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

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.

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

[2]  A. Nayfeh,et al.  Nonlinear normal modes of buckled beams , 1999 .

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

[4]  Alexander F. Vakakis,et al.  An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes , 1996 .

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

[6]  Ali H. Nayfeh,et al.  Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances , 1999 .

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

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

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

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

[11]  Ali H. Nayfeh,et al.  On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems , 1995 .

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

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

[14]  Christophe Pierre,et al.  Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes , 2002 .

[15]  Richard H. Rand,et al.  A direct method for non-linear normal modes , 1974 .

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

[17]  R. Rand,et al.  Normal modes and global dynamics of a two-degree-of-freedom non-linear system—I. Low energies , 1992 .

[18]  Ali H. Nayfeh,et al.  On Nonlinear Modes of Continuous Systems , 1994 .

[19]  Ali H. Nayfeh,et al.  Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities , 1995 .

[20]  R. Rand,et al.  Bifurcation of nonlinear normal modes in a class of two degree of freedom systems , 1992 .

[21]  R. M. Rosenberg,et al.  On Nonlinear Vibrations of Systems with Many Degrees of Freedom , 1966 .