# New Insights into the Short Pitch Corrugation Enigma Based on 3D-FE Coupled Dynamic Vehicle-Track Modeling of Frictional Rolling Contact

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. FE Model

_{c}supported by the primary suspension, which is represented by spring-damper elements. The sprung mass above a half wheelset is 1/8 of the sprung dynamic load of a whole vehicle, which is approximately a quarter of the sprung mass carried by a bogie. The fastening system and the ballast are also modeled as spring-damper elements. The track parameters are taken from [29] as shown in Table 1 and represent the typical Dutch railway system. In the FE model, the wheel and rail are meshed with 8-node solid elements. To achieve a solution of sufficient accuracy and an acceptable computation time, only the size of the elements in the solution zone is refined (0.8 mm × 0.8 mm in the longitudinal and lateral directions). The elements far from the solution zone are meshed at an element size up to 7.5 cm. These choices are based on [24], which concluded that the contact mechanics solution with an element size of 1.3 mm × 1.3 mm is sufficiently accurate when the FE approach used here is implemented for engineering applications. The total number of elements in the model is 1,135,384; the number of nodes is 1,297,900; and the model length is 18 m. In [30], a track length of 10 m was sufficient for problems of similar frequency and wavelength. The damage mechanism studied in this paper is wear; thus, the wheel and rail materials are assumed to be elastic. A Coulomb friction law is employed with a friction coefficient f

_{C}of 0.6 as in [31]. In the literature, the friction coefficient of dry wheel-rail contact is reported to be between 0.4 and 0.65 [32].

^{−8}s in this model) is smaller than the critical time step (5 × 10

^{−8}s) determined by the Courant criterion [33], convergence is guaranteed. By keeping the time step sufficiently small, the model can include all necessary vibration modes. In the explicit analysis, the frictional rolling is modeled using a surface-to-surface algorithm with the penalty method described in [34]. Because of the nature of explicit integration, the effect of transient rolling and the high-frequency dynamic behavior of the vehicle-track system excited by the moving wheel are automatically included in the solution.

_{1}is used during the explicit process to diminish the effect of vibration excited by imperfect initial equilibrium because of numerical errors from the implicit solution. A total of 10 waves of corrugation with length L

_{2}are introduced after L

_{1}. The traction coefficient μ is defined as follows:

#### 2.2. Corrugation Model

_{2}equals 300 mm. The corrugation considered is located above a sleeper support (as shown in Figure 1b), which has been reported as a position where corrugation is more likely to develop [22]. To examine the wheel-rail contact during the growth process of the corrugation, the amplitudes of A = 0 µm for smooth rail and A = 2.5 µm, 5 µm, 10 µm and 20 µm for corrugated rail are modeled (the peak-to-trough distance is twice the amplitude). By maintaining $\mathsf{\theta}$ between −π/2 and π, contact solutions can be studied at different locations within one complete corrugation wavelength, i.e., the falling edge (P1), the trough (P2), the rising edge (P3) and the peak (P4), as shown in Figure 3b. Figure 3c shows a magnified 3D configuration of the corrugation in the rail surface.

#### 2.3. Validity of the Model

- (1)
- With respect to the contact problems, a method similar to that of [24] is used herein. This method is suitable for resolving dynamic contact problems, and its validity has been demonstrated by the close reproduction of the evolution of squats [7,48]. Moreover, this method has been verified as suitable for resolving static contact problems based on the established solutions of Hertz, Spence, Cataneo, Mindlin and Kalker [24,49].
- (2)
- To validate the structural dynamics, (a) the approach used in this paper can reproduce the hammer test [50]; thus, the model can simulate measured track receptance based on identified track parameters. (b) Furthermore, the model can simulate axle-box acceleration (ABA) measurements [30,51]. Consequently, the model can capture the dynamics of wheels and tracks and the interaction between the vehicle and the track in the relevant frequency range.
- (3)
- With respect to assessing the validity of the model for a vehicle-track system with direct coupling between the contact problem and structural dynamics, the model exhibits a good representation of the dynamic response (spatial and frequency) of corrugation induced by squats. As noted in [21], “the model provides a good explanation for the development of corrugation initiated from isolated railhead irregularities”. Thus, in this paper, the challenge is to extend the model to the study of the more general type of corrugation that does not present clear local irregularities as the source of corrugation initiation.

## 3. Contact Solutions at Corrugation

#### 3.1. Normal Contact

^{2}(A = 20 µm) at P2, i.e., the corrugation trough, whereas it is approximately 170 mm

^{2}when the rail is smooth. Figure 4b displays the maximum contact pressure along the longitudinal axis (y = 0 mm) for different corrugation amplitudes A. As the corrugation amplitude increases from 0–20 µm, the contact pressure at P2 drops significantly to 56% of the original level, i.e., from 1320–745 MPa. This decrease is partly because of the 15% increase in the contact patch size (Figure 4a). At P1 and P3, the maximum pressure declines slightly to 78% of the original level. An increase in pressure of 6% is observed at P4 for the case of A = 20 µm.

