In this paper, we study transverse and longitudinal oscillations and resonances in a hoisting system induced by boundary disturbances. The dynamics can be described by an initial-boundary value problem for a coupled system of nonlinear wave equations on a slowly time-varying spatial domain. It will be shown how the boundary excitations and the nonlinear terms influence transverse and longitudinal vibrations of the system. Firstly, due to the 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 three-timescales 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 of the longitudinal stiffness and the conveyance mass, many oscillation modes jump up from small to large amplitudes in the transverse and longitudinal directions. Finally, numerical simulations are presented to verify the obtained analytical results.

The paper deals with the input-to-state stabilization for the 2×2 system of first-order hyperbolic equations, which convect in opposite directions cascaded with an unstable ODE equation. First, an inverse backstepping transformation is introduced to obtain a target system. Then, by active disturbance rejection control (ADRC) method, the disturbance is estimated via a disturbance estimator with time-varying gain. When the unmatched disturbances are absent, the disturbance estimator is exponentially convergent to the matched disturbance. Furthermore, in order to reject the matched disturbance and obtain the input-to-state stability of the system, the controller is proposed by using the disturbance estimator. Finally, numerical simulations are presented to validate theoretical results.

Varying-length cable systems are widely applied in a vast class of engineering problems which arise in industrial, civil, aerospatial, mechanical, and automotive applications. Due to external excitations, large oscillations can occur when cables are lifted up or down. This phenomenon is caused by resonance. In general, resonance is harmful, and can cause significient deformations and dynamic stresses in machinery and structures, and even can lead to accidents. Therefore, this doctoral dissertation is devoted to the study of transverse and longitudinal resonance phenomena and output feedback stabilization of varying-length cables....

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

In this paper, we consider the output feedback stabilisation of an axially moving string system subject to a spring-mass-dashpot boundary condition. By constructing an invertible backstepping transformation, we design a state feedback controller to stabilise the system. Next, we present an observer to estimate the states of the system, and based on the estimated states, we design an output-feedback controller. The closed-loop system is proved to be exponentially stable by Lyapunov analysis. Numerical simulations are presented to verify the effectiveness of the proposed controller.

In this paper an initial–boundary value problem on a bounded, fixed interval is considered for a one-dimensional and forced string equation subjected to a Dirichlet boundary condition at one end of the string and a Robin boundary condition with a slowly varying time-dependent coefficient at the other end of the string. This problem may serve as a simplified model describing transverse or longitudinal vibrations as well as resonances in axially moving cables for which the length changes in time. By introducing an adapted version of the method of separation of variables, by using averaging and singular perturbation techniques, and by finally using a three time-scales perturbation method, resonances in the problem are detected and accurate, analytical approximations of the solutions of the problem are constructed. It will turn out that small order ɛ excitations can lead to order ɛ responses when the frequency of the external force satisfies certain conditions. Finally, numerical simulations are presented, which are in full agreement with the obtained analytical results.