Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams

Abstract

This article attempts to investigate the dynamical analysis of beam vibrations in the presence of preload discontinuity and proposes an innovative accurate equivalent function for this well-known nonlinearity. This approach enables us to overcome the inherent computational difficulty of the preload nonlinearity in the analytical investigations. At first, the nonlinear equation of beam vibration with preload boundary condition is considered and analytical solution is obtained using He’s parameter expanding method. The precision of the proposed equivalent function has been elucidated by comparison of our results with the obtained solutions using numerical method. Finally, the accuracy of the obtained results in the vibration analysis of suspension bridges as a realistic problem, verifies the strength of the presented modeling.

Accurate investigation of lateral vibrations of a quintic nonlinear beam on an elastic foundation: Using an exact formulation of the beam curvature

“…Xu [31] applied the PEM for strongly nonlinear oscillators. The PEM application to the cantilever beam vibration problem with dead-zone and preload boundary conditions was investigated by Sedighi et al [6][7][8].…”

“…From the engineering point of view, structures like helicopter rotor blades, spacecraft antennae, flexible satellites, airplane wings, robot arms, high-rise buildings, long-span bridges, drill strings, etc., can be modeled as a beam-like member. The problem of a transversely vibrating beam was formulated in terms of the partial differential equations of motion with different boundary conditions by many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. These investigations predicted the nonlinear frequency of the beam, which is very important for the design of many engineering structures.…”

High precise analysis of lateral vibration of quintic nonlinear beam

“…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].…”

“…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.…”

“…There have been several classical approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations including perturbation methods [18], He's Max-Min Approach (MMA) [2], He's Energy Balance Method [19], Combined Homotopy Variational Approach [20], Iteration perturbation method [21], Homotopy perturbation method (HPM) [22,23], Multistage Adomian Decomposition Method [24], Variational iteration method [3], Multiple scales method [25], Monotone iteration schemes [26], ADM-Padé technique [27], Navier and Levy-type solution [28], Hamiltonian approach [29], Parameter Perturbation Method [30], Differential Transform method [31], Laplace Transform method [32] . The application of new equivalent function for deadzone and preload nonlinearities on the dynamical behavior of beam vibration using PEM has been investigated by [6][7][8]. Therefore, many different methods have recently introduced various ways to eliminate the small parameter.…”