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Longitudinal Train Dynamics 243
By including the F t/db in each equation, thus on every vehicle, the equations can be applied
to any locomotive placement or system of distributed power. For unpowered vehicles F t/db is set
to zero.
For nonlinear modelling of the system, the stiffness and damping constants are replaced with
functions. It is usual to express stiffnessasafunction of displacement and incorporate coupler slack
and piece-wise-linear approximations ofdraft gear response. Damping is usually expressed as
afunction of velocity. More complex functions, incorporating asecond independent variable,
(i.e., displacement and velocity for astiffness function), can also be used. Thegeneralised nonlinear
equations are therefore:
For the lead vehicle:
For the i th vehicle:
For the n th or last vehicle:
m 1 a 1 þ f wcð v 1 ; v 2 ; x 1 ; x 2 Þ¼F t = db1 2 F r1 2 F g1
m i a i þ f wcð v i ; v i 2 1 ; x i ; x i 2 1 Þþf wcð v i ; v i þ 1 ; x i ; x i þ 1 Þ¼F t = dbi 2 F r i 2 F g i
m n a n þ f wcð v n ; v n 2 1 ; x n ; x n 2 1 Þ¼F t = dbn 2 F r n 2 F g n
where f wc is the nonlinear function describing the full characteristics of the wagon connection.
Solution and simulation of the above equation set is further complicated by the need to
calculate the forcing inputs to the system, i.e., F t/db, F r ,and F g .The traction-dynamic brake force
term F t/db must be continually updated for driver control adjustments and any changes to
locomotive speed. The retardation forces, F r , are dependent on braking settings, velocity,
curvature, and rollingstock design. Gravity force components, F g ,are dependent on track grade
and, therefore, the position of the vehicle on the track. Approaches to the nonlinear modelling of
the wagon connection and modelling ofeach of the forcing inputs are included and discussed
in the following sections.
B . W AGON C ONNECTION M ODELS
Perhaps the most important component in any longitudinal train simulation is the wagon connection
element. The autocoupler with friction type draft gearsisthe mostcommon wagon connection in the
Australian and North American freight train systems. It also, perhaps, presents the mostchallenges
for modellingand simulation due to the nonlinearities of air gap (or coupler slack), draft gear spring
characteristic, (polymer or steel),and stick–slip friction provided by awedge system. Due to these
complexities, the common autocoupler-frictiontype draft gear wagon connection will be examined
first. Other innovations such as slackless packages, drawbars, and shared bogies are then more
easily considered.
© 2006 by Taylor & Francis Group, LLC
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