A train air brake force model: Locomotive automatic brake valve and brake pipe flow formulations

Abstract

The goal of this study is to integrate an air brake model with efficient algorithms for train longitudinal force calculation that are based on trajectory coordinate formulations. The air brake model, developed in this investigation and presented in this paper and a companion paper, consists of the locomotive automatic brake valve, air brake pipe and car control unit (CCU). The proposed air brake force model accounts for the effect of the air flow in long train pipes as well as the effect of leakage and branch pipe flows. This model can be used to study the dynamic behavior of the air flow in the train pipe and its effect on the longitudinal train forces during brake application and release. The governing equations of the air pressure flow are developed using the general fluid continuity and momentum equations, simplified using the assumptions of one-dimensional isothermal flow. Using these assumptions, one obtains two coupled air velocity/pressure partial differential equations that depend on time and the longitudinal coordinate of the brake pipes. The partial differential equations are converted to a set of first-order ordinary differential equations using the finite element method. The resulting air brake ordinary differential equations are solved simultaneously with the train’s second-order non-linear dynamic differential equations of motion that are based on the trajectory coordinates. The train car non-linear dynamics is defined using a body track coordinate system that follows the car motion. The body track coordinate system translation and orientation are defined in terms of one parameter that describes the distance traveled by the car. The configuration of the car with respect to its track coordinate system is described using two translation coordinates and three Euler angles. The operation modes of the brake system considered in this investigation are the brake release mode and the brake application mode that includes service and emergency brakes. A detailed model of the locomotive automatic brake valve is presented in this investigation and used to define the inputs to the air brake pipe during the simulation. A simplified model of this valve is also proposed in order to reduce the computational time of the simulation. In a companion paper, the detailed CCU formulation is presented. The coupling between the air brake, locomotive automatic brake valve, CCUs and train equations is established and used in the companion paper in the simulation of the non-linear dynamics of long trains. The air brake formulations presented in the two companion papers are implemented in a computer code called Analysis of Train/Track Interaction Forces (ATTIF) which is developed for the analysis of longitudinal train forces.