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

Abstract

This paper provides numerical and exact solutions for an axially moving string
system with variable length by both a space-time finite element and 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.
Keywords: moving string; space-time finite element; changing mass; time-varying spatial domain; propagating wave

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ORCID for Neil S. Ferguson: orcid.org/0000-0001-5955-7477

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