HARMONIC RESPONSE ANALYSIS OF DAMPING CHARACTERISTICS OF PLATE VIBRATION REDUCTION TRACK
Shujuan LIANG，Liang GAO，Yanglong ZHONG，Dacheng LI
School of Civil Engineering，Beijing Jiaotong University，Beijing 100044
Abstract：In this paper，a three-dimension finite element dynamics simulation model is constructed to analyze the normal models of floating slab track with ABAQUS finite element software.The influence of track slab thickness and rubber support pad stiffness on vibration-reduction characteristics of precast slab track system is also made clear.The analysis results can provide theoretical basis for the optimal design of the precast slab track structure.
Keywords：urban rail transit，precast vibration damping slab track，vibration characteristics，dynamics
Email：14121188@bjtu.edu.cn
1 Introduction
Urban rail transit plays an important role in alleviating the pressure of urban traffic and promoting the development of urbanization.At present，China has become the most rapidly developing country of urban rail transit in the world.With the rapid development of urban rail transit，it is convenient for the passengers to travel.It also brings many problems，such as environmental vibration，construction speed，quality of construction，operation noise and maintenance[1].Therefore，it is urged to change the way of toughly design，construction and maintenance for urban rail transit to high standard and precision of the high-speed railway[2]，which can improve the quality of urban rail transit and ensure the foundation safe，smooth and durable.In view of the questions above，a high-precision reinforced concrete slab is invented drawing lessons from the ballastless track of high-speed railway technology，which can realize the precast of the slab and the installation of the rubber damping pad.
2 Development Status of Rubber Floating Slab Track
The structure of the precast slab track is similar to the rubber floating slab track.Supporting ways of the rubber bearing floating slab track are divided into 3 kinds，namely overall supporting type，linear supporting type，and distribution supporting type，as shown in Figure 1.
Figure 1 Supporting ways of rubber bearing floating slab track
At present domestic and foreign scholars adopt a variety of research methods on vibration characteristics of floating slab track，such as modal analysis method，the transfer matrix method，frequency response function method and finite element method analysis.The track analysis model has developed from the single degree of freedom system to beam model of three-dimensional multilayer structure，and introduce the method of vehicle-track coupling dynamic analysis.In 1977，Grootenhuis[3]first analyzed floating slab track using a single degree of freedom vibration isolation mechanism and the design principle is given；In 2003，Hussein and Hunt[4]based on Eider-Bernoulli beam theory，established a three-dimensional model of floating slab track；In 2006，they further put forward the double deck beam system，and analyzed the floating slab track system[5]；In 2008，Duschlbauer and Dominik[6]combined the modal test and finite element method，analyzed the mode of floating slab track system from 5 Hz to 250 Hz，and discussed the floating slab vibration level when the train passes.
In China，Xu[7]established vehicle-track rubber supporting floating structure coupling dynamic model of the system，analyzed the dynamics performance of the vehicle and floating slab track system，and vibration isolation effect of track structure；Zhang[8]based on FORTRAN language established the vehicle-track vertical coupling model，analyzed the vibration characteristics of floating slab track under the action of the train load；Liu[9]established a three-dimensional finite element analysis model of track-tunnel-soil environment system，analyzed vibration transfer characteristics of the track structure，discussed ground vibration condition under the influence of different parameters and the insertion loss change.
3 Calculation Parameters and Analysis Model of Vibration Damping Track
3.1 The main calculation parameters
The main calculation parameters of vibration damping track are shown in Table 1.
Table 1 Calculation parameters of vibration damping track
3.2 Coupled finite element model of vibration damping track
The rail is considered as Euler beam，and the beam element is used to discrete.Track slab，rubber pad and the foundation are simulated by solid element，and the geometry and physical properties of the structure can be fully considered.The foundation is simplified to the U type slot.The fastening system is simulated by linear spring damper element.The rail is longitudinally constrained；both ends of the foundation are exerted symmetric boundary condition.Spring damping boundary is adopted at the bottom and both sides of the foundation.Figure 3 and Figure 2 is the coupled model of the vibration damper.
Figure 2 Longitudinal section of spatial coupling model of vibration damper track
Figure 3 Cross section of spatial coupling model of vibration damping track
Harmonic response analysis is based on the modal analysis model，and the length of the model is chosen to be 5 blocks.The spatial coupling model of vibration damper track is shown in Figure 4.
Figure 4 Vibration transfer characteristic analysis model of vibration damper track
4 Analysis of Vibration Characteristics of Track System under Different Working Conditions
4.1 Slab size
The influence of slab thickness on the vibration characteristics of the track system is discussed.
4.1.1 Influence of slab thickness on natural frequency of track system
According to the line condition，the research scope of the slab thickness is 0.20-0.34 m.
