NONLINEAR DYNAMIC RESPONSE AND ACTIVE VIBRATION CONTROL OF THE VISCOELASTIC CABLE WITH SMALL SAG

The problem considered is an initially stressed viscoelastic cable with small sag.The cable material is assumed to be constituted by the hereditary differential type. Thepartial differential equations of motion is derived first. Then by applying Galerkin'smethod, the governing equations are reduced to a set of third order nonlinear ordinarydifferential equations which are solved by Runge-Kutta numerical integration procedures.Only after the transverse vibration of the plane is considered and the nonlinear terms areneglected, can the nonlinear ordinary differential equations be expressed as a continuousstate equation in the state space. The matrix of state transition is approximated stepwise bythe matrix exponential; in addition, the state equation is discretized to a difference equationto improve the computing efficiency. Furthermore, an optimal control of procedure systembased on the minimization of a quadratic performance index for state vector and controlforces is developed. Finally, the effect of dynamic response of the cable, which is produced by viscoelastic parameters, is testified by the research of digital simulation.