With the assumption that a load varies linearly within each time step,an analytical solution was derived for a single-degree-of-freedom(SDOF) vibration system with low damping.Based on this solution,the modeling of a train-bridge system and its solving procedure were deduced.The train-bridge system model consisted of a train subsystem and a bridge one.The motion equation of the train subsystem was derived using the rigid component assumption and D′Alembert’s principle,and that of the bridge subsystem was derived using the finite element method.The mode superposition method was applied to uncouple the equations of motion of the two subsystems.The effects of the non-orthogonal damping of the train subsystem and the dynamic interaction between two subsystems were treated as nonlinear virtual forces.A 4-axle vehicle passing through a simply-supported beam of 32 m span at a constant speed was taken as a case study.The dynamic analysis of the coupled system was performed using the proposed analytical method,Newmark-β method and Gauss precise integration method,respectively.The results showed that compared with Newmark-β method and Gauss precise integration method,the analytical method can not only improve the time step of numerical integration but also avoid the computation of complex exponential matrices,so it has a good applicability in engineering.
Key words
train-bridge system /
dynamic interaction /
analytical solution to a single-degree-of-freedom system /
mode superposition method /
non-orthogonal damping
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