ON THE SLOW TRANSITION ACROSS INSTABILITIES IN NONLINEAR DISSIPATIVE SYSTEMS

Non-linear vibratory systems are often characterized by external or ex citation parameters which vary with time (i.e., are ''non-stationary'' ). A general methodology is presented to predict analytically the resp onse of some weakly non-linear dissipative systems as an excitation pa rameter varies slowly across points of instability corresponding to co -dimensional-l bifurcations. It is shown that the motion near the bifu rcation/critical point can be approximated by motion along a center ma nifold, and can be represented by a 1-dimensional dynamical system wit h a slowly varying parameter. Techniques expounded by Haberman [1] for analyzing such 1-dimensional equations using matched asymptotic expan sions and non-linear boundary layers are summarized. The results are t hen used to obtain responses of some classical non-linear vibratory sy stems in the presence of non-stationary excitation. The problem of tra nsition across saddle-node bifurcations or jumps during passage throug h primary resonance in the forced Duffing's oscillator is studied. The n, the transition across the points of dynamic instability (pitchfork bifurcations) in the parametrically excited non-linear Mathieu equatio n is analyzed. Lastly, the transition across a Hopf bifurcation in the Parkinson-Smith model for galloping of bluff bodies is discussed. The methodology described here is found to be effective in approximating the behavior of the systems in the vicinity of bifurcation points. The solutions and their qualitative features predicted by the analysis ar e in good agreement with those obtained from direct numerical integrat ion of the equations. (C) 1996 Academic Press Limited