Research On The Effect Of End Excitation Phase Difference On The Response Characteristics Of Stay Cable

The nonlinear vibration of the stay cable is one of the current key research topics.The research shows that there are different initial phases between the modes of the cable and between the excitations in each direction,and it has a significant effect on the excitation frequency and the nonlinear vibration characteristics of the cable.At present,the study of cable stays considering the phase difference of cable stays is still in its infancy,most of which are limited to the analysis of a specific set of parameters.This paper mainly uses the method of theoretical analysis,parameter analysis and comparative induction to study the effect of the end excitation phase difference on the amplitude of the cable response.The main contents are as follows:（1）The nonlinear vibration theoretical model when the two ends of the inclined cable are excited by three directions is established,and the initial phase of the excitation is considered.Galerkin is used to perform the discretization to obtain the generalized cableless dimensionless discrete control equation.On this basis,the equation is degraded to study the free vibration characteristics of the cable,and the influence characteristics of the sag and bending stiffness on the natural frequencies of each mode are analyzed.The research shows that the cable sag only affects the fundamental frequency of the low-order odd mode;in addition,when considering the influence of the bending stiffness of the cable on the fundamental frequency,when the modal order is the low-order odd mode,the cable The greater the cable-span ratio,the more obvious the influence of the cable sag on the fundamental frequency.At this time,the impact of the small bending stiffness on the fundamental frequency is negligible;however,as the modal order increases,the vertical-span ratio decreases The influence of bending stiffness on the fundamental frequency cannot be ignored.（2）Using multi-scale method,the effect of excitation amplitude,frequency and initial phase on the first-order amplitude of cable stay under the action of horizontal excitation combination（Ua+Ub）and horizontal-vertical excitation combination（Ua+Wb）was studied.The research shows that when the excitation frequencyΩis greater than the first-order natural frequencyω₂ outside the plane of the cable,the cable is coupled by nonlinear parameters,and the in-plane excitation combination can greatly stimulate the first-order vibration of the cable outside the plane;Under the effect of gravity sag,the first-order natural frequencyω₂ of the cable will be slightly smaller than the first-order natural frequencyω₁ of the in-plane.At this time,the out-of-plane first-order vibration is easier to be greatly excited,and in the in-plane and out-of-plane first-order vibration It is dominant;whenω₂≤ω₁<Ω,the excitation combination in the cable end surface can cause strong internal resonance of the cable.With the emergence of the phase difference between excitations,whenω₂=ω₁ orΩ<ω₂<ω₁,the influence of the phase difference between excitations on the steady-state amplitude under the combination of two in-plane excitations takes 2πas the change period,and the excitation tends to In the reverse phase,the amplitude is increased or weakened to maximize the intensity.（3）The numerical analysis method was used to study the linear and nonlinear steady-state responses of the main resonance area and the main parametric resonance area respectively,whenω₂=ω₁ and the cable end is subjected to the same excitation combination,and the phase difference of different excitations;Parametric analysis is carried out on the combined conditions between the out-of-plane transverse bridge excitation at the lower end and the in-plane longitudinal bridge and vertical excitation.The research shows that the calculation results of the two solving methods are basically consistent.Excitation phase difference can not only cause the quantitative change of the non-linear vibration of the cable,but also change the manifestation of internal resonance.Under the in-plane excitation combination,the change period will be 2π.When the phase difference isπ+2kπ（k=0,1,2,…）,the effect on the steady-state amplitude is most significant;under the in-plane and out-of-plane excitation combination,The influence of the phase difference on the amplitude of the cable response changes periodically withπ,and the effect is most prominent when the phase difference tends to π/2+kπ（k=0,1,2...）.