Research Article Effect of Track Support Structures on Friction-Induced Vibrations of Wheelset-Track Systems

. Te aim of the present investigation is to study the efect of track support structures on the metro 5th to 8th-order out-of-round (OOR) wheel caused by friction-induced vibrations. Tree fnite element models of metro-leading wheelset-track systems with three types of track support structures were established. Te motion stabilities of these systems were studied using the complex feature value method. Tree unstable vibrations that may cause the formation of the 5th to 8th order wheel OOR profles were predicted. Te simulation results show that the saturated creep forces can lead to unstable vibrations of wheelset-track systems, which are capable of inducing the 5th to 8th-order OOR wheel shapes. Choosing the appropriate value of the fastener damping and using the rail dampers are helpful to reduce or even eliminate wheel polygonal wear caused by these vibrations.


Introduction
In recent years, many cities in China have built new subway lines. However, due to the complexity of some subway lines and the diversity of train operating environments, the wheelrail interaction in some sections is extremely intense, resulting in the formation of out-of-round (OOR) wheels. Wheel polygonization is a typical periodic wheel out-ofroundness, and the order of this OOR wheel refers to the number of peaks of the wheel polygonal profle. When a metro train with OOR wheels runs, there are severe vibrations and noises making passengers uncomfortable. Tese vibrations also can cause fatigue fractures of some vehicle and track components [1,2]. So far, there is no agreement on the generation mechanism of OOR wheels [3][4][5][6][7]. Meinke et al. [8] established a 40-DOF dynamical model with a high-speed wheelset. In this model, the wheelset was regarded as a spring-supported rotor, and the wheel-rail system was regarded as a roller bearing. Te efect of imbalances on wheelset vibrations and wheel tread wear was studied. Te results showed that the wheel tread sufered from polygonal wear when the unbalanced wheelset operated at a high speed. Morys [9] studied the development of an initial wheel corrugation using an ICE-1 vehicle-track dynamical model. A wear model was established to simulate the evolution of radius deviations between the two wheels of the same wheelset over a long-term operation. Te results showed that when a wheelset with an initial wheel corrugation ran on a track supported by a high stifness fastener, wheel-rail normal contact forces varied acutely, thus inspiring the bending resonance of the wheelset. Tis resonance can cause lateral slip and material wear of the wheels, leading to the generation of OOR wheel profles. Based on the high-frequency vibrations of the wheel-rail system and track irregularity, Nielsen et al. [10] studied the short-pitch rail corrugation occurring on smooth curved tracks or on straight tracks, wheel uneven wear caused by pad-wheel braking, and polygonalization of wheel treads. Some inhibition measures were proposed. Johansson and Andersson [11] used a multibody system model to simulate the wheelrail interaction, using an iterative approach to input the feedback from these simulation results into a long-term wear model and proposed a method for predicting the wheel polygonal wear. Simulation results showed that the frequency-fxing mechanisms causing the development of OOR wheels were the wheel-rail vertical resonance of around 40 Hz, the P2 resonance, and the vertical track antiresonance of frequency of about 165 Hz. Dekker [12] studied the phenomenon of wheel polygonal wear by building a nonrigid vehicle-track vibrational model, and he found that the enhancement of a vibrational wheel instability led to the increase of the polygonization of wheels. Increasing the train mass can reduce this nuisance. Wei et al. [13] established a railway vehicle dynamics model with fexible wheelsets to study the generation mechanism of metro OOR wheels. Te bending mode of the wheelset coupling with the bounce motion of the liner motor can lead to wheel polygonal wear. Many vibration tests of vehicles and tracks have been performed by Tao et al. [14] in Chinese subway lines. Te results showed that the direction of the curved track as well as track support structures had important efects on the formation of the 5th to 8th-order OOR wheels. Wu et al. [15] analyzed the stability of a wheelsettrack system with axle-mounted disc brake units by using the complex eigenvalue method. Simulation results showed that the unstable vibrations of frequencies from 580 Hz to 650 Hz caused by axle-mounted disc braking may be transmitted to wheel-rail systems, thus inducing the 20th to 23rd-order polygonal wear on high-speed train trailer wheels. Zhao et al. [16] established a fnite element model of a subway wheelsetrail system. When the wheelset is operated on a tightly curved track, frictional self-excited vibrations that may occur on this system were predicted, causing the formation of OOR wheels. Te relative error of this prediction method was analyzed by comparing the results of feld measurement and model simulation [17].
Some subway lines in China use diferent track support structures [18] due to diferent operating environments, including diferent fasteners and sleepers. Previous feld measurements [17] have shown that in Beijing Metro Line 4, the main vibrational frequencies of the three track test sites supported by diferent track support structures are diferent when a metro train is passing. In addition, changing the track support structure, such as reducing the fastener stifness [14], may be efective in suppressing the 5th to 8th-order OOR wheel. Terefore, it is helpful for understanding the formation mechanism of metro OOR wheels to study the efect of track support structures on the wheel-rail interaction.
When a metro train runs on a tightly curved track, wheel-rail creep forces of the leading wheelset easily reach saturation [19]. Te authors believe that these forces may lead to unstable self-excited vibrations in wheel-rail systems, causing the periodical lateral slip between the rail head and the wheel tread, thus inducing the wheel polygonal wear. Te fnite element model is usually used to study the system stability [20], wheel-rail contact [21], and railway noise and vibrations [22]. Te diferences in the research objects lead to diferences in modelling and analysis methods. In this paper, the fasteners were simulated by a new method in fnite element models. According to the frequency-fxing mechanism and the mode analysis, it can be found that frictional self-excited vibrations may be a cause of metro wheel polygon wear. Figure 1 shows the three types of sharp curved ballastless tracks supported by diferent track support structures in a metro line (line A) [18], including a track with DTVI fasteners and short sleepers (type 1), a track with cologne-egg fasteners (type 2), a track with type-II fasteners, and rubber-booted short sleepers (type 3). Te rail profle is the CN60, and the wheel profle is the LM worn-type. Te values of the stifness and damping of these fasteners [18] are shown in Table 1. Figure 2 shows the fnite element models of the leading wheelset-track systems with diferent types of tracks [23]. In these models, a series of equivalent spring and viscous damping elements are, respectively, established to simulate the support stifness and damping of fasteners, slabs, and the subgrade. In the model with the track type 1, as shown in Figure 2(a), the short sleepers are fxed on slabs by using the tie constraint. Tis constraint is also used to simulate bolts connecting iron pads of fasteners with sleepers. Tere is no sleeper in the model with track type 2 [24], as shown in Figure 2(b), so the bolts connect iron pads to slabs. Te model with track type 3 is shown in Figure 2(c), and in this model, rubber boots connecting the short sleepers to the slabs are modelled as the spring and damping elements [25]. Values of the main parameters of these fnite element models are shown in Table 2 [19].

