research article vibration characteristics of a mistuned

Research ArticleVibration Characteristics of a Mistuned Bladed Disk consideringthe Effect of Coriolis Forces

Xuanen Kan and Bo Zhao

State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Xuanen Kan kanxuanenstuxjtueducn

Received 27 March 2016 Revised 12 July 2016 Accepted 14 July 2016

Academic Editor Jorg Wallaschek

Copyright copy 2016 X Kan and B ZhaoThis 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

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk a bladed disk with 22 bladesis employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response arediscussed considering the effects of Coriolis forces The results show that if there are frequency veering regions the largest splitof double natural frequencies of eachmodal family considering the effects of Coriolis forces appears at frequency veering region Inaddition the amplitude magnification factor considering the Coriolis effects is increased by 102 compared to the system withoutconsidering the Coriolis effects as the rotating speed is 3000 rpm while the amplitude magnification factor is increased by 276as the rotating speed is 10000 rpm The results indicate that the amplitude magnification factor may be moderately enhanced withthe increasing of rotating speed Moreover the position of the maximum forced response of bladed disk may shift from one bladeto another with the increasing of the rotational speed when the effects of Coriolis forces are considered

1 Introduction

Generally when the vibration characteristics of bladed diskare designed and analyzed the bladed disk is assumed to betuned [1ndash4] However practical experience shows that thereare always deviations of blade-to-blade caused by manufac-turing tolerances and wears during operation These smalldeviations are often called mistuning of blades Mistuningof bladed disk may cause vibration localization which mayaccelerate high cycle fatigue [5] Wei and Pierre [6 7] devel-oped perturbationmethods to investigate the vibration local-ization and pointed out that a small mistuning may result instrong vibration localization for weakly coupled systems Yooet al [8] established a simplified pendulummodel to researchthe influence of stiffness coupling damping parameters onthe vibration localization Chiu and Huang [9] investigatedthe influence of mistuning caused by bladersquos stagger angleon the stability of mistuned bladed disk Bladh et al [10]researched the relationship between mistuning strength andamplitude magnification factor Their results indicated thatforced response amplitude and stress of mistuned bladed diskare related to mistuning strength and coupling strength

In the previous works the effects of Coriolis force areusually negligible However blades of jet engine systembecome thinner and geometrical shapes become more andmore complicated [11] in particular the rotational speedbecomes higher and higher The model without consideringthe effects of Coriolis force cannot accurately describe thevibration characteristics of mistuned bladed disksThereforeHuang and Kuang [12] used Galerkinrsquos method to researchthemode localization of bladed disk considering the effects ofCoriolis forces Their results pointed out that rotating speedhas a significant effect on the mode localization of mistunedbladed disk Nikolic et al [13] established an experiment forthe first time to validate the split of double natural frequen-cies considering the effect of Coriolis forces and used alumped parameters mass-spring model to investigate theforced response localizationThe Coriolis forces of blades arerelated to the geometry of bladed disk Therefore Xin andWang [14] used realistic bladed disk to investigate the effectof Coriolis forces on vibration characteristics and pointedout that the localization of mode shapes was significantlyinfluenced by the Coriolis effects

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4656032 9 pageshttpdxdoiorg10115520164656032

2 Shock and Vibration

On the other hand if there are frequency veering regionsin curves of natural frequencies versus nodal diameters thedegree of double natural frequencies split of every modalfamily is not thoroughly discussed in previous work It isimportant for avoiding resonance of bladed disk In additionthe effects of Coriolis forces are affected by rotational speedbut the different rotational speeds on the maximum forcedresponsewith andwithout considering theCoriolis effects arenot thoroughly investigated in previous works

In this paper if there are frequency veering regions thesensitivity of degree of double natural frequencies split ofevery modal family to the Coriolis effects is discussed Inaddition the effects of different rotational speeds and excita-tion engine orders on the maximum forced response consid-ering the Coriolis effects are investigated

The remaining parts of this study are organized as followsIn Section 2 the theory of rotating bladed disk with consid-ering the effects of Coriolis force is presented In Section 3 ifthere are frequency veering regions the sensitivity of degreeof double natural frequencies split of every modal familyto the Coriolis effects is discussed In Section 4 the effectsof different rotational speeds and excitation engine orderson the maximum forced response considering the Corioliseffects are investigated In Section 5 main conclusions aresummarized

2 Theory of Rotating Blade Disk System withConsidering the Coriolis Effects

The general differential equation of motions of forcedresponse for rotating bladed disk is described as

Mx (119905) + (G + C) x (119905) + Kx (119905) = F (119905) (1)

where M is mass matrix G is Coriolis matrix C is dampingmatrixK = Ko+KC+Km whereKo is stiffnessmatrixesKC isstress stiffening matrix (it is described in detail in Appendix)andKm is spin softeningmatrix x(119905) si displacement responsevector of the system F(119905) is vector of external forces TheCoriolis matrix is generated

