Non-linear dynamic response of a cable system with a tuned mass damper to stochastic base excitation via equivalent linearization technique

Abstract: Non-linear dynamic model of a cable–mass system with a transverse tuned mass damper is considered. The system is moving in a vertical host structure therefore the cable length varies slowly over time. Under the time-dependent external loads the sway of host structure with low frequencies and high amplitudes can be observed. That yields the base excitation which in turn results in the excitation of a cable system. The original model is governed by a system of non-linear partial differential equations with corresponding boundary conditions defined in a slowly time-variant space domain. To discretise the continuous model the Galerkin method is used. The assumption of the analysis is that the lateral displacements of the cable are coupled with its longitudinal elastic stretching. This brings the quadratic couplings between the longitudinal and transverse modes and cubic nonlinear terms due to the couplings between the transverse modes. To mitigate the dynamic response of the cable in the resonance region the tuned mass damper is applied. The stochastic base excitation, assumed as a narrow-band process mean-square equivalent to the harmonic process, is idealized with the aid of two linear filters: one second-order and one first-order. To determine the stochastic response the equivalent linearization technique is used. Mean values and variances of particular random state variable have been calculated numerically under various operational conditions. The stochastic results have been compared with the deterministic response to a harmonic process base excitation

Effects of modeling nonlinearity in cross-ties on the dynamics of simplified in-plane cable networks

Cross-ties are often employed as passive devices for the mitigation of stay-cable vibrations, which have been observed in the field under the excitation of wind and windrain. In-plane cable networks are structural systems derived by interconnecting several stays through transverse cross-ties. This study was motivated by a recent research activity aimed at the study of the free-vibration dynamics for in-plane cable networks. Even though dynamic models for the analysis of the network vibration had been proposed by one of the writers in previous studies, a linear dynamic modeling of the system had been utilized. In this paper, the use of a nonlinear element was introduced to describe the nonlinear behavior of the cross-tie and to account for an 'imperfect' transfer of the restoring force mechanism at the anchorages (collars) between the cross-tie and the stay. The goal of this model is to simulate, perhaps more realistically, failure onset at the anchorages, sometimes experienced on some bridges. The solution to the free-vibration problem for two simplified cable networks was determined by energy-based 'equivalent linearization' (EL), which simulates the nonlinear response in the restrainer through an equivalent linear restoring force component (amplitude-dependent). The first system consists of one stay with one cross-tie, anchoring the cable to the deck; the second system is a double-cable network with nonlinear cross-tie. Performance of both systems was analyzed. Investigation was restricted to the fundamental mode and some of the higher ones. A time-domain lumped-mass algorithm was utilized for the validation of the EL method

Generalized power-law stiffness model for nonlinear dynamics of in-plane cable networks

Cross-ties are used for mitigating stay-cable vibration, induced by wind and wind-rain on cable-stayed bridges. In-plane cable networks are obtained by connecting the stays by transverse cross-ties. While taut-cable theory has been traditionally employed for simulating the dynamics of cable networks, the use of a nonlinear restoring-force discrete element in each cross-tie has been recently proposed to more realistically replicate the network vibration when snapping or slackening of the restrainer may be anticipated. The solution to the free-vibration dynamics can be determined by "equivalent linearization method". In an exploratory study by the authors a cubic-stiffness spring element, in parallel with a linear one, was used to analyze the restoring-force effect in a cross-tie on the nonlinear dynamics of two simplified systems. This preliminary investigation is generalized in this paper by considering a power-law stiffness model with a generic integer exponent and applied to a prototype network installed on an existing bridge. The study is restricted to the fundamental mode and some of the higher ones. A time-domain lumped-mass algorithm is used for validating the equivalent linearization method. For the prototype network with quadratic-stiffness spring and a positive stiffness coefficient, a stiffening effect is observed, with a ten percent increment in the equivalent frequency for the fundamental mode. Results also show dependency on vibration amplitude. For higher modes the equivalent nonlinear effects can be responsible for an alteration of the linear mode shapes and a transition from a "localized mode" to a "global mode"

Nonlinear Computer Model for the Simulation of Lock-in Vibration on Long-Span Bridges

The susceptibility of modern bridges to vortex-shedding-induced vibration is a major concern for researchers and designers. The relevance of this phenomenon is associated with the onset of large-amplitude aeroelastic vibration at moderate wind-velocity regimes due to synchronization, that is, lock-in, of the vortex shedding frequencies with those corresponding to the natural modes of the structure. Recent observations, either recorded during the monitoring of full-scale bridges or during experimental tests of deck models in wind tunnels, confirm the importance of these aspects during the operational life of the structure. In this article, a computer model for the simulation of the aeroelastic loading associated with vortex shedding in lock-in regime is presented, for a direct application to dynamic analysis of long-span bridges. This approach is based on earlier work focused on the response of slender vertical cylindrical chimneys to vortex-shedding excitation, which is here extended to noncircular cross sections. The numerical model was employed in conjunction with a finite-element code for time-domain nonlinear simulation of the structural dynamic response. The validation of the procedure is performed through numerical simulation, conducted on two specific bridge examples