Vibration isolation analysis of elastic bearing based on train-track-bridge coupled vibration model

Transcription

1 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Porto, Portugal, 3 June - 2 July 214 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: ; ISBN: Vibration isolation analysis of elastic bearing based on train-track-bridge coupled vibration model Zhijun Zhang, Xiaozhen Li, Xun Zhang, Quanmin Liu School of Civil Engineering, Southwest Jiaotong University, 6131 Chengdu, China zhijun973@163.com, zhli@swjtu.cn, zhunun@126.com, qmlau7@126.com ABSTRCT: Elastic bearings have been used on simply supported bridges on high-speed railway in Taiwan to isolate train-induced vibration. However, their influence on dynamic behavior of the vehicle and bridge has not been fully investigated, and their vibration attenuation has not been quantificationally discussed. In this study, spring-damper elements with 12 degrees of freedom are established to simulate elastic bearings, and added to Train-Track-Bridge System. To eplore the influence of elastic bearings on isolation performance, a simply supported bo-girder bridge with a span of 32 m, was simulated as a case study. Based on the Train-Track-Bridge coupled vibration system, the influence of the elastic bearing on the response of the vehicle and bridge are firstly discussed. Wheel unloading rate and vertical acceleration of the vehicle are hardly influenced by the vertical stiffness of the bearing, yet influenced by the vehicle speed significantly. With the vertical stiffness of the bearing decreasing, the static component of the bridge response increases gradually, while the dynamic component of the bridge response firstly decreases and then increases. The analysis shows that the elastic bearing with suitable vertical stiffness won t have negative effects on the response of the vehicle and bridge. Then the influence of elastic bearings on vibration isolation is researched. The analysis shows that, besides affected by the vertical stiffness of the bearing, the vibration isolation performance of elastic bearings might be also influenced by the vehicle speed. For the elastic bearing with vertical stiffness of kn/mm, the vibration attenuation is db in the frequency band of ~8 Hz, and the vibration attenuation is 15.5 db in the frequency band of 2~8 Hz, which makes the significant contributions to the vibration attenuation in the whole range. Because the natural frequency of the elastic bearing vibration isolation system eists in the frequency band of 1~2 Hz, vibration in the middle and high frequency band can be absorbed, and the vibration in its natural frequency range can be amplified. Based on the results, it can be seen that elastic bearings can be used to isolate vibration in the region where the environmental vibration needs to be controlled rigorously. KEY WORDS: elastic bearing; train-track-bridge system; coupled vibration; 1/3 octave; vibration isolation 1. INTRODUCTION In recent years, environmental influence problems due to traffic-induced vibration at high-speed railway bridges, such as ground vibration, have occurred here and there. Elastic bearings have been used in high-speed railway bridges in Taiwan to reduce the environmental influence of traffic-induced vibration[1]. For the same reason, elastic bearings have also been used in Yichang Yangtze River bridge of Yiwan railway and the Minjiang River bridge of Fuia railway in China[2]. Japanese scholar Mitsuo Kawatani analysed dynamic response of a highway bridge with elastic bearings by combining theoretical analysis with eperiments. Elastic bearings performance was evaluated[3-5]. It was then reported that dynamic response analysis of vehicle-bridge system with elastic supports was conducted by Jiangtong[6]. Recent years, the 3-D seismic isolation bearings, which offer certain fleibility at vertical direction to isolate some vertical vibration, were studied [7-9]. As described above, elastic bearings have been utilized in highspeed railway bridges to isolate vibration induced by trains traveling across bridges. Then some accompanying questions appear. What abut the vibration isolation effects of elastic supports? How the reduced vertical stiffness influences the dynamic response of train-track-bridge system and the running safety of trains? These are worth eploring problems. In this study, three-dimensional analysis of the dynamic response of train-track-bridge system while moving vehicle across the bridge is carried out due to replacing steel bearings with elastic bearings. Elastic bearings are simulated as spring-damper elements. To eplore the influence of elastic bearings on its vibration isolation performance, a typical double tracks simply supported bo-girder bridge with a span of 32 m, was simulated as a case study. 2. VIBRATION ISOLATION THEORY AND ANALYSIS MODEL 2.1 Vibration isolation theory of elastic bearings Bridges with elastic bearings can be regarded as mass-spring system. Its vibration isolation theory is analogous to steel spring floating slabs[1-12]. And vibration isolation model has been shown in Figure 1. The mass block m and the foundation were assumed to be body. The mass block m is subjected to the eternal load F() t = F sinθt. Then the vibration transmission coefficient of the spring-damper system can be derived based on theory of structure dynamics[13] (2 ξβ ) T = (1) (1 β ) + (2 ξβ ) Where m is the mass of the block, k is the spring stiffness of the vibration isolation system, c is its damping, ω is natural circular frequency of the system, θ is frequency of the eciting force, β = θ / ω is the frequency ratio, ξ is the damping ratio. 1517

