Dynamic Coefficient Method And Its Applications In Dynamic Analysis Of Bridge Structures

With high accuracy and efficiency in computation,the dynamic stiffness matrix method and the transfer matrix method have been widely used.However,these traditional exact methods often encounter numerical instability.Although the non-dimensionlization of parameters can partially address this difficulty,they cannot fundamentally solve the problem.Particularly,in some specific bridge dynamic calculation,such as high-order free vibration analysis of composite continuous beam,coupled vibration analysis of vehicle and composite beam bridge,dynamic analysis of damaged composite beam and inverse problem in dynamic analysis of beam with non-uniform cross-sections,the numerical instability seriously restricts the application of the traditional exact methods.This paper proposes a numerically stable and theoretically exact method to analyze these dynamic problems of bridges:(1)A numerically stable dynamic coefficient method is proposed to analyze the dynamic responses of the composite continuous beam.To tackle the numerical instability,the undetermined coefficients in the general solutions are used as the fundamental unknowns instead of displacements or internal forces of the beam to avoid the matrix inversion,and some modified exponential functions are employed to replace the hyperbolic functions and prevent the floating-point overflow of the exponential functions.The method can also be extended to the dynamic analysis of the structures that composed of different materials or members and combined by shear connectors(i.e.,generalized composite structures).(2)According to the orthogonality of the vibration modes of composite beams and the mode superposition method,the closed form approximate solution of the coupled vibration responses of vehicle and composite beam under arbitrary boundary conditions and typical boundary conditions(simply supported beams,equal-span continuous beams)are given,respectively,and the numerical instability encountered in the computation of bridges with too many segments is solved by using the dynamic coefficient method.The natural frequencies of the composite beam are included in the dynamic responses of the composite beam and the vehicle,providing a theoretical basis for the method to obtain the dynamic characteristics of the composite beam from the vehicle-bridge coupling vibration analysis.The method has been used to detect the dynamic characteristics of two practical composite continuous beam bridges with non-uniform cross-sections.(3)Based on the dynamic coefficient method,a method is proposed to simulate the composite beam with the damage in beam-body and connectors,which solves the numerical instability in analysis of bridges with too many segments in the traditional exact methods.Subsequently,based on the moving average filter and empirical mode decomposition,the speed pseudo-frequency signal,which is sensitive to damage,is separated from the acceleration signal of vehicle.Compared the difference between the speed pseudo-frequency signal separated from the coupled vibration response of the composite beam with and without damage and the vehicle,and then the damage location of the composite beam can be identified.This method is easy to use and has high calculation efficiency,which has a bright future in damage identification of practical bridge with composite beams.(4)The dynamic coefficient method is applied in the dynamic analysis of beam with non-uniform cross-sections to solve the numerical instability encountered when using the traditional exact methods,which is caused by the great difference in sectional characteristics of each segment of the beam.On this basis,a method to inversely analyze the boundary condition parameters of the beam with non-uniform cross-sections by using the measured axial force and the first two frequencies(or only the first-order frequency)is given.Based on this method,the boundary condition parameters at the two ends of the hangers with non-uniform cross-sections on an actual tied-arch bridge are inversely analyzed,and the measurement accuracy for the axial forces of the hanger is evaluated according to the inverse analysis results,which provide a method to precisely control the axial forces of hangers.