Distributed Control of a Vibrating String in Response to Pointwise Force Application

Abstract

The problem of controlling the vibrations of a string by a discrete applied force is considered. The vibrations of the string are modeled by the linear wave equation and the control is provided by an added force term. The wave equation is solved for controlled and uncontrolled cases with and without control force term. The applied force is chosen to be proportional to string displacement at some specified point. In the controlled case; the wave equation involves a control parameter (gain) and related terms involving the value of the displcement at a single point and a delta function. This makes the equation quite different from the usual wave equation. The problem is solved analytically using a modified (compared to usual wave equation) solution procedure and an equation relating the string eigenfrequencies to the proportionality constant (gain) is derived. This allows the observation of the change in eigenfrequencies with the gain. Finally, examples of uncontrolled and controlled responses are presented, graphically. The results show that the resonances can be avoided by the applied control procedure.