The Method of Strained Coordinates for Vibrations with Weak Unilateral Springs

TL;DR

This paper develops an asymptotic method using strained coordinates to analyze vibrations in structures with weak unilateral springs, providing theoretical insights and a numerical algorithm for computing nonlinear normal modes.

Contribution

It introduces a new asymptotic expansion approach for weak unilateral springs and a numerical algorithm to compute nonlinear normal modes, effective even for larger spring rigidity.

Findings

Asymptotic expansion valid for small epsilon with long-time effects

Numerical algorithm accurately computes nonlinear normal modes

Results confirmed by numerical simulations

Abstract

We study some spring mass models for a structure having a unilateral spring of small rigidity $\epsilon$. We obtain and justify an asymptotic expansion with the method of strained coordinates with new tools to handle such defects, including a non negligible cumulative effect over a long time: $T_\eps \sim \eps^{-1}$ as usual; or, for a new critical case, we can only expect: $T_\eps \sim \eps^{-1/2}$. We check numerically these results and present a purely numerical algorithm to compute "Non linear Normal Modes" (NNM); this algorithm provides results close to the asymptotic expansions but enables to compute NNM even when $\epsilon$ becomes larger.