2 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Figure 2 shows the vibration transmission coefficient T changing with the frequency ratio β and the damping ratio ξ. It can be seen that the transmission coefficient T < 1, while β > 2. So the fundamental principle of isolating vibration is to increase frequency ratio β as far as possible, namely reducing natural frequency ω of the vibration system while the eciting frequency θ is not easy to be changed. in the time domain. Then the dynamic response of the train-track-bridge system, the running safety and the ride comfort of trains passing through bridges can be analyzed. The matri equations of motion of the vehicle, track and bridge subsystem can be formed as follows: Mu&& v v + Cu& v v + Ku v v = R v (2) Mu&& + Cu& + Ku = R (3) t t t t t t t Mu&& b b + Cu& b b + Ku b b = R b (4) where M, C and K are the mass, damping, and stiffness matrices, R, u, &u and u&& are the vectors of generalized load, displacement, velocity and acceleration, and the subscripts v, t and b denote the vehicle, track and bridge respectively. Figure 1. Isolation system of single degree of freedom Figure 2. Transfer coefficients of isolation system Although the above vibration isolation theory is derived on the basis of assuming the mass block and the foundation to be, which is not so accordant with the practical situation, it has still been widely used and provides clear ideas for elastic vibration isolation research. Based on the thought, the vibration isolation is eplored by properly reducing stiffness of bridges bearings. 2.2 Dynamic analysis model Although specific models for different high-speed trains or tracks and for different bridge structures can be very different, they all have the same basic framework, which takes into account the train, track, bridge subsystem components coupled with the wheel-rail interaction and the track-bridge interaction[14-16]. For eample, Figure 3 shows the schematic diagram of a high-speed train-track-bridge dynamic interaction model for simply supported composite bo-girder bridge with slab track structure on it. Research techniques of vehicle dynamics, track dynamics and bridge dynamics are utilized to establish equations of motion of the vehicle, track and bridge subsystem separately. Then a fundamental model is established for analyzing the train-track-bridge dynamic interactions, in which the vehicle subsystem is coupled with the track subsystem through a spatially interacted wheel-rail model; and the track subsystem is coupled with the bridge subsystem by a track-bridge dynamic interaction model. An eplicit-implicit integration scheme is adopted to numerically solve the equations of motion of the large non-linear dynamic system Figure 3. Dynamic model of train-track-bridge system Bridges supports are simulated as couplings in the traditional train-track-bridge coupled vibration system. In practical engineering, bridges bearings are not generally. And elastic bearings stiffness is even lower than the common ones. So it s necessary to establish elastic bearing model. Then the spring-damper element is introduced in train-track-bridge system. The entire system, as shown in Figure 3, is still composed of vehicle, track and bridge model with elastic bearings as a whole. Spring-damper element contains two nodes, connecting the girder and the pier. There are three directions of linear displacement and three directions of angular displacement. The entire bearing element contains 12 degrees of freedom. The stiffness matri of spring-damper elements under the local coordinate system is shown as equation (5). k k y k z kr k ry krz K e = k k ky ky kz kz k r kr kry kry krz krz (5) where k, ky and k z are three directions linear ity, k r, kry and krz are three directions angular ity respectively. If k, ky and k z are replaced by c, cy and c z,and k r, kry and k rz are replaced by c r, cry and c rz separately, then the damping matri of spring-damper element is derived. c, c and c are three directions linear damping factors. y z c r, cry and c rz are three directions angular damping factors. 1518

