The electrostatic actuation with its several advantages is the main principle for micro-electro-mechanical systems (MEMS). One major drawback is the nonlinear behavior, which results into instability, known as the electrostatic pull-in effect. This effect might also push a closed-loop configuration into instability and thus makes a linear time-invariant control inapplicable to the system. The paper investigates the stability of an acceleration sensor in closed-loop operation with this setting. A simplified controller adjustment gives a first insight into this topic. Practical implementations saturate on the quantizer's full-scale value, which is also considered in the stability analysis. Numerical phase-plane analysis verifies the stability and shows further surprising results.

Snap-through is a striking instability in which an elastic object rapidly jumps from one state to another. It is seen in the leaves of the Venus flytrap plant and umbrellas flipping on a windy day among many other examples. Similar structures that snap-through are used to generate fast motions in soft robotics, switches in micro-scale electronics and artificial heart valves. Despite the ubiquity of snap-through in nature and engineering, its dynamics is usually only understood qualitatively. In this thesis we develop analytical understanding of this dynamics, focussing on how the mathematical structure underlying the snap-through transition controls the timescale of instability. We begin by considering the dynamics of 'pull-in' instabilities in microelectromechanical systems (MEMS) - a type of snap-through caused by electrostatic forces in which the motions are dominated by fluid damping. Using a lumped-parameter model, we show that the observed time delay near the pull-in transition is a type of critical slowing down - a so-called 'bottleneck' due to the 'ghost' of a saddle-node bifurcation. We obtain a scaling law describing this slowing down, and, in the process, unify a large range of experiments and simulations that exhibit delay phenomena during pull-in. We also investigate the pull-in dynamics of MEMS microbeams, extending the lumped-parameter approach to incorporate the details of the beam geometry. This provides a model system in which to understand snap-through of a continuous elastic structure due to external loading. We develop a perturbation method that systematically exploits the proximity to pull-in to reduce the governing equations to a simpler evolution equation, with a structure that highlights the saddle-node bifurcation. This allows us to analyse the bottleneck dynamics in detail, which we compare with previous experimental and numerical data. The remainder of the thesis is concerned with the dynamics of snap-through in macroscopic systems. In particular, we explore the extent to which dissipation is required to explain anomalously slow snap-through. Considering an elastic arch as an archetype of a snapping system, we use the perturbation method developed earlier to show that two bottleneck regimes are possible, depending delicately on the relative importance of external damping. In particular, we show that critical slowing down occurs even in the absence of damping, leading to a new scaling law for the snap-through time that is confirmed by elastica simulations and experiments. In many real systems material viscoelasticity is present to some degree. Finally, we examine how this influences the snap-through dynamics of a simple truss-like structure. We present a regime diagram that characterises when the timescale of snap-through is controlled by viscous, elastic or viscoelastic effects.

The electrostatic actuation with its several advantages is the main principle for micro-electro-mechanical systems (MEMS). One major drawback is the nonlinear behavior, which results into instability, known as the electrostatic pull-in effect. This effect might also push a closed-loop configuration into instability and thus makes a linear time-invariant control inapplicable to the system. The paper investigates the stability of an acceleration sensor in closed-loop operation with this setting. A simplified controller adjustment gives a first insight into this topic. Practical implementations saturate on the quantizer's full-scale value, which is also considered in the stability analysis. Numerical phase-plane analysis verifies the stability and shows further surprising results.