Stable and Unstable Nonlinear Resonant Response of Hanging Chains: Theory and Experiment

The three-dimensional nonlinear dynamics of a hanging chain, driven by harmonic excitation at the top, are studied first analytically and numerically, and then experimentally. Asymptotic results demonstrate a sensitive dependence on excitation frequency and amplitude. For moderately large excitation amplitudes there are distinct regions of stable two-dimensional and stable three-dimensional response as function of frequency, as well as a distinct region in which all steady-state solutions are unstable. Numerical results were obtained to verify the asymptotic solutions and investigate the dynamics within the irregular response region. Numerical results for even larger excitation amplitudes showed that large impulse-like tension forces cause the chain to lose tension over a region adjacent to its freely hanging end, and then collapse. Following the collapse, the chain configuration intersects itself. Experimental results confirm qualitatively and quantitatively the theoretical predictions. The experimental results also demonstrate the loss of tension and subsequent collapse of the chain at the predicted excitation amplitudes, as well as the intersection of the chain with itself.