Super and Combinatorial Harmonic Response of Flexible Elastic Cables With Small Sag

Abstract

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude ;. Such cables are characterized by a sag-to-chord-length ratio below 0)02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, and , are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the order n, supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency K /n. At moderate ;, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a "rst order perturbation solution to the superharmonic response is performed based on the averaging method. For K(m/n), m(n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m) . Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n"3 and combinatorial harmonics of the order (n, m)"(3,2). All analytical solutions have been veri"ed by direct numerical integration of the modal equations of motion.

Optimal Damping of Stays in Cable-Stayed Bridges for in-Plane Vibrations

“…In case of harmonic resonant excitation, the parametric term will not a!ect the "rst order perturbational solution of the harmonic response [5]. For moderate amplitudes, the response to an in-plane excitation is in plane.…”

Statics of Shallow Inclined Elastic Cables under General Vertical Loads: A Perturbation Approach

“…In this equation, we have used Equation (10); moreover, s (x; y(x)) must be expressed by Equation (11). Equation (13) is known as the funicular equation of the vertical loads (although, usually, the elastic term appearing in the denominator of q is ignored).…”

“…The dynamics of inclined cables are addressed in [10] in the case of taut conditions, using as a key factor the small sag-to-span ratio. In [11], the static balance equations for an inclined cable are addressed in order to obtain the condition for which the dynamic and static axial strains remain sufficiently small, as a prerequisite to study the superharmonic response of a reduced 2 degrees of freedom (d.o.f.) model.…”

The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces

“…Some interesting work on the nonlinear dynamics of cables to the harmonic excitations can be found in the review articles by Rega [2,3]. Nielsen and Kierkegaard [4] investigated simplified models of inclined cables under super and combinatorial harmonic excitation and gave analytical and purely numerical results. Zheng, Ko and Ni [5] considered the super-harmonics and internal resonance of a suspended cable with almost commensurable natural frequencies.…”