3 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN CASE STUDY 3.1 Project profile A typical double tracks simply supported bo-girder bridge with a span of 32 m used on high-speed railway is a single bo single chamber structure. The overall length is 32.6 m, and the computed span is 31.5 m. Depth at the centre of the beam is 3.78 m. The beam width is 12 m. Distance between centers of tracks is 5 m. Center distance of bearings in the transverse direction of the bridge is 4.5 m. Bridge beam structure is constructed with C5 concrete. The secondary dead load is 14 kn/m. The pier is uniform cross section. The section dimension is 68 cm 33 cm, and the thickness is.5 m. The pier s height is 2 m. The typical cross sections of the bridge beam and pier are shown as Figure 4. (a) Typical cross section of the girder Influence of bearings vertical ity on vehicle running characteristics To investigate the influence of bearings vertical stiffness on vehicle s running characteristics, wheel unloading rate and vertical acceleration of the first locomotive and the last carriage are analyzed. Figure 6 shows the isolines of the wheel unloading rate amplitude of the locomotive and carriage while the train driving across bridges with different stiffness s bearings in different speeds. No matter the locomotive or the carriage, the wheel unloading rate changes with vehicle speeds, and increases with the speeds rising. By contrast, Influence of bearings vertical stiffness can be almost neglected. Under the same conditions, the carriage s wheel unloading rate is a little higher than the locomotive s. Figure 7 shows the isolines of the vertical acceleration amplitude of the locomotive and carriage while the train driving across bridges with different stiffness s bearings in different speeds. The vertical acceleration of both the locomotive and the carriage are mainly influenced by vehicle speeds, and increases with the speed raising. Slight fluctuations can be found from the maimum acceleration isolines. Influence of bearings stiffness is rather small compared with vehicle speeds. Generally, vehicle s wheel unloading rate can reflect its running safety. And its vertical acceleration affects riding comfort of passengers. From the above analysis, we know that vehicle s running safety and passengers riding comfort are mainly influenced by train s travelling speed. In contrast, influence of bearings vertical stiffness can be almost neglected. (b) Cross section of the pier Figure 4. Typical cross section of the girder and pier unit:mm) The dynamic analysis model of bridges, as shown in Figure 5, contains 5 spans beam and 6 piers. Each pier and the bridge beam are connected by two bearing elements. The vertical stiffness of bearings is decided by reference to the parameters of Taiwan high-speed railway elastic bearings. In this study, 5 kinds of bearings parameters are discussed. They are, 3, 24,, 1 kn/mm respectively. Figure 5. Dynamic analysis model of the bridge 3.2 Results and analysis The train model contains 16 Germany ICE3 cars. Train formation is 2 (locomotive +carriage +4 locomotive +carriage+locomotive). The vehicle speeds discussed are 2, 25, 3, 35 and 4 km/h separately. Trains running across the bridge on the left track is simulated. And travelling distance is 15 m before and after the train travelling on bridges. Stiffness of bearings [kn/mm] Stiffness of bearings [kn/mm] (a) Locomotive (b) Carriage Figure 6. Isolines of wheel unloading rate amplitude

4 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Stiffness of bearings [kn/mm] Stiffness of bearings [kn/mm] (a) Locomotive (b) Carriage Figure 7. Isolines of vertical acceleration amplitude (unit: m/s 2 ) Influence of bearings vertical ity on bridge s dynamic response To investigate the influence of bearings vertical stiffness on bridge s dynamic response, vertical displacement and acceleration of midspan cross section at the third span are analysed. Figure 8 shows the vertical displacement amplitude of the bridge middle span. It can be seen that vertical displacement amplitude increases with vehicle speed raising from 2 km/h to 4km/h, ecept for the condition when the bearings stiffness is 1 kn/mm. In addition, vertical displacement amplitude increases with bearings vertical stiffness decreasing due to the natural frequency of the bridge diminishing. Figure 9 shows time histories of vertical acceleration root mean square(rms) at bridge s midspan section while the train travelling with the design speed of 35 km/h. Vertical acceleration response firstly decreases with bearings vertical stiffness varying from ity to 24 kn/mm. However, it gradually increases with bearings vertical stiffness continuing decreasing from 24 kn/mm to 1 kn/mm. To analyse the influence of bearings vertical stiffness on bridge s acceleration response any further, Figure 1 shows acceleration Fourier spectrum of the bridge s midspan section while the train travelling with the design speed of 35 km/h. No matter what kind of bearings discussed, the vertical acceleration response of bridge s midspan section is rather small while the frequency greater than Hz. The low frequency vibration response is the dominant part in the acceleration response of bridges. Peak frequency of Fourier spectrum is 3.91 Hz or its multiples. It can be found the cyclical loading phenomenon of train