Analytical study of nonlinear oscillatory systems using the Hamiltonian approach technique

Abstract

In this article, we investigate and apply Hamiltonian approach method as one of the analytical approximate techniques, for studying the strongly nonlinear dynamical systems such as the motion of a rigid rod rocking back on the circular surface without slipping and the free vibrations of an autonomous conservative oscillator with inertia and static-type fifth-order nonlinearities. To illustrate the applicability and accuracy of the method, the approximate solution results are compared with exact and numerical solutions.

Study of autonomous conservative oscillator using an improved perturbation method

“…Therefore, the frequency from the zeroth order approximation is, which is same as the frequency from the first order approximation in ATHPM [1] and HT [28]. The remaining part of x 1 (τ ) in equation (28) gives the first order approximation of x,…”

“…The coefficient of τ sin τ in equation (28) needs to be equal to zero for avoiding the secular term, i.e., 8a + 4 a 3 + 3αa 5 − Λ 0 (8aλ + 6βa 3 + 5γa 5 ) = 0, (…”

“…The percentage error in ω are 2.385 and 0.584 for AT HP M and LT HP M h, respectively, for a = 1 which reduce to 0.167 if the second order term in LT HP M h is considered. We plot the displacements obtained from HT (x HT , green cross) [28] , AT HP M (x AT , red diamond) [1] and LT HP M h (x L , blue circles) expressed in equation ( 35) respectively, with increasing t for four sets of parameters A(a = 0.2, OP = 0.2), B(0.2, 1.0), C(1.0, 0.2) and D(1.0, 1.0) in the left column of figure 2, and compared with the same obtained by numerical solution of equation ( 21) employing the fourth order Runge-Kutta (RK4) method (x RK4 (t), black solid line). It is seen that for small values of the parameters, approximate displacements match well with x RK4 ) but a significant deviations for x AT and x HT from x RK4 are noticed for large values of force parameters.…”

“…LT HP M h is used to get the displacement (x) and ω for strongly nonlinear ACO. We compare LT HP M h results to those obtained from AT HP M and Hamiltonian approach technique (HT ) [28] where the results from numerical calculations (RK4) are considered as the benchmark for checking the accuracy. This paper is organized as follows.…”

An Iterative Approach to Obtaining the Nonlinear Frequency of a Conservative Oscillator with Strong Nonlinearities

“…Recently, many different methods have been developed to obtain the approximate solutions of such problems. Some of them are: harmonic balance method [4][5][6], energy balance method [7,8], Hamiltonian approach [9], He's amplitude-frequency formulation [10] and variational iteration method [11]. In these methods, unlike the classical perturbation method, the presence of small parameter is not required.…”