Vibration Analysis of a Nonlinear Beam Under Axial Force by Homotopy Analysis Method

Document Type : Research Paper


1 Department of Mechanical Engineering, Imam Hossein University

2 Civil Engineering, Imam Hossein University


In this paper, Homotopy Analysis Method is used to analyze free non-linear vibrations of a beam simply supported by pinned ends under axial force. Mid-plane stretching is also considered for dynamic equation extracted for the beam. Galerkin decomposition technique is used to transform a partial dimensionless nonlinear differential equation of dynamic motion into an ordinary nonlinear differential equation. Then Homotopy Analysis Method is employed to obtain an analytic expression for nonlinear natural frequencies. Effects of design parameters including axial force and slenderness ratio on nonlinear natural frequencies are studied. Moreover, effects of factors of nonlinear terms on the general shape of the time response are taken into account. Combined Homotopy-Pade technique is used to reduce the number of approximation orders without affecting final accuracy. The results indicate increased speed of convergence as Homotopy and Pade are combined. The obtained analytic expressions can be used for a vast range of data. Comparison of the results with numerical data indicated a good conformance. Having compared accuracy of this method with that of the Homotopy perturbation analytic method, it is concluded that Homotopy Analysis Method is a very strong method for analytic and vibration analysis of structures.


[1] Ahmadian M.T., Mojahedi M., 2009, Free vibration analysis of a nonlinear beam using homotopy and modified lindstedt-poincare methods, Journal of Solid Mechanics 2(1): 29-36.
[2] Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, New York, Wiley, First Edition.
[3] Shames I.H., Dym C.L., 1985, Energy and Finite Element Methods in Structural Mechanics, New York, McGraw-Hill, First Edition.
[4] Malatkar P., 2003, Nonlinear Vibrations of Cantilever Beams and Plates, Virginia, Virginia Polytechnic Institute , PhD thesis.
[5] Pillai S.R.R., Rao B.N., 1992, On nonlinear free vibrations of simply supported uniform beams, Sound and Vibration 159(3): 527-531.
[6] 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(1): 663-670.
[7] 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.
[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(1): 371-380.
[9] Sedighi.H.M., Shirazi.K.H., 2012, An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method, Non-Linear Mechanics 47: 777-784.
[10] Hoseinia S.H., Pirbodaghi T., 2008, Nonlinear free vibration of conservative oscillators with inertia and static type cubic nonlinearities using homotopy analysis method, Sound and Vibration 316: 263-273.
[11] Samir A., Emam A., 2002, Theoretical and Experimental Study of Nonlinear Dynamics of Buckled Beams, Virginia, Virginia Polytechnic Institute, PhD thesis.
[12] Liao S.J., 1992, On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and its Application , Shanghai, Jiao Tong University, PhD thesis.
[13] 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.
[14] Saff E.B., Varga R.S., 1977, Pade´ and Rational Approximation, Academic Press, New York.
[15] Wuytack L., 1979, Pade´ Approximation and its Applications, Lecture Notes in Mathematics, Springer, Berlin.
[16] Liao S.J., Cheung K.F., 2003, Homotopy analysis of nonlinear progressive waves in deep water, Journal of Engineering Mathematics 45(1): 105-116.