Laying shock absorber fasteners is one of the effective countermeasures used to reduce the ground vibration induced from urban rail transit. However, this kind of fasteners could cause severe rail corrugation. Based on the “wheel-rail dynamic flexibility difference” mechanism, the optimization and further research of fastener stiffness were performed. With the finite element method, the simple beam and board model of the rail system is established to study the vertical and lateral dynamic flexibility characteristics of rails below 1,200 Hz. Within 5–40 kN/mm, a comparison is made between wheel-rail dynamic flexibility differences corresponding to the vertical stiffness and lateral stiffness of different fasteners. The results show that 20 kN/mm and 10 kN/mm are the least and most suitable vertical stiffness values of fasteners, respectively; 40 kN/mm and 5–10 kN/mm are, respectively, the least and most suitable lateral stiffness values of fasteners. The research and analysis results can be adopted as references for deciding the fastener stiffness of urban track.
Urban rail transit provides extremely convenient transportation. At the same time, the environmental impact of vibration and noise has caused extensive concern in society [
Rail corrugation on the track of shock absorber fastener.
It is well known that many factors affect the emergence and development of rail corrugation; different corrugation has different mechanisms, and it is difficult to use one method to completely cure rail corrugation [
The fixed wavelength mechanism is obtained from the dynamic interaction between wheel set and track. The damage mechanism is obtained from plastic deformation, rolling contact fatigue, and wear. Most of these theoretical and experimental researches are around wheel and track contact, train structure, track structure, and other aspects. However, another key factor influencing the rail corrugation is the relationship between change of discontinuous support stiffness of the track system and the dynamic flexibility of wheel and track [
The formation and development process of rail corrugation is the cyclic process of the interaction between dynamic behaviour of vehicle track and corrugation. When a vehicle passes an irregular track, vibration occurs between the vehicle and track, resulting in uneven wear and plastic deformation of the wheel-rail contact surface; when another vehicle passes, the irregular track and the accumulated uneven wear and plastic deformation of rail contact surface in turn increase the vibration of vehicle and track and result in corrugation on the rail contact surface after repeated cycles [
Due to the discontinuous support characteristic of the rail system, changes occur in the dynamic flexibility of rail in the direction of the track. Especially at the pinned-pinned frequency, the dynamic flexibility of the rail above the fastener is quite different from that in the mid-span position. The instability of the track itself is greatly increased. When the wheels roll along the rail, the wheel-rail dynamic flexibility varies by the positions of the actions points, forming the wheel-rail dynamic flexibility difference. This means under the action of the same exciting force, the vibration displacement of the rail is different from that of the wheel. In the process of wheel-rail contact, such vibration displacement difference will lead to the change and cyclical fluctuation of wheel-rail contact force. The larger the wheel-rail dynamic flexibility difference, the bigger the cyclical fluctuation and the more serious the corrugation is [
The new rail corrugation mechanism, “wheel-rail dynamic flexibility difference” mechanism, proposed by studying the relationship between the change of support stiffness of the rail system and the dynamic flexibility of the wheel-rail system mainly has two aspects: (
Because when rolling along the rail wheels have different positions of action point and the wheel-rail dynamic flexibility difference exists, an unsteady dynamic force is produced on the wheel-rail contact surface mainly in the vertical direction, as shown in Figure
Interaction of wheel-rail rolling.
Vertical-longitudinal model
Vertical-lateral model
When wheels roll along the rail, due to the slope design of wheel-rail contact surface, there is lateral sliding on the contact surface in addition to the longitudinal rolling and longitudinal sliding along the track. Laterally, there exists difference between the wheel and track in dynamic flexibility, especially in the p-p frequency section and longitudinal dynamic flexibility along the rail. Due to the lateral periodic relative vibration displacement difference, sliding friction and wear occur on the wheel-rail contact surface. When the vehicle speed is constant, the component of lateral vibration and vertical vibration of the rail leads to contact force fluctuation and lateral alternating sliding, as shown in Figure
The actual rail system is a structure with infinite length, but computer ability is limited and the calculation model cannot be expanded endlessly. Therefore, a section of the track system is cut for the finite element analysis of the infinitely long rail system. To eliminate the boundary effect and consider the calculation speed, in this paper the model length is a spacing of 131 fasteners. Among them, the fastener spacing is 0.625 m, which means the model is 81.875 m long.
The track structure adopted in the urban rail traffic in China is simplified; a simple beam and board model of the track system is established, as shown in Figure
Simple beam and slab model of the track system.
Dynamic flexibility is the vibration response under the action of unit force and one of the transmission response functions. In this paper, it is used to reflect the vibration characteristics and transfer rules of the track structure.
The dynamic equilibrium equation of the track system is described with the following formula:
where
Assume that vibration displacement is
Then
Therefore, dynamic flexibility can be expressed as follows:
Dynamic flexibility
where
Based on the beam and board model established in this paper, with two adjacent fasteners in the middle of the model as the research object, the unit resonant force is applied on the rail, and the harmonic response analysis is conducted. The dynamic flexibility at the frequency of 0~1,200 Hz [
Main parameters of beam and board model of the track structure.
