Parametric instability in a stationary string with a periodically varying length is investigated. The free linear vibration of a stationary string with a constant tension, a fixed boundary, and a periodically moving boundary is studied first. 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. 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. A corresponding nonlinear model that consider coupled transverse and longitudinal vibrations of the string are introduced next. Responses and vibratory energies of the linear and nonlinear models are calculated for both stable and unstable cases using the finite difference method. The mechanism of the parametric instability is explained in the linear model and through axial strains in the nonlinear model.

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