Mechanism of Inner Rail Corrugation on Large-Radius Curves in Metro Systems

Abstract

This paper investigates the underlying cause of inner rail corrugation on large-radius curved tracks in metro systems. A dynamic model of the vehicle–track system (VTS) was developed to analyze the creep characteristics between the guiding wheelset and the rails when the vehicle negotiates large-radius curves under coasting, traction, and braking conditions. A finite element-based complex eigenvalue analysis was conducted to evaluate the stability of the wheel–rail frictional system. The results reveal that under coasting conditions, the wheel–rail creep forces on large-radius curves remain unsaturated, substantially reducing the likelihood of corrugation formation. In contrast, during braking, the creep force may approach saturation on the guiding inner wheel, increasing the possibility of wheel–rail sliding. This braking-induced sliding can trigger friction-induced self-excited vibrations at the wheel–rail interface, leading to the development of inner rail corrugation on large-radius curves.

1. Introduction

Despite extensive research spanning several decades, rail corrugation in metro systems remains an unresolved challenge. Due to the prevalence of curved tracks and the frequent traction and braking of trains, corrugation on metro lines tends to be particularly severe, as shown in Figure 1. As a major contributor to failures in both vehicles and track components [1,2], it demands urgent attention. Over the past few decades, systematic research has established a theoretical framework for corrugation studies, consisting of the material damage mechanism and the wavelength-fixing mechanism [3,4,5]. Since material removal in metro rail corrugation primarily results from wear [6], current investigations predominantly emphasize the wavelength-fixing mechanism, which identifies the vibration sources responsible for corrugation formation. At present, three wavelength-fixing mechanisms are recognized.

The first mechanism attributes rail corrugation to the resonant vibration at the rail–wheel interface excited by rail surface irregularities [7,8,9,10]. Specifically, several studies have interpreted different corrugation patterns in terms of P2 resonance [11,12], pinned–pinned resonance [13], vibration wave interference [14,15], and the third-order bending vibration of rails [16,17,18]—all triggered by surface irregularities. It is worth noting that most theoretical models do not take the curve radius into account. As a result, the predicted likelihood of corrugation exhibits a weak dependence on the curve radius, which contradicts field observations.

The second mechanism attributes rail corrugation to stick–slip oscillations arising in the wheel–rail interface [19,20,21,22], induced by the negative friction–velocity characteristic—namely, the decrease in the friction coefficient with increasing sliding speed when creepage exceeds the saturation limit. This theoretical model links corrugation primarily to the stick–slip motion of the wheelset and shows only a weak dependence on the track structure. In practice, however, the corrugation wavelength has been found to vary with the track structure [23].

The third mechanism attributes rail corrugation primarily to friction-induced self-excited vibrations in the wheel–rail system, triggered by saturated creep forces [24]. Unlike earlier models, this theory incorporates the curve radius as a key input parameter. Theoretical analyses have demonstrated a strong correlation between rail corrugation and curve radius [25]. Within this framework, researchers have clarified the mechanisms responsible for corrugation on the inner rail of sharp curves [26], as well as on tangent tracks equipped with Cologne Egg fastening systems [27]. Moreover, a novel wheelset topology has been proposed that can suppress or even eliminate corrugation, providing valuable insights for active control strategies [28]. In recent years, this theory has gained growing recognition and application among researchers worldwide [29,30,31].

According to statistics, inner rail corrugation is a deterministic phenomenon on tracks with a curve radius R ≤ 350 m, whereas its likelihood on tracks with R ≥ 650 m is much lower [32]. In this study, subway tracks with R ≥ 650 m are defined as large-radius curves. Although extensive research has been conducted on inner rail corrugation in small-radius curves, studies focusing on large-radius curves remain limited. Tao et al. [16] proposed that third-order rail bending vibrations play a key role in the development of inner rail corrugation on large-radius curves, while Li et al. [33] attributed corrugation development primarily to the dynamic interaction between the flexible wheelset and track. Overall, the formation mechanisms and corresponding control strategies for inner rail corrugation in large-radius curves are still not fully understood. This study investigates the phenomenon from the perspective of wheel–rail friction-induced self-excited oscillation, examining the conditions under which creep forces reach saturation and analyzing the system’s dynamic stability to elucidate the mechanisms responsible for inner rail corrugation in large-radius curves.

The structure of this paper is as follows: Section 2 analyzes the wheel–rail creep characteristics when a vehicle negotiates large-radius curves under traction, braking, and coasting conditions. Section 3 examines the occurrence of self-excited oscillations triggered by braking-induced sliding through a practical case study. Finally, the conclusions are presented in Section 4.

