Tractive effort

Tractive effort is the pulling force exerted, by a locomotive or other vehicle. The term is used specifically in Railway terminology.

The tractive effort value can be either a theorectical or experimentally obtained value, and will usually be quoted under normal operating conditions. The actual value for a particular locomotive vaaries depending on speed and track conditions, and is influenced by a number of other facotrs.

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.

Introduction

For a long, heavy train to accelerate from a stationary position at a satisfactory rate of acceleration, the locomotive must apply a large force, that is, a large tractive effort (because from Newton's laws of motion Force = mass x acceleration). The tractive effort minus the additional frictional and other resistive forces (eg incline, air resistance) gives the force available to make the train accelerate.

In general the resistive forces increase with velocity, so at a some given rate of movement the tractive effort will equal the resistive forces and the train will not be able to accelerate further - this gives rise to a limit in any trains tops speed.

For a train running at a desired velocity, the locomotive needs only to provide enough forward force to counteract the counteracting forces of friction (wheels on rails, axles in bearings) and wind resistance (a small force compared to the other forces at work) on level track, plus the parallel-to-track vector component of gravity's acceleration of the train's mass on grades (which is fighting against the locomotive on uphill grades, and pushing it forward on downhill grades).

As well as been calculated theorectically from the characteristics of the engine, transmission system and the wheel diameter and mass of a locomotive, the tractive effort can also be obtained experimental through combinations of drawbar strain sensors and a dynamometer car.

Maximum Tractive effort

The maximum tractive effort^[1] is the maximum pulling force a locomotive 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 excede the 'Tractive mass (m)' x 'the coefficient of friction' (μ) . If a vehicle attempts to supply more force (F_tractive>μm) this will cause Wheelspin^[2].
The maximum torque capable of being generated by the traction device (either piston/connecting rod, or electric motor, or hydraulic transmission). For example: a high powered locomotive such as an electric locomotive could supply more current than the motors are capable of handling without being damaged.
The safe working torques of the drive system components - in general a locomotive will be engineered so that the transmission components do not limit the tractive effort of a locomotive.

Which ever of the above variables is least will limit the maximum tractive effort.

Continuous Tractive effort

The continuous tractive effort is the the tractive effort which is supplied at a given velocity. It may refer to the tractive effort required to keep a train rolling without acceleration or the maximum force that can be produced at at given speed.^[3]

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

P=vF or P/v=F

the continous tractive effort is inversly proportional to the velocity for constant power; the continous tractive effort is therefor dependent on the power at rail^[4]

Because locomotives have a power source (diesel engine, electrical supply etc) which is limited in terms of maximum total power (including steam engines^[5]) the maximum continuous tractive effort is limited by the engine's power.

Continous 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

Tractive effort curves

Technical specifications of locomitives often include tractive effort curves^[6]^[7]^[8]^[9], 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 ~7000kW

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.^[10]

Tractive effort curves will often have graphs of rolling resistance superimposed on them - the intersection of the rolling resistance graph^[11] and tractive effort graph gives the maximum velocity (ie when the net tractive effort is zero).

Steam locomotives

An approximate theoretical value for the tractive effort of a single cylinderd steam can be obtained by considering the cylinder pressure, cylinder area, and stroke of the piston^[12] and the diameter of the wheel. The torque developed by the action of the linear motion of the piston depends on the angle that the driving rod makes with the tangent of the radius on the driving wheel.^[13] For a more useful value an average value over the rotation of the wheel is used. The driving force is simply the torque divided by the wheel radius.

For a two cylinder locomotive the average force is twice that of a single cylinder locomoitive.

Thus as an approximation the following equation can be obtained (for a 2 cylinder locomotive)^[14] :

t={\frac {cPd^{2}s}{D}}

where

t is tractive effort
c is a constant representing losses in pressure and friction; normally 0.85 is used^[15]
P is the boiler pressure^[16]
d is the piston diameter (bore)
s is the piston stroke
D is the driving wheel diameter

The constant 0.85 was the Association of American Railroads (AAR) standard for such calculations, and certainly over-estimated the efficiency of some locomotives and underestimated that of others. Modern, roller bearing fitted locomotives were probably underestimated in this calculation.

European designers used a constant of 0.6 instead of 0.85, so the two cannot be directly compared without a conversion factor. In Britain, the main-line railways generally used a constant of 0.85 but builders of industrial locomotives often used a lower figure, typically 0.75.

