tractiveeffortaccelerationandbraking

Transport: Railways Tractive effort, acceleration, and braking

© The Mathematical Association 2004 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 Figure 1 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-

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50

Speed (m/s)

Tra

ctive

effo

rt o

r D

rag

(kN

)

TE Measurement

TE Approximation

Drag

Figure 1

Tractive effort and drag

as a function of speed

Algebra and functions Differentiation Integration

Tractive effort, acceleration, and braking Transport: Railways

2 © The Mathematical Association 2004

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:

],459.24[52533300

]9.242.4[144056100

]2.40[50000)(

<≤−=

<≤−=

<≤=

vv

vv

vvP

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:

,5.3202000)( 2vvvQ ++=

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:

1. the adhesion between the rail and the wheels being braked, and

2. 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 (β, say) of the total weight of the train:

βmgB =

A typical value for β is 0.09

Train dynamics

The dynamics of a train moving with speed v along a track inclined at an angle α to

the horizontal are determined by the forces shown in Figure 2.

Here,

Algebra and functions

Quadratic functions and their graphs

mg

Q(v) + B

Nv

f

P(v)

α

Figure 2

Forces acting on a train

on a track with

inclination α



)(vP 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.

By Newton’s second law of motion, the acceleration f is given by:

αsin)()( mgBvQvPmf −−−=

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:

0)()( =−− γmgvQvP

where αγ sin≡ is the gradient.

Since the approximation to )(vP is linear within each segment, and that for )(vQ 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 }{ iγ whose corresponding maximum speeds are

equal to the transition speeds iv between segments. Specifically:

( ) mgvQvP iii )()( −=γ

Then:

],[ ncalculatiofor segment use 11 iiii vv −− ⇒<< γγγ

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%).

Figure 3

Maximum speed as a

function of track

gradient

20

25

30

35

40

45

50

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Gradient (%)

Ma

xim

um

sp

ee

d (

m/s

)



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

0)( =vP . Since acceleration, f, is rate of change of velocity, a differential equation:

γmgvQBdt

dvm −−−= )(

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 (= mgβ ), independent of speed, the

differential equation can be integrated by separation of variables, leading to:

.)()(

0

0

∫∫ −=++

T

V

dtvQmg

mdv

γβ

Remembering that the drag Q(v) is approximated by a quadratic function of speed:

,)( 2

210 vqvqqvQ ++=

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

integral:

∫ ++=

v

cbuau

duvT

0

2)(

where:

.)(/;/;/ 012 γβ ++=== gmqcmqbmqa

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:

ds

dvv

dt

ds

ds

dv

dt

dvf ===

in the original equation, to give:

γmgvQBds

dvmv −−−= )(

which is a relation between distance s and speed v.

This differential equation can also be integrated by separation of variables, leading to:

.)()(

0

0

∫∫ −=++

S

V

dsvQmg

mvdv

γβ

and hence the braking distance S required from speed v is obtained as the integral:

∫ ++=

v

cbuau

uduvS

0

2)(

where again

.)(/;/;/ 012 γβ ++=== gmqcmqbmqa

Appendix 2 shows how this integral can be expressed in terms of standard functions.

Integration Analytic solution of first order differential equation with separable variables

Integration Analytic solution of first order differential equation with separable variables

Differentiation Chain rule



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 0=B produces the differential equation:

γmgvQvPdt

dvm −−= )()(

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

Since the tractive effort )(vP is a function of speed only, the differential equation can

be integrated by separation of variables, leading to:

.)()(

00

∫∫ =−−

TV

dtmgvQvP

mdv

γ

Because the approximation to )(vP 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 )(vP .

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

∫ ++=

f

s

v

vcbuau

duvT

2)( ∫ ++

=f

s

v

vcbuau

uduvS

2)(

0

200

400

600

800

1000

1200

1400

1600

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Gradient (%)

Sto

pp

ing

dis

tan

ce

(m

)

45

40

35

30

25

Initial Speed

(m/s)

Figure 4

Stopping distance as a

function of gradient for a

range of initial speeds.



where sv and fv are, respectively, starting and finishing speeds for the segment, and

the parameters:

( ) ( ) ./;/;/ 00112 γgmqpcmqpbmqa −−=−=−=

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 γ 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.

