Application of Recent Powerful Analytical Approaches on the Non-Linear Vibration of Cantilever Beams

Abstract

This paper presents the advantages of some effective analytical approaches applied on the governing equation of transversely vibrating cantilever beams. Six studied methods are Min-Max Approach, Parameter Expansion Method, Hamiltonian Approach, Variational Iteration Method, Bubnov-Galerkin and Energy Balance Method. The powerful analytical approaches are used to obtain frequency-amplitude relationship for dynamic behavior of nonlinear vibration of cantilever beams. It is demonstrated that one term in series expansions of all methods are sufficient to obtain a highly accurate solution. Finally, a numerical example is conducted to verify the accuracy of these methods.

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

“…Sedighi et al [1] presented the advantages of recent modern analytical approaches such as the PEM, VIM, HA, MMA, and EBM applied to the governing equation of transversely vibrating cantilever beams. Liu [30] studied the approximated period of nonlinear oscillators with discontinuities by a modified PEM.…”

“…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.…”

Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

“…In this paper, the frequency-amplitude relationship of the conservative Helmholtz-Duffing oscillator is obtained by means of the max-min [22][23][24][25][26], Hamiltonian [27][28][29][30][31], and global error minimization methods [32][33][34][35]. The Hamiltonian approach is a kind of energy method and is proposed by He [27].…”

High precise analysis of lateral vibration of quintic nonlinear beam

“…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.…”