Wheel–rail contact simulation with lookup tables and KEC profiles: a comparative study


This paper describes and compares the use and limitations of two constraint-based formulations for the wheel–rail contact simulation in multibody dynamics: (1) the use of contact lookup tables and (2) the Knife-edge Equivalent Contact constraint method (KEC-method). Both formulations are presented and an accurate procedure to interpolate within the data in the lookup table is also described. Since the wheel–rail constraint contact approach finds difficulties at simultaneous tread and flange contact scenarios, the lookup table method is implemented with a penetration-based elastic contact model for the flange, turning the method into a hybrid (constant in the tread and elastic in the flange) approach. To deal with the two-point contact scenario in the KEC-method, a regularisation of the tread–flange transition allows the use of the constraint approach in the tread and also in the flange. To show the applicability and limitations of both methods, they are studied and compared with special emphasis in the calculation of normal and tangential contact forces. Numerical results are based on the simulation of a two-wheeled bogie vehicle in different case studies that consider irregular tracks and two wheel–rail profiles combinations: profiles that do not show two-point wheel–rail contacts and profiles that do show two-point wheel–rail contacts. Although results show a good agreement between both approaches, the use of the KEC-method is more extensive since it allows to reproduce the wheel-climbing scenario that cannot be simulated with the lookup table method with the hybrid contact approach. It is concluded that simulations with this later method may not be on the safe side.

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\(\bar{\mathbf {r}}^{lir}\), \(\bar{\mathbf {r}}^{rir}\) :

The relative position vector of the irregular rail centreline with respect to the ideal rail centreline.

\(\bar{\mathbf {r}}^{lrp}\), \(\bar{\mathbf {r}}^{rrp}\) :

The relative position vector of the ideal rail centreline with respect to the the track frame.

\(\bar{\mathbf {r}}^{wi}_{c}\), \(\bar{\mathbf {r}}^{rp}_{c}\) :

The position vectors of contact points on the wheel and rail in track frame.

\(\bar{\mathbf {t}}^{wi}_{1,c}\), \(\bar{\mathbf {n}}_{c}^{rp}\) :

The unit-tangent vector and normal vector at the contact point in track frame.

\(\beta\) :

The orientation angle of the rail profiles

\(\boldsymbol{\lambda}\) :

The array of Lagrange multipliers.

\(\delta\) :

The linearised rotation angle due to the irregularity

\(\delta^{wi}\), \(\dot{\delta}^{wi}\) :

The wheel-rail penetration at the flange contact and its time derivative.

\(\hat{\mathbf {u}}^{rrp}_{P}\), \(\hat{\mathbf {u}}^{lrp}_{Q}\) :

The position vector of points \(P\) and \(Q\) in the rail profiles.

\(\hat{\mathbf {u}}^{wIi}_{R}\), \(\hat{\mathbf {u}}^{wIi}_{L}\) :

The position of the points in the wheel surface with respect to the wheelset intermediate frame.

\(\mathbf {A}^{t,lrp}\), \(\mathbf {A}^{t,rrp}\) :

The rotation matrix from the railhead frame with respect to the track frame.

\(\mathbf {A}^{t}\) :

The rotation matrix from the track frame to the global frame.

\(\mathbf {A}^{wti,wi}\), \(\mathbf {A}^{wti,wIi}\), \(\mathbf {A}^{wIi,wi}\) :

The rotation matrices from wheel frame to the wheelset track frame, from wheelset intermediate frame to the wheelset track frame and from wheel frame to the wheelset intermediate frame.

\(\mathbf {C}^{clt}\) :

The wheel-rail contact constraint equations modelled with lookup tables.

\(\mathbf {C}^{clt}_{\mathbf {q}}\), \(\dot{\mathbf {C}}^{clt}_{\mathbf {q}}\) :

The Jacobian matrix and its time derivative of all wheel-rail contact constraints modelled with lookup tables.

\(\mathbf {C}^{KEC}\) :

The contact constraint equations of a wheelset with KEC profiles.

\(\mathbf {C}^{KEC}_{\mathbf {q}}\), \(\dot{\mathbf {C}}^{KEC}_{\mathbf {q}}\) :

The Jacobian matrix and its time derivative of KEC contact constraint equations with respect to generalised coordinates \(\mathbf {q}\).

