| Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. In this work, parametric instability regions of a second-order non-dispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions considered here are classified as period-1 and period-i (i > 1) instability regions, where period-1 parametric instability regions are analytically obtained and period-i (i > 1) parametric instability regions can be numerically calculated using bifurcation diagrams. The parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy, due to formation of infinitely compressed shock-like waves. Five non-trivial cases that involve different combinations of string and boundary motions are investigated for their instability regions, wave patterns, and vibratory energy growth. The stable and unstable responses of the linear model in the first case are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models with a slower energy growth rate. |
