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Impact of non-elliptic contact modelling in wheel wear simulation
Wear 265 (2008) 1532–1541
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Wear journal homepage: www.elsevier.com/locate/wear
Impact of non-elliptic contact modelling in wheel wear simulation Roger Enblom a,b,∗ , Mats Berg a a b
Royal Institute of Technology (KTH), Division of Rail Vehicles, SE-10044 Stockholm, Sweden Bombardier Transportation, Mainline and Metros Division, SE-72173 V¨ aster˚ as, Sweden
a r t i c l e
i n f o
Article history: Accepted 16 January 2008 Available online 2 June 2008 Keywords: Wheel–rail contact Non-elliptic contact Railway wheel wear Wheel profile Simulation
a b s t r a c t Advances in simulation of railway wheel wear in the sense of material removal have drawn the attention to the importance of wheel–rail contact modelling. As a further step of enhancing the used simulation procedure in direction of increased generality and reduced need for application-dependent calibration, the focus of this investigation is the influence of non-elliptic contact models on the wheel wear rate and profile shape. To facilitate evaluation the semi-Hertzian contact procedure Stripes, developed by INRETS in France, has been implemented. To investigate the capabilities of Stripes to assess the contact area and pressure, shape comparisons have been made with other numerical methods for a set of wheel–rail contact situations. The referenced results are based on the linear elastic half-space assumption, elastic finite element analysis, and elastic–plastic finite element analysis. For reference also the elliptic contact area according to Hertz is shown as given by the contact data table of the multi-body simulation code. After exploring the properties of the Stripes procedure with respect to contact area estimation and pressure distribution, the focus is moved to the influence on wear rate, being the principal objective of this investigation. First the wear distribution over the contact patch is studied and compared to results using the elliptic model from the MBS code Gensys and the non-elliptic approach with Kalker’s code Contact. Finally the evolution of the wheel profile is simulated for a few typical cases. This investigation of wear distributions over non-elliptic patches under different operating conditions indicates significant differences compared to both Contact and the applied Hertzian approach. The expansion from single contact occasions to complete simulations indicates comparable material removal rates but relocation towards the flange side. This tendency is apparent in all of the cases shown, however limited to initial wear in tangent run or reasonably mild curve negotiation. © 2008 Roger Enblom. Published by Elsevier B.V. All rights reserved.
1. Introduction Raising performance requirements and increased emphasis on maintenance and life cycle cost for both rolling stock and infrastructure have drawn attention to the possibility of predicting wheel and rail wear by simulation. A successful wear prediction procedure needs to be cross-disciplinary in addressing vehicle dynamics, contact mechanics, and tribology. The focus of this investigation is on the wheel–rail contact modelling and its influence on the predicted wheel wear rate. 1.1. Wear simulation procedure The objective of the current research is to arrive at numerical procedures able to predict wheel and rail profile evolution related
∗ Corresponding author at: Royal Institute of Technology (KTH), Division of Rail Vehicles, Teknikringen 8, SE-10044, Stockholm, Sweden. E-mail address: [email protected] (R. Enblom).
to uniform wear for arbitrary railway vehicle operations [1,2]. The quality of the simulated profiles shall be sufficient for use in MBS simulations. The core technology in the present approach is the combination of Archard’s wear model [3] and the simplified contact theory by Kalker [4] together with a deterministic definition of operating conditions. The vehicle dynamics is simulated using a traditional three-dimensional multi-body model in the commercial MBS code Gensys [5]. The wheel profile evaluation is determined through an iterative procedure looping over a set of simulations representing the variations in operating conditions. Those are typically curve distribution, friction, rail wear status, and track alignment. After each execution of the simulation set the wheel profiles are updated and the procedure continues until the desired mileage is achieved. In the original version the normal contact problem is solved by Hertz’s theory and the tangential one by Kalker’s simplified theory. The wear calculation is carried out subsequent to each transient simulation, evaluating one contact occasion per wheel revolution. The wear is assumed to be constant around the wheel perimeter.
0043-1648/$ – see front matter © 2008 Roger Enblom. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.01.027
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541
Fig. 1. Wear map for dry wheel–rail contact [2]. ki : Archard’s wear coefficients; H: material hardness (GPa).
Archard’s wear model states the dependency between the wornoff material volume V and normal contact force N, sliding distance s, and material hardness H as V =k
Ns , H
where the wear coefficient k depends on the prevailing wear mechanism, depending on contact conditions and tribological environment. Here the governing parameters for assessment of wear mechanism are taken to be contact pressure and relative sliding velocity (Fig. 1). Possible temperature dependence is assumed to be inherently covered by the sliding velocity. Kalker’s algorithm Fastsim [4] for solution of the tangential contact stress problem has been extended with functionality for slip velocity, slip distance, and wear depth. 1.2. Non-Hertzian contact modelling In railway vehicle dynamics simulations the wheel–rail contact is usually modelled relying on Hertz’s theory for calculation of contact patch size, ellipticity, and pressure. For assessment of the tangential properties Fastsim is the prevailing approach. These methods facilitate swift evaluation of the contact conditions and are thus suitable for use in transient simulations. Both theories are based on some fundamental assumptions and simplifications, most often sufficiently fulfilled in the wheel–rail contact. The geometric conditions of constant curvature and infinite half-space may however be violated at certain locations, in particular towards the rail gauge corner and wheel flange root or in case of worn profiles. When it comes to wear simulation the details of contact modelling become more critical since not only integrated quantities, like contact forces, come into play but also stress and slip distributions over the patch. The boundary between adhesion and slip areas needs to be determined as well as the distributions of contact pressure, shear stress, and sliding speed. To improve the contact modelling, several more or less approximate alternative procedures have been proposed, reviewed by Piotrowski and Chollet [6]. One class of methods takes a semi-Hertzian approach by dividing the contact patch in longitudinal strips. The degree of contact for the different strips is evaluated with respect to the profile interpenetration with its maximum depth determined by some preliminary application of Hertz’s theory. The procedures proposed by Knothe and Le-The [7], Kik and Piotrowski [8], Linder and Brauchli [9], and Ayasse and Chollet [10] are of this kind. Knothe and Le-The use an iterative method to determine the contact area by means of the virtual approach of an equivalent
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body. To reduce the number of unknowns while still satisfying the contact condition, an elliptic stress distribution along each strip is assumed and the advantage of neighbouring strip lengths not being independent is recognised. Kik and Piotrowski propose a noniterative approach to estimate the contact patch, defined by the interpenetration area at an approach taken as the elastic deformation calculated using Boussinesq’s influence function. The contact stress distribution is assumed to be elliptical in the direction of rolling. The method by Linder and Brauchli is similar, the difference being the calculation of interpenetration depth taken as a given fraction of the Hertzian body approach. Finally, the development by Ayasse and Chollet as well relies on the interpenetration approach, assigning a virtual ellipse to each strip. The ellipticity is compensated due to the differing geometry relations compared to Hertz and the orientation of the strips takes the local contact angle into consideration. The resulting contact patch is thus not flat but follows the lateral curvature of the profile combination. The contact stress distribution is assumed to be parabolic with reference to Kalker’s simplified theory [4], used for the tangential problem. This method, labelled Stripes, has been used here to investigate the impact on wear calculation of this class of contact models. A different approach is to employ a Winkler bed of separate independent springs. The properties of these springs need however to be calibrated, requiring some supporting theory. The Fastsim algorithm for the tangential contact is of this kind, calibrated with Kalker’s linear theory [4]. An application to the normal contact with ´ non-constant curvatures is proposed by Alonso and Gimenez [11]. Calibration is done with respect to the non-linear stiffness property of Hertzian contact using patch area and ellipticity as parameters. A generalised formulation for solution of both the normal and tangential problem is proposed by Telliskivi [12]. Laterally the springs may be made dependent by selecting a suitable level of neighbourhood influence. The calibration is done with finite element analysis and thus not restricted to the half-space assumption. The local wear depth calculation strategy used in earlier work along with Hertzian contact and the Fastsim algorithm [2,13], may be generalised to non-elliptic contacts. In case of a semi-Hertzian approach the only difference is the integration limits, determined by the shape of the interpenetrated profiles. For a Winkler bed the deflection of the discrete springs serve the same purpose as the surface flexibility in Fastsim in determining the slip state. 2. Evaluation of Stripes To facilitate evaluation of contact modelling influence on calculated wear, the Stripes procedure [10] has been implemented both as a stand-alone module and as a post-processor to the transient vehicle simulation. In the contact patch related diagrams to follow, the reference frame is defined with the longitudinal axis positive pointing to the rear and the lateral axis positive towards the field side. 2.1. Implementation strategy With the purpose of investigating single contact occasions the procedure was implemented as a stand-alone module in Matlab. Thus the implementation as such could be verified with known cases as well as patch wear be calculated and compared to other approaches. This implementation features: • Determination of the rigid contact point at given lateral wheelset displacement. • Evaluation of local profile radii and calculation of initial interpenetration by Hertzian theory.
