Abstract Summary
In this paper, transverse and longitudinal vibrations and resonances in an elevator cable system induced by boundary excitations are studied. The dynamics can be described by an initial-boundary value problem for a (coupled system of) nonlinear wave-equation(s) on a relatively slowly time-varying spatial domain. It will be shown how boundary excitations and nonlinear terms influence transverse and longitudinal oscillations in the system. Firstly, due to the relatively slow variation of the cable length, a singular perturbation problem arises. By using an interior layer analysis many resonance manifolds are detected. Secondly, it will be shown that resonances in the system are caused not only by boundary disturbances but also by nonlinear interactions. Based on these observations, a multiple time-scales perturbation method is used to approximate the solution of the initial-boundary value problem analytically. It turns out that for special frequencies in the boundary excitations and for certain parameter values for the longitudinal stiffness and the conveyance mass, many oscillation modes jump up from small to large amplitudes in the transverse and longitudinal directions. Numerical simulations are presented to verify the obtained analytical results.
