Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity

A new parametric instability phenomenon characterized by infinitely compressed, shocklike waves with a bounded displacement and an unbounded vibratory energy is discovered in a translating string with a constant length and tension and a sinusoidally varying velocity. A novel method based on the wave solutions and the fixed point theory is developed to analyze the instability phenomenon. The phase functions of the wave solutions corresponding to the phases of the sinusoidal part of the translation velocity, when an infinitesimal wave arrives at the left boundary, are established. The period number of a fixed point of a phase function is defined as the number of times that the corresponding infinitesimal wave propagates between the two boundaries before the phase repeats itself. The instability conditions are determined by identifying the regions in a parameter plane where attracting fixed points of the phase functions exist. The period-1 instability regions are analytically obtained, and the period-i $(i>1)$ instability regions are numerically calculated using bifurcation diagrams. The wave patterns corresponding to different instability regions are determined, and the strength of instability corresponding to different period numbers is analyzed.