G = 2 int

119881

N119879ΩN 119889119881 (2)

where N is the shape function matrix and Ω the rotationalmatrix

Ω =

[

[

[

[

0 minusΩ

119911Ω

119910

Ω

1199110 minusΩ

119909

minusΩ

119910Ω

1199090

]

]

]

]

(3)

An engine order forcing is introduced

F (120579 119905) = F0119890

119894EO(Ω119905+120579)

(4)

where F0 is the force amplitude EO is the engine order Ω

is the rotational speed 119905 is the time 120579 is the circumferentialposition

Generally the amplitude magnification factor is used todescribe the change of forced response betweenmistuned and

tuned bladed disk and it is a critical parameter for evaluatinghigh cycle fatigue of bladed disk Quantitatively amplitudemagnification factor is the ratio between the maximumforced response of mistuned bladed disk and tuned bladeddisk [15]

Amplitude magnification factor =

119909maxmis119909maxtune

(5)

where 119909maxmis and 119909maxtune are themaximum forced responseof mistuned and tuned bladed disk respectively

3 Sensitivity of Degree of Double NaturalFrequencies Split of Every Modal Family tothe Coriolis Effects

The effects of Coriolis forces of bladed disk are influencedby the geometrical shapes of the blade so a realistic bladeddisk of jet engine is employed in this paper The bladed diskcontains 22 blades with 330659 eight-node solid elementsThe mesh of finite element of the bladed disk is shown inFigure 1 The working rotational speed is 16100 rpm Youngrsquosmodulus is 1172 GPa mass density is 45395 Kgm3 Poissonrsquosratio is 03 and the structural damping is 0002 Finiteelement method is used in the simulation

The curves of nodal diameters versus natural frequenciesof the tuned bladed disk are shown in Figure 2 It can beobserved that some eigenvalues converge and then veer apartwith the increasing of number of nodal diameters Thisphenomenon is called frequency veering as shown in Figure 2which is marked with dotted line It can be seen that thefirst frequency veering region locates between the first andsecond modal family and the second locates between thethird and fourth modal family as shown in Figure 2 Modeshapes which locate in the frequency veering regions aremixed blade disk motion [16ndash18]

The Coriolis effects leads to the double natural frequen-cies split of the tuned bladed disk Double natural frequenciessplit of the first modal family is shown in Figure 3(a) It canbe seen that the degree of double natural frequencies split isdifferent with the increasing of number of nodal diametersand the largest double natural frequencies split emerges in thefirst order of the first nodal diameter At the same time thedouble natural frequencies split of the secondmodal family isshown in Figure 3(b) It shows that the largest double naturalfrequencies split emerges in the second order of the first nodaldiameter The first order of the first nodal diameter and thesecond order of the first nodal diameter locate in the firstfrequency veering region as shown in Figure 2

In addition it is interesting that the largest split of doublenatural frequencies of the third modal family appears atthe third order of the third nodal diameter as shown inFigure 3(c) The largest split of double natural frequencies ofthe fourth modal family appears at the fourth order of thethird nodal diameter as shown in Figure 3(d)The third orderof the third nodal diameter and the fourth order of the thirdnodal diameter locate in the second frequency veering regionas shown in Figure 2

The results show that if there are frequency veeringregions the largest split of double natural frequencies of

Shock and Vibration 3

Figure 1 Model and mesh of finite element of the bladed disk

0

500

1000

1500

2000

2500

3000

3500

4000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110

Number of nodal diameters

1

2

Figure 2 Natural frequencies versus nodal diameters

every modal family considering the effects of Coriolis forcesappears at frequency veering region The results indicate thatthe effects of Coriolis force should be specially considered foravoiding resonance if the resonance region appears nearbythe frequency veering regions and it is critical for strengthand vibration design of jet engine system

The vibration displacements of bladed disk can be dividedinto three types as shown in Figure 4 [19] The first typeis described as disk-dominated modes in which the bladeand disk have large displacements while the blade acts likerigid plate without relevant deformations The second typeis described as coupled modes in which both the blade anddisk participate in the vibrationThe third type is described asblade-dominatedmodes in which the blade has deflect whilethe disk does not participate in the vibration

The largest split of double natural frequencies of everymodal family considering the effects of Coriolis forcesappears at frequency veering region since the effects of Cori-olis forces are influenced by the mode shapes The modes inthe veering regions tend to feature mixed disk-blade motionThe degree of coupling of blade and disk is higher in the veer-ing regions Firstly the effects of Coriolis forces will influencethe vibration of blade Secondly the effects of Coriolis forcesinfluence the vibration of disk and then the vibration of diskwill influence the vibration of blades Therefore the effectsof Coriolis forces have a significant influence on vibrationof modes in the veering regions The largest split of doublenatural frequencies of every modal family considering theeffects of Coriolis forces appears at frequency veering region

