Scientia Iranica B (2015) 22(5), 1911{1917

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Research Note

Application of variational iteration method to largevibration analysis of slenderness beams consideringmid-plane stretching

M. Daeichi and M.T. Ahmadian�

Center of Excellence in Design, Robotics and Automation (CEDRA), School of Mechanical Engineering, Sharif University ofTechnology, P.O. Box: 11155-9567, Iran.

Received 15 June 2014; received in revised form 11 November 2014; accepted 4 May 2015

KEYWORDSLarge vibration;Variational iterationmethod;Nonlinear naturalfrequency;Galerkin technique.

Abstract. In this paper, the nonlinear vibration of transversely vibrated beams withlarge slenderness and immovable ends is analyzed using the Variational Iteration Method(VIM). The nonlinear partial di�erential equation of motion is converted to a set of coupledordinary di�erential equations using the Galerkin technique. A two mode expansion forthe system response is considered and a second order analytical solution of the equationsis obtained using VIM. Two algebraic coupled equations are also developed to �nd thenonlinear frequencies of the system. Numerical simulations are performed for various initialconditions. Close agreement between the numerical and analytical results is achieved. Also,a frequency analysis is performed for a range of initial amplitudes of vibration.© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Due to the wide application of beam-type structuresin mechanical, civil and railway engineering, inves-tigation of the dynamic behavior of such structureshas become of signi�cant scienti�c and technologicalimportance [1-2]. The large vibration of beams withlarge slenderness and immovable ends is a nonlinearproblem, due to the mid-plane stretching e�ect, andlinearization in the governing equations leads to re-markable errors in predicting the dynamic behaviorof the system [3-5]. Surveying literature reveals thata large number of research papers are dedicated tothe analysis of the nonlinear vibration of beams us-ing analytic and semi-analytic approaches. Nayfehimplemented multiple scales, normal forms, and Shawand Pierre methods, to investigate the free vibrationof beams resting on nonlinear foundations [6]. A

*. Corresponding author. Tel.: +98 21 66165503;E-mail address: ahmadian@sharif.edu (M.T. Ahmadian)

harmonically exited beam with a nonlinear elasticfoundation was studied by Santee and Goncalves [7].They applied the Melnikov method to obtain thefrequency response, as well as bifurcations of thestructure. Kargarnovin et al. obtained the responseof beam subjected to a harmonic moving load whileresting on a nonlinear viscoelastic foundation [8].Malekzadeh and Vosoughi investigated a restrainededge composite beam vibrating with large amplitudeand resting on a nonlinear elastic foundation [9].Vibration of a Timoshenko beam on a random vis-coelastic foundation under a harmonic moving loadwas analyzed by Younesian et al. using perturbationtechniques [10].

Previous studies are mostly based on classicalmethods. However, in recent years, novel analyticalapproximate methods have been introduced to analyzethe dynamic behavior of nonlinear oscillatory systemswith simpler and more e�cient procedures. TheHomotopy Analysis Method (HAM) [11], the EnergyBalance Method (EBM) [12], Frequency-Amplitude

1912 M. Daeichi and M.T. Ahmadian /Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1911{1917

Formulation (FAF) [13] and the Variational IterationMethod (VIM) [14] are some of these methods. Forexample, Liao et al. introduced the homotopy analysismethod and employed it for various systems, suchas fractional partial di�erential equations, nonlineareigenvalue problems, and wave motion in deep wa-ter [15-17]. The energy balance method has beenutilized e�ectively to �nd periodic solutions for thegeneralized nonlinear equations of oscillators contain-ing fraction order elastic force [18]. Jamshidi andGanji employed EBM to investigate the oscillationof a mass attached to a stretched elastic wire [19].Younesian et al. also obtained analytical approximatesolutions for the generalized nonlinear oscillator usingthis method [20]. Frequency-amplitude formulationwas implemented in many equations containing dis-continuity [21], irrational force [22] and the nonlin-ear Schrodinger equation [23]. The variational iter-ation method is a quite e�cient method applicablefor nonlinear ordinary and partial di�erential equa-tions. Wazwaz solved linear and nonlinear Volterraintegral and integro-di�erential equations using thismethod [24]. Investigation of linear and nonlinearSchrodinger equations was carried out by him usingVIM [25]. Odibat and Momani implemented thismethod for fractional partial di�erential equations in uid mechanics [26]. Analytical solutions for nonlinearwave propagation in shallow media were obtained byYounesian et al. applying VIM [27]. Di�usion equationswith local and nonlocal functional were analyzed usingVIM by Guo-Cheng Wu [28]. He also employed VIMfor Burgers' ow with fractional derivatives using thenew Lagrange multiplier [29].

