ࡱ> VXU ,objbj ,/~( ( kk8#|D;<!(G!G!!2.N..;;;;;;;<3?;Q.+\2...;kkG!! S;222.kG!!;2.;22RT9<):!?|2p::i;0;:?2?<:?: ..2.....;;2...;....?.........( 1: Tractive effort, acceleration and braking Context For a railway to operate efficiently and safely, its locomotives should be powerful enough 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 or signal, even on an adverse gradient. Railway operators need to calculate train accelerations and decelerations in order to plan their timetables, and signals must be sited so as to allow adequate stopping distances for all the various passenger and goods services that they are required to control. In practice there are many different and complex considerations that must be included in a realistic model of railway operation. Here, just some of the simpler main issues are identified and examined, in order to show how mathematical analysis can be used to provide an indication of expected performance. The data values used in the examples (from [1]) do not refer to any specific operating company, locomotive or rolling stock, but are chosen to give realistic illustrations of how practical equipment might behave. Tractive effort The force which a locomotive can exert when pulling a train is called its tractive effort, 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 the weight per wheel, and determines the force that can be applied before the wheels begin to slip; speed up to a certain speed, the tractive effort is almost constant. As speed increases further, the current in the traction motor falls, and hence so does the tractive effort. To characterise the power of their locomotives, manufacturers measure tractive effort as a function of speed. Tests are often performed with the locomotive stationary but resting on rollers, thereby avoiding the effects of air resistance and any imperfections in the track. The data points in Figure1 show an example of the tractive effort of an electric locomotive. In order to use this information easily in calculations of acceleration and deceleration, it is helpful to develop an approximation which covers the speed range of interest, but has a simple mathematical form. One possible technique is piecewise-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 the example shown, a good representation can be obtained by using three speed segments, and a linear approximation for tractive effort on each: EMBED Equation.3 where P is the tractive effort in newtons, and v is the speed in metres per second. This is shown as a solid line in the Figure. Drag Inevitably, a moving train exerts a drag on the locomotive propelling it. This force, which opposes the motion, comes from a variety of sources, the most important being friction in the axle bearings, air resistance, and resistance from the rail as the wheels roll along it. Railway operators estimate drag from experiments which measure the force needed to keep a train moving at a constant speed. Polynomials can again be used to approximate the variation of drag with speed, and it is generally agreed in the railway industry that a quadratic function often suffices over the full range, although the coefficients used will vary from railway to railway and with train type. As an example, the drag might be given approximately by: EMBED Equation.3 where Q is the drag in newtons, and v is the speed in metres per second. This is shown as the dashed line in Figure 1. Brake force The brake force available depends on two factors: the adhesion between the rail and the wheels being braked, and the normal reaction of the rail on the wheels being braked (and hence on the weight per braked wheel) Generally, it is specified as a fraction (b, say) of the total weight of the train:  EMBED Equation.3  A typical value for b is 0.09 Train dynamics The dynamics of a train moving with speed v along a track inclined at an angle EMBED Equation.3 to the horizontal are determined by the forces shown in Figure 2. Here, EMBED Equation.3 is the tractive effort of the locomotive; EMBED Equation.3 is the drag; B is the brake force; mg is the weight of the train; N is the reaction of the track. By Newtons second law of motion, the acceleration f is given by: EMBED Equation.3 This equation can be used to derive a number of relationships that are important to different aspects of railway operation. Some of these are considered in the following sections. Maximum speed as a function of gradient A train reaches its maximum speed when available tractive effort just balances the sum of drag and downhill gravitational force, reducing the acceleration to