Civil Engineering Reference
In-Depth Information
The horizontal reaction at the wheel axle, P ,is
Wa r
r
P
=
R H (t)
=
,
(4.20)
=
where W
m F g is the weight of the concentrated force, m F is the mass of the concen-
trated force, r is the wheel radius, and R V (t) and R H (t) are the vertical and horizontal
components, respectively, of the reaction force due to rolling friction. The resultant
reaction force, R(t) , is located at a horizontal distance, a r , from the wheel centroid
as a result of rolling friction (McLean and Nelson, 1962). The distance a r is often
referred to as the coefficient of rolling resistance. Rolling friction is small at constant
train speed and greater at nonuniform train speeds. The horizontal component of the
reaction, R H (t) , is generally small because the applied vertical forces greatly exceed
applied horizontal forces. Neglecting axle bearing and wheel rim friction again, the
force equilibrium relating to the horizontal effects of rolling motion, considering
complete adhesion, yields (Figure 4.10b)
d 2 x(t)
d t 2
H LF (t)
R T (t)
+
T F (t)
m F
=
0,
(4.21)
where H LF (t) is the longitudinal force transferred to rails and deck/superstructure and
R T (t) is the resistance to horizontal movement (primarily air resistance or vehicle drag
forces since axle bearing and wheel flange friction is considered negligible). R T (t)
is generally relativity small in comparison to other horizontal forces and it is not too
conservative to neglect this force. T F (t) is the locomotive traction force and is equal to
( M T (t)c /r ), where M T (t) is the driving torque applied to wheel, and c is a constant
depending on locomotive engine characteristics and gear ratio.
Therefore, Equation 4.21 may be simplified to
m F d 2 x(t)
d t 2
H LF (t)
+
T F (t)
=
0.
(4.21a)
Also, neglecting axle bearing and wheel rim friction, the force equilibrium relating
to the rotational effects of rolling motion, considering complete adhesion, provides
(Figure 4.10c)
d 2
θ (t)
d t 2
rH LF (t) + M T (t) a r R V (t) + rR H (t) I p
=
0,
(4.22)
where I p is the rotational moment of the inertia of mass.
Since the distance, a r , is small, the moment from rolling friction, a r R V (t) , may
be neglected. In addition, because R H (t) is relatively small, Equation 4.22 may be
simplified to
d 2
(t)
d t 2
θ
rH LF (t)
+
M T (t)
I p
=
0.
(4.23)
 
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