| The arch structure is widely used in bridges since it has beautiful shape and excellent mechanical properties.The arch structure bears the action of axial force when it vibrates.Based on this,when the arch structure vibrates,it will begin to deform from the initial deformation state generated by the axial force,so that the subsequent deformation of the arch structure decreases,which will inevitably affect the natural vibration characteristics of the arch.At present,the research on the in-plane free vibration of arch structure mainly focuses on the numerical method,and lacks the analysis of the in-plane free vibration of non-circular arch structure.In this paper,the non-circular arch in Cartesian coordinate system is taken as the research object.Based on the Hamilton principle and the strain expression of arch structure,the equilibrium differential equation of linear free vibration in non-circular arch plane and its practical analysis of natural frequency are deduced.Based on this,the equilibrium differential equation of free vibration in parabolic arch plane considering axial force and its approximate analysis of natural frequency are deduced.On this basis,the influence of parabolic arch considering axial force on natural vibration characteristics under different initial conditions is analyzed and studied.The main work and achievements are as follows:(1)The variable coefficient equilibrium differential equation of linear free vibration in non-circular arch plane and its practical approximate analysis of natural frequency are deduced.Based on the Hamilton principle and the linear strain of the arch structure in Cartesian Cartesian coordinate system,the variable coefficient equilibrium differential equation of in-plane free vibration of non-circular arch structure is deduced.Based on the similarity of the vibration modes of steep arch and shallow arch,the high-precision practical analysis of the in-plane natural frequency of non-circular arch is deduced,and the proportional relationship between the natural frequency of non-circular arch and the non-first-order in-plane natural frequency of straight beam with the same parameters is revealed.The accuracy of the method is verified by finite element numerical results.(2)The equilibrium differential equation of in-plane free vibration of parabolic arch considering axial force and its approximate analysis of natural frequency are deduced.Based on the in-plane equilibrium differential equation of non-circular arch structure in Cartesian coordinate system,the in-plane free vibration equilibrium differential equation of parabolic arch considering axial force is further deduced,and the approximate analysis of in-plane free vibration frequency of parabolic pinned arch and fixed arch considering axial force is obtained.Finally,by comparing with the results calculated by the finite element method,the accuracy of the theoretical deduction results and the approximate analytical method is verified.(3)The influence of parabolic arch considering axial force on natural vibration characteristics under different initial conditions is explored.By studying the natural vibration characteristics of parabolic arch under different rise-span ratio,slenderness ratio and axial force,it is found that when the axial force of parabolic arch reaches the critical load,its vibration mode characteristics will change,and the natural vibration frequency is zero. |
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