This section reintroduces and describes a series of mathematical symbols used through-out the text. This section concentrates solely on symbols based on the Latin alphabet.
If a symbol lacks a value, it generally means that value is variable.
132 Symbols
A.1.1 A to B
Symbol Description Value Units
a Lateral distance from center of mass of wheelset to suspension elements.
1.074 [m]
a1..6 Distances from the center of mass of the wheelset to the contact forces τ1..6, re-spectively.
[m]
ae Contact ellipse major semi-axis. [m]
ae,0 Contact ellipse major semi-axis under static load from wheelset.
[m]
ae,dyn Contact ellipse major semi-axis under dynamic load from wheelset.
[m]
al Distance from center of mass of the wheelset to the left contact point.
[m]
ar Distance from center of mass of the wheelset to the right contact point.
[m]
b Lateral distance from center of mass of railcar to wheelset suspension elements.
1.074 [m]
b Longitudinal clearance between wheel-set and wheelwheel-set guidance.
0.025 [m]
be Contact ellipse minor semi-axis. [m]
be,0 Contact ellipse minor semi-axis under static load from wheelset.
[m]
be,dyn Contact ellipse minor semi-axis under dynamic load from wheelset.
[m]
A.1 Latin Symbols 133
A.1.2 C to E
c Lateral clearance between wheelset and wheelset guidance.
0.02 [m]
c Average stiffness [N m−1]
c% Loading stiffness. [N m−1]
c. Unloading stiffness. [N m−1]
C11 Kalker creepage coefficient. [1]
C22 Kalker creepage coefficient. [1]
C23 Kalker creepage coefficient. [1]
Clr· Left wheel-rail contact force in the · -direction of the rail coordinate system.
[N]
Crr· Right wheel-rail contact force in the · -direction of the rail coordinate system.
[N]
Clw· Left wheel-rail contact force in the · -direction of the wheelset coordinate system.
[N]
Crw· Right wheel-rail contact force in the
·-direction of the wheelset coordinate system.
[N]
d Damping constant. [N s m−1]
d3 Height from the wagon underframe to the wagon floor.
0.265 [m]
d4 Height from the wagon floor to the top of vertical wagon walls.
2.35 [m]
d8 Car body cargo space length. 16.01 [m]
d9 Car body cargo space width. 2.9 [m]
e Unit vector. See appendix C. [1]
134 Symbols
A.1.3 F to L
F·r,ext Exterior forces in the ·=x, y, z coordi-nate direction of the inertial rail refer-ence frame.
[N]
Fres Restoring force. [N]
F˜x Creep force in the x-direction of the contact coordinate system.
[N]
F˜y Creep force in the y-direction of the contact coordinate system.
[N]
Fτ Adjusted tangential creep force after adjustment by Shen-Hedrick-Elkins.
[N]
F˜τ Resulting tangential creep force prior to Shen-Hedrick-Elkins adjustment.
[N]
g Acceleration due to gravity. 9.82 [m s−2] G Shear modulus of steel. 8.27·1010 [N m−2]
h Vertical distance from center of mass of railcar to wheelset suspension elements
0.802∗ [m]
Ic· Moment of inertia for the car body about the ·=x, y, z axis.
[kg m2] Iw· Moment of inertia for the wheelset
about the ·=x, y, z axis.
[kg m2] Il1..2 Moment of inertia for the car body
un-der roll and yaw. Used in the Maple code.
[kg m2]
Jij The (i, j)’th element in the Jacobi ma-trix.
[1]
k Spring constant. [N m−1]
k1..4 Spring constants in the UIC suspen-sion’s lateral and longitudinal dynam-ics.
[N m−1]
l Longitudinal distance from center of mass of railcar to wheelset suspension elements.
5.0 [m]
LC Angular momentum about the center of mass and principal axes of the wheelset.
[kg m2 s−1]
A.1 Latin Symbols 135
A.1.4 M to Q
mi The i’th particle’s mass. [kg]
mc Car body mass. Includes freight. 13563∗ [kg]
mw Wheelset mass. 1022 [kg]
M Mass. [kg]
M0 Mass of the empty car body. [kg]
Mfreight Mass of freight. [kg]
aMb Coordinate transformation matrix from a coordinates to b coordinates.
[1]
N Normal force. [N]
N0 Normal force from static wheelset load. [N]
Ndyn Normal force from dynamic wheelset load.
[N]
px,1..2 Penetrations of the wheelset longitudi-nally, that give rise to contact forces τ1..2.
[m]
py,3..6 Penetrations of the wheelset laterally, that give rise to contact forces τ3..6.
[m]
q Total penetration of a wheelset into the rail at a contact point.
[m]
q0 Penetration of a wheelset into the rail at a contact point corresponding to the static load of the wheelset.
[m]
qdyn Penetration of a wheelset into the rail at a contact point corresponding to the dynamic load of the wheelset.
[m]
∆q The difference in penetration (addi-tional penetration) between a static and a dynamic load of the wheelset.
[m]
∆ql The left wheel’s additional penetration. [m]
∆qr The right wheel’s additional penetra-tion.
[m]
136 Symbols
A.1.5 R to V
r0 Basic rolling radius of the wheelset. [m]
rl Rolling radius of the left wheel. [m]
rr Rolling radius of the right wheel. [m]
r Vector from the center of mass of the wheelset to the contact point.
[m]
rC Vector from the origin to the center of mass of an object.
[m]
ri Vector from the origin to the i’th par-ticle’s center of mass.
[m]
R Radius. Distance from point to center of consideration (moment of inertia cal-culations).
[m]
R· Vector to the contact point on the · = w, r (wheel, rail).
[m]
s Parametric variable used in differential succesion.
[1]
Slr· Spring forces from the UIC suspension on the left side in the rail coordinate system in the ·=x, y, z-directions.
- [N]
Sr·r Spring forces from the UIC suspension on the right side in the rail coordinate system in the ·=x, y, z-directions.
- [N]
T Dry friction slider force. [N]
T0,1..4 Dry friction slider threshold forces for lateral and longitudinal UIC suspen-sion models.
[N]
vs Rail car velocity. [m s−1]
V Rail car velocity. [m s−1]
Vc Velocity of a contact point. [m s−1] Vtrans Translational velocity of the wheelset. [m s−1] Vrot Rotational velocity of the wheelset. [m s−1]
A.1 Latin Symbols 137
A.1.6 X to Y
xd Displacement of the wheelset in the x-coordinate of the rail x-coordinate sys-tem.
[m]
(Xr, Yr, Zr) Euclidean coordinates in the rail refer-ence frame. Inertial.
[1]
(Xw, Yw, Zw) Euclidean coordinates in the wheelset reference frame. Not inertial.
[1]
(Xb, Yb, Zc) Euclidean coordinates in the car body reference frame. Not inertial.
[1]
(Xc, Yc, Zc) Euclidean coordinates in the contact point reference frame. Not inertial.
[1]
yd Displacement of the wheelset in the y-coordinate of the rail y-coordinate sys-tem.
[m]
138 Symbols