Part | Item | Unit | Value |
---|---|---|---|
Rail | Section | kg/m | 60 |
Elasticity modulus | MPa | 2.06 × 105 | |
Poisson’s ratio | — | 0.3 | |
|
|||
Fastener | Vertical stiffness | kN⋅mm−1 | 40 |
Vertical damp | kN⋅s⋅m−1 | 10 | |
Lateral stiffness | kN⋅mm−1 | 10 | |
Lateral damp | kN⋅s⋅m−1 | 10 | |
Fastener spacing | m | 0.625 | |
|
|||
Track bed slab | Elasticity modulus | MPa | 3.35 × 104 |
Poisson’s ratio | — | 0.2 | |
Thickness | m | 0.26 |
There are two kinds of working conditions above the rail fasteners and on the cross section 1/2 of the fastener span, the unit resonant force is applied, respectively, in the vertical and lateral directions, and vertical and lateral dynamic flexibility spectrum of the rail is obtained, as shown in Figures
Rail dynamic flexibility spectrums.
Vertical dynamic flexibility spectrum
Lateral dynamic flexibility spectrum
The first peak shown in Figure
Vibration mode of rail resonance.
Vertical rail resonance
Lateral rail resonance
First-order pinned-pinned resonance of the rail.
First-order vertical pinned-pinned resonance
First-order lateral pinned-pinned resonance
In order to further study the vertical and lateral dynamic flexibility change on different positions on the longitudinal beam of the rail, with adjacent four fasteners in the middle of the model as the research object, the coordinate system is established, as shown in Figure
Schematic diagram of loading position of unit resonant force.
Half of the fastener spacing is a cycle. Each cycle is equally divided to select loading points. There are a total of nine different loading positions, as shown in Figure
Dynamic flexibility of different positions on the longitudinal beam of rail at pinned-pinned frequency.
Vertical dynamic flexibility at the pinned-pinned frequency of 1,020 Hz
Lateral dynamic flexibility at the pinned-pinned frequency of 440 Hz
Figure
Dynamic flexibility spectrums on different positions of the rail.
Vertical dynamic flexibility spectrum
Lateral dynamic flexibility spectrum
In general, the wheel-rail contact spring is simply considered to only relate to the displacement of wheel and track. However, the wheel-rail dynamic flexibility differences are only studied in this study, not the displacement of wheel and track. And, therefore, the wheel is considered as a single-mass system without spring. Under the action of unit resonant force, its vibration equation is as follows:
Based on the definition of dynamic flexibility of the wheel:
Then:
Assume that the nonsuspension equivalent mass of the vehicle is 800 kg, typical of most vehicles. The vertical and lateral dynamic flexibility spectrum of wheel is shown in Figure
Wheel dynamic flexibility spectrum.
Figures
Wheel-rail dynamic flexibility spectrums.
Vertical dynamic flexibility spectrum
Lateral dynamic flexibility spectrum
Figure
Figures
Dynamic flexibility of wheel and track on different positions along the rail at the pinned-pinned frequency.
Vertical dynamic flexibility of wheel and track when the pinned-pinned frequency is 1,020 Hz
Lateral dynamic flexibility of wheel and track when the pinned-pinned frequency is 440 Hz
In addition, the wheel-rail dynamic flexibility difference is the difference between dynamic flexibility of the wheel and dynamic flexibility of the rail. The curve of vertical and lateral wheel-rail dynamic flexibility difference with half of fastener spacing on different positions along the rail at pinned-pinned frequency is shown in Figure
The rail wheel-rail dynamic flexibility difference on different positions along the rail at the pinned-pinned frequency.
At the pinned-pinned frequency of 1,020 Hz vertical wheel-rail dynamic flexibility difference
At the pinned-pinned frequency of 440 Hz lateral wheel-rail dynamic flexibility difference
As shown in Figure
As shown in Figure
At the pinned-pinned frequency, the lateral wheel-rail dynamic flexibility difference is much greater than the vertical wheel-rail dynamic flexibility difference. Short-pitch corrugation mainly appears on the curve track with large radius; this may be associated with the large lateral wheel-rail exciting force.
The wheel-rail dynamic flexibility difference rate is the change rate of wheel-rail dynamic flexibility difference, which controls, to some extent, the speed of formation and development of corrugation. This paper uses the cubic function for the fitting of wheel-rail dynamic flexibility difference curve. By solving the slope of fitting function, the wheel-rail dynamic flexibility difference rate on each position is obtained.
The fitting function of vertical wheel-rail dynamic flexibility difference curve is as follows:
The fitting function of lateral wheel-rail dynamic flexibility difference curve is as follows:
In these,
The slope of Formula (
The slope of Formula (
In these,
Therefore, the vertical and lateral wheel-rail dynamic flexibility difference rate curve on different positions along the rail within half of fastener spacing at the pinned-pinned frequency is as shown in Figures
Wheel-rail dynamic flexibility difference rates on different positions along the rail at the pinned-pinned frequency.