2. Analysis of Wheel–Rail Creep Characteristics on Large-Radius Curves

Friction-induced self-excited vibrations may arise once the wheel–rail creep force becomes saturated. Such vibrations cause synchronous fluctuations in the frictional power, ultimately leading to the development of rail corrugation [34]. To investigate the link between friction-induced self-excited vibrations and the corrugation observed on the inner rail of large-radius curves, it is essential to identify the conditions under which the wheel–rail creep forces become saturated. This section develops a dynamic model of the VTS and computes the corresponding creep characteristics under three operating conditions—coasting, traction, and braking—on large-radius curves. Based on these results, the saturation conditions of the creep forces are subsequently determined.

2.1. Development of a Vehicle–Track Dynamic Model

A dynamic model of the VTS (Figure 2) was developed in the commercial software Universal Mechanism 9.0, with vehicle parameters adopted from reference [35]. The vehicle subsystem consists of rigid-body components—the wheelsets, bogie frames, and car body—together with the primary and secondary suspensions. Each rigid-body component has six DOFs. The rails are modeled as Timoshenko beams, with fasteners represented by viscoelastic spring–damper elements. The wheel–rail contact forces are calculated using the Kik–Piotrowski method with a constant friction coefficient of 0.35 [20]. It should be noted that friction is assumed to be constant in this study, whereas the wheel–rail friction coefficient is known to depend on factors such as relative sliding velocity and temperature. Future work will incorporate velocity-dependent or temperature-dependent friction laws to more accurately characterize wheel–rail frictional behavior. The rail adopts the standard 60 kg/m profile, while the wheelset features an LM-type tread profile. The track layout consists of a 100 m initial straight section, a 60 m entry transition curve, a 200 m circular curve, a 60 m exit transition curve, and a 100 m final straight section. The subway vehicles comprise both motor and trailer cars, with only the motor cars capable of providing traction and electric braking forces. Although gearboxes are commonly used in motor cars to transmit motor torque to the wheelsets, the drivetrain is simplified in this study by directly applying the torque to the wheelset axles, omitting the mechanical transmission components.

2.2. Creep Characteristics of the Wheel–Rail Interface During Coasting

Figure 3, Figure 4, Figure 5 and Figure 6 present the time–history curves of the wheel–rail contact forces for the leading wheelset as the vehicle coasts through curves of different radii. It can be observed that when the vehicle negotiates the R = 300 m and 350 m curves, the resultant creep forces ($T=\sqrt{{T_x}^2+{T_y}^2}$) on both wheels approach the sliding friction forces ($F=\mu N$), indicating that the creep forces saturate. Here, T_x and T_y represent the longitudinal and lateral creep forces, respectively; N denotes the normal force, and μ is the coefficient of friction. In contrast, on the R = 800 m and 1000 m curves, the resultant creep forces are markedly lower than the sliding friction forces, suggesting that the creep forces in the contact patches remain unsaturated. Overall, when the train coasts through large-radius curves (R ≥ 650 m), the creep forces at various wheel–rail interfaces of the bogie are unsaturated, making corrugation formation on the rail surface unlikely. Conversely, creep force saturation is intrinsic to small-radius curves (R ≤ 350 m), rendering these sections more susceptible to corrugation. Line-tracing measurements confirmed that corrugation developed on the inner rail of the R = 350 m curve during the trial operation phase, whereas little to no corrugation was observed on the R = 600 m curve or straight track sections [36], thereby supporting this conclusion.

2.3. Creep Characteristics of the Wheel–Rail Interface During Traction

Figure 7 illustrates the traction characteristic curve of a Type B metro train with a 4M2T configuration. Under the AW2 load condition and a supply voltage of 1500 V, the train delivers a maximum tractive force of 348 kN. This corresponds to 87 kN per motor car and a tractive torque of 9135 N·m per axle.

Figure 8, Figure 9 and Figure 10 present the wheel–rail contact forces when the train negotiates large-radius curves under traction conditions, with an initial speed of 60 km/h and a tractive torque of 4000 N·m. Following the application of tractive torque, the resultant creep force increases on the guiding outer wheel, whereas it decreases on the inner wheel. This variation arises from the redistribution of longitudinal creep forces between the inner and outer wheels under traction conditions. Furthermore, with increasing curve radius, the creep force diverges further from its saturation value. Notably, when the vehicle negotiates large-radius curves under traction, the creep force on the inner wheel remains unsaturated.

2.4. Creep Characteristics of the Wheel–Rail Interface During Braking

Figure 11 depicts the electric braking characteristic curve of a Type B metro train. Under the AW2 load condition and a supply voltage of 1650 V, the train produces a maximum electric braking force of 340 kN. This corresponds to 85 kN per motor car and a braking torque of 8925 N·m per axle.

Figure 12, Figure 13 and Figure 14 present the wheel–rail contact forces when the vehicle negotiates large-radius curves under braking conditions. Owing to braking action, the resultant creep force decreases on the guiding outer wheel, whereas it increases on the inner wheel and approaches saturation.

