨

⎪⎪⎩

(16)

where fd and fc represent the interpolation functions for thedrive and coast sides which are the same in this work εd minus ε0and εd minus ε0 + b express the dynamic mesh deformation along

the line of action for the drive and coast sides respectivelyand θt is the mesh position of a tooth pair

5 Gear Pair Dynamic Model

51 A 12-DOF Lumped-Parameter Model e gear pairmodel is illustrated in Figure 15(a) Each gear with itssupporting shaft is modeled as a single rigid body e gear-shaft bodies are each supported by two compliant rollingelement bearings placed at arbitrary axial locations In thecurrent case the pinion is overhung supported while thegear is simply supported

As shown in Figure 15(b) the position of each gear-shaftbody in the space is determined by its centroid position Cj

and the attitude of the body relative to the centroid whichare defined by the translational coordinates (xj yj zj) andthe Cardan angles (αj βj cj) respectively e rotationprocess of the body about the centroid described by theCardan angles is as follows the body firstly rotates about thex1

j-axis with the angle αj from the original position indicated

18000

16000

14000

12000

10000

8000

6000

4000

2000

0024 0018Deformation along LOA (mm) Tooth pair mesh position (rad)

0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)

Figure 14 Relationship between the tooth contact force and its corresponding deformation along the line of action at each mesh position

A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)

Figure 15 (a) Gear pair model and (b) coordinate systems of each gear-shaft body


by the coordinate system Cjx1jy1

jz1j to the position repre-

sented by the coordinate system Cjx2jy2

jz2j and then it ro-

tates about the y2j-axis with the angle βj reaching the

location shown by the coordinate system Cjx3jy3

jz3j e last

rotation is about the z3j-axis with the angle cj and the body

reaches the final instantaneous position represented by thecoordinate system Cjx

4jy4

jz4j Generally αj and βj are small

angles while cj is the large rolling angle of the gear In all thecoordinates mentioned above and the following equationsthe subscript j p g indicates that the quantities belong tothe pinion or gear component e derivation of theequations of motion is demonstrated in Appendix B Herethe detailed equations of motion of the gear pair with 12DOFs are given directly as follows

mp euroxp minus xp 1113944lAB

klpxt minus βp 1113944

lAB

klpxtZ

lp minus _xp 1113944

lAB

clpxt minus _βp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)

mp euroyp minus yp 1113944lAB

klpxt + αp 1113944

lAB

klpxtZ

lp minus _yp 1113944

lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)

mp eurozp minus zp 1113944lAB

klpzt minus _zp 1113944

lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp

_βp _cp + Jzpβpeurocp minus θpx 1113944

lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J

zpαpeurocp minus θpy 1113944

lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)

Jzpeurocp Tp + 1113944

N

i1λpziFmi (17f)

mg euroxg minus xg 1113944lAB

klgxt minus βg 1113944

lAB

klgxtZ

lg minus _xg 1113944

lAB

clgxt minus _βg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)

mg euroyg minus yg 1113944lAB

klgxt + αg 1113944

lAB

klgxtZ

lg minus _yg 1113944

lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)

mg eurozg minus zg 1113944lAB

klgzt minus _zg 1113944

lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg

_βg _cg + Jzgβgeurocg minus θgx 1113944

lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J

zgαgeurocg minus θgy 1113944

lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)

Jzgeurocg minus Tg + 1113944

N

i1λgziFmi (17l)


where N is the maximum number of tooth pairs that si-multaneously participate in the meshing Tp and Tg are thedriving torque acting on the pinion body and the torque loadapplied to the gear body respectively Fmi is the DMF ofeach tooth pair and can be calculated by equations (13) (14)or (16) according to the needs and nji(njxi njyi njzi) is thecontact force directional cosine vector of eachmeshing toothpair i e calculation of the directional rotation radius λjxiλjyi and λjzi is given by

1113882λjxi λjyi λjzi1113883 1113882njziyjmi minus njyizjmi njxizjmi

minus njzixjmi njyixjmi minus njxiyjmi1113883

(18)

e evaluation of sliding friction is beyond the scope ofthis article and thereby the current dynamic model isformulated without consideration of sliding frictionHowever it should be noted that friction is a significantexternal source of excitation influencing the dynamic be-havior of gearing systems Explanations for other symbolsused above are given in Appendix B and nomenclature eequations of motion given above are expressed based on theMMTP If the MMGP is applied the DMF and mesh pa-rameters of the individual tooth pair should be replaced bythose of the gear pair e following description is also basedon the MMTP e related information about the MMGP isintroduced in the study by Peng [13]

Geometric modeling of spiral bevel gears TCA andLTCA are performed in the global fixed reference frameOxyz (as shown in Figure 2) so it is necessary to transformthemesh point position rmi and the line of action vector Li tothe local inertial reference frames of the pinion and gearOjx

0jy0

jz0j in which the dynamic analysis is performed e

relationship between rpmi(xpmi ypmi zpmi) and rmi(xmi

ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)

e directional cosine vector of the mesh force acting onthe pinion npi(npxi npyi npzi) is

npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

e relationship between rgmi(xgmi ygmi zgmi) andrmi(xmi ymi zmi) is

xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0

0 minus 1 0 minus OgO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

e directional cosine vector of the mesh force acting onthe gear ngi(ngxi ngyi ngzi) is

ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)

It is noted that the direction of the mesh force acting onthe pinion is opposite to the direction of the line of actionvector Li

52 Dynamic Transmission Error Calculation e trans-lational dynamic transmission error vector is defined as thedifference between the effective mesh point position vectorsof the pinion and gear e dynamic transmission erroralong the line of action εd can be obtained by projecting thetranslational dynamic transmission error vector along theline of action vector

For the MMGP the time-dependent mesh parametersare often treated as the function of the position on the meshcycle (PMC) [29] while for theMMTP the mesh parametersof each tooth pair should be treated as the function of theposition on the one tooth engagement cycle (PTEC) Forcurrent formulation all the teeth are assumed to be equallyspaced and have the same surface topology e PMC andPTEC are calculated according to the following equations

PMC θpz + θpz0

11138681113868111386811138681113868

11138681113868111386811138681113868 minus floorθpz + θpz0

11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868

11138681113868111386811138681113868 minus floorθpz + θt0

11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)

where θpz is the pinion roll angle which approximatelyequals cp as shown in equation (B11) θpz0 is the initialangular position of the pinion which can be arbitrarily se-lected and θpzp is the angular pitch If equation (24) isapplied to calculate the PTEC of the reference tooth pair θt0is the initial angular position of the reference tooth pairwhich can also be chosen discretionarily and N is themaximum number of tooth pairs that participate in themeshing simultaneously e PTEC of other simultaneouslyengaged tooth pairs can be deduced by the PTEC of thereference tooth pair and the number of angular pitchesbetween them In both equations the ldquofloorrdquo functionrounds a number to the nearest smaller integer


On the pinion side it is convenient to express the in-stantaneous position vector of the effective mesh point in thecoordinate system Cpx3

py3pz3

p

r3pmi xpmi PTECi( 1113857i3p + ypmi PTECi( 1113857j3p + zpmi PTECi( 1113857k3p

(25)

To consider the effect of large rotational angles θpz(orcp)

and θgz(orcg) on the dynamic transmission error animaginary coordinate system CgxI

gyIgzI

g considering theinstantaneous angular TE about the z-axis on the gear side(θgz minus (θpzR)) is established e effective mesh pointposition vector of the gear expressed in CgxI

gyIgzI

g is

rIgmi xgmi PTECi( 1113857iIg + ygmi PTECi( 1113857jIg + zgmi PTECi( 1113857kI

g

(26)

Transforming the coordinates to the global fixed refer-ence frame Oxyz we obtain

rOpmi MOOpprimeMOpCpprimer

3pmi

rOgmi MOOgprimeMOgCgprimer

Igmi

(27)

where the transformationmatrices forCpx3py3

pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)

e transformation matrices for CgxIgyI

gzIg to Oxyz are

MOOgprime

0 1 0 0

0 0 minus 1 minus OgO

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime

1 minus θgz minus θpzR1113872 11138731113872 1113873 βg xg

θgz minus θpzR1113872 1113873 1 minus αg yg

minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)

e dynamic transmission error along the line of actionfor each pair of teeth in contact is calculated by

εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)

6 Numerical Simulation Analysis

e proposed equations of motion are solved by MATLABrsquosODE45 [38] which implements the 45th order RungendashKutta

integration routine with the adaptive step size e second-order differential form of equations (17a)ndash(17l) must berewritten in the state space (first-order) form before they canbe solved by the solver Step speed sweep simulations areperformed to obtain the dynamic response in a prescribedrange of mesh frequency Numerical integration at eachmesh frequency is performed up to the time when thesteady-state response is reached e variable values of thefinal state at this mesh frequency are applied as the initialconditions for dynamic analysis at the next one Time-do-main DMF response at a specified mesh frequency iscomputed directly from the solution of numerical in-tegration Frequency response is obtained by root meansquare (RMS) values of the time-domain response over onetooth engagement period at each frequency Note that timehistories of DMF on each pair of teeth over one tooth en-gagement cycle are not the same at the frequencies wheresubharmonic or chaotic motion occurs which results indifferent RMS values for different tooth pairs In such casesthe average value is adopted for graphing

e system parameters are listed in Table 2 e ratedtorque load of the gear pair is about 700Nm Considering thedynamic characteristics formedium- and heavy-load cases aresimilar to but different from those for the light-load case [13]we performed dynamic simulations at 50Nm and 300Nmtorque loads to examine themodeling effects for the light-loadand medium-to-heavy-load conditions respectively

61 Effects ofMeshModel To preclude the influence of meshstiffness on the calculation results of DMF the more oftenused ASMS is adopted for both the MMTP and the MMGP

Figure 16 compares the DMF responses with respect tothe mesh frequency from the two models It can be seen thatthe simulation results from the two models are overallconsistent which partly verifies the validity of the MMTPproposed in this work Differences exist in the responseamplitudes particularly at higher mesh frequency but theresonance frequencies estimated by the two models areapproximately the same In the lightly loaded condition(50Nm) jump discontinuities in the vicinity of the reso-nance frequencies are predicted by both models e toothseparation and impact occur in the frequency range betweenthe upward and downward jump discontinuities e jumpfrequencies near the resonance frequency of about 3000Hzare different for speed sweep up and down simulationsowing to the strong nonlinearity of the system For themedium-to-heavy-loaded case (300Nm) the nonlineartime-varying responses from both models are continuouscurves and the results for speed sweep up and down casesare just the same e disappearance of nonlinear jumpsmeans the load has become large enough to prevent the lossof contact and the backlash nonlinearity is ineffectualGenerally the increase of load tends to weaken the effect ofbacklash nonlinearity not only because of the higher meanload which plays the role in keeping tooth surfaces in contact[11] but also because the variations of mesh stiffness meshvector and LTE which all aggravate the degree of backlashnonlinearity are alleviated with the increase of load


e essential differences between the MMTP and theMMGP can be explored further by comparing the timehistories of DMF per tooth pair Analysis results that candescribe some typical phenomena of gear meshing are il-lustrated as follows

Figure 17 compares the response histories at 50Hz meshfrequency and 300Nm torque load As can be seen globallythe DMF predicted by the MMGP waves smoothly aroundthe static MF obtained through LTCA By contrast the DMFfrom the MMTP seems to fluctuate more frequently withdramatic oscillation (due to the tooth engaging-in impact)occurring at the start of tooth engagement which howevercannot be observed in the results from the MMGP eoverall difference in the predicted DMFs can be attributed tothe fact that the MMGP uses the mesh parameters whichreflect the gentle overall effect of time-varying mesh char-acteristics on the entire gear pair while the more stronglychanged mesh parameters of a tooth pair are applied in theMMTP

It is known that tooth impact always occurs at themoment of the first contact of each tooth pair is phe-nomenon can be explained by the short infinite high ac-celeration appearing in the acceleration graph (which is thesecond derivative of the motion or KTE graph) at thechangeover point between two adjacent pairs of teeth [39]e results shown in Figure 17 indicate that the MMTP iscapable of describing tooth impact at the instant of en-gagement and disengagement but the MMGP is in-competent in this respect No impact occurs in theestimation from the MMGP because the DMF on each toothpair in simultaneous contact can only be roughly estimatedthroughmultiplying the net mesh force by the quasistaticallydefined load distribution factor which gradually increasesfrom zero in the engaging-in stage and decreases to zerowhen the tooth pair leaves the contact

Another interesting phenomenon shown in the graph isthat the DMF from the MMTP oscillates above the staticmesh force in the engaging-in stage but below the staticmesh force in the engaging-out stage To find the reason weapplied the MMTP to a two-DOF dynamic model whichtakes into account only the torsional motions of the pinionand gear It is shown in Figure 17 that the predicted DMFfluctuates around the static mesh force from LTCA isseems to be reasonable because in LTCA the pinion and

gear are restricted to rotating about their rolling axes as welle results thereby indicate that the deviation between theDMF and the static mesh force is because of the change ofthe relative position of the pinion and gear due to the releaseof additional lateral degrees of freedom For the MMGP themesh force acting on each pair of teeth obtained according tothe principle of load distribution waves around the staticmesh force e estimation however may deviate from theactuality if the pose of the gear can change in any degree offreedom e simulation results also indicate that the meshparameters obtained by the current LTCA method are moresuitable for the two-DOF dynamic model

At the light torque load of 50Nm (as shown in Fig-ure 18) the deviation between the DMF and the static meshforce is reduced and the engaging-in impact is less signif-icant is is because the change of relative position of thepinion and gear caused by the load is relatively small at thelightly loaded condition erefore we can find that thetransmission performance of a spiral bevel gear pair issusceptible to the relative position of the two gears and areasonable design of the external support structure plays anessential role in improving the stability of the gear trans-mission Tooth flank modification is also necessary to reducethe sensitivity of transmission performance to the inevitablechange of relative position caused by the load or installationerrors

It is noteworthy that the prediction from the meshmodelwith predefined mesh parameters has an unavoidable de-viation from the reality as the mesh parameters under thedynamic condition differ from those for the static caseerefore the impact force sometimes may be overpredictedbecause of the inaccurate initial contact position of a toothpair e inaccuracy leads to overestimation of the contactdeformation at the start of tooth engagement Howeveraccording to the contact conditions of gear teeth discussedbefore the tooth pair will get into contact ahead of thequasistatically defined beginning of contact because the largedeformation evaluated at the point of the predefined start ofcontact means the profile separation between the matingflanks of the extended contact area has been closed

Figure 19 compares the DMF histories per tooth pair atthe mesh frequency of 400Hz where loss of contact occursAs displayed near resonant frequency DMF varies drasti-cally with the peak value much higher than static mesh forcee two mesh models provide almost identical results with aslight difference near the end of tooth engagement wheretransient loss of contact only appears in the results from theMMTP It is worth noting that the differences between thecalculations from the MMTP and MMGP usually appearnear the beginning and end of tooth contact since in thesetwo regions the KTE curve of the tooth pair and the gearpair which are separately applied in the two models isdistinct as shown in Figure 1

62 Effects of Mesh Stiffness Here the MMTP is applied forall analyses e dynamic responses and time histories ofDMF per tooth pair predicted by using the ASMS and LTMSare compared with the results from the mesh force

Table 2 System parameters

Parameters Pinion GearMass m (kg) 182 671Torsional moment of inertia Jz (kgmiddotm2) 1065E minus 3 394E minus 2Bending moment of inertia Jx (kgmiddotm2) 7064E minus 4 2E minus 2Centroid position of the gear L (mm) 97507 minus 41718Bearing A axial position LA (mm) 120 200Bearing B axial position LB (mm) 80 80Translational bearing stiffness kxt (Nm) 28E8 28E8Axial bearing stiffness kzt (Nm) 8E7 8E7Tilting bearing stiffness kxr (Nmrad) 6E6 8E6Mesh damping ratio 006Support component damping ratio 002Gear backlash (mm) 015


