More Than One Degree of Freedom Systems and the Euler-Lagrange Process

This chapter expands the analysis to systems with more than one degree of freedom. I present a formal analysis of degrees of freedom in two dimensions. I introduce the Euler-Lagrange method of finding the equations of motion for the mechanical systems, the rate of work function for the calculation of generalized forces (and torques), and the Rayleigh dissipation function for including viscous damping. I discuss linearization and linear stability. I introduce simple DC motors as prime movers for mechanical systems. I combine these elements in several examples of systems with two and more degrees of freedom. This chapter introduces some systems that will recur throughout: the overhead crane, a simple servo system, and a magnetic suspension.