A plane halftrack model and a periodic tracksubstructure model are established. The spectral element method and spectral transfer matrix method are developed and applied to investigate the track decay rate (TDR) and transmission rate (TR) of the vertical rail vibrations, which can reflect the transmission characteristics in the longitudinal and downward directions, respectively. Furthermore, the effects of different track parameters on TDR and TR are investigated. The results show that the antiresonance frequency of the rail and the outofphase resonance frequency of the rail and sleeper form the boundary frequencies of the highattenuation zone for longwise vibration transmission, where the vibration absorption of the sleeper is significant. The downward transmissibility of vertical rail vibrations is greatest around the antiresonance frequency of the rail. Vertical rail vibrations are primarily transmitted in the downward direction at low frequencies, while they are mainly transmitted along the rail at high frequencies. Stiffer rail pads can make more vibrations transmitted downwards to the sleeper above the antiresonance frequency of the rail, while the changes of other track parameters have different effects on the transmission characteristics. Additionally, a field measurement is performed for verification, and the simulations are well consistent with measurements.
The ballast track has been widely used throughout the world for years. In highspeed operation conditions, the ballast track will face many problems, such as large noises and groundborne vibrations. They can greatly discomfort the residents. Because the energy of noises and vibrations is transmitted from the rail and vertical rail vibrations are dominant vibration sources in the straight ballast track, it is greatly significant to study the transmission characteristics of vertical rail vibrations for solving these problems.
There are two transmission directions of vertical rail vibrations: along the rail and downwards to the sleeper. TDR [
Most of the studies on the transmission of vertical rail vibrations in the ballast track only pay attention to one transmission direction. As a result, they cannot explain the interaction between vibration transmissions in different directions. Jones et al. [
The spectral element method (SEM) is a highly precise and efficient frequencydomain solution method where the spectral element equation is formulated in the frequencydomain and solved by using the spectral analysis method [
In this paper, a plane halftrack model and a periodic tracksubstructure model are established. The SEM and STMM are developed and applied to investigate the TDR and TR of the vertical rail vibrations, and the transmission characteristics are studied in the longitudinal and downward directions, simultaneously. Furthermore, the effects of different track parameters on TDR and TR are investigated. Additionally, a field measurement is performed for verification.
The straight ballast track structure is used for modeling.
To investigate the downward transmission characteristic of vertical rail vibrations, a plane halftrack model is established to obtain the TR, as shown in Figure
Plane halftrack model.
The model consists of a long straight rail, rail pads, sleepers (of half length), and ballast. The model length is 33 m, and the sleeper spacing
The SEM is applied to solve the plane halftrack model. Due to the fact that the calculation precision of SEM will not be affected by the element size, the rail between two adjacent sleepers can be modeled as only one SEM Timoshenko beam element, which will greatly reduce the calculation time. To simulate the nonreflecting boundary conditions, two Timoshenko beam throwoff elements are built at two ends of the long rail, respectively, as shown in Figure
Elements of SEM.
The free vibration of a uniform Timoshenko beam is represented by [
The solutions of (
Spectral nodal displacements and forces of twonode SEM Timoshenko beam element are shown in Figure
Spectral nodal displacements and forces of SEM Timoshenko beam element.
Substituting (
The general solutions to (
Equation (
Solving (
From the first line of (
By combining (
The boundary conditions on the element are
From (
By combining (
When
Since the sign convention at the left end of the Timoshenko beam element in the SEM is different from that in the strength of materials, there are minus
In order to reduce the impact of wave reflection on the calculation precision, throwoff elements are set at two ends of the model. In this section, we derive the spectral element stiffness matrix of Timoshenko beam throwoff element which generates no reflection waves travelling to the left. Figure
Spectral nodal displacements and forces of Timoshenko beam throwoff element.
By letting
The boundary conditions
From (
In this paper, the derivations of the spectral element stiffness matrixes of the mass element and the spring element [
In this section, the spectral structural stiffness matrix of the whole model will be assembled. The spectral element stiffness matrix in the element coordinate can be transformed for the global coordinate using the same way as the finite element method [
To investigate the transmission characteristics of vertical rail vibrations in the longitudinal direction, a periodic tracksubstructure model with the length of
Periodic tracksubstructure model.
The periodic tracksubstructure model is solved by using the STMM, and a fivenode calculation model is built, as shown in Figure
STMM elements.
The spectral structural stiffness matrix of this model is obtained by the SEM. The relation between spectral nodal forces and displacements of the whole model is shown as
Since nodes 2, 4, and 5 are free from external loads,
Let us transform (
For the onedimensional periodic track structure, Bloch’s theorem [
Thus, the dispersion relation between the wave vector
Finally, the TDR can be obtained by [
The same parameters are adopted in the plane halftrack model and periodic tracksubstructure model, as shown in Table
Model parameters.
Track component  Parameter  Symbol  Value 