travelling Hz corresponds to the train s loading frequency, that is f = 35 / 3.6 / Hz. Displacement amplitude [mm] Figure 8. The vertical displacement amplitude of the bridge middle span RMS acceleration [m/s 2 ] Time [s] Figure 9. The time-history of the vertical root-mean-square acceleration at the middle span Fourier amplitude [m/s 2 ] Frequency [Hz] Figure 1. Acceleration Fourier spectrum at the bridge middle span Combining Yang s idea of studying beam s displacement response with elastic bearings by superposition method[17], some results can be derived through the above analysis. With bearings vertical stiffness decreasing, the static part of bridge s response gradually increases, while the dynamic part of which firstly decreases and then increases gradually. The results agree well with study conclusions from Jiangtong [6] Vibration isolation effects of elastic bearings To investigate vibration isolation effects of elastic bearings, vibration responses of piers are discussed. The 1/3 octave bands is applied to acceleration vibration signals of piers. Since the vibration signal contains various frequency components, Z weighting processing has been applied. Figure 11 shows comparison of the third pier s vibration level in the frequency band from to 4 Hz for different 152

5 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 vehicle speeds. To illustrate vibration isolation effects of elastic bearings with different stiffness, Figure 12 shows vibration attenuation corresponding to conditions as shown in Figure 11. And vibration attenuation is defined as the difference value of the pier s vibration level while bearings are and elastic. If vibration attenuation is positive value, then vibration response of the pier is decreased by elastic bearing, and vice versa Figure 11. Vibration level in the frequency band of ~4 Hz Vibration attenuation [db] Vertical stiffness of bearings [kn/mm] Figure 12. Vibration attenuation of elastic bearings in the frequency band of ~4 Hz Figure 11 and 12 show that vibration isolation effects of elastic bearings are not only affected by vertical stiffness of bearings, but also affected by vehicle speeds. Compared with the above-mentioned single degree of freedom system, vibration isolation effects are not better with lower vertical stiffness. Since the bridge and the pier are not body, it s different from the single degree of freedom system. Train-track-bridge system is a comple time-varying system. The ecitation frequency changes while the train travelling across bridges with different speeds and different vehicles. The natural frequency of the bridge is influenced by vertical stiffness of elastic bearings. So in Figure 11, the pier s vibration level is different with bearings stiffness varying. And in Figure 12, vibration attenuation is quite different with different vehicle speeds although the bearing s vertical stiffness is the same. In Figure 12, the pier s vibration attenuation appears positive value while the bearing s vertical stiffness is kn/mm while the train speed less than 4 km/h. Based on the above analysis, elastic bearings with stiffness of kn/mm, have no negative influence on dynamic response of vehicles and bridges and offer obvious vibration isolation effects. Figure 13 shows 1/3 octave spectrum comparison of the pier s vertical acceleration while the bearings stiffness is and kn/mm respectively (a) At speed of 2 km/h (b) At speed of 25 km/h (c) At speed of 3 km/h (d) At speed of 35 km/h (e) At speed of 4 km/h Figure 13. 1/3 octave of the Z direction vibration level at the pier top for different vehicle speeds 1521

6 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 In Figure 13, vibration frequency spectrums of the pier with the same bearings are quite different because of different speeds. However, Z vibration level of the pier s acceleration appears peak values in the frequency band of 2~8 Hz. Then the conclusion can be obtained that the frequency band of 2~8 Hz is the dominant band of the pier s vertical vibration. In frequency band of 1~4 Hz, Z vibration level of the pier s acceleration appears a fast attenuation, and the attenuation is even faster with elastic bearings of kn/mm than the bearings. It is observed that vibration isolation effects of elastic bearings in high frequency band are rather remarkable. However, the concerned frequency band for environmental vibration is usually.1~8 Hz on the basis of Environmental vibration standard in urban region (GB17-88). Figure 14 shows comparison of Z vibration level of the pier s acceleration in frequency band of ~8 Hz. To analyze the dominant band of elastic bearings vibration isolation effects, Figure 15~17 show comparison of Z vibration level of the pier s acceleration in frequency band of ~1 Hz, 1~2 Hz and 2~8 Hz respectively Figure 14. Z direction vibration level of the vertical acceleration in the frequency band of ~8 Hz Figure 15. Z direction vibration level of the vertical acceleration in frequency band of ~1 Hz Figure 16. Z direction vibration level of the