Analytic Solution for Nonlinear Multimode Beam Vibration ...

Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified HarmonicBalance Approach and Vietas Substitution

Lee Y Y

Published inShock and Vibration

Published 01072016

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Published version (DOI)10115520163462643

Publication detailsLee Y Y (2016) Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified Harmonic BalanceApproach and Vietas Substitution Shock and Vibration 2016 [3462643] httpsdoiorg10115520163462643

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Research ArticleAnalytic Solution for Nonlinear MultimodeBeam Vibration Using a Modified HarmonicBalance Approach and Vietarsquos Substitution

Y Y Lee

Department of Architecture and Civil Engineering City University of Hong Kong Kowloon Tong Kowloon Hong Kong

Correspondence should be addressed to Y Y Lee bcrayleecityueduhk

Received 29 June 2015 Accepted 7 October 2015

Academic Editor Dumitru I Caruntu

Copyright copy 2016 Y Y Lee This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a modified harmonic balance solution method incorporated with Vietarsquos substitution technique for nonlinearmultimode damped beam vibrationThe aim of the modification in the solution procedures is to develop the analytic formulationswhich are used to calculate the vibration amplitudes of a nonlinearmultimode dampedbeamwithout the need of nonlinear equationsolver for the nonlinear algebraic equations generated in the harmonic balance processes The result obtained from the proposedmethod shows reasonable agreement with that from a previous numerical integration method In general the results can show theconvergence and prove the accuracy of the proposed method

1 Introduction

Over the past decades many solution methods were devel-oped for various engineering modelling problems (eg [1ndash4]) The two well-known solution methods the perturbationmethod and multiple scale method (eg [5ndash7]) are widelyadopted In these two methods coupled nonlinear algebraicequations are generated and required to be solved by nonlin-ear equation solverThere are two harmonic balancemethodsalso widely employed (ie total harmonic balance methodand incremental harmonic balance method eg [8 9])Theyalso generate coupled nonlinear algebraic equations in theharmonic balance processes In the previous works [10 11]the multilevel residue harmonic balance method was modi-fied from the total harmonic balance method and developedfor nonlinear vibrations Although the accuracies of thesesolutionswere good and verified by other solutionmethods itwas quite time consuming to develop the nonlinear equationsolver for those nonlinear algebraic equations generated inthe solution procedures On the other hand there havebeen numerous beamplate related research problems solvedusing various numerical and classical methods (eg [12ndash15]) That is the motivation in this study to develop theanalytic formulations to calculate the vibration responses of

a nonlinearmultimode beamwithout using a nonlinear equa-tion solver for the nonlinear algebraic equations generated inthe harmonic balance procedures The proposed method ismodified from the previous harmonic balance method andincorporated with Vietarsquos substitution technique Using theanalytic calculation formulations the results can be obtainedwithout the need of nonlinear equation solver

2 Governing Equation

The governing equation of motion of nonlinear vibration ofthe Euler-Bernoulli theory is as follows [16]

1198641198681198821015840101584010158401015840

+ 120588119860 minus (

119864119860

2119871int

119871

0

(1198821015840)2

119889119909)11988210158401015840= 119865 (119909 119905) (1)

where 119909 is longitudinal coordinate and 119882 is transversedisplacement 1198821015840 11988210158401015840 and 119882

1015840101584010158401015840 are 1st 2nd and 4thderivatives with respect to 119909 respectively and are 1stand 2nd derivatives with respect to time 119905 119871 is length ℎ isthickness 119864119860 is Youngrsquos modulus times cross-sectional area 120588

119860

is material density per unit length and 119865(119909 119905) is externalharmonic excitation For uniformly distributed excitation119865(119909 119905) = 119865

119900sin(120596119905) 119865

119900= 120581120588119860119892 = excitation magnitude

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3462643 6 pageshttpdxdoiorg10115520163462643

2 Shock and Vibration

120581 is a dimensionless excitation parameter 119892 =

981msminus2 and 120596 is excitation frequencyThen the governing equation is discretized using the

modal reduction approach

119882(119909 119905) =

119899

sum

119894=1

119902119894(119905) 120593119894(119909) (2)

where 119902119894is the modal amplitude of the 119894th mode 120593

119894is the 119894th

structural mode shape and 119899 is number of modes usedConsider substituting (2) into (1) andmultiplying 120593

119898to it

and taking integration over the beam length

120588119860

119899

sum

119894=1

11990211989412057200

119894119898+ 119864119868

119899

sum

119894=1

11990211989412057240

119894119898minus119864119860

2119871

119899

sum

119894=1

119899

sum

119895=1

119899

sum

119896=1

11990211989411990211989511990211989612057220

11989411989812057211

119895119896

minus 119865119898sin (120596119905) = 0

(3)

where 12057200

119894119898= int119871

0120593119894120593119898119889119909 12057240119894119898

= int119871

01205931015840101584010158401015840

119894120593119898119889119909 12057220119894119898

= int119871

012059310158401015840

119894

120593119898119889119909 12057211119895119896

= int119871

01205931015840

1198951205931015840

119896119889119909 and 119865

119898sin(120596119905) is modal force For

uniformly distributed excitation 119865119898

= int119871

0119865119900120593119898119889119909 int

119871

0120593119898

120593119898119889119909If the beam is simply supported then 120593

119898(119909) = sin((119898120587

119871)119909) and 12057200119894119898

= 12057240

119894119898= 12057220

119894119898= 12057211

119894119898= 0 for 119894 = 119898 Then (3) can

be reduced into the following form

120588119860

119902119898+ 1205881198601205962

119898119902119898+

119899

sum

119895=1

120573119898119895

1199021198981199022

119895minus 119865119898sin (120596119905) = 0 (4)

where 120573119898119895

= minus(1198641198602119871)(12057220

11989811989812057200

119898119898)12057211

119895119895and120596

119898is the natural

frequency of the119898th mode

3 Modified Solution Procedure UsingVietarsquos Substitution

The solution form of themodal amplitudes is given by [10 17]

