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 a nonlinear spring composed of cubic and constant terms are established. With the use of a certain bilinear transformation, the equation of motion is successfully converted into a regular Duffing equation whose exact solution already exists. The transformation and the reduced Duffing's nonlinear spring are identified by solving simultaneous nonlinear algebraic equations along with the given initial displacement. The waveform of the solution resembles a suspension bridge. The so-called skeleton curve is also asymmetric, 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 verify the accuracy of an analytical approximate solution obtained by the perturbation method, as well as of the numerical integration by the Runge-Kutta-Gill scheme.