Numerical calculation of nonlinear normal modes in structural systems

Abstract

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).

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

“…Typically the modes with the lowest frequencies are retained in the reduced-order model. To construct the linear-based reduced model, let the n × m transformation matrix be defined as = I m T (4) in which I m is the m × m identity matrix and the (n − m) × m matrix T is found by iterating the following equation [11,12] …”

“…It was shown in [11,12] that T preserves the exact eigenstructure of Equation (3). Equation (7) is thus a Guyan-like order reduction transformation which accounts for the inertia as well as stiffness effects.…”

“…The linear mode shapes in the first subregion are φ 1 = (1.0 0.618) T and φ 2 = (1.0 −1.618) T and the modal frequencies are ω 1 = 0.618 and ω 2 = 1.618. The linear-based order reduction transformation of Equation (11) results in the reduced model of Equation (12) where β 1 = 0.724 in the first mode and β 2 = 0.276 in the second. We compare the various reduced models with the exact BNM frequencies as a function of the dimensionless parameter ρ.…”

“…Rhee and Burton [10] applied a previously developed linear-based Guyan-like reduction procedure which preserves the exact eigenstructure of the linearized model [11,12] to the cases of deadzone and bang-bang nonlinearities in a two-degree-of-freedom vibrating system. The frequencies of the reduced models were compared with those of the actual NNMs in the full model obtained by direct numerical simulation.…”

“…This is done in an effort to retain the advantages of the linear-based reduction method used in [10][11][12] while improving on the accuracy of the reduced model. This is accomplished by approximating the amplitude-dependent NNM frequencies and mode shapes of the full model using linear approximation methods based on extensions of the bilinear frequency relation.…”