Free Vibration Analysis of a Conservative Two-Mass System with General Odd Type Nonlinear Connection

Abstract

In this paper, free vibration of a two-mass system with nonlinear connection investigated. The connections considered as sums of linear and odd type nonlinear springs. The mathematical model of this system is derived by two differential equations with nonlinear coupling. By introducing the intermediate variables equations of motion transformed into a single one-variable differential equation which known as the generalized Duffing equation. The energy balance method based on Galerkin–Petrov approach is applied to solve the generalized Duffing equation and new amplitude-frequency relationships obtained. Comparisons between results obtained from exact approach and results obtained from energy balance method based on Galerkin–Petrov approach show obtained solutions reach better accuracy rather than other solutions that obtained for the generalized Duffing equation and obtained solutions can be served as a most accurate solution for this problem. Then using the intermediate variables obtained solution extended to analysis of a two-mass system with general odd type nonlinear connection. Three examples with different types of the nonlinear connection between masses are considered. Comparisons show very good agreement between obtained analytical solutions and numerical ones. Obtained solution is capable to derive an accurate analytical solution for free vibration of a two-mass with any odd type of nonlinear connection.