Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order nondispersive 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 without spatially discretizing the governing partial differential equation. There are five nontrivial cases that involve different combinations of string and boundary motions: (I) a translating string with a sinusoidally varying velocity and two stationary boundaries; (II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; (III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; (IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and (V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-$i$ (⁠$i>1$⁠) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-$i$ (⁠$i>1$⁠) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-$i$ (⁠$i>1$⁠) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, 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. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-$i$ (⁠$i≥1$⁠) parametric instability region corresponding to an integer $k$⁠, an initial smooth wave will be infinitely compressed to $k$ shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I 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.