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Table 1 Verification cases for Stripes implementation Case
Radii (mm) Wheel
Rail
Hertz 1 Hertz 2 A–C [10]a
10000/460 30/460 S1002/460
300 13 UIC60 1:40
a
Load (kN)
78 78 78
Zero lateral displacement.
• Assessment of virtual ellipses and determination of the length of each stripe. The longitudinal semi-axis is compensated and the lateral curvature smoothed as recommended by the authors. • Refinement of the interpenetration by iterating the normal problem part of Stripes to achieve the correct normal force. • Executing the tangential problem part of Stripes calculating shear stresses and slip area boundary. • Calculation of sliding velocity, incremental sliding distance, and wear depth distribution according to the KTH model. For evaluation in conjunction with transient simulation in the complete wear simulation procedure, the extended Fastsim-based post-processor was replaced by Stripes. Here the initial interpenetration was calculated from single point Hertzian contact parameters taken from the standard MBS solution. To overcome a possible inconsistency in using different contact models in the MBS simulation and wear calculation respectively, the interpenetration was iteratively adapted to ensure correct contact force at the given wheelset lateral displacement. This approach is considered justified facilitating this kind of tentative investigation using available MBS simulation software. 2.2. Validation examples
Fig. 2. Stripes implementation verification comparison for Hertzian cases. Solid line: Stripes result. Circles: Hertzian ellipse.
Fig. 3. Stripes implementation verification comparison for the Ayasse–Chollet reference case. Left: contact patch comparison with solid line showing actual implementation and circles the Ayasse–Chollet reference case [10]. Right: contact stress distribution for the actual implementation.
To verify the implementation the contact area has been calculated for three reference cases (Table 1), two purely Hertzian and the Ayasse–Chollet reference case [10]. The contact patch shapes calculated with Stripes compare well with the reference cases (Figs. 2 and 3). As can be seen adequate agreement has been obtained for these elementary cases. The patch shape agrees well for the two implementations. The slight difference is attributed to the determination and smoothing of local curvatures where the Stripes documentation leaves some freedom.
For reference also the elliptic contact area according to Hertz is shown as given by the MBS code contact data table. This table calculation includes an estimation of the elastic deformation by a Winkler approach. In the contact pressure comparisons it should be noted that the Hertzian stress distribution is elliptical while Stripes uses a parabolic distribution. This gives for the same contact area and normal force by definition a higher maximum stress in the latter case.
3. Contact area and pressure distribution
To calculate the contact area on an assumed elastic half-space the complete elastic theory of Kalker, realised in the software code Contact, has been used [4]. Initially three contact positions [14] corresponding to the lateral wheelset displacements 3, 0, and 5 mm have been investigated (Table 2, Fig. 4). Also with respect to contact patch shape Stripes and contact agree well, while the normal stress distributions differ to some extent. Example distributions are given for the neutral and gauge corner cases (Fig. 5).
To explore the capabilities of Stripes to assess the contact area, shape comparisons have been made with other numerical methods, being linear elastic half-space, linear elastic finite element analysis, and elastic–plastic finite element analysis. In the area comparison pictures the patches are scaled to approximately 1:1 size. The lateral extension calculated by Stripes follows the rail envelope surface.
3.1. Comparison with linear elastic half-space
Table 2 Basic contact cases Case
Load (kN)
Field side Neutral Gauge corner
90 90 90
Profile pairing: wheel S1002 on rail UIC60 1:40.
Contact area (mm2 )
Maximum contact pressure (MPa)
Elliptic
Contact
Stripes
Elliptic
Contact
Stripes
129 167 145
151 210 122
149 196 137
1049 806 929
934 849 2327
1202 784 1539
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Fig. 4. Contact area comparison with wheelset lateral positioning. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40. (a) 3 mm offset towards the field side, (b) zero lateral offset, and (c) 5 mm offset towards the gauge corner.
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Fig. 6. Contact area comparison during curving. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:30. (a) tangent, 120 km/h, (b) R = 600 m, wheelset displacement 5.9 mm, (c) R = 400 m, wheelset displacement 6.3 mm, and (d) R = 300 m, wheelset displacement 6.5 mm.
Fig. 7. Contact pressure comparison in a 300-m curve. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:30.
Fig. 5. Contact pressure comparison. Upper: zero lateral offset. Lower: 5 mm offset towards gauge corner. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40.
Reasonably good agreement between Stripes and contact is obtained. It is noteworthy that the relative rank switches towards the gauge corner, likely because of the increasing curvature. A similar comparison has been performed for four simulated cases at different curve radii (Table 3, Fig. 6). The contact conditions have been taken from simulations with a commuter vehicle [13] for the leading outer wheel during quasistatic curve negotiation.
It shall be noted that for the R600 case the contact position is close to the rail head radius transition from 80 to 13 mm. The large area predicted by Stripes for the R300 case includes a low stress connection between the two major contact regions. For the tangent case the contact area sizes agree well for all three cases. Due to the parabolic distribution Stripes however predicts higher maximum stress for the same normal force. For the R600 and R400 cases the Hertzian approach and Stripes both indicate larger contact areas than contact. The non-constant rail curvature in this region together with the different influence on the geometry by the elastic deformation may have an influence. Again Stripes shows higher maximum stresses than the elliptic case. For the final R300 case the comparison appears differently, likely due to the smaller lateral radii and the reduced applicability of the half-space assumption. The contact area shapes agree well as long as the curvatures are shallow enough. For the R300 case the influence of the curvature becomes evident as also illustrated by the stress distribution (Fig. 7).
Table 3 Curving cases Case
Radius (m)
Load (kN)
Speed (km/h)
Displacement (mm)
Tangent R600 R400 R300
∞ 600 400 300
78 99 98 101
120 120 93 80
0 5.9 6.3 6.5
Profile pairing: wheel S1002 on rail UIC60 1:30. a Two-point contact.
Contact area (mm2 )
Maximum contact pressure (MPa)
Elliptic
Contact
Stripes
Elliptic
Contact
Stripes
135 74 72 77a
130 51 59 77
125 70 84 155
867 1994 2038 2132
929 3242 2694 3114
1229 2995 2417 1782
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Table 4 Basic contact cases—FE comparison Case
Field side Neutral Gauge corner
Maximum contact pressure (MPa) FEA
Stripes
825 810 630
1202 784 1539
Fig. 9. Contact area comparison during curving. R = 303 m, 75 km/h. Left: finite element, elastic–plastic [12], right: Stripes. Worn S1002 wheel on worn UIC60 rail. (a) second axle outer wheel and (b) leading axle outer wheel.