4 Forced Response of Mistuned Blade Diskconsidering the Coriolis Effects

The harmonic analysis is applied and the full method is usedin the analysis of the forced response of mistuned bladed diskconsidering the Coriolis effects In this section the influencesof the Coriolis effects on the maximum forced response ofmistuned bladed disk are investigated

The first blade mode family is considered in this paperThe frequency response of tuned bladed disk under engineorder 3 without considering the Coriolis effects is shown inFigure 5 We can see that the frequency response of bladeis identical with each other and there is only one peak It isbecause that tuned bladed disk is cyclic symmetric structureand energy can uniformly transmit between each bladeHence the frequency response of blade is identical with eachother Moreover the engine order 3 will excite the vibrationwith the third nodal diameter in the tuned bladed disk [20]Therefore the frequency response curve has only one peak


000

005

010

015

020

025

030

035

040

Freq

uenc

y sp

lit (

)

83 4 5 6 7 9 1021

Number of nodal diameters(a) The first modal family

2 3 4 5 6 7 8 9 101


00

05

10

15

20

25

30

35

40

Freq

uenc

y sp

lit (

)

(b) The second modal family

00

02

04

06

08

10

12

14

16

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101

Number of nodal diameters(c) The third modal family

000

005

010

015

020

025

030

035

040

045

050

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101

Number of nodal diameters(d) The fourth modal family

Figure 3 Double natural frequencies split

There is always mistuning of blade due to manufacturingtolerances and wear in operation Mistuning of blades willbreak the cyclic symmetric properties andwill lead to the splitof double natural frequencies of tuned bladed diskThe effectof Coriolis forces can also lead to the split of double naturalfrequencies of tuned bladed disk So it is interesting to knowthe change of the maximum forced response of mistunedbladed disk with and without considering the Coriolis effects

Mistuning of the blades is introduced by allowing eachblade to have different Youngrsquos modulus In the simulationYoungrsquos modulus of the blades satisfies the following relation-ship

119864

119895= (1 + 120575

119895) 119864

0 119895 = 1 22 (6)

where 119864

0and 119864

119895are Youngrsquos modulus for the 119895th blade of

tuned and mistuned bladed disk 120575

119895is the mistuning ratio

and the dimensionless mistuning parameter 120575

119895is randomly

obtained from a normal distribution with standard deviationof 1 percent and mean value 0 as shown in Figure 6

The frequency response of the mistuned bladed diskwithout considering the Coriolis effects under EO 3 forcing

is shown in Figure 7 It can be observed that each blade has adifferent frequency response curve andmultipeaks emergingMistuning of blade destroys the cyclic symmetric propertiesof blade disk system and the energy cannot uniformlytransmit Energy is confined to a few blades which will leadto vibration localization of bladed disk

The frequency response of mistuned bladed disk consid-ering the effect of Coriolis forces under EO 3 forcing is shownin Figure 8 It can be seen that each blade has a differentfrequency response curve and multipeaks emerging

The amplitude magnification factor of mistuned bladeddisk considering the Coriolis effects is increased by 213compared to the system without considering the Corioliseffects At the same time the maximum forced response ofeach blade will vary when the Coriolis effects is consideredas shown in Figure 9 The maximum variation of forcedresponse appears at the 6th blade and it is increased by 346when the Coriolis effects are considered

41 Effects of Excitation Engine Order on the MaximumForced Response considering the Coriolis Effects The vector ofexternal forces are related to engine order so in this section


Disk-dominated modes

ldquoCoupledrdquo modes

udisk

ublade

udisk

ublade

udisk

ublade

Blade-dominated modes

Figure 4 Blade and disk deflection for different vibration mode

Table 1 The maximum forced response of mistuned bladed diskwith and without considering the Coriolis effects

Engine order 119860

119873119862119860

119862|(119860

119862minus 119860

119873119862)119860

119873119862| ()

1 29209 27608 5481minus1 30343 26367 131043 27349 27932 21317minus3 26121 26739 2365911 25679 26927 486minus11 25679 26927 486

the effects of engine order on the maximum forced responseare discussed with considering the Coriolis effects

The maximum forced response with and without consid-ering the Coriolis effects under different excitation engineorders is shown in Table 1 119860

119873119862is the maximum forced

response without considering the Coriolis effects and 119860

119862is

the maximum forced response with considering the Corioliseffects It can be observed that the maximum forced responseof EO 11 is identical with EO minus11 while the maximum forcedresponses of mistuned bladed disk excited by other engineorders are different

