Simple mechanical control systems
The preceding two chapters were preparatory material, and were presented fairly hastily, with the reader being directed to references for further details. In this chapter, we begin the presentation of the material that can be considered the core of the book. The aim in this chapter is to provide the methodology for going from a physical problem to a mathematical model. What is more, the mathematical model we consider uses the tools described in Chapter 3, particularly the affine connection formalism of Section 3.8. We are extremely systematic in our presentation, far more so than is the norm. This has the disadvantage of making the presentation lengthy. To this end, we should mention that it is not normal, when modeling a mechanical system, to go systematically through the steps we outline in this chapter. Indeed, someone experienced with the process can perform many of the steps we describe “just by looking at the system.” The systematic presentation we provide has two advantages to counterbalance its length.