The construction of non-linear normal modes for systems with internal resonance

D Jiang,C Pierre,S.W Shaw

doi:10.1016/j.ijnonlinmec.2004.08.010

Abstract

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.

Highlights

IntroductionIn order to obtain accurate reduced order models for non-linear systems, “non-linear modal analysis”
In order to obtain accurate reduced order models for non-linear systems, “non-linear modal analysis”has been proposed as an analogy to its linear counterpart
Time responses for the displacements of the master and slave coordinates are shown and compared in Figs. 8 and 9 using three different simulation approaches: (i) direct time simulations based on the 36-state reference model, with initial conditions that satisfy the constraint relationships; (ii) time simulations for the master coordinates using the 4-state reduced-order model, along with the reconstruction of the slave coordinate responses using the constraint functions; and (iii) simulations based on the reducedorder model obtained by the asymptotic expansion method described in Pesheck et al [18], wherein the invariant manifold and the corresponding reducedξ1(t)

Summary

Introduction

In order to obtain accurate reduced order models for non-linear systems, “non-linear modal analysis”. A computational approach is proposed, which combines Galerkin projections in the phase coordinates and finite difference discretizations in the amplitude coordinates Using this methodology, a 2M-dimensional invariant manifold can, in principle, be constructed for the system, and motions on this manifold are governed by a set of 2M first-order differential equations in the master coordinates. The multi-NNM approach is applied to a simple three-DOF system, as well as to a rotating blade model in which transverse motions are non-linearly coupled with axial extensions of the blade For the latter system, an 18-DOF discretized model derived from linear modal analysis is examined, which features an internal resonance between the first and second flapping modes.

Multi-mode invariant manifolds

A three-DOF example system

The rotating beam system

Conclusions