119902119898(119905) cong 120576

01199021198980

(119905) + 12057611199021198981

(119905) + 12057621199021198982

(119905) + sdot sdot sdot (5)

where 120576 is an embedding parameter 1199021198980

(119905) 1199021198981

(119905) and1199021198982

(119905) are the zero 1st and 2nd level modal amplitudesolutions

By inputting (5) into (4) consider those terms associatedwith 120576

0 and set up the zero level governing equations

120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980minus 119865119898sin (120596119905)

+

119899

sum

119895=1119898 =119895

120573119898119895

1199021198980

1199022

1198950= Δ1198980

(119905)

(6)

where 1199021198980

= 11986011989801

sin(120596119905) and the first second and thirdsubscripts in 119860

11989801are the mode number zero level and 1st

harmonic number Δ1198980

is the zero level residual

For simplicity let

1198771198980

(119905) = 120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980

minus 119865119898sin (120596119905)

(7a)

1198781198980

(119905) =

119899

sum

119895=1

120573119898119895

1199021198980

1199022

1198950 (7b)

Then put (7a)-(7b) into (6)

1198771198980

(119905) + 1198781198980

(119905) = Δ1198980

(119905) (8)

Consider the harmonic balance of sin(120596119905) in (7a)

1198603

11989801+

4

3120573119898119898

120588119860(1205962

119898minus 1205962)11986011989801

+4

3120573119898119898

119865119898= 0 (9)

When the damped vibration is considered the zero levelmodal vibration amplitude is rewritten from (9) and itscomplex form is given by

1003816100381610038161003816119860119898011003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119865119898

prod119898(120596) + (34) 120573

119898119898

1003816100381610038161003816119860119898011003816100381610038161003816

2

+ 119869 (2120585120596120596119898)

1003816100381610038161003816100381610038161003816100381610038161003816

(10a)

Then (10a) can be rewritten as

(3

4120573119898119898

)

2

1003816100381610038161003816119860119898011003816100381610038161003816

6

+3

2120573119898119898

prod

119898

(120596)100381610038161003816100381611986011989801

1003816100381610038161003816

4

+ ((prod

119898

(120596))

2

+ (2120585120596120596119898)2

)100381610038161003816100381611986011989801

1003816100381610038161003816

2

minus (119865119898)2

= 0

(10b)

Consider a polynomial form

1198861199103+ 1198871199102+ 119888119910 + 119889 = 0 (10c)

where for simplicity let prod119898(120596) = 120588

119860(minus1205962+ 1205962

119898) +

119869(2120585120588119860120596120596119898) 119869 = radicminus1 120585 is damping ratio and 120596

119898is nonlin-

ear peak frequency (note that in case of linear vibration120596119898=

120596119898) 119910 = |119860

11989801|2 119886 = ((34)120573

119898119898)2 119887 = (32)120573

119898119898prod119898(120596)

119888 = (prod119898(120596))2+ (2120585120596120596

119898)2 119889 = minus(119865

119898)2

Using the substitution of119910 = 120591minus1198873119886 (10c) can be furthersimplified as

1205913+ 119906120591 + V = 0 (11)

where 119906 = (3119886119888 minus 1198872)31198862 V = (2119887

3minus 9119886119887119888 + 27119886

2119889)27119886

3120591 = 119861 minus 1199063119861 (ie Vietarsquos substitution [18]) Equation (10c)can be rewritten as

1198616+ V1198613 minus

1199063

27= 0 (12)

If 1198613 is considered as an independent unknown (12) is aldquomodifiedrdquo quadratic equation Hence the analytic formula

Shock and Vibration 3

for the solutions can be obtained without the need of nonlin-ear equation solverThen the 1st level governing equation canbe set up by inputting (5) into (4) and picking up those termsassociated with 120576

1

120588119860

1199021198981

+ 2120585120588119860120596119898

1199021198981

+ 1205881198601205962

1198981199021198981

+

119899

sum

119895=1

120573119898119895

[(1199021198950)2

1199021198981

+ 21199021198980

11990211989501199021198951]

+ Δ1198980

(119905) = Δ1198981

(119905)

(13)

where 1199021198981

(119905) =11986011989811

sin(120596119905) + 11986011989813

sin(3120596119905)Note that 119860

11989801has been found from the zero level

solution procedure Thus 1199021198950

and Δ1198980

(119905) in (13) which arein terms of 119860

11989801 are known Therefore (13) is a linear

equationConsider the harmonic balances of sin(120596119905) and sin(3120596119905)

in (13) to set up two algebraic linear equations

int

2120587

0

Δ1198981

(119905) sin (120596119905) 119889119905 = 0 (14a)

int

2120587

0

Δ1198981

(119905) sin (3120596119905) 119889119905 = 0 (14b)