(1)Thenatural frequencies are double natural frequenciesof the tuned bladed disk except for the 0 and 1198732 (119873 is even)nodal diameters There are 22 blades in our model of bladeddisk system Hence the mode shapes at nodal diameter 11 aresingle mode Moreover the back traveling excitation by EO 11is coincident with the forward traveling excitation by EO minus11

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Am

plitu

de (m

m)

04

06

08

10

12

14

16

18

20

22

02

00

Figure 5 Frequency response of tuned bladed disk without consid-ering the Coriolis effects under EO 3

minus003

minus002

minus001

000

001

002

003

Mist

unin

g ra

tio

3 5 7 9 11 13 15 17 191 21

Blade number

Figure 6 Mistuning pattern

The forced response curve of the mistuned bladed diskexcited by EO 11 is identical with the forced response curveexcited by EO minus11 Therefore the maximum forced responseis identical with each other excited by EO 11 and EO minus11

(2) The mode shapes at 1 to 10 nodal diameters aredouble modes The effect of Coriolis force will lead to split ofthe double natural frequencies Moreover the back travelingexcitation EO 1 and EO 3 are different with the forwardtraveling excitation EO minus1 and EO minus3 Therefore frequencyresponse curves excited by EO 1 and EO 3 are different withthe frequency response curves excited by EO minus1 and EO minus3respectively

42 Effects of Different Rotational Speeds on the MaximumForced Response with and without Considering the CoriolisEffects The Coriolis forces are affected by the different rota-tional speeds In this section the effects of different rotational

6 Shock and VibrationA

mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Figure 7 The frequency response of mistuned bladed disk withoutconsidering the Coriolis effects under EO 3

Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600

Frequency (Hz)Blade number

2

0333

69

1215

18 520

540560

580

6

Frequency (Hz)Blade numbe

Figure 8 The frequency response of mistuned bladed disk consid-ering the Coriolis effects under EO 3

speeds on the maximum forced response are investigatedconsidering the Coriolis effects

The forced response amplitude magnification factor withand without the effect of Coriolis forces under differentrotational speeds is shown in Figure 10 The amplitude mag-nification factor considering the Coriolis effects is increasedby 102 compared to the system without considering theeffect of Coriolis forces as the rotating speed is 3000 rpmThe amplitude magnification factor considering the effect ofCoriolis forces is increased by 276 compared to the systemwithout considering the Coriolis effects as the rotating speedis 10000 rpm

At the same time the maximum forced response of eachblade varies when the effect of Coriolis force is considered

06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number

Without Coriolis forceWith Coriolis force

Figure 9 The maximum forced response of each blade with andwithout considering the Coriolis effects


126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)

Figure 10 The amplitude magnification factor with and withoutconsidering the effect of Coriolis forces under different rotationalspeeds

The largest change of the maximum forced response consid-ering the effect of Coriolis forces appears at the 6th blade andthemaximum forced response of the 6th blade is increased by656 compared to the system without considering the effectof Coriolis forces when the rotational speed is 3000 rpm asshown in Figure 11(a)The largest change of maximum forcedresponse considering the effect of Coriolis forces appears atthe 6th blade and the maximum forced response of the 6thblade is increased by 255 compared to the system withoutconsidering the effect of Coriolis forces when the rotationalspeed is 10000 rpm as shown in Figure 11(b)

Another interesting phenomenon found that the positionof themaximum forced response of themistuned bladed diskshifts from the 11th blade to the 8th blade with the effects



5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se

(a) Rotating speed is 3000 rpm


5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se

(b) Rotating speed is 10000 rpm

Figure 11 The maximum forced response of each blade with and without considering the Coriolis effects

3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number

Figure 12The average change ratio of amplitude of each bladewhenthe rotating speed is 3000 rpm and 10000 rpm

of Coriolis forces being considered when the rotating speedis 10000 rpm as shown in Figure 11(b) The average changeratio of amplitude of each blade with the rotational speed3000 rpm and 10000 rpm is shown in Figure 12 In order toquantitatively describe the change of each blade with andwithout considering the Coriolis effects a new parameter isintroduced

119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875

119903is the average change of amplitude of each blade

when the rotating speed is 119903 rpm 119860

119862

119903119894and 119860

119873119862

119903119894are the ampli-

tude of 119894th blade with and without considering the Coriolis

effects and119873 is the total number of bladesWhen the rotatingspeed is 3000 rpm and 10000 rpm 119875

119903is 277 and 964

respectivelyPrevious work [12] indicated that rotating speed has a sig-

nificant effect on mode localization of mistuned bladed diskThe results of the present paper give some new conclusions asfollows

(1) Themaximum forced response of bladed disk may beenhanced with the increasing of rotating speed

