Vibration of wires used in electro-discharge machining (EDM)

Electro-discharge machining (EDM) is a noncontact process of electrically removing (cutting) material from conductive workpieces. In this process, an axially moving wire melts metallic workpieces by extremely high temperatures caused by sparks between the wire and the workpiece. The transversal vibration of EDM wires can be represented by a nonlinear partial differential equation. The nonlinear mathematical model is that of an axially moving Kirchhoff wire plus terms that take into account the temperature variation along the wire. Using this mathematical model, it is shown that the straight configuration of the wire is stable, i.e., the transversal vibration of the wire decays to zero when the wire axial speed is less than a critical value. The stability of the wire is established by showing that an energy-like (Lyapunov) function corresponding to the wire decays to zero exponentially.