As (14a)-(14b) are linear the unknown modal amplitudes11986011989811

and 11986011989813

can be found analyticallyAgain the 2nd level governing equation can be set up by

inputting (5) into (4) and picking up those terms associatedwith 120576

2

120588119860

1199021198982

+ 2120585120588119860120596119898

1199021198982

+ 1205881198601205962

1198981199021198982

+

119899

sum

119895=1

120573119898119895

[1199021198980

(1199021198951)2

+ 21199021198980

11990211989501199021198952

+ 1199021198982

(1199021198950)2

+ 21199021198981

11990211989501199021198951] + Δ

1198981(119905) = Δ

1198982(119905)

(15)

where 1199021198982

(119905) = 11986011989821

sin(120596119905) + 11986011989823

sin(3120596119905) +

11986011989825

sin(5120596119905)Note that 119860

11989811and 119860

11989813have been found from the 1st

level solution procedureThus 1199021198951

and Δ1198981

(119905) in (15) whichare in terms of 119860

11989811and 119860

11989813 are known Therefore (15)

is a linear differential equationConsider the harmonic balances of sin(120596119905) sin(3120596119905) and

sin(5120596119905) in (15) to set up three algebraic linear equations

int

2120587

0

Δ1198982

(119905) sin (120596119905) 119889119905 = 0 (16a)

int

2120587

0

Δ1198982

(119905) sin (3120596119905) 119889119905 = 0 (16b)

int

2120587

0

Δ1198982

(119905) sin (5120596119905) 119889119905 = 0 (16c)

As (16a)ndash(16c) are linear the unknown modal amplitudes11986011989821

11986011989823

and 11986011989825

can be found analytically Similarlythe 119903th level governing equations can be set up by inputting

Table 1 (a) Convergence study for various excitation magnitudes120596 = 15120596

1 (b) Convergence study for various excitation frequencies

120581 = 75

(a)

Normalized overallamplitude 120581 = 1 75 15 25

Zero level solution 9992 10100 10193 102691st level solution 10000 10003 10013 100322nd level solution 10000 10000 10000 10000

(b)

Normalized overallamplitude 120596 = 05120596

11205961

31205961

61205961


(5) into (4) and picking up those terms associated with1205763 1205764 1205765 and consider the higher level solution form

119902119898119903

(119905) = 1198601198981199031

sin (120596119905) + 1198601198981199033

sin (3120596119905)

+ 1198601198981199035

sin (5120596119905) + sdot sdot sdot

(17)

Then consider the harmonic balances of sin(120596119905)sin(3120596119905) sin(5120596119905) to set up the linear algebraic equa-tions and solve for the unknown modal amplitudes 119860

1198981199031

1198601198981199033

1198601198981199035

Finally the overall amplitude and overall modal ampli-

tude are defined as

119860over = radicsum

119898

sum

119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18a)

119860over119898 = radicsum

119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18b)

where 119898 is mode number (ie 1 2 3 ) 119903 is solution level(ie 0 1 2 ) and 119904 is harmonic number (ie 1 3 5 )

4 Results and Discussions

In this section the material properties of the simply sup-ported beams in the numerical cases are considered asfollows Youngrsquos modulus 119864 = 71 times 109Nm2 mass density120588 = 2700 kgm3 beam dimensions = 05m times 02m times

5mm Poissonrsquos ratio = 03 and damping ratio 120585 = 002Tables 1(a)-1(b) show the convergence studies of normalizedbeam vibration amplitude for various excitation magnitudesand frequencies The two-mode approach is adopted Theharmonic excitation is uniformly distributed The 2nd levelsolutions are normalized as one hundred It is shown thatthe 1st level solutions are accurate enough for the excitationfrequency range from 05120596

1to 61205961and excitation parameter

from 120581 = 1 to 25 The differences between the zero


PresentLee et al (2009)

Nor

mal

ized

ampl

itude

A

over

thic

knes

s

2 2002Normalized excitation frequency 1205961120596

001

01

1

10

(a)



Nor

mal

ized

ampl

itude

A

over

thic

knes

s

001

01

1

10

(b)

Figure 1 (a) Comparison between the results from the proposed and numerical integration methods 119865 = 75120588119860119892 sin(120596119905) (b) Comparison

between the results from the proposed and numerical integration methods 119865 = 75(01 sin((120587119871)119909) + 09 sin((3120587119871)119909)) 120588119860119892 sin(120596119905)

75

200

Nor

mal

ized

ampl

itude

120581 = 25


001

01

1

10

Aov

erth

ickn

ess

Figure 2 Frequency-amplitude curve for various excitation levels

Table 2 (a) Modal contributions for various excitation frequencies120596 = 15120596

1 (b) Modal contributions for various excitation frequen-

cies 120581 = 75

(a)

Normalized modalcontribution 120581 = 1 75 15 25

1st mode 9994 9988 9967 99542nd mode 005 010 029 0403rd mode 001 001 004 006

(b)

Normalized modalcontribution 120596 = 05120596

11205961

31205961

61205961


level and 2nd level solutions are bigger when the excitationlevel or excitation frequency is higher It is implied thatthe higher harmonic components in the vibration responses

are more important when the excitation level or excitationfrequency is higher Tables 2(a)-2(b) show the contributionsof the first three symmetric modes for various excitationmagnitudes and frequencies It can be seen that the 2 modesolutions are accurate enough for the excitation frequencyrange from 05120596

