Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems

TL;DR

This paper introduces an efficient integral equation method using Green's functions for rapid steady-state response computation in nonlinear high-degree-of-freedom mechanical systems, improving speed and robustness over traditional techniques.

Contribution

It presents a novel integral equation approach with explicit Green's functions and combined iterative schemes for fast, reliable response analysis of complex nonlinear systems.

Findings

Fast computation of steady-state responses achieved

Picard iteration effective for non-smooth systems

Newton-Raphson iteration ensures convergence near resonance

Abstract

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green's function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton--Raphson iteration instead, obtaining robust convergence. We further show that this integral-equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response.