Nonlinear Stay Cable - Bridge Deck Interaction

Nonlinear coupling between stay cables and bridge deck can cause internal resonance. This coupling is due to the length change of the cables when vibrating. During one cycle of transverse motion, the cable length is reduced twice, once for the positive and once for negative displacement. This internal resonance occurs when the natural frequency of the deck is twice the natural frequency of the cable. It is thus also called 2: 1 resonance. Tests have been performed at Empa on a full-size model bridge in the laboratory. The cable frequency has been tuned to half of the deck frequency by attaching additional masses. The bridge was excited by an electromechanical shaker with a frequency near the resonance frequency of the deck. Some interesting phenomena can be observed: (1) Internal resonance only occurs if the natural frequency of the cables is about half the natural frequency of the deck. (2) Internal resonance only occurs for excitation frequencies near the resonance frequency of the deck. Outside a certain frequency range, the cables mainly act as flexible supports for the deck. (3) The transition from one regime to the other takes place at different frequencies for an ascending or a descending frequency sweep. (4) Internal resonance only occurs above a critical level of excitation. (5) Increasing the excitation force above the critical value mainly increases the displacement of the cables while the deck movement is hardly affected (saturation phenomenon). To better understand this unexpected behavior, the system is modelled as a two-degree-of-freedom system including nonlinear coupling. For the steady-state case, this nonlinear system is then solved by perturbation methods. Although the mathematics are a bit involved, the analytical solutions are very helpful for understanding and interpreting the experimental results, which otherwise appear strange sometimes.