chapter  5
18 Pages

Estimating the Behavior of Coupled Discrete Systems

In this chapter we will talk about modeling the behavior of coupled discrete linear systems, first introducing an extension of the famous Castigliano’s theorems to show that we can (1) formally apportion the loads among the elements of a coupled parallel system by minimizing that parallel system’s displacement as determined by Castigliano’s second theorem, and (2) formally apportion the displacement among the elements of a coupled series system by minimizing that series system’s load as determined from Castigliano’s first theorem. These extensions provide a means for apportioning loads in coupled continuous systems, which we will show in Chapter 6 for two systems of coupled Timoshenko beams supporting discrete and continuous loads. We then conclude this chapter by describing a second classic mechanics result, the Rayleigh quotient, and using it to lay the foundation for estimating the natural frequencies of elastic elements, which we will do in Chapter 7.