Theoretical Research Of Synchronous And Asynchronous Vibrations Of Double-Cable

For the load-bearing structure of cable,the vibration of cable brings great harm and attract much attention of researchers.For the vibration phase of the cable,if each cable is in the synchronous vibration state(in phase),that is,the direction of the inertia force of the cable is coincident,then the effect of the cable on the anchoring structure will instantly increase several times,which causes serious damage to the anchorage structure;if each cable is in asynchronous vibration state(out of phase),it is easy to cause the collision between adjacent cables and reduced the service life of the cable structure greatly.Therefore,our paper focuses on the transient phase-frequency characteristics of cables when they are disturbed by external interference from the theoretical point of view.By varying the parameters,different types of cable are studied.The research results are presented as follows:(1)For horizontal cables,the nonlinear equations of in-plane and out-of-plane motion of the horizontal string structure without considering sag,the horizontal suspension cable structure with considering sag,and the horizontal suspension cable structure with considering bending stiffness are respectively established by using Hamilton’s principle.Firstly,the equation of motion is treated dimensionless,and Galerkin method is used to obtain the discrete ordinary differential equation without considering the internal resonance.The first two approximate solutions under the resonance response of the recipient are obtained by the multi-scale perturbation method.Finally,Hilbert transform is applied to the approximate solution to obtain the instantaneous phase of response.The results show that there is only one order in the in-plane and out-plane approximation without considering the sag of the horizontal string.The instantaneous phase difference between the response and the excitation is constant.Its phase frequency curve is bent due to the influence of non-linearity,and there are two jump points of phase difference,the corresponding frequency of which is the same as that of the jump points of the amplitude-frequency curve.When the effect of sag is considered,there are two orders in the approximate solution.Under certain parameters,the influence of the higher order approximation term is great,which makes the instantaneous phase difference between the response and the excitation change periodically with time.The magnitude of the variation amplitude is related to the drift term in the higher order approximation term.The instantaneous phase difference is larger when the excitation is distributed in a certain Irvine parameter region.However,under the end excitation,the amplitude of the instantaneous phase difference is smaller in the larger parameter region,and increases sharply to more than 0.5π only under the specific excitation frequency and cable parameters.The bending stiffness Narrows the Irvine parameter range corresponding to the area where the amplitude of the instantaneous phase difference increases sharply under distributed excitation,and has little effect on the value of the instantaneous phase difference under end excitation.(2)For the stay cables,the distributed excitation and the end excitation are respectively studied.The nonlinear motion equations of single cable under two excitation conditions are established.Based on the multi-scale method,the law and reason of the instantaneous phase difference caused by the higher order approximation term are introduced.The results show that the phase difference between the out-of-plane primary resonance response and the excitation is constant,while the instantaneous phase difference between the in-plane response and the excitation is related to the elastic parameters and sag of the cable,etc.Small parameter changes may lead to obvious changes in the instantaneous phase-frequency characteristics.High order approximation effect on instantaneous phase,mainly reflected in two paragraphs frequency doubling and drift and drift amplitude ratio,the former makes response instantaneous phase appears twice in a single cycle of plus or minus alternately,the latter decided to in-plane response and motivate the maximum instantaneous phase difference and its change law of the ratio can be used as the parameters of the qualitative analysis to the instantaneous phase difference;In the case of distributed excitation,the variation trend of instantaneous phase difference of the stay cable is similar to that of the horizontal cable,but there is a certain difference in the case of end excitation.(3)Based on the study of instantaneous phase frequency of single cable,a mechanical model of mass block of double horizontal cable under axial parameter excitation is established.The nonlinear motion equations of double cable and mass are obtained by Newton’s theorem.The first two order approximate solutions of double cable and mass block are obtained by using the multi-scale method.The transient phase-frequency characteristics when only linear approximation solutions are considered,the transient phase-frequency characteristics when only higher order approximation solutions are considered,and the transient phase-frequency characteristics when linear solution terms and higher order approximation terms are combined are studied.The results show that the jump point of the phase frequency curve changes obviously due to the difference of the sag span ratio when only the linear approximate solution is considered.The larger phase difference between two cables is usually between the jump points produced successively by two cables.The instantaneous phase difference between two cables caused by the higher order approximation term is related to the ratio of the amplitude of drift and no drift.The larger the ratio difference,the greater the instantaneous phase difference,and the smaller the vice versa;By comparing the instantaneous phase difference when both the linear solution term and the higher order approximation term are considered with that when only the linear approximation term is considered,it is found that the maximum increase of the instantaneous phase difference of the former is 0.45π.Therefore,the instantaneous phase caused by the higher order approximation term in the approximate solution cannot be ignored in the study of phase frequency characteristics.For the reliability of the results,the fourth-order Runge-Kutta direct integration method and the nonlinear finite element analysis software ANSYS were used to verify the results.