Forced Response of Translating Media With Variable Length and Tension and a Mass-Spring Termination

The lateral response of vertically translating media with variable length and tension and a mass-spring termination is determined for general initial conditions and external excitation. The translating media are modeled as a taut string and tensioned beams with pinned and fixed boundary at the upper end. In each case a spring-mass sub-system is attached at the lower end of the translating medium. The rates of change of the energies in each case are analyzed and interpreted from the control volume and system viewpoints. The models are used to predict the forced response of cable-car systems in high-speed elevators due to building sway, pulley eccentricity, and guide-rail irregularity. The optimal design of the suspension stiffness of the car is investigated. A resonance condition y(x,t) that leads to dramatically increasing vibratory energy of the cable-car system is identified.