Nonsmooth Modal Analysis: From the Discrete to the Continuous Settings

Abstract

This chapter addresses the prediction of vibratory resonances in nonsmooth structural systems via Nonsmooth Modal Analysis. Nonsmoothness in the trajectories is induced by unilateral contact conditions in the governing (in)equations. Semi-analytical and numerical state-of-the-art solution methods are detailed. The significance of nonsmooth modal analysis is illustrated in simplified one-dimensional space semi-discrete and continuous frameworks whose theoretical and numerical discrepancies 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. Terminology and acronyms 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 dedicated 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. Unless otherwise stated, the epithet discrete (as in "discrete systems" or "discrete setting") designates semidiscretization in space, while continuous refers to everything else.

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

“…Non-smooth modal analysis (NSA) is used to obtain resonant vibratory responses and frequencies for structures prone to unilateral contact conditions [6,18]. While linear modes of vibration are obtained by solving eigenvalue problems, non-smooth modes are seen as invariant manifolds supporting periodic autonomous solutions [16].…”

Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

“…Such frequencies are actually avoided in the remainder since the solutions of interest lie "between" these two extreme bar configurations where the contact gap is either always open or always closed. It is worth to state that Equation (23), which dictates relationships between Fourier Transforms evaluated at x D 1, stems from a boundary condition at x D 0. Inserting ( 21) into ( 23) leads to a system of linear equations in the coefficients .a n ; b n /, n D 0; 1; 2; : : : of the form…”

“…The corresponding strain field is u f . !/, expressions which agree with (23). The same procedure applies to the Robin-Signorini system and Expression ( 27) would be retrieved.…”

“…However, in the framework of continuum mechanics, it is now established that, at least for the bouncing bar example [4], neither impulsive dynamics nor chattering exist in the solution. Accordingly, it is crucial to develop solution methods capable of handling the above considerations, which discards the classical Finite Element Method (FEM) because of two major difficulties: the need of an impact law which generates chattering [23] or the implementation of a penalization technique with questionable residual penetrations and the difficulty to properly quantify the penalty parameter. The issue stems from the distribution of mass notably at the contact interface.…”

Nonsmooth Modal Analysis of a Non-internally Resonant Finite Bar Subject to a Unilateral Contact Constraint

“…To some extent, nonlinear modal analysis can be employed for predicting vibratory resonances, computing the nonlinear spectra of vibration or performing model-order reduction. Techniques traditionally employed for nonlinear modal analysis require a certain degree of smoothness in the nonlinearities [12] and thus fail for systems with nonsmooth nonlinearities such as unilateral contact constraints. Certainly, an accurate characterization of the vibratory response of these systems is essential to achieving enhanced and safer engineering applications [11].…”