Exact Solutions of the Free Vibration of a System with Asymmetrical Single-Term Cubic Spring.

Abstract

Exact solutions of the free vibration in a single-degree-of-freedom system having an offset forced single-term cubic spring are established. By employing a certain bilinear transformation, the equation of motion is successfully converted into a regular Duffing equation whose exact solution already exists. The constants included in the transformation and Duffing's nonlinear spring are determined by solving simultaneous nonlinear algebraic equations along with the given initial displacement. The waveform of the solution is composed of even, as well as odd, order harmonics and is distorted so as to resemble a suspension bridge. The skeleton curve is also asymmetrical and the maximum and minimum amplitudes must be distinguished. The response reveals combined soft and hard spring characteristics and possesses a two-branched property within a certain frequency range. The exact solution is successfully applied to check the accuracy of an analytical approximate solution by the perturbation method as well as of the numerical integral by the Runge-Kutta-Gill scheme.