A Unified Procedure for Deriving the Differential Equations of Electrical and Mechanical Systems

Abstract

Based on energy considerations, it is possible to obtain the differential equations of motion of any physical system. A statement of equilibrium involving operations on energy functions is the Lagrange equation d / OT a7' D aOV dt (aq,) -aq aq + aq. Providing that the kinetic energy T, potential energy V, and dissipation function D can be written, the differential equations of the system are obtained by following a straightforward systematic procedure. It is not necessary to employ Kirchhoff's laws or Newton's force law to obtain the equations of electrical and mechanical systems. Rather, the two kinds of systems fall within the scope of this general method. The energy method is particularly useful in dealing with electromechanical systems and with mechanical systems that combine rotation and translation. Nonlinear as well as linear systems can be handled with equal ease. Versatility of the method is shown by its application to various examples, chosen in more or less increasing order of complexity. A set of tables is provided, listing the energy functions for each basic type of electrical, mechanical and electromechanical element. Those charged with teaching students the different disciplines of dynamics and electric circuits should find herein a common meeting ground wherein one general method suffices to yield the necessary differential equations.