Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable.
Structures with underlying snap-through static behavior may exhibit highly nonlinear and unpredictable oscillations. Such systems rarely lend themselves to investigation by analytical means. This is not surprising as nonlinear phenomena such as chaos run counter to the predictability of an analytical closed form solution. However, many unexpected analytical approximations of global stability may be obtained for simple systems using the harmonic balance method.
In this paper a simple single-degree-of-freedom arch is studied using the harmonic balance method. The equations developed with the harmonic balance approach are then solved using an arc-length method and an approximate snap-through boundary in forcing parameter space is obtained. The method is shown to exhibit excellent agreement with numerical results. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many applications in both civil engineering, where arches are a canonical structural element, and mechanical/aerospace engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.
Higher Approximation of Steady Oscillations in Nonlinear Systems with Single Degree of Freedom : Suggested Multi-Harmonic Balance Method