The value of the constant c also will depend on the cylinder dimensions, and the time at which the steam inlet valves are open; if the steam inlet valves are closed immediately after obtaining full cylinder pressure the piston force can be expected to have dropped to less than half the initial force.^[17] giving a low c value. If the cylinder valves are left open for longer the value of c will rise nearer to 1.

For other numbers and combinations of cylinders, including double and triple expansion engines the tractive effort can be estimated by adding the tractive efforts due to the individual cylinders at their respective pressures and cylinder strokes.^[18]

Values and comparisons for Steam locomotives

Tractive effort is the figure most often quoted when people are comparing the power of different steam locomotives, but the use can be misleading, because tractive effort shows the ability to start a train, not the ability to do work by hauling it. Possibly the highest figure for starting tractive effort ever recorded was for the Virginian Railway's 2-8-8-8-4 Triplex locomotive, which in simple expansion mode had a starting T.E. of 199,560 lbf (888 kN) — but this did not translate into power, for the boiler was undersized and could not produce enough steam to haul at speeds over 5 mph (8 km/h).

Of more successful large steam locomotives, those with the highest rated starting tractive effort were the Virginian Railway AE-class 2-10-10-2s, at 176,000 lbf (783 kN) in simple-expansion mode. The Union Pacific's famous Big Boys had a starting T.E. of 135,375 lbf (602 kN); the Norfolk & Western's Y5, Y6, Y6a, and Y6b class 2-8-8-2s had a starting T.E. of 152,206 lbf (677 kN) in simple expansion mode (later modified, resulting in a claimed T.E. of 170,000 lbf (756 kN)); and the Pennsylvania Railroad's freight Duplex Q2 attained 114,860 lbf (511 kN) — the highest for a rigid framed locomotive. Later two cylinder passenger locomotives were generally 70,000 to 80,000 lbf (300 to 350 kN) of T.E.

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, which can be very large.

Examples

A table to illustrate the speed the maximum tractive effort, continuous tractive effort and the speed at which the tractive effort should be reduced on a selection of trains operating in the United Kingdom:

Class	Type	Top speed		Maximum tractive effort	Speed to reduce tractive effort	Continuous tractive effort	Maximum power at rail	Mass
Class	Type	mph	km/h	Maximum tractive effort	Speed to reduce tractive effort	Continuous tractive effort	Maximum power at rail	Mass
Class 08	Shunter	15		156 kN	8.8 mph	49 kN	194 kW	49.6 - 50.4 t
Class 33	Passenger	85		200 kN	17.5 mph	116 kN	906 kW	77.7 t
Class 56	Light freight	80		275 kN	16.8 mph	240 kN	1790kW	125.2 t
Class 58	Light freight	80		275 kN	17.4 mph	240 kN	1780 kW	130 t
Class 59	Heavy freight	60 or 75		506 kN	14.3 mph	291 kN	1889 kW	121 t
Class 60	Heavy freight	60		500 kN	17.4 mph	336 kN	1800 kW	129-131 t
Class 66	Heavy freight	75		409 kN	15.9 mph	260 kN	1850 kW	126 t
Class 67	Light freight	125	200	141 kN	?? mph	90 kN	1860 kW	90 t

The power at rail of a train follows the equation power (kW) = force (kN) x speed (m/s)

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. Light freight trains (such as Class 56, Class 58 and Class 67 locomotives) and passenger trains (such as Class 33 and Class 43 / Intercity High Speed Train locomotives) usually have much lower maximum tractive efforts.