0

5

10

15

20

25

30

35

40

45

50

0 100 200 300 400 500 600 700 800

Time (s)

Sp

ee

d (

m/s

)

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35

Distance from reference point (km)

He

igh

t a

bo

ve

re

fere

nce

po

int (m

)

3931098

508

338

335

L

333

812

Figure 5

Vertical profile of track.

Each segment is labelled

with its reciprocal

gradient.

Figure 6

Speed against time for

given length of track.



Sources

1. Data provided by Vince Barker, Modelling Consultant, formerly at Alstom

Transport

2. 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

][),,,,(2 SF

x

x

FS xxcbxax

dxxxcbaI

S

F

>++

= ∫

Step 1: Write the denominator in the form:

( )( ) acbaabxa 4with,4/2/ 222−=∆∆−+

and check the value of the discriminant ∆ .

i) 0<∆ complex roots; no singularities

ii) 0=∆ double real root; 2)( −

singularity at abx 2/1 −=

iii) 0>∆ real roots; two 1)( − singularities at ( ) abx 22,1 ∆±−=

In case (iii), for the location of the singularities, use:

( )( )∆−−

=∆−−

=>b

cx

a

bxb

2;

2:0 21

( )( )

a

bx

b

cxb

2;

2:0 21

∆+−=

∆+−=< ,

to minimise loss of accuracy through numerical cancellation.

Step 2: Check that the range of integration does not include a singularity.

In case (ii): SF xxxx << 11 or

In case (iii): 2121 oror xxxxxxxx FSSF <<<<<

Step 3: Carry out the integration by making the substitution:

abxu 2/+= .

Putting ∆=R , the results are:

i)

F

S

x

xR

bax

RI

+=

2arctan

2

ii)

F

S

x

xbax

I

+

−=

)2(

2

iii)

F

S

x

xRbax

Rbax

RI

++

−+=

2

2ln

1

Integration Integration using partial fractions

Algebra and functions Completing the square for a quadratic function. Algebra and functions The discriminant of a quadratic function.

Integration Integration by substitution.



2 Integral of x times reciprocal quadratic polynomial

∫ ++=

F

S

x

x

FScbxax

xdxxxcbaJ

2),,,,(

For this integral, carry out the checks in steps 1 and 2 above, and then write:

ababaxx 2/2/)2( −+=

to give:

),,,,(2

)log(2

1 2

FS

x

x

xxcbaIa

bcbxax

aJ

F

S

−

++=



Tractive force

The tractive force is the pulling force exerted by a vehicle, or machine or

body.

Tractive effort is a synonym of tractive force, used in railway engineering

terminology when describing the pulling power of a locomotive.

The tractive force value can be either a theoretically or experimentally

obtained value, and will usually be quoted under normal operating conditions.

The actual value for a particular locomotive varies depending on speed and

track conditions, and is influenced by a number of other factors.

Types of tractive efforts

When a figure for tractive effort is quoted in technical documentation it is

either for the starting tractive effort (at a dead start with the wheels not

turning) or as the continuous tractive effort which will be quoted at a

particular speed.

Maximum tractive effort

The maximum tractive effort is the maximum pulling force a vehicle or

machine can exert under any (non-damaging) conditions. In general the

maximum tractive effort will be obtained at a standstill and/or low speeds.

A variety of factors limit the maximum value:

• The maximum tractive effort cannot exceed the tractive mass (m) times

the coefficient of friction (µ) . If a vehicle attempts to supply more force

(Ftractive > µm) this will cause wheel spin.

• The gear ratios of drive components.

• The maximum power capable of being supplied to the drive systems.

• The safe working torques of the drive system components.