zero. Consequently, the maximum speed is found by solving:  EMBED Equation.3  where  EMBED Equation.3  is the gradient. Since the approximation to  EMBED Equation.3  is linear within each segment, and that for  EMBED Equation.3  is quadratic, the calculation of maximum speed for a particular gradient reduces to the solution of a quadratic equation. However, in order to determine which segment of the tractive effort approximation should be used for a given gradient, it is useful first to establish a set of gradient values  EMBED Equation.3  whose corresponding maximum speeds are equal to the transition speeds  EMBED Equation.3  between segments. Specifically:  EMBED Equation.3  Then:  EMBED Equation.3  Figure 3 shows the results of calculations for a train of total weight 865 tons. Here, gradient is given in percent the amount in metres the track rises for every hundred metres traversed. An alternative convention is to specify it reciprocally the distance in metres along the track for a rise of one metre (e.g. 1 in 50 is equivalent to 2%). Braking distance To 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, set  EMBED Equation.3 . Since acceleration, f, is rate of change of velocity, a differential equation:  EMBED Equation.3  describes the motion, and, once the initial speed is given, defines v as a function of time t. Since the braking force B is essentially a constant (= mgb ), independent of speed, the differential equation can be integrated by separation of variables, leading to:  EMBED Equation.3  Remembering that the drag Q(v) is approximated by a quadratic function of speed:  EMBED Equation.3  it becomes clear that the braking time T required from speed v is obtained as the integral:  EMBED Equation.3  where:  EMBED Equation.3  Appendix 1 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 a function of time. A simpler alternative is to calculate the braking distance directly by writing:  EMBED Equation.3  in the original equation, to give:  EMBED Equation.3  which is a relation between distance s and speed v. This differential equation can also be integrated by separation of variables, leading to:  EMBED Equation.3  and hence the braking distance S required from speed v is obtained as the integral:  EMBED Equation.3  where again  EMBED Equation.3  Appendix 2 shows how this integral can be expressed in terms of standard functions. Since braking time and distance depend both on initial speed and the gradient of the track, there are various summary presentations that provide useful information. As an example, Figure 4 shows the distance needed to brake to a standstill as a function of the track gradient, calculated for a range of different initial speeds. Time spent accelerating to required speed Each stop that a train makes during its journey involves three phases: braking to a standstill, remaining stationary to set down and pick up passengers, and accelerating to the required line speed. An 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. The previous section considered time taken for braking; calculation of the time taken in acceleration is similar, but somewhat more involved because of the piecewise-linear approximation to the variation of tractive effort with speed. Setting  EMBED Equation.3  produces the differential equation:  EMBED Equation.3  which, once the initial speed is given, defines v as a function of time t. Since the tractive effort  EMBED Equation.3 is a function of speed only, the differential equation can be integrated by separation of variables, leading to:  EMBED Equation.3  Because the approximation to  EMBED Equation.3 is a piecewise-linear function of speed, and the drag Q(v) is approximated by a quadratic function of speed, the time T required to accelerate to speed v can be obtained by splitting the motion into segments. A transition between segments is required when the speed reaches one of the breakpoint speeds in the piecewise-linear approximation for  EMBED Equation.3 . For each segment, the elapsed time and the distance travelled can be expressed as:  EMBED Equation.3   EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3  are, respectively, starting and finishing speeds for the segment, and the parameters:  EMBED Equation.3  all remain constant throughout the segment. The two integrals are again of the type considered in Appendices 1 and 2, and so can be expressed in terms of standard functions. The total time or distance needed to accelerate to a given speed is found by summing over the segments. Dealing with changes in track gradient Generally, the gradient  EMBED Equation.3  is a