Solutions for the vibration of an axially moving variable length string system: Wave propagation versus space–time finite element predictions

This paper provides numerical and exact solutions for an axially moving string system with variable length by a space–time finite element and a propagating wave model, respectively. Firstly, from the variational form, the dynamic problem for the continuum possessing changing mass is solved by a space–time finite element method. For the problem of a time-varying spatial domain, this finite element method discretizes the spatial and temporal domains simultaneously. Secondly, according to the regularity of propagating wave reflection, an exact solution for a variable-length moving string under uniform motion is derived by a propagating wave method. Subsequently, these two methods proposed are applied to a real-life example, i.e., a high-speed elevator cable. The vibration characteristics of the variable-length moving string with different boundary conditions are analyzed. Compared to the propagating wave method, the space–time finite element method has universality and low computational cost.