Nonlinear analysis of forced piecewise-linear vibration system

Abstract

This study concerns the strong nonlinear dynamic behavior of mechanical systems in presence of a clearance, in which the nonlinearity is induced from the assumed piecewise-linear characteristics of the variation of stiffness and dam ping. The steady-state solutions are found by two newly developed methods, a direct method and am FFT method. The direct method is derived from satisfying the boundary conditions at the contact points. The formulation allows for inclusion of subharmonics using Stokers assertion. The resulting nonlinear algebraic equations are solved by setting the duration of contact as a parameter, which leads to a set of linear algebraic equations. The FFT method is based on adoption of Fast Fourier Transformation by expressing the unknown steady-state solution and the nonlinear force by Fourier series expansion and solving for the resulting nonlinear algebraic equations by Newton-Raphson iteration. The newly developed techniques are applied to the important practical problems of link mechanisms using the direct method and to a nonlinear rotor-bearing dynamic problem using the FFT method. The results show good agreement with those obtained through numerical integration. Exploitation of the piecewise-linearity of the system is made to conduct the stability analysis on a Poincare section. The domains of attraction are determined for the nonautonomous systems involved using the recently introduced cell-to-cell mapping which was previously applied to autonomous systems.