Equilibrium

Abstract

In this chapter, we will apply what we have learned so far to the analysis of equilibrium configurations and stability of mechanical structures and electrical networks. Both physical problems fit into a common, and surprisingly general, mathematical framework. The physical laws of equilibrium mechanics and circuits lead to linear algebraic systems whose coefficient matrix is of positive (semi-)definite Gram form. The positive definite cases correspond to stable structures and networks, which can support any applied forcing or external current, producing a unique, stable equilibrium solution that can be characterized by an energy minimization principle. On the other hand, systems with semi-definite coefficient matrices model unstable structures and networks that are unable to remain in equilibrium except under very special configurations of external forces. In the case of mechanical structures, the instabilities are of two types: rigid motions, in which the structure moves while maintaining its overall geometrical shape, and mechanisms, in which it spontaneously deforms in the absence of any applied force. The same linear algebra framework, but now reformulated for infinite-dimensional function space, also characterizes the boundary value problems for both ordinary and partial differential equation that model the equilibria of continuous media, including bars, beams, solid bodies, and many other systems arising throughout physics and engineering, [61, 79].