Numerical Method for Friction-Induced Vibration
Te motion equation of a wheel-rail system without friction is expressed by the following formula [26]: where M, C, and K are, respectively, the mass, damping, and stifness symmetric matrixes, and x is the nodal displacement vector. Tere is no positive real part eigenvalue in the solution of the corresponding eigenvalue equation, so this system is stable. Te friction equation is as follows: where F is the friction, μ is the friction coefcient, and N is the normal contact force. Te value of μ can be obtained by using the following formula: where μ s is the static friction coefcient, u is the frictionvelocity slope, and v 1 is the wheel-rail relative sliding speed. Ten, the motion equation of the system with friction becomes where M f is the friction infuence matrix of mass, C f is the friction infuence matrix of damping, C u is the friction-velocity slope infuence matrix of damping, and K f is the friction 2 Shock and Vibration infuence matrix of stifness. ΔN is the normal force disturbance vector. Te simplifed motion equation is as follows: where M r , C r , and K r are the mass, damping, and stifness asymmetric matrixes, respectively. Te eigenvalue equation of equation (5) can be solved according to the QZ method, and the general solution is where α i + jω i is the eigenvalue and t is the time. As can be seen from the general solution, the nodal displacement will increase over time if α i is greater than 0, and an unstable vibration may occur in the system. Te efective damping ratio ξ for quantifying the tendency of the unstable vibration is defned as when ξ is less than −0.001, the unstable vibration can overcome the damping of the system. Te smaller the effective damping ratio, the stronger the trend of the unstable vibration occurring in the system.