Based on the above analysis, when a vehicle coasts through large-radius curves, the wheel–rail creep forces remain unsaturated. Since metro trains operate in coasting mode for roughly half of their running time, this unsaturated state on large-radius curves substantially reduces the likelihood of rail corrugation. Under traction conditions, the creep force on the guiding inner wheel is less likely to reach saturation, whereas under braking conditions, it tends to saturate more readily, potentially causing relative sliding at the contact interface. Therefore, the formation of corrugation on the inner rail of large-radius curves may be attributed to wheel–rail sliding induced by braking.

3. Case Study: Inner Rail Corrugation on Long Downhill Track

3.1. Phenomenon Description

Figure 15 shows the variation in train speed with distance along the Henggang–Tangkeng section of Shenzhen Metro Line 3. After departing from Henggang Station, the train accelerates to 85 km/h at K32+200 and maintains this speed until K31+550. Between K31+550 and K31+050, the track features a 29‰ downhill gradient, a curve radius of 700 m, and a superelevation of 119 mm. To counteract the effect of gravitational acceleration and comply with the speed limit (not exceeding 70 km/h) at the entrance to the next speed-controlled section (K31+050), the train must apply considerable electric braking force while traversing this section. Field investigations revealed that corrugation had developed on the inner rail of this long downhill segment, with a wavelength of approximately 65 mm. When trains passed through this section, pronounced wheel–rail noise was produced, and frequent breakage of fastener clips clamping the inner rail was observed.

3.2. Study on the Self-Excited Vibrations on the Long Downhill Curved Track

Inner rail corrugation on long, downhill curved tracks leads to squealing noise and accelerates the fracture of fastener clips. This raises the question: What type of vibration induces this inner rail corrugation? As described in Section 2.4, during braking on a large-radius curve, the creep force on the guiding inner wheel reaches saturation. This section examines the dynamic stability of the wheel–rail system under braking-induced creep force saturation, aiming to elucidate the relationship between friction-induced self-excited vibrations and inner rail corrugation formation.

3.2.1. Finite Element Modeling

Figure 16 illustrates the wheel–rail contact configuration when a vehicle passes through a long, downhill and curved section. The high rail’s gauge corner contacts the outer wheel flange, while the low rail head contacts the inner wheel tread. F_SVL, F_SVR, F_SLL, and F_SLR represent the primary suspension forces at the axle ends. The wheel–rail contact angles are denoted by δ_L and δ_R, with F_L and F_R representing the lateral creep forces, and N_L and N_R corresponding to the normal forces. The fastener system and foundation support are modeled using spring–damper elements. Based on this configuration, a numerical model of the guiding wheelset–track system was developed in ABAQUS 2020, as illustrated in Figure 17. The wheel–rail contact interaction is modeled using a contact pair algorithm, where normal contact is defined as “hard contact” and tangential interaction follows the Coulomb friction model. The wheelset, rails, sleepers, and track slab are meshed using C3D8I elements. The use of C3D8I elements helps avoid hourglass modes and shear locking. To improve computational accuracy, a refined mesh with an element size of 2 mm × 2 mm is employed in the wheel–rail contact region. The rail has a total length of 36 m, and the rail ends are pinned with all three translational degrees of freedom constrained.

Table 1. Modeling parameters [23,26].

Parameter Identifier	Value
Lateral suspension force at the left axle end FSLL (N)	3240
Lateral suspension force at the right axle end FSLR (N)	3300
Vertical suspension force at the right axle end FSVR (N)	50,658
Vertical suspension force at the left axle end FSVL (N)	45,300
Contact angle at the outer wheel δL	11°
Contact angle at the inner wheel δR	3.1°
Vertical stiffness of the fastener KRV (MN/m)	40.73
Vertical damping of the fastener CRV (N·s/m)	9900
Lateral stiffness of the fastener KRL (MN/m)	8.79
Lateral damping of the fastener CRL (N·s/m)	1927.96
Foundation support stiffness KF (MN/m)	170
Foundation support damping CF (N·s/m)	31,000

lists the relevant modeling parameters.

3.2.2. Complex Eigenvalue Analysis

Frictional self-excited oscillations may arise when the creep force saturates, potentially leading to dynamic instability in the wheel–rail system. Complex eigenvalue analysis is a widely used method for evaluating the dynamic stability of frictional systems [38,39,40]. The basic principles of this approach are briefly outlined below [29].

