A New Analytical Approach for Strongly Nonlinear Vibration of a Microbeam Considering Structural Damping Effect

Document Type : Research Article


1 university of tabriz

2 Azarbaijan Shahid madani University


In this study, strongly nonlinear free vibration behavior of a microbeam considering the structural damping effect is investigated analytically on the basis of modified couple stress theory. Employing Von Karman’s strain-displacement relations and implementing the Galerkin method, the governing nonlinear partial differential equation is reduced to a nonlinear ordinary differential equation which is related to the size effect of the beam. Because of large coefficient of nonlinear term and due to existence of the damping effect, none of the traditional perturbation methods leads to a valid solution. Also, there are many difficulties encountered in applying homotopy techniques when the damping effect is taken in to account in the strongly nonlinear damped system. To overcome these limitations, here, a new analytical method is presented which is based on classical perturbation methods and fundamentals of Fourier expansion with an embedding nondimensional parameter. To solve the equation, the nonlinear frequency is assumed to be time dependent. The comparison between time responses of the system obtained by the presented approach and numerical method indicates the high accuracy of the new method. To validate the results of the presented method with those available in the literatures which are obtained for a special case of an undamped system, the damping coefficient is set to zero. The comparison shows a good agreement between the results for a wide range of vibration amplitudes.


Main Subjects

[1] M.I. Younis, MEMS linear and nonlinear statics and dynamics, Springer Science & Business Media, 2011.
[2] M. Ghadiri, N. Shafiei, S.A. Mousavi, Vibration analysis of a rotating functionally graded tapered microbeam based on the modified couple stress theory by DQEM, Applied Physics A, 122(9) (2016) 837.
[3] A.M. Bataineh, M.I. Younis, Dynamics of a clamped–clamped microbeam resonator considering fabrication imperfections, Microsystem Technologies, 21(11) (2015) 2425-2434.
[4] M. Rezaee, M. Minaei, A theoretical and experimental investigation on large amplitude free vibration behavior of a pretensioned beam with clamped–clamped ends using modified homotopy perturbation method, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 230(10) (2016) 1615-1625.
[5] P. Roura, J.A. Gonzalez, Towards a more realistic description of swing pumping due to the exchange of angular momentum, European Journal of Physics, 31(5) (2010) 1195.
[6] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, 1979.
[7] A.H. Nayfeh, Introduction to perturbation techniques, 1981.
[8] S.J. Liao, A.T. Chwang, Application of homotopy analysis method in nonlinear oscillations, ASME Journal of Applied Mechanics 65 (1998) 914.
[9] J.H. He, An elementary introduction to the homotopy perturbation method, Computers and Mathematics with Applications, 57 (2009) 410–412.
[10] J.H. He, Homotopy perturbation method with an auxiliary term, Hindawi Publishing Corporation-Abstract and Applied Analysis,  (2012).
[11] J.H. He, New interpretation of homotopy perturbation method, International Journal of Modern Physics B, 20(18) (2006) 2561–2568.
[12] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanical Engineering, 178 (1999) 257–262.
[13] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics 35 (1999) 37–43.
[14] S. Momani, G.H. Erjaee, M.H. Alnasr, The modified homotopy perturbation method for solving strongly nonlinear oscillators, Computers and Mathematics with Applications, 58 (2009) 2209–2220.
[15] J.H. He, Comment on He’s frequency formulation for nonlinear oscillators, European Journal of Physics, 29 (2008) 19–22.
[16] R.E. Mickens, Mathematical and numerical study of the Duffing-harmonic oscillator, Journal of Sound and Vibration, 244 (2001) 563–567.
[17] M.K. Yazdi, A. Mirzabeigy, H. Abdollahi, Nonlinear oscillators with non-polynomial and discontinuous elastic restoring forces, Nonlinear Science Letters A, 3 (2012) 48–53.