Unstable Vibration of Leading Wheelset-Track Systems.
When a metro train passes through a tightly curved track in China, the speed V c of this train changes from 45 to 55 km/h [19], the wheel diameter is about 0.84 m, and the dominating wavelengths λ of the wheel of the 5th to 8th-order polygon are from 330 to 530 mm [14]. Terefore, the OOR wheel passing frequency range can be computed as follows: Distributions of unstable vibrations of leading wheelset-track systems within the frequency domain were obtained by complex eigenvalue analysis, as shown in Figure 3. In the frequency range of 0∼600 Hz, fve unstable vibrations occur in the system with track type 1, as shown in Figure 3(a). Te minimum frequency of these vibrations is about 51 Hz which is larger than 46 Hz, so it is clear that all vibrations are not within the frequency range of 24∼46 Hz which may cause the 5th to 8th-order OOR wheel. However, the vibrations of frequencies of about 51 Hz, 60 Hz, and 69 Hz may be related to the 9th-order' OOR wheel [19,27]. Figure 3(b) shows the distribution of unstable vibrations of the system with track type 2. Tere is a vibration of frequency of about 41 Hz within the 5th to 8th-order OOR wheel passing frequency range, and the efective damping ratio of this vibration is the smallest, with a value of −0.05243. It means that this unstable vibration of frequency of about 41 Hz is more likely to occur than other unstable vibrations in this system. Figure 3(c) shows the distribution of unstable vibrations of the system with track type 3. Te unstable vibrations of frequencies of about 35 Hz and 42 Hz may cause the 5th to 8th-order wheel polygonal wear. Te vibration of frequency of about 42 Hz has the smallest efective damping ratio with a value of −0.09092, which means that it is more likely to occur than other unstable vibrations in this system.
In summary, there are three unstable vibrations that may lead to the formation of the 5th to 8th-order wheel OOR shapes, which have the frequencies of about 35 Hz, 41 Hz, and 42 Hz, respectively. Modes of these unstable vibrations are, respectively, shown in Figures 4(a)-4(c), and these vibrations mainly occur in wheels and rails. As can be seen from the details (the direction of the red arrow is the direction of the nodal displacement in the fnite element model, and the length of the arrow represents the size of the nodal displacement), each low rail is obviously deformed. Te deformation of the low rail can be described by the low rail outlines indicated by the broken line in Figure 4(d). Te inner wheel tread slips laterally on the low rail head due to this deformation, which causes wear at the contact position of the wheel treads away from the wheel back. It should be noted that the slip is periodical, and the period is the same as that of the unstable vibration. Moreover, from Figure 4, it can be seen that the rail rigidly rotates around the z-axis instead of any elastic deformation (like at curve squealing). As a result, the 5th to 8th-order polygonal wear is formed on the inner wheel tread. In addition, there is no obvious lateral slip between the high rail head and the outer wheel tread, and the contact point between them is near the wheel fange. Terefore, it is difcult for the unstable vibration to cause polygonal wear on the outer wheel tread.
Te speed V t of the metro train operating on a tangential track often changes from 60 to 75 km/h in line A, and the range of the vibration frequency caused by the 5th to 8thorder OOR wheel shapes can be computed as follows:  Te P2 resonance with a frequency of 50∼80 Hz may be excited by this vibration, thus exacerbating the wheel's polygonal wear [14]. After a long-time service, the 5th to 8thorder OOR wheel is gradually formed.

Effect of Fastener Damping
Te lateral damping and vertical damping of fasteners are mainly provided by rail rubber pads, and they vary over time due to rubber ageing. In order to study the efect of fastener damping on the unstable vibrations which may cause the formation of the 5th to 8th-order wheel OOR shapes, different values of fastener damping were, respectively, set in the wheelset-track systems with track type 2 and track type 3. When the value of one damping (lateral or vertical) changes within the range from 0 to 90 kNs/m, the value of another damping remains unchanged.
As shown in Figure 5(a), when the value of the lateral damping increases to 45 kNs/m from 0 kNs/m in the wheelset-track system with track type 2, the efective damping ratio of unstable vibration of frequency of about 41 Hz rises to −0.00386 from −0.05980. When the value of the lateral damping increases to 60 kNs/m or greater, this unstable vibration disappears. When the value of the lateral damping increases to 15 kNs/m from 0 kNs/m in the system with track type 3, as shown in Figure 5(b), the efective damping ratio of unstable vibration of frequency of about 35 Hz rises to −0.00098 from −0.00453, and this vibration disappears when the value of the lateral damping increases to 30 kNs/m or greater. In addition, the efective damping ratio of unstable vibration of frequency of about 42 Hz rises to −0.01348 from −0.08032 when the value of the lateral damping increases to 60 kNs/m from 0 kNs/m. However, when the value of the lateral damping increases continually to 75 kNs/m, the efective damping ratio decline slightly to −0.02165, and then it increases to −0.01626 when the value of the lateral damping increases to 90 kNs/m. Tis means that when the value of the lateral damping is 60 kNs/m, the occurrence likelihood of this unstable vibration is the lowest. Terefore, it can be concluded that increasing the value of the lateral damping to 60 kNs/m is helpful to reduce or even eliminate the 5th to 8th-order OOR wheel caused by the friction-induced vibrations.
When the value of the vertical damping increases to 7.5 kNs/m from 0 kNs/m in the system with track type 2, as shown in Figure 6(a), the efective damping ratio of unstable vibration of frequency of about 41 Hz rises to −0.00544 from −0.0685. Tis unstable vibration disappears when the value of the vertical damping increases to 10 kNs/m or greater. In Figure 6(b), when the value of the vertical damping increases to 7.5 kNs/m from 0 kNs/m in the system with track type 3, the efective damping ratio of unstable vibration of frequency of about 35 Hz increases to −0.00411 from −0.00955, and this vibration disappears when the value of the vertical damping increases to 15 kNs/m or greater. Te efective damping ratio of another unstable vibration of frequency of about 42 Hz rises to −0.00825 from −0.11468 when the value of the vertical damping increases to 37.5 kNs/m from 0 kNs/ m. When the value of the vertical damping increases to 45 kNs/m or greater, this unstable vibration disappears. Terefore, properly choosing the value of the vertical damping can suppress the formation of the 5th to 8th-order OOR wheel.

