Evading instabilities in spring-mass chains with time-modulated stiffnesses

The vibrations of a one-dimensional elastic continuum or a string under tension are classic examples of harmonic analysis, with normal modes taking the form of standing waves which are sinusoidal in space and time. Less widely known are the modes of systems in which the global stiffness or tension is sinusoidally varied in time. In such systems, standing waves still exist, but their temporal evolution is described by Mathieu functions which become unstable at certain wavelengths. We investigate the vibrations of a periodic spring-mass chain with time-modulated spring stiffnesses, using classical dynamics simulations implemented in HOOMD-Blue software. Upon initializing the system in standing waves of different wavelengths and tracking the subsequent evolution, we obtain quantitative agreement of trajectories with predicted Mathieu functions. By analyzing the relationships between system properties and stability of standing waves, we find parameter combinations for which the discrete system has no unstable modes, in contrast to the continuum equivalents which generically harbor instabilities. Our results show that discrete time-modulated systems can be dynamically stable without requiring losses or damping.