Generalized-α time integration solutions for hanging chain dynamics

In this paper, we study numerically the two- and three-dimensional nonlinear dynamic response of a chain hanging under its own weight. Previous authors have employed the box method, a finite-difference scheme popular in cable dynamics problems, for this purpose. The box method has significant stability problems, however, and thus is not well suited to this highly nonlinear problem. We illustrate these stability problems and propose a new time integration procedure based on the generalized-alpha method. The new method exhibits superior stability properties compared to the box method and other algorithms such as backward differences and trapezoidal rule. Of four time integration methods tested, the generalized-alpha algorithm was the only method that produced a stable solution for the three-dimensional whirling motions of a hanging chain driven by harmonic linear horizontal motion at the top.