NON-LINEAR VIBRATIONS OF HARDENING SYSTEMS: CHAOTIC DYNAMICS AND UNPREDICTABLE JUMPS TO AND FROM RESONANCE

Abstract The non-linear resonance behaviour of two typical hardening systems subjected to various forms of excitation is examined. For the system with a linear stiffness component it is shown that for small forcing levels the system behaves like a linear system with resonance occurring when the forcing frequency is approximately equal to the linearized natural frequency. As the forcing amplitude is increased the steady state response peaks towards higher frequencies leading to the well knownjump phenomenon. Often such jumps to (and from) resonance are a purely deterministic event in which the system settles on to the solution lying on the (non-)resonant branch of the response curve. At higher, but still moderate forcing values, it is shown that such jumps can beindeterminatein the sense that one cannot predict whether the system re-stabilizes or not; indeed, their outcome may go to another coexisting solution at the bifurcation. Examples of hardening systems exhibiting unpredictable jumpstoandfromresonance are presented.