1to 6120596

1and excitation parameter from 120581

= 1 to 25 For the excitation frequency near or less than the1st resonant frequency the 1st mode response is dominant Itis implied that the single mode solution is accurate enoughWhen the excitation frequency is set higher and closer tothe 2nd resonant frequency the 2nd mode contributionis more significant Figures 1(a)-1(b) show the frequency-amplitude curves for two excitation cases (one is uniformlydistributed 119865(119905) = 75120588

119860119892 sin(120596119905) the other is 119865(119909 119905) =

(01 sin((120587119871)119909) + 09 sin((3120587119871)119909))75120588119860119892 sin(120596119905)) It can be

seen that the modified harmonic balance solutions well agreewith those obtained from the numerical integration methodused in [11] The well-known jump phenomenon can be seenat the 1st peak There are only small deviations observationsaround 120596 = 05120596

1and 2120596

1 In Figure 1(a) the 2nd resonant

peak looks like more linear (no jump phenomenon) becausethe 2nd modal force is quite small Thus it can be consideredas linear vibration In Figure 1(b) the 2nd modal force is sethigher to induce the jump phenomenon Figure 2 shows theoverall vibration amplitude plotted against the excitation fre-quency for various excitation levels The excitation functionis 119865(119909 119905) = (01 sin((120587119871)119909) + 09 sin((3120587119871)119909))120581120588

119860119892 sin(120596119905)

The jump phenomenon is seen at each peak of the threeexcitation cases except the 2nd peak of 120581 = 25 Besides thereis a very small damped nonlinear peak observed around 120596 =

041205961due to the high excitation level or high nonlinearity

Figures 3(a)ndash3(c) show the 1st level residues remained atthe 1st and 2nd modal equations for various excitationfrequencies It can be seen that the residues aremonotonicallyincreasing with the excitation parameter the residues in the1st modal equation are bigger than those in the 2nd modalequation and the residues of 120596 = 120596

1are the highest among

the three cases while the residues of 120596 = 61205961(the excitation


1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

40 80 120 160 2000Excitation parameter 120581

001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

Excitation parameter 120581

40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)

Figure 3 (a) The 1st level residues remained at the 1st and 2nd modal equations 120596 = 1205961 (b) The 1st level residues remained at the 1st and

2nd modal equations 120596 = 61205961 (c) The 1st level residues remained at the 1st and 2nd modal equations 120596 = 9120596

1

frequency is in between the two resonant frequencies) are thesmallest In general the results have shown the convergedsolutions and proven the accuracy of the proposed method

5 Conclusions

In this study the analytic solution steps for nonlinear mul-timode beam vibration using a modified harmonic balanceapproach andVietarsquos substitution have been developed Usingthe proposed method the nonlinear multimode beam vibra-tion results can be generated without the need of nonlinearequation solver The standard simply supported beam casehas been considered in the simulation The solution con-vergences and modal contributions have been checked Thetheoretical result obtained from the proposed method showsreasonable agreement with that from the previous numericalintegration method

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work described in this paper was fully supported bya grant from City University of Hong Kong (Project no7004362ACE)

References

[1] R A Roberts and S M Eastbourn ldquoModeling techniquesfor a computational efficient dynamic turbofan engine modelrdquoInternational Journal of Aerospace Engineering vol 2014 ArticleID 283479 11 pages 2014

[2] L Maxit C Yang L Cheng and J-L Guyader ldquoModeling ofmicro-perforated panels in a complex vibro-acoustic environ-ment using patch transfer function approachrdquo Journal of theAcoustical Society of America vol 131 no 3 part 1 pp 2118ndash2130 2012

[3] M Luczak S Manzato B Peeters K Branner P Berring andM Kahsin ldquoUpdating finite element model of a wind turbineblade section using experimental modal analysis resultsrdquo Shockand Vibration vol 2014 Article ID 684786 12 pages 2014

[4] G De Gaetano D Mundo F I Cosco C Maletta and SDonders ldquoConcept modelling of vehicle joints and beam-likestructures through dynamic FE-based methodsrdquo Shock andVibration vol 2014 Article ID 303567 9 pages 2014

[5] I V Andrianov V I OlevsrsquoKyy and J Awrejcewicz ldquoAnalyticalperturbation method for calculation of shells based on 2D Pade


approximantsrdquo International Journal of Structural Stability andDynamics vol 13 no 7 Article ID 1340003 2013

[6] A Marathe and A Chatterjee ldquoWave attenuation in nonlin-ear periodic structures using harmonic balance and multiplescalesrdquo Journal of Sound andVibration vol 289 no 4-5 pp 871ndash888 2006

[7] B-Y Moon and B-S Kang ldquoVibration analysis of harmonicallyexcited non-linear system using the method of multiple scalesrdquoJournal of Sound and Vibration vol 263 no 1 pp 1ndash20 2003

[8] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[9] J L Huang R K L Su Y Y Lee and S H Chen ldquoNonlinearvibration of a curved beam under uniform base harmonicexcitation with quadratic and cubic nonlinearitiesrdquo Journal ofSound and Vibration vol 330 no 21 pp 5151ndash5164 2011

[10] A Y T Leung H X Yang and Z J Guo ldquoThe residue harmonicbalance for fractional order van der Pol like oscillatorsrdquo Journalof Sound and Vibration vol 331 no 5 pp 1115ndash1126 2012

[11] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[12] C Y Lee and C Y Chen ldquoExperimental application of avibration absorber in structural vibration reduction usingtunable fluid mass driven by micropumprdquo Journal of Sound andVibration vol 348 pp 31ndash40 2015