#### 3.2. Tangential Contact

^{2}, which is 40% of the whole contact patch (170 mm

^{2}). After the initial fluctuations at A = 2.5 µm and 5 µm, the value at P2 presents the largest change and increases up to 98 mm

^{2}at A = 30 µm, which is 57% of the total smooth contact zone. At the other three positions, the change in the slip zone size is small. Specifically, at P3, this value remains almost at the same level, whereas at P1 and P4, a 10% decrease is observed compared with that of the smooth rail (A = 30 µm).

## 4. Wear and Corrugation Simulation

#### 4.1. Wear Model

_{2}in Figure 2a is extended along the rail to start at x = 0.87 m to cover both the middle sleeper span and sleeper support. The frictional work is calculated in the contact patch for each element for a whole wheel passage, i.e., from element entry until leaving contact. The wear of an element is as follows:

_{f}(x, y) is the frictional work, τ

_{i}(x, y) and v

_{i}(x, y) are the local tangential stress and slip, respectively, and N is the number of time steps ∆t during which the element passes through the contact patch. The wear coefficient k is a constant that depends on the material, lubrication and temperature among other factors [8,9,53,54,55]. In this work, the calculated wear is normalized with respect to the average value of the wear under the smooth rail condition. This normalization is performed because the phase angle between wear and corrugation determines whether the corrugation will grow, which represents our main concern in this paper and is not directly affected by the wear coefficients. It is part of further research to include measurements of the wear coefficients to guide maintenance.

#### 4.2. Prediction of Major Field Observations

#### 4.3. Analysis of Longitudinal and Vertical Rail Modes

#### 4.4. Additional Comments

## 5. Relationship between Contact Forces and Wear as well as New Insights

#### 5.1. Normal and Longitudinal Forces Do Not Exactly Follow Corrugation in Wavelength and Phase

_{N}) and longitudinal contact force (F

_{L}) when the rail is smooth and when corrugation is present. Although the wavelength of the modeled corrugation is constant (30 mm), the wavelengths of the resulting dynamic wheel-rail contact forces vary along the rail (Figure 11a). Specifically, F

_{N}initially lags behind the corrugation, i.e., the peak of F

_{N}appears to the left side of the corresponding corrugation peak. When approaching the sleeper, which has a width of 140 mm and is centered at 1.20 m, the force catches up and becomes in phase with the corrugation at approximately 1.1 m, i.e., near the edge of the sleeper. The force subsequently tends to lead the corrugation. A similar situation applies to F

_{L}(Figure 11b). However, F

_{N}leads F

_{L}initially, and the two forces later tend to be in phase at approximately 1.23 m, with subsequent lagging of F

_{N}. Note that the wear is also strongest at approximately 1.23 m (according to Section 4.2).

_{N}and F

_{L}are 1257 Hz and 1226 Hz, respectively. The corresponding vertical and longitudinal rail vibration modes closest to the corrugation passing frequency of 1297 Hz are 1185 and 1291 Hz, which means that the normal and longitudinal contact forces do not exactly follow the excitation and have frequencies that are different from the natural vibration modes. Previous studies have either explicitly or implicitly assumed that the frequencies of the longitudinal contact force, the normal contact force and the vertical natural mode are the same as the corrugation passing frequency; however, the longitudinal mode has not been previously considered.

