Transverse Vibration Of An Axially Moving String With Direct Time Delayed Position Feedback

Transverse vibration of axially moving strings is involved in many engineering devices such as power transmission belts, thread lines, aerial cable tramways and aether lifts. It is shown that time delay is inevitably in controllers. And the time delay often affects the dynamics of systems essentially which can not only affect the stability of systems, but also lead to the complex dynamics of systems. However, an artificially introduced delay in the feedback can play an essential role in stabilizing the transverse vibration of axially moving strings.The local dynamics of an axially moving string under aerodynamic forces are investigated with a time-delayed position feedback controller. The research work can be summarized as follows:(1) The dynamical model of transverse vibration of axially moving strings with a time-delayed position feedback controller is established. The difference-differential governing equation is obtained in modal coordinates of a two-degree-of-freedom system through the Galerkin's discrete procedure.(2) The Belair Theorem is advanced to a more generalized theorem for any polynomial-exponential equations with constant time delay. It is proved that as the time delay varies, the number of solutions of the characteristic equation can only be changed when the eigenvalue passes through the imaginary axis. The Hopf bifurcation curves are presented in the space of controlling parameter. Two different kinds of periodic solutions are reported.(3) A new delayed system is obtained by adding nonlinear delayed position feedback and external excitation to the original system. With the aid of the center manifold reduction, a functional analysis is carried out to reduce the modal equation to a single ordinary differential equation in one complex variable on the center manifold. The approximate analytical solutions in the vicinity of Hopf bifurcations are derived in the case of primary resonance. A periodic solution expressed in the closed form is found to be in good agreement with that obtained by numerical simulation. A Poincare section is defined to find the stability of periodic solutions. Two different kinds of quasi-periodic solutions are reported.