Parametric Instability in a Taut String With a Periodically Moving Boundary

Parametric instability in a taut string with a periodically moving boundary, which is governed by a one-dimensional wave equation with a periodically varying domain, is investigated. Parametric instability usually occurs when coefficients in governing differential equations of a system periodically vary, and the system is said to be parametrically excited. Since the governing partial differential equation of the string with a periodically moving boundary can be transformed to one with a fixed domain and periodically varying coefficients, the string is parametrically excited and instability caused by the periodically moving boundary is classified as parametric instability. The free linear vibration of a taut string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. The exact response of the linear model is obtained using the wave or d'Alembert solution. The parametric instability in the string features a bounded displacement and an unbounded vibratory energy, and parametric instability regions in the parameter plane are classified as period-$i$ (⁠$i≥1$⁠) parametric instability regions, where period-1 parametric instability regions are analytically obtained using the wave solution and the fixed point theory, and period-$i$ (⁠$i>1$⁠) parametric instability regions are numerically calculated using bifurcation diagrams. If the periodic boundary movement profile of the string satisfies certain condition, only period-1 parametric instability regions exist. However, parametric instability regions with higher period numbers can exist for a general periodic boundary movement profile. Three corresponding nonlinear models that consider coupled transverse and longitudinal vibrations of the string, only the transverse vibration, and coupled transverse and axial vibrations are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using three numerical methods: Galerkin's method, the explicit finite difference method, and the implicit finite difference method; advantages and disadvantages of each method are discussed. Numerical results for the linear model can be verified using the exact wave solution, and those for the nonlinear models are compared with each other since there are no exact solutions for them. It is shown that for parameters in the parametric instability regions of the linear model, the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models. The mechanism of the parametric instability is explained in the linear model and through axial strains in the third nonlinear model.