[13] D D Zhang and L Zheng ldquoActive vibration control of platepartly treated with ACLD using hybrid controlrdquo InternationalJournal of Aerospace Engineering vol 2014 Article ID 43297012 pages 2014

[14] S W Yuen and S L Lau ldquoEffects of inplane load on nonlinearpanel flutter by incremental harmonic balance methodrdquo AIAAJournal vol 29 no 9 pp 1472ndash1479 1991

[15] F Alijani M Amabili and F Bakhtiari-Nejad ldquoOn the accuracyof the multiple scales method for non-linear vibrations ofdoubly curved shallow shellsrdquo International Journal of Non-Linear Mechanics vol 46 no 1 pp 170ndash179 2011

[16] A Barari H D Kaliji M Ghadim and G Domairry ldquoNon-linear vibration of Euler-Bernoulli beamsrdquo Latin AmericanJournal of Solids and Structures vol 8 no 2 pp 139ndash148 2011

[17] A Y T Leung and Z J Guo ldquoFeed forward residue harmonicbalance method for a quadratic nonlinear oscillatorrdquo Interna-tional Journal of Bifurcation and Chaos vol 21 no 6 pp 1783ndash1794 2011

[18] E W Weisstein ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolframWebResource httpmathworldwolframcomVieta-sSubstitutionhtml

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research ArticleAnalytic Solution for Nonlinear MultimodeBeam Vibration Using a Modified HarmonicBalance Approach and Vietarsquos Substitution

Y Y Lee

Department of Architecture and Civil Engineering City University of Hong Kong Kowloon Tong Kowloon Hong Kong

Correspondence should be addressed to Y Y Lee bcrayleecityueduhk

Received 29 June 2015 Accepted 7 October 2015

Academic Editor Dumitru I Caruntu

Copyright copy 2016 Y Y Lee This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a modified harmonic balance solution method incorporated with Vietarsquos substitution technique for nonlinearmultimode damped beam vibrationThe aim of the modification in the solution procedures is to develop the analytic formulationswhich are used to calculate the vibration amplitudes of a nonlinearmultimode dampedbeamwithout the need of nonlinear equationsolver for the nonlinear algebraic equations generated in the harmonic balance processes The result obtained from the proposedmethod shows reasonable agreement with that from a previous numerical integration method In general the results can show theconvergence and prove the accuracy of the proposed method

1 Introduction

Over the past decades many solution methods were devel-oped for various engineering modelling problems (eg [1ndash4]) The two well-known solution methods the perturbationmethod and multiple scale method (eg [5ndash7]) are widelyadopted In these two methods coupled nonlinear algebraicequations are generated and required to be solved by nonlin-ear equation solverThere are two harmonic balancemethodsalso widely employed (ie total harmonic balance methodand incremental harmonic balance method eg [8 9])Theyalso generate coupled nonlinear algebraic equations in theharmonic balance processes In the previous works [10 11]the multilevel residue harmonic balance method was modi-fied from the total harmonic balance method and developedfor nonlinear vibrations Although the accuracies of thesesolutionswere good and verified by other solutionmethods itwas quite time consuming to develop the nonlinear equationsolver for those nonlinear algebraic equations generated inthe solution procedures On the other hand there havebeen numerous beamplate related research problems solvedusing various numerical and classical methods (eg [12ndash15]) That is the motivation in this study to develop theanalytic formulations to calculate the vibration responses of

a nonlinearmultimode beamwithout using a nonlinear equa-tion solver for the nonlinear algebraic equations generated inthe harmonic balance procedures The proposed method ismodified from the previous harmonic balance method andincorporated with Vietarsquos substitution technique Using theanalytic calculation formulations the results can be obtainedwithout the need of nonlinear equation solver

2 Governing Equation

The governing equation of motion of nonlinear vibration ofthe Euler-Bernoulli theory is as follows [16]

1198641198681198821015840101584010158401015840

+ 120588119860 minus (

119864119860

2119871int

119871

0

(1198821015840)2

119889119909)11988210158401015840= 119865 (119909 119905) (1)

where 119909 is longitudinal coordinate and 119882 is transversedisplacement 1198821015840 11988210158401015840 and 119882

1015840101584010158401015840 are 1st 2nd and 4thderivatives with respect to 119909 respectively and are 1stand 2nd derivatives with respect to time 119905 119871 is length ℎ isthickness 119864119860 is Youngrsquos modulus times cross-sectional area 120588

119860

is material density per unit length and 119865(119909 119905) is externalharmonic excitation For uniformly distributed excitation119865(119909 119905) = 119865

119900sin(120596119905) 119865

119900= 120581120588119860119892 = excitation magnitude

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3462643 6 pageshttpdxdoiorg10115520163462643





119882(119909 119905) =

119899

sum

119894=1

119902119894(119905) 120593119894(119909) (2)


