An investigation into wave propagation in hanging chains

Abstract

The study introduced in this work is motivated by the prospect of using a hanging chain as an Acoustic Black Hole (ABH) for passive vibration control. An ABH is effectively a waveguide in which a wave slows progressively as it propagates away from the source enabling it to be extinguished with modest damping. The effect can be achieved by engineering inhomogeneity into a structure's geometry or material, the most common realisation being a beam of tapered thickness. This paper proposes an alternative realisation, that of a chain hanging under its own weight. Such a system has a wave speed that naturally decreases to zero, owing to its linear variation in tension, thus overcoming the challenges of constructing precisely shaped beams with vanishingly thin tips. The study of transverse vibration of hanging chains is a classical problem in structural dynamics. The motion of the chain can be described in terms of Bessel or Hankel functions, which are needed to account for the variation in tension along the chain. In this work, the hanging chain problem is revisited from a wave propagation perspective. An expression is derived for the amplitude of the waves in an infinite chain due a point excitation. From which, the spatial behaviour and the receptances of the waves are evaluated, revealing differing characteristics of upward and downward propagating waves. Some experimental results are presented to support the theoretical analysis.