NLTV analysis with MMGP(speed sweep up)NLTV analysis with MMGP(speed sweep down)

NLTV analysis with MMTP(speed sweep up)NLTV analysis with MMTP(speed sweep down)

1000 2000 3000 4000 5000 60000Mesh frequency (Hz)

400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)

NLTV analysis with MMTPNLTV analysis with MMGP


2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)

Figure 16 RMS values of the DMF acting on the individual tooth pair evaluated at (a) 50Nm and (b) 300Nm torque loads

0 01 02 03 04 05 06 07 08 09 1ndash01Pinion roll angle (rad)

0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)

Static MF from LTCADMF from MMGP

DMF from MMTPDMF from MMTP2-DOF

Figure 17 Time histories of DMF per tooth pair at 50Hz mesh frequency and 300Nm torque load


interpolation function which in theory are believed to bemore accurate

Figure 20 shows the comparison of the DMF responses at300Nm torque load Differences occur in both responseamplitudes and resonant frequencies and become noticeablein the range of high mesh frequency Overall both theapproaches using the ASMS and LTMS tend to un-derestimate the effective values of DMF and the predictedresonance frequencies consistently shift to the lower meshfrequency range in comparison with the results from themesh force interpolation function Generally the resultsfrom the model applying the LTMS seem to be closer to theinterpolated results is finding can also be observed fromthe time histories of the individual DMF illustrated inFigure 21

As shown in Figure 21 DMF predicted by the mesh forceinterpolation function changes most smoothly and thetooth engaging-in impact is the minimum among the threeapproaches It manifests that if the nonlinear relationshipbetween the mesh force and the mesh deflection (ie thevariation of mesh stiffness with mesh deflection) is

considered the fluctuation of DMF is actually moderateFrom the graph we can also observe that the DMF estimatedwith the LTMS varies smoothly relative to the DMF pre-dicted by the ASMS and the tooth engaging-in impact is lesspowerful Although the two methods do not give sub-stantially different results the approach using the LTMS ispreferred for vibration analysis of gears around the nominalstatic deformation is is not only because the resultspredicted with the LTMS aremore similar to the results fromthe mesh force interpolation function which precisely de-scribes the force versus deformation behavior but also be-cause the calculation approach using the LTMS shown inequation (14) better represents the relationship between theDMF and the nominal static mesh force when gears vibratearound the nominal static deformation

Similar to the results shown in Figure 21 for the examplegear pair in the predefined mesh frequency range (0ndash6000Hz) the impact rarely occurs when the tooth pair leavesthe contact It should be noted that whether impact occurs atthe instant of tooth engagement and disengagement and themagnitude of the impact largely depends on the

0100200300400500600700800900

Mes

h fo

rce (

N)

ndash01 0 02 03 04 05 06 07 08 0901Pinion roll angle (rad)

Static MF from LTCADMF from MMGPDMF from MMTP

Figure 18 Time histories of DMF per tooth pair at the mesh frequency of 50Hz and 50Nm torque load

0 01 02

Loss ofcontact

ndash04 ndash03 ndash02 ndash01 0 01 02 03 04 05 06 07 08 09ndash05 111Pinion roll angle (rad)

0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800




transmission error which can be modified through toothsurface modification

7 Conclusion

In this paper two critical factors influencing the calculationof dynamic mesh force (DMF) for spiral bevel gears ie themeshmodel and themesh stiffness representation have beeninvestigated An improved mesh model of a tooth pair(MMTP) which better describes the mesh characteristics ofeach pair of teeth is proposed As the difference between thekinematic transmission error (KTE) of a tooth pair and theloaded transmission error (LTE) of a gear pair represents theactual deformation of the tooth pair caused by the load itbears instead of the KTE of a gear pair which is commonlyused in the single-point and multipoint coupling meshmodels the KTE of a tooth pair is applied in the MMTP forthe calculation of the mesh stiffness of individual tooth pairs

the identification of tooth contact and the estimation ofdynamic mesh deflection of each pair of teeth in contactemesh parameters and dynamic responses are comparedbetween the proposed model and the mesh model of a gearpair (MMGP) (ie the single-point coupling mesh model)Overall the twomodels predict the similar variation trend ofdynamic responses with the mesh frequency Althoughnoticeable differences can be observed in the responseamplitudes particularly in the range of high mesh frequencythe resonance frequencies estimated by the two models arealmost the same e time histories of dynamic mesh force(DMF) per tooth pair from the two models are significantlydifferent since the variations of mesh parameters which actas the parametric excitations in the dynamic model aredramatic for the MMTP but moderate for the MMGPCompared with previous mesh models the MMTP is su-perior in simulating the dynamic contact behavior of eachpair of teeth especially at the instant of tooth engagement

2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)

2000 60000 1000 500040003000Mesh frequency (Hz)

ASMSLTMSMF interpolation function

Figure 20 RMS values of the DMF acting on each tooth pair evaluated by the approaches using the ASMS LTMS and mesh forceinterpolation function at 300Nm toque load

0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05

Figure 21 Time histories of DMF per tooth pair estimated by the approaches using the ASMS LTMS andmesh force interpolation functionat the mesh frequency of 50Hz and 300Nm torque load


and disengagement e results from the MMTP also showthat the DMF will deviate from the static mesh force if thepose of the gear can change in any degree of freedom isphenomenon however has never been observed in previousstudies where the single-point or multipoint coupling meshmodel is applied

For the MMGP the total mesh stiffness of all contactingtooth pairs is applied while for the MMTP the stiffness ofthe individual tooth pair is used Both stiffnesses can becalculated by the average slope and local slope approachese average secant mesh stiffness (ASMS) derived from theaverage slope approach substantially differs from the localtangent mesh stiffness (LTMS) estimated by the local slopeapproach owing to their definitions and calculationmethods and we cannot make a conclusion about whichcalculation is more appropriate Besides because of theparticularity of the tooth shape the ASMS for the MMGPcannot be directly computed by the summation of the ASMSof each tooth pair but the LTMS of a gear pair approxi-mately equals the sum of the LTMS of all tooth pairs incontact Both the ASMS and LTMS can be used to calculateDMF but the calculation formulas are rather different emisuse of mesh stiffness for a given method will lead toerrors in the physical sense

DMF calculations from the method applying the ASMSand LTMS are compared with those from the mesh forceinterpolation function which accurately represents thenonlinear force-deformation relationship Although theresults given by the methods using the LTMS and ASMS arenot substantially different the former approach is recom-mended to calculate the DMF when simulating the vibrationof gears about the nominal static deformation One of thereasons is that the results predicted with the LTMS are closerto the results from the mesh force interpolation functionAnother reason is that the model using the LTMS betterrepresents the relationship between the DMF and thenominal static mesh force when gears vibrate around thenominal static deformation e third reason is that theassumption which believes the position of the effective meshpoint does not change before and after the tooth de-formation is more reasonable for the calculation of LTMSis is because the change of the effective mesh point po-sition caused by the mean tooth deflection relative to theundeformed state is noticeable but the change caused by thesmall deformation relative to the nominal state is negligiblee approach applying the mesh force interpolation func-tion can provide the most accurate prediction However thedevelopment of the mesh force interpolation function re-quires multiple time-consuming loaded tooth contact ana-lyses at different load levels and the interpolation in solvingdynamic problems needs more computational efforts thanthe other two approaches

Abbreviations

ASMS Average secant mesh stiffnessDMF Dynamic mesh forceFEM Finite element methodKTE Kinematic transmission error

LTCA Loaded tooth contact analysisLTE Loaded transmission errorLTMS Local tangent mesh stiffnessMMGP Mesh model of a gear pairMMTP Mesh model of a tooth pairRMS Root mean squareTCA Tooth contact analysis

