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Longitudinal Train Dynamics 263
G . C URVING R ESISTANCE
Curving resistance calculations are similar to propulsion resistance calculations in that empirical
formulae must be used. Rollingstock design and condition, cant deficiency, rail profile, rail
lubrication, and curve radius will all affect the resistance imposed onavehicle on the curve. As
rollingstock design and condition, rail profile, and cant deficiency can vary, it is usual to estimate
curving resistance by afunction relating only to curve radius. The equation commonly used is 14 :
F cr ¼ 6116= R ð 9 : 22Þ
where F cr is in Newtons per tonne of wagon mass and R is curve radius in metres.
Rail flange lubrication is thought to be capable of reducing curving resistance by50%. The
curving resistance ofawagon that is stationary on acurve is thought to be approximately double,
i.e., 200% of the value given byEquation 9.22.
H . T RAIN D YNAMICS M ODEL D EVELOPMENT AND S IMULATION
As can be seen from the preceding sections, the modelling of the train as alongitudinal system
involves arange of modelling challenges for the dynamicist. The basic interconnected mass–
damper–spring type model, representing the train vehicle masses and wagon connections, is
complicated by nonlinear gap, nonlinear spring, and stick slip friction elements. The complexity
and detail, which is chosenfor models such as the wagon connection element, may limit the choices
available in the modelling and simulation software used. Software packages with predefined model
blocks and look up tables etc. can usually be used, sometimes with difficulty, to model systems of
this complexity. Thewagon connection models used by Duncan 1 and Cole 9 were both implemented
only as code subroutines. In some cases, asubroutine or function written in aprogramming
language will be easier to develop than acomplex combination of re-existing stiffness’ and dampers
from asoftware library.
Having developed asuitable connector for the mass–damper–spring, i.e., f wcð v i ; v i þ 1 ; x i ; x i þ 1 Þ ;
the remaining subsystems for traction, braking, resistance forces, and control inputs require
modelling and data bases must be provided. Again, software packages with predefined modelling
features can be used, but code scripts will also usually be required towork with track databases or
for more complex models. The pneumatic braking system, not treated in detail in this chapter, will
require acomplete time stepping simulation of its fluid flow dynamics. The pneumatic braking
model must interface with the train simulation model at the locomotive control input subsystem to
receive brake control inputs. The output from the brake model, cylinder pressures, must bescaled
by cylinder sizes, brake rigging, and brake shoe friction coefficients to give retardation forces
which are applied to the vehicle masses.Ifthe brake model is afully detailed gas dynamics model,
it will usually require amuch smaller time step than the train mass–damper–spring model. It is
not unusual for this problem to be solved by completing several integration steps of brake pipe
simulation for every one integration step ofthe train mass–damper–spring model. Such models
are computationally expensive and until recently would only be found in engineering analysis
simulators. Many existing rail industry specific train simulation software packages, because of
the era in which they were developed, utilise some simplification of the brake model to allow
reasonable run times for simulation studies. This is particularly the case for driver training
simulators where adesign criteria is that the graphics and experience of the simulator must be at
real time speed.
Train simulators with highly nonlinear and hard limited connections, as described in this
section, can be simulated successfully with explicit schemes such as Fourth Order Runge Kutta.
The simulation examples presented in this chapter utilise this solver with a10m/sec time step.
Some simulations and variations of the wagon connection model have been found to require a
slightly smaller step. Adiscussion of numerical methods is given in Ref. 8. The advantage of the
© 2006 by Taylor & Francis Group, LLC