Equilibrium
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 frame
- PDF / 664,838 Bytes
- 40 Pages / 504.547 x 719.979 pts Page_size
- 36 Downloads / 199 Views
Equilibrium 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]. The starting point is a linear chain of masses interconnected by springs and constrained to move only in the longitudinal direction. Our general mathematical framework is already manifest in this rather simple mechanical system. In the second section, we discuss simple electrical networks consisting of resistors, current sources and/or batteries, interconnected by a network of wires. Here, the resulting Gram matrix is known as the graph Laplacian, which plays an increasingly important role in modern data analysis and network theory. Finally, we treat small (so as to remain in a linear modeling regime) displacements of two- and three-dimensional structures constructed out of elastic bars. In all cases, we consider only the equilibrium solutions. Dynamical (time-varying) processes for each of these physical systems are governed by linear systems of ordinary differential equations, to be formulated and analyzed in Chapter 10.
6.1 Springs and Masses A mass–spring chain consists of n masses m1 , m2 , . . . mn arranged in a straight line. Each mass is connected to its immediate neighbor(s) by springs. Moreover, the chain may be connected at one or both ends to a fixed support by a spring — or may even be completely free, e.g., floating in outer space. For specificity, let us first look at the case when both ends of the chain are attached to unmoving supports, as illustrated in Figure 6.1 We assume that the masses are arranged in a vertical line, and order them from top to bottom. For simplicity, we will only allow the masses to move in the vertical direction, that is, we re
Data Loading...