#### 5.2. Preferred Frequency of Contact Forces

- (1)
- The frequencies of the normal and longitudinal forces are different. The frequencies of the contact forces are different from that of the excitation, i.e., the passing frequency of the corrugation, and they are sensitive to and change with the corrugation wavelengths. The dynamic forces are stronger at certain wavelengths than others. With the current track parameters, both the longitudinal and vertical contact forces are strongest at λ = 30 mm. This again is in agreement with the observed corrugation wavelength of 30 mm.
- (2)
- The bandwidth of the frequency change of the longitudinal force is between 1116 Hz and 1288 Hz (Figure 12a), which is narrower than that of the normal force (between 1073 and 1361 Hz). The frequency band of the vertical force is broader because it follows the change in the corrugation wavelength, as shown in Figure 12b for the relatively constant ratio of the corrugation passing frequency to the contact force frequency. The frequency of the longitudinal contact force is lower than that of the normal contact force when the corrugation wavelength is short, i.e., between λ = 28 mm and 31 mm (Figure 12b), and vice versa when the corrugation wavelength is longer than 31 mm.
- (3)
- The frequency of the normal contact force is always lower than that of the excitation (Figure 12b). The presence of corrugation is an excitation mainly in the normal direction; thus, the response always follows the excitation. However, the frequency of the longitudinal contact force can be lower or higher than that of the excitation, which likely depends on the nearest natural frequency, as well as the complex relationship between the tangential and normal contact forces. This pattern reveals a strong dependence of the normal contact force on the excitation and a relatively weaker dependence of the longitudinal contact force on the excitation. These dependencies are in line with the narrower band of the frequency change of the longitudinal force compared with the normal force.
- (4)
- In Figure 12b, the frequency of the longitudinal contact force has the largest deviation from the corrugation passing frequency when the wavelength is shorter, i.e., at λ = 28 mm. With increasing λ, the deviation decreases. At approximately λ = 31.5 mm, the frequency of the longitudinal contact force equals the passing frequency. The longitudinal contact force subsequently follows the excitation closely.
- (5)
- The frequency curves (Figure 12b) of the normal and longitudinal contact forces cross each other at a wavelength of 31 mm, where the frequencies of the normal and longitudinal forces are equal. Is an equal frequency of the two forces a condition for the corrugation to initiate, grow and become a wavelength-fixing mechanism? As shown in Section 4.2 and Section 5.1, the wear is strongest when the normal and longitudinal forces F
_{N}and F_{L}are in phase at 1.23 m. Because an equal frequency is a necessary condition for F_{N}and F_{L}to be in phase over many wavelengths, it is indeed a favorable condition for corrugation development.

#### 5.3. Frequencies Converge to Develop Uniform Corrugation

#### 5.4. Importance of the Proposed Modeling Approach and Track Parameters

#### 5.5. Additional Discussions

_{0}, with f

_{0}representing the fundamental frequency, the frequency will jump to harmonics of a higher frequency (n + i)f

_{0}, i ≥ 1 when the speed is increased so that the wavelength remains at the fixed wavelength.

## 6. Conclusions and Future Work

- -
- Along the longitudinal centerline of the contact patch, increases in the contact area, maximum pressure, shear stress and micro-slips at the corrugation crest are small, and some of them even decrease with increases in the corrugation amplitude. However, changes at the trough are large. The large micro-slip and the significantly reduced contact pressure at the trough are the major contributions to the differential wear, which causes corrugation initiation and growth. The dependence of the normal contact force on corrugation excitation is strong, and the dependence of the longitudinal contact force on excitation is relatively weak.
- -
- In addition to the commonly-accepted hypothesis for corrugation studies, i.e., the vertical vibration modes of the vehicle-track system determine the development of corrugation, it is found that the longitudinal vibration modes are also important. Longitudinal modes are likely to be important for the initiation of corrugation, and when the corrugation amplitude is sufficiently large, the vertical modes will be dominant. For intermediate situations, the longitudinal and vertical modes together determine whether the corrugation will grow or be suppressed by wear depending on whether the wear is of the necessary consistent frequency and phase.
- -
- The main frequencies of the vertical and longitudinal vibration modes and contact forces, as well as the resulting wear are different. Consequently, a condition (that might not be unique) for corrugation to consistently initiate and grow should be that the longitudinal and vertical main frequencies are consistent. This consistency may be achieved by the control of certain track parameters, for instance by properly constraining the rail fastening.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Short pitch corrugation and the resulting squats in the Dutch railway network. Wavelength and periodicity are distinguished (wavelength is the distance between two adjacent shining spots of ripples, and periodicity refers to the periodic pattern that contains multiple wavelengths of non-uniform amplitude of corrugation). (

**a**) is at a gentle curve, and (

**b**,

**c**) are on straight tracks. (

**a**) Uniform corrugation with a wavelength of approximately 30 mm. Squats have not yet developed, and a ballasted track with mono-block sleepers and fastenings with a W-shaped tension clamp is shown. Photo taken near Assen, the Netherlands. (

**b**) Non-uniform corrugation of a constant periodicity of a sleeper span. The corrugation wavelength varies largely within a period. Squats have not yet developed, and a ballasted track with duo-block sleepers and fastenings with Deenik clips is shown. Photo taken near Steenwijk, the Netherlands. (

**c**) Non-uniform corrugation with a periodicity shorter than a sleeper span. The corrugation wavelength is approximately 30 mm. The squats were caused by corrugation, and a ballasted track with duo-block sleepers and fastenings with Deenik clips is shown. Photo taken near Steenwijk, The Netherlands.

**Figure 2.**Vehicle-track frictional rolling model in 3D. (

**a**) Schematic diagram of the model. (

**b**) FE model in 3D.

**Figure 3.**Modeled corrugation: (

**a**) field measurement of corrugation (27–33 mm bandpass filtering); (

**b**) schematic diagram of the corrugation and 4 positions (blue line: P1, red line: P2, green line: P3 and magenta line: P4); and (

**c**) illustration of the applied corrugation (Corrugation 2 with P2 at 0.6 m) in the rail surface with 5× magnification (only 3 complete waves are plotted).

