State Variable Approach to a Second-Order System

The main reason for presenting this paper is to stress for educators the interpretive information that can be gained concerning the response of a second-order system when the solution is cast in a state variable form, with results plotted as state plane trajectories. Although the treatment is more exhaustive than usual for a beginning undergraduate, a wealth of engineering insight becomes available which is normally lost. Two such insights may be noted: the fact that the underdamped response is closely related to the geometry of an ellipse whose dimensions are determined by the system parameters and the initial conditions; and the fact that critical damping is geometrically more logically viewed as a limit of the underdamped response, although in a purely formal sense the critically damped solution is derived equally well from either the under-damped or overdamped solution. Other interesting information easily obtained concerns certain restrictive rules governing the locations and shapes of the trajectories for the overdamped case; rules which become clearly evident in the state plane portrayal, but which are not evident from an analytical formulation.