RESUMEN
Elastic strips provide a general motif for studying shape transitions. When actuated through rotation of its boundaries, a buckled strip exhibits, depending on the direction of rotation, three types of shape transitions: buckling, algebraic snap-through, or exponential snap-through. The transition dynamics is linked to the character of the bifurcation, which, in turn, is disclosed by the normal form of the system, but deriving normal forms is challenging. Recent work has used asymptotic methods to obtain this form for algebraic snap-through, but, to date, there is no methodology for extending this analysis to other transitions. Here we introduce a method to analyze the dynamic characteristics of an elastic strip near a transition and extend, in a straightforward manner, the previously proposed asymptotic analysis to exponential snap-through and buckling transitions. Importantly, we show that these normal forms dictate all the dynamic characteristics of the elastic strip near a shape transition. Our analysis provides reliable tools to diagnose and anticipate elastic shape transitions.
RESUMEN
We study the influence of gravity on the dynamics of upward propagating premixed flames. We show that the role of gravity on the dispersion relation is small, but that the nonlinear effects are large. Using a Michelson Sivashinsky equation modified with a gravity term, it can be observed that the nonlinear dynamics of the crests is greatly influenced by gravity, as well as the final amplitude of the flame. A simple model is proposed to explain the role of gravity on the amplitude.
RESUMEN
Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside out. This snap through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.
RESUMEN
We investigate the influence of gravity and heat loss on the long-time nonlinear dynamics of premixed flames. We show that even when their influence remains weak in the linear regime they can significantly modify the long-time behavior. We suggest that the presence of such a large-scale stabilizing effect could be responsible for the creation of new cells on the front and the appearance of the strong persistent patterns observed in several recent experimental and numerical studies. It could also explain some statistical anomalies observed in the topology of flame fronts.