(2) The position of the maximum of forced responseconsidering the Coriolis effects may shift from oneblade to another with the increasing of rotationalspeed

(3) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects

5 Conclusions

The influences of Coriolis effects on vibration characteristicsof a mistuned bladed disk have been investigated and themajor conclusions of this paper are as follows

(1) If there are frequency veering regions in curvesof natural frequencies versus nodal diameters thelargest double natural frequencies split of everymodalfamily appears at frequency veering regions Theresults indicate that the effect of Coriolis force shouldbe specially considered for avoiding resonance ofbladed disk if the resonance region appears nearbythe frequency veering regions

(2) The maximum forced response of bladed disk andthe maximum forced response of each blade willvary when the effects of Coriolis force are considered


Z w

Y

X u

t

KL

P

s

J

M N

r

I

Figure 13 Eight-node brick solid element

and this variation is related to the excitation engineorders Furthermore the position of the maximumforced response of bladed disk will shift from oneblade to another as the effects of Coriolis forces areconsidered

(3) The amplitude magnification factor considering theeffect of Coriolis forces is increased by 102 com-pared to the system without considering the effectsof Coriolis forces as the rotating speed is 3000 rpmwhile the amplitude magnification factor is increasedby 276 as the rotating speed is 10000 rpm Theresults indicate that the maximum forced responsemay be moderately enhanced with the increasing ofrotational speed Furthermore the position of themaximum forced response considering the effectsof Coriolis forces may shift with the increasing ofrotational speed


Appendix

In this paperKC is the stress stiffening matrix 119873 is the shapefunction of 8-node brick solid element as shown in Figure 13

KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2

120597zsdot sdot sdot

120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906

119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (

1 minus 119904) (1 minus 119905) (1 minus 119903)

+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests

The authors declare that they have no competing interests

References

[1] I Y Shen ldquoVibration of rotationally periodic structuresrdquoJournal of Sound and Vibration vol 172 no 4 pp 459ndash4701994

[2] J Tang and K W Wang ldquoVibration control of rotationallyperiodic structures using passive piezoelectric shunt networksand active compensationrdquo Journal of Vibration and AcousticsTransactions of the ASME vol 121 no 3 pp 379ndash390 1999

[3] J Y Chang and J A Wickert ldquoResponse of modulated doubletmodes to travelling wave excitationrdquo Journal of Sound andVibration vol 242 no 1 pp 69ndash83 2001

[4] J Y Chang and J A Wickert ldquoMeasurement and analysisof modulated doublet mode response in mock bladed disksrdquoJournal of Sound and Vibration vol 250 no 3 pp 379ndash4002002

[5] M P Castanier and C Pierre ldquoModeling and analysis ofmistuned bladed disk vibration current status and emergingdirectionsrdquo Journal of Propulsion and Power vol 22 no 2 pp384ndash396 2006

[6] S-T Wei and C Pierre ldquoLocalization phenomena in mistunedassemblies with cyclic symmetrymdashpart I free vibrationsrdquo Jour-nal of Vibration Acoustics Stress and Reliability in Design vol110 no 4 pp 429ndash438 1988

[7] S-T Wei and C Pierre ldquoLocalization phenomena in mistunedassemblies with cyclic symmetry Part II forced vibrationsrdquoJournal of Vibration Acoustics Stress and Reliability in Designvol 110 no 4 pp 439ndash449 1988

[8] H H Yoo J Y Kim and D J Inman ldquoVibration localizationof simplified mistuned cyclic structures undertaking externalharmonic forcerdquo Journal of Sound and Vibration vol 261 no5 pp 859ndash870 2003

[9] Y-J Chiu and S-C Huang ldquoThe influence of amistuned bladersquosstaggle angle on the vibration and stability of a shaft-disk-bladeassemblyrdquo Shock and Vibration vol 15 no 1 pp 3ndash17 2008

[10] R Bladh C Pierre M P Castanier and M J Kruse ldquoDynamicresponse predictions for a mistuned industrial turbomachinery

rotor using reduced-order modelingrdquo Journal of Engineering forGas Turbines and Power vol 124 no 2 pp 311ndash324 2002

[11] C Gibert V Kharyton F Thouverez and P Jean ldquoOn forcedresponse of a rotating integrally bladed disk predictions andexperimentsrdquo in Proceedings of the ASME Turbo Expo Power forLand Sea and Air pp 1103ndash1116 Glasgow UK June 2010

[12] B W Huang and J H Kuang ldquoMode localization in a rotatingmistuned turbo disk with Coriolis effectrdquo International Journalof Mechanical Sciences vol 43 no 7 pp 1643ndash1660 2001

