aceleracion y frenado

8/11/2019 Aceleracion y Frenado

1/8

Transport: Railways Tractive effort, acceleration, and braking

The Mathematical Association 200 !

Esfuerzo Tractivo, aceleracion y frenadoContextFor a railway to operate efficiently and safely, its locomotives should be powerfulenough to accelerate their trains rapidly to the maximum allowed line speed, and the

braking systems must be able to bring a train reliably to a standstill at a station orsignal, even on an adverse gradient. Railway operators need to calculate train

accelerations and decelerations in order to plan their timetables, and signals must besited so as to allow adequate stopping distances for all the various passenger andgoods services that they are required to control.

In practice there are many different and complex considerations that must be includedin a realistic model of railway operation. Here, ust some of the simpler main issuesare identified and examined, in order to show how mathematical analysis can be usedto provide an indication of expected performance. !he data values used in theexamples "from #$%& do not refer to any specific operating company, locomotive orrolling stock, but are chosen to give realistic illustrations of how practical equipmentmight behave.

Tractive effort!he force which a locomotive can exert when pulling a train is called its tractiveeffort , and depends on various factors. For electric locomotives, which obtain their

power by drawing current from an external supply, the most important are'

weight the adhesion between the driving wheels and the track depends on theweight per wheel, and determines the force that can be applied beforethe wheels begin to slip(

speed up to a certain speed, the tractive effort is almost constant. )s speedincreases further, the current in the traction motor falls, and hence sodoes the tractive effort.

!o characterise the power of their locomotives, manufacturers measure tractive effortas a function of speed. !ests are often performed with the locomotive stationary butresting on rollers, thereby avoiding the effects of air resistance and any imperfections

in the track.

!he data points in Figure $ show an example of the tractive effort of an electriclocomotive. In order to use this information easily in calculations of acceleration anddeceleration, it is helpful to develop an approximation which covers the speed range ofinterest, but has a simple mathematical form. *ne possible technique is piecewise-

0

!0

20

"0

0

#0

$0

0 # !0 !# 20 2# "0 "# 0 # #0

%peed &m's(

T r a c

t i v e e

f f o r t o r

) r a g

& k * (

T+ Meas rement T+ Appro-imation )rag

Figure 1

!ractive effort and dragas a function of speed

Algebra and functionsDifferentiationIntegration


2/8

Tractive effort, acceleration, and braking Transport: Railways

2 The Mathematical Association 200

polynomial approximation + the speed range is split into several contiguous intervals,in each of which the tractive effort is represented by a polynomial function. For theexample shown, a good representation can be obtained by using three speed segments,and a linear approximation for tractive effort on each'

%,-./#-/-00011

%.//.#$ 1-2$11

%/.1#-1111&"


3/8


The Mathematical Association 200 "

&"v P is the tractive effort of the locomotive(&"vQ is the drag(

B is the brake force(mg is the weight of the train(

N is the reaction of the track.

7y 8ewton9s second law of motion, the acceleration f is given by'

sin&"&" mg BvQv P mf = !his equation can be used to derive a number of relationships that are important todifferent aspects of railway operation. :ome of these are considered in the followingsections.

Maximum speed as a function of gradient) train reaches its maximum speed when available tractive effort ust balances thesum of drag and downhill gravitational force, reducing the acceleration to 5ero.;onsequently, the maximum speed is found by solving'

1&"&" = mg vQv P

where sin is the gradient .

:ince the approximation to &"v P is linear within each segment, and that for &"vQ isquadratic, the calculation of maximum speed for a particular gradient reduces to thesolution of a quadratic equation. However, in order to determine which segment ofthe tractive effort approximation should be used for a given gradient, it is useful firstto establish a set of gradient values


4/8


The Mathematical Association 200

Braking distance!o calculate how long it will take for a train to come to rest when the locomotive

power is cut off and the brakes are applied, and how far it will travel in this time, set1&" =v P . :ince acceleration, f, is rate of change of velocity, a differential equation'

mg vQ Bdt dv

m = &"

describes the motion, and, once the initial speed is given, defines v as a function oftime t.

:ince the braking force B is essentially a constant "A mg &, independent of speed, thedifferential equation can be integrated by separation of variables, leading to'

.&"&" 1

1

=++T

V

dt vQmg

mdv

Remembering that the drag Q"v& is approximated by a quadratic function of speed'

,&" //$1 vqvqqvQ ++=

it becomes clear that the braking time T required from speed v is obtained as the

integral'

++=v

c!a!d!

vT 1

/&"

where'

.&"B(B(B 1$/ ++=== g mqcmqmqa

)ppendix $ shows how this integral can be expressed in terms of standard functions.

From this result, a further integration is needed to recover the distance travelled as afunction of time. ) simpler alternative is to calculate the braking distance directly bywriting'

dsdvv

dt ds

dsdv

dt dv f ===

in the original equation, to give'

mg vQ Bdsdv

mv = &"

which is a relation between distance s and speed v.

