Nonsmooth modal analysis: From the discrete to the continuous settings

This chapter addresses the prediction of vibratory resonances in nons- mooth structural systems via Nonsmooth Modal Analysis. Nonsmoothness in the tra- jectories is induced by unilateral contact conditions in the governing (in)equations. Semi-analytical and numerical state-of-the-art solution methods are detailed. The sig- nificance of nonsmooth modal analysis is illustrated in simplified one-dimensional space semi-discrete and continuous frameworks whose theoretical and numerical dis- crepancies are explained. This contribution establishes clear evidence of correlation between periodically forced and autonomous unilaterally constrained oscillators. It is also shown that strategies using semi-discretization in space are not suitable for nonsmooth modal analysis. The spectrum of vibration exhibits an intricate network of backbone curves with no parallel in nonlinear smooth systems. The purpose of this chapter is to provide a general picture of the state-of-the-art vibratory analysis of nonsmooth systems. This topic lies at the interface between modal analysis of smooth nonlinear systems and nonsmooth contact dynamics dedi- cated to the time-evolution of nonsmooth systems, undergoing impact or dry friction, for instance. Some elementary concepts are succinctly recalled for the purpose of completeness. Terminology Unless otherwise stated, the epithet discrete (as in "discrete systems" or "discrete set- ting") designates semi-discretization in space, while continuous refers to everything else.