Continuous tractive effort

The continuous tractive effort is the tractive effort which is supplied at a given

velocity. It may refer to the tractive effort required to keep a vehicle rolling

without acceleration or the maximum force that can be produced at given

speed.[2]

Because of the relationship between Power (P), velocity (v) and force (F) of:

P = vF or P/v = F

the continuous tractive effort is inversely proportional to the velocity for

constant power; the continuous tractive effort is therefore dependent on the

power at rail

In vehicles which have a power source (diesel engine, electrical supply etc)

which is limited in terms of maximum total power (including steam engines)

the maximum continuous tractive effort at a given speed is limited by the

engine's power.



Continuous tractive effort is quoted as a force at a given speed, and may be

presented in graph form at a range of speeds as part of a tractive effort curve

Maximum continuous tractive effort

For vehicles propelled by electric motors the maximum continuous tractive

effort can be less than the short term maximum tractive effort at a given speed.

The maximum continuous tractive effort is defined as:

"The tractive force delivered at full throttle notch (power) after the traction

system has heated to maximum operating temperature"

Similar considerations also apply to hydrodynamic transmissions such as fluid

couplings and torque converters which create more heat at stall than when free

running. (See also Stall torque).

Tractive effort curves

Technical specifications of locomotives often include tractive effort curves,

which show the relationship between tractive effort and velocity.

Schematic diagram of tractive effort vs. speed for a hypothetical locomotive

with power at rail of ~7000 kW

The basic shape of the graph is shown schematically (diagram right). The line

AB shows the operation at the maximum tractive effort, the line BC shows the

relationship of continuous tractive effort being inversely proportional to speed.

Tractive effort curves will often have graphs of rolling resistance superimposed

on them—the intersection of the rolling resistance graph and tractive effort

graph gives the maximum velocity (i.e. when the net tractive effort is zero).



Diesel and electric locomotives

For a diesel-electric locomotive or electric locomotive, starting tractive effort

can be calculated from the stall torque of the traction motors (the turning force

it can produce while at a dead stop), the gearing, and the wheel diameter. For a

diesel-hydraulic locomotive the starting tractive effort depends on the stall

torque of the torque converter.

In general, it is more common for heavy freight trains (such as Class 59, Class

60 and Class 66 locomotives) to have a high maximum tractive effort due to

the mass which they haul. Passenger trains (such as Class 43/Intercity High

Speed Train locomotives) usually have much lower maximum tractive efforts

due to the higher gear ratio required for a higher top speed.

Stall torque

Stall torque is the torque which is produced by a device when the output

rotational speed is zero, it may also mean the torque load that causes the output

rotational speed of a device to become zero - i.e. to cause stalling

Devices such as electric motors, steam engines and hydrodynamic

transmissions produce torque under these conditions.

Electric motors continue to provide torque when stalled. However, electric

motors left in a stalled condition are prone to overheating and possible damage

since the current flowing is maximum under these conditions.

The maximum torque an electric motor can produce in the long term when

stalled without causing damage is called the maximum continuous stall

torque.

Torque converter

A torque converter is a modified form of fluid coupling that is used to transfer

rotating power from a prime mover, such as an internal combustion engine or

electric motor, to a rotating driven load. Like a basic fluid coupling, the torque

converter normally takes the place of a mechanical clutch, allowing the load to

be separated from the power source. As a more advanced form of fluid

coupling, however, a torque converter is able to multiply torque when there is a

substantial difference between input and output rotational speed, thus providing

the equivalent of a reduction gear.

Function

Torque converter elements

A fluid coupling is a two element drive that is incapable of multiplying torque,

while a torque converter has at least one extra element—the stator—which

alters the drive's characteristics during periods of high slippage, producing an

increase in output torque.

In a torque converter there are at least three rotating elements: the pump, which

is mechanically driven by the prime mover; the turbine, which drives the load;

and the stator, which is interposed between the pump and turbine so that it can

alter oil flow returning from the turbine to the pump. The classic torque

converter design dictates that the stator be prevented from rotating under any

condition, hence the term stator. In practice, however, the stator is mounted on



an overrunning clutch, which prevents the stator from counter-rotating with

respect to the prime mover but allows forward rotation.

