A Linear Model of Stationary Elevator Traveling and Compensation Cables

Based on a recent asymptotic analysis of a nonlinear model of elevator traveling and compensation cables, a computationally efficient, linear model is developed for calculating the natural frequencies, mode shapes, and dynamic responses of stationary elevator traveling and compensation cables. The linear cable model consists of two vertical cable segments connected by a half circular loop at the bottom. The two vertical cable segments are modeled as a string with a variable tension due to the weight of the cable. The horizontal displacements of the cable segments consist of boundary induced displacements and internal displacements, where the boundary induced displacements are interpolated from the displacements of the two ends of the cable segments, and the internal displacements satisfy the corresponding homogeneous boundary conditions of the cable segments. The horizontal displacement of the lower loop is interpolated from those of the two lower ends of the two cable segments, and the bending stiffness of the lower loop is represented by a spring with a constant stiffness, which can be calculated from the nonlinear model. Given a car position, the natural frequencies and mode shapes of an elevator traveling or compensation cable are calculated using the linear model and compared with those from the nonlinear model. The calculated natural frequencies are also compared with those from a full-scale experiment. In addition, the dynamic responses of a cable under a boundary excitation are calculated and compared with those from the nonlinear model. There is a good agreement between the predictions from the linear and nonlinear models and between the measured natural frequencies from a full-scale experiment and the corresponding calculated ones.