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conveniently grouped with locomotive traction in the F t/db term). Rolling and air resistances are
usually grouped asaterm known as propulsion resistance, F pr, making the equation for F r as
follows:
F r ¼ F pr þ F cr þ F b
where F pr is the propulsion resistance; F cr is the curving resistance; and F b is the braking
resistance due to pneumatic braking.
The three mass train allows the three different differential equations to be developed. With
linear wagon connection models the equations can be written as:
m 1 a 1 þ c 1 ð v 1 2 v 2 Þþk 1 ð x 1 2 x 2 Þ¼F t = db 2 F r1 2 F g1
m 2 a 2 þ c 1 ð v 2 2 v 1 Þþc 2 ð v 2 2 v 3 Þþk 1 ð x 2 2 x 1 Þþk 2 ð x 2 2 x 3 Þ¼2 F r2 2 F g2
m 3 a 3 þ c 2 ð v 3 2 v 2 Þþk 2 ð x 3 2 x 2 Þ¼2 F r3 2 F g3
Note that apositive value of F g is taken as an upward grade, i.e., aretarding force.
Allowing for locomotives to be placed at any train position and extending equationnotation for
atrain of any number of vehicles, amore general set of equations can be written as:
For the lead vehicle:
For the i th vehicle:
a 3
m 1 a 1 þ c 1 ð v 1 2 v 2 Þþk 1 ð x 1 2 x 2 Þ¼F t = db1 2 F r 1 2 F g 1
m i a i þ c i 2 1 ð v i 2 v i 2 1 Þþc i ð v i 2 v i þ 1 Þþk i 2 1 ð x i 2 x i 2 1 Þþk i ð x i 2 x i þ 1 Þ¼F t = dbi 2 F r i 2 F g i
For the n th or last vehicle:
m k
3 2 ,c2 m k
2 1 ,c1 m 1
F r3
F g3
a 2
F r2
F g2
FIGURE 9.1 Three mass train model, where: a is vehicle acceleration, m/sec 2 ; c is damping constant, Nsec/m;
k is spring constant, N/m; m is vehicle mass, kg; v is vehicle velocity, m/sec; x is vehicle displacement, m; F g is
gravity force components due to track grade, N; F r is sum of retardation forces, N; and F t/db is traction and
dynamic brake forces from alocomotive unit, N.
© 2006 by Taylor & Francis Group, LLC
Handbook of Railway Vehicle Dynamics
m n a n þ c n 2 1 ð v n 2 v n 2 1 Þþk n 2 1 ð x n 2 x n 2 1 Þ¼F t = dbn 2 F r n 2 F g n
a 1
F r1
F g1
F t/db
ð 9 : 1 Þ
ð 9 : 2 Þ
ð 9 : 3 Þ
ð 9 : 4 Þ
ð 9 : 5 Þ
ð 9 : 6 Þ