In this paper, the variational iteration method isemployed to investigate the nonlinear transverse freevibration of beams with large slenderness and immov-able ends. The governing partial di�erential equationof motion is discretized using the Galerkin technique. Acoupled set of nonlinear ordinary di�erential equationsis obtained considering two mode shapes. Closedform expressions for the system response are obtainedand nonlinear frequencies are evaluated from algebraicequations. Obtained analytical approximate results arecompared with the numerical results of di�erent initialconditions to examine the accuracy and validity of thesolution procedures. The sensitivity of the nonlinearfrequencies, with respect to initial conditions, is studiedfor a wide range of vibration amplitudes. Findingsindicate that the solution procedure is valid, whilesimple to use for nonlinear systems.

2. Mathematical modeling

The nonlinear free vibration of a beam is considered inthe section, considering the mid-plane stretching e�ect.The beam is clamped-hinged, and the corresponding

governing equation of motion is derived as [1]:

�A@2w@t2

+ EI@4w@x4 �

24EA2l

lZ0

�@w@x

�2

dx

35 @2w@x2 = 0;

(1)

where w shows the transverse de ection; �, E, I, A andl are density, modulus of elasticity, moment of inertia,cross section area, and length of the beam, respectively.Also, x is the axial axis and t represents the time. Theboundary conditions of the clamped-hinged beam areexpressed as:

w = 0;@w@x

= 0 at x = 0; (2)

w = 0;@2w@x2 = 0 at x = l: (3)

The following two mode discretized response is consid-ered by employing the Galerkin method:

w(x; t) = u(t)'1(x) + v(t)'2(x); (4)

where 'i(x), (i = 1; 2), are linear mode shapes ofa clamped-hinged beam. Substituting Eq. (4) intoEq. (1), and using the orthogonality principle of themode shapes, one can arrive at the following coupledordinary di�erential equations:

�u+ !21u+ �1u3 + �2u2v + �3uv2 + �4v3 = 0; (5)

�v + !22v + �1v3 + �2v2u+ �3vu2 + �4u3 = 0; (6)

where the linear frequencies, !1 and !2, and coe�-cients, �i;i=1;2;3;4 and �i;i=1;2;3;4, are non-dimensionalconstants, which can be computed according to thephysical and geometrical parameters of the beam pre-sented in the Appendix.

3. Variational iteration method

In this section, a brief review of VIM is presented�rst [14] and then VIM is employed to obtain ap-proximate solutions of the coupled system in Eqs. (5)and (6). Consider the following general di�erentialequation:

Ly(t) +Ny(t) = 0; (7)

where L and N represent the linear and nonlinear op-erators, respectively. Based on the variational iterationmethod, the iteration formulation is constructed asfollows [14]:

yn+1 = yn +Z t

0� (Lyn(s) +N ~yn(s)) ds; (8)

where � denotes a general Lagrange multiplier which

M. Daeichi and M.T. Ahmadian /Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1911{1917 1913

can be identi�ed via the variational theory; yn is thenth approximate solution for Eq. (7), and ~yn is arestricted variation in which �~yn = 0 [14]. Eq. (8)is called a correction functional. Making the abovecorrection functional stationary, noticing that �yn(0) =0, the following iteration can be written:

�yn+1(t) = �yn(t) + �tZ

0

���yn + !2

1yn +Neyn� ds= �yn(t) + �(s)� _yn(s)js=t

� _�(s)�yn(s)���s=t

+tZ

0

���+!2

1���ynds = 0; (9)

which yields the following stationary conditions:

��+ !21� = 0; (10a)

1� _����s=t

= 0; (10b)

�js=t = 0: (10c)

Therefore, in this case, the Lagrange multiplier can beidenti�ed as follows:

� =1!1

sin [!1(s� t)] : (11)

Applying VIM on Eqs. (5) and (6) and using Eq. (11),the following iterations are constructed:

un+1 =un +1!1

tZ0

sin!1(s� t)f�un + !21un

+ �1u3n + �2u2

nvn + �3unv2n

+ �4v3ngds; (12)

vn+1 =vn +1!2

tZ0

sin!2(s� t)f�vn + !22vn

+ �1v3n + �2v2

nun + �3vnu2n

+ �4u3ngds: (13)

First approximate solutions for u(t) and v(t) are con-sidered as follows:

u0 = A cos 1t; (14)

v0 = B cos 2t; (15)