Modifications to the basic three element design have been periodically

incorporated, especially in applications where higher than normal torque

multiplication is required. Most commonly, these have taken the form of

multiple turbines and stators, each set being designed to produce differing

amounts of torque multiplication. For example, the Buick Dynaflow automatic

transmission was a non-shifting design and, under normal conditions, relied

solely upon the converter to multiply torque. The Dynaflow used a five

element converter to produce the wide range of torque multiplication needed to

propel a heavy vehicle.

Although not strictly a part of classic torque converter design, many

automotive converters include a lock-up clutch to improve cruising power

transmission efficiency and reduce heat. The application of the clutch locks the

turbine to the pump, causing all power transmission to be mechanical, thus

eliminating losses associated with fluid drive.

Operational phases

A torque converter has three stages of operation:

• Stall. The prime mover is applying power to the pump but the turbine

cannot rotate. For example, in an automobile, this stage of operation

would occur when the driver has placed the transmission in gear but is

preventing the vehicle from moving by continuing to apply the brakes.

At stall, the torque converter can produce maximum torque

multiplication if sufficient input power is applied (the resulting

multiplication is called the stall ratio). The stall phase actually lasts for

a brief period when the load (e.g., vehicle) initially starts to move, as

there will be a very large difference between pump and turbine speed.

• Acceleration. The load is accelerating but there still is a relatively large

difference between pump and turbine speed. Under this condition, the

converter will produce torque multiplication that is less than what could

be achieved under stall conditions. The amount of multiplication will

depend upon the actual difference between pump and turbine speed, as

well as various other design factors.

• Coupling. The turbine has reached approximately 90 percent of the

speed of the pump. Torque multiplication has essentially ceased and the

torque converter is behaving in a manner similar to a plain fluid

coupling. In modern automotive applications, it is usually at this stage

of operation where the lock-up clutch is applied, a procedure that tends

to improve fuel efficiency.

The key to the torque converter's ability to multiply torque lies in the stator. In

the classic fluid coupling design, periods of high slippage cause the fluid flow

returning from the turbine to the pump to oppose the direction of pump

rotation, leading to a significant loss of efficiency and the generation of

considerable waste heat. Under the same condition in a torque converter, the

returning fluid will be redirected by the stator so that it aids the rotation of the

pump, instead of impeding it. The result is that much of the energy in the

returning fluid is recovered and added to the energy being applied to the pump



by the prime mover. This action causes a substantial increase in the mass of

fluid being directed to the turbine, producing an increase in output torque.

Since the returning fluid is initially traveling in a direction opposite to pump

rotation, the stator will likewise attempt to counter-rotate as it forces the fluid

to change direction, an effect that is prevented by the one-way stator clutch.

Unlike the radially straight blades used in a plain fluid coupling, a torque

converter's turbine and stator use angled and curved blades. The blade shape of

the stator is what alters the path of the fluid, forcing it to coincide with the

pump rotation. The matching curve of the turbine blades helps to correctly

direct the returning fluid to the stator so the latter can do its job. The shape of

the blades is important as minor variations can result in significant changes to

the converter's performance.

During the stall and acceleration phases, in which torque multiplication occurs,

the stator remains stationary due to the action of its one-way clutch. However,

as the torque converter approaches the coupling phase, the energy and volume

of the fluid returning from the turbine will gradually decrease, causing pressure

on the stator to likewise decrease. Once in the coupling phase, the returning

fluid will reverse direction and now rotate in the direction of the pump and

turbine, an effect which will attempt to forward-rotate the stator. At this point,

the stator clutch will release and the pump, turbine and stator will all (more or

less) turn as a unit.

Unavoidably, some of the fluid's kinetic energy will be lost due to friction and

turbulence, causing the converter to generate waste heat (dissipated in many

applications by water cooling). This effect, often referred to as pumping loss,

will be most pronounced at or near stall conditions. In modern designs, the

blade geometry minimizes oil velocity at low pump speeds, which allows the

turbine to be stalled for long periods with little danger of overheating.

Efficiency and torque multiplication

A torque converter cannot achieve 100 percent coupling efficiency. The classic

three element torque converter has an efficiency curve that resembles an

inverted "U": zero efficiency at stall, generally increasing efficiency during the

acceleration phase and low efficiency in the coupling phase. The loss of

efficiency as the converter enters the coupling phase is a result of the

turbulence and fluid flow interference generated by the stator, and as

previously mentioned, is commonly overcome by mounting the stator on a one-

way clutch.

