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A nonlinear approach for modeling rail flexibility using the absolute nodal coordinate formulation

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Abstract

This paper describes a new nonlinear formulation based on the absolute nodal coordinate formulation (ANCF) for modeling the dynamic interaction between rigid wheels and flexible rails. The generalized forces and spin moments at the contact points are formulated in terms of the absolute coordinates and gradients of ANCF finite elements used to model the rail. To this end, a new procedure for formulating the generalized ANCF applied moment based on a continuum mechanics approach is introduced. The generalized moment is calculated using the spin tensor defined in terms of the gradients at the contact points, and therefore, the use of this approach does not require the use of angles. In order to have an accurate definition of the creepages, the location and velocity of the contact points are updated online using the rail deformations. An elastic contact formulation is used to define the contact forces that enter into the dynamic formulation of the system equations of motion. An elastic line approach is used to define the rail stress forces, and the relative slip between the rigid wheel and the flexible rail is iteratively updated using the deformations of the ANCF finite elements. The formulation proposed in this investigation is demonstrated using a five-body railroad vehicle negotiating flexible rails. In order to validate the ANCF rail model, the obtained results are compared with previously published results obtained using the floating frame of reference formulation that employs eigenmodes. The comparative study presented in this paper shows that there is, in general, a good agreement between the results obtained using the two different formulations.

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Abbreviations

\(\xi _x \) :

Longitudinal creepage

\(\xi _y \) :

Lateral creepage

\(\xi _s \) :

Spin creepage

\(\mathbf{v}_c^\mathrm{{w}}\) :

Wheel’s contact point c velocity vector

\(\mathbf{v}_c^\mathrm{{r}}\) :

Rail’s contact point c velocity vector

\(\mathbf{t}_1^\mathrm{{r}}\) :

Longitudinal tangent to the rail surface

\(\mathbf{t}_2^\mathrm{{r}}\) :

Lateral tangent to the rail surface

\(\mathbf{n}_1^\mathrm{{r}}\) :

Normal vector to the rail surface

\(\mathbf{r}\) :

Global position of a point in a rigid/flexible body

\(\mathbf{r}_0\) :

Global position of the body frame of reference

\(\mathbf{A}\) :

Orthonormal orientation matrix

\(\bar{\mathbf{u}}\) :

Local position vector of a deformed point

\(\bar{\mathbf{u}}_0\) :

Local undeformed position vector

\(\bar{\mathbf{u}}_\mathrm{{f}}\) :

Local, linear elastic displacement vector

\({\varvec{\Phi }}\) :

Modal matrix for FFR

\(\mathbf{q}_\mathrm{{r}}\) :

Reference coordinates in FFR

\(\mathbf{q}_\mathrm{{f}}\) :

Elastic coordinates in FFR

\({\varvec{\upomega }}\) :

Angular velocity vector

\({\varvec{\uptheta }}_\mathrm{{f}}\) :

Linearized set of angles describing body’s flexibility

\({\varvec{\uptheta }}\) :

Finite rotational parameters

\(\mathbf{S}(x,y,z)\) :

ANCF shape function matrix

\(\mathbf{e}\) :

Absolute nodal coordinates vector

\(\mathbf{L}\) :

Velocity gradient tensor

\(\mathbf{W}\) :

Spin tensor

\(\mathbf{D}\) :

Rate of deformation tensor

\(\mathbf{J}\) :

Position vector gradient tensor

\({\bar{\mathbf{R}}}\) :

Orthogonal tensor from polar decomposition of \(\mathbf{J}\)

\(\mathbf{U}\) :

Symmetric stretch tensor from polar decomposition of \(\mathbf{J}\)

\({\varvec{\upomega }}_\mathrm{{r}}\) :

Rigid body angular velocity vector

\({\varvec{\upomega }}_\mathrm{{f}}\) :

Spin deformation vector

\(\tilde{\varvec{\upomega }}_\alpha \) :

Skew symmetric tensor associated with vector \({\varvec{\upomega }}_\alpha \)

\(\mathbf{G}\) :

Linear operator to obtain ANCF generalized moment

\(\mathbf{M}\) :

External Cartesian moment

\(\mathbf{F}\) :

External force

\(s_1^\mathrm{{r}} ,s_2^\mathrm{{r}}\) :

Parameters defining rail surface

\(s_1^\mathrm{{w}} ,s_2^\mathrm{{w}}\) :

Parameters defining wheel surface

\(\mathbf{B}\) :

Velocity transformation matrix

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Acknowledgments

This research was supported by the National University Rail (NURail) Center, a US DOT-OST Tier 1 University Transportation Center and the Spanish Ministry of Science and Innovation (MICINN) under Project Reference TRA2010-16715. These supports are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 W. Taylor St., Chicago, 60607, USA

    Antonio M. Recuero & Ahmed A. Shabana

  2. Department of Mechanical and Manufacturing Engineering, University of Seville, Camino de los Descubrimientos s/n, 41092, Seville, Spain

    Javier F. Aceituno & José L. Escalona

Authors
  1. Antonio M. Recuero
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  2. Javier F. Aceituno
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  3. José L. Escalona
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  4. Ahmed A. Shabana
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Correspondence to Ahmed A. Shabana.

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Recuero, A.M., Aceituno, J.F., Escalona, J.L. et al. A nonlinear approach for modeling rail flexibility using the absolute nodal coordinate formulation. Nonlinear Dyn 83, 463–481 (2016). https://doi.org/10.1007/s11071-015-2341-5

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