Nonlinear normal modes for vibratory systems under harmonic excitation

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to harmonic excitation. The approach is an extension of the nonlinear normal mode (NNM) formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree-of-freedom (dof), whose response is known. A reduced-order model for the forced system is then determined by the usual NNM procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple 2-dof mass–spring system with cubic nonlinearities, and for a discretized beam model with 12 dof. The results show that this method provides very accurate responses over a range of frequencies near resonances.

[1]  Christophe Pierre,et al.  Normal modes of vibration for non-linear continuous systems , 1994 .

[2]  Daniel J. Inman,et al.  Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes , 2001 .

[3]  Christophe Pierre,et al.  Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method , 2002 .

[4]  Christophe Pierre,et al.  Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems , 1995, Nonlinear Dynamics.

[5]  C. Pierre,et al.  A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS , 2002 .

[6]  Christophe Pierre,et al.  The construction of non-linear normal modes for systems with internal resonance , 2005 .

[7]  S. Tzeng,et al.  Pumping Speed Measurement and Analysis for the Turbo Booster Pump , 2004 .

[8]  Alexander F. Vakakis,et al.  NON-LINEAR NORMAL MODES (NNMs) AND THEIR APPLICATIONS IN VIBRATION THEORY: AN OVERVIEW , 1997 .

[9]  A. H. Nayfeh,et al.  On Nonlinear Normal Modes of Systems With Internal Resonance , 1996 .

[10]  P. Sundararajan,et al.  An algorithm for response and stability of large order non-linear systems : Application to rotor systems , 1998 .

[11]  Christophe Pierre,et al.  Modal analysis-based reduced-order models for nonlinear structures : An invariant manifold approach , 1999 .

[12]  C. Pierre,et al.  Large-amplitude non-linear normal modes of piecewise linear systems , 2004 .

[13]  Christophe Pierre,et al.  Normal Modes for Non-Linear Vibratory Systems , 1993 .

[14]  Christophe Pierre,et al.  Non-linear normal modes and invariant manifolds , 1991 .

[15]  Philip Rabinowitz,et al.  Numerical methods for nonlinear algebraic equations , 1970 .

[16]  Alexander F. Vakakis,et al.  An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems , 1994 .

[17]  Alexander F. Vakakis,et al.  Normal modes and localization in nonlinear systems , 1996 .

[18]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[19]  Christophe Pierre,et al.  Finite-Element-Based Nonlinear Modal Reduction of a Rotating Beam with Large-Amplitude Motion , 2003 .