Nomenclature

ΔB Composite deformation of the toothand base

b Gear backlashcm Mesh dampingcxt Translational motion dampingczt Axial motion dampingcxr Tilting motion dampingek Angular kinematic transmission errorFm Mesh force of the gear pairFmi Mesh force of the tooth pair i

Fm d Magnitude of dynamic mesh forceFnet Magnitude of net mesh forceFma Magnitude of contact force calculated

with the average secant mesh stiffnessFml Magnitude of contact force calculated

with the local tangent mesh stiffnessFc Viscous damping forceFr Reference force used to determine the

operational conditionΔH Contact deformationJz

p and Jzg Torsional moment of inertia of the

pinion and gearJx

p and Jxg Bending moment of inertia of the

pinion and gearkm Mesh stiffnesskma Average secant mesh stiffnesskxt Translational bearing stiffnesskzt Axial bearing stiffnesskxr Tilting bearing stiffnessLm LOA vector of the gear pairLmi LOA vector of the tooth pair i

LA and LB Axial position of bearings A and BN Number of engaged tooth pairsNp Tooth number of the pinionNg Tooth number of the gearnjxi njyi and njzi Contact force directional cosine of the

tooth pair i

rm Effective mesh position of the gear pairrmi Effective mesh position of the tooth pair

i

ΔS Profile separationTp Driving torque acting on the pinion

bodyTg Torque load applied to the gear bodyαj βj and cj Cardan anglesθpz Pinion roll angleθgz Gear roll angleθpz0 Initial angular position of the pinion


θpzp Angular pitchθt0 Initial angular position of the reference

tooth pairδ Mesh deflection along the line of actionδd Dynamic mesh deflection along the line

of actionδT Static mesh deflection along the line of

action caused by nominal load T

ε0 Translational kinematic transmissionerror along the line of action

εd Translational dynamic transmissionerror along the line of action

λ Directional rotation radius about therolling axis

λjxi λjyi and λjzi Directional rotation radius of the toothpair i

Appendix

A Explanations of Why the ASMS of the GearPair Is Not the Summation of the ASMS of EachMeshing Tooth Pair and Why the LTMS of theGear Pair Approximately Equals the Sum of theLTMS of Each Tooth Pair in Contact

In the multiple-tooth contact zone the total applied torqueTt must be the summation of the individual torque Ti

carried by each meshing tooth pair i For the example in thisstudy where two pairs of teeth are in contact

Tt T1 +T2 (A1)

us

kma e0 minus e( 1113857λ2 kma1 e01 minus e1113872 1113873λ21 + kma2 e02 minus e1113872 1113873λ22

(A2)

where kma is the ASMS of the gear pair kmai(i 1 2) de-notes the ASMS of each engaged tooth pair e0 and e are theangular KTE and LTE of the gear pair e0i(i 1 2) repre-sents the angular KTE of the tooth pair λ is the equivalentdirectional rotation radius of the gear pair and λi(i 1 2)

indicates the directional rotation radius of each meshingtooth pair

At any rolling position the angular KTE of the gear pairis equal to the angular KTE of either tooth pair 1 or toothpair 2 Assuming at some position e0 e01 equation (A2)can be transformed to

kma e01 minus e1113872 1113873λ2 kma1 e01 minus e1113872 1113873λ21 + kma2 e01 minus S2 minus e1113872 1113873λ22

(A3)

where S2(S2 gt 0) is the angular profile separation betweenthe matting flanks of tooth pair 2 If the differences among λλ1 and λ2 are ignored namely λ asymp λ1 asymp λ2 equation (A3)can be simplified to

kma asymp kma1 + kma21113872 1113873 minuskma2S2

e01 minus e1113872 1113873 (A4)

Similarly at any position where e0 e02


e02 minus e1113872 1113873 (A5)

erefore it can be seen that apart from the positionwhere the initial profile separations of tooth pair 1 (S1) andtooth pair 2 (S2) in the no-loading condition are both zero(at which kma asymp kma1 + kma2 such as position A in Fig-ure 12) the sum of the ASMS of each meshing tooth pair isgreater than the ASMS of the gear pair

As illustrated in Figure 8 the LTMS describes the changeof contact force relative to the nominal static contact forceFT

m when the mesh deflection changes slightly relative to thenominal static deflection δT In the multiple-tooth contactzone the change of total applied torque ΔTt must be thesummation of the change of the individual torque ΔTi

carried by each engaged tooth pair i For two pairs of teeth incontact

ΔTt ΔT1+ΔT2 (A6)

us

kml eT minus eT+ΔT( 1113857λ2 kml1 eT1 minus eT+ΔT1113872 1113873λ21

+ kml2 eT2 minus eT+ΔT1113872 1113873λ22(A7)

where kml is the LTMS of the gear pair kmli(i 1 2) rep-resents the LTMS of each engaged tooth pair eT and eT+ΔTare the angular LTE of the gear pair at the torque load T andT + ΔT respectively eTi(i 1 2) indicates the LTE of eachtooth pair λ is the equivalent directional rotation radius ofthe gear pair and λi(i 1 2) denotes the directional rotationradius of each meshing tooth pair

Since the LTEs of all pairs of teeth in contact are the sameand equal the LTE of the gear pair if the differences amongλ λ1 and λ2 are neglected namely λ asymp λ1 asymp λ2 equation(A7) can be reduced to

kml asymp kml1 + kml2 (A8)

erefore the summation of the LTMS of each pair ofteeth in contact is approximately equal to the LTMS of thegear pair

B Dynamic Modeling

In this section iij jij an d ki

j are the triad of base vectors ofthe coordinate system Cjx

ijy

ijz

ij(i 1 2 3) e subscript

j p g refers to the driving pinion and driven gear and thesuperscript l A B denotes bearings A and B

As shown in Figure 15(b) the instantaneous angularvelocity of each gear-shaft body expressed by the derivativesof Cardan angles is

ωj _αji1j + _βjj

2j + _cjk

3j (B1)

e ωj can be expressed in the body-fixed coordinatesystem Cjx

4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

_αj cos βj cos cj + _βj sin cj

minus _αj cos βj sin cj + _βj cos cj

_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B2a)whereM21jM32j andM43j are the rotation transformationmatrices

M21j

1 0 0

0 cos αj sin αj

0 minus sin αj cos αj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j

cos βj 0 minus sin βj

0 1 0sin βj 0 cos βj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j

cos cj sin cj 0minus sin cj cos cj 0

0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)

Considering αj and βj are small angles (the assumptionis applied in the derivations of the following equations aswell) the gear angular velocity vector is

ωj _αj cos cj + _βj sin cj1113872 1113873i4j+ minus _αj sin cj + _βj cos cj1113872 1113873j4j

+ _cj + βj _αj1113872 1113873k4j

(B3)Likewise ωj can also be expressed in the local inertial

reference frame Ojx0jy0

jz0j as

ωj _αj + βj _cj1113872 1113873i0j+ _βj minus αj _cj1113872 1113873j0j + _cj + αj_βj1113872 1113873k0j (B4)

e gear translational velocity vector is

_rj _xji0j+ _yjj

0j + _zjk

0j (B5)

e mounting positions of bearings are measured in thefixed reference frame Oxyz by Ll

j as shown in Figure 15(a)(LA

p LBp and LA

g are positive while LBg is negative) e

position coordinates of bearings expressed in their body-fixed reference frames are (0 0 Zl

j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA

g minus LAg + Lg and ZB

g minus LBg + Lg where Lp

(positive value) and Lg (negative value) are the centroidpositions of the supported gear-shaft bodies measured in thefixed reference frame Oxyz under the initial static equi-librium condition In the state of working the inner ring ofthe bearing moves with the gear-shaft body and its in-stantaneous position expressed in the local inertial referenceframe Ojx

0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOj1M12jprimeM23jprimeM34jprime

0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)

where the rotation transformation matrices M12jprime M23jprimeand M34jprime are

M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)

e translation transformation matrix MOj1 is

MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)

e outer ring of the bearing is fixed to the base andthus the inner ring translational displacement vector rela-tive to the outer ring is

dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)

