Study On Nonlinear Vibration Of Cable-stayed Bridges Based On The Cable-beam/Shallow Arch Model

Cable-stayed bridges as a composite structure consist of the bridge tower,bridge deck and cables,which has advantages of good stress system,mature construction technique,good economic performance.Hence,cable-stayed bridges are widely built and have the pivotal position in bridge engineering in the world.In order to increase the span,some new materials that are light-weight and high-strength are used to design the cable and the bridge deck,which induces the decreasing of entire stiffness of the bridge.For such a flexible system,the cable and bridge deck may generate the internal resonance when subjected to a small disturbance,and further induce the large vibration and threat the safety of traffic and bridge.So far,the theoretical research work on the cable-stayed bridge are mainly focused on the single cable model and the cable-beam model consisting of a single cable and beam,while lack of the research on more complicated model,such as multi-cable bridge deck model.The theoretical mechanism of many nonlinear dynamic behaviors between different cables and bridge deck are unknown.Moreover,the geometrical nonlinearity of bridge deck has not been considered.To bridge this research gap,the following works are carried out:(1)Based on the classic vibration theory of cable and beam,combining with the coupling conditions between the cable and the beam,the in-plane and out-of-plane vibration equation of the multi-cable beam are established.The eigen-frequency and modal expression are obtained,and the parametric study of tension,inclined degree,mass ratio and stiffness ration between cable and beam are conducted.The Veering phenomenon between the adjacent frequencies are observed.The occurrence of global,local and coupled mode are affected by the mass ratio and stiffness ration between cable and beam are conducted.(2)By using the stationary functional method,the in-plane single-degree-offreedom vibration equations of a multi-cable beam model is established.The Galerkin method and multi-scale method are applied to deduce the modulation equations governing the amplitude and phase.The dynamic responses of the system subjected to single and multiple frequency harmonic excitations are investigated.The frequency response,amplitude response,phase plane,time history and power spectrum are presented to explore the system's dynamic response.The system's frequency response has the harden spring property and affected by the quadratic and cubic nonlinearities caused by the cable sag and the cable stretch.(3)Considering the geometrical nonlinearity of the bridge deck,a multi-cable shallow arch model is established.The in-plane eigen-frequency and modal expression are obtained.The parametric study of arch rise,inclined degree and materials of cable on the eigen-value problem are investigated.The results show that increasing the rise in a certain range can only affect a certain frequency and others are not affected.(4)Based on the vibration equations of the cable and shallow arch,applying the Galerkin discretization and multi-scale method,an in-plane nonlinear multi-cable shallow arch model is established.Three excitation cases that loads are acted on the shallow arch,cable and cable's boundary are considered respectively.The one-to-oneto-one and two-to-one-to-two internal resonance among cable to shallow arch to cable are studied.The frequency response,amplitude response,phase plane,bifurcation diagram and time history are presented to explore the dynamic response of the system.The results show that the frequency response of the shallow arch always behaves the soften spring property,while the cable has the soften and harden spring properties.At the saddle-node bifurcation,the shallow arch and cables have converse jump phenomena.Moreover,the cable's response decreases with the increase in amplitude of the external excitation,which shows the complex dynamic behaviors.(5)The in-plane nonlinear dynamic response of the multi-cable shallow arch model is investigated when the arch is subjected to the double-frequency excitation.Three simultaneous resonance cases that primary resonance with the addition of Order 1/3,1/2 subharmonic resonance and Order 3 super-harmonic resonance are considered.The results show that the addition of a subharmonic or super-harmonic resonance to a primary resonance can increase the arch's response,which depends on the external detuning parameter,but hardly affect the cable's response.