*Author for correspondence Indian Journal of Science and Technology, Vol 9(43), DOI: 10.17485/ijst/2016/v9i43/104968, November 2016 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 Strongly Nonlinear Free Vibration Analysis of Beams using Modified Homotopy Perturbation Method subjected to the Nonlinear Thermal Loads Masoud Minaei * Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran; masoud_minaei@tabrizu.ac.ir Keywords: Euler-Bernoulli Beam, Modified Homotopy Perturbation Method, Nonlinear Thermal Load, Pasternak Foundation, Strongly Nonlinear Vibration Abstract Objectives: In this study, large amplitude e free vibration behavior of Euler-Bernoulli beam subjected to the nonlinear thermal loads and resting on a Pasternak foundation is investigated. Methods: The Hamilton’s principle is used to derive the beam governing partial differential equation of motion. By implementing the Galerkin’s method and applying the clamped-clamped boundary condition, the partial differential equation is converted to an ordinary nonlinear differential equation. Results: Because of the large coefficient of the nonlinear term, the Modified Homotopy Perturbation Method (MHPM) is used to solve the obtained equation. The effect of nonlinear thermal load on the system nonlinear vibration behavior is studied. Applications: The results show that although increasing the nonlinear thermal load coefficients decreases both linear and nonlinear frequency, but it increases the frequency ratio. 1. Introduction Most of the physical phenomena and engineering problems occur in nature in the forms of nonlinear dif- ferential systems. Many structures such as high-rise buildings, long span bridges and aerospace vehicles can be modeled as a beam and by increasing the amplitude of oscillations, the governed equation of motion can be obtained as a nonlinear ODE. e common techniques for constructing the analytical approximate solutions to the nonlinear oscillator equations are the perturbation methods. Some well-known perturbation methods are the Krylov Bogoliubov Mitropolskii (KBM) 1–5 method, the Lindstedt-Poincare (LP) method 6–8 and the method of multiple time scales 9 . All of these classical perturbation methods are based on assuming a small parameter which exists in the equation. In 10 has investigated the homo- topy perturbation technique. In another paper, in 11 has developed a coupling method of a homotopy perturba- tion technique and a perturbation technique for strongly nonlinear problems. Recently, in 12 has also presented a new interpretation of homotopy perturbation method for strongly nonlinear differential systems. In 13 proposed a new perturbation technique to solve the nonlinear un-damped Duffing equation in which the maximum relative error at the first order approximation is less than 7%. In 14 presented a method called MHPM which can solve strongly nonlinear problems more accurately. ey show that the maximum relative error at the first order and second order of MHPM is less than 2.22% and 0.03%, respectively. Newly, in 15 studied nonlinear free vibration of laminated composite thin beams on nonlinear elastic foundation with elastically restrained against rotation edges by Differential Quadrature (DQ) approach. ey developed a finite element program to verify the results of the DQ approach and also they studied the effects of different parameters on the ratio of nonlinear to linear natural frequency. In 16 investigated the large amplitude