Editing Tractive effort (section)

==Rail vehicles==

In order to start a train and accelerate it to a given speed, the locomotive(s) must develop sufficient tractive force to overcome the train's [[Rail vehicle resistance|resistance]], which is a combination of axle [[bearing (mechanical)|bearing]] friction, the friction of the wheels on the rails (which is substantially greater on curved track than on tangent track), and the force of [[gravity]] if on a [[grade (slope)|grade]].  Once in motion, the train will develop additional drag as it accelerates due to [[aerodynamic force]]s, which increase with the square of the speed.  Drag may also be produced at speed due to [[hunting oscillation|truck (bogie) hunting]], which will increase the rolling friction between wheels and rails.  If acceleration continues, the train will eventually attain a speed at which the available tractive force of the locomotive(s) will exactly offset the total drag, causing acceleration to cease.  This top speed will be increased on a downgrade due to gravity assisting the motive power, and will be decreased on an upgrade due to gravity opposing the motive power.

Tractive effort can be theoretically calculated from a locomotive's mechanical characteristics (e.g., steam pressure, weight, etc.), or by actual testing with [[strain gauge|strain sensor]]s on the [[drawbar (haulage)|drawbar]] and a [[dynamometer car]].  Power at rail is a railway term for the available power for traction, that is, the power that is available to propel the train.

===Steam locomotives===
An estimate for the tractive effort of a single cylinder steam locomotive can be obtained from the cylinder pressure, cylinder bore, [[stroke (engines)|stroke]] of the piston<ref group="note">Half the stroke distance is about the same as the radial distance from the coupling of the driving rod to the centre of the driven wheel</ref> and the diameter of the wheel. The torque developed by 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.<ref group="note">The relationship is: Torque = Force<sub>piston</sub> 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</ref> For a more useful value an average value over the rotation of the wheel is used. The driving force is the torque divided by the wheel radius.

As an approximation, the following formula can be used (for a two-cylinder locomotive):<ref group="note">As with any physical formula, [[units of measurement]] must be consistent: 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.</ref>

:<math>\{t\}_\mathrm{lbf} = \frac {\{d\}_\mathrm{in}^2 \{s\}_\mathrm{in} \{p\}_\mathrm{psi}} {\{w\}_\mathrm{in}} \times 0.85,</math> <ref>{{cite book | last = Allan| first = Ian | title = British Railways Locomotives Combined Volume | publisher = Ian Allan Ltd | date = 1957}}</ref>

where
* ''t'' is tractive effort in pounds-force
* ''d'' is the [[piston]] diameter in inches ([[bore (engine)|bore]])
* ''s'' is the piston stroke in inches
* ''p'' is the working pressure in [[pounds per square inch]]
* ''w'' is the diameter of the driving wheels in inches

The constant 0.85 was the [[Association of American Railroads]] (AAR) standard for such calculations, and overestimated the efficiency of some locomotives and underestimated that of others. Modern locomotives with [[roller bearings]] were probably underestimated.

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

The constant ''c'' also depends 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.<ref group="note">See [[Gas laws]] for an explanation.</ref> giving a low ''c'' value. If the cylinder valves are left open for longer the value of ''c'' will rise nearer to one.

;Three or four cylinders (simple)
The result should be multiplied by 1.5 for a three-cylinder locomotive and by two for a four-cylinder locomotive.<ref>''Ian Allan ABC of British Railways Locomotives'', winter 1960/61 edition, part 1, p. 3</ref>

Alternatively, tractive effort of all "simple" (i.e. non-compound) locomotives can be calculated thus:

<math>\{t\}_\mathrm{lbf} = \frac{0.85 \{d\}_\mathrm{in}^2 n \{s\}_\mathrm{in} \{p\}_\mathrm{psi}} {2 \{w\}_\mathrm{in}},</math> <ref>{{cite book | last = Phillipson| first = E.A. | title = Steam Locomotive Design: Data and Formulae | publisher = The Locomotive Publishing Company | date = 1936}}</ref>

where
* ''t'' is tractive effort in pounds-force
* ''n'' is the number of cylinders
* ''d'' is the piston diameter in inches
* ''s'' is the piston stroke in inches
* ''p'' is the maximum rated boiler pressure in psi
* ''w'' is the diameter of the driving wheels in inches


