Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Abstract

This paper addresses the model reduction and the simulation of a damped Euler -Bernoulli -von Kármán pinned beam excited by a distributed force. This nonlinear problem is formulated as a PDE and reformulated as a well-posed state-space system. The model order reduction and simulation are derived by combining two approaches: a Volterra series expansion and truncation; a pseudo-modal truncation defined from the eigenbasis of the linearized problem. The interest of this approach lies in the large class of input waveshapes that can be considered and in the simplicity of the simulation structure. This structure only involves cascades of finite dimensional decoupled linear systems and multilinear functions. Closed-form bounds depending on the model coefficients and the truncation orders are provided for the Volterra convergence domain and the approximation error. These theoretical results are generalized to a large class of nonlinear models and refinement of bounds are also proposed for a large subclass. Numerical experiments confirm that the beam model is well approximated by the very first Volterra terms inside the convergence domain.

Interactions between nonlinear resonators represented in Volterra series

“…The convergence has not been studied in this paper and is not guaranteed since the excitation and interaction forces should be lower than the radius of convergence of the series. Results on convergence for partial differential equations can be found in [37] and [38].…”