Since cj is a large rolling angle the inner ring rotationaldisplacement vector relative to the outer ring cannot beexpressed by the base vectors of Ojx

0jy0

jz0j Considering the fact

that the large rolling angle θjz of the inner ring relative to theouter ring carries subtle effect in terms of causing bearing forcewe only take the rotation angles about x- and y-axes ofOjx

0jy0

jz0j θjx and θjy into consideration As θjx and θjy are

small rotation angles the rotational displacement vector of theinner ring can be expressed as

Γlj θjxi0j + θjyj

0j (B10)

Because θjx θjy and θjz are also gear rotation anglesabout the coordinate axes of Ojx

0jy0

jz0j the differential

equations can be obtained from equation (B4)_θjx _αj + βj _cj

_θjy _βj minus αj _cj

_θjz _cj + αj_βj asymp _cj

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)

_θjz asymp _cj because αj_βj is a small quantity relative to _cj

e angular coordinates (θjx θjy θjz) can be acquired bysolving equation (B11) and the equations of motionsimultaneously

Since the axes of the body-fixed coordinate system areassumed to coincide with the principal axes of inertia of each


body the inertia tensor of each body is Jj diag[Jxj Jx

j Jzj]

(the first and second inertia terms are the same because thegear-shaft body is axisymmetric) e kinetic energy is givenby

T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)

e bearing supporting forces are regarded as externalforces so the potential energy V is zero According to theLagrange equation

ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)

where qj and Qj are the generalized coordinates (namely xjyj zj αj βj and cj) and the corresponding generalizedforces e resulting equations of translational and rota-tional motions of each body are

mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr

j and Qcrj are generalized torques about x1

j-y2

j- and z3j-axes ey can be expressed by the torques Qxr

j Q

yrj and Qzr

j about x0j- y0

j- and z0j-axes of the local inertial


jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr

j βjQxrj minus αjQ

yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)

where Qcrj is approximately equal to Qzr

j because the ori-entation variation of the rolling axis z3

j relative to z0j is

extremely small and it can be observed that βjQxrj minus αjQ

yrj is

small relative to Qzrj Substituting equation (B16) into

equation (B15) the equations of rotational motion can bewritten as

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz

j _αj _cj minus Jzjαjeurocj Q

yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)

e system equations of motion can be written in thematrix form as

Meuroq + Gv _q + Gaq fQ (B18)

whereGv andGa are gyroscopic matrices associated with therotation speeds and angular accelerations of gears re-spectively e generalized force vector fQ includes elasticbearing supporting forces Kq damping forces C _q and other

sources of forces fext including external loading and internalexcitations (namely the DMF) Hence the completeequations of motion in the matrix form are

Meuroq + C _q + Kq + Gv _q + Gaq fext (B19)

q xp yp zp αp βp cp xg yg zg αg βg cg1113872 1113873T (B20)

All bearings are isotropic in the x-y plane giving thestiffness matrices for translation and rotation asKl

jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl

jxr 0] re-spectively Accordingly the damping matrices for trans-lation and rotation are Cl

jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl

jxr 0]Confined to the length of this paper the matrices in

equation (B19) are not listed here while the detailedequations of motion are given as equations (17a)ndash(17l)

Data Availability

e data used and analyzed to support the findings of thisstudy are available from the corresponding author uponrequest

Conflicts of Interest

e authors declare no potential conflicts of interest withrespect to the research authorship andor publication ofthis article

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (No 51505100) and the ScienceFoundation for Young Scientists of Heilongjiang Province(No QC2015046)

References

[1] F L Litvin A Fuentes Q Fan and R F HandschuhldquoComputerized design simulation of meshing and contactand stress analysis of face-milled formate generated spiralbevel gearsrdquo Mechanism and Machine Aeory vol 37 no 5pp 441ndash459 2002

[2] C Gosselin L Cloutier and Q D Nguyen ldquoA general for-mulation for the calculation of the load sharing and trans-mission error under load of spiral bevel and hypoid gearsrdquoMechanism and Machine Aeory vol 30 no 3 pp 433ndash4501995

[3] Q Fan and L Wilcox New Developments in Tooth ContactAnalysis (TCA) and Loaded TCA for Spiral Bevel and HypoidGear Drives AGMA Technical Paper Alexandria VA USA2005

[4] C Zhou C Tian W Ding L Gui and Z Fan ldquoAnalysis ofhypoid gear time-varying mesh characteristics based on thefinite element methodrdquo Journal of Mechanical Engineeringvol 52 no 15 pp 36ndash43 2016

[5] S Vijayakar ldquoA combined surface integral and finite elementsolution for a three-dimensional contact problemrdquo In-ternational Journal for Numerical Methods in Engineeringvol 31 no 3 pp 525ndash545 1991


[6] S Vijayakar Tooth Contact Analysis Software CALYX Ad-vanced Numerical Solutions Hilliard OH USA 1998

[7] S Kiyono Y Fujii and Y Suzuki ldquoAnalysis of vibration ofbevel gearsrdquo Bulletin of JSME vol 24 no 188 pp 441ndash4461981

[8] M G Donley T C Lim and G C Steyer ldquoDynamic analysisof automotive gearing systemsrdquo SAE International Journal ofPassenger CarsmdashMechanical Systems vol 101 pp 77ndash871992

[9] Y Cheng and T C Lim ldquoVibration analysis of hypoidtransmissions applying an exact geometry-based gear meshtheoryrdquo Journal of Sound and Vibration vol 240 no 3pp 519ndash543 2001

[10] Y Cheng and T C Lim ldquoDynamics of hypoid gear trans-mission with nonlinear time-varying mesh characteristicsrdquoJournal of Mechanical Design vol 125 no 2 pp 373ndash3822003

[11] J Wang T C Lim and M Li ldquoDynamics of a hypoid gearpair considering the effects of time-varying mesh parametersand backlash nonlinearityrdquo Journal of Sound and Vibrationvol 308 no 1-2 pp 302ndash329 2007

[12] J Wang and T C Lim ldquoEffect of tooth mesh stiffnessasymmetric nonlinearity for drive and coast sides on hypoidgear dynamicsrdquo Journal of Sound and Vibration vol 319no 3ndash5 pp 885ndash903 2009

[13] T Peng CoupledMulti-Body Dynamic and Vibration Analysisof Hypoid and Bevel Geared Rotor system PhD thesisUniversity of Cincinnati Cincinnati OH USA 2010

[14] Z Feng S Wang T C Lim and T Peng ldquoEnhanced frictionmodel for high-speed right-angle gear dynamicsrdquo Journal ofMechanical Science and Technology vol 25 no 11pp 2741ndash2753 2011

[15] I Karagiannis S eodossiades and H Rahnejat ldquoOn thedynamics of lubricated hypoid gearsrdquo Mechanism and Ma-chine Aeory vol 48 pp 94ndash120 2012

[16] J Yang and T Lim ldquoDynamics of coupled nonlinear hypoidgear mesh and time-varying bearing stiffness systemsrdquo SAEInternational Journal of Passenger CarsmdashMechanical Systemsvol 4 no 2 pp 1039ndash1049 2011

[17] J Yang and T C Lim ldquoNonlinear dynamic simulation ofhypoid gearbox with elastic housingrdquo in Proceedings of theASME International Design Engineering Technical Conferencesand Computers and Information in Engineering Conferencepp 437ndash447 ASME Washington DC USA August 2011

[18] H Wang Gear Mesh Characteristics and Dynamics of HypoidGeared Rotor system PhD thesis University of AlabamaTuscaloosa AL USA 2002

[19] T C Lim and Y Cheng ldquoA theoretical study of the effect ofpinion offset on the dynamics of hypoid geared rotor systemrdquoJournal of Mechanical Design vol 121 no 4 pp 594ndash6011999

[20] Y Wang T C Lim and J Yang ldquoMulti-point mesh modelingand nonlinear multi-body dynamics of hypoid geared sys-temrdquo SAE International Journal of PassengerCarsmdashMechanical Systems vol 6 no 2 pp 1127ndash1132 2013

