ON THE SOLUTIONS OF DIFFERENTIAL EQUATIONS DESCRIBING THE MOTION OF NON-LINEAR SPRING PENDULUM

Abstract

The mathematical formulation of many vibrating physical systems often results in a system of non-linear differential equations. Such systems represent many engineering applications. One of the important subjects expressed here is the dynamical system, which is governed by different types of non-linear differential equations. The harmonically excited spring-pendulum •system is one of the famous dynamically systems simulating many engineering applications. One of these applications is the ship roll motion. The spring-pendulum system, which simulates the ship roll motion, is considered as a two-degree-of­ freedom system. It can be described by two coupled, second-order, non-linear, ordinary differential equations. Of course, the spring stiffness or the damping coefficients may be linear or non-linear..

In this study we consider the non-linear spring-pendulum system, subject to either single-excitation force for each mode or multi-excitation forces for the first mode and the excitation frequencies are different. The method of multiple time scale perturbation is applied to solve the non-linear differential equations up to and including the third-order approximation. All possible resonance cases were extracted at this approximation order and investigated numerically. The effects of the different parameters such as, natural frequencies, damping coefficients, • amplitudes of excitation forces, single and multi-excitation forces and non-linear parameters on system behavior are studied numerically. The

stability of the system is investigated using both frequency response equations and phase-plane method. The solutions of the frequency response equations regarding the stability of the system are shown graphically. Phase-plane is shown for the steady state amplitudes as a criterion for system stability and chaos presence or not. At the end of the work some recommendations regarding the system with different