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RESEARCH PAPERS

# Effect of the Linearization of the Kinematic Equations in Railroad Vehicle System Dynamics

[+] Author and Article Information
Ahmed A. Shabana, Mahmoud Tobaa

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

Brian Marquis

Volpe National Transportation Systems Center Kendall Square Cambridge, MA 02142

Magdy El-Sibaie

Office of Research & Development Federal Railroad Administration 1120 Vermont Avenue Washington, DC 20590

J. Comput. Nonlinear Dynam 1(1), 25-34 (May 05, 2005) (10 pages) doi:10.1115/1.1951783 History: Received March 24, 2005; Revised May 05, 2005

## Abstract

The sensitivity of the wheel/rail contact problem to the approximations made in some of the creepage expressions is examined in this investigation. It is known that railroad vehicle models that employ kinematic linearization can predict, particularly at high speeds, significantly different dynamic response as compared to models that are based on fully nonlinear kinematic and dynamic equations. In order to analytically examine this problem and numerically quantify the effect of the approximations used in the linearized railroad vehicle models, the fully nonlinear kinematic and dynamic equations of a wheel set are presented. The linearized kinematic and dynamic equations used in some railroad vehicle models are obtained from the fully nonlinear model in order to shed light on the assumptions and approximations used in the linearized models. The assumptions of small angles that are often made in developing railroad vehicle models and their effect on the angular velocity, angular acceleration, and the inertia forces are investigated. The velocity creepage expressions that result from the use of the assumptions of small angles are obtained and compared with the fully nonlinear expressions. Newton-Euler equations for the wheel set are presented and their dependence on Euler angles and their time derivatives is discussed. The effect of the linearization assumptions on the form of Newton-Euler equations is examined. A suspended wheel set model is used as an example to obtain the numerical results required to quantify the effect of the linearization. The results obtained in this investigation show that linearization of the creepages can lead to significant errors in the values predicted for the longitudinal and tangential forces as well as the spin moment. There are also significant differences between the two models in the prediction of the lateral and vertical forces used to evaluate the $L∕V$ ratios as demonstrated by the results presented in this investigation.

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## Figures

Figure 19

Error in the lateral contact force

Figure 18

Error in the vertical contact force

Figure 17

Spin moment of the right wheel

Figure 16

Tangential contact force of the right wheel

Figure 15

Longitudinal contact force of the right wheel

Figure 14

Normal contact force of the right wheel

Figure 13

Lateral creepage of the right wheel

Figure 12

Longitudinal creepage of the right wheel

Figure 11

Error estimates

Figure 10

Comparison between the components of the local angular velocity vector and the time derivatives of the angles

Figure 9

Pitch velocity of the wheel set

Figure 8

Roll velocity of the wheel set

Figure 7

Yaw velocity of the wheel set

Figure 6

Pitch rotation of the wheel set

Figure 5

Roll rotation of the wheel set

Figure 4

Yaw rotation of the wheel set

Figure 3

Spin creepage for the right wheel

Figure 2

Lateral displacement of the wheel set

Figure 1

Suspended wheel set and the track deviation

## Errata

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