| As a kind of cable,the sling(or the vertical cable)is widely used in various engineering structures such as suspension bridges and arch bridges.With the increasing span of engineering structures,the length of the sling is also increasing.Due to its small slenderness ratio and large flexibility,the long sling leads to a series of vibration problems,which increases the probability of large-amplitude vibration.Besides,there are also some phenomena that linear vibration theory cannot solve,such as parametric vibration,etc.Via the observation of some engineering structures that have been built or are being built,we find that song long slings will vibrate violently even in the case of light rain or breeze.And the violent vibration may leads to fatigue of the sling?fatigue crack of the connection of the sling and anchor?damage of the corrosion resistant system of the sling.If the situation is more serious,the sling will be out of use and increases the stress concentration of other slings and finally leads to the destroy of the whole engineering structures.Therefore,it is very important to study the nonlinear vibration of structure with sling.Based on the previous studies of cable and cable-beam coupled structure at home and abroad,the article mainly studies: 1:2 internal resonance?summation resonance? difference resonance of suspension-sling-beam coupled structure;the influence of other sling on the resonance of suspension-multiple slings-beam coupled structure,which occurs between the suspension?beam and one of the sling;the experiment of the suspension-sling-beam coupled structure and suspension-multiple slings-beam coupled structure.The specific research contents are as follows:(1)Firstly,I conclude the research methodology of nonlinear vibration,including the qualitative method?quantitative method and experimental method;secondly,I introduce the research status of nonlinear vibration of cable,emphasizing on parametric vibration;thirdly,I introduce the research status of cable-beam coupled model?summation resonance and difference resonance.(2)Firstly,a three degree of freedom suspension-sling-beam coupled model is built;secondly,the transverse vibration differential equation of coupled model is obtained by D'Alembert's principle and Galerkin's method;thirdly,I discuss the feasibility of using sinusoidal functional modal equations of suspension?sling and beam;finally,I compare the analytical value?finite element value?measured value of the slings and get some conclusions.(3)The free vibration of 1:2 internal resonance of the suspension-sling-beam coupled structure is investigated.Firstly,based on the transverse vibration differential equation of the coupled model got in Section 2,I get the first-order approximate analytical situation of the differential equation by multiple method,and by analyzing the first-order approximate analytical situation,I obtain that if the frequencies of the suspension and sling?beam and sling satisfy the ratio 1:2,the coupled model will occur the kind of 1:2 internal resonance;secondly,placing the 1:2 internal resonance of beam and sling as object,the first-order approximate analytical solution of free vibration of the 1:2 internal resonance of the coupled model is got;thirdly,by numerical integration method,time history of amplitude curve with damping and without damping of the coupled model under the situation of free vibration of 1:2 internal resonance are analyzed;finally,by numerical integration method,time history of amplitude curve with damping and without damping of the coupled model under the situation of free vibration of 1:2 internal resonance are analyzed.Besides,the two kinds of numerical integration method can get the same conclusions and well verify the conclusions got from the analytical solutions.(4)The free vibration of summation resonance and difference resonance of suspension-sling-beam coupled structure is investigated.Firstly,based on the first-order approximate analytical situation got in Section 3 and assuming the 1:2 internal resonance can not occur,I get the second-order approximate analytical situation of the differential equation by multiple method,and by analyzing the second-order approximate analytical situation,I obtain that if the sum of the frequencies of the suspension and beam approaches two times of the frequency of the sling,the coupled model will occur the kind of summation resonance,and if the difference of the frequencies of the suspension and beam approaches two times of the frequency of the sling,the coupled model will occur the kind of difference resonance;secondly,by multiple method,the second-order approximate analytical solutions of the free vibration of summation resonance and difference resonance are obtained;thirdly,by two kinds of numerical integration method,the nonlinear vibration characteristics of the coupled model under the situation of free vibration of summation resonance are analyzed,including time history of displacement curve and time history of amplitude curve with damping and without damping,besides,the two kinds of numerical integration method can get the same conclusions and well verify the conclusions got from the analytical solutions;finally,I discuss the influence of some parameters of the coupled model and frequency combination on the time history of amplitude of coupled model.(5)The forced vibration of summation resonance and difference resonance of suspension-sling-beam coupled structure is investigated.At first,by adding the lateral excitation term to the coupled structure,the second-order approximate analytical solutions of the forced vibration of summation resonance and difference resonance are obtained,furthermore,in the solution of the approximate analytical solution of the summation resonance,it is divided into two cases: lateral load acting on the sling and lateral load acting on the main girder,in the solution of the approximate analytical solution of the difference resonance,it is divided into three cases: lateral load acting on the suspension,lateral load acting on the sling and lateral load acting on the beam;then,by numerical integration method,the nonlinear vibration characteristics of the coupled model under the situation of forced vibration of summation resonance are analyzed,including amplitude-excitation frequency curve?amplitude-excitation amplitude curve,and steady-state solution.(6)Based on the suspension-sling-beam coupled structure and considering multiple slings,the suspension-multiple slings-beam coupled structure is built,and then the influence of other sling on the resonance is discussed if the 1:2 internal resonance?summation resonance and difference resonance occur between suspension,one of the slings and beam.Firstly,the suspension-multiple slings-beam coupled structure is built;secondly,by D'Alembert's principle and Galerkin's method,the transverse vibration differential equation of coupled structure is obtained;thirdly,by multiple method,the transverse vibration differential equation is solved;finally,placing the 1:2 internal resonance of beam and sling as object and by numerical integration method,the influence of other sling on the resonance is discussed.(7)The nonlinear dynamic experimental study of the suspension-sling-main beam coupled structure and the suspension-multiple slings-beam coupled structure are carried out.Firstly,the experimental system and experimental procedure of the coupled structure are introduced,and the experimental system includes three components: expentiental components,excitation system,text and analysis system;secondly,the expentiental study of summation resonance of the suspension-sling-main beam coupled structure is carried out,including transfer function,amplitude-excitation frequency curve and amplitude-excitation amplitude curve;finally,the expentiental study of summation resonance of the suspension-multiple slings-main beam coupled structure is carried out,including transfer function,amplitude-excitation frequency curve and amplitude-excitation amplitude curve.The conclusions obtained from the experimental study of the coupled structure well prove the correctness of the conclusions obtained from the analytical and numerical analysis in the previous chapters. |
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