Variational Iteration Method for Free Vibration Analysis of a Timoshenko Beam under Various Boundary Conditions


1 Department of Mechanical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran

2 School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

3 Department of Mechanical Engineering, University of Kashan, Kashan, Iran


In this paper, a relatively new method, namely variational iteration method (VIM), is developed for free vibration analysis of a Timoshenko beam with different boundary conditions. In the VIM, an appropriate Lagrange multiplier is first chosen according to order of the governing differential equation of the boundary value problem, and then an iteration process is used till the desired accuracy is achieved. Solution of VIM for natural frequencies and mode shapes of a Timoshenko beam is compared to the available exact closed-form solution and numerical results of differential quadrature method (DQM). The accuracy of VIM is approximately the same as exact solution and much better than the DQM for solving the free vibration of a Timoshenko beam. Also, convergence speed and simplicity of this method is more than the other two methods because it works with polynomial at the first iteration. Thus, VIM can be used for solving the complicate engineering problems which do not have analytical solution.


1.     Nesterenko, V., "A theory for transverse vibrations of the timoshenko beam", Journal of Applied Mathematics and Mechanics,  Vol. 57, No. 4, (1993), 669-677.

2.     Horr, A. and Schmidt, L., "Closed-form solution for the timoshenko beam theory using a computer-based mathematical package", Computers & Structures,  Vol. 55, No. 3, (1995), 405-412.

3.     Lin, S. and Hsiao, K., "Vibration analysis of a rotating timoshenko beam", Journal of Sound and Vibration,  Vol. 240, No. 2, (2001), 303-322.

4.     Torabi, K., Jazi, A.J. and Zafari, E., "Exact closed form solution for the analysis of the transverse vibration modes of a timoshenko beam with multiple concentrated masses", Applied Mathematics and Computation,  Vol. 238, (2014), 342-357.

5.     Zhong, H. and Guo, Q., "Nonlinear vibration analysis of timoshenko beams using the differential quadrature method", Nonlinear Dynamics,  Vol. 32, No. 3, (2003), 223-234.

6.     Karami, G., Malekzadeh, P. and Shahpari, S., "A dqem for vibration of shear deformable nonuniform beams with general boundary conditions", Engineering Structures,  Vol. 25, No. 9, (2003), 1169-1178.

7.     He, J., "Variational iteration method for delay differential equations", Communications in Nonlinear Science and Numerical Simulation,  Vol. 2, No. 4, (1997), 235-236.

8.     He, J.-H., "Variational iteration method—some recent results and new interpretations", Journal of Computational and Applied Mathematics,  Vol. 207, No. 1, (2007), 3-17.

9.     Wazwaz, A.-M., "Partial differential equations and solitary waves theory, Springer Science & Business Media,  (2010).

10.   Liu, Y. and Gurram, C.S., "The use of he’s variational iteration method for obtaining the free vibration of an euler–bernoulli beam", Mathematical and Computer Modelling,  Vol. 50, No. 11, (2009), 1545-1552.

11.   Hasseine, A., Barhoum, Z., Attarakih, M. and Bart, H.-J., "Analytical solutions of the particle breakage equation by the adomian decomposition and the variational iteration methods", Advanced Powder Technology,  Vol. 26, No. 1, (2015), 105-112.

12.   Xu, L., "Variational iteration method for solving integral equations", Computers & Mathematics with Applications,  Vol. 54, No. 7, (2007), 1071-1078.

13.   Noor, M.A., Noor, K.I. and Mohyud-Din, S.T., "Modified variational iteration technique for solving singular fourth-order parabolic partial differential equations", Nonlinear Analysis: Theory, Methods & Applications,  Vol. 71, No. 12, (2009), e630-e640.

14.   Olayiwola, M., Akinpelu, F. and Gbolagade, A., "Modified variational iteration method for the solution of a class of differential equations", American Journal of Computational and Applied Mathematics,  Vol. 2, No. 5, (2012), 228-231.

15.   Altıntan, D. and Ugur, O., "Solution of initial and boundary value problems by the variational iteration method", Journal of Computational and Applied Mathematics,  Vol. 259, (2014), 790-797.

16.   Wazwaz, A.-M., "Linear and nonlinear integral equations", Springer,  Vol. 639, ISBN 978-3-642-21449-3, (2011).

17.   Ding, H., Shi, K., Chen, L. and Yang, S., "Adomian polynomials for nonlinear response of supported timoshenko beams subjected to a moving harmonic load", Acta Mechanica Solida Sinica,  Vol. 27, No. 4, (2014), 383-393.

18.   Berkani, S., Manseur, F. and Maidi, A., "Optimal control based on the variational iteration method", Computers & Mathematics with Applications,  Vol. 64, No. 4, (2012), 604-610.

19.   Baghani, M., Fattahi, M. and Amjadian, A., "Application of the variational iteration method for nonlinear free vibration of conservative oscillators", Scientia Iranica,  Vol. 19, No. 3, (2012), 513-518.

20.   Rezazadeh, G., Madinei, H. and Shabani, R., "Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method", Applied Mathematical Modelling,  Vol. 36, No. 1, (2012), 430-443.

21.   He, J.-H. and Wu, X.-H., "Variational iteration method: New development and applications", Computers & Mathematics with Applications,  Vol. 54, No. 7, (2007), 881-894.

22.   Sherafatnia, K., Farrahi, G. and Faghidian, S.A., "Analytic approach to free vibration and buckling analysis of functionally graded beams with edge cracks using four engineering beam theories", International Journal of Engineering-Transactions C: Aspects,  Vol. 27, No. 6, (2013), 979-990.

23.   Rakideh, M., Dardel, M. and Pashaei, M., "Crack detection of timoshenko beams using vibration behavior and neural network", International Journal of Engineering-Transactions C: Aspects,  Vol. 26, No. 12, (2013), 1433-1441.

24.   Sadeghian, M. and Ekhteraei Toussi, H., "Frequency analysis for a timoshenko beam located on an elastic foundation", International Journal of Engineering,  Vol. 24, (2011), 87-105.

25.   Chen, Y., Zhang, J. and Zhang, H., "Free vibration analysis of rotating tapered timoshenko beams via variational iteration method", Journal of Vibration and Control,  Vol. 23, No. 2, (2017), 220-234.

26.   Khaji, N., Shafiei, M. and Jalalpour, M., "Closed-form solutions for crack detection problem of timoshenko beams with various boundary conditions", International Journal of Mechanical Sciences,  Vol. 51, No. 9, (2009), 667-681.

27.   Torabi, K., Afshari, H. and Aboutalebi, F.H., "A dqem for transverse vibration analysis of multiple cracked non-uniform timoshenko beams with general boundary conditions", Computers & Mathematics with Applications,  Vol. 67, No. 3, (2014), 527-541.