Even with the benefit of the one-way stator clutch, a converter cannot achieve

the same level of efficiency in the coupling phase as an equivalently sized fluid

coupling. Some loss is due to the presence of the stator (even though rotating

as part of the assembly), as it always generates some power-absorbing

turbulence. Most of the loss, however, is caused by the curved and angled

turbine blades, which do not absorb kinetic energy from the fluid mass as well

as radially straight blades. Since the turbine blade geometry is a crucial factor

in the converter's ability to multiply torque, trade-offs between torque

multiplication and coupling efficiency are inevitable. In automotive

applications, where steady improvements in fuel economy have been mandated

by market forces and government edict, the nearly universal use of a lock-up

clutch has helped to eliminate the converter from the efficiency equation

during cruising operation.



The maximum amount of torque multiplication produced by a converter is

highly dependent on the size and geometry of the turbine and stator blades, and

is generated only when the converter is at or near the stall phase of operation.

Typical stall torque multiplication ratios range from 1.8:1 to 2.5:1 for most

automotive applications (although multi-element designs as used in the Buick

Dynaflow and Chevrolet Turboglide could produce more). Specialized

converters designed for industrial or heavy marine power transmission systems

are capable of as much as 5.0:1 multiplication. Generally speaking, there is a

trade-off between maximum torque multiplication and efficiency—high stall

ratio converters tend to be relatively inefficient below the coupling speed,

whereas low stall ratio converters tend to provide less possible torque

multiplication.

While torque multiplication increases the torque delivered to the turbine output

shaft, it also increases the slippage within the converter, raising the temperature

of the fluid and reducing overall efficiency. For this reason, the characteristics

of the torque converter must be carefully matched to the torque curve of the

power source and the intended application. Changing the blade geometry of the

stator and/or turbine will change the torque-stall characteristics, as well as the

overall efficiency of the unit. For example, drag racing automatic transmissions

often use converters modified to produce high stall speeds to improve off-the-

line torque, and to get into the power band of the engine more quickly.

Highway vehicles generally use lower stall torque converters to limit heat

production, and provide a more firm feeling to the vehicle's characteristics.

A design feature once found in some General Motors automatic transmissions

was the variable-pitch stator, in which the blades' angle of attack could be

varied in response to changes in engine speed and load. The effect of this was

to vary the amount of torque multiplication produced by the converter. At the

normal angle of attack, the stator caused the converter to produce a moderate

amount of multiplication but with a higher level of efficiency. If the driver

abruptly opened the throttle, a valve would switch the stator pitch to a different

angle of attack, increasing torque multiplication at the expense of efficiency.

Some torque converters use multiple stators and/or multiple turbines to provide

a wider range of torque multiplication. Such multiple-element converters are

more common in industrial environments than in automotive transmissions, but

automotive applications such as Buick's Triple Turbine Dynaflow and

Chevrolet's Turboglide also existed. The Buick Dynaflow utilized the torque-

multiplying characteristics of its planetary gearset in conjunction with the

torque converter for low gear and bypassed the first turbine, using only the

second turbine as vehicle speed increased. The unavoidable trade-off with this

arrangement was low efficiency and eventually these transmissions were

discontinued in favor of the more efficient three speed units with a

conventional three element torque converter.

[edit] Lock-up torque converters

As described above, pumping losses within the torque converter reduce

efficiency and generate waste heat. In modern automotive applications, this

problem is commonly avoided by use of a lock-up clutch that physically links

the pump and turbine, effectively changing the converter into a purely

mechanical coupling. The result is no slippage, and virtually no power loss.



The first automotive application of the lock-up principle was Packard's

Ultramatic transmission, introduced in 1949, which locked up the converter at

cruising speeds, unlocking when the throttle was floored for quick acceleration

or as the vehicle slowed down. This feature was also present in some Borg-

Warner transmissions produced during the 1950s. It fell out of favor in

subsequent years due to its extra complexity and cost. In the late 1970s lock-up

clutches started to reappear in response to demands for improved fuel

economy, and are now nearly universal in automotive applications.

