Effects of Base Points and Normalization Schemes on the Non-Linear Normal Modes of Conservative Systems

Zhang, X.H.

doi:10.1006/jsvi.2001.4241

Cited by 2 publications

(1 citation statement)

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“…A nonlinear normal mode has been defined by Shaw and Pierre [6] as a two-dimensional invariant manifold in the 2n-dimensional state space, such that the eigenspace of the associated linear mode is tangent to the NNM at the origin of the state space. Studies of NNM properties and methods for calculation of the modal manifolds have appeared in many works, including those by Vakakis and coworkers [7][8][9][10][11][12], Shaw and coworkers [13][14][15][16][17], Nayfeh and coworkers [18][19][20][21][22][23][24][25], and others [26][27][28][29][30][31][32][33][34][35][36]. These works may be roughly divided into several overlapping categories: (1) studies of basic phenomena associated with NNMs, including bifurcation, mode localization, and internal resonance; (2) methodologies for calculating individual NNMs; (3) applications of NNMs to physically motivated problems; and (4) obtaining reduced order models (ROMs) of nonlinear structural systems.…”

Section: Introductionmentioning

confidence: 99%

Numerical calculation of nonlinear normal modes in structural systems

Burton

2007

Nonlinear Dyn

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This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to "artificially destabilize" the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).

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