An Explanation for Why Natural Frequencies Shifting in Structures with Membrane Stresses, Using Backbone Curve Models

Abstract

In this paper, the phenomenon of natural frequencies shifting due to the nonlinear stiffness effects from membrane stress is studied using a nonlinear reduced order model based on backbone curves. The structure chosen for study in this paper is a rectangular plate with a pinned constraint along all edges. To analytically explore the frequency varying phenomenon, a four nonlinear-mode based reduced-order model that contains both single-mode and coupled-mode nonlinear terms is derived. The process of deriving the reduced order model is based on a normal form transformation, combined with a Galerkin type decomposition of the governing partial differential equation of the plate. This allows a low number of ordinary differential equations to be obtained, which in turn can be used to derive backbone curves that relate directly to the nonlinear normal modes (NNMs). The frequency shifting is then investigated relative to the backbone curves. Modal interactions, caused by nonlinear terms are shown to cause the frequency shifts. In the final part of the paper, an attempt is made to quantify the frequency shifting due to different nonlinear effects.