Multiple resonances in suspended cables: direct versus reduced-order models

We apply two analytical approaches to construct asymptotic models for the n on-linear three-dimensional responses of an elastic suspended shallow cable to a harmonic excitation. We investigate the case of primary resonance of the first in-plane symmetric mode when it is involved in a one-to-one inter nal resonance with the first antisymmetric in-plane and out-of-plane modes and a two-to-one internal resonance with the first symmetric out-of-plane m ode. First, we apply the method of multiple scales directly to the governin g two integral-partial-differential equations and associated boundary condi tions. Reconstitution of the solvability conditions at second and third ord ers leads to a system of four coupled non-linear complex-valued equations d escribing the modulation of the amplitudes and phases of the interacting mo des. The homogeneous solutions associated with the first in-plane and out-o f-plane modes in the second-order problem are needed to make the reconstitu ted modulation equations derivable from a Lagrangian. However, this procedu re leads to an indeterminacy, indicating a likely inconsistency with this s pecific application of the method of multiple scales. Then, we apply the me thod to a four-degree-of-freedom Galerkin discretized model obtained using the pertinent excited eigenmodes. Again, the homogeneous solutions associat ed with the first in-plane and out-of-plane modes in the second-order probl ems are required to make the reconstituted modulation equations derivable f rom a Lagrangian. Frequency-response curves obtained using the two generate d asymptotic models, for a specific choice of the arbitrary constant appear ing in both models, show different qualitative as well as quantitative pred ictions for some classes of motions. The effects of an inconsistent reconst itution in the direct approach are also investigated. (C) 1999 Published by Elsevier Science Ltd. All rights reserved.