[21] C G Cooly C Liu X Dai and R G Parker ldquoGear toothmesh stiffness a comparison of calculation approachesrdquoMechanism and Machine Aeory vol 105 pp 540ndash553 2016

[22] O Lundvall N Stromberg and A Klarbring ldquoA flexiblemulti-body approach for frictional contact in spur gearsrdquoJournal of Sound and Vibration vol 278 no 3 pp 479ndash4992004

[23] T Eritenel and R G Parker ldquoree-dimensional nonlinearvibration of gear pairsrdquo Journal of Sound and Vibrationvol 331 no 15 pp 3628ndash3648 2012

[24] Karlsson amp Sorensen Inc ABAQUS Userrsquos Manual Version614 Hibbitt Karlsson amp Sorensen Inc Providence RI USA2014

[25] F L Litvin and A Fuentes Gear Geometry and AppliedAeory Cambridge University Press Cambridge UK 2ndedition 2004

[26] F Hou C Deng and Z Ning ldquoContact analysis of spiral bevelgears based on finite element modelrdquo Journal of HarbinEngineering University vol 36 no 6 pp 826ndash830 2015

[27] R G Parker S M Vijayakar and T Imajo ldquoNon-lineardynamic response of a spur gear pair modelling and ex-perimental comparisonsrdquo Journal of Sound and Vibrationvol 237 no 3 pp 435ndash455 2000

[28] V K Tamminana A Kahraman and S Vijayakar ldquoA study ofthe relationship between the dynamic factors and the dynamictransmission error of spur gear pairsrdquo Journal of MechanicalDesign vol 129 no 1 pp 75ndash84 2007

[29] A Palermo D Mundo R Hadjit and W Desmet ldquoMulti-body element for spur and helical gear meshing based ondetailed three-dimensional contact calculationsrdquo Mechanismand Machine Aeory vol 62 pp 13ndash30 2013

[30] S He R Gunda and R Singh ldquoInclusion of sliding friction incontact dynamics model for helical gearsrdquo Journal of Me-chanical Design vol 129 no 1 pp 48ndash57 2007

[31] K D Brabanter P Karsmakers F Ojeda et al LS-SVMlabToolbox Userrsquos Guide Version 18 Katholieke UniversiteitLeuven Leuven Belgium 2011

[32] Q Fan ldquoEnhanced algorithms of contact simulation forhypoid gear drives produced by face-milling and face-hobbingprocessesrdquo Journal of Mechanical Design vol 129 no 1pp 31ndash37 2007

[33] L Chang G Liu and LWu ldquoA robust model for determiningthe mesh stiffness of cylindrical gearsrdquo Mechanism andMachine Aeory vol 87 pp 93ndash114 2015

[34] N L Pedersen and M F Joslashrgensen ldquoOn gear tooth stiffnessevaluationrdquo Computers amp Structures vol 135 pp 109ndash1172014

[35] H Ma R Song X Pang and B Wen ldquoTime-varying meshstiffness calculation of cracked spur gearsrdquo EngineeringFailure Analysis vol 44 pp 179ndash194 2014

[36] K Chen Y Huangfu H Ma Z Xu X Li and B WenldquoCalculation of mesh stiffness of spur gears consideringcomplex foundation types and crack propagation pathsrdquoMechanical Systems and Signal Processing vol 130 pp 273ndash292 2019

[37] X Dai C G Cooley and R G Parker ldquoAn efficient hybridanalytical-computational method for nonlinear vibration ofspur gear pairsrdquo Journal of Vibration and Acoustics Trans-actions of the ASME vol 141 pp 0110061ndash12 2019

[38] e MathWorks Inc MATLAB Version 2016b e Math-Works Inc Natick MA USA 2016

[39] H J Stadtfeld and U Gaiser ldquoe ultimate motion graphrdquoJournal of Mechanical Design vol 122 no 3 pp 317ndash3222000


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well as other driveline components such as the bearing [16]and elastic housing [17]

In all contributions mentioned above the single-pointcoupling mesh model where the gear mesh coupling ismodeled by a single spring-damper unit located at the ef-fective mesh point and acting along the instantaneous line ofaction was applied Although the model is believed to be ableto catch the overall effect of the mesh characteristics of a gearpair it cannot reflect the mesh properties of each meshinterface when more than one tooth pair is engaged inmeshing and hence the DMF for each couple of teeth incontact cannot be predicted directly Karagiannis et al [15]utilized the load distribution factor defined and calculated inquasistatic LTCA to estimate the dynamic contact load pertooth when load sharing occurs is method may be fea-sible but the results cannot accurately reflect the real fea-tures of the contact force under the dynamic condition Toovercome the deficiency of the single-point coupling meshmodel Wang [18] proposed a multipoint coupling meshmodel in which each mesh interface is represented by therespective effective mesh point line of action and meshstiffness e mesh properties and dynamic responses fromthe model were then compared with those from a meshmodel whose parameters are obtained from pitch conedesign [19]e differences between them stem from the factthat the mesh positions and line of action vectors which aretime-variant for the multipoint coupling mesh model areconstant for the mesh model based on the pitch cone designconcept e authors also believed that the former is capableof describing the exact mesh properties of loaded gearswhile the latter is only valid for no-load and light-loadedconditions as the mesh parameters of them are obtainedunder loaded and unloaded conditions respectively So farthe multipoint coupling mesh model has not been used aswidely as the single-point coupling mesh model for dynamicanalysis probably because most studies focus on the overalleffect of mesh characteristics on dynamics of a gear pairrather than the detailed dynamic responses of each meshingtooth pair Among recent studies maybe the multipointcoupling mesh model was only applied by Wang et al [20]eir research proved that the model has the capability ofpredicting the DMF-time history for each tooth pair within afull tooth engagement cycle Even so because the model stilluses the transmission error of a gear pair to determine thetooth contact and contact deformation of each engagedtooth pair the results cannot properly reflect the real contactstate of each pair of teeth when multiple pairs of teeth are inthe zone of contact

e accuracy of DMF calculation results is to a largeextent dependent on the accuracy of mesh stiffness edeformation of an engaged gear pair or an individualcontacting tooth pair is nonlinear with the applied loadbecause of the nonlinearity of the local contact deformationwith the contact force It means that mesh stiffness shouldvary with mesh deformation to accurately describe thenonlinear force-deflection relationship However in pre-vious studies although the variation of mesh stiffness withthe mesh position was considered the change in meshstiffness with deformation at a given mesh position was

rarely taken into consideration in DMF calculation ereare generally two methods to calculate mesh stiffness thecommonly used one is called the average slope approach andthe other which is seldom applied but may be more ap-propriate for dynamic analysis is called the local slopeapproach [21] e mesh stiffness evaluated by the twoapproaches can both be applied in DMF estimation but toaccurately calculate the DMF an error-prone point whichshould be paid enough attention is that the calculationformulas of DMF are entirely different for different types ofmesh stiffness applied It is worth mentioning that not alldynamic models utilize prespecified mesh stiffness for thecalculation of DMF In researches on the dynamics ofparallel axis gears [22 23] for example the mesh stiffness isevaluated according to instantaneous mesh deflection ateach time step of the dynamic simulationis kind of modelis referred to as the coupled dynamics and contact analysismodel as the dynamic and contact behaviors are mutuallyinfluential e variation of mesh stiffness with deformationis realized in this kind of model However the model iscurrently not computationally feasible for spiral bevel gearsbecause of the more complex and time-consuming TCAcomputation

is study aims to explore appropriate methods for moreaccurate prediction of the DMF of each meshing tooth pairof spiral bevel gears through comparative analyses consid-ering the two factors mentioned above First an improvedmesh model of a tooth pair (MMTP) which represents thevariation of the mesh properties of a tooth pair throughout afull engagement cycle was proposed e model applies thetransmission error curve of a tooth pair instead of that of agear pair to estimate mesh stiffness and identify the contactfor each couple of teeth is improvement enables themodel to capture more realistic mesh characteristics of eachcontacting tooth pair than the multipoint coupling meshmodel Second we compared the dynamic responses fromthe proposed mesh model and the widely used single-pointcoupling mesh model for a generic spiral bevel gear pair onthe one hand to validate the proposed one and on the otherhand to demonstrate the superiority of the proposed modelin the evaluation of DMF per tooth pair It is shown that thetwo models predict the similar variation tendencies of DMFresponse with the mesh frequency Although some differ-ences exist in the response amplitudes particularly in ahigher mesh frequency range the resonance frequenciesestimated by the two models are almost the same Besidesthe time histories of DMF per tooth pair predicted by the twomodels are significantly different but the results from theproposed model can provide much better insight into dy-namic contact behavior of each contacting tooth pair irdwe performed a comparative study on three calculationapproaches of DMF to further investigate the effect of meshstiffness representations on DMF calculations One methoduses the mesh stiffness estimated by the average slope ap-proach another one applies the mesh stiffness evaluated bythe local slope approach and the third method utilizes aquasistatically defined interpolation function indexed by themesh deflection and the pinion roll angle Weighing theaccuracy and cost of the computation we found that the