119894is the 119894th


119898to it


120588119860

119899

sum

119894=1

11990211989412057200

119894119898+ 119864119868

119899

sum

119894=1

11990211989412057240

119894119898minus119864119860

2119871

119899

sum

119894=1

119899

sum

119895=1

119899

sum

119896=1

11990211989411990211989511990211989612057220

11989411989812057211

119895119896

minus 119865119898sin (120596119905) = 0

(3)

where 12057200

119894119898= int119871

0120593119894120593119898119889119909 12057240119894119898

= int119871

01205931015840101584010158401015840

119894120593119898119889119909 12057220119894119898

= int119871

012059310158401015840

119894

120593119898119889119909 12057211119895119896

= int119871

01205931015840

1198951205931015840

119896119889119909 and 119865



= int119871

0119865119900120593119898119889119909 int

119871

0120593119898


119898(119909) = sin((119898120587

119871)119909) and 12057200119894119898

= 12057240

119894119898= 12057220

119894119898= 12057211

119894119898= 0 for 119894 = 119898 Then (3) can


120588119860

119902119898+ 1205881198601205962

119898119902119898+

119899

sum

119895=1

120573119898119895

1199021198981199022

119895minus 119865119898sin (120596119905) = 0 (4)

where 120573119898119895

= minus(1198641198602119871)(12057220

11989811989812057200

119898119898)12057211

119895119895and120596





119902119898(119905) cong 120576

01199021198980

(119905) + 12057611199021198981

(119905) + 12057621199021198982



(119905) 1199021198981

(119905) and1199021198982




120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980minus 119865119898sin (120596119905)

+

119899

sum

119895=1119898 =119895

120573119898119895

1199021198980

1199022

1198950= Δ1198980

(119905)

(6)

where 1199021198980

= 11986011989801





For simplicity let

1198771198980

(119905) = 120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980

minus 119865119898sin (120596119905)

(7a)

1198781198980

(119905) =

119899

sum

119895=1

120573119898119895

1199021198980

1199022

1198950 (7b)


1198771198980

(119905) + 1198781198980

(119905) = Δ1198980

(119905) (8)


1198603

11989801+

4

3120573119898119898

120588119860(1205962

119898minus 1205962)11986011989801

+4

3120573119898119898

119865119898= 0 (9)


1003816100381610038161003816119860119898011003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119865119898

prod119898(120596) + (34) 120573

119898119898

1003816100381610038161003816119860119898011003816100381610038161003816

2

+ 119869 (2120585120596120596119898)

1003816100381610038161003816100381610038161003816100381610038161003816

(10a)


(3

4120573119898119898

)

2

1003816100381610038161003816119860119898011003816100381610038161003816

6

+3

2120573119898119898

prod

119898

(120596)100381610038161003816100381611986011989801

1003816100381610038161003816

4

+ ((prod

119898

(120596))

2

+ (2120585120596120596119898)2

)100381610038161003816100381611986011989801

1003816100381610038161003816

2

minus (119865119898)2

= 0

(10b)


1198861199103+ 1198871199102+ 119888119910 + 119889 = 0 (10c)


119860(minus1205962+ 1205962

119898) +


119898is nonlin-


120596119898) 119910 = |119860

11989801|2 119886 = ((34)120573

119898119898)2 119887 = (32)120573

119898119898prod119898(120596)

119888 = (prod119898(120596))2+ (2120585120596120596

119898)2 119889 = minus(119865

119898)2


1205913+ 119906120591 + V = 0 (11)

where 119906 = (3119886119888 minus 1198872)31198862 V = (2119887

3minus 9119886119887119888 + 27119886

2119889)27119886


1198616+ V1198613 minus

1199063

27= 0 (12)




1

120588119860

1199021198981

+ 2120585120588119860120596119898

1199021198981

+ 1205881198601205962

1198981199021198981

+

119899

sum

119895=1

120573119898119895

[(1199021198950)2

1199021198981

+ 21199021198980

11990211989501199021198951]

+ Δ1198980

(119905) = Δ1198981

(119905)

(13)

where 1199021198981

(119905) =11986011989811

sin(120596119905) + 11986011989813

sin(3120596119905)Note that 119860



and Δ1198980





int

2120587

0

Δ1198981

(119905) sin (120596119905) 119889119905 = 0 (14a)

int

2120587

0

Δ1198981

(119905) sin (3120596119905) 119889119905 = 0 (14b)


and 11986011989813



2

120588119860

1199021198982

+ 2120585120588119860120596119898

1199021198982

+ 1205881198601205962

1198981199021198982

+

119899

sum

119895=1

120573119898119895

[1199021198980

(1199021198951)2

+ 21199021198980

11990211989501199021198952

+ 1199021198982

(1199021198950)2

+ 21199021198981

11990211989501199021198951] + Δ

1198981(119905) = Δ

1198982(119905)

(15)

where 1199021198982

(119905) = 11986011989821

sin(120596119905) + 11986011989823

sin(3120596119905) +

11986011989825

sin(5120596119905)Note that 119860

11989811and 119860



and Δ1198981


11989811and 119860




int

2120587

0

Δ1198982

(119905) sin (120596119905) 119889119905 = 0 (16a)

int

2120587

0

Δ1198982

(119905) sin (3120596119905) 119889119905 = 0 (16b)

int

2120587

0

Δ1198982

(119905) sin (5120596119905) 119889119905 = 0 (16c)


11986011989823

and 11986011989825




120581 = 75

(a)



(b)


11205961

31205961

61205961



119902119898119903

(119905) = 1198601198981199031

sin (120596119905) + 1198601198981199033

sin (3120596119905)

+ 1198601198981199035


(17)


1198981199031

1198601198981199033

1198601198981199035


tude are defined as


119898

sum

119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18a)


119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18b)









Nor

mal

ized

ampl

itude

A

over

thic

knes

s


001

01

1

10

(a)



Nor

mal

ized

ampl

itude

A

over

thic

knes

s

001

01

1

10

(b)



