M. Pakdemirli et al., ANALYSIS OF ONE-TO-ONE AUTOPARAMETRIC RESONANCES IN CABLES - DISCRETIZATION VS DIRECT TREATMENT, Nonlinear dynamics, 8(1), 1995, pp. 65-83
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.