| At present,China’s rail transportation,especially high-speed railway are entering a new epoch of rapid development,with the increase in train running speed,the dynamic interaction between the train and the structure under the rail is significantly intensified,the vibration and noise problems are increasingly prominent.The track structure in railway usually has long laying length,so it can be regarded as infinite length structure,which can be regarded as phononic crystal structure due to the periodic arrangement of its fastener and sleepers.According the phononic crystal theory,which is used in solid physics to analyze the vibration problems of periodic structures in recent years,this paper analyzes elastic wave transmission characteristic from the viewpoint of band gap characteristics,and the vibration response of the track structure under the action of moving load or harmonic load is also studied,to proposes a new analysis method of the vibration characteristics of periodic track structures which is more comprehensive in consideration,and can reflected the actual engineering.The main work is as follows.(1)The transfer matrix method is used to establish four 1-D phononic crystal theoretical analysis model of periodic track on subgrade,they are the phononic crystal theoretical analysis models of single-layer Euler beam,single-layer Timoshenko beam,double-layer Euler beam and double-layer Timoshenko beam.The analysis results showed that when not considering effects of damping,there is no obvious difference between band gap positions of single layer Euler beam model and single-layer Timoshenko one within 0-250 Hz,but they are significantly different within the range of larger than1000 Hz,the band gap positions within 0-250 Hz of double-layer Euler beam model and double-layer Timoshenko one are quite different,while their band gap positions within the range of larger than 250 Hz are basically the same,this phenomenon is significantly different from that of single-layer beam models.When considering effects of damping,there is a phenomenon of the passing band becoming an incomplete band gap and the band width of the forbidden band being slightly widened,effects of damping on the center position of the forbidden band can be ignored.Using the transfer matrix method and Fourier transform method,the vibration response characteristics between the single-layer beam model and the double-layer beam model are quite different.According the theoretical analysis and onsite testing results,when using the phononic crystal theory to analyze track vibration and noise control,Timoshenko beam model is more accurate for medium and high frequency vibration within the range of larger than 250Hz;when analyzing low frequency vibration within the range of less than 250 Hz,single-layer Euler beam model or single-layer Timoshenko one can be used for ballastless track,double-layer Timoshenko beam model can be adopted for ballast track.(2)To investigate the properties of elastic waves propagating in the periodic ballasted track on subgrade,a double-rail phononic crystal spatial model which consists of two parallel rails fastened on equally spaced sleepers supported by ballast is proposed in this study.The presented numerical results show that there are twelve characteristic waves occurring in the model.These waves can be divided into three kinds.Within the calculated frequency range,the first kind of characteristic waves always propagate except an initial lower frequency range.The second kind of characteristic waves have alternative pass bands and stop bands,showing obvious phononic property of the model.The wavenumbers of the third kind of characteristic waves have larger imaginary parts,implying that this kind of waves are evanescent waves,and can only propagate a few spans of the track away from the source.For explaining the elastic wave propagation mechanism in the ballasted track,and antisymmetric vibration mode,as well as investigating the vibration response under moving load,The numerical results show more advantages of the spatial model than the 1-D single-rail phononic crystal model,for the spatial model is more comprehensive,which can account for the coupling between the two rails of the track.(3)Based on the energy generalized function principle,the 1-D phononic crystal model of the PC multi-span simply supported box girder bridge and its track structure commonly used in railway is proposed.The whole structures are simplified as a triple girder periodic structure with infinite periodic support in the model.The unique vibration characteristics of the model is analyzed through representative span(cell)of dynamical equations,combined with periodic boundary conditions,and Bloch’s theorem.Being quite different from the periodic track structure on subgrade,the energy band structure of the rail on the bridge occurs many energy band "strips" compared with the rail on on subgrade,due to the influence of the track plate and the bridge,the band gap of the rail in the low frequency ranged of 0-50 Hz depends on the bridge structure.In the 0-50 Hz range,the band gap in the bridge is generated by the inherent overall vibration mode and its track structure,and the elastic wave propagation in the bridge is limited affect by the track-slab and rails on the bridge in the low frequency range.Lots of the elastic waves in the range of 50-250 Hz in the rails can propagated to the bridge downward,but the band gap track structure on the bridge is affected by the bridge and track plate.The conclusion of the analysis also indicates that the vibration waves in the rails caused by harmonic load are mainly transmitted to the bridge structure when they propagate forward along the rails longitudinally,because the periodic bridge is mainly connected by rails from span to span,the vibration displacement attenuation zones of the bridge is related to the inherent vibration characteristics of the bridge,and also are affected by the vibration displacement attenuation zone of the rails in some respects.(4)To verify the above theoretical analysis results,onsite testing research were conducted in the actual railway.The measured vibration transmission characteristics of ballast rail on subgrade are consistent with the results of the double-layered Timoshenko beam model with damping in frequency range of 0-250 Hz.The measured rail vibration transmission characteristics of ballastless track on subgrade and the analysis results of single-layer Timoshenko beam model considering damping are well coincident in frequency range of0-250 Hz.The results of the periodic bridge field test also show that the propagation speed of longitudinal waves in the bridge is larger than that of transverse waves and vertical bending waves.When the first wheel pair of the train is about to arrive,more low-frequency inherent vibration peaks of the testing span bridge are excited,and when multiple wheel pairs of the train arrive at the testing span bridge,more medium-high frequency vibration peaks of the bridge and vehicle-bridge coupling vibration peaks are excited.The general trend of the periodic measured bridge transmission rate curve in 0-7.4Hz,46.3-48.2Hz,107-113 Hz,130-130-154 Hz and 172-185 Hz is consistent with the theoretical value.The vibration attenuation zones of the measured bridge did not clearly show the starting and ending boundary as the theoretical analysis results,but the existence of the the theoretical vibration attenuation zones of 9.7-16 Hz and 107-126 Hz were confirmed from onsite testing.It is worth noted that the field tests show that the low frequency band vibration wave propagation caused by the train excitation can be transmitted to the adjacent pier,but the vibration wave energy transmitted to the adjacent pier is small,which can be mainly attributed to the adoption of the spherical vibration damping steel bearings for the effect of their vibration isolation,so the vibration waves caused by the train excitation propagate between the adjacent span bridge mainly along the rails on the bridge,while the vibration energy transmitted through the ground-foundation-abutment is smaller. |
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