Free Vibration Analysis of a Nonlinear Beam Using Homotopy and Modified Lindstedt-Poincare Methods

Document Type : Research Paper


1 Center of Excellence in Design, Robotics and Automation, School of Mechanical Engineering, Sharif University of Technology

2 School of Mechanical Engineering, Sharif University of Technology


In this paper, homotopy perturbation and modified Lindstedt-Poincare methods are employed for nonlinear free vibrational analysis of simply supported and double-clamped beams subjected to axial loads. Mid-plane stretching effect has also been accounted in the model. Galerkin's decomposition technique is implemented to convert the dimensionless equation of the motion to nonlinear ordinary differential equation. Homotopy and modified Lindstedt-Poincare (HPM) are applied to find analytic expressions for nonlinear natural frequencies of the beams. Effects of design parameters such as axial load and slenderness ratio are investigated. The analytic expressions are valid for a wide range of vibration amplitudes. Comparing the semi-analytic solutions with numerical results, presented in the literature, indicates good agreement. The results signify the fact that HPM is a powerful tool for analyzing dynamic and vibrational behavior of structures analytically.


[1] Pirbodaghi T., Ahmadian M.T., Fesanghary M., 2008, On the homotopy analysis method for non-linear vibration of beams, Mechanics Research Communications, in press, doi:10.1016/j.mechrescom.2008.08.001.
[2] Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, Wiley, New York.
[3] Shames I.H., Dym C.L., 1985, Energy and Finite Element Methods in Structural Mechanics, McGraw-Hill, New York.
[4] Malatkar P., 2003, Nonlinear Vibrations of Cantilever Beams and Plates, Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering Mechanics.
[5] Pillai S.R.R., Rao B.N., 1992, On nonlinear free vibrations of simply supported uniform beams, Journal of Sound and Vibration 159(3): 527-531.
[6] Ramezani A., Alasty A., Akbari J., 2006, Effects of rotary inertia and shear deformation on nonlinear free vibration of microbeams, ASME Journal of Vibration and Acoustics 128(5): 611-615.
[7] Foda M.A., 1999, Influence of shear deformation and rotary inertia on nonlinear free vibration of a beam with pinned ends, Computers and Structures 71: 663-670.
[8] Liao S.J., 1995, An approximate solution technique which does not depend upon small parameters: a special example, International Journal of Nonlinear Mechanics 30: 371-380.
[9] He J.H., 2000, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics 35: 37-43.
[10] Abbasbandy S., 2006, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A 360: 109-113.
[11] Abbasbandy S., 2007, The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation, Physics Letters A 361: 478-483.
[12] Hayat T., Sajid M., 2007, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A 361: 316–322.
[13] HE J.H., 2000, A new perturbation technique which is also valid for large parameters, Journal of Sound and Vibration 229(5): 1257-1263.
[14] He J.H., 2003, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135: 73-79.
[15] Belendez A., Hernandez A., Belendez T., Neipp C., Marquez A., 2007, Application of the homotopy perturbation method to the nonlinear Pendulum, European Journal of Physics 28: 93-104.
[16] Belendez A., Belendez T., Marquez A., Neipp C., 2006, Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators, Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2006.09.070.