Abstract
Non-linear vibratory systems are often characterized by external or excitation parameters which vary with time (i.e., are “non-stationary”). A general methodology is presented to predict analytically the response of some weakly non-linear dissipative systems as an excitation parameter varies slowly across points of instability corresponding to co-dimensional-1 bifurcations. It is shown that the motion near the bifurcation/critical point can be approximated by motion along a center manifold, and can be represented by a 1-dimensional dynamical system with a slowly varying parameter. Techniques expounded by Haberman [1] for analyzing such 1-dimensional equations using matched asymptotic expansions and non-linear boundary layers are summarized. The results are then used to obtain responses of some classical non-linear vibratory systems in the presence of non-stationary excitation. The problem of transition across saddle-node bifurcations or jumps during passage through primary resonance in the forced Duffing's oscillator is studied. Then, the transition across the points of dynamic instability (pitchfork bifurcations) in the parametrically excited non-linear Mathieu equation 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 are in good agreement with those obtained from direct numerical integration of the equations.
Published Version
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