General perturbation correction: full-decomposition and physics-based elimination of non-secular terms

Abstract

Discretized perturbation analysis based upon a structure's single (finite) mode Galerkin-truncated model may lead to erroneous results in comparison with full-basis discretization (or, equivalently, direct perturbation method). This error is due to the two involved analytical steps, i.e., multi-scale expansion and mode truncation, being non-commutative with each other. However, the key underlying physical origin for this error (or non-commutativity) is still unclear. The novelty of the current work is to propose a new physics-based error source interpretation and, accordingly, a general perturbation correction procedure. Explicitly, a general physics-based interpretation for the error source is referred to the incomplete characterization of low-order non-secular effects due to both spectrum mixture and single (finite) mode truncation. Three typical dynamical scenarios, i.e., hard external excitation, quadratic nonlinearity, and hard boundary motion, are addressed in the same unified framework. By introducing a full spectrum decomposition and a complete elimination of the non-secular term in a physically equivalent manner, a general correction procedure of the finite mode truncation often used in perturbation analysis is proposed and then validated through numerical examples.