The dynamic behavior of the frictional system is governed by the following equation of motion:

$$[{\mathit{M}}_{\mathit{f}}]\ddot{\mathit{u}}+[{\mathit{C}}_{\mathit{f}}]\dot{\mathit{u}}+[{\mathit{K}}_{\mathit{f}}]\mathit{u}=0$$

(1)

where [M_f], [C_f], and [K_f] denote the system’s mass, damping, and stiffness matrices, and u denotes the displacement vector. The characteristic equation corresponding to Equation (1) is formulated as:

$$({\lambda }^2[{\mathit{M}}_{\mathit{f}}]+\lambda [{\mathit{C}}_{\mathit{f}}]+[{\mathit{K}}_{\mathit{f}}])\psi =0$$

(2)

The general solution of Equation (1) can be expressed as:

$$u(t)={\displaystyle \sum {\psi }_i{e}^{{\lambda }_{i}t}}={\displaystyle \sum {\psi }_i{e}^{({\alpha }_i+\mathrm{j}{\omega }_i)t}}$$

(3)

When the real part of an eigenvalue (α_i) is positive, the displacement increases exponentially with time, indicating that the system becomes unstable [35]. The equivalent damping ratio is commonly employed as an indicator of system stability and is given by:

$${\xi }_i=-2\times {\alpha }_i/\left|{\omega }_i\right|$$

(4)

A positive real part of the eigenvalue yields a negative equivalent damping ratio, indicating that the frictional system is dynamically unstable. Generally, a smaller equivalent damping ratio reflects a higher propensity for self-excited vibrations.

3.2.3. Solution Procedure

The procedure for predicting the self-excited vibration of a wheel–rail system is as follows:

Step 1: Apply the vertical and lateral suspension forces to the axle ends of the guiding wheelset.

Step 2: Introduce frictional coupling at the wheel–rail interface by using the keyword *MOTION, TRANSLATION to define the relative sliding velocity between the wheel and rail.

Step 3: Extract the natural frequencies of the wheel–rail system using the AMS solver, with the frequency range specified as 0–1200 Hz.

Step 4: Perform complex eigenvalue analysis to obtain the self-excited vibration frequencies and mode shapes.

3.2.4. Numerical Simulation Results

The dynamic stability of the guiding wheelset–track system under braking-induced wheel–rail sliding conditions was investigated using complex eigenvalue analysis. The calculation results are presented in Figure 18. As shown in the figure, the guiding wheelset–track system exhibits a total of seven self-excited vibration modes. Among these, the mode at 313 Hz (highlighted by the red box) has the smallest equivalent damping ratio, indicating that it is the most likely to be excited. The corresponding mode shape is shown in Figure 19. This self-excited vibration is mainly concentrated on the inner rail, which is consistent with the observed phenomenon. According to the formula v = λf, the passing frequency of the inner rail corrugation can be calculated to be 299–363 Hz, given a corrugation wavelength of 65 mm and a train speed of 70–85 km/h. The predicted self-excited vibration frequency thus falls within the characteristic frequency range of the inner rail corrugation. Consequently, the mechanism underlying inner rail corrugation on long downhill curved tracks is clarified. As a train negotiates such a section, an excessive electric braking force may cause the leading inner wheel to slide, thereby triggering friction-induced self-excited vibrations that ultimately result in corrugation development. Therefore, a key strategy for mitigating this issue on large-radius curves is to regulate the electric braking force to ensure that the wheel–rail creep force remains unsaturated. This conclusion is consistent with the findings of Liu [37], who reported that rail corrugation was completely suppressed after the speed profile was adjusted and electric braking was deactivated on a long downhill section.

4. Conclusions

The underlying mechanism of corrugation in large-radius curved tracks remains not fully clarified, and further investigation is still required. In this study, the onset of inner rail corrugation in large-radius curves is examined from the perspective of wheel–rail friction-induced self-excited vibrations, and the main conclusions are summarized as follows:

(1)

During coasting on a large-radius curve, the creep forces at the contact interfaces generally remain unsaturated. This characteristic accounts for the markedly lower incidence of rail corrugation on large-radius curves in metro systems compared with that on small-radius curves.

(2)

During braking on a large-radius curve, the creep force on the guiding inner wheel may reach saturation, causing relative sliding between the wheel and rail. This braking-induced sliding can excite friction-induced self-excited oscillations, thereby promoting corrugation formation on the inner rail.

(3)

The train braking zone is a high-incidence area for rail corrugation on large-radius curves. Regulating the braking torque to prevent creep force saturation can effectively mitigate the progression of rail corrugation. This strategy offers a valuable reference for track maintenance.

It should be acknowledged that when a train passes over rail weld zones, the P2 resonance of the vehicle–track system may be excited, leading to localized corrugation typically a few meters in length. However, of greater concern are the sections where corrugation extends continuously for several tens or even hundreds of meters. The formation of such long rail corrugation requires sustained vibrational excitation. Friction-induced self-excited vibration triggered by creep force saturation provides the continuous excitation that drives corrugation to extend along the rail. The mechanism by which friction-induced self-excited vibration drives the development of rail corrugation warrants further investigation.