References

^ Article : "So just what do terms used to describe the performance of locomotives and multiple units like Maximum Tractive Effort, Power At Rail, and Continuous Power mean?" Tony Woof B.Eng C.Eng MIEE
^ Wheel spin can damage the wheel and rail. Low frictional coefficients can be a problem for rail vehicles - eg see Slippery rail. Most locomotives carry a sandbox for use when the wheels are likely to slip
^ Handbook of Railway Vehicle Dynamics , Simon Iwnicki , page 256 , Google books books.google.com
^ quoted figures will usually refer to the maximum continuous tractive effort - ie when the engine or other power source is operating at its maximum. ie at the maximum available power at rail.
^ Although it may seem that the maximum power of a conventional steam engine is limited at the rate at which the fireman can shovel coal into the steam engine, in fact the power will be limited by a variety of other factors, including - the rate of combustion of the fuel, the rate at which heat can be transferred across the heat exchanging mechansism from fire to water boiler etc
^ XPT: Delivery, test runs and demonstration runs railpage.au.org see graph
^ The Gravita Locomotive Family voithturbo.de (page 2)
^ EURO 4000 Freight Diesel-Electric Locomotives vossloh-espana.com (page 2)
^ Eurorunner ER20 BF and ER20 BU, Diesel electric platform locomotives for Europe siemens.dk (page 3)
^ Marks' Standard Handbook for Mechanical Engineers By Eugene A. Avallone, Theodore Baumeister, Ali Sadegh, Lionel Simeon Marks page 166 Google Books books.google.com
^ The graphs will typically show rolling resistance for standard train lengths or weights, both on the level or on an uphill gradient
^ It can be shown as a first approximation that half the stroke distance is approximately the same as the radial distance from the coupling of the driving rod to the centre of the driven wheel
^ The relationship is simply Torque = Force_piston x R (the radial distane to the point of connection of the driving rod) x cos(A) where A is the angle the driving rod makes with the tangent to the radius from wheel centre to driving rod attatchment
^ As with any physical formula, consistent units of measurement are required: pressure in psi and lengths in inches give tractive effort in lbf, while pressure in Pa and lengths in metres give tractive effort in N.
^ For a 'perfect' locomotove with cylinder piston pressure equal to boiler pressure (idenpendant of stroke) and with no frictional losses the constant c can be taken as 1
^ note that the boiler pressure may be greater than the cylinder pressure
^ See Gas laws for an explanation.
^ The value of the constant c for a low-pressure cylinder is taken to be 0.80 (when the value for a high pressure cylinder is taken to be 0.85

Additional references and further reading

[tonywoof-1] Article : "So just what do terms used to describe the performance of locomotives and multiple units like Maximum Tractive Effort, Power At Rail, and Continuous Power mean?" Tony Woof B.Eng C.Eng MIEE

[2] Wheel spin can damage the wheel and rail. Low frictional coefficients can be a problem for rail vehicles - eg see Slippery rail. Most locomotives carry a sandbox for use when the wheels are likely to slip

[3] Handbook of Railway Vehicle Dynamics , Simon Iwnicki , page 256 , Google books books.google.com

[4] quoted figures will usually refer to the maximum continuous tractive effort - ie when the engine or other power source is operating at its maximum. ie at the maximum available power at rail.

[5] Although it may seem that the maximum power of a conventional steam engine is limited at the rate at which the fireman can shovel coal into the steam engine, in fact the power will be limited by a variety of other factors, including - the rate of combustion of the fuel, the rate at which heat can be transferred across the heat exchanging mechansism from fire to water boiler etc

[6] XPT: Delivery, test runs and demonstration runs railpage.au.org see graph

[7] The Gravita Locomotive Family voithturbo.de (page 2)

[8] EURO 4000 Freight Diesel-Electric Locomotives vossloh-espana.com (page 2)

[9] Eurorunner ER20 BF and ER20 BU, Diesel electric platform locomotives for Europe siemens.dk (page 3)

[10] Marks' Standard Handbook for Mechanical Engineers By Eugene A. Avallone, Theodore Baumeister, Ali Sadegh, Lionel Simeon Marks page 166 Google Books books.google.com

[11] The graphs will typically show rolling resistance for standard train lengths or weights, both on the level or on an uphill gradient

[12] It can be shown as a first approximation that half the stroke distance is approximately the same as the radial distance from the coupling of the driving rod to the centre of the driven wheel

[13] The relationship is simply Torque = Force_piston x R (the radial distane to the point of connection of the driving rod) x cos(A) where A is the angle the driving rod makes with the tangent to the radius from wheel centre to driving rod attatchment

[14] As with any physical formula, consistent units of measurement are required: pressure in psi and lengths in inches give tractive effort in lbf, while pressure in Pa and lengths in metres give tractive effort in N.

[15] For a 'perfect' locomotove with cylinder piston pressure equal to boiler pressure (idenpendant of stroke) and with no frictional losses the constant c can be taken as 1

[16] te that the boiler pressure may be greater than the cylinder pressure

[17] See Gas laws for an explanation.

[18] The value of the constant c for a low-pressure cylinder is taken to be 0.80 (when the value for a high pressure cylinder is taken to be 0.85

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