Analysis of the spatial non-linear vibrations of a string

Abstract

The problem of the free and forced quasilinear spatial vibrations of a string is investigated in the single-mode approximation of wave processes. A mathematical model is constructed in which geometric non-linearity, caused by the linear extensibility of a string, is the major source of non-linearity. As a result. the tension turns out to be variable both with respect to time and length. An asymptotic analysis of the free vibrations is carried out and the phenomenon of the instability of plane vibrations is investigated. The resonance curves corresponding to a plane harmonic excitation are constructed in terms of the system parameters and analysed. The stability of steady vibrations is completely investigated using Lyapunov's first method. Qualitative effects. associated with the stability and instability of plane and spatial forced vibrations. are detected and studied within the framework of the spatial model.