ANALYSIS OF ONE-TO-ONE AUTOPARAMETRIC RESONANCES IN CABLES - DISCRETIZATION VS DIRECT TREATMENT

We discuss solution methods for nonlinear vibrations of cables having small initial sag-to-span ratios. One-to-one internal resonances betwe en the in-plane and out-of-plane modes as well as primary resonances o f the in-plane mode are considered. Approximate solutions are obtained by two different approaches. In the first approach, the method of mul tiple scales is applied directly to the governing partial-differential equations and boundary conditions. In the second approach, the equati ons are first discretized, and then the method of multiple scales is a pplied to the resulting ordinary-differential equations. It is shown t hat treatment of the discretized system is inaccurate compared to dire ct treatment of the partial-differential system. Discrepancies between the two solutions appear even at the first level of approximation. St ability analyses of the amplitude and phase modulation equations for b oth methods are also performed.