Substituting Eqs. (14) and (15) into Eqs. (12) and (13),one can arrive at:

u1 =A cos 1t+1!1

tZ0

sin!1(s�t)fA(!21�2

1) cos 1s

+ �1A3 cos3 1s+ �2BA2 cos2 1s cos 2s

+ �3B2A cos 1s cos2 2s

+ �4B3 cos3 2sgds; (16)

v1 =B cos 2t+1!2

tZ0

sin!2(s�t)fB(!22�2

2) cos 2s

+ �1B3 cos3 2s+ �2B2A cos 1s cos2 2s

+�3A2B cos2 1s cos 2s+�4A3 cos3 1sgds; (17)

where 1 and 2 represent the nonlinear naturalfrequencies. Implementing sort of appropriate math-ematical operations and after some simpli�cations, onecan reach the second approximate solutions as:

u1 =�A+

�3AB2

2(!21 � 2

1)+

�2BA2

2(!21 � 2

1)+�3AB2

4�1

!21�(22+1)2 +

1!2

1�(22�1)2

��2BA2

4�1

!21 � (21 + 2)2 +

1!2

1 � (21 � 2)2

�+�1A3

4

�3

!21 � 2

1+

1!2

1 � 921

��4B3

4

�3

!21 � 2

2+

1!2

1 � 922

��cos!1t

� �3AB2

4

�2 cos 1t!2

1 � 21

+cos(22 + 1)t!2

1 � (22 + 1)2

+cos(22 � 1)t!2

1 � (22 � 1)2

�� �1A3

4

�3 cos 1t!2

1 � 21

+cos 31t!2

1 � 921

�� �4B3

4

�3 cos 2t!2

1 � 22

+cos 32t!2

1 � 922

�� �2BA2

4

�2 cos 2t!2

1 � 22

+cos(2 + 21)t!2

1 � (21 + 2)2

+cos(21 � 2)t!2

1 � (21 � 2)2

�; (18)

and:

1914 M. Daeichi and M.T. Ahmadian /Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1911{1917

v1 =�B +

�3BA2

2(!22 � 2

2)+

�2AB2

2(!22 � 2

2)+�3BA2

4�1

!22�(21+2)2 +

1!2

2�(21�2)2

��2AB2

4�1

!22 � (22 + 1)2 +

1!2

2 � (22 � 1)2

�+�1B3

4

�3

!22 � 2

2+

1!2

2 � 922

��4A3

4

�3

!22 � 2

1+

1!2

2 � 921

��cos!2t

� �3BA2

4

�2 cos 2t!2

2 � 22

+cos(21 + 2)t!2

2 � (22 + 1)2

+cos(21 � 2)t!2

2 � (21 � 2)2

���1B3

4

�3 cos 2t!2

2 � 22

+cos 32t!2

2 � 922

���4A3

4

�3 cos 1t!2

2 � 21

+cos 31t!2

2 � 921

���2AB2

4

�2 cos 1t!2

2 � 21

+cos(22 + 1)t!2

2 � (22 + 1)2

+cos(22 � 1)t!2

2 � (22 � 1)2

�: (19)

Elimination of secular terms in the above equationsleads to the following algebraic coupled system:(

A+�3AB2

2(!21 � 2

1)+

�2BA2

2(!21 � 2

1)

+�3AB2

4

�1

!21�(22+1)2 +

1!2

1�(22�1)2

��2BA2

4

�1

!21�(21+2)2 +

1!2

1�(21�2)2

�+�1A3

4

�3

!21 � 2

1+

1!2

1 � 921

��4B3

4

�3

!21 � 2

2+

1!2

1 � 922

�)= 0; (20)

and:(B +

�3BA2

2(!22 � 2

2)+

�2AB2

2(!22 � 2

2)

+�3BA2

4

�1

!22�(21+2)2 +

1!2

2�(21�2)2

��2AB2

4

�1

!22�(22+1)2 +

1!2

2�(22�1)2

�+�1B3

4

�3

!22 � 2

2+

1!2

2 � 922

��4A3

4

�3

!22 � 2

1+

1!2

2 � 921

�)= 0: (21)

So, unknown natural frequencies can be obtainedfrom the above coupled system, with respect to ini-tial conditions, linear frequencies and constant coe�-cients.