[18] J.H. He, Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communications, 29 (2002) 107-111.
[19] J.H. He, Max-min approach to nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 9 (2008) 207–210.
[20] J.H. He, Modified Lindstedt-Poincare methods for some strongly non-linear oscillations Part I: expansion of a constant, International Journal of Non-Linear Mechanics, 37 (2002) 309-314.
[21] J.H. He, A variational iteration method to Duffing equation (in Chinese), Chinese Journal of Computational Physics, 16 (1999) 121–133.
[22] J.H. He, Variational iteration method: A kind of nonlinear analytical technique, International Journal of Nonlinear Mechanics, 34 (1999) 699–708.
[23] M.A. Noor, W.A. Khan, New iterative methods for solving nonlinear equation by using homotopy perturbation method, Applied Mathematics and Computation, 219 (2012) 3565–3574.
[24] J.H. He, A review on some new recently developed nonlinear analytical techniques, International Journal of Nonlinear Science and Numerical Simulation 1(1) (2000) 51–70.
[25] A.H. Nayfeh, Problems in pertubation, 1985.
[26] J.H. He, A new perturbation technique which is also valid for large parameters, Journal of Sound and Vibration, 229(5) (2000) 1257-1263.
[27] Y. Khan, F. Austin, Application of the Laplace decomposition methodto nonlinear homogeneous and non-homogenous advection equations, Zeitschriftfür Naturforschung A, 65a (2010) 849–853.
[28] Y. Khan, Q. Wu, Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers and Mathematics with Applications, 61 (2011) 1963–1967.
[29] S. Nourazar, A. Mirzabeigy, Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method, Scientia Iranica, 20(2) (2013) 364–368.
[30] M. Akbarzadea, Y. Khan, Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: Analytical solutions, Mathematical and Computer Modelling, 55 (2012) 480–489.
[31] M. Ghadiri, N. Shafiei, Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method, Microsystem Technologies, 22(12) (2016) 2853-2867.
[32] F. Yang, A. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39(10) (2002) 2731-2743.
[33] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science, 46(5) (2008) 427-437.
[34] M.H. Ghayesh, H. Farokhi, M. Amabili, Coupled nonlinear size-dependent behaviour of microbeams, Applied Physics A, 112(2) (2013) 329-338.
[35] R. Sourki, S. Hoseini, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory, Applied Physics A, 122(4) (2016) 413.
[36] A.E. Abouelregal, A.M. Zenkour, Generalized thermoelastic vibration of a microbeam with an axial force, Microsystem Technologies, 21(7) (2015) 1427-1435.
[37] H.M. Sedighi, A. Reza, J. Zare, Study on the frequency-amplitude relation of beam vibration, International Journal of the Physical Sciences, 6(36) (2011) 8051-8056.
[38] Z.K. Peng, Z.Q. Lang, S.A. Billings, G.R. Tomlinson, Comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis, Journal of Sound and Vibration, 311(1) (2008) 56–73.
[39] C. Hayashi, Subharmonic oscillations in nonlinear systems, Journal of Applied Physics, 24(5) (1953) 521–529.
[40] M. Urabe, Numerical investigation of subharmonic solutions to Duffing's equation, Publications of the Research Institute for Mathematical Sciences, 5(1) (1969) 79–112.
[41] L. Azrar, R. Benamar, R.G. White, A semi-analytical approach to the nonlinear dynamic response problem of S–S and C–C beams at large vibration amplitudes part I: General theory and application to the single mode approach to free and forced vibration analysis, Journal of Sound and Vibration 224 (1999) 183–207.
[42] T. Pirbodaghi, M.T. Ahmadian, M. Fesanghary, On the homotopy analysis method for nonlinear vibration of beams, Mechanics Research Communications 36 (2009) 143–148.
[43] M.I. Qaisi, Application of the harmonic balance principle to the nonlinear free vibration of beams, Applied Acoustics 40 (1993) 141–151.
[44] R. Lewandowski, Application of the Ritz method to the analysis of nonlinear free vibrations of beams, Journal of Sound and Vibration 224 (1987) 183–207.
[45] J.H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20(10) (2006) 1141–1199.