vertical acceleration in frequency band of 1~2 Hz Figure 17. Z direction vibration level of the vertical acceleration in frequency band of 2~8 Hz In Figure 14, in the frequency band of ~8 Hz, Z vibration level of the pier s vertical acceleration is decreased compared elastic bearings with bearings while the vehicle speed varying from 2 to 35 km/h, yet increased when the vehicle speed is 4 km/h. In Figure 15, in the frequency band of ~1 Hz, Z vibration levels of the pier s vertical acceleration with elastic bearings and ones are close to each other. In Figure 16, in the frequency band of 1~2 Hz, Z vibration level of the pier s vertical acceleration with elastic bearings makes a little increase than bearings. In Figure 17, in the frequency band of 2~8 Hz, Z vibration level of the pier s vertical acceleration with elastic bearings appears obvious attenuation than bearings. Comparing Figure 14~17, the value of the pier s vibration level in the frequency band of ~8 Hz, is closest to that in the frequency band of 2~8 Hz. It can be concluded that the vibration level in the frequency band of 2~8 Hz has the maimum contribution to that in the frequency band of ~8 Hz. From the above analysis, the frequency band of 2~8 Hz is the dominant band of the pier s vertical vibration. The vibration contribution of the dominant frequency band is the most remarkable to the whole frequency band. Figure 18 shows vibration attenuation in different frequency bands. 1522

7 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 214 Vibration attenuation (db) ~1 1~2 2~8 ~8 Frequency band (Hz) Figure 18. Vibration attenuation in different frequency bands In Figure 18, vibration attenuation effects of elastic bearings in different frequency bands have much difference. When the train speed is 35 km/h, vibration attenuation in frequency band of ~8 Hz is db, and that in the frequency band of 2~8 Hz is 15.5 db. Since the frequency band of 2~8 Hz is the dominant band, vibration isolation effect in this band makes the most contribution to that in the whole frequency band of ~8 Hz. On the basis of the above analysis, vibration attenuation effects of elastic bearings is obvious in the frequency band of 2~8 Hz. This is because that the natural frequency of the bridge with elastic bearings is = ω Hz 2π = K f 2π M = ( K is the stiffness of the bearings, and M is the mass sum of the bridge s deadweight and secondary dead load). Vibration energy in middle and high frequency band can be isolated to some etent. Vibration attenuation effects mainly depend on that in the dominant band. So as for vibration isolation analysis, what needs to do firstly is to determine the dominant frequency band, then to reduce vibration in this band. 4. CONCLUSIONS In this study, a 3-D train-track-bridge interaction solution was used to investigate the dynamic response of the train and a simply supported bo-girder high-speed railway bridge seated on elastic bearings. Then vibration isolation effects of elastic bearings were discussed in detail. The results are summarized as follows: 1) The wheel unloading rate and vertical acceleration of the vehicle are mainly affected by vehicle speeds. The influence of bearings vertical stiffness, by contrast, can be almost ignored. 2) With the vertical stiffness of bearings decreasing, the static component of bridge s response increases gradually, yet the dynamic component of which firstly decreases and then increases by degrees. 3) Vibration isolation effects of elastic bearings are not only influenced by its vertical stiffness, but also affected by vehicle speeds. Although the stiffness of bearings is the same, obvious differences eist in the vertical acceleration frequency spectrum as a result of different vehicle speeds. When the vertical stiffness is kn/mm, elastic bearings can obtain reasonable vibration isolation effects in a relatively wide speed range. 4) Through the analysis of the pier s Z vibration level by comparing elastic bearings and bearings, we can get to know that the difference of the pier s vibration is really small in the frequency band of ~1 Hz. The pier s vibration in the band of 1~2 Hz is increased, yet is decreased in the band of 2~8 Hz by replacing bearings with elastic bearings. Since 2~8 Hz is the dominant frequency band, vibration isolation effects in this band make a greater contribution. The natural frequency of elastic bearings eists in the band of 1~2 Hz. Elastic bearings are able to decrease vibration in the middle and high frequency band, yet vibration in their natural frequency band is usually increased. AKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos , ), Program for New Century Ecellent Talents in University of China (Grant No. NCET-1-71), the National High Technology Research and Development Program of China ( 863 Program) (Grant No. 211AA11A13), and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No ). REFERENCES [1] Marioni, L. Chen, J. T. Hu. Application of EBP on Taiwan high-speed railway. 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