Transverse Vibration of a Moving String: A Comparison Between the Closed-Form Solution and the Normal-Mode Solution

摘要

There are two physical explanations of the transverse vibration of axially moving strings. One explanation, based on wave propagation theory, leads to a closed-form solution, and the other explanation, based on the interactions of forces on the string, leads to a series solution comprising normal modes of the stationary taut string. This paper presents a quantitative analysis of the relationship between these two solutions, which enables a numerical study on the accuracies of the natural frequencies and vibratory forms of the series solution to be carried out, and hence the accuracies of Galerkin and Rayleigh-Ritz methods which are based upon these series. A reduced-mode method is suggested as an approximate method for obtaining the natural frequencies of the higher order modes of the moving strings. Forced vibration using the closed form solution and the series solution is discussed.