Rail  Young’s modulus (GPa) 

210 
Shear modulus (GPa) 

80.8  
Crosssectional area (m^{2}) 

7.745 × 10^{−3}  
Mass density (kg/m^{3}) 

7750  
Shear correction factor 

0.5329  
Area moment of inertia (m^{4}) 

3.217 × 10^{−5}  


Rail pad  Vertical stiffness (kN/mm) 

225 
Damping loss factor 

0.25  
Length (m) 

0.17  


Sleeper  Sleeper spacing (m) 

0.55 
Mass (half sleeper) (kg) 

170  
Moment of inertia (kg·m^{2}) 

1.22  
Length (m) 

0.275  


Ballast  Vertical stiffness (per half sleeper) (kN/mm) 

20 
Damping loss factor 

1 
Since the crosssectional deformation of rails occurs above 1500 Hz [
In this section, the transmission characteristics of vertical rail vibrations in ballast track are studied.
A field measurement was carried out for the verification. The measurement was conducted in an existing railway line with CHN60 rails and concrete sleepers. The sleeper spacing is 55 cm. In addition, the ballast tamping was just finished, and the track was in good conditions. The section selected for the measurement was far away from rail joints. The layout of the measurement is shown in Figure
Layout of measurement sensors.
The sensor 1 and sensor 2 were accelerometers with the operating frequency range of 1~15000 Hz, the nominal sensitivity of 5 mV/g, and the measurement range of 1000 g; the sensor 3 was an accelerometer with the operating frequency range of 0.2~2500 Hz, the nominal sensitivity of 500 mV/g, and the measurement range of 10 g. The sensor 1 was placed on the midspan railhead. The sensor 2 was placed on the onsupport railhead. And the sensor 3 was placed on the sleeper. The hammer had the measurement range of 125 kN and the sensitivity of 0.0417 mV/N. The sampling frequency of the measurement was 12800 Hz.
By vertically exciting the middle of the midspan railhead and acquiring the signals of the sensor 1 and hammer, the direct vertical accelerance of the midspan rail can be obtained; by vertically exciting the middle of the onsupport railhead and acquiring the signals of the sensor 2 and hammer, the direct vertical accelerance of the onsupport rail can be obtained; by vertically exciting the middle of the onsupport railhead and acquiring the signals of the sensor 3 and hammer, the transfer vertical accelerance of the sleeper can be obtained. Thus, the TR can be calculated from the measurements of the direct vertical accelerance of the onsupport rail and the transfer vertical accelerance of the sleeper.
The TDR is measured according to the standard BS EN15461:2008+A1:2010 [
Five effective impacts are made at each exciting point, and the averages are taken for the results of each accelerance. The lower limit of the effective frequency range of the measurement depends on the coherent coefficient of the accelerance, and the coherent coefficient is required to be over 0.8 [
The accelerance behaviors are of great use to explain the trends of TDR and TR curves. The accelerances of the rail and sleeper are shown in Figure
Vertical accelerances of (a) rail and (b) sleeper.
Figure
TDR of vertical rail vibrations is shown in Figure
TDR of vertical rail vibration.
As the solutions of the periodic tracksubstructure model contain two pairs of Bloch wave vectors, two TDRs can be obtained in the simulation. The first wave, known as the nearfiled wave, maintains a high decay rate in the whole frequency range. The second wave, known as the propagating wave, has a lower decay rate. Because the nearfiled wave is attenuated greatly, the longwise transmission characteristics are reflected by the propagating wave. Figure
In the low frequency range, simulated TDR decreases with the increase of the frequency, until it starts increasing at 100 Hz. The antiresonance frequency
The transmission rate of vertical rail vibration is shown in Figure
TR of vertical rail vibration.