Fig. 8. Contact area comparison with wheelset lateral positioning. Left: finite element [14], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40. (a) 3 mm offset towards the field side, (b) zero lateral offset, and (c) 5 mm offset towards the gauge corner.
3.2. Comparison with linear elastic finite element analysis Further the basic contact cases (Table 2) have been compared to linear finite element results with respect to contact patch shape and maximum pressure (Table 4, Fig. 8). The referenced finite element analysis [14] features linear elastic material properties. Other properties, for instance a contact formulation including stiffness reduction due to surface roughness are however nonlinear. As would be expected with the formulation indicated, the finite element analysis predicts lower stresses, mainly due to larger contact areas. In particular the gauge corner case shows a significant difference since the Stripes formulation predicts a more distinguished two-point contact with a sharp stress peak at the slender second contact zone (Fig. 5). It is obvious that the finite element modelling predicts significantly larger contact areas despite the moderate lateral curvature at these locations and the elastic material behaviour. The difference is to a large extent attributed to the reduced contact stiffness due to surface roughness. 3.3. Comparison with elastic–plastic finite element analysis The final contact area comparison reference is a narrow curving case analysed with an elastic–plastic finite element model considering worn wheel and rail profiles (Table 5, Fig. 9). The application is a commuter vehicle in a 303 m radius curve [15]. As can be expected the elastic–plastic analysis predicts larger contact area and lower stresses, here clearly for the 2nd wheel tread
contact. For the leading wheel contact close to the flange the results are less obvious, however showing similar values. The shapes of the contact patches in the finite element analysis have been estimated from the rather coarse description in the reference. 4. Wear rate and distribution After exploring the properties of the Stripes procedure with respect to contact area estimation and pressure distribution, the focus is moved to the influence on wear rate, being the principal objective of this investigation. First the wear distribution over the contact patch is studied, followed by simulation of wheel profile evolution. 4.1. Patch wear evaluation The significance of non-elliptic properties is illustrated by a few basic cases comparing the elliptic case as calculated by Gensys [5], the Contact [4] solution, and the actual Stripes implementation. Graphs showing sliding velocity and wear depth distributions are collected in Appendix A. The Stripes reference case [10] (Fig. 3) is first shown. Under those mild operating conditions the creepages remain small and the adhesion zone dominates the contact patch. Some typical features may however be identified. The shapes of the sliding zones towards the rear edge of the patches do differ significantly due to the non-constant curvature. The sliding velocities (Fig. A.1a) based on Kalker’s simplified theory show in general somewhat smaller slip area but higher magnitudes towards the rear edge compared to the corresponding calculation with Contact, based on Kalker’s complete theory. Another typical effect comparing Stripes with Contact, relying on the half-space assumption, is in the former case the relocation of maxima towards increasing curvature, apparent for both sliding velocity and wear (Fig. A.1b). To obtain adequate wear rates of interest for comparison the selected cases need to have a significant slip zone. It is also the expe-
Table 5 R303 curving cases Case
Outer wheel 2nd axle Outer leading wheela
Load (kN)
66 73
Speed (km/h)
75 75
Displacement (mm)
5.4 11.1
Contact area (mm2 )
Maximum contact pressure (MPa)
e-p FEA
Stripes
e-p FEA
Stripes
173 135
115 149
665 1500
1240 1397
Profile pairing: worn S1002 wheel on slightly worn rail UIC60 1:30. a For this case also linear elastic FEA results are reported: contact area = 48 mm2 , maximum contact pressure = 2500 MPa.
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541 Table 7 Twin-disc wear comparison at 1600 N and 1.5% creep
Table 6 Contact forces Case
Force (kN)
Elliptic
Contact
Stripes
Tangent+
Longitudinal Lateral Normal
21.4 0.6 78.3
21.9 0.7 78.3
21.2 0.6 78.3
R600
Longitudinal Lateral Normal
10.9 6.0 98.8
29.3 0.9 98.7
13.3 13.0 98.7
R300
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Longitudinal Lateral Normal
Method
12.3 26.8 101.3
29.8 4.8 101.2
14.6 16.7 101.2
rience that partial slip conditions are more critical to model than full slip. Here the wear rates for a tangent case with increased longitudinal creep (Tangent+), corresponding to for instance braking, and the curving cases R600 and R300 (Table 3) are compared. The general response and consistency of the three methods to common conditions in terms of normal force and creep rates are illustrated by the resulting contact forces (Table 6). In the tangent case all methods agree very well. In the curving cases principal differences are revealed, in particular the tendency by Contact to predict a high longitudinal force. The shear stresses are related to the sliding velocities and thus to the wear coefficients while the normal stresses influence the amount of worn-off material. The sliding velocity distributions for the Tangent+ case (Fig. A.2a) basically confirm the difference in the tangential contact theory applied. Thus the maximum sliding velocity at the rear edge of the contact patch is smaller for Contact than for the other methods. The effect on the wear distribution (Fig. A.2b) is that the transition to the severe wear regime at 0.2 m/s (Fig. 1) is just reached for this case but significantly exceeded for the other cases. In the R600 case, with contact location in the vicinity of the rail head radius transition, Contact predicts full slip while the other methods show partial slip conditions (Fig. A.3a). This principal difference is attributed to the higher shear stresses in the Contact case. The effect on the wear distribution (Fig. A.3b) is dramatic. Both Contact and Stripes show contact stresses (Table 3) in the seizure region, above 2.2 GPa (Fig. 1). This portion of the patch lies however in the Stripes case within the adhesion zone while the Contact case experiences excessive wear in the middle region. In the R300 case all methods predict a full slip situation (Fig. A.4a). Again Contact predicts higher sliding velocities. In addition the contact pressure at the gauge corner contact enters the seizure region, giving a very high wear rate (Fig. A.4b). The rear half of the tread contact region reaches the mild wear region above 0.7 m/s (Fig. 1). The other two methods show differences in location of severe wear but none of them reaches seizure region. This kind of single contact occasion investigations is rarely found in the literature. The predicted initial wear rate may however be compared to simulated wear using the semi-Winkler approach as well as twin-disc test results reported by Telliskivi [12]. The case chosen for comparison is the most severe with the properties: normal load 1600 N, corresponding nominal Hertzian pressure 1400 MPa, creep 1.5%, nominal coefficient of friction 0.6, disc radius 32.5 mm, and circumferential speed 1.0 m/s (Table 7). It shall be noted that in this scale 1.5% creep corresponds to a sliding velocity of 0.015 m/s, indicating mild wear. Furthermore, at the initial stage the contact is basically elliptic. The principal difference between Stripes and the elliptic case is thus the stress distribution model. Despite that, good agreement with the experimental wear rate is observed.
Method
Pressure (MPa)
Traction coefficient
Wear rate (g/rev)
Elliptic Stripes Semi-Winkler (no stiffness scaling) Semi-Winkler (stiffness scaling /2) Twin-disc test
1400 1870a 1360
0.59 0.59 0.59
6.3b 6.1b 17.1c
1360
0.60
39.1c
a b c
–
6.7c
0.6–0.55
Parabolic stress distribution. Initial wear rate based on nominal geometry. Average wear rate over 30,000 revolutions.
Table 8 Wear illustration cases Case
Radius (m)
Speed (km/h)
Cant (mm)
Lubr
Tangent R1500 R600
∞ 1500 600
120 120 104
0 150 150
Nat Nat Lub
Profile pairing: wheel S1002 on rail UIC60 1:30 coefficient of friction: 0.3.
