Document Type: Research Paper


Aerospace Engineering Department, Faculty of New Technologies and Engineering, Shahid Beheshti University, GC


An efficient and accurate analytical solution is provided using the homotopy-Pade technique for the nonlinear vibration of parametrically excited cantilever beams. The model is based on the Euler-Bernoulli assumption and includes third order nonlinear terms arisen from the inertial and curvature nonlinearities. The Galerkin’s method is used to convert the equation of motion to a nonlinear ordinary differential equation, which is then solved by the homotopy analysis method (HAM). An explicit expression is obtained for the nonlinear frequency amplitude relation. It is found that the proper value of the so-called auxiliary parameter for the HAM solution is dependent on the vibration amplitude, making it difficult to rapidly obtain accurate frequency-amplitude curves using a single value of the auxiliary parameter. The homotopy-Pade technique remedied this issue by leading to the approximation that is almost independent of the auxiliary parameter and is also more accurate than the conventional HAM. Highly accurate results are found with only third order approximation for a wide range of vibration amplitudes.

Graphical Abstract


Main Subjects

[1]     M. N. Hamdan, A. A. Al-Qaisia, “B. O. Al-Bedoor, Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever”, International Journal of Mechanical Sciences, Vol. 43, No. 6, pp. 1521-1542, (2001).

[2]     E. C. Haight, W.W. King, “Stability of nonlinear oscillations of an elastic rod”, Journal of the Acoustical Society of America, Vol. 52, No. 3B, pp. 899-911, (1971).

[3]     R. S. Haxton, A. D. S. Barr, “The autoparametric vibration absorber”, Transactions of the ASME, Journal of Engineering for Industry, Vol. 94, No. 1, pp. 119-23, (1972).

[4]     K. Sato, H. Saito, K. Otomi, “The parametric response of a horizontal beam carrying a concentrated mass under gravity”, Transactions of the ASME Journal of Applied Mechanics, Vol. 44, No. 3, pp. 643-8, (1978).

[6]     F.C. Moon, Experiments on chaotic motion of a forced nonlinear oscillator a strange attractors, Journal of Applied Mechanics, Vol. 47, No. 3, pp. 639-44, (1980).

[7]     F. Pai, A.H. Nayfeh, “Nonlinear non-planar oscillations of a cantilever beam under lateral base excitations”, Journal of Sound and Vibration, Vol. 25, No. 5, pp. 455-74, (1990).

[8]     L. D. Zavodney, A. H. Nayfeh, “The nonlinear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment”, International Journal of Nonlinear Mechanics, Vol. 24, No. 2, pp. 105-25, (1989).

[9]     T. D. Burton, M. Kolowith, “Nonlinear resonance and chaotic motion in a flexible parametrically excited beam”, Proceedings of the Second Conference on Nonlinear Vibrations, Stability and Dynamics of Structures and Mechanisms, Blacksburg, VA, (1988).

[10]   H. M. Sedighi, K. H. Shirazi and A. Noghrehabadi, “Application of recent powerful analytical approaches on the non-linear vibration of cantilever beams”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 13, No. 7, pp. 487-494, (2012).

[11]   H. M. Sedighi, K. H. Shirazi, “A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition”, Journal of Computational and Nonlinear Dynamics, Vol. 7, No. 3, pp. 1-4, (2012).

[12]   H. M. Sedighi, K. H. Shirazi, A. Reza and J. Zare, “Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 226, No. 10, pp. 2474-2484, (2012).

[13]   H. M. Sedighi, K. H. Shirazi, M. A. Attarzadeh, A study on the quantic nonlinear beam vibrations using asymptotic approximate approaches, Acta Astronautica, Vol. 91, pp. 245-250, (2013).

[14]   H. M. Sedighi, A. Reza, “High precise analysis of lateral vibration of quintic nonlinear beam”, Latin American Journal of Solids and Structures, Vol. 10, No. 2, pp. 441- 452, (2013).

[15]   H. M. Sedighi, F. Daneshmand, “Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term”, Journal of Applied and Computational Mechanics, Vol. 1, No. 1, pp. 1-9 , (2015).

[16]   S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, (2003).

[17]   S. J. Liao, K. F. Cheung, “Homotopy analysis of nonlinear progressive waves in deep water”, Journal of Engineering Mathematics, Vol. 45, No. 2, pp. 105-116, (2003).

[18]   S. J. Liao, A. Campo, Analytic solutions of the temperature distribution in Blasius viscous flow problems, Journal of Fluid Mechanics. Vol. 453, No. 1, pp. 411-425, (2002).

[19]   T. Pirbodaghi, M. T. Ahmadian, M. Fesanghary, “On the homotopy analysis method for non-linear vibration of beams”, Mechanics Research Communications, Vol. 36, No. 2, pp. 143-148, (2009).

[20]   R. Wu, J. Wang, J. Du, Y. Hu, H. Hu, “Solutions of nonlinear thickness-shear vibrations of an infinite isotropic plate with the homotopy analysis method”, Numerical Algorithms, Vol. 59, No. 2, pp. 213-226, (2012).

[21]   M. Poorjamshidian, J. Sheikhi, S. Mahjoub-Moghadas, M. Nakhaie, “Nonlinear vibration analysis of the beam carrying a moving mass using modified homotopy”, Journal of solid mechanics, Vol. 6, No. 4, pp. 389-396, (2014).

[22]   H. M. Sedighi, K. H. Shirazi, J. Zare, “An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method”, International Journal of Non-Linear Mechanics, Vol. 47, No. 7, pp. 777-784, (2012).

[23]   S. H. Hoseini, T. Pirbodaghi, M. T. Ahmadian, G.H. Farrahi, “On the large amplitude free vibrations of tapered beams: an analytical approach”, Mechanic Research Communication, Vol. 36, No. 8, pp. 892-897, (2009).

[24]   M. R. M. Crespo da  Silva, C.  C.  Glynn,



          “Nonlinear flexural-flexural-torsional dynamics of inextensible beams, I: equations of motion”, Journal of Structural Mechanics, Vol. 6, No. 4, pp. 437-48, (1978).

[25]   A. H. Nayfeh, P. F. Pai, Linear and Nonlinear Structural Mechanics, John Wiley & Sons, Weinheim, (2004).

[26]   E. B. Saff, R. S. Varga, Pade and Rational Approximation, Academic Press, New York, (1977).

[27]   J. Kallrath., On Rational Function Techniques and Pade Approximants. An Overview, Report, Ludwigshafen, Germany, (2002).