Modelling Freight Wagons
Mark Hoffmann
mh@imm.dtu.dk
Institute of Informatics and Mathematical Modelling Technical University of Denmark
Overview
about 30 min. presentation Freight wagons today
Mathematical model of a freight wagon Simulating the freight wagon
Future work
Introduction
Freight Wagons Today
Problems with stability at moderate speeds
Economical restriction in the production phase
Poor dynamic behaviour of freight wagons leads to an unnecessary amount of wear and damaged freight
Lack of a fundamental understanding of the dynamics and stability
Hard competition due to the flexibility with trucks and the low transportation time with air planes
Goal of Research
Mathematical formulation of the dynamics of the freight wagon
Achieve a fundamental understanding of the dynamic behaviour by simulating different configurations of the system, e.g.
different suspension characteristics
symmetrically and asymmetrically loaded wagon various rail gauge
Improve the dynamic behaviour of freight wagons
Mathematical Model – diagram
0000 0000 0000 1111 1111 1111
00 00 00 00
11 11 11 11 00 00 00 00
11 11 11 11
00 00 00 00
11 11 11 11 00 00 00 00
11 11 11 11
Center of Mass b
l Center of Mass
Guidance
Center of Mass
0000 0000 0000 1111 1111 1111 00 00 00 00
11 11 11 11
00 00 00 00
11 11 11 11 00
00 00 00
11 11 11 11
00 00 00 00
11 11 11 11
0000 0000 0000 00
1111 1111 1111 11
000000 111111 000000 111111
000000 111111 000000 111111 0000
0000 0000 00
1111 1111 1111 11
000000 111111
000000 111111
0000 0000 0000 00
1111 1111 1111 11
0000 0000 0000 00
1111 1111 1111 11
Center of Mass
h
Guidance
Center of Mass
00 00 00 00 00 00 00 00 00 0
11 11 11 11 11 11 11 11 11 1
00 00 00 00 00 00 00 00 00 0
11 11 11 11 11 11 11 11 11 1
000000 111111 000000
111111
Mechanical Analysis
Model and interacting forces:
Multibody system (3 bodies)
The motion of these bodies is wanted
The dynamic equations of motion are derived
classically with Newton-Euler equations of motion The interacting forces is a challenge due to its
non-smooth nature, e.g.
Dry friction Impacts
Main assumptions
The bodies are assumed to be rigid except for a local elasticity in the contact patch between the wheel and rail
The wagon is running at a constant speed
Straight and perfect track with no irregularities No wind
Gyroscopic forces have minor influence on the dynamics at low velocities (i.e. v ≤ 40 m/s)
Kinematics
Reference frame Body Description
Kinematics
Reference frame Body Description
(O; x, y, z) – Global frame
Kinematics
Reference frame Body Description
(O; x, y, z) – Global frame
(Owe; xwe, ywe, zwe) Wheelset Equilibrium frame (Owb; xwb, ywb, zwb) Wheelset Body frame
(Owcl; xwcl, ywcl, zwcl) Wheelset Left w/r-contact frame (Owcr; xwcr, ywcr, zwcr) Wheelset Right w/r-contact frame
Kinematics
Reference frame Body Description
(O; x, y, z) – Global frame
(Owe; xwe, ywe, zwe) Wheelset Equilibrium frame (Owb; xwb, ywb, zwb) Wheelset Body frame
(Owcl; xwcl, ywcl, zwcl) Wheelset Left w/r-contact frame (Owcr; xwcr, ywcr, zwcr) Wheelset Right w/r-contact frame
(Ocbe; xcbe, ycbe, zcbe) Car body Equilibrium frame (Ocbb; xcbb, ycbb, zcbb) Car body Body frame
Dynamics
Newton-Euler equations of motions
˙
q = f(q) q = [q1, . . . , q12]T f = [f1(q), . . . , f12(q)]T
Dynamics
Newton-Euler equations of motions
˙
q = f(q) q = [q1, . . . , q12]T f = [f1(q), . . . , f12(q)]T
6 equations to determine the position and velocity of the center of mass of the body
3 equations to determine the Euler angles of the body frame
3 equations to determine the angular velocity of the principal axes of the body w.r.t. the global frame
Interacting Forces
Interacting Forces
Wheel/rail contact forces
Important w.r.t. the dynamics of the vehicle
Data from realistic wheel and rail profiles are tabulated Forces are calculated using the SHE approximation
Interacting Forces
Wheel/rail contact forces
Important w.r.t. the dynamics of the vehicle
Data from realistic wheel and rail profiles are tabulated Forces are calculated using the SHE approximation Suspension forces
Vertical suspension is modelled linearly
Horizontal suspension is modelled by Piotrowski’s model
Interacting Forces
Wheel/rail contact forces
Important w.r.t. the dynamics of the vehicle
Data from realistic wheel and rail profiles are tabulated Forces are calculated using the SHE approximation Suspension forces
Vertical suspension is modelled linearly
Horizontal suspension is modelled by Piotrowski’s model Impact forces
The guidances are modelled by springs
UIC Suspension
UIC Suspension
k
k1 T01 m
Parameter Identification
Solving the Model
The complete model consist of 50 nonlinear ODE’s The system is stiff
Implicit method needed in order to ensure numerical stability
SDIRK (Singly Diagonally Implicit Runge Kutta)
Implementation
Implementation
main program is written in c++
Speed
Object oriented
SDIRK is written in c++
Implementation
main program is written in c++
Speed
Object oriented
SDIRK is written in c++
Optimizations are important
Avoid redundant computations
Avoid computing something that is 0
0
5
10
Suspension Type 2
Results – Stability
0 2 4 6 8 10 12 14 16 18
−2
−1 0 1 2 3
x 10−4
Time [s]
Displacement [m]
Lateral Displacements
Front Wheelset Rear Wheelset Car Body
0 2 4 6 8 10 12 14 16 18 20
−0.015
−0.01
−0.005 0 0.005 0.01 0.015
Time [s]
Displacement [m]
Lateral Displacements
Front Wheelset Rear Wheelset Car Body
v = 15 m/s v = 25 m/s
Future Work
Future Work
Improve model for the parabolic leaf spring providing the vertical suspension
Future Work
Improve model for the parabolic leaf spring providing the vertical suspension
Introduce different impact model longitudinally
Future Work
Improve model for the parabolic leaf spring providing the vertical suspension
Introduce different impact model longitudinally Analysis
Thorough bifurcation analysis for various configurations
Investigate different suspension models
Suggestions on how to improve the suspension
