Explicit solution of the large amplitude transverse vibrations of a flexible string under constant tension

Abstract

This paper presents the analytical simulation of string with large amplitudes using the Variational Iteration Method (VIM) and Hamiltonian Approach (HA). In order to verify the precision of the presented methods, current results were compared with He's Variational Approach and Runge-Kutta 4 th order. It has been found that these methods are well suited for a range of parameters and the approximate frequencies and periodic solutions show a good agreement with other techniques. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering.

Approximate solutions of large amplitude vibration of a string

“…The obtained results are compared in the next section. Obtained solutions using AFF, IAFF, MMA and Runge-Kutta are compared to the results of variational Iteration Method (VIM) (Taghipour et al, 2014), Hamiltonian Approach (HA) (Taghipour et al, 2014) and Variational approach method (VAM) which is reproduced using the derived equations in (Omran et al, 2013) in Table 1 and Table 2. Comparing the results in tables 1 and 2, the computed response of the cable under the large amplitude vibration using VAM, AFF and MMA are near eachother, while those of the IAFF is far apart from other methods.…”

Modeling of Surface Stress Effects on the Dynamic Behavior of Actuated Non-Classical Nano-Bridges

“…The current work attempts to compute the frequency-amplitude relation of doubly-clamped actuated nano-beams by employing a modern asymptotic approach namely Homotopy Perturbation Method with an auxiliary term. In recent times, several approaches have been proposed to solve the nonlinear differential equations such as Homotopy analysis Method [29], Variational Iteration Method [30], Variational Approach [31], Method of Multiple Scales [32], Max-Min approach [33], Iteration Perturbation Method [34], Hamiltonian Approach [35], Enriched Multiple Scales Method [36] and Homotopy Perturbation Method with an auxiliary term [37,38]. Recently a new powerful method namely Homotopy Perturbation Method with an auxiliary term proposed by He [39] has proven to be a very effective and convenient way of handling nonlinear problems.…”

Assesment of New Analytical Method for Solving the Foam Drainage Equation

“…In recent years, several such techniques have drawn special attention, such as inverse scattering method [19], Adomian decomposition method [20,21], Hamiltonian approach [22], variational iteration method [23][24][25], homotopy analysis method [26,27], variational approach [28], and homotopy perturbation method [29][30][31][32][33][34][35][36][37][38].…”