[13] M Nikolic E P Petrov and D J Ewins ldquoCoriolis forces inforced response analysis of mistuned bladed disksrdquo Journal ofTurbomachinery vol 129 no 4 pp 730ndash739 2007

[14] J Xin and J Wang ldquoInvestigation of coriolis effect on vibrationcharacteristics of a realistic mistuned bladed diskrdquo in Proceed-ings of the ASME Turbo Expo Turbine Technical Conference andExposition pp 993ndash1005 Vancouver Canada June 2011

[15] M P Castanier and C Pierre ldquoInvestigation of the com-bined effects of intentional and random mistuning on theforced response of bladed disksrdquo in Proceedings of the34th AIAAASMESAEASEE Joint Propulsion Conference andExhibit vol 1001 Cleveland Ohio USA 1998

[16] J L Du Bois S Adhikari and N A J Lieven ldquoOn thequantification of eigenvalue curve veering a veering indexrdquoJournal of Applied Mechanics vol 78 no 4 Article ID 0410072011

[17] J A Kenyon J H Griffin and N E Kim ldquoSensitivity oftuned bladed disk response to frequency veeringrdquo Journal ofEngineering for Gas Turbines and Power vol 127 no 4 pp 835ndash842 2005

[18] I Lopez R R J J van Doorn R van der Steen N B Roozenand H Nijmeijer ldquoFrequency loci veering due to deformationin rotating tyresrdquo Journal of Sound and Vibration vol 324 no3-5 pp 622ndash639 2009

[19] T Klauke U Strehlau and A Kuhhorn ldquoInteger frequencyveering of mistuned blade integrated disksrdquo Journal of Turbo-machinery vol 135 no 3 Article ID 061004 2013

[20] S J Wildheim ldquoExcitation of rotationally periodic structuresrdquoJournal of Applied Mechanics vol 46 no 4 pp 878ndash882 1979

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



On the other hand if there are frequency veering regionsin curves of natural frequencies versus nodal diameters thedegree of double natural frequencies split of every modalfamily is not thoroughly discussed in previous work It isimportant for avoiding resonance of bladed disk In additionthe effects of Coriolis forces are affected by rotational speedbut the different rotational speeds on the maximum forcedresponsewith andwithout considering theCoriolis effects arenot thoroughly investigated in previous works

In this paper if there are frequency veering regions thesensitivity of degree of double natural frequencies split ofevery modal family to the Coriolis effects is discussed Inaddition the effects of different rotational speeds and excita-tion engine orders on the maximum forced response consid-ering the Coriolis effects are investigated

The remaining parts of this study are organized as followsIn Section 2 the theory of rotating bladed disk with consid-ering the effects of Coriolis force is presented In Section 3 ifthere are frequency veering regions the sensitivity of degreeof double natural frequencies split of every modal familyto the Coriolis effects is discussed In Section 4 the effectsof different rotational speeds and excitation engine orderson the maximum forced response considering the Corioliseffects are investigated In Section 5 main conclusions aresummarized

2 Theory of Rotating Blade Disk System withConsidering the Coriolis Effects

The general differential equation of motions of forcedresponse for rotating bladed disk is described as

Mx (119905) + (G + C) x (119905) + Kx (119905) = F (119905) (1)

where M is mass matrix G is Coriolis matrix C is dampingmatrixK = Ko+KC+Km whereKo is stiffnessmatrixesKC isstress stiffening matrix (it is described in detail in Appendix)andKm is spin softeningmatrix x(119905) si displacement responsevector of the system F(119905) is vector of external forces TheCoriolis matrix is generated

G = 2 int

119881

N119879ΩN 119889119881 (2)

where N is the shape function matrix and Ω the rotationalmatrix

Ω =

[

[

[

[

0 minusΩ

119911Ω

119910

Ω

1199110 minusΩ

119909

minusΩ

119910Ω

1199090

]

]

]

]

(3)

An engine order forcing is introduced

F (120579 119905) = F0119890

119894EO(Ω119905+120579)

(4)

where F0 is the force amplitude EO is the engine order Ω

is the rotational speed 119905 is the time 120579 is the circumferentialposition

Generally the amplitude magnification factor is used todescribe the change of forced response betweenmistuned and

tuned bladed disk and it is a critical parameter for evaluatinghigh cycle fatigue of bladed disk Quantitatively amplitudemagnification factor is the ratio between the maximumforced response of mistuned bladed disk and tuned bladeddisk [15]

Amplitude magnification factor =

119909maxmis119909maxtune

(5)

where 119909maxmis and 119909maxtune are themaximum forced responseof mistuned and tuned bladed disk respectively

