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Torque and power

by François Dovat

A torque consists of two forces diametrically opposed which tend to rotate an object. The moment of a force about a point is its rotating effect about the point. A couple is a pair of equal and parallel but opposite forces. The moment of a force or couple is known as torque; we usually consider the two constituents of the couple as a single one using the axis of rotation as fulcrum

A spanner tightening a nut exerts a torque on the nut. Similarly while trying to turn a key in a lock, a wheel, a tap, one exerts a torque on these objects. If the object turns, a certain work is carried out. The power is larger if the time to achieve a given work is shorter.

A cyclist who lays all his weight on a pedal produces a torque that was measured in pound-inch (lb-in), pound-foot (lb-ft) or kilogram-meter (mkg). Nowadays a SI unit is preferably used: the Newton-meter or Nm.

Newton meter (Nm) : 1 Nm = 0.738 lb-ft

Pound-foot (lb-ft): 1 lb-ft = 1.356 Nm

Kilogram-meter (mkg) : 1 mkg = 9.807 Nm

If this cyclist weighs 900 N (202 lbs) and the crank is 0.17 m long, the torque obtained is 900 x 0.17 = 153 Nm, which is that of a common 1.6 liters engine. But as this engine rotates much faster, its power is larger than that of the cyclist, the torque being only one of the two constituents of the power, the other one being the speed of revolution. The climb of a grade implies a given work; the faster the pedals or the engine spins, the more quickly that work will be carried out – for an identical torque.

One thus calculates the power by multiplying the torque by the angular velocity – also known as "revs". A horsepower (hp) is the power necessary to lift 33 000 lbs of one foot in one minute.

If the torque is given in lb-ft and the revs in rpm, the relation is obtained by the constant 5252 (result of 33 000 divided by 2 π). So, by multiplying the torque by the rpm and dividing by 5252 we obtain the power; or conversely, by dividing the power by the rpm and multiplying by 5252 we know the torque. This constant is due to the nature of the units employed. When S.I. units are used (Nm, kW), the constant becomes 60 x 1000 divided by 2π : 9549.

Power (kW) = torque (Nm) x rpm / 9549

Power (hp). = torque (lbs-ft) x rpm / 5252

The torque developed by an engine is measured on a test bench at various revs. By drawing a line between the points of measurement, one obtains the torque curve. The power curve is then calculated with the help of one of the two constants.

If the gears of a transmission have teeth, it is not to bite, but to transmit a power. Even if two meshing gears have a different number of teeth, the power recovered on the output gear is identical to the input power (less about 2% lost in friction), but the torque and revs will have changed into proportion of the respective numbers of teeth. E.g., if a gear of 10 teeth drives a gear of 40 teeth, the ratio of reduction is 4 (or sometimes of 0.25, if we take it the other way around). The output rpm will be divided by 4 while the torque it multiplied by 4. Gears are comparable with two levers in interaction; by changing the respective numbers of teeth, one modifies the length of the lever.

Speed characteristic curves of a truck engine. A constant torque (horizontal line Nm) corresponds to a linear rise in power (oblique line). A constant power (horizontal line kW) comes from a hyperbolic torque curve, known as hyperbole of constant power. (here between 1750 and 2200 rpm). The lines inside the torque curve indicate the specific fuel consumption (sfc or bsfc) - which decreases with an increase of the load.

(© François Dovat)
#6

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