Analysis of longitudinal oscillations in a vertically moving cable subject to nonclassical boundary conditions

• Published in: Applied mathematical modelling Vol. 111; pp. 44 - 62
• Main Authors: Wang, Jing, van Horssen, Wim T.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.11.2022
• Subjects: Averaging; Boundary excitation; Resonance; Singular perturbation; Time-varying length; Resonance; Averaging; Boundary excitation; Singular perturbation; Time-varying length
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2022.04.021

•A PDE on a time-varying spatial interval with a small harmonic disturbance and a moving mass.•An adapted version of the method of separation of variables.•Averaging methods, singular perturbation techniques, and multiple timescales perturbation methods.•Accurate approximations of the solutions are constructed.•For a given arbitrary excitation frequency, oscillation modes jump up from O(ε)− to O(ε)− amplitudes. In this paper, we study a model of a flexible hoisting system, in which external disturbances exerted on the boundary can induce large vibrations, and so damage to the performance of the system. The dynamics is described by a wave equation on a slow time-varying spatial domain with a small harmonic boundary excitation at one end of the cable, and a moving mass at the other end. Due to the slow variation of the cable length, a singular perturbation problem arises. By using an averaging method, and an interior layer analysis, many resonance manifolds are detected. Further, a three time-scales perturbation method is used to construct formal asymptotic approximations of the solutions. It turns out that for a given boundary disturbance frequency, many oscillation modes jump up from order ε amplitudes to order ε amplitudes, where ε is a small parameter with 0<ε<<1. Finally, numerical simulations are presented to verify the obtained analytical results.