2 Mesh Model



py0pz0

p and Ogx0gy0



py4pz4

p andCgx4

gy4gz4






Fnet F2

mx + F2my + F2

mz

1113969

Fml 1113944

N

i1Fmli l x y z

(1)


nl Fml

Fnet

l x y z (2)


Mml 1113944

N

i1Mmli l x y z (3)


xm 1113936

Ni1xmiFmi

1113936Ni1Fmi

zm Mmy + Fmzxm1113872 1113873

Fmx

ym Mmx + Fmyzm1113872 1113873

Fmz

(4)











024

Ang

ular

TE

(rad

)

times10ndash5

DA B C



Contact zone 2





O

y

x

z

Og

Cg

Cp

x0g

x0p

z0p

y0p

x4p

x4g

z4p

z4g

z0g y4

py4g

Op

y0g






Gear

Pinion

km (θp)

ek (θp)

Lm (θp)

b

rpm (θp)

cm (θp) rgm (θp)






Lmi (θp)

bi eki (θp)

kmi (θp)

cmi (θp)

(a) (b)


x

yz

Mm


Fm (Fmx Fmy Fmz)

rm (xm ym zm)O

(a)

O

x

y zMmi




rmi (xmi ymi zmi)


(b)



40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)





ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



2 Mesh Model



py0pz0

p and Ogx0gy0



py4pz4

p andCgx4

gy4gz4






Fnet F2

mx + F2my + F2

mz

1113969

Fml 1113944

N

i1Fmli l x y z

(1)


nl Fml

Fnet

l x y z (2)


Mml 1113944

N

i1Mmli l x y z (3)


xm 1113936

Ni1xmiFmi

1113936Ni1Fmi

zm Mmy + Fmzxm1113872 1113873

Fmx

ym Mmx + Fmyzm1113872 1113873

Fmz

(4)











024

Ang

ular

TE

(rad

)

times10ndash5

DA B C



Contact zone 2





O

y

x

z

Og

Cg

Cp

x0g

x0p

z0p

y0p

x4p

x4g

z4p

z4g

z0g y4

py4g

Op

y0g






Gear

Pinion

km (θp)

ek (θp)

Lm (θp)

b

rpm (θp)

cm (θp) rgm (θp)






Lmi (θp)

bi eki (θp)

kmi (θp)

cmi (θp)

(a) (b)


x

yz

Mm


Fm (Fmx Fmy Fmz)

rm (xm ym zm)O

(a)

O

x

y zMmi




rmi (xmi ymi zmi)


(b)



40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)





ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









024

Ang

ular

TE

(rad

)

times10ndash5

DA B C



Contact zone 2





O

y

x

z

Og

Cg

Cp

x0g

x0p

z0p

y0p

x4p

x4g

z4p

z4g

z0g y4

py4g

Op

y0g






Gear

Pinion

km (θp)

ek (θp)

Lm (θp)

b

rpm (θp)

cm (θp) rgm (θp)






Lmi (θp)

bi eki (θp)

kmi (θp)

cmi (θp)

(a) (b)


x

yz

Mm


Fm (Fmx Fmy Fmz)

rm (xm ym zm)O

(a)

O

x

y zMmi




rmi (xmi ymi zmi)


(b)



40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)





ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Lmi (θp)

bi eki (θp)

kmi (θp)

cmi (θp)

(a) (b)


x

yz

Mm


Fm (Fmx Fmy Fmz)

rm (xm ym zm)O

(a)

O

x

y zMmi




rmi (xmi ymi zmi)


(b)



40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)





ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

300Nm50Nm

(a)

300Nm50Nm

40

30

20

10

0

ndash10ndash25

ndash30ndash35

ndash40 50 60 70 80 90 100 110

z m (m

m)

ym (mm) xm (mm)

(b)





ndash067

ndash066

ndash065

ndash064

ndash063

ndash062

LOA

dire

ctio

nal c

osin

e nx

(a)




ndash017

ndash0165

ndash016

ndash0155

ndash015

ndash0145

ndash014

LOA

dire

ctio

nal c

osin

e ny

(b)

Figure 7 Continued





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0735

074

0745

075

0755

076

0765

LOA

dire

ctio

nal c

osin

e nz

(c)




80

85

90

95

100

105

EMP

coor

dina

te x

m (m

m)

(d)




ndash32

ndash31

ndash30

ndash29

ndash28

ndash27

EMP

coor

dina

te y m

(mm

)

(e)

Figure 7 Continued











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











ndash6

ndash4

ndash2

0

2

4

6

EMP

coor

dina

te z m

(mm

)

(f )


Fme

Fma

Fml

Fm

FmT

LTMS

ASMS

Local tangent

Average secant

0

2000

4000

6000

8000

10000

12000

14000

16000

18000M

esh

forc

e (N

)


δd

δT

kTma = Fm

T δT






kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




kma Fm

δ (5)


δ ε0 minus ε (6)



ε0 e0λy

ε eλy(7)



e0 Δθg0 minusNp

Ng

Δθp0

e Δθg minusNp

Ng

Δθp

(8)






λi


λi+1 Δθ

(9)




λi

ΔSi+1 + Fmi+1kmai+11113872 1113873

λi+1 Δθ

(10)


kml dFm

dδ δδT

1113868111386811138681113868 asymp12


m


m minus FTm

δT+ΔT minus δT1113890 1113891 (11)



kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


kml asymp12


m


FT+ΔTm minus FT

m

εT minus εT+ΔT1113890 1113891 (12)













5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







5500500045004000350030002500200015001000

5000

0 02 04 06 08 1 12

Mes

h fo

rce (

N)

X 0207Y 6387

X 0171Y 1618

X 0171Y 4436

X 0711Y 4416

X 0783Y 7205

X 0747Y 2304 X 1107

Y 2556

X 1071Y 3584






Tooth pair 2

Tooth pair 1

Profile separation

Deformed gear

Undeformedgear

Pinion

Gear





DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




DTC STC DTC

0

0002

0004

0006

0008

001

0012

0014

Mes

h de

flect

ion

alon

gth

e LO

A (m

m)





DTC STC DTC


Gear pair

times105


1

2

3

4

5

6

ASM

S (N

mm

)

A

A

A

X 0351Y 5783e + 05

X 0351Y 3227e + 05

X 0351Y 267e + 05





times105

STC DTCDTC

1

2

3

4

5

6

7

LTM

S (N

mm

)




4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



4 DMF Calculation


Fmd




⎧⎪⎪⎨

⎪⎪⎩

(13)





Fmd




⎧⎪⎪⎨

⎪⎪⎩

(14)






dramatically





Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Fm d




⎧⎪⎪⎨

⎪⎪⎩

(16)








18000

16000

14000

12000

10000

8000

6000

4000

2000


0012 0006 0

0

02 03 04 05 06 07 08 09

0 01

Elas

tic co

ntac

t for

ce (N

)


