Exact Response of a Translating String With Arbitrarily Varying Length Under General Excitation

Abstract

The exact response of a translating string with constant tension and arbitrarily varying length is determined under general initial conditions and external excitation. The governing equation is transformed to a standard hyperbolic equation using characteristic transformation. The domain of interest for the transformed equation is divided into groups of subdomains according to the properties of wave propagation. d’Alembert’s solution for any point in the zeroth subdomain group is obtained by using the initial conditions. The solution is extended to the whole domain of interest by using the boundary conditions, and a recursive mapping is found for the solution in the second and higher groups of subdomains. The least upper bound of the displacement of the freely vibrating string is obtained for an arbitrary movement profile. The forced response of the string with nonhomogeneous boundary conditions is obtained using a transformation method and the direct wave method. A new method is used to derive the rate of change of the vibratory energy of the translating string from the system viewpoint. Three different approaches are used to derive and interpret the rate of change of the vibratory energy of the string within a control volume, and the energy growth mechanism of the string during retraction is elucidated. The solution methods are applied to a moving elevator cable with variable length. An interesting parametric instability phenomenon in a translating string with sinusoidally varying length is discovered.