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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.

[edit] 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.

[edit] Maximum tractive effort

The maximum tractive effort^[1] 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)' x 'the coefficient of friction' (μ) . If a vehicle attempts to supply more force (F_tractive>μm) this will cause Wheel spin^{[note 1]}.
• 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.

[edit] 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^{[note 2]}

In vehicles which have a power source (diesel engine, electrical supply etc) which is limited in terms of maximum total power (including steam engines^{[note 3]}) 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^[3]

[edit] 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^{[note 4]} 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"^[3]

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).

[edit] Tractive effort curves

Technical specifications of locomotives often include tractive effort curves^[4][5][6][7], 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.^[8]

Tractive effort curves will often have graphs of rolling resistance superimposed on them - the intersection of the rolling resistance graph^{[note 5]} and tractive effort graph gives the maximum velocity (ie when the net tractive effort is zero).

[edit] Rail vehicles

For a long, heavy train to accelerate from a stationary position at a satisfactory rate of acceleration, the locomotive must apply a large force. 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 theoretically 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.

Power at rail is a railway term for the available power for traction.

[edit] Steam locomotives

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An approximate theoretical value for the tractive effort of a single cylindered steam can be obtained by considering the cylinder pressure, cylinder area, and stroke of the piston^{[note 6]} 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.^{[note 7]} 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 locomotive.

Thus as an approximation the following equation can be obtained (for a 2 cylinder locomotive)^{[note 8]} :

where

• t is tractive effort
• c is a constant representing losses in pressure and friction; normally 0.85 is used^{[note 9]}
• P is the boiler pressure^{[note 10]}
• 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.^{[note 11]} 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.^{[note 12]}

[edit] 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.

[edit] 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 asClass 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.

[edit] See also

• factor of adhesion, which is simply the weight on the locomotive's driving wheels divided by the starting tractive effort
• Tractor pulling,Bollard pull - articles relating to tractive effort for other forms of vehicle
• Rail adhesion

• power classification - British Railways and London, Midland and Scottish railway classification scheme

[edit] References and notes

[edit] Notes

1. ^ 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
2. ^ 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.
3. ^ 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 mechanism from fire to water boiler etc
4. ^ For electric motors higher torques require higher currents, which also produce greater resistive heating, the term maximum continuous current is a related figure for traction motors. In the short term currents higher than the maximum continuous current may not cause damage by overheating.
5. ^ The graphs will typically show rolling resistance for standard train lengths or weights, both on the level or on an uphill gradient
6. ^ 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
7. ^ The relationship is simply Torque = Force_piston x R (the radial distance 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 attachment
8. ^ 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.
9. ^ For a 'perfect' locomotive with cylinder piston pressure equal to boiler pressure (independent of stroke) and with no frictional losses the constant c can be taken as 1
10. ^ note that the boiler pressure may be greater than the cylinder pressure
11. ^ See Gas laws for an explanation.
12. ^ 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

[edit] References

[edit] Additional references and further reading

• A simple guide to train physics
• Tractive effort, acceleration and braking