Table 2，Figure 5 and Figure 6 show that influence of the slab thickness on the natural frequency of track system is as follows：
Figure 5 Natural frequency of track under different slab thickness
Figure 6 Relationship between thickness of slab and natural frequency of track
（1）The mass of participating of the unit area increases as the increasing of the slab thickness，and the natural frequency of the track system decreases in the low frequency phase of the“rigid body motion”.In this stage，the thicker of the slab，the smaller the natural frequency，and the better the effect of vibration reduction.For example，for the natural frequency of the first order vertical vibration mode，the slab thickness is 0.20 m，the natural frequency is 32.530 Hz；while the slab thickness is 0.34 m，the natural frequency is 25.672 Hz，and the latter has better vibration damping effect than the former.
（2）From the nineteenth order frequency，deformation vibration of the track slab appears，and at this time，bending stiffness of the track increases as the increasing of slab thickness，the natural frequency increases.Therefore，from the eighteenth order frequency to the nineteenth order frequency，the change rule in Figure 5 is opposite.Before the 18th order frequency，the smaller，the slab thickness，the lower，the natural frequency.After the 19th order frequency，the larger，the slab thickness，the higher，the natural frequency.
（3）In the low order frequency，the influence of slab thickness on the natural frequency is smaller than that of the higher order frequency.For example，in the 5th order frequency，the natural frequency of 0.20 m thickness slab is 32.530 Hz，and that of the 0.34 m slab is 25.699Hz，which reduces by 6.83 Hz（21％）.While in the 24th order frequency，the natural frequency of 0.20 m thickness slab is 98.178 Hz，and that of the 0.34 m slab is 146.52 Hz，which increases by 48.34 Hz（49％）.
（4）The influence of subway vibration on the surrounding is mainly low order frequency.Therefore，in this frequency range，the thicker of the slab，the better of the vibration isolation effect.
Table 2 Comparison of vertical vibration of track under different track slab thickness
4.1.2 Effect of slab thickness on transmission characteristics of track vibration
The characteristics of the track vibration transmission are analyzed by the harmonic response analysis method[10]，the results are shown in Figure 7 and Figure 8.
Figure 7 Vertical displacement admittance of foundation under different slab thickness
Figure 8 Vertical acceleration admittance of foundation under different slab thickness
The results show that the increasing of slab thickness can improve the effect of vibration reduction.Taking the frequency 100 Hz as an example，when the slab thickness increases from 0.20 m to 0.34 m，the vibration acceleration level decreases by 10 dB.
4.2 Stiffness of rubber pad
Static stiffness of the several kinds of rubber pats used on the subway respectively is 0.020N/mm3，0.024 N/mm3，0.030 N/mm3.Figure 9and Figure 10 show that the lower the slab stiffness，the better the effect of vibration reduction.At 60 Hz，the vibration acceleration level of rigidity 0.02 N/mm3is reduced by 2.42dB than that of rigidity 0.024 N/mm3.And stiffness 0.024 N/mm3compared with stiffness 0.030 N/mm3，vibration acceleration level decreases by 3.31 dB.
Figure 9 Vertical displacement admittance of foundation with different rubber pad static stiffness
Figure 10 Vibration acceleration admittance of the foundation with different rubber pad stiffness
5 Conclusions
In this paper，the reasonable size and structure of the track are researched in terms of the stability of the rail system and the effect of vibration reduction.Through systematic analysis，the main conclusions are as follows：
（1）Slab size：the increase of slab thickness can effectively reduce the structure of the first order vertical vibration natural frequency.The acceleration level of foundation vibration decreases with the increase of the slab thickness，therefore，increasing the slab thickness can improve the effect of vibration reduction.
（2）Stiffness of rubber pad：the vertical displacement and acceleration of the rail and track slab decrease with the increase of the stiffness of the rubber pad.While the acceleration law of foundation is just the opposite.Except in the vicinity of the natural frequency，the vertical acceleration level of foundation reduces with the decrease of the stiffness of the rubber pad.The results show that the smaller the slab stiffness is，the better the vibration reduction effect is.
To sum up，to get a better effect of vibration reduction，it is needed to reduce the natural frequency of the track down to the natural frequency.Through the analysis above，it is known that the natural frequency of the track can be reduced by increasing the thickness of the slab and reducing the stiffness of the slab.In order to ensure that the rail system does not appear large vertical displacement，the stiffness under slab can’t be too small.
Therefore，it is suggested that change slab thickness and other ways to improve the vibrating mass first and then reasonably choose rubber pads of small stiffnessin the condition of certain track slab type.If the stiffness of the rubber pad is small，the overall stiffness of the slab can be further reduced by changing the overall supporting type to the linear or distribution supporting.It should be noted that，ensure that the system dynamic displacement can’t be too large while meeting the needs of vibration reduction.
In addition，the vertical and horizontal stiffness of the system can be effectively improved by using the position limiting structure.The stability of the track system can be improved by the longitudinal link of the track slab.And these ways can be reasonably adopted when designing.
Acknowledgement
The authors wish to acknowledge the support and motivation provided by Major issues in the science and technology research and development program of China Railway Corporation（No.2014G001-F，No.2015G0001-A），and scientific fund of Beijing Municipal Science and Technology Commission（No.Z151100001615005，No.Z151100001315008）.
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