Strong nonlinear modal interaction in shallow suspended cables with oscillating ends

This paper examines essentially nonlinear regimes of the stretching-bending interactions in suspended cables with horizontally vibrating ends. Using a special nonlinear transformation of coordinates in the configuration space, the Lagrange function of the cable is written in terms of physical coordinates such as stretching and bending-swinging which possess different time scales. The transformed system is considered as a quasi-linear one with respect to the fast varying stretching generalized coordinate. The relatively slow bending-swinging component of the motion is described by strongly nonlinear equations on an ‘unstretched manifold’. It has been shown that the stretching natural frequency strongly depends on the bending position of the cable. As a result, the cable performs a chaos-like stretching-bending interaction around the resonance region, i.e. when the frequency of shaking is equal to the natural frequency of stretching computed for an undeformed position of the cable. Outside the resonance region the motion has a regular character.