75

200

Nor

mal

ized

ampl

itude

120581 = 25


001

01

1

10

Aov

erth

ickn

ess




cies 120581 = 75

(a)



(b)


11205961

31205961

61205961




1to 6120596



119860119892 sin(120596119905) the other is 119865(119909 119905) =



1and 2120596



119860119892 sin(120596119905)







1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)



1


5 Conclusions




Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119882(119909 119905) =

119899

sum

119894=1

119902119894(119905) 120593119894(119909) (2)


119894is the 119894th


119898to it


120588119860

119899

sum

119894=1

11990211989412057200

119894119898+ 119864119868

119899

sum

119894=1

11990211989412057240

119894119898minus119864119860

2119871

119899

sum

119894=1

119899

sum

119895=1

119899

sum

119896=1

11990211989411990211989511990211989612057220

11989411989812057211

119895119896

minus 119865119898sin (120596119905) = 0

(3)

where 12057200

119894119898= int119871

0120593119894120593119898119889119909 12057240119894119898

= int119871

01205931015840101584010158401015840

119894120593119898119889119909 12057220119894119898

= int119871

012059310158401015840

119894

120593119898119889119909 12057211119895119896

= int119871

01205931015840

1198951205931015840

119896119889119909 and 119865



= int119871

0119865119900120593119898119889119909 int

119871

0120593119898


119898(119909) = sin((119898120587

119871)119909) and 12057200119894119898

= 12057240

119894119898= 12057220

119894119898= 12057211

119894119898= 0 for 119894 = 119898 Then (3) can


120588119860

119902119898+ 1205881198601205962

119898119902119898+

119899

sum

119895=1

120573119898119895

1199021198981199022

119895minus 119865119898sin (120596119905) = 0 (4)

where 120573119898119895

= minus(1198641198602119871)(12057220

11989811989812057200

119898119898)12057211

119895119895and120596





119902119898(119905) cong 120576

01199021198980

(119905) + 12057611199021198981

(119905) + 12057621199021198982



(119905) 1199021198981

(119905) and1199021198982




120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980minus 119865119898sin (120596119905)

+

119899

sum

119895=1119898 =119895

120573119898119895

1199021198980

1199022

1198950= Δ1198980

(119905)

(6)

where 1199021198980

= 11986011989801





For simplicity let

1198771198980

(119905) = 120588119860

1199021198980

+ 1205881198601205962

1198981199021198980

+ 120573119898119898

1199023

1198980

minus 119865119898sin (120596119905)

(7a)

1198781198980

(119905) =

119899

sum

119895=1

120573119898119895

1199021198980

1199022

1198950 (7b)


1198771198980

(119905) + 1198781198980

(119905) = Δ1198980

(119905) (8)


1198603

11989801+

4

3120573119898119898

120588119860(1205962

119898minus 1205962)11986011989801

+4

3120573119898119898

119865119898= 0 (9)


1003816100381610038161003816119860119898011003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119865119898

prod119898(120596) + (34) 120573

119898119898

1003816100381610038161003816119860119898011003816100381610038161003816

2

+ 119869 (2120585120596120596119898)

1003816100381610038161003816100381610038161003816100381610038161003816

(10a)


(3

4120573119898119898

)

2

1003816100381610038161003816119860119898011003816100381610038161003816

6

+3

2120573119898119898

prod

119898

(120596)100381610038161003816100381611986011989801

1003816100381610038161003816

4

+ ((prod

119898

(120596))

2

+ (2120585120596120596119898)2

)100381610038161003816100381611986011989801

1003816100381610038161003816

2

minus (119865119898)2

= 0

(10b)


1198861199103+ 1198871199102+ 119888119910 + 119889 = 0 (10c)


119860(minus1205962+ 1205962

119898) +


119898is nonlin-


120596119898) 119910 = |119860

11989801|2 119886 = ((34)120573

119898119898)2 119887 = (32)120573

119898119898prod119898(120596)

119888 = (prod119898(120596))2+ (2120585120596120596

119898)2 119889 = minus(119865

119898)2


1205913+ 119906120591 + V = 0 (11)

where 119906 = (3119886119888 minus 1198872)31198862 V = (2119887

3minus 9119886119887119888 + 27119886

2119889)27119886


1198616+ V1198613 minus

1199063

27= 0 (12)




1

120588119860

1199021198981

+ 2120585120588119860120596119898

1199021198981

+ 1205881198601205962

1198981199021198981

+

119899

sum

119895=1

120573119898119895

[(1199021198950)2

1199021198981

+ 21199021198980

11990211989501199021198951]

+ Δ1198980

(119905) = Δ1198981

(119905)

(13)

where 1199021198981

(119905) =11986011989811

sin(120596119905) + 11986011989813

sin(3120596119905)Note that 119860



and Δ1198980





int

2120587

0

Δ1198981

(119905) sin (120596119905) 119889119905 = 0 (14a)

int

2120587

0

Δ1198981

(119905) sin (3120596119905) 119889119905 = 0 (14b)


and 11986011989813



2

120588119860

1199021198982

+ 2120585120588119860120596119898

1199021198982

+ 1205881198601205962

1198981199021198982

+

119899

sum

119895=1

120573119898119895

[1199021198980

(1199021198951)2

+ 21199021198980

11990211989501199021198952

+ 1199021198982

(1199021198950)2

+ 21199021198981

11990211989501199021198951] + Δ

1198981(119905) = Δ

1198982(119905)

