Fast Computation of Steady-State Response for Nonlinear Vibrations of High-Degree-of-Freedom Systems
This work addresses the computational bottleneck of simulating high-degree-of-freedom nonlinear mechanical systems for engineers, offering a faster alternative to existing methods.
The paper presents an integral equation method for fast computation of steady-state responses of nonlinear multi-degree-of-freedom systems under periodic and quasi-periodic excitation, achieving robust convergence via Picard and Newton-Raphson iterations and enhanced performance with continuation schemes.
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.