4.2. Profile evolution It is obvious that the different methods may show significant differences at the detailed level as represented by single contact patch examples above. To investigate the effect on the profile evolution, the accumulated wheel wear for a few type cases has been simulated (Table 8) comparing the elliptic approach with Stripes. To account for natural lubrication by the presence of for instance moist and contamination, the wear map for dry conditions (Fig. 1) has been scaled down by a factor 5.5 as previously estimated [2]. The 600 m curve is assumed to be equipped with lubrication devices and the corresponding wear downscaling factor 11 is used. The curve passing cases include entrance and exit transition curves and a typical circular portion. Equal amount of right-hand and left-hand curves as well as forward–backward symmetry is assumed. To achieve reasonable wear and accommodate at least a few wear steps, the total running distance for each type case was set to a few thousand km. The commuter car model from previous work has been used in these simulations as well. This vehicle has a reasonably soft wheelset guidance. The results are presented as worn-off area (Table 9) and wear depth distributions across the outer axle wheel profile. In the diagrams the lateral co-ordinate is taken positive towards the field side (Fig. 10). In general the results from these type cases confirm the tendency seen at the patch level that Stripes redistributes stresses and consequently wear towards increasing curvature. For the tangent case (Fig. 11) the maximum radial depth is increased by about three times although the worn-off area remains the same. It shall also be noted that the difference in perpendicular wear depth is slightly smaller due to the different contact angles, since the Stripes contact patch follows the curvature. Table 9 Simulated wear comparison Case
Running distance (km)
Tangent R1500 R600
6000 6000 3000
Worn-off area (mm2 ) Elliptic
Stripes
0.19 0.97 1.5
0.19 0.89 2.4
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Fig. 10. Wheel profile co-ordinates.
Fig. 13. Radial wear distribution across the profile for the R600 case. Running distance 3000 km. Bold line: Stripes. Thin line: elliptic.
Fig. 11. Radial wear distribution across the profile for the tangent case. Running distance 6000 km. Bold line: Stripes. Thin line: elliptic.
The outcome of the mild 1500 m curving case (Fig. 12) is essentially the same. The relative difference in peak depth at the flange root is however slightly smaller since the elliptic method starts to develop a peak towards the flange root. At the opposite side, showing the wear running as inner wheel, the elliptic approach predicts about 50% larger depth. Narrowing the curve to 600 m (Fig. 13) reveals somewhat larger differences. The difference in peak depth is again about three times but in the Stripes case more pronounced. The worn-off area is in this case significantly larger. Towards the field side both methods show similar results. The differences observed so far between wear calculations based on the elliptic approach and the Stripes method, respectively, would suggest higher flange wear in the latter case. In addition the difference in tread wear distribution shape suggests differences in the equivalent conicity.
Fig. 12. Radial wear distribution across the profile for the R1500 case. Running distance 6000 km. Bold line: Stripes. Thin line: elliptic.
An early simulation using non-elliptic contact is reported by Knothe and Le-The [16], comparing the initial wear distribution calculated applying their strip method [7] with an elliptical approach. The case studied is a single wheelset running in sinusoidal motion on tangential track. The profile combination is nominal S1002 wheel on UIC60 rail. Due to the prescribed lateral motion the contact locations are determined by the profile geometry and the wear rates in the boundary regions do not differ significantly. The principal difference is found in the central region where the elongated shape of the non-elliptic contact area predicts more wear than the elliptic approach. Linder and Brauchli [9] apply their strip-based contact model to simulation of wheel wear at a Eurofima car running in the 300-m curves on the Gotthard line. The conclusion is that the non-elliptic approach improves the results in particular in the tread—flange transition zone due to better modelling of conformance. 5. Conclusions The focus of this investigation is the influence of non-elliptic contact models on the wheel wear rate and profile shape intended as a further step of enhancement towards increased generality of the emerging simulation procedure [1,13]. The possible need for application-dependent calibration is as a consequence anticipated to be reduced. To facilitate evaluation the semi-Hertzian procedure Stripes [10], has been implemented. The results may be considered somewhat tentative since Stripes is applied as a post-processor to the MBS simulation, the latter using its standard contact model. The investigations of wear distributions over the non-elliptic patches under different operating conditions indicate significant differences compared to both Contact [4] and the applied Hertzian approach [5]. The expansion from single contact occasions to complete simulations indicates comparable predicted material removal rates but relocation towards the flange side. The anticipated consequences would be higher flange root wear rate and differences in equivalent conicity. This tendency is apparent in all of the cases shown here, however limited to initial wear in tangent run or reasonably mild curve negotiation. Qualitatively similar observations have been made by Linder and Brauchli [9], simulating wear in 300-m curves. Simulated initial wear rate agrees well with twin-disc test results reported by Telliskivi [12]. The results and experience of this initial study suggest several paths for further research. An evident continuation step is application of this kind of nonelliptic contact modelling to the wear simulation of a complete operation. This would cover wide variety of contact conditions rather than a few single curves.
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Investigation of the performance of a Winkler bed approach of the kind proposed by Telliskivi [12] would be of great interest as an alternative. Both elementary cases and complete operations should be investigated. The aim of this investigation has been impact of contact modelling on the wear distribution. The results however also indicate interesting differences between contact model properties, for instance the determination of creep forces in case of a curved contact patch (Table 6). Investigation of the limitations of the half-space assumption, usually applied, should be done.
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Acknowledgements The authors gratefully acknowledge the financial support from Swedish National Rail Administration (Banverket), Stockholm Transport (SL), Bombardier Transportation, The Association ¨ ˚ of Swedish Train Operators (Tagoperat orerna), and Interfleet Technology, making this research possible. Appendix A See Figs. A.1–A.4.
Fig. A.1. (a) Sliding velocity comparison for the Stripes reference case. (b) Wear depth comparison for the Stripes reference case.
Fig. A.2. (a) Sliding velocity comparison for the tangent case. (b) Wear depth comparison for the tangent case.
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Fig. A.3. (a) Sliding velocity comparison for the R600 case. (b) Wear depth comparison for the R600 case.
Fig. A.4. (a) Sliding velocity comparison for the R300 case. (b) Wear depth comparison for the R300 case.
References [1] R. Enblom, Simulation of wheel and rail profile evolution—wear modelling and validation, Royal Institute of Technology (KTH) Report TRITA AVE 2004:19, ISBN 91-7283-806-X, Stockholm, 2004. [2] T. Jendel, Prediction of wheel profile wear—methodology and verification. Royal Institute of Technology (KTH) Report TRITA FKT 2000:49, ISSN 1103-470X, Stockholm, 2000. [3] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (1953) 981–988. [4] J.J. Kalker, Rolling contact phenomena—linear elasticity, in: CISM International Centre for Mechanical Sciences, vol. 411, Springer, New York, 2001.
¨ [5] www.gensys.se: Homepage of DEsolver. Supplier of Gensys, Ostersund, 2006. [6] J. Piotrowski, H. Chollet, Wheel–rail contact models for vehicle system dynamics including multi-point contact, Vehicle Syst. Dyn. 43 (6–7) (2005) 455–483. [7] K. Knothe, H. Le-The, A contribution to the calculation of the contact stress distribution between two elastic bodies of revolution with non-elliptical contact area, Comput. Struct. 18 (1984) 1025–1033. [8] W. Kik, J. Piotrowski, A fast, approximate method to calculate normal load at contact between wheel and rail and creep forces during rolling, in: Proceedings of the 2nd Mini Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Budapest, 1996. [9] C. Linder, H. Brauchli, Prediction of wheel wear, in: I. Zobory (Ed.), Proceedings of 2nd Mini Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Budapest, 1996, pp. 215–223.
