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Research Papers

# Use of the Non-Inertial Coordinates in the Analysis of Train Longitudinal Forces

[+] Author and Article Information
Ahmed A. Shabana, Ahmed K. Aboubakr

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

Lifen Ding

School of Mechanical, Electronic and Control Engineering,  Beijing Jiaotong University, Beijing, China

J. Comput. Nonlinear Dynam 7(1), 011001 (Jul 22, 2011) (10 pages) doi:10.1115/1.4004122 History: Received September 01, 2010; Revised April 23, 2011; Published July 22, 2011; Online July 22, 2011

## Abstract

In this investigation, a new three-dimensional nonlinear train car coupler model that takes into account the geometric nonlinearity due to the coupler and car body displacements is developed. The proposed nonlinear coupler model allows for arbitrary three-dimensional motion of the car bodies and captures kinematic degrees of freedom that are not captured using existing simpler models. The coupler kinematic equations are expressed in terms of the car body coordinates, as well as the relative coordinates of the coupler with respect to the car body. The virtual work is used to obtain expressions for the generalized forces associated with the car body and coupler coordinates. By assuming the inertia of the coupler components negligible compared to the inertia of the car body, the system coordinates are partitioned into two distinct sets: inertial and noninertial coordinates. The inertial coordinates that describe the car motion have inertia forces associated with them. The noninertial coupler coordinates; on the other hand, describe the coupler kinematics and have no inertia forces associated with them. The use of the principle of virtual work leads to a coupled system of differential and algebraic equations expressed in terms of the inertial and noninertial coordinates. The differential equations, which depend on the coupler noninertial coordinates, govern the motion of the train cars; whereas the algebraic force equations are the result of the quasi-static equilibrium conditions of the massless coupler components. Given the inertial coordinates and velocities, the quasi-static coupler algebraic force equations are solved iteratively for the noninertial coordinates using a Newton–Raphson algorithm. This approach leads to significant reduction in the numbers of state equations, system inertial coordinates, and constraint equations; and allows avoiding a system of stiff differential equations that can arise because of the relatively small coupler mass. The use of the concept of the noninertial coordinates and the resulting differential/algebraic equations obtained in this study is demonstrated using the knuckle coupler, which is widely used in North America. Numerical results of simple train models are presented in order to demonstrate the use of the formulation developed in this paper.

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Copyright © 2012 by American Society of Mechanical Engineers
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## Figures Figure 1

Knuckle coupler Figure 2

Conventional autocoupler package Figure 3

Two-car system Figure 4

Concept of the noninertial coordinates Figure 5

Forward position [car i (▪), slider block i (•), slider block j (▴), and car j(▾)] Figure 6

EOC and shank knuckle spring deformations [ddi (▪), ddj (▾), and ddij(•)] Figure 7

EOC and shank knuckle spring deformation rates [d·di (▪),d·dj (▾), and d·dij (•)] Figure 8

Draft gear and shank knuckle spring deformations [ddi (▪), ddj (▾), and ddij (•)] Figure 9

Draft gear and shank knuckle spring deformation rates [d·di (▪), d·dj (▾), and d·dij (•)] Figure 10

Curved track Figure 11

EOC spring deformation [ddi (▪), ddj (▾)] Figure 12

EOC spring deformation rates [d·di (▪), d·dj (▾)] Figure 13

Torsional spring deformation of car i Figure 14

Torsional spring deformation rate of car i Figure 15

Torsional spring deformation of car j Figure 16

Torsional spring deformation rate of car j

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