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Research Papers

# Directions of the Tangential Creep Forces in Railroad Vehicle Dynamics

[+] Author and Article Information
Ali Afshari, Ahmed A. Shabana

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607-7022

J. Comput. Nonlinear Dynam 5(2), 021006 (Feb 18, 2010) (10 pages) doi:10.1115/1.4000796 History: Received February 23, 2009; Revised July 13, 2009; Published February 18, 2010; Online February 18, 2010

## Abstract

In some of the wheel/rail creep theories used in railroad vehicle simulations, the direction of the tangential creep forces is assumed to be the wheel rolling direction (RD). When the Hertz theory is used, an assumption is made that the rolling direction is the direction of one of the axes of the contact ellipse. In principle, the rolling direction depends on the wheel motion while the direction of the axes of the contact ellipse (CE) are determined using the principal directions, which depend only on the geometry of the wheel and rail surfaces and do not depend on the motion of the wheel. The RD and CE directions can also be different from the direction of the rail longitudinal tangent (LT) at the contact point. In this investigation, the differences between the contact frames that are based on the RD, LT, and CE directions that enter into the calculation of the wheel/rail creep forces and moments are discussed. The choice of the frame in which the contact forces are defined can be determined using one longitudinal vector and the normal to the rail at the contact point. While the normal vector is uniquely defined, different choices can be made for the longitudinal vector including the RD, LT, and CE directions. In the case of pure rolling or when the slipping is small, the RD direction can be defined using the cross product of the angular velocity vector and the vector that defines the location of the contact point. Therefore, this direction does not depend explicitly on the geometry of the wheel and rail surfaces at the contact point. The LT direction is defined as the direction of the longitudinal tangent obtained by differentiation of the rail surface equation with respect to the rail longitudinal parameter (arc length). Such a tangent does not depend explicitly on the direction of the wheel angular velocity nor does it depend on the wheel geometry. The CE direction is defined using the direction of the axes of the contact ellipse used in Hertz theory. In the Hertzian contact theory, the contact ellipse axes are determined using the principal directions associated with the principal curvatures. Therefore, the CE direction differs from the RD and LT directions in the sense that it is function of the geometry of the wheel and rail surfaces. In order to better understand the role of geometry in the formulation of the creep forces, the relationship between the principal curvatures of the rail surface and the curvatures of the rail profile and the rail space curve is discussed in this investigation. Numerical examples are presented in order to examine the differences in the results obtained using the RD, LT and CE contact frames.

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## Figures

Figure 2

(a) The coordinate systems used to define track geometry, (b) the surface parameters at the contact point of the right rail, and (c) the space curve and the horizontal plane

Figure 3

(a) Wheel surface parameters and (b) CE frame axes

Figure 4

Lateral displacement of the wheelset (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 5

Angle of attack at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 6

Normal force at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 18

The angle between the RD and the CE frames

Figure 19

The angle between the LT and the CE frames

Figure 1

Surface geometry

Figure 7

Longitudinal force at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 8

Lateral force at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 9

The angle between the RD and the CE frames

Figure 10

The angle between the LT and the CE frames

Figure 11

X component of the right contact force in the global coordinate system (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 12

Y component of the right contact force in the global coordinate system (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 13

Z component of the right contact force in the global coordinate system (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 14

Longitudinal creepage at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 15

Lateral creepage at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

Figure 16

Right rail profiles (Profile #1: ———, Profile #2: – – – –, Profile #3: - - - - - -)

Figure 17

Lateral force at the right wheel (CE frame: ———, LT frame: – – – –, RD frame: - - - - - -)

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