;Multiple cylinders (compound)
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.<ref group="note">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</ref>

====Values and comparisons for steam locomotives====

Tractive effort is the figure often quoted when comparing the powers of steam locomotives, but is misleading because tractive effort shows the ability to start a train, not the ability to haul it. Possibly the highest tractive effort ever claimed was for the [[Virginian Railway]]'s [[2-8-8-8-4]] triplex locomotive, which in [[Expansive working|simple expansion]] mode had a calculated starting T.E. of 199,560&nbsp;lbf (887.7&nbsp;kN)&mdash;but the boiler could not produce enough steam to haul at speeds over 5&nbsp;mph (8&nbsp;km/h).

Of more successful steam locomotives, those with the highest rated starting tractive effort were the Virginian Railway AE-class [[2-10-10-2]]s, at 176,000&nbsp;lbf (783&nbsp;kN) in simple-expansion mode (or 162,200&nbsp;lb if calculated by the usual formula). The [[Union Pacific]] [[Union Pacific Big Boy|Big Boys]] had a starting T.E. of 135,375&nbsp;lbf (602&nbsp;kN); the [[Norfolk & Western]]'s Y5, Y6, Y6a, and Y6b class  [[2-8-8-2]]s had a starting T.E. of 152,206&nbsp;lbf (677&nbsp;kN) in simple expansion mode (later modified to 170,000&nbsp;lbf (756&nbsp;kN), claim some enthusiasts); and the [[Pennsylvania Railroad]]'s freight [[Duplex (locomotive)|duplex]] [[PRR Q2|Q2]] attained 114,860&nbsp;lbf (510.9&nbsp;kN, including booster)&mdash;the highest for a rigid-framed locomotive. Later two-cylinder passenger locomotives were generally 40,000 to 80,000&nbsp;lbf (170 to 350&nbsp;kN) of T.E.

===Diesel and electric locomotives===
For an [[electric locomotive]] or a [[diesel-electric locomotive]], starting tractive effort can be calculated from the amount of weight on the driving wheels (which may be less than the total locomotive weight in some cases), combined [[stall torque]] of the [[traction motor]]s, the [[Spur gears|gear ratio]] between the traction motors and axles, and driving wheel [[diameter]].  For a [[diesel-hydraulic locomotive]], the starting tractive effort is affected by the stall torque of the [[torque converter]], as well as gearing, wheel diameter and locomotive weight.

The relationship between power and tractive effort was expressed by Hay (1978) as

:<math>t = \frac{PE}{v},</math> <ref>{{cite book | last = Hay| first = William | title = Railroad Engineering | publisher = Wiley, New York | date = 1978 |page=100}}</ref>

where
* ''t'' is tractive effort, in [[newton (unit)|newton]]s (N)
* ''P'' is the power in [[watt]]s (W)
* ''E'' is the efficiency, with a suggested value of 0.82 to account for losses between the motor and the rail, as well as power diverted to auxiliary systems such as lighting
* ''v'' is the speed in [[metres per second]] (m/s)

Freight locomotives are designed to produce higher maximum tractive effort than passenger units of equivalent power, necessitated by the much higher weight that is typical of a freight train.  In modern locomotives, the gearing between the traction motors and axles is selected to suit the type of service in which the unit will be operated.  As traction motors have a maximum speed at which they can rotate without incurring damage, gearing for higher tractive effort is at the expense of top speed.  Conversely, the gearing used with passenger locomotives favors speed over maximum tractive effort.

Electric locomotives with [[monomotor]] bogies are sometimes fitted with two-speed gearing.  This allows higher tractive effort for hauling freight trains but at reduced speed. Examples include the SNCF classes [[SNCF Class BB 8500|BB 8500]] and [[SNCF Class BB 25500|BB 25500]].