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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
4 Shock and Vibration
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
Number of nodal diameters
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
Shock and Vibration 5
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
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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
4 Shock and Vibration
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
Number of nodal diameters
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
Shock and Vibration 5
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal 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
4 Shock and Vibration
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
Number of nodal diameters
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
Shock and Vibration 5
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Shock and Vibration
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
Number of nodal diameters
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
Shock and Vibration 5
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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DistributedSensor Networks
International Journal of
Shock and Vibration 5
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
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
Without Coriolis forceWith Coriolis force
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
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 7
Without Coriolis forceWith Coriolis force
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
Without Coriolis forceWith Coriolis force
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
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Shock and Vibration
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
(4) A new parameter is introduced to quantitativelydescribe the average change of amplitude of eachblade with and without considering the Corioliseffects
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)
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 9
+ V119874 (1 + 119904) (1 + 119905) (1 + 119903)
+ V119875 (1 minus 119904) (1 + 119905) (1 + 119903))
119908 =
1
8
(119908
119868 (1 minus 119904) (1 minus 119905) (1 minus 119903)
+ 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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of