The study of the non-stationary response of systems has many applications in problems related to transition through resonance in rotating machinery, aerospace structures and other physical systems.
In this paper, we present methods to analytically predict the response of some weakly nonlinear systems to slowly varying parameter changes. We consider systems which can be averaged and represented as two first order equations. The evolution of the solutions of such systems through critical (jump or bifurcation) points is studied using the method of matched asymptotic expansions. As an example, the method is used to predict the response of the forced Duffing’s oscillator during passage through resonance.
Starting with a general system of two, first-order equations, we set up a slowly varying equilibrium or ‘outer’ solution as an asymptotic expansion about the stationary solution. This solution is seen to be invalid in a small neighborhood of the critical points — the ‘inner’ region. In this inner layer, the system of equations is transformed into the Jordan canonical form, which is easier to study. Using approximations from the center manifold theory, the problem is reduced to one first-order equation. By making appropriate scale changes, an ‘inner’ solution is developed. This solution is asymptotically matched with the outer expansion to yield a unified solution valid for all time.