Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Abstract

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for 294 Numer Algor (2011) 58:293-314 different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.

On the Approximate Analytical Solution to Non-Linear Oscillation Systems

“…The values of parameters ε 1 , ε 2 , ε 3 , ε 4 associated with each of the six calculation modes are shown in Table 1. Table 1 gives the comparison of the obtained results with those obtained by Qian et al [36] and Runge-Kutta algorithm for different values of λ, ε 1 , ε 2 , ε 3 , ε 4 and different initial conditions. It can be observed from Table 1 that there is high level of agreement between the results obtained from the variational approach and those by Qian et al [36] and Runge-Kutta algorithm.…”

The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

“…In order to make sure of the effectiveness of the current technique, we compare the results from the second-order spreading residue harmonic balance approach SRHB with the Advances in Mathematical Physics 5 energy balance method EBM [34], the homotopy analysis method HAM [35], and the exact solution ex [33], which are presented in Table 1, for different parameters ( = 1, 2, 3, 4) and amplitudes of vibration , where the exact solution ex is computed using the numerical technique. The relative errors of vibration frequency are tabulated in Table 2.…”

“…They used these methods to solve this strongly nonlinear problem, and lower-order approximate solutions are yielded. Qian et al [35] studied the nonlinear vibrations of cantilever beam by the HAM. Latterly, Guo et al [36,37] have presented the residue harmonic balance solution procedure to approximate the periodic behavior of different oscillation systems and they have obtained some more accurate results.…”

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).…”