Nonlinear Vibration Analysis of Flexible Hoisting Rope with Time-Varying Length

The nonlinear vibration of a flexible hoisting rope with time-varying length and axial velocity is investigated. The flexible hoisting rope is modeled as a taut translating string with a rigid body attached at its low end. A systematic procedure for deriving the system model of a flexible hoisting rope with time-varying length and axial velocity is presented. The governing equations were developed by employing the extended Hamilton's principle considering coupling of axial movement and flexural deformation of the rope. The derived governing equations are nonlinear partial differential equations(PDEs) with time-varying coefficients. The Galerkin's method and the 4th Runge-Kutta method were employed to numerically analyze the resulting equations. Further, the dynamic stability of the flexible hoisting rope was investigated according to the Lyapunov stability theory. The motions of an elevator hoisting system were presented to illustrate the proposed mathematical models. The results of simulation show that the dynamic motions of the flexible hoisting string are stable during downward movement but are unstable during upward movement. The proposed systematic procedures in analyzing the dynamic stability can facilitate further development in dynamic control of the flexible hoisting system in practice.