[edit] Capacity and failure modes

As with a basic fluid coupling the theoretical torque capacity of a converter is

proportional to , where r is the mass density of the fluid, N is the impeller

speed (rpm), and D is the diameter. In practice, the maximum torque capacity

is limited by the mechanical characteristics of the materials used in the

converter's components, as well as the ability of the converter to dissipate heat

(often through water cooling). As an aid to strength, reliability and economy of

production, most automotive converter housings are of welded construction.

Industrial units are usually assembled with bolted housings, a design feature

that eases the process of inspection and repair, but adds to the cost of

producing the converter.

In high performance, racing and heavy duty commercial converters, the pump

and turbine may be further strengthened by a process called furnace brazing, in

which molten brass is drawn into seams and joints to produce a stronger bond

between the blades, hubs and annular ring(s). Because the furnace brazing

process creates a small radius at the point where a blade meets with a hub or

annular ring, a theoretical decrease in turbulence will occur, resulting in a

corresponding increase in efficiency.

Overloading a converter can result in several failure modes, some of them

potentially dangerous in nature:

• Overheating: Continuous high levels of slippage may overwhelm the

converter's ability to dissipate heat, resulting in damage to the elastomer

seals that retain fluid inside the converter. This will cause the unit to

leak and eventually stop functioning due to lack of fluid.

• Stator clutch seizure: The inner and outer elements of the one-way

stator clutch become permanently locked together, thus preventing the

stator from rotating during the coupling phase. Most often, seizure is

precipitated by severe loading and subsequent distortion of the clutch

components. Eventually, galling of the mating parts occurs, which

triggers seizure. A converter with a seized stator clutch will exhibit very

poor efficiency during the coupling phase, and in a motor vehicle, fuel

consumption will drastically increase. Converter overheating under

such conditions will usually occur if continued operation is attempted.

• Stator clutch breakage: A very abrupt application of power can cause

shock loading to the stator clutch, resulting in breakage. When this

occurs, the stator will freely counter-rotate the pump and almost no

power transmission will take place. In an automobile, the effect is

similar to a severe case of transmission slippage and the vehicle is all

but incapable of moving under its own power.



• Blade deformation and fragmentation: Due to abrupt loading or

excessive heating of the converter, the pump and/or turbine blades may

be deformed, separated from their hubs and/or annular rings, or may

break up into fragments. At the least, such a failure will result in a

significant loss of efficiency, producing symptoms similar (although

less pronounced) to those accompanying stator clutch failure. In

extreme cases, catastrophic destruction of the converter will occur.

• Ballooning: Prolonged operation under excessive loading, very abrupt

application of load, or operating a torque converter at very high RPM

may cause the shape of the converter's housing to be physically

distorted due to internal pressure and/or the stress imposed by

centrifugal force. Under extreme conditions, ballooning will cause the

converter housing to rupture, resulting in the violent dispersal of hot oil

and metal fragments over a wide area.



FAILURE ANALYSIS RAIL CONTACT

Rail contact: plastic deformation (Ratcheting behavior).

Repeated rolling or sliding contacts stresses the material cyclically. Dependant on the contact stresses the material responds in one of the following four ways,

1. Perfectly elastic behavior if the contact pressure does not exceed the elastic limit, i.e. for a line contact pmax/τc= 0.31 or expressed in the load intensity τc/pmax=1/0.31.

2. Elastic shakedown behavior, in which plastic deformation takes place during running in, while due to residual stresses or strain hardening the steady state is perfectly elastic.

3. Plastic shakedown behavior, in which the steady state is a closed elastic plastic loop, with no net accumulation of plastic deformation.

4. Ratcheting behavior, in which the steady state is an open elastic plastic loop, the material accumulates a net strain during each cycle.

Railcontact: rolling contact fatigue, crack initiated at the surface.

For full description of failure analysis and remedy see reference .

tractiveeffortaccelerationandbraking

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