4. Results and discussions

In this section, obtained analytical results for the timeresponse of the system are shown and compared withthe numerical solutions for di�erent initial conditionsof two mode shapes. Also, a frequency analysis ispresented here for a range of initial amplitudes ofvibration. It should be noted that the non-dimensionalconstant of the system is considered for a practical sys-tem as the following numerical values, and computedby the formulations in the Appendix [5].8>>>>>>>>><>>>>>>>>>:

!1 = 1; !2 = 3:24064;

�1 = 0:278769; �2 = �0:311074;�3 = 1:11585; �4 = �0:386361;

�1 = 3:87030; �2 = �1:159083;�3 = 1:11585; �4 = �0:1036913;

(22)

Figures 1 to 4 show the time histories of two modeshapes for both analytical and numerical solutions forsome values of initial amplitudes of vibration. It isobvious that the analytical approximate solutions havea reliable agreement with the numerical results for both

Figure 1. Time history of the dynamic response for the�rst mode (A = 0:3, B = 0:4).

M. Daeichi and M.T. Ahmadian /Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1911{1917 1915

Figure 2. Time history of the dynamic response for thesecond mode (A = 0:3, B = 0:4).

Figure 3. Time history of the dynamic response for the�rst mode (A = 0:3, B = 0:1).

Figure 4. Time history of the dynamic response for thesecond mode (A = 0:3, B = 0:1).

modes. Consequently, our solution procedures obtainappropriate approximate expressions in analytical formfor analysis of the proposed system.

Variations of the �rst and second nonlinear nat-ural frequencies are illustrated in the counter mapsshown in Figures 5 and 6, and the nonlinear sensitivityof the frequencies to the initial conditions is representedby the color gradient. Also, the frequency ratio, whichis nonlinear to linear versus the initial amplitudes, isrepresented in Figures 7 and 8. It is seen that for smallinitial amplitudes, the ratio is close to one. However,by increasing this value, the ratio increases nonlinearlyand the growth is higher for the �rst mode frequency asthe corresponding initial condition is further e�ectivein the nonlinear frequency.

Figure 5. First nonlinear natural frequency contour map.

Figure 6. Second nonlinear natural frequency contourmap.

Figure 7. Frequency ratio of the �rst mode versus initialamplitudes.

Figure 8. Frequency ratio of the second mode versusinitial conditions.

1916 M. Daeichi and M.T. Ahmadian /Scientia Iranica, Transactions B: Mechanical Engineering 22 (2015) 1911{1917

5. Conclusion

In this paper, the nonlinear dynamic behavior of beamswith large slenderness and immovable ends is analyzedusing the variational iteration method. The beam isconsidered to be clamped-clamped and free to vibratetransversely. The Galerkin method is applied to thegoverning equation of motion to obtain a set of ordinarydi�erential equations, and the approximate analyticalsolutions of the corresponding equations were achievedusing VIM. The accuracy of the results for time re-sponses is examined by comparing them with numerical�ndings. The nonlinear frequencies of the beam are an-alyzed for di�erent initial conditions. E�ects of initialconditions on the natural frequencies and the frequencyratio are demonstrated. It is shown that the solutionprocedure is accurate enough, and, in comparison withnumerical �ndings, no signi�cant di�erence is observed.The procedure is simple and applicable for analysis andmodeling of nonlinear systems.

References

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16. Liao, S.J. and Cheung, K.F. \Homotopy analysis ofnonlinear progressive waves in deep water", Journal ofEngineering Mathematics, 45, pp. 105-116 (2003).

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Appendix

Constant non-dimensional coe�cients introduced inSection 2 can be obtained from the following formu-lations. The linear frequencies are computed as:

!1 = 1; !2 =��2

�1

�2

;

�2i = !LiL

2i

��AEI

�1=2

; i = 1; 2; (A.1)

where !Li is the ith linear frequency of the beam.Other coe�cients are:8><>:�1 = 1

�41 1111; �2 = 1

�41(3 1121);

�3 = 1�4

1( 1122 + 2 1221); �4 = 1

�41 2221

(A.2)

and:8><>:�1 = 1�4

1 2222; �2 = 1

�41(3 2212)

�3 = 1�4

1( 2211 + 2 2112); �4 = 1

�41 1112

(A.3)

where:

nmpq =12

1Z0

@'n@x1

@'m@x1

dx1

1Z0

@'p@x1

@'q@x1

dx1;

(x1 = x=L; n;m; p; q = 1; 2): (A.4)

Biographies

Meysam Daeichi obtained his MS degree from theDepartment of Mechanical Engineering at Sharif Uni-versity of Technology, Tehran, Iran, in 2013. Hisresearch interests include nonlinear dynamics and vi-bration.

Mohamamd Taghi Ahmadian received a PhD de-gree in Mechanical Engineering from the University ofKansas, USA, in 1986, and is currently Professor in theDepartment of Mechanical Engineering at Sharif Uni-versity of Technology, Tehran, Iran. He has publishedmore than 100 journal papers and over 250 conferencepapers.