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Implementation of a non-Hertzian contact model for railway dynamic application

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Abstract

The development of wheel–rail contact models is an active topic of railway research with the dual objective of improving the accuracy of multibody simulations and reducing its computational effort. This paper extends the online Hertzian contact model, proposed by Pombo et al. (Veh. Syst. Dyn. 45: 165–189, 2007) to propose a non-Hertzian contact model. The new methodology presented here includes the following steps: (i) search of the points of contact; (ii) identification of the undeformed distance function; (iii) evaluation of the contact patch; (iv) calculation of the normal and tangential contact forces; (v) application of the contact forces in the multibody vehicle model. Among several contact models available in the literature, this non-Hertzian contact approach uses the Kik–Piotrowski model for the normal contact force, while the tangential forces are obtained from the interpolation of the available Kalker Book of Tables for non-Hertzian (KBTNH) contact. With the purpose to demonstrate the proper implementation and selection of parameters that define this new model, several contact analysis and dynamic simulations are performed in which the wheel S1002 and the rail UIC50 are considered. First, the contact analyses that determine the contact condition of different wheel–rail interactions serve to assess the accuracy of the Hertzian and non-Hertzian models with respect to the software of reference CONTACT. Second, the Hertzian and non-Hertzian models are utilised to perform dynamic simulations of a wheelset, a bogie and a vehicle running in tangent and curved tracks. In short, this work provides, not only a complete description of the implementation of a non-Hertzian contact model in a multibody code, but also suggests for the proper selection of the parameters that promote better accuracy and optimal computational efficiency.

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Abbreviations

\((.)_{\mathrm{cp}}\) :

Subscript to identify the contact patch

\((.)_{w}\) :

Subscript to identify the wheelset \(w\)

\((.)_{\mathrm{r}}\) :

Subscript to identify the rail

\((.)^{\mathit{side}}\) :

Superscript to identify the left and right side

\(a\) :

Length of semi-axes of SDEC or elliptical contact patch in longitudinal direction

\(A\) :

Curvature of the contact point in the lateral direction

\(A_{\mathrm{cp}}\) :

Area of the contact patch

\(\mathbf{A}\) :

Transformation matrix

\(b\) :

Length of semi-axes of SDEC or elliptical contact patch in longitudinal direction

\(\mathbf{b}\) :

Binormal vector

\(B\) :

Curvature of the contact point in the longitudinal lateral

\(\mathbf{d}\) :

Distant vector

\(D\) :

Damping coefficient for the normal contact force

\(e\) :

Restitution coefficient

\(E\) :

Young modulus

\(f_{{r}}\) :

Ordinate of the profile that represents the rail cross section

\(f_{w}\) :

Ordinate of the profile that represents the wheelset \(w\) cross section

\(f_{{x}}\) :

Normalised longitudinal creep force

\(f_{{y}}\) :

Normalised lateral creep force

\(F_{{x}}\) :

Longitudinal creep force

\(F_{{y}}\) :

Lateral creep force

\(g\) :

Aspect ratio

\(g_{\mathrm{und}}\) :

Undeformed distance function

\(\mathbf{g}\) :

External generalised forces vector

\(G\) :

Shear Modulus

H:

Hertzian

\(H\) :

Distance between the left and right wheel profiles

\(K\) :

Contact stiffness

KBTNH:

Kalker Book of Tables for non-Hertzian

L:

Left side

\(m_{{z}}\) :

Normalised creep moment

\(M_{{z}}\) :

Spin creep moment

\(\mathbf{M}\) :

Mass matrix

\(n\) :

Hertz nonlinear exponent

\(\mathbf{n}\) :

Normal unit vector

NH:

non-Hertzian

\(N\) :

Normal force magnitude

PS:

Primary Suspension

\(P\) :

Potential point of contact in the rail

\(p_{0}\) :

Maximum normal pressure of the KP model

\(p_{\max}\) :

