The Vibration Research Of The Cable-stayed Beam Systems

Long-span cable-stayed bridges have been extensively employed in the last fourdecades due to their inherent mechanical effectiveness, economical design, andesthetic appearance. However, long cable-stayed bridges with high flexibility and lowstructural damping are prone to wind-, rain/wind-, earthquake-, and traffic-inducedvibrations. Determination of the dynamic behavior of these structures undertime-varying loads has become an important research area. There have been twoattempts to explain the large-amplitude vibrations of the stay cables. First,rain-/wind-induced galloping was suggested. The excitation mechanism of therain-/wind-induced vibration was subsequently investigated by many researchers. Aseries of wind tunnel tests have been conducted to investigate the excitationmechanism and to find an efficient countermeasure to suppress it. Second, when thegirders and/or the towers vibrate, the stay cables are excited by the support motions.A cable-stayed bridge consists of many cable-beam structures. Based on theHamilton principle, the continuous dynamical model of cable-stayed beam are derived.Thus, the equations of motion together with the boundary and connection conditionsexpressed in a non-dimensional form, linearized around the trivial equilibrium con-figuration neglecting the structural damping, are reduced to a complete self-adjointedboundary-value problem. Then, using a transfer matrix method, we can obtain anefficient and simplified computation of the eigensolutions of the model of acable-stayed bridge. According to the introduction of the piecewise function of themode shapes, the occurrence of global and local modes is influenced by themechanical characteristics. Then, considering the different subcases of thecable-stayed bridge, a parametric investigation is studied. The contents of this thesisare as follows:1. Based on the extensively reviewing literatures, the modeling and nonlineardynamics of the cable-beam structures (the representative of the flexible engineeringstructure) are stated.2. Considering the continuity conditions, the3D nonlinear motion equations ofthe cable-stayed beam are obtained via the Hamilton principle, and the in-plane andout-of-plane eigenvalue problems are investigated. Applying the quasi-staticassumption and the static configuration, the condensed model of the cable-stayed beam is derived. Then, the free vibration analysis is performed to obtain theclosed-form eigenvalue solutions for the linear problems. The effects of the stiffnessratio and sag-to-span ratio on the in-plane and out-of-plane natural frequencies of thecable-stayed beam are systematically investigated.3. In order to reflect the properties of the cable-stayed bridge better, a morecomplex cable-stayed bridge model with eight cables is investigated. We can treat thecable-stayed bridge as a continuous multi-span beam with multi-cable. By introducingthe transfer matrix to the cable-stayed bridges firstly, we can solve the eigenvalueproblem. In particular, the order of the determinant to obtain the eigenvalues does notincrease as the number of cables of the cable-stayed bridge increases. With thetransfer matrix, the eigenvalue problems of both the in-plane and out-of-planemotions can be not only solved, but also simplified. We can obtain the mode shapes ofthe cables and beam conveniently according to the introduction of the piecewisefunction. Following, the effect of the parameters and the close natural frequencies arediscussed according to the mode shapes.4. The model of suspension bridge is considered to the cable and beam withmany discrete springs. Based on Hamilton’s variational principle, the motions of thesuspension bridge are investigated. Considering the static and dynamicconfiguration of the system and using the transfer matrix for the suspension bridge,the eigenvalue problem can be solved. In particular, there are only six unknownconstants of the system to be determined. Following, the effects of the parameters arediscussed. Finally, a comparison of the obtained values with the finite element methodresults is made.5. We investigate the nonlinear modal properties of the cable-stayed beam withthe direct approach and the discretization approach. The effects of the nonlinearcoupling term are also studied. It is worth noting that the coupling term make ameaningful contribution of the nonlinear characteristics of the modal properties.Following, a discussion about the existence and stability of coupled nonlinearnormal modes is presented.The results show that: The advantage of the transfer matrix method is to decreasethe undetermined coefficients of the matrix comparing with other methods, such asthe finite element methods. Moreover, the theoretical results also show that thereduced characteristic equation, which corresponds to the frequency crossover, isindependent of the parameters. Remarkably, there are some differences between thecurve veering phenomenon here and that in other reference, is that two frequency loci are close and paralleled for a relatively big range. When coming to the repeatedfrequency, the middle curve veer away from the beginning curve abruptly and close tothe third curve. Due to the existence of the nonlinear coupling term of thecable-stayed beam, the coupling term affect not only the natural frequency of thesystem but also the amplitude. This study can be theoretical basis in practicalengineering applications.