In the low frequency range, the TR increases with the increase of the frequency. At the antiresonance frequency
As simulations and measurements of the TDR and TR coincide well with each other, the presented models and methods in this paper are sufficient for studying the transmission characteristics of vertical rail vibrations.
In this section, influences of different parameters on TDR and TR are investigated to further reveal the transmission characteristics of vertical rail vibrations.
Vertical rail pad stiffness determines the coupling degree between the rail and sleeper, directly affecting TDR and TR. The simulation results are shown for vertical rail pad stiffness values of 60, 120, 220, and 320 kN/mm in Figure
Influences of vertical rail pad stiffness on (a) TDR and (b) TR.
With the increase of vertical rail pad stiffness, the coupling between the rail and sleeper is strengthened, and both the antiresonance frequency
In the low frequency range, the TR is slightly influenced by the vertical rail pad stiffness. With the increase of the vertical rail pad stiffness, the frequency and value of the TR curve peak both get higher, and the TR value significantly increases at high frequencies.
It can be seen that the increase of vertical rail pad stiffness has a little effect on the transmission characteristics of vertical rail vibration in the low frequency range, but it can make more vibrations transmitted downwards to the sleeper above the antiresonance frequency of the rail.
Vertical ballast stiffness reflects the coupling degree between the sleeper and subgrade and also affects the transmission characteristics of vertical rail vibration. The simulation results are shown for vertical rail pad stiffness values of 20, 40, 60, and 80 kN/mm in Figure
Influence of vertical ballast stiffness on (a) TDR and (b) TR.
Figure
The existence of damping will dissipate the mechanical energy of the vibration system, which directly influences the transmission of vibration. The simulation results are shown for rail pad damping loss factor values of 0.1, 0.25, 0.5, and 1 in Figure
Influence of rail pad damping loss factor on (a) TDR and (b) TR.
With the increase of the rail pad damping loss factor, the resonance of the sleeper will be mitigated, and therefore it will weaken the vibration absorption of the sleeper. As a result, the TDR in the highattenuation zone is decreased. However, the TDR above the outofphase resonance frequency
The increase of the rail pad damping loss factor will decrease the TR near the antiresonant frequency
The simulation results are shown for ballast damping loss factor values of 0.5, 1, and 2 in Figure
Influence of ballast damping loss factor on (a) TDR and (b) TR.
The ballast damping loss factor mainly affects the TDR below the antiresonant frequency
The ballast damping loss factor mainly affects TR near the antiresonant frequency
The simulation results are shown for sleeper mass values of 125, 150, and 170 kg in Figure
Influence of sleeper mass on (a) TDR and (b) TR.
The main influence of sleeper mass on the TDR and TR consists in changing the characteristic frequencies, leading to approximate transverse translations of TDR and TR curves. With the increase of sleeper mass, the characteristic frequencies
This paper investigates the transmission characteristics of vertical rail vibrations in the ballast track. The main conclusions can be drawn as follows.
The work in this paper can be helpful to find the reasons for problems of large noises and vibrations for the ballast track.
The authors declare that they have no conflicts of interest.
The work in this paper was supported by the National Natural Science Foundation of China (nos. 51425804 and 51508479) and the Research Fund for the key research and development projects in the Sichuan Province (2017GZ0373).