Abstract Summary
The study introduced in this work focuses on the possible use of a hanging chain as an Acoustic Black Hole (ABH) for a passive vibration control solution. An acoustic black hole is effectively a waveguide in which a wave propagates away from the source and does not return. It can be achieved by exploiting the effect of inhomogeneity obtained by a variation in the structure geometry or material. One way of achieving the ABH effect in waveguide is by a local stiffness reduction combined with a local increase in damping. This can be achieved, for instance, by reducing the thickness of a waveguide, such as in a beam accompanied by a gradual increase in damping along the beam. This results in a significant reduction in the speed of a damped propagating wave. Hence there is very little reflection of the wave at the end of the beam. An alternative to using a beam is to use a hanging chain. This is a system that has a decreasing the wave speed naturally without having to change its geometry and thus it overcomes the constructive challenges inherent to an ABH realized by using a beam. The reduction of the local stiffness is obtained by the effect of gravity. This effect decreases linearly from the top, where the chain is fixed, to the bottom, where the wave speed tends to zero. The study of transverse vibration on hanging chains is a classical problem in structural dynamics that can be analysed as a string-like structure in which the tension is a function of gravity, and hence reduces from the top to the bottom of the chain. The motion of the chain can be described in terms of Bessel or Hankel functions, which are needed to account for the variation of tension along the position in the chain. However, both Bessel and Hankel functions have numerical singularities that generally affect the solutions at the boundary’s conditions, which must be handled carefully. In this work, the hanging chain problem has been revisited in terms of free and forced vibration. Using the wave approach, infinite and finite chains are investigated. The receptance and dynamic stiffness matrices are obtained, which facilitate the understanding of the asymptotic behaviour at low and high frequencies. The formulation presented allows other mechanical elements to be attached to the chain using the dynamic stiffness approach. Some experimental results are presented to support the theoretical analysis.
