High precise analysis of lateral vibration of quintic nonlinear beam

Abstract

This article intends to achieve a new formulation of beam vibration with quintic nonlinearity, including exact expressions for the beam curvature. To attain a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. In this direction, new powerful analytical method called Parameter Expansion Method (PEM) is employed to obtain the exact solution of frequency-amplitude relationship. Afterwards, it is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of mentioned system. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results. Nonlinear vibration of beams is of substantial interest for engineers and has been studied, considerably. From the engineering outlook, structures like helicopter rotor blades, space craft antennae, flexible satellites, airplane wings, robot arms, high-rise buildings, long-span bridges, drill strings and vibratory drilling can be modelled as a beam-like member. The problem of the transversely vibrating beam was recently formulated in terms of the partial differential equation of motion by many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] with different boundary conditions. Sedighi et al. [1] presented the advantages of recent modern analytical approaches applied on the governing equation of transversely vibrating cantilever beams. A comprehensive study on the analytical investigation of cubic nonlinear vibrating tapered beams has been conducted by Bayat et al. [2]. The analytical expression for geometrically non-linear vibration of clamped-clamped Euler-Bernoulli beams including cubic non-linear strain-displacement relationship has been obtained by Barari et al. [3]. The application of new analytical approaches on the dynamical behavior of beams vibration with different boundary conditions has been investigated by Sedighi et al. [5][6][7][8]. Closed form solution to free vibration of beams with mixed boundary conditions has been proposed by Motaghian et al. [9]. Nikkhah Bahrami et al.[10] used modified wave approach for calculation of natural frequencies and mode shapes of arbitrary non-uniform beams. Non-linear modal analysis of a rotating beams studied by [11,12]. When the vibration amplitudes are moderate or large, the geometric nonlinearity must be included.It is very important to provide an accurate analysis towards the understanding of the non-linear vibration characteristics of these structures. Most models dealing with nonlinear dynamics of flexible

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