Resonance And Cancellation In Torsional Vibration Of Monosymmetric ?-Sections Under Moving Loads

With the rapid development of high-speed railways,the speed of the train is getting higher and higher,and the load bearing capacity of the supporting bridge is getting larger and larger.The vibration problem of the bridge caused by the moving loads is more obvious than before.In the past,the study of vehicle-bridge interaction was mostly directed toward the vertical bending vibration of the bridge,that is to say,the moving loads wereassumed to be driven along the central line of the bridge.In fact,the train or vehicle loads may move along a line with an offset from the central line of the bridge.In such a situation,it will not only cause the vertical bending vibration of the beam,but also cause the lateral bending vibration and torsional vibration of the beam.For beams with monosymmetrical sections,the bending and torsional vibrations in the asymmetric direction are coupled.This paper is concerned with the lateral and torsional coupled vibration of monosymmetric I-beams under the moving loads.To this end,a train is modeled as two subsystems of eccentric wheel loads of constant intervals to account for the front and rear wheels.In order to obtain the closed form solution for revealing key parameters involved,both the inertial and damping effects of the bridge are neglected.In addition,by assuming the lateral and torsional displacements to be restrained at the two ends of the beamand considering the warping deformation of the section,the vibration governingequation of the simply supported beaminduced by the eccentric moving loads is determined.Both the lateral and torsional displacements are approximated by a series of sine functions.The method of variation of constants is adopted to derive the closed-form solution.The resonance and cancellation conditions of the torsional vibration of the beam are determined,which was further utilized to derive the optimal design criteria for the simple beam under torsion vibration.The finite element model of the I-section beam bridge is established,with the warping deformation of the cross section simulated.For the example selected,the results obtained by the finite element method are compared with those of the analytical method.On the basis of closed-form and finite element analyses,the main conclusions are drawn as follows.First,since the best design rule for the simply supported beam under torsional vibration is just the same as the one for vertical vibration,this offers a great advantage in the design of monosymmetric I-beams,as no distinction needs to be made between the suppression of vertical or torsional resonance.Second,when the ratio of beam span to vehicle length satisfies certain conditions,the response of simply supported beam will almost be free of resonanance.There are three innovative points in this paper.First,considering the actual situation of train running,that is,the train may move along a line deviating from the central line of the bridge and causes both the torsional and lateral vibrations of the bridge.Particularly,the torsional and lateral vibrations are coupled for singly axisymmetric I-beams.Second,assuming that the torsional response is a superposition of simple harmonic waves,the closed-form solution of the torsional response is derived.Third,both the resonance and cancellation conditions of the torsional response are derived from the closed-form solution,from which the optimal design rule is establishedfor the simply supported beam under torsional vibration.