On the longitudinal vibration of a moving elevator cable-car system

The vibration and noise problem has been a major challenge in the design and installation of an elevator system. While the transverse vibration of a moving hoist cable has been recently studied, the longitudinal vibration of a moving elevator cable-car system has not been studied. In this work, a new mathematical model for the longitudinal vibration of a moving elevator cable-car system with an arbitrary movement profile is developed. The car is modeled as a mass-spring-damper subsystem attached to the lower end of the cable. Both the Newtonian method and Hamilton's principle are used to derive the governing equations of the moving cable-car system. The assumed modes method and a new spatial discretization method are used to discretize either the governing partial differential equation of the moving cable or the Lagrangian of the system before the application of Lagrange's equations. The rate of change of the total mechnaical energy is analyzed from the system viewpoint and the control volume viewpoint. An energy flux is defined to describe the rate of energy transfer at the upper boundary of the cable. Numerical simulation of the responses from three spatial discretization schemes are obtained and compared. A definition of the equilibrium position of the system is discussed, and the dynamic responses around the equilibrium position are obtained. The new spatial discretization method can achieve uniform convergence and overcome the drawbacks of the classical assumed modes method