Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses

Abstract

An analytical technique has been developed based on the harmonic balance method to obtain approximate angular frequencies. This technique also offers the periodic solutions to the nonlinear free vibration of a conservative, couplemass-spring system having linear and nonlinear stiffnesses with cubic nonlinearity. Two real-world cases of these systems are analysed and introduced. After applying the harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of the complicated higher-order nonlinear algebraic equations is cumbersome, especially in the case when the vibration amplitude of the oscillation is large. The proposed technique overcomes this limitation to utilize the iterative method based on the homotopy perturbation method. This produces desired results for small as well as large values of vibration amplitude of the oscillation. In addition, a suitable truncation principle has been used in which the solution achieves better results than existing solutions. Comparing with published results and the exact ones, the approximated angular frequencies and corresponding periodic solutions show excellent agreement. This proposed technique provides results of high accuracy and a simple solution procedure. It could be widely applicable to other nonlinear oscillatory problems arising in science and engineering.

Periodic Property and Instability of a Rotating Pendulum System

“…The main objective of this section is to study the stability of the considered dynamical model by examining its EOM (16), in which it isn't useful to study stability through the obtained solution (32). Therefore, we are going to consider the linear and nonlinear parts of Equation (16) in which they are represented by Equations ( 18) and ( 19) respectively. It is worthy to mention that, the stability of the linear part depends on the frequency term (1 − Λ) which is always positive.…”

“…The stability of an excited delayed Mathieu equation using the He-multiple-scales perturbation method [14] was investigated in [15]. HPM was utilized in [16] to deal with a cubic nonlinearity problem of a conservative couple mass-spring system dynamical system in which the periodic solutions are obtained. This problem was examined also in [17] through the development of an iteration technique based on the method of Mickens iteration to get the asymptotic angular frequencies.…”

Analysis of nonlinear vibration of couple-mass–spring systems using iteration technique

“…In the recent past, many new analytical techniques have been investigated to overcome these limitations. These include the Newton harmonic balance method (Lai and Lim, 2007), He's variational approach (Hashemi Kachapi et al, 2010), the energy balance method (Bayata et al, 2011), the max-min approach (Ganji et al, 2011), He's improved amplitude-frequency formulation (Bayat et al, 2011;Wang and An, 2018a;Ren and Hu, 2018) and the harmonic balance method (Hosen and Chowdhury, 2019). All have been used to derive approximate angular frequencies and corresponding periodic solutions to the TDOF system.…”