**Figure 4.**Influence of corrugation amplitude on the size of the contact patch and the maximum contact pressure along the longitudinal axis y = 0 mm. (

**a**) Contact size; (

**b**) contact pressure.

**Figure 5.**Magnitude of the normal force, contact size and maximum pressure along the longitudinal axis at (y = 0 mm) at the four analyzed positions P1, P2, P3 and P4 (A = 20 μm).

**Figure 6.**Contact pressure, shear stress, contact patch, adhesion-slip distributions and vector graphs of micro-slips (vectors in Figure 6

**b**,

**d**) when A = 20 µm along one corrugation wavelength (P1–P4) (projection onto the xOy plane; Border 1: contact patch border; Border 2: adhesion-slip distribution border).

**Figure 7.**Size of the slip zone as a function of the corrugation amplitude and slip distribution along the longitudinal axis y = 0 mm at the four positions. (

**a**) Size of slip zone; (

**b**) slip along the longitudinal axis y = 0 mm.

**Figure 8.**Distribution of wear at different corrugation amplitudes obtained for the corrugation in Figure 3b with $\mathsf{\theta}$ = 0 and P4 at 1.2 m. The middle of a sleeper is at 1.2 m, and the midpoint between two sleepers is at 0.9 m. (

**a**) High-pass filtered at λ = 20 mm. (

**b**) High-pass filtered at λ = 30 mm.

**Figure 9.**Power spectral density (PSD) of wear λ = 30 mm, with a major peak at 1296 Hz and a secondary peak at 1185 Hz, when A = 20 µm.

**Figure 10.**Modal analysis of the two major frequencies (1185 and 1296 Hz) shown in Figure 9: (

**a**) rail vertical mode at 1185 Hz and (

**b**) rail longitudinal mode at 1291 Hz, in comparison with the closest vertical pin-pin mode at 1100 Hz shown in (

**c**).

**Figure 11.**Dynamic wheel-rail contact forces obtained for the corrugation in Figure 3 with $\mathsf{\theta}$ = 0 and P4 at 1.2 m (1.2 m is in the middle of a sleeper and 0.9 m is at the midpoint between two sleepers.). (

**a**) Normal contact force. (

**b**) Longitudinal contact force.

**Figure 12.**Relationships between the PSD and frequencies of the contact forces and corrugation wavelength obtained for the corrugation in Figure 3b with $\mathsf{\theta}$ = 0, P4 at 1.2 m and A = 20 μm. Note that in (

**b**), f

_{corrugation}is not constant, but changes with the corrugation wavelength. Thus, although the PSD of F

_{L}varies in a narrower band than F

_{N}in (

**a**), the opposite pattern is observed in (

**b**). (

**a**) PSD of F

_{N}(upper) and F

_{L}(lower). (

**b**) Ratio of corrugation over contact force frequency.

Parameters | Values | Parameters | Values | ||
---|---|---|---|---|---|

Wheel load | 116.8 kN | Wheel and rail material | Young’s modulus | 210 GPa | |

Primary suspension | Stiffness | 1.15 MN/m | Poisson’s ratio | 0.3 | |

Damping | 2.5 kNs/m | Density | 7800 kg/m^{3} | ||

Rail pad | Stiffness | 1300 MN/m | Sleeper | Young’s modulus | 38.4 GPa |

Damping | 45 kNs/m | Poisson’s ratio | 0.2 | ||

Ballast | Stiffness | 45 MN/m | Mass density | 2520 kg/m^{3} | |

Damping | 32 kNs/m | Spacing (L) | 0.6 m |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, S.; Li, Z.; Núñez, A.; Dollevoet, R.
New Insights into the Short Pitch Corrugation Enigma Based on 3D-FE Coupled Dynamic Vehicle-Track Modeling of Frictional Rolling Contact. *Appl. Sci.* **2017**, *7*, 807.
https://doi.org/10.3390/app7080807

**AMA Style**

Li S, Li Z, Núñez A, Dollevoet R.
New Insights into the Short Pitch Corrugation Enigma Based on 3D-FE Coupled Dynamic Vehicle-Track Modeling of Frictional Rolling Contact. *Applied Sciences*. 2017; 7(8):807.
https://doi.org/10.3390/app7080807

**Chicago/Turabian Style**

Li, Shaoguang, Zili Li, Alfredo Núñez, and Rolf Dollevoet.
2017. "New Insights into the Short Pitch Corrugation Enigma Based on 3D-FE Coupled Dynamic Vehicle-Track Modeling of Frictional Rolling Contact" *Applied Sciences* 7, no. 8: 807.
https://doi.org/10.3390/app7080807