Nonlinear Vibration Responses Of Suspended Cables Under Non-Uniform End Excitations

The suspended cable shows a complex nonlinear dynamic response under external excitation,in which the parametric vibration caused by the displacement excitation of the bridge deck and bridge tower is one of the main forms of vibration.In this thesis,the mechanism of cable vibration under non-uniform end excitation is studied by combining theoretical analysis with numerical analysis.The main work is as follows:（1）After summarizing the relevant research at home and abroad,in order to study the influence of non-uniform end excitation on the response characteristics of the suspended cable,the theoretical model of a suspended cable subjected to three-dimensional non-uniform end excitation is established,and the discrete modal equation is obtained by Galerkin discretization after the motion equation is derived by Hamilton principle.The in-plane and out-of-plane frequency equations of the primary resonance and the subharmonic resonance are obtained by using two different Multiple Scales Methods（MSM）.The results show that the parametric resonance cannot be induced outside the subharmonic plane,and the average equations of the primary resonance under the two different MSM perturbations are obviously different.（2）The qualitative analysis of the response of the suspended cable mechanical model is carried out.Due to the different initial phases of the non-uniform excitations,the general expressions of the equivalent excitation amplitude and the response phase of the multi-excitation system are obtained.The effect of the phase difference of different excitation combinations on the response characteristics of the nonlinear system is qualitatively analyzed,which provides a more general theoretical model for the numerical analysis later.The results show that whenΩ≈2ω_m andΩ≈ω_m,the multi-excitation system can be regarded as a nominal single-excitation system（NSES）,and it makes the response phase shift（phase shift value）based on the single-excitation system.At the same time,the non-uniform excitation will affect the equivalent excitation amplitude of the cable,and both the phase shift value and the equivalent excitation amplitude depend on the excitation combination and the modal parity.（3）The influence of the excitation phase difference on the primary resonance response is obtained by combining the general theoretical model of the primary resonance with the analytical solutions of different MSMs.After the main resonance frequency response equations of two kinds of suspended cables in and out of the plane are obtained under two different MSM perturbations,the same set of cable parameters is determined for numerical comparative analysis.The results show that the response phase γ can be shifted from the response phase （?） of the nominal single excitation system by the initial excitation phase （?）_s for the primary resonance multi-excitation system,and the phase shift is related to the type of excitation combination and the excitation mode.In the excitation combination,when the excitation phase difference changes from 0 toπgradually,the nominal single excitation phase frequency curve also moves to different degrees,and the final phase distance change isπor 0.For the two different MSM perturbation methods,the response amplitude approximate solution of MSM＿Case2 is better than that of MSM＿Case1 as a whole,where the positive and negative values of the parameters Γ_em and Γ₁ can reflect the"soft and hard"properties of the nonlinear system（NPE）.However,the parameters Γ₂ and Γ₃ in MSM＿Case2 change suddenly in a certain range ofσ,which leads to the phenomenon of"soft and hard"transition in the amplitude-frequency response curve of the nonlinear system,and the Runge-Kutta numerical integration under some f also has a similar phenomenon,but they are not simultaneous.（4）The general theoretical model of subharmonic resonance is compared with the numerical analysis of the frequency response equation.The specific physical parameters of the cable are cited to analyze the example,and the subharmonic resonance under different excitation combination conditions is studied.The Runge-Kutta numerical integration is used to verify the analytical solution of MSM.Finally,the qualitative analysis is compared with the numerical analysis,and the conclusion is drawn that the influence of the excitation phase difference on the response amplitude of the suspended cable changes with a period of 2π,which is symmetrical aboutπand does not change the soft and hard properties of the system,but affects the response amplitude and the width of the resonance region,which is symmetrical aboutσ=0.The system is mainly excited by parametric vibration,and the effect of horizontal excitation is much greater than that of vertical excitation,but vertical excitation in a certain phase difference interval can inhibit the effect of horizontal excitation.In the excitation combination,when the excitation phase difference is gradually changed from 0 toπ,the response phase distance change of the nominal single excitation phase-frequency curve isπor 0.