On low-frequency vibrations of a composite string with contrast properties for energy scavenging fabric devices

Free vibrations of a two-component string with high-contrast material parameters are considered at different boundary conditions to illustrate the very low-frequency energy harvesting capability of fabric devices. It is revealed that, only for the case of mixed boundary conditions, low-frequency (locally) almost rigid-body vibrations are admissible, provided that material parameter ratios lie in some well defined interval. A low-frequency perturbation procedure is carried out to determine the eigenfrequencies as well as the eigenforms. The analysis is extended to a piecewise inhomogeneous string and to a string supported on an elastic foundation. It is shown that both situations may still admit low-frequency vibrations, under certain restrictions on the material properties. This is particularly remarkable given that the situation of elastic support normally possesses two nonzero cutoff frequencies. The results may be especially relevant for energy scavenging fabric devices, where very low-frequency (< 10 Hz) mechanical vibrations of textile fibers are harvested through friction.