piecewise-constant function of distance along the track an example is shown in Figure 5, which refers to part of the UK West-Coast main line [2]. To deal with this, the analysis for both braking and acceleration calculations can be further segmented, with transitions between segments corresponding to instants when the train reaches a position on the track at which the gradient changes. As an example, Figure 6 shows a graph of speed against time for acceleration from rest over the given track profile, calculated using the tractive effort of Figure 1. Sources Data provided by Vince Barker, Modelling Consultant, formerly at Alstom Transport BR main-line gradient profiles, ISBN 0-7110-0875-2 Acknowledgement Thanks to Richard Stanley and colleagues at Alstom Transport for their comments that helped correct a draft version of the article. Appendices: Evaluation of integrals 1 Integration of reciprocal quadratic polynomial  EMBED Equation.3  Step 1: Write the denominator in the form:  EMBED Equation.3  and check the value of the discriminant  EMBED Equation.3 .  EMBED Equation.3  complex roots; no singularities  EMBED Equation.3  double real root;  EMBED Equation.3 singularity at  EMBED Equation.3   EMBED Equation.3  real roots; two  EMBED Equation.3  singularities at  EMBED Equation.3  In case (iii), for the location of the singularities, use:  EMBED Equation.3   EMBED Equation.3 , to minimise loss of accuracy through numerical cancellation. Step 2: Check that the range of integration does not include a singularity. In case (ii):  EMBED Equation.3  In case (iii):  EMBED Equation.3  Step 3: Carry out the integration by making the substitution:  EMBED Equation.3 . Putting  EMBED Equation.3 , the results are:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Integral of x times reciprocal quadratic polynomial  EMBED Equation.3  For this integral, carry out the checks in steps 1 and 2 above, and then write:  EMBED Equation.3  to give:  EMBED Equation.3      Tractive effort, acceleration, and braking Transport: Railways Transport: Railways Tractive effort, acceleration, and braking  PAGE 8 The Mathematical Association 2004 The Mathematical Association 2004  PAGE 7 Integration Analytic solution of first order differential equation with separable variables Algebra and functions Differentiation Integration Figure  SEQ Figure \* ARABIC 6 Speed against time for given length of track. Figure  SEQ Figure \* ARABIC 5 Vertical profile of track. Each segment is labelled with its reciprocal gradient. Figure  SEQ Figure \* ARABIC 4 Stopping distance as a function of gradient for a range of initial speeds. 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FMicrosoft Equation 3.0 DS Equation Equation.39q߄mII f=dvdt=dvdsdsdt=vdvdsObjInfoqEquation Native _1129142406tF@?@?Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q$\IPI mvdvds="B"Q(v)"mg FMicrosoft Equation 3.0 DS EqCompObjsufObjInfovEquation Native x_1129142430hyF@?@?Ole CompObjxzfObjInfo{Equation Native uation Equation.39q$ޠIdI mvdvmg(+)+Q(v) V0 +" ="ds 0S +" .0z FMicrosoft Equation 3.0 DS Equation Equation.39q_1122109254~F@?@?Ole CompObj}fObjInfoxmII S(v)=uduau 2 +bu+c 0v +" FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1129142454F@??Ole CompObjfObjInfoEquation Native _1122130488cF??Ole $ެII a=q 2 /m;b=q 1 /m;c=q 0 /m+g(+). FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native 4_1129142512F??mII B=0= FMicrosoft Equation 3.0 DS Equation Equation.39q$`TItI mdvdt=P(v)"Q(v)"mgHOle CompObjfObjInfoEquation Native |_1122130692F??Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q II P(v) FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native 8_1129302221F??Ole CompObjfObjInfoEquation Native _1122130912F??Ole 'IPI mdvP(v)"Q(v)"mg 0V +" =dt 0T +" . FMicrosoft Equation 3.0 DS Equation Equation.39qII P(v)CompObjfObjInfoEquation Native 8_1124035092F??Ole CompObjfObjInfoEquation Native 8 FMicrosoft Equation 3.0 DS Equation Equation.39qmII P(v) FMicrosoft Equation 3.0 DS Equation Equation.39q_1129142673F??Ole CompObjfObjInfo$ތIPI T(v)=duau 2 +bu+c v s v f +" FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1129142706F??Ole CompObjfObjInfoEquation Native _11240359061F??Ole $ސI@I S(v)=uduau 2 +bu+c v s v f +" FMicrosoft Equation 3.0 DS Equation Equation.39qII v sCompObjfObjInfoEquation Native 8_1124035919F??Ole CompObjfObjInfoEquation Native 8 FMicrosoft Equation 3.0 DS Equation Equation.39qTIPI v fa FMicrosoft Equation 3.0 DS Equation Equation.39q_1129142733F??Ole CompObjfObjInfo$̯II a="q 2 /m;b=p 1 "q 1 ()/m;c=p 0 "q 0 ()/m"g. 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