Double scale analysis of periodic solutions of some non linear vibrating systems

TL;DR

This paper develops a double scale analytical method to approximate and analyze the stability of small periodic solutions in nonlinear vibrating systems, including a rigorous proof of the expansion's convergence.

Contribution

It introduces a double scale analysis approach with a convergence proof for nonlinear vibrating systems near primary resonance.

Findings

Successful approximation of small solutions using double scale analysis

Proof of convergence for the approximate solutions

Stability results for forced responses near resonance

Abstract

We consider {\it small solutions} of a vibrating system with smooth non-linearities for which we provide an approximate solution by using a double scale analysis; a rigorous proof of convergence of a double scale expansion is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance.