Three-Dimensional Oscillations of Suspended Cables Involving Simultaneous Internal Resonances

Abstract

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

“…The results are generalized and thorough parametric studies in each case allow one to get a complete picture of the dynamical solutions. For the 1:2:2 case, our results also complement those of Lee and Perkins [16] by considering all possible excitation frequencies. We also consider the case where both high-frequency modes are excited simultaneously, which renders the analysis more complex as the fundamental solution involves two directly excited modes.…”

“…Complications to the classical 1:2 case have already been considered as it appears in many physical systems such as strings, cables, plates, and shells. Lee and Perkins [16] reported a study on a 1:2:2 resonance occurring in suspended cables between in-plane and out-of-plane modes, and denoted that resonance as a 2:1:1 case. In their study, only one of the two high-frequency modes was excited, and the coupling with the two other modes was studied.…”

A linear curved-beam model for the analysis of galloping in suspended cables

“…First, the classical hypothesis of small sag-to-span ratio d/ is introduced [Luongo et al 1984;Lee and Perkins 1992;Irvine and Caughey 1984], commonly accepted for cables falling into the technical range. Then, advantage is drawn from the fact that the cable is a very slender body; hence, the flexural-torsional effects are expected to be smaller than the funicular effects, except close to the boundaries.…”

“…The first three equations in (16) are identical to those of the flexible cable [Lee and Perkins 1992]. The fourth (16) is also known in the literature, since it represents the moment equilibrium around the tangent of a planar circular arch [Lee and Chao 2000].…”

“…The forces are usually modeled referring to the quasisteady theory, and they depend on the mean wind speed and on the angle of attack, which in turn depends on the velocity of the structure and on its surrounding flow. The structure is generally modeled as a perfectly flexible cable, that is as a one-dimensional continuum capable of translational displacements only [Luongo et al 1984; Lee and Perkins 1992]. This assumption is reliable, since the torsion stiffness of the single cable is usually high and the bending stiffness is negligible, compared to the geometric one, because of the slenderness of the structure.…”