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Summary Motivated by an engineering application in cable mining elevators, we address a new problem on stabilization of 2×2 coupled linear first‐order hyperbolic PDEs sandwiched between 2 ODEs. A novel methology combining PDE backstepping and ODE backstepping is proposed to derive a state‐feedback controller without high differential terms. The well‐posedness and invertibility properties of the PDE backstepping transformation are proved. All states, including coupled linear hyperbolic PDEs and 2 ODEs, are included in the closed‐loop exponential stability analysis. Moreover, boundedness and exponential convergence of the designed controller are proved. The performance is investigated via numerical simulation.

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Lifting up a cage with miners via a mining cable causes axial vibrations of the cable. These vibration dynamics can be described by a coupled wave partial differential equation-ordinary differential equation (PDE-ODE) system with a Neumann interconnection on a time-varying spatial domain. Such a system is actuated not at the moving cage boundary, but at a separate fixed boundary where a hydraulic actuator acts on a floating sheave. In this paper, an observer-based output-feedback control law for the suppression of the axial vibration in the varying-length mining cable is designed by the backstepping method. The control law is obtained through the estimated distributed vibration displacements constructed via available boundary measurements. The exponential stability of the closed-loop system with the output-feedback control law is shown by Lyapunov analysis. The performance of the proposed controller is investigated via numerical simulation, which illustrates the effective vibration suppression with the fast convergence of the observer error.

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The aim of this article is to investigate hysteretic damping characteristics of a typical tensioner used in engine accessory drive systems. An experiment device is developed to measure the friction coefficients of three contact pairs within the tensioner. Statistic results of test data show that the friction coefficient is linearly dependent on normal forces, and thus a linear function is used to describe it. An exact mathematical model and an accurate three-dimensional finite element model are proposed in this study to calculate the relationship of friction torque and rotation angle as well as the damping characteristics of the tensioner. The mathematical model and three-dimensional finite element model are verified through an experiment. Comparison indicates that both the mathematical and finite element model can accurately predict the working torque of the tensioner during operation process, while the finite element model has better accuracy in predicting the damping characteristics than the mathematical model.

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