Maximum normal pressure

\(\mathbf{q}\) :

System generalised coordinates

\(Q\) :

Potential point of contact in the wheel

\(r\) :

Radial coordinate

\(\mathbf{r}\) :

Position vector

\(s_{\mathrm{r}}\) :

Arc-length coordinate of the rail surface

\(s_{w}\) :

Angular coordinate of the wheel surface

SS:

Secondary Suspension

SDEC:

Single Double Elliptical Contact

R:

Right side

\(\mathbf{v}\) :

Velocity vector

\(\mathbf{t}\) :

Tangential vector

\(u_{\mathrm{r}}\) :

Lateral coordinate of the rail surface

\(u_{w}\) :

Lateral coordinate of the wheel surface

\(x_{\mathrm{L}}\) :

Length of the strip

\(x,y,z\) :

Cartesian coordinates

\(y_{0}\) :

One dimension of the SDEC

\(\alpha \) :

Direction of the linear creepage

\(\gamma \) :

Tangent angle of the cross section

\(\boldsymbol{\gamma }\) :

Right-hand side of the acceleration constraint equations vector

\(\delta \) :

Penetration magnitude

\(\dot{\delta }^{\max } \) :

Maximum penetration velocity

\(\Delta F_{{x}}\) :

Deviation of the longitudinal creep force

\(\Delta F_{{y}}\) :

Deviation of the lateral creep force

\(\Delta M_{{z}}\) :

Deviation of the spin creep moment

\(\Delta r \) :

Step size for the radial coordinate

\(\Delta s\) :

Width of the strip

\(\Delta \theta \) :

Step size for the angular coordinate

\(\varepsilon \) :

Parameter that takes into account the existing deformation

\(\eta \) :

Normalised lateral creepage

\(\theta \) :

Angular coordinate

\(\kappa \) :

Curvature

\(\boldsymbol{\lambda }\) :

Lagrange multipliers vector

\(\mu \) :

Friction coefficient

\(\nu \) :

Magnitude of the linear creepages

\(\xi \) :

Normalised longitudinal creepage

\(\sigma \) :

Poisson ratio

\(\upsilon _{{x}}\) :

Longitudinal creepage

\(\upsilon _{{y}}\) :

Lateral creepage

\(\varphi \) :

Spin creepage

\(\boldsymbol{\Phi }_{\mathbf{q}}\) :

Jacobian matrix of the constraint equations

\(\chi \) :

Normalised spin creepage

\(\psi \) :

Shape factor of SDEC

\(\boldsymbol{\upomega}\) :

Angular velocity vector

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Acknowledgements

The first and second authors express their gratitude to the Portuguese Foundation for Science and Technology (Fundação para a Ciência e a Tecnologia) through the PhD grants SFRH/BD/96695/2013 and PD/BD/114154/2016, respectively. This work was supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2019.

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Authors and Affiliations

  1. Institute of Railway Research, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK

    Hugo Magalhães, Pedro Antunes & João Pombo

  2. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Hugo Magalhães, Pedro Antunes, João Pombo & Jorge Ambrósio

  3. CMEMS-UMinho, Departamento de Engenharia Mecânica, Universidade do Minho, Guimarães, Portugal

    Filipe Marques & Paulo Flores

  4. Dipartimento di Meccanica, Politecnico di Milano, Milan, Italy

    Binbin Liu & Stefano Bruni

  5. ISEL, Intituto Politecnico de Lisboa, Lisboa, Portugal

    João Pombo

  6. Institute of Vehicles, Warsaw University of Technology, Warsaw, Poland

    Jerzy Piotrowski

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  1. Hugo Magalhães
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  2. Filipe Marques
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Correspondence to Hugo Magalhães.

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Magalhães, H., Marques, F., Liu, B. et al. Implementation of a non-Hertzian contact model for railway dynamic application. Multibody Syst Dyn 48, 41–78 (2020). https://doi.org/10.1007/s11044-019-09688-y

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