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541 [10] J.B. Ayasse, H. Chollet, Determination of the wheel rail contact patch in semiHertzian conditions, Vehicle Syst. Dyn. 43 (3) (2005) 161–172. ´ [11] A. Alonso, J.G. Gimenez, A new method for the solution of the normal contact problem in the dynamic simulation of railway vehicles, Vehicle Syst. Dyn. 43 (2) (2005) 149–160. [12] T. Telliskivi, Simulation of wear in a rolling-sliding contact by a semi-Winkler model and Archard’s wear law, Wear 256 (2004) 817–831. [13] R. Enblom, M. Berg, Simulation of railway wheel profile development due to wear—influence of disc braking and contact environment, Wear 258 (2005) 1055–1063.
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[14] S. Damme, U. Nackenhorst, A. Wetzel, B.W. Zastrau, On the numerical analysis of the wheel–rail system in rolling contact, in: K. Popp, W. Schielen (Eds.), System Dynamics and Long-Term Behaviour of Railway Vehicles, Track and Sub-grade, Lecture Notes in Applied Mechanics, vol. 6, Springer, 2002, pp. 155–174. [15] T. Telliskivi, Contact mechanics analysis of measured wheel–rail profiles using the finite element method, J. Rail Rapid Transit 215 (F2) (2001) 65–72. [16] K. Knothe, H. Le-The, A method for the analysis of the tangential stresses and the wear distribution between two elastic bodies of revolution in rolling contact, Int. J. Solids Struct. 21 (1985) 889–906.
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Impact of non-elliptic contact modelling in wheel wear simulation Roger Enblom a,b,∗ , Mats Berg a a b
Royal Institute of Technology (KTH), Division of Rail Vehicles, SE-10044 Stockholm, Sweden Bombardier Transportation, Mainline and Metros Division, SE-72173 V¨ aster˚ as, Sweden
a r t i c l e
i n f o
Article history: Accepted 16 January 2008 Available online 2 June 2008 Keywords: Wheel–rail contact Non-elliptic contact Railway wheel wear Wheel profile Simulation
a b s t r a c t Advances in simulation of railway wheel wear in the sense of material removal have drawn the attention to the importance of wheel–rail contact modelling. As a further step of enhancing the used simulation procedure in direction of increased generality and reduced need for application-dependent calibration, the focus of this investigation is the influence of non-elliptic contact models on the wheel wear rate and profile shape. To facilitate evaluation the semi-Hertzian contact procedure Stripes, developed by INRETS in France, has been implemented. To investigate the capabilities of Stripes to assess the contact area and pressure, shape comparisons have been made with other numerical methods for a set of wheel–rail contact situations. The referenced results are based on the linear elastic half-space assumption, elastic finite element analysis, and elastic–plastic finite element analysis. For reference also the elliptic contact area according to Hertz is shown as given by the contact data table of the multi-body simulation code. After exploring the properties of the Stripes procedure with respect to contact area estimation and pressure distribution, the focus is moved to the influence on wear rate, being the principal objective of this investigation. First the wear distribution over the contact patch is studied and compared to results using the elliptic model from the MBS code Gensys and the non-elliptic approach with Kalker’s code Contact. Finally the evolution of the wheel profile is simulated for a few typical cases. This investigation of wear distributions over non-elliptic patches under different operating conditions indicates significant differences compared to both Contact and the applied Hertzian approach. The expansion from single contact occasions to complete simulations indicates comparable material removal rates but relocation towards the flange side. This tendency is apparent in all of the cases shown, however limited to initial wear in tangent run or reasonably mild curve negotiation. © 2008 Roger Enblom. Published by Elsevier B.V. All rights reserved.
1. Introduction Raising performance requirements and increased emphasis on maintenance and life cycle cost for both rolling stock and infrastructure have drawn attention to the possibility of predicting wheel and rail wear by simulation. A successful wear prediction procedure needs to be cross-disciplinary in addressing vehicle dynamics, contact mechanics, and tribology. The focus of this investigation is on the wheel–rail contact modelling and its influence on the predicted wheel wear rate. 1.1. Wear simulation procedure The objective of the current research is to arrive at numerical procedures able to predict wheel and rail profile evolution related
∗ Corresponding author at: Royal Institute of Technology (KTH), Division of Rail Vehicles, Teknikringen 8, SE-10044, Stockholm, Sweden. E-mail address: [email protected] (R. Enblom).
to uniform wear for arbitrary railway vehicle operations [1,2]. The quality of the simulated profiles shall be sufficient for use in MBS simulations. The core technology in the present approach is the combination of Archard’s wear model [3] and the simplified contact theory by Kalker [4] together with a deterministic definition of operating conditions. The vehicle dynamics is simulated using a traditional three-dimensional multi-body model in the commercial MBS code Gensys [5]. The wheel profile evaluation is determined through an iterative procedure looping over a set of simulations representing the variations in operating conditions. Those are typically curve distribution, friction, rail wear status, and track alignment. After each execution of the simulation set the wheel profiles are updated and the procedure continues until the desired mileage is achieved. In the original version the normal contact problem is solved by Hertz’s theory and the tangential one by Kalker’s simplified theory. The wear calculation is carried out subsequent to each transient simulation, evaluating one contact occasion per wheel revolution. The wear is assumed to be constant around the wheel perimeter.
0043-1648/$ – see front matter © 2008 Roger Enblom. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2008.01.027
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541
Fig. 1. Wear map for dry wheel–rail contact [2]. ki : Archard’s wear coefficients; H: material hardness (GPa).
