Dynamic analysis of piecewise linear multi-degree-of-freedom systems subjected to arbitrary general loads

Abstract

A novel method is presented for the response quantification of piecewise linear multi-degree-of-freedom systems to arbitrary general loadings. The dynamical system in each piecewise linear branch is recast, through a real, nonlinear, and invertible transformation into an undamped and decoupled system posed in canonical form. Two procedures are then presented for the response quantification of the resulting independent equations. The step-by-step piecewise exact method is first adopted to compute the response in the time-domain by means of a recurrence formula derived from exact evaluation of the equation of motion. A frequency-domain approach is then developed whereby the closed-form solution to harmonic components of the excitation, decomposed by means of discrete Fourier transform, is obtained. The solutions associated with the piecewise linear segments are synthesised to construct the time history of the response. Numerical investigations on the analysis of a block rocking on a linear viscoelastic isolation system and the analysis of a bilinear secondary oscillator vibrating on a linear structure are presented to indicate the utility of the method.