Vertical wheel-rail dynamic flexibility difference rate at the pinned-pinned frequency of 1,020 Hz
Lateral wheel-rail dynamic flexibility difference rate at the pinned-pinned frequency of 440 Hz
According to the “wheel-rail dynamic flexibility difference” mechanism and the formation and development mechanism of rail corrugation, in order to reduce and control the vibration noise at the source, research on the optimal selection of parameters for the urban rail system is crucial based on the principle of lowering the wheel-rail dynamic flexibility difference. In this paper, the optimal selection is mainly implemented for the fastener stiffness.
Fastener stiffness in the urban rail structure is usually 5–40 kN/mm [
Design of cases for the vertical stiffness of the fastener.
Fastener parameter | Case |
Case |
Case |
Case |
Case |
---|---|---|---|---|---|
Vertical stiffness/kN⋅mm−1 | 5 | 10 | 20 | 25 | 40 |
Vertical damp/kN⋅s⋅m−1 | 10 | 10 | 10 | 10 | 10 |
Lateral stiffness/kN⋅mm−1 | 10 | 10 | 10 | 10 | 10 |
Lateral damp/kN⋅s⋅m−1 | 10 | 10 | 10 | 10 | 10 |
Fastener spacing/m | 0.625 | 0.625 | 0.625 | 0.625 | 0.625 |
Design of cases for the lateral stiffness of the fastener.
Fastener parameter | Case |
Case |
Case |
Case |
Case |
---|---|---|---|---|---|
Vertical stiffness/kN⋅mm−1 | 40 | 40 | 40 | 40 | 40 |
Vertical damp/kN⋅s⋅m−1 | 10 | 10 | 10 | 10 | 10 |
Lateral stiffness/kN⋅mm−1 | 5 | 10 | 20 | 25 | 40 |
Lateral damp/kN⋅s⋅m−1 | 10 | 10 | 10 | 10 | 10 |
Fastener spacing/m | 0.625 | 0.625 | 0.625 | 0.625 | 0.625 |
Under five kinds of working conditions shown in Table
Different vertical stiffness of fastener.
Wheel-rail dynamic flexibility difference
Characteristic value of wheel-rail dynamic flexibility difference
Wheel-rail dynamic flexibility difference rate
From Figure
In order to further compare different vertical stiffness of fastener, the change rule of the wheel-rail dynamic flexibility difference rate is as shown in Figure
Based on the wheel-rail dynamic flexibility difference rate corresponding to different vertical stiffness of fastener in Figure
The change rule of wheel-rail dynamic flexibility difference under the five kinds of working conditions shown in Table
Lateral stiffness of different fasteners.
Wheel-rail dynamic flexibility difference
Characteristic value of wheel-rail dynamic flexibility difference
Wheel-rail dynamic flexibility difference rate with lateral stiffness of different fasteners
From Figure
In order to further compare lateral stiffness of different fasteners, the change rule of wheel-rail dynamic flexibility difference rate is as shown in Figure
Based on the “wheel-rail dynamic flexibility difference” mechanism, through the research on the vertical and lateral dynamic flexibility characteristics within the range of 0~1200 Hz along the longitudinal beam of the rail on different positions, this paper analyses the impact of vertical stiffness and lateral stiffness of different fasteners within half of fastener spacing on different positions along the rail at the pinned-pinned frequency on the wheel-rail dynamic flexibility difference. The following conclusion is reached.
Due to discontinuous bearing of the rail, at the pinned-pinned frequency, the dynamic flexibility of rail above the fastener and on the mid-span section varies considerably. Above the fastener its dynamic flexibility significantly decreases and the valley is presented. At mid-span its dynamic flexibility significantly increases, and the peak is presented. When the rail is at pinned-pinned frequency, the dynamic flexibility on different positions of the rail increases at first and decreases later above the fastener in the mid-span above the fastener. The change is cyclical. Such dramatic dynamic flexibility change has a great influence on rail corrugation.
At the pinned-pinned frequency, the lateral wheel-rail dynamic flexibility difference is much greater than the vertical wheel-rail dynamic flexibility difference. The short-pitch corrugation mainly appears on the curve track with large radius. It may be associated with the large lateral wheel-rail exciting force. To reduce the wheel-rail dynamic flexibility difference and rail vibration, so as to slow down and control the angle of the rail corrugation, when vertical stiffness of the fastener is 10 kN/mm, the wheel-rail dynamic flexibility difference is the minimum. When the lateral stiffness of fastener is 5 kN/mm, the wheel-rail dynamic flexibility difference is the minimum.
We should reduce the wheel-rail dynamic flexibility difference, and the wheel-rail dynamic flexibility difference rate should not be too high. The most unfavourable value of vertical stiffness of fastener is 20 kN/mm, and the best value of vertical stiffness of fastener is 10 kN/mm; the most unfavourable value of lateral stiffness of the fastener is 40 kN/mm, and the best values of lateral stiffness of the fastener are 5–10 kN/mm.
The authors declare that they have no conflicts of interest.
Thanks are due to the NSFC (National Natural Science Foundation of China, no. 51425804) for the research grant awarded to the corresponding author. The work described in this paper was supported by the National Natural Science Foundation of China (51508479) and the Research Fund for the Key Research and Development Projects in Sichuan Province (2017GZ0373).