UC Berkeley

Engineering and everyday experience alike provide countless examples of deformable objects coming into contact with effectively rigid objects, from a belt stretched between pulleys in an automobile engine to a ball bouncing on a blacktop. But despite the ubiquity of such systems, we have little in the way of a fundamental understanding of the dynamics they can be expected to display. The main obstacle to deeper understanding is mathematical: the unilateral constraint of contact creates free boundaries, and free-boundary-value problems are inherently nonlinear, even if the underlying differential equations are themselves linear. This dissertation explores the dynamics of contact in several prototypical problems inspired by engineering applications such as indentation testing, offshore oil exploration and production, and microelectromechanical systems (MEMS) design.

The first major thrust of the present work is the linear vibration analysis of a beam contacting a flat surface due to combined gravity and adhesion. In analyzing this problem we develop a technique for appropriately linearizing the conditions at a free boundary. We then extend the technique to study the nonlinear vibrations of the gravity-only problem using a second-order perturbation analysis, which shows that contact generates a rich array of nonlinear resonances. The results show good agreement with numerical solutions from a special finite element method that precisely tracks the free boundary. We conclude by using the gap function formalism common in computational contact mechanics to demonstrate that the same pattern of resonances can be expected to arise in a more general class of problems.

Having studied a problem involving a one-dimensional continuum, we shift attention to one with a two-dimensional continuum: an electrostatically actuated plate contacting a charged, dielectric-coated substrate. We use a variational method to derive one model in which the dielectric thickness is finite, and the other in which the thickness is small. The latter corresponds to a common type of surface adhesion. We formulate the linear stability equations for both models and characterize their equilibria in the case of a circular actuator before adopting an approximation from dynamic fracture mechanics to study the dynamics of the thin-dielectric model, which display some unique mathematical properties, including finite-time blow-up. Finally, we discuss application of the model to the problem of vibration-assisted stiction repair.

This work incorporates many analytical and computational tools, from configurational mechanics, to perturbation theory, to arbitrary Lagrangian-Eulerian finite element methods, and represents a first step toward the development of rules-of-thumb for the design of vibration-critical mechanical systems with contact constraints.