Free-Wave Dispersion Curves of a Multi-Supported String

Abstract

Free-wave propagation of an infinite, tensioned string, supported along its length by repeating segments of multiple spring-mass connections, is examined. The segments can consist of an arbitrary number of different support sets and be of any overall length. Periodicity is intrinsic, since the segments repeat; the goal, though, is to examine what effect variations within the segments have on dispersion. The formulation reveals an unexpected amount of complexity for such a simply posed system. Each support set has independent mass, stiffness, and viscous damping, and the sets are allowed to be offset from one another. A free-wave dispersion formula is derived for two sets of supports (Q = 2) and compared to the well-known ideally periodic expression (Q = 1). A means to obtain general dispersion formulas, for any Q, is discussed. It is shown that the systems’ dispersion curves are primarily governed by the material properties of the string and by the location of the supports.