An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method

High precise analysis of lateral vibration of quintic nonlinear beam

“…1. Denoting by w the transverse deflection, the differential equation governing the equilibrium in the deformed situation is derived as: where is the "exact" expression for the curvature, using the approximation (2) which the nonlinear term has been extracted from [5]. The governing quintic nonlinear equation (3) can be expressed as: (3) which is subjected to the following boundary conditions (4) Assuming , where is the first eigenmode of the simply supported beam and can be expressed as:…”

“…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.…”

“…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].…”

“…However, research on flexible beams has so far been restricted to cubic nonlinearity. Literature which considered high order of nonlinearities is very limited [5].…”

Nonlinear Vibrations of Cantilever Timoshenko Beams: A Homotopy Analysis

“…Its freedom to choose different base functions to approximate a nonlinear problem and its ability to control the convergence of the solution series have been very advantageous in solving highly nonlinear problems in science and engineering (Hoseini et al, 2008;Hoshyar et al, 2015;Mastroberardino, 2011;Mehrizi et al, 2012;Mustafa et al, 2012;Pirbodaghi and Hoseini, 2009;Qian et al, 2011;Ray and Sahoo, 2015;Sedighi et al, 2012;Wen and Cao, 2007;Wu et al, 2012).…”