Archard’s wear model states the dependency between the wornoff material volume V and normal contact force N, sliding distance s, and material hardness H as V =k
Ns , H
where the wear coefficient k depends on the prevailing wear mechanism, depending on contact conditions and tribological environment. Here the governing parameters for assessment of wear mechanism are taken to be contact pressure and relative sliding velocity (Fig. 1). Possible temperature dependence is assumed to be inherently covered by the sliding velocity. Kalker’s algorithm Fastsim [4] for solution of the tangential contact stress problem has been extended with functionality for slip velocity, slip distance, and wear depth. 1.2. Non-Hertzian contact modelling In railway vehicle dynamics simulations the wheel–rail contact is usually modelled relying on Hertz’s theory for calculation of contact patch size, ellipticity, and pressure. For assessment of the tangential properties Fastsim is the prevailing approach. These methods facilitate swift evaluation of the contact conditions and are thus suitable for use in transient simulations. Both theories are based on some fundamental assumptions and simplifications, most often sufficiently fulfilled in the wheel–rail contact. The geometric conditions of constant curvature and infinite half-space may however be violated at certain locations, in particular towards the rail gauge corner and wheel flange root or in case of worn profiles. When it comes to wear simulation the details of contact modelling become more critical since not only integrated quantities, like contact forces, come into play but also stress and slip distributions over the patch. The boundary between adhesion and slip areas needs to be determined as well as the distributions of contact pressure, shear stress, and sliding speed. To improve the contact modelling, several more or less approximate alternative procedures have been proposed, reviewed by Piotrowski and Chollet [6]. One class of methods takes a semi-Hertzian approach by dividing the contact patch in longitudinal strips. The degree of contact for the different strips is evaluated with respect to the profile interpenetration with its maximum depth determined by some preliminary application of Hertz’s theory. The procedures proposed by Knothe and Le-The [7], Kik and Piotrowski [8], Linder and Brauchli [9], and Ayasse and Chollet [10] are of this kind. Knothe and Le-The use an iterative method to determine the contact area by means of the virtual approach of an equivalent
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body. To reduce the number of unknowns while still satisfying the contact condition, an elliptic stress distribution along each strip is assumed and the advantage of neighbouring strip lengths not being independent is recognised. Kik and Piotrowski propose a noniterative approach to estimate the contact patch, defined by the interpenetration area at an approach taken as the elastic deformation calculated using Boussinesq’s influence function. The contact stress distribution is assumed to be elliptical in the direction of rolling. The method by Linder and Brauchli is similar, the difference being the calculation of interpenetration depth taken as a given fraction of the Hertzian body approach. Finally, the development by Ayasse and Chollet as well relies on the interpenetration approach, assigning a virtual ellipse to each strip. The ellipticity is compensated due to the differing geometry relations compared to Hertz and the orientation of the strips takes the local contact angle into consideration. The resulting contact patch is thus not flat but follows the lateral curvature of the profile combination. The contact stress distribution is assumed to be parabolic with reference to Kalker’s simplified theory [4], used for the tangential problem. This method, labelled Stripes, has been used here to investigate the impact on wear calculation of this class of contact models. A different approach is to employ a Winkler bed of separate independent springs. The properties of these springs need however to be calibrated, requiring some supporting theory. The Fastsim algorithm for the tangential contact is of this kind, calibrated with Kalker’s linear theory [4]. An application to the normal contact with ´ non-constant curvatures is proposed by Alonso and Gimenez [11]. Calibration is done with respect to the non-linear stiffness property of Hertzian contact using patch area and ellipticity as parameters. A generalised formulation for solution of both the normal and tangential problem is proposed by Telliskivi [12]. Laterally the springs may be made dependent by selecting a suitable level of neighbourhood influence. The calibration is done with finite element analysis and thus not restricted to the half-space assumption. The local wear depth calculation strategy used in earlier work along with Hertzian contact and the Fastsim algorithm [2,13], may be generalised to non-elliptic contacts. In case of a semi-Hertzian approach the only difference is the integration limits, determined by the shape of the interpenetrated profiles. For a Winkler bed the deflection of the discrete springs serve the same purpose as the surface flexibility in Fastsim in determining the slip state. 2. Evaluation of Stripes To facilitate evaluation of contact modelling influence on calculated wear, the Stripes procedure [10] has been implemented both as a stand-alone module and as a post-processor to the transient vehicle simulation. In the contact patch related diagrams to follow, the reference frame is defined with the longitudinal axis positive pointing to the rear and the lateral axis positive towards the field side. 2.1. Implementation strategy With the purpose of investigating single contact occasions the procedure was implemented as a stand-alone module in Matlab. Thus the implementation as such could be verified with known cases as well as patch wear be calculated and compared to other approaches. This implementation features: • Determination of the rigid contact point at given lateral wheelset displacement. • Evaluation of local profile radii and calculation of initial interpenetration by Hertzian theory.
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Table 1 Verification cases for Stripes implementation Case
Radii (mm) Wheel
Rail
Hertz 1 Hertz 2 A–C [10]a
10000/460 30/460 S1002/460
300 13 UIC60 1:40
a
Load (kN)
78 78 78
Zero lateral displacement.
• Assessment of virtual ellipses and determination of the length of each stripe. The longitudinal semi-axis is compensated and the lateral curvature smoothed as recommended by the authors. • Refinement of the interpenetration by iterating the normal problem part of Stripes to achieve the correct normal force. • Executing the tangential problem part of Stripes calculating shear stresses and slip area boundary. • Calculation of sliding velocity, incremental sliding distance, and wear depth distribution according to the KTH model. For evaluation in conjunction with transient simulation in the complete wear simulation procedure, the extended Fastsim-based post-processor was replaced by Stripes. Here the initial interpenetration was calculated from single point Hertzian contact parameters taken from the standard MBS solution. To overcome a possible inconsistency in using different contact models in the MBS simulation and wear calculation respectively, the interpenetration was iteratively adapted to ensure correct contact force at the given wheelset lateral displacement. This approach is considered justified facilitating this kind of tentative investigation using available MBS simulation software. 2.2. Validation examples
Fig. 2. Stripes implementation verification comparison for Hertzian cases. Solid line: Stripes result. Circles: Hertzian ellipse.
Fig. 3. Stripes implementation verification comparison for the Ayasse–Chollet reference case. Left: contact patch comparison with solid line showing actual implementation and circles the Ayasse–Chollet reference case [10]. Right: contact stress distribution for the actual implementation.
To verify the implementation the contact area has been calculated for three reference cases (Table 1), two purely Hertzian and the Ayasse–Chollet reference case [10]. The contact patch shapes calculated with Stripes compare well with the reference cases (Figs. 2 and 3). As can be seen adequate agreement has been obtained for these elementary cases. The patch shape agrees well for the two implementations. The slight difference is attributed to the determination and smoothing of local curvatures where the Stripes documentation leaves some freedom.
For reference also the elliptic contact area according to Hertz is shown as given by the MBS code contact data table. This table calculation includes an estimation of the elastic deformation by a Winkler approach. In the contact pressure comparisons it should be noted that the Hertzian stress distribution is elliptical while Stripes uses a parabolic distribution. This gives for the same contact area and normal force by definition a higher maximum stress in the latter case.
3. Contact area and pressure distribution
To calculate the contact area on an assumed elastic half-space the complete elastic theory of Kalker, realised in the software code Contact, has been used [4]. Initially three contact positions [14] corresponding to the lateral wheelset displacements 3, 0, and 5 mm have been investigated (Table 2, Fig. 4). Also with respect to contact patch shape Stripes and contact agree well, while the normal stress distributions differ to some extent. Example distributions are given for the neutral and gauge corner cases (Fig. 5).
To explore the capabilities of Stripes to assess the contact area, shape comparisons have been made with other numerical methods, being linear elastic half-space, linear elastic finite element analysis, and elastic–plastic finite element analysis. In the area comparison pictures the patches are scaled to approximately 1:1 size. The lateral extension calculated by Stripes follows the rail envelope surface.
3.1. Comparison with linear elastic half-space
Table 2 Basic contact cases Case
Load (kN)
Field side Neutral Gauge corner
90 90 90
Profile pairing: wheel S1002 on rail UIC60 1:40.
Contact area (mm2 )
Maximum contact pressure (MPa)
Elliptic
Contact
Stripes
Elliptic
Contact
Stripes
129 167 145
151 210 122
149 196 137
1049 806 929
934 849 2327
1202 784 1539
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541
Fig. 4. Contact area comparison with wheelset lateral positioning. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40. (a) 3 mm offset towards the field side, (b) zero lateral offset, and (c) 5 mm offset towards the gauge corner.
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Fig. 6. Contact area comparison during curving. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:30. (a) tangent, 120 km/h, (b) R = 600 m, wheelset displacement 5.9 mm, (c) R = 400 m, wheelset displacement 6.3 mm, and (d) R = 300 m, wheelset displacement 6.5 mm.
Fig. 7. Contact pressure comparison in a 300-m curve. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:30.
Fig. 5. Contact pressure comparison. Upper: zero lateral offset. Lower: 5 mm offset towards gauge corner. Left: Contact [4], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40.
Reasonably good agreement between Stripes and contact is obtained. It is noteworthy that the relative rank switches towards the gauge corner, likely because of the increasing curvature. A similar comparison has been performed for four simulated cases at different curve radii (Table 3, Fig. 6). The contact conditions have been taken from simulations with a commuter vehicle [13] for the leading outer wheel during quasistatic curve negotiation.