3 Sensitivity of Degree of Double NaturalFrequencies Split of Every Modal Family tothe Coriolis Effects

The effects of Coriolis forces of bladed disk are influencedby the geometrical shapes of the blade so a realistic bladeddisk of jet engine is employed in this paper The bladed diskcontains 22 blades with 330659 eight-node solid elementsThe mesh of finite element of the bladed disk is shown inFigure 1 The working rotational speed is 16100 rpm Youngrsquosmodulus is 1172 GPa mass density is 45395 Kgm3 Poissonrsquosratio is 03 and the structural damping is 0002 Finiteelement method is used in the simulation

The curves of nodal diameters versus natural frequenciesof the tuned bladed disk are shown in Figure 2 It can beobserved that some eigenvalues converge and then veer apartwith the increasing of number of nodal diameters Thisphenomenon is called frequency veering as shown in Figure 2which is marked with dotted line It can be seen that thefirst frequency veering region locates between the first andsecond modal family and the second locates between thethird and fourth modal family as shown in Figure 2 Modeshapes which locate in the frequency veering regions aremixed blade disk motion [16ndash18]

The Coriolis effects leads to the double natural frequen-cies split of the tuned bladed disk Double natural frequenciessplit of the first modal family is shown in Figure 3(a) It canbe seen that the degree of double natural frequencies split isdifferent with the increasing of number of nodal diametersand the largest double natural frequencies split emerges in thefirst order of the first nodal diameter At the same time thedouble natural frequencies split of the secondmodal family isshown in Figure 3(b) It shows that the largest double naturalfrequencies split emerges in the second order of the first nodaldiameter The first order of the first nodal diameter and thesecond order of the first nodal diameter locate in the firstfrequency veering region as shown in Figure 2

In addition it is interesting that the largest split of doublenatural frequencies of the third modal family appears atthe third order of the third nodal diameter as shown inFigure 3(c) The largest split of double natural frequencies ofthe fourth modal family appears at the fourth order of thethird nodal diameter as shown in Figure 3(d)The third orderof the third nodal diameter and the fourth order of the thirdnodal diameter locate in the second frequency veering regionas shown in Figure 2

The results show that if there are frequency veeringregions the largest split of double natural frequencies of



0

500

1000

1500

2000

2500

3000

3500

4000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110


1

2









000

005

010

015

020

025

030

035

040

Freq

uenc

y sp

lit (

)

83 4 5 6 7 9 1021


2 3 4 5 6 7 8 9 101


00

05

10

15

20

25

30

35

40

Freq

uenc

y sp

lit (

)


00

02

04

06

08

10

12

14

16

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101


000

005

010

015

020

025

030

035

040

045

050

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101





119864

119895= (1 + 120575

119895) 119864

0 119895 = 1 22 (6)

where 119864

0and 119864





119895is randomly










udisk

ublade

udisk

ublade

udisk

ublade




Engine order 119860

119873119862119860

119862|(119860

119862minus 119860

119873119862)119860

119873119862| ()






119862is



36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Am

plitu

de (m

m)

04

06

08

10

12

14

16

18

20

22

02

00


minus003

minus002

minus001

000

001

002

003

Mist

unin

g ra

tio

3 5 7 9 11 13 15 17 191 21

Blade number






mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)


Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600


2

0333

69

1215

18 520

540560

580

6






06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number




126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)






5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

500

1000

1500

2000

2500

3000

3500

4000

Freq

uenc

y (H

z)

1 2 3 4 5 6 7 8 9 10 110


1

2









000

005

010

015

020

025

030

035

040

Freq

uenc

y sp

lit (

)

83 4 5 6 7 9 1021


2 3 4 5 6 7 8 9 101


00

05

10

15

20

25

30

35

40

Freq

uenc

y sp

lit (

)


00

02

04

06

08

10

12

14

16

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101


000

005

010

015

020

025

030

035

040

045

050

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101





119864

119895= (1 + 120575

119895) 119864

0 119895 = 1 22 (6)

where 119864

0and 119864





119895is randomly










udisk

ublade

udisk

ublade

udisk

ublade




Engine order 119860

119873119862119860

119862|(119860

119862minus 119860

119873119862)119860

119873119862| ()






119862is



36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Am

plitu

de (m

m)

04

06

08

10

12

14

16

18

20

22

02

00


minus003

minus002

minus001

000

001

002

003

Mist

unin

g ra

tio

3 5 7 9 11 13 15 17 191 21

Blade number






mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)


Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600


2

0333

69

1215

18 520

540560

580

6






06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number




126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)






5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000

005

010

015

020

025

030

035

040

Freq

uenc

y sp

lit (

)

83 4 5 6 7 9 1021


2 3 4 5 6 7 8 9 101


00

05

10

15

20

25

30

35

40

Freq

uenc

y sp

lit (

)


00

02

04

06

08

10

12

14

16

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101


000

005

010

015

020

025

030

035

040

045

050

Freq

uenc

y sp

lit (

)