(15)

where 1199021198982

(119905) = 11986011989821

sin(120596119905) + 11986011989823

sin(3120596119905) +

11986011989825

sin(5120596119905)Note that 119860

11989811and 119860



and Δ1198981


11989811and 119860




int

2120587

0

Δ1198982

(119905) sin (120596119905) 119889119905 = 0 (16a)

int

2120587

0

Δ1198982

(119905) sin (3120596119905) 119889119905 = 0 (16b)

int

2120587

0

Δ1198982

(119905) sin (5120596119905) 119889119905 = 0 (16c)


11986011989823

and 11986011989825




120581 = 75

(a)



(b)


11205961

31205961

61205961



119902119898119903

(119905) = 1198601198981199031

sin (120596119905) + 1198601198981199033

sin (3120596119905)

+ 1198601198981199035


(17)


1198981199031

1198601198981199033

1198601198981199035


tude are defined as


119898

sum

119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18a)


119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18b)









Nor

mal

ized

ampl

itude

A

over

thic

knes

s


001

01

1

10

(a)



Nor

mal

ized

ampl

itude

A

over

thic

knes

s

001

01

1

10

(b)



75

200

Nor

mal

ized

ampl

itude

120581 = 25


001

01

1

10

Aov

erth

ickn

ess




cies 120581 = 75

(a)



(b)


11205961

31205961

61205961




1to 6120596



119860119892 sin(120596119905) the other is 119865(119909 119905) =



1and 2120596



119860119892 sin(120596119905)







1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)



1


5 Conclusions




Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1

120588119860

1199021198981

+ 2120585120588119860120596119898

1199021198981

+ 1205881198601205962

1198981199021198981

+

119899

sum

119895=1

120573119898119895

[(1199021198950)2

1199021198981

+ 21199021198980

11990211989501199021198951]

+ Δ1198980

(119905) = Δ1198981

(119905)

(13)

where 1199021198981

(119905) =11986011989811

sin(120596119905) + 11986011989813

sin(3120596119905)Note that 119860



and Δ1198980





int

2120587

0

Δ1198981

(119905) sin (120596119905) 119889119905 = 0 (14a)

int

2120587

0

Δ1198981

(119905) sin (3120596119905) 119889119905 = 0 (14b)


and 11986011989813



2

120588119860

1199021198982

+ 2120585120588119860120596119898

1199021198982

+ 1205881198601205962

1198981199021198982

+

119899

sum

119895=1

120573119898119895

[1199021198980

(1199021198951)2

+ 21199021198980

11990211989501199021198952

+ 1199021198982

(1199021198950)2

+ 21199021198981

11990211989501199021198951] + Δ

1198981(119905) = Δ

1198982(119905)

(15)

where 1199021198982

(119905) = 11986011989821

sin(120596119905) + 11986011989823

sin(3120596119905) +

11986011989825

sin(5120596119905)Note that 119860

11989811and 119860



and Δ1198981


11989811and 119860




int

2120587

0

Δ1198982

(119905) sin (120596119905) 119889119905 = 0 (16a)

int

2120587

0

Δ1198982

(119905) sin (3120596119905) 119889119905 = 0 (16b)

int

2120587

0

Δ1198982

(119905) sin (5120596119905) 119889119905 = 0 (16c)


11986011989823

and 11986011989825




120581 = 75

(a)



(b)


11205961

31205961

61205961



119902119898119903

(119905) = 1198601198981199031

sin (120596119905) + 1198601198981199033

sin (3120596119905)

+ 1198601198981199035


(17)


1198981199031

1198601198981199033

1198601198981199035


tude are defined as


119898

sum

119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18a)


119903

100381610038161003816100381610038161003816100381610038161003816

sum

119904

119860119898119903119904

100381610038161003816100381610038161003816100381610038161003816

2

(18b)









Nor

mal

ized

ampl

itude

A

over

thic

knes

s


001

01

1

10

(a)



Nor

mal

ized

ampl

itude

A

over

thic

knes

s

001

01

1

10

(b)



75

200

Nor

mal

ized

ampl

itude

120581 = 25


001

01

1

10

Aov

erth

ickn

ess




cies 120581 = 75

(a)



(b)


11205961

31205961

61205961




1to 6120596



119860119892 sin(120596119905) the other is 119865(119909 119905) =



1and 2120596



119860119892 sin(120596119905)







1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)



1


5 Conclusions




Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nor

mal

ized

ampl

itude

A

over

thic

knes

s


001

01

1

10

(a)



Nor

mal

ized

ampl

itude

A

over

thic

knes

s

001

01

1

10

(b)



75

200

Nor

mal

ized

ampl

itude

120581 = 25


001

01

1

10

Aov

erth

ickn

ess




cies 120581 = 75

(a)



(b)


11205961

31205961

61205961




1to 6120596



119860119892 sin(120596119905) the other is 119865(119909 119905) =



1and 2120596



119860119892 sin(120596119905)







1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)



1


5 Conclusions




Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


001

01

1

10Δ21(120588A1205962 2A

over2

)

(a)

1st mode equation

2nd mode equation

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and


40 80 120 160 2000

000001

00001

0001

001

01

Δ21(120588A1205962 2A

over2

)

(b)

1st

leve

l res

idue

stimes100

Δ11(120588A1205962 1A

over1

) and

1st mode equation

2nd mode equation

001

01

1

10

Δ21(120588A1205962 2A

over2

)


(c)



1


5 Conclusions




Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Analytic Solution for Nonlinear Multimode Beam Vibration ...

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