Efect of Rail Dampers.
Field tests and simulations have proved that the installation of rail dampers can efectively inhibit the formation of noise and rail corrugation [28][29][30], such as the Beijing Metro Line 6 and Shenzhen Metro Line 1 [31] in China. In order to study the efect of rail dampers on unstable vibrations, the fnite element model of rail dampers was built. Te rail damper consists of a mass bar and the elastomeric material layer. Te mass bar was established as an elastomer, and its density, Young's modulus, and Poisson's ratio are 7850 kg/m 3 , 1.9 × 10 11 N/m 2 , and 0.29, respectively. Te elastomeric material layer between the damper and the rail web was simulated by spring and damping elements, as shown in Figure 7. Te lateral and vertical stifness of the elastomeric material layer are 15 MN/ m and 30 MN/m, respectively, and its lateral and vertical dampings are 20 kNs/m and 40 kNs/m, respectively. Parameters of rail dampers in the fnite element model were obtained by feld measurement [32].
As shown in Figure 8(a), the minimum efective damping ratio in the system with track type 2 increases signifcantly to −0.01341, compared to −0.05243, when the rail dampers are not mounted. In the leading wheelset-track system with track type 3, as shown in Figure 8(b), the minimum efective damping ratio also rises to −0.03168 from −0.09092. Tis means that the stability of the system is improved when the rail dampers are installed [33]. Due to the tuned frequency of rail dampers being far higher than the frequency of the unstable vibration which may lead to the formation of the 5th to 8th-order wheel OOR shapes, the  inhibition efect of the rail dampers on the 5th to 8th-order wheel polygonal wear is mainly because it is attached on the rail as an extra mass. Terefore, using the rail dampers can slow down the formation of the 5th to 8th-order OOR wheel induced by these unstable vibrations.

Conclusions
In the present work, three unstable vibrations that may cause the 5th to 8th-order OOR shapes on the metro leading wheel are predicted. Te efect of track support structures on wheel polygonal wear is discussed, and some mitigation measures are proposed. According to simulation results, the following conclusions can be drawn: (1) When a metro train runs on a tightly curved track, unstable vibrations may take place in the leading wheelset-track system due to the saturated wheel-rail creep forces. In the wheelset-track systems with track type 2 and type 3, the unstable vibrations of frequencies of about 35 Hz, 41 Hz, and 42 Hz may cause the periodical slide between the inner wheel tread and the low rail head, producing the 5th to 8th-order wheel OOR shapes. (2) In the wheelset-track system with track type 2, increasing the lateral damping to 60 kNs/m or the vertical damping to 10 kNs/m can eliminate the 5th to 8thorder polygonal wear caused by the unstable vibration of frequency of about 41 Hz. In the system with track type 3, when the lateral damping rises to 60 kNs/m, wheel polygonal wear is the lowest likely to generate. When the vertical damping increases to 45 kNs/m, the 5th to 8th-order polygonal wear can be eliminated. (3) Installing the rail dampers on the track can improve the stability of the wheelset-track system, and the rail dampers can suppress the unstable vibrations that cause the 5th to 8th-order wheel polygonal wear by adding extra mass to the rails.
Tis article preliminarily explores the impact of fastener damping changes in a single direction on wheel polygonal wear. In future work, the strong coupling relationship between the lateral and vertical fastener damping during the ageing process of the rail rubber pad will be studied, and its impact on wheel polygonal wear will be further analyzed.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.  Shock and Vibration