2 3 4 5 6 7 8 9 101





119864

119895= (1 + 120575

119895) 119864

0 119895 = 1 22 (6)

where 119864

0and 119864





119895is randomly










udisk

ublade

udisk

ublade

udisk

ublade




Engine order 119860

119873119862119860

119862|(119860

119862minus 119860

119873119862)119860

119873119862| ()






119862is



36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Am

plitu

de (m

m)

04

06

08

10

12

14

16

18

20

22

02

00


minus003

minus002

minus001

000

001

002

003

Mist

unin

g ra

tio

3 5 7 9 11 13 15 17 191 21

Blade number






mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)


Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600


2

0333

69

1215

18 520

540560

580

6






06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number




126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)






5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












udisk

ublade

udisk

ublade

udisk

ublade




Engine order 119860

119873119862119860

119862|(119860

119862minus 119860

119873119862)119860

119873119862| ()






119862is



36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)

Am

plitu

de (m

m)

04

06

08

10

12

14

16

18

20

22

02

00


minus003

minus002

minus001

000

001

002

003

Mist

unin

g ra

tio

3 5 7 9 11 13 15 17 191 21

Blade number






mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)


Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600


2

0333

69

1215

18 520

540560

580

6






06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number




126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)






5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










mpl

itude

(mm

)

04

0608

1012

14

16

18

20

22

24

26

28

02

00

36

912

1518

21

Blade number500

520

540

560

580

600

Frequency (Hz)


Am

plitu

de (m

m)

04

0608

1012

14

16

18

20

22

24

26

28

02

003

69

1215

1821

500

520

540560

580

600


2

0333

69

1215

18 520

540560

580

6






06

08

10

12

14

Nor

mal

ized

forc

ed re

spon

se

5 10 15 200

Blade number




126

128

130

132

134

136

138

Am

plitu

de m

agni

ficat

ion

fact

or

4000 6000 8000 10000 12000 14000 16000 180002000

(rpm)






5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











5 10 15 200

Blade number

05

06

07

08

09

10

11

12

13

Nor

mal

ized

forc

ed re

spon

se



5 10 15 200

Blade number

04

05

06

07

08

09

10

11

12

13

14

Nor

mal

ized

forc

ed re

spon

se



3000 rpm10000 rpm

0

5

10

15

20

25

Chan

ge ra

tio (

)

5 10 15 200

Blade number



119875

119903=

sum

119873

119894=1

1003816

1003816

1003816

1003816

1003816

(119860

119862

119903119894minus 119860

119873119862

119903119894) 119860

119873119862

119903119894

1003816

1003816

1003816

1003816

1003816

119873

(7)

where 119875



119862

119903119894and 119860

119873119862




119903is 277 and 964






5 Conclusions





Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Z w

Y

X u

t

KL

P

s

J

M N

r

I





Appendix


KC =

[

[

[

S0

S0

S0

]

]

]

(A1)

where

S0

= int

VSg

TSmSg119889V

Sm =

[

[

[

[

120590x 120590xy 120590xz

120590xy 120590y 120590yz

120590xz 120590yz 120590z

]

]

]

]

Sg =

[

[

[

[

[

[

[

[

[

120597119873

1

120597119909

120597119873

2

120597119909

sdot sdot sdot

120597119873

8

120597119909

120597119873

1

120597119910

120597119873

2

120597119910

sdot sdot sdot

120597119873

8

120597119910

120597119873

1

120597z120597119873

2


120597119873

8

120597z

]

]

]

]

]

]

]

]

]

119873 =

[

[

[

119906

V

119908

]

]

]

(A2)

where

119906 =

1

8

(119906


+ 119906

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119906

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119906

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119906

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119906

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119906

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119906

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

V =

1

8

(V119868 (


+ V119869 (

1 + 119904) (1 minus 119905) (1 minus 119903)

+ V119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ V119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ V119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ V119873 (1 + 119904) (1 minus 119905) (1 + 119903)


+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))

119908 =

1

8

(119908


+ 119908

119869 (1 + 119904) (1 minus 119905) (1 minus 119903)

+ 119908

119870 (1 + 119904) (1 + 119905) (1 minus 119903)

+ 119908

119871 (1 minus 119904) (1 + 119905) (1 minus 119903)

+ 119908

119872 (1 minus 119904) (1 minus 119905) (1 + 119903)

+ 119908

119873 (1 + 119904) (1 minus 119905) (1 + 119903)

+ 119908

119874 (1 + 119904) (1 + 119905) (1 + 119903)

+ 119908

119875 (1 minus 119904) (1 + 119905) (1 + 119903))

(A3)

Competing Interests


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article vibration characteristics of a mistuned

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