\(\mathbf {C}^{KEC}_{\mathbf {s}}\), \(\dot{\mathbf {C}}^{KEC}_{\mathbf {s}}\) :

The Jacobian matrix and its time derivative of KEC contact constraint equations with respect to KEC surface parameters.

\(\mathbf {n}_{c}^{rp}\) :

The normal vector to the rail surface at the contact point in global frame.

\(\mathbf {Q}\) :

The force vectors of generalised applied forces and generalised quadratic-velocity inertia forces.

\(\mathbf {Q}_{fla}^{nor}\), \(\mathbf {Q}^{tang}\), \(\mathbf {Q}_{tread}^{nor}\) :

The force vectors of generalised wheel-rail normal flange forces, generalised tangential tread and flange forces, and generalised normal forces at the wheel tread.

\(\mathbf {R}^{t}\) :

The absolute position vector of an arbitrary point on the ideal track centreline with respect to a global frame.

\(\mathbf {R}^{wi}_{c}\), \(\mathbf {R}^{rp}_{c}\) :

The position vectors of contact points on the wheel and rail in global frame.

\(\mathbf {t}^{wi}_{1,c}\), \(\mathbf {t}^{wi}_{2,c}\) :

The two unit-tangent vectors to the wheel surface at the contact point in global frame.

\(\psi^{t}\), \(\theta^{t}\), \(\varphi^{t}\) :

The Euler angles that describe the orientation of the track frame with respect to a global frame.

\(al\), \(vp\), \(gv\), \(cl\) :

Alignment, vertical profile, gauge variation and cross level.

\(f^{lk}\), \(f^{rk}\) :

The value of the equivalent profiles at the lateral positions of \(s^{lk}\) and \(s^{rk}\).

\(K_{hertz}\), \(C_{damp}\) :

The Hertzian stiffness and the constant that introduces non-linear damping.

\(L_{w}\) :

The lateral distance of the wheel frames with respect to the wheelset frame.

\(R^{t}_{x}\), \(R^{t}_{y}\), \(R^{t}_{z}\) :

The absolute position of an arbitrary point on the ideal track centreline with respect to a global frame in \(X\), \(Y\) and \(Z\) direction.

\(r_{0}\) :

The rolling radius of the wheel when centred in the track.

\(s\) :

The arc-length along the track.

\(s^{lk}\), \(s^{rk}\) :

The lateral positions of the contact point in the left and right KEC profiles.

\(y^{lir}\), \(z^{lir}\), \(y^{rir}\), \(z^{rir}\) :

The track irregularities in \(Y\) and \(Z\) direction.

\({h}^{r}\), \({h}^{w}\) :

The functions that define the railhead and wheel profiles.

\({s}^{r}_{1}\), \({s}^{r}_{2}\), \({s}^{w}_{1}\), \({s}^{w}_{2}\) :

The surface parameters of the railhead and wheel profiles.

\(\bar{F}^{wi}_{z}\), \(\hat{M}^{wi}_{x}\) :

The vertical force and roll torque at the wheelset due to the normal contact forces.

\(\nu\) :

The Poisson’s ratio.

\(A\), \(B\) and \(\beta_{h}\) :

The parameters to compute Hertzian stiffness which depend on the curvatures of rail/wheel surfaces.

\(E\) :

The Young’s modulus of the surface.


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The first and third authors thank the Department of Economy, Science, Enterprise and University of the Andalusian Regional Government, in Spain, under the PAIDI 2020 program with project reference P18-RT-1772. The second author thanks for the support given by Business of Finland under the SmartTram-LUT project with reference 6292/31/2018. All this support is gratefully acknowledged.

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Correspondence to Javier F. Aceituno.

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Escalona, J.L., Yu, X. & Aceituno, J.F. Wheel–rail contact simulation with lookup tables and KEC profiles: a comparative study. Multibody Syst Dyn (2020). https://doi.org/10.1007/s11044-020-09773-7

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  • Contact lookup table
  • KEC-method
  • Interpolation
  • Wheel–rail contact
  • Wheel-climbing