!his differential equation can also be integrated by separation of variables, leading to'

.&"&" 1

1

=++"

V

dsvQmg

mvdv

and hence the braking distance " required from speed v is obtained as the integral'

++=v

c!a!!d!

v" 1

/&"

where again

.&"B(B(B 1$/ ++=== g mqcmqmqa

)ppendix / shows how this integral can be expressed in terms of standard functions.

Integration Analytic solution of firstorder differential equationwith separable variables

Integration Analytic solution of firstorder differential equationwith separable variables

Differentiationhain rule


5/8


The Mathematical Association 200 #

:ince braking time and distance depend both on initial speed and the gradient of thetrack, there are various summary presentations that provide useful information.

)s an example, Figure shows the distance needed to brake to a standstill as afunction of the track gradient, calculated for a range of different initial speeds.

Time spent accelerating to required speedCach stop that a train makes during its ourney involves three phases' braking to astandstill, remaining stationary to set down and pick up passengers, and accelerating tothe required line speed. )n appropriate allowance for the time taken for each of these

phases, as well as other braking and acceleration manoeuvres "e.g. to traverse a set of points& must be included when drawing up realistic timetables. !he previous sectionconsidered time taken for braking( calculation of the time taken in acceleration issimilar, but somewhat more involved because of the piecewiseDlinear approximationto the variation of tractive effort with speed.

:etting 1= B produces the differential equation'

mg vQv P dt dv

m = &"&"

which, once the initial speed is given, defines v as a function of time t.

:ince the tractive effort &"v P is a function of speed only, the differential equation can be integrated by separation of variables, leading to'

.&"&" 11

=T V

dt mg vQv P

mdv

7ecause the approximation to &"v P is a piecewiseDlinear function of speed, and thedrag Q"v& is approximated by a quadratic function of speed, the time T required toaccelerate to speed v can be obtained by splitting the motion into segments. )transition between segments is required when the speed reaches one of the breakpointspeeds in the piecewiseDlinear approximation for &"v P .

For each segment, the elapsed time and the distance travelled can be expressed as'

++= f

s

v

v c!a!d!

vT /&" ++= f

s

v

v c!a!!d!

v" /&"

0

200

00

$00

/00

!000

!200

! 00

!$00

2.0 !.# !.0 0.# 0.0 0.# !.0 !.# 2.0

radient &1(

% t o p p

i n g

d i s t a n c e

& m (

#

0

"#

"0

2#

3nitial %peed&m's(

Figure 4

:topping distance as afunction of gradient for arange of initial speeds.


6/8


$ The Mathematical Association 200

where sv and f v are, respectively, starting and finishing speeds for the segment, andthe parameters'

( ) ( ) .B(B(B 11$$/ g mq pcmq pmqa === all remain constant throughout the segment. !he two integrals are again of the typeconsidered in )ppendices $ and /, and so can be expressed in terms of standardfunctions. !he total time or distance needed to accelerate to a given speed is found by

summing over the segments.Dealing wit c anges in track gradient4enerally, the gradient is a piecewiseDconstant function of distance along the track

+ an example is shown in Figure -, which refers to part of the E GestD;oast mainline #/%.

!o deal with this, the analysis for both braking and acceleration calculations can befurther segmented, with transitions between segments corresponding to instants whenthe train reaches a position on the track at which the gradient changes. )s an example,Figure 2 shows a graph of speed against time for acceleration from rest over the giventrack profile, calculated using the tractive effort of Figure $.

0

#

!0

!#

20

2#

"0

"#

0

#

#0

0 !00 200 "00 00 #00 $00 400 /00

Time &s(

% p e e d

& m ' s (

20

0

20

0

$0

/0

!00

0 # !0 !# 20 2# "0 "#

)istance from reference point &km(

5 e

i g h t a

b o v e r e

f e r e n c e

p o

i n t & m (

"6"!06/

#0/""/

""#

7

"""

/!2

Figure 5

ertical profile of track.Cach segment is labelledwith its reciprocalgradient.

Figure 6

:peed against time forgiven length of track.


7/8


The Mathematical Association 200 4

!ources$. ata provided by ince 7arker, @odelling ;onsultant, formerly at )lstom

!ransport

/. B# main-line gradient profiles, I:78 1DJ$$1D1>J-D/

"cknowledgement!hanks to Richard :tanley and colleagues at )lstom !ransport for their comments that

helped correct a draft version of the article."ppendices# Evaluation of integrals

$ %ntegration of reciprocal quadratic polynomial

%#&,,,," / " $ x

x $ " x xcxax

dx x xca %

"

$

>++

=

"tep &' Grite the denominator in the form'

( ) acaa xa with,B/B /// =+ and check the value of the discriminant .

i& 1


8/8


/ The Mathematical Association 200

& %ntegral of x times reciprocal quadratic polynomial

++= $

"

x

x $ " cxax

xdx x xca * /&,,,,"

For this integral, carry out the checks in steps $ and / above, and then write'

aaax x /B/B&/" += to give'

&,,,,"/

&log"/$ /

$ "

x

x

x xca % a

cxaxa

* $

"

++=

aceleracion y frenado

Documents