A

A

B

B

x

y

O

y0g

z0g

z0p

y0p

LAp

LAg

Lp

LBp

LBg

LgOg

Op

(a)

y1j

y0j

z0j

x0j

x3j

Cj

Oj

x1j (x2

j )

z3j (z4

j )

y2j (y3

j )

z2j

z1j

γj

αj

αj

βj

βj

(b)









jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








jz3j e last



4jy4





lAB

klpxtZ


lAB


lAB

clpxtZ

lp + 1113944

N

i1npxiFmi (17a)


klpxt + αp 1113944

lAB

klpxtZ


lAB

clpxt + _αp 1113944

lAB

clpxtZ

lp + 1113944

N

i1npyiFmi (17b)



lAB

clpzt + 1113944

N

i1npziFmi (17c)

Jxpeuroαp + J

zp


lAB

klpxr + 1113944

lAB

klpxtZ

lp minus αpZ

lp + yp1113872 1113873 minus _θpx 1113944

lAB

clpxr + 1113944

lAB

clpxtZ

lp minus _αpZ

lp + _yp1113872 1113873 + 1113944

N

i1λpxiFmi

(17d)

Jxpeuroβp minus J

zp _αp _cp minus J


lAB

klpxr minus 1113944

lAB

klpxtZ

lp βpZ

lp + xp1113872 1113873 minus _θpy 1113944

lAB

clpxr minus 1113944

lAB

clpxtZ

lp

_βpZlp + _xp1113872 1113873 + 1113944

N

i1λpyiFmi

(17e)


N

i1λpziFmi (17f)



lAB

klgxtZ


lAB


lAB

clgxtZ

lg + 1113944

N

i1ngxiFmi (17g)


klgxt + αg 1113944

lAB

klgxtZ


lAB

clgxt + _αg 1113944

lAB

clgxtZ

lg + 1113944

N

i1ngyiFmi (17h)



lAB

clgzt + 1113944

N

i1ngziFmi (17i)

Jxgeuroαg + J

zg


lAB

klgxr + 1113944

lAB

klgxtZ

lg minus αgZ

lg + yg1113872 1113873 minus _θgx 1113944

lAB

clgxr + 1113944

lAB

clgxtZ

lg minus _αgZ

lg + _yg1113872 1113873 + 1113944

N

i1λgxiFmi

(17j)

Jxgeuroβg minus J

zg _αg _cg minus J


lAB

klgxr minus 1113944

lAB

klgxtZ

lg βgZ

lg + xg1113872 1113873 minus _θgy 1113944

lAB

clgxr minus 1113944

lAB

clgxtZ

lg

_βgZlg + _xg1113872 1113873 + 1113944

N

i1λgyiFmi

(17k)


N

i1λgziFmi (17l)





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(18)



0jy0



ymi zmi) is

xpmi

ypmi

zpmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpOprimermi

0 0 minus 1 0

0 1 0 0

1 0 0 minus OpO

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

ymi

xmi minus OpO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(19)


npxi

npyi

npzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOpO minus Li( 1113857

0 0 minus 1

0 1 0

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nxi

minus nyi

minus nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nzi

minus nyi

minus nxi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)


xgmi

ygmi

zgmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgOprimermi

0 0 minus 1 0

1 0 0 0


0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

xmi

ymi

zmi

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus zmi

xmi

minus ymi minus OgO

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)


ngxi

ngyi

ngzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ MOgOLi

0 0 minus 1

1 0 0

0 minus 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

nxi

nyi

nzi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus nzi

nxi

minus nyi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)




PMC θpz + θpz0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

θpzp

⎛⎝ ⎞⎠θpzp (23)

PTEC θpz + θt0

11138681113868111386811138681113868


11138681113868111386811138681113868

11138681113868111386811138681113868

Nθpzp

⎛⎝ ⎞⎠Nθpzp (24)




py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



py3pz3

p


(25)



gyIgzI


gyIgzI

g is


g

(26)



3pmi


Igmi

(27)


pz3p toOxyz are

MOOpprime

0 0 1 OpO

0 1 0 0

minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOpCpprime

1 0 βp xp

0 1 minus αp yp

minus βp αp 1 zp

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(28)


gzIg to Oxyz are

MOOgprime

0 1 0 0


minus 1 0 0 0

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

MOgCgprime



minus βg αg 1 zg

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(29)


εdi rOpmi minus rO

gmi1113872 1113873 middot Li (30)























400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

















400

600

800

1000

1200

1400

RMS

valu

e of D

MF

(N)

(a)



2900

3000

3100

3200

3300

3400

3500

3600

RMS

valu

e of D

MF

(N)

(b)



0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)










0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0100200300400500600700800900

Mes

h fo

rce (

N)




0 01 02

Loss ofcontact


0

500

1000

1500

2000

2500

Mes

h fo

rce (

N)

0200400600800





7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



7 Conclusion



2800

3000

3200

3400

3600

3800

RMS

valu

e of D

MF

(N)




0

1000

2000

3000

4000

5000

Mes

h fo

rce (

N)


1500

1000

50006


4400

4200

03 04 05






Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Abbreviations



Nomenclature








pinion and gearJx




tooth pair i


i












Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










Appendix




Tt T1 +T2 (A1)

us


(A2)





(A3)



e01 minus e1113872 1113873 (A4)



e02 minus e1113872 1113873 (A5)





ΔTt ΔT1+ΔT2 (A6)

us







B Dynamic Modeling



ijy

ijz




ωj _αji1j + _βjj

2j + _cjk

3j (B1)


4jy4

jz4j as follows


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ωx4j

ωy4j

ωz4j

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M43j M32j M21j

_αj

0

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

0_βj

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ +

0

0

_cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠



_αj sin βj + _cj

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


M21j

1 0 0

0 cos αj sin αj


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2b)

M32j



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2c)

M43j


0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (B2d)



+ _cj + βj _αj1113872 1113873k4j



jz0j as



_rj _xji0j+ _yjj

0j + _zjk

0j (B5)



p LBp and LA



j) with ZAp LA

p minus LpZB

p LBp minus Lp ZA




0jy0

jz0j is derived by

xlj

ylj

zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


0

0

Zlj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

βjZlj + xj

minus αjZlj + yj

Zlj + zj

1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B6)


M12jprimeMT

21j 0

0 1⎡⎣ ⎤⎦

M23jprime MT

32j 0

0 1⎡⎣ ⎤⎦

M34jprime MT

43j 0

0 1⎡⎣ ⎤⎦

(B7)


MOj1

1 0 0 xj

0 1 0 yj

0 0 1 zj

0 0 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(B8)


dlj βjZ

lj + xj1113960 1113961i0j + minus αjZ

lj + yj1113960 1113961j0j + zjk

0j (B9)


0jy0



0jy0




0j (B10)


0jy0





⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B11)






j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



j Jzj]


T 12

ωTj Jjωj + _rT

j mj _rj1113872 1113873 (B12)


ddt

z(T minus V)

z _qj

1113888 1113889 minusz(T minus V)

zqj

Qj (B13)


mj euroxj Qxtj

mj euroyj Qyt

j

mj eurozj Qztj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B14)

Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qαrj

Jxjeuroβj minus Jz

j _αj _cj Qβr

j

Jzjeurocj Q

crj

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(B15)

where Qαrj Q

βr


j-y2


j Q

yrj and Qzr

j about x0j- y0



jz0j

Qαrj Qxr

j

Qβr

j Qyrj + αjQ

zrj

Qcr


yr

j + Qzrj asymp Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B16)





yrj is



Jxj euroαj + Jz

j_βj _cj + Jz

jβjeurocj Qxrj

Jxjeuroβj minus Jz


yrj

Jzjeurocj Qzr

j

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(B17)








jt diag[kljxt kl

jxt kljzt] and Kl

jr diag[kljxr kl


jt diag[cljxt cl

jxt cljzt] and Cl

jr

diag[cljxr cl



Data Availability




Acknowledgments


References












































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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