It shall be noted that for the R600 case the contact position is close to the rail head radius transition from 80 to 13 mm. The large area predicted by Stripes for the R300 case includes a low stress connection between the two major contact regions. For the tangent case the contact area sizes agree well for all three cases. Due to the parabolic distribution Stripes however predicts higher maximum stress for the same normal force. For the R600 and R400 cases the Hertzian approach and Stripes both indicate larger contact areas than contact. The non-constant rail curvature in this region together with the different influence on the geometry by the elastic deformation may have an influence. Again Stripes shows higher maximum stresses than the elliptic case. For the final R300 case the comparison appears differently, likely due to the smaller lateral radii and the reduced applicability of the half-space assumption. The contact area shapes agree well as long as the curvatures are shallow enough. For the R300 case the influence of the curvature becomes evident as also illustrated by the stress distribution (Fig. 7).
Table 3 Curving cases Case
Radius (m)
Load (kN)
Speed (km/h)
Displacement (mm)
Tangent R600 R400 R300
∞ 600 400 300
78 99 98 101
120 120 93 80
0 5.9 6.3 6.5
Profile pairing: wheel S1002 on rail UIC60 1:30. a Two-point contact.
Contact area (mm2 )
Maximum contact pressure (MPa)
Elliptic
Contact
Stripes
Elliptic
Contact
Stripes
135 74 72 77a
130 51 59 77
125 70 84 155
867 1994 2038 2132
929 3242 2694 3114
1229 2995 2417 1782
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Table 4 Basic contact cases—FE comparison Case
Field side Neutral Gauge corner
Maximum contact pressure (MPa) FEA
Stripes
825 810 630
1202 784 1539
Fig. 9. Contact area comparison during curving. R = 303 m, 75 km/h. Left: finite element, elastic–plastic [12], right: Stripes. Worn S1002 wheel on worn UIC60 rail. (a) second axle outer wheel and (b) leading axle outer wheel.
Fig. 8. Contact area comparison with wheelset lateral positioning. Left: finite element [14], right: Stripes. S1002 wheel on UIC60 rail inclined 1:40. (a) 3 mm offset towards the field side, (b) zero lateral offset, and (c) 5 mm offset towards the gauge corner.
3.2. Comparison with linear elastic finite element analysis Further the basic contact cases (Table 2) have been compared to linear finite element results with respect to contact patch shape and maximum pressure (Table 4, Fig. 8). The referenced finite element analysis [14] features linear elastic material properties. Other properties, for instance a contact formulation including stiffness reduction due to surface roughness are however nonlinear. As would be expected with the formulation indicated, the finite element analysis predicts lower stresses, mainly due to larger contact areas. In particular the gauge corner case shows a significant difference since the Stripes formulation predicts a more distinguished two-point contact with a sharp stress peak at the slender second contact zone (Fig. 5). It is obvious that the finite element modelling predicts significantly larger contact areas despite the moderate lateral curvature at these locations and the elastic material behaviour. The difference is to a large extent attributed to the reduced contact stiffness due to surface roughness. 3.3. Comparison with elastic–plastic finite element analysis The final contact area comparison reference is a narrow curving case analysed with an elastic–plastic finite element model considering worn wheel and rail profiles (Table 5, Fig. 9). The application is a commuter vehicle in a 303 m radius curve [15]. As can be expected the elastic–plastic analysis predicts larger contact area and lower stresses, here clearly for the 2nd wheel tread
contact. For the leading wheel contact close to the flange the results are less obvious, however showing similar values. The shapes of the contact patches in the finite element analysis have been estimated from the rather coarse description in the reference. 4. Wear rate and distribution After exploring the properties of the Stripes procedure with respect to contact area estimation and pressure distribution, the focus is moved to the influence on wear rate, being the principal objective of this investigation. First the wear distribution over the contact patch is studied, followed by simulation of wheel profile evolution. 4.1. Patch wear evaluation The significance of non-elliptic properties is illustrated by a few basic cases comparing the elliptic case as calculated by Gensys [5], the Contact [4] solution, and the actual Stripes implementation. Graphs showing sliding velocity and wear depth distributions are collected in Appendix A. The Stripes reference case [10] (Fig. 3) is first shown. Under those mild operating conditions the creepages remain small and the adhesion zone dominates the contact patch. Some typical features may however be identified. The shapes of the sliding zones towards the rear edge of the patches do differ significantly due to the non-constant curvature. The sliding velocities (Fig. A.1a) based on Kalker’s simplified theory show in general somewhat smaller slip area but higher magnitudes towards the rear edge compared to the corresponding calculation with Contact, based on Kalker’s complete theory. Another typical effect comparing Stripes with Contact, relying on the half-space assumption, is in the former case the relocation of maxima towards increasing curvature, apparent for both sliding velocity and wear (Fig. A.1b). To obtain adequate wear rates of interest for comparison the selected cases need to have a significant slip zone. It is also the expe-
Table 5 R303 curving cases Case
Outer wheel 2nd axle Outer leading wheela
Load (kN)
66 73
Speed (km/h)
75 75
Displacement (mm)
5.4 11.1
Contact area (mm2 )
Maximum contact pressure (MPa)
e-p FEA
Stripes
e-p FEA
Stripes
173 135
115 149
665 1500
1240 1397
Profile pairing: worn S1002 wheel on slightly worn rail UIC60 1:30. a For this case also linear elastic FEA results are reported: contact area = 48 mm2 , maximum contact pressure = 2500 MPa.
R. Enblom, M. Berg / Wear 265 (2008) 1532–1541 Table 7 Twin-disc wear comparison at 1600 N and 1.5% creep
Table 6 Contact forces Case
Force (kN)
Elliptic
Contact
Stripes
Tangent+
Longitudinal Lateral Normal
21.4 0.6 78.3
21.9 0.7 78.3
21.2 0.6 78.3
R600
Longitudinal Lateral Normal
10.9 6.0 98.8
29.3 0.9 98.7
13.3 13.0 98.7
R300
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Longitudinal Lateral Normal
Method
12.3 26.8 101.3
29.8 4.8 101.2
14.6 16.7 101.2
rience that partial slip conditions are more critical to model than full slip. Here the wear rates for a tangent case with increased longitudinal creep (Tangent+), corresponding to for instance braking, and the curving cases R600 and R300 (Table 3) are compared. The general response and consistency of the three methods to common conditions in terms of normal force and creep rates are illustrated by the resulting contact forces (Table 6). In the tangent case all methods agree very well. In the curving cases principal differences are revealed, in particular the tendency by Contact to predict a high longitudinal force. The shear stresses are related to the sliding velocities and thus to the wear coefficients while the normal stresses influence the amount of worn-off material. The sliding velocity distributions for the Tangent+ case (Fig. A.2a) basically confirm the difference in the tangential contact theory applied. Thus the maximum sliding velocity at the rear edge of the contact patch is smaller for Contact than for the other methods. The effect on the wear distribution (Fig. A.2b) is that the transition to the severe wear regime at 0.2 m/s (Fig. 1) is just reached for this case but significantly exceeded for the other cases. In the R600 case, with contact location in the vicinity of the rail head radius transition, Contact predicts full slip while the other methods show partial slip conditions (Fig. A.3a). This principal difference is attributed to the higher shear stresses in the Contact case. The effect on the wear distribution (Fig. A.3b) is dramatic. Both Contact and Stripes show contact stresses (Table 3) in the seizure region, above 2.2 GPa (Fig. 1). This portion of the patch lies however in the Stripes case within the adhesion zone while the Contact case experiences excessive wear in the middle region. In the R300 case all methods predict a full slip situation (Fig. A.4a). Again Contact predicts higher sliding velocities. In addition the contact pressure at the gauge corner contact enters the seizure region, giving a very high wear rate (Fig. A.4b). The rear half of the tread contact region reaches the mild wear region above 0.7 m/s (Fig. 1). The other two methods show differences in location of severe wear but none of them reaches seizure region. This kind of single contact occasion investigations is rarely found in the literature. The predicted initial wear rate may however be compared to simulated wear using the semi-Winkler approach as well as twin-disc test results reported by Telliskivi [12]. The case chosen for comparison is the most severe with the properties: normal load 1600 N, corresponding nominal Hertzian pressure 1400 MPa, creep 1.5%, nominal coefficient of friction 0.6, disc radius 32.5 mm, and circumferential speed 1.0 m/s (Table 7). It shall be noted that in this scale 1.5% creep corresponds to a sliding velocity of 0.015 m/s, indicating mild wear. Furthermore, at the initial stage the contact is basically elliptic. The principal difference between Stripes and the elliptic case is thus the stress distribution model. Despite that, good agreement with the experimental wear rate is observed.
Method
Pressure (MPa)
Traction coefficient
Wear rate (g/rev)
Elliptic Stripes Semi-Winkler (no stiffness scaling) Semi-Winkler (stiffness scaling /2) Twin-disc test
1400 1870a 1360
0.59 0.59 0.59
6.3b 6.1b 17.1c
1360
0.60
39.1c
a b c
–
6.7c
0.6–0.55
Parabolic stress distribution. Initial wear rate based on nominal geometry. Average wear rate over 30,000 revolutions.
Table 8 Wear illustration cases Case
Radius (m)
Speed (km/h)
Cant (mm)
Lubr
Tangent R1500 R600
∞ 1500 600
120 120 104
0 150 150
Nat Nat Lub
Profile pairing: wheel S1002 on rail UIC60 1:30 coefficient of friction: 0.3.
4.2. Profile evolution It is obvious that the different methods may show significant differences at the detailed level as represented by single contact patch examples above. To investigate the effect on the profile evolution, the accumulated wheel wear for a few type cases has been simulated (Table 8) comparing the elliptic approach with Stripes. To account for natural lubrication by the presence of for instance moist and contamination, the wear map for dry conditions (Fig. 1) has been scaled down by a factor 5.5 as previously estimated [2]. The 600 m curve is assumed to be equipped with lubrication devices and the corresponding wear downscaling factor 11 is used. The curve passing cases include entrance and exit transition curves and a typical circular portion. Equal amount of right-hand and left-hand curves as well as forward–backward symmetry is assumed. To achieve reasonable wear and accommodate at least a few wear steps, the total running distance for each type case was set to a few thousand km. The commuter car model from previous work has been used in these simulations as well. This vehicle has a reasonably soft wheelset guidance. The results are presented as worn-off area (Table 9) and wear depth distributions across the outer axle wheel profile. In the diagrams the lateral co-ordinate is taken positive towards the field side (Fig. 10). In general the results from these type cases confirm the tendency seen at the patch level that Stripes redistributes stresses and consequently wear towards increasing curvature. For the tangent case (Fig. 11) the maximum radial depth is increased by about three times although the worn-off area remains the same. It shall also be noted that the difference in perpendicular wear depth is slightly smaller due to the different contact angles, since the Stripes contact patch follows the curvature. Table 9 Simulated wear comparison Case
Running distance (km)
Tangent R1500 R600
6000 6000 3000
Worn-off area (mm2 ) Elliptic
Stripes
0.19 0.97 1.5
0.19 0.89 2.4
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Fig. 10. Wheel profile co-ordinates.
Fig. 13. Radial wear distribution across the profile for the R600 case. Running distance 3000 km. Bold line: Stripes. Thin line: elliptic.
Fig. 11. Radial wear distribution across the profile for the tangent case. Running distance 6000 km. Bold line: Stripes. Thin line: elliptic.
The outcome of the mild 1500 m curving case (Fig. 12) is essentially the same. The relative difference in peak depth at the flange root is however slightly smaller since the elliptic method starts to develop a peak towards the flange root. At the opposite side, showing the wear running as inner wheel, the elliptic approach predicts about 50% larger depth. Narrowing the curve to 600 m (Fig. 13) reveals somewhat larger differences. The difference in peak depth is again about three times but in the Stripes case more pronounced. The worn-off area is in this case significantly larger. Towards the field side both methods show similar results. The differences observed so far between wear calculations based on the elliptic approach and the Stripes method, respectively, would suggest higher flange wear in the latter case. In addition the difference in tread wear distribution shape suggests differences in the equivalent conicity.
Fig. 12. Radial wear distribution across the profile for the R1500 case. Running distance 6000 km. Bold line: Stripes. Thin line: elliptic.
An early simulation using non-elliptic contact is reported by Knothe and Le-The [16], comparing the initial wear distribution calculated applying their strip method [7] with an elliptical approach. The case studied is a single wheelset running in sinusoidal motion on tangential track. The profile combination is nominal S1002 wheel on UIC60 rail. Due to the prescribed lateral motion the contact locations are determined by the profile geometry and the wear rates in the boundary regions do not differ significantly. The principal difference is found in the central region where the elongated shape of the non-elliptic contact area predicts more wear than the elliptic approach. Linder and Brauchli [9] apply their strip-based contact model to simulation of wheel wear at a Eurofima car running in the 300-m curves on the Gotthard line. The conclusion is that the non-elliptic approach improves the results in particular in the tread—flange transition zone due to better modelling of conformance. 5. Conclusions The focus of this investigation is the influence of non-elliptic contact models on the wheel wear rate and profile shape intended as a further step of enhancement towards increased generality of the emerging simulation procedure [1,13]. The possible need for application-dependent calibration is as a consequence anticipated to be reduced. To facilitate evaluation the semi-Hertzian procedure Stripes [10], has been implemented. The results may be considered somewhat tentative since Stripes is applied as a post-processor to the MBS simulation, the latter using its standard contact model. The investigations of wear distributions over the non-elliptic patches under different operating conditions indicate significant differences compared to both Contact [4] and the applied Hertzian approach [5]. The expansion from single contact occasions to complete simulations indicates comparable predicted material removal rates but relocation towards the flange side. The anticipated consequences would be higher flange root wear rate and differences in equivalent conicity. This tendency is apparent in all of the cases shown here, however limited to initial wear in tangent run or reasonably mild curve negotiation. Qualitatively similar observations have been made by Linder and Brauchli [9], simulating wear in 300-m curves. Simulated initial wear rate agrees well with twin-disc test results reported by Telliskivi [12]. The results and experience of this initial study suggest several paths for further research. An evident continuation step is application of this kind of nonelliptic contact modelling to the wear simulation of a complete operation. This would cover wide variety of contact conditions rather than a few single curves.
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Investigation of the performance of a Winkler bed approach of the kind proposed by Telliskivi [12] would be of great interest as an alternative. Both elementary cases and complete operations should be investigated. The aim of this investigation has been impact of contact modelling on the wear distribution. The results however also indicate interesting differences between contact model properties, for instance the determination of creep forces in case of a curved contact patch (Table 6). Investigation of the limitations of the half-space assumption, usually applied, should be done.
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Acknowledgements The authors gratefully acknowledge the financial support from Swedish National Rail Administration (Banverket), Stockholm Transport (SL), Bombardier Transportation, The Association ¨ ˚ of Swedish Train Operators (Tagoperat orerna), and Interfleet Technology, making this research possible. Appendix A See Figs. A.1–A.4.
Fig. A.1. (a) Sliding velocity comparison for the Stripes reference case. (b) Wear depth comparison for the Stripes reference case.
Fig. A.2. (a) Sliding velocity comparison for the tangent case. (b) Wear depth comparison for the tangent case.
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Fig. A.3. (a) Sliding velocity comparison for the R600 case. (b) Wear depth comparison for the R600 case.
Fig. A.4. (a) Sliding velocity comparison for the R300 case. (b) Wear depth comparison for the R300 case.
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