Gyro Torques

In this section all units will be converted to SI units (Systeme Internationale) . The moment of inertia is  I = m r_g² , so if a 10 in. diameter prop weighing 16 grams in considered, we can estimate:

• r_g = 0.225 * (10.0/12.0) * 0.3048 =0.05715   (meters)
• Then compute   I = 0.01600 * 0.05715²=52.258E-6   (kilogram-meter².)

At 16000 rpm the angular momentum of the propeller is

• @ 16000 rpm   w= (16000.0/60.0)*2*Pi = 1675.5  (radians/second)
• H = wI =1675.5 * 52.258E-8 = 0.08756   (kilogram-meter²/second)

Suppose the aircraft is executing an inside loop in about 3 seconds, say rotating at q=2*pi/3.0  =2.094 rad./sec.. The gyro torque is nose RIGHT and has a magnitude Hq = qwI= 2.095*0.08756= 0.1834  (Newton-meters)

Torque in an outside loop is nose left.

"What's that in real units?" an American might ask. Well, duuh. -- OK,say what -- inch ounces??

• 1 Newton = 2.698 oz
• 1 meter = 39.37 inches
• Torque = 0.1834*2.698*39.37 = 19.5 inch-ounces

Note that the gyro torque increases rapidly with diameter. A 12 inch diameter with weight increased accordingly would have about 2.5 times more torque.

A note in passing is that the ounce (avoirdupois), pound and so-on are units of force. When they are used as mass in the "English" or customary US system they are typically converted to slugs. In the SI system grams are mass units and Newtons are force units. This in spite of the fact that weights are often given in kilograms, but this implies the weight that a kilogram of mass would have in a standard gravity field.

Another example is a control line aircraft flying upright and wings level in the normal counterclockwise direction. The rate of rotation here is a yaw rate. In a manner similar to the pitch rate case examined above, the moment generated is nose up. An AMA scale racing ("Goodyear") example would be an aircraft with a lap time of 2.1 seconds and a 6.5 inch diameter propeller weighing 5 grams and turning at 28000 rpm.

The yaw rate is r=pi/2.1= 3.00 radians/second.

The propeller angular rate is w = 2*pi*28000/60=2930 radians/second.

The radius of gyration is r_g=6.5*25.4*0.225 = 37.1 mm.

The moment of inertia is I=5.0E-3*(37.1E-3)²= 6.88E-6 kilogram-meters.

The gyro torque is nose up and is T = r I w =  3.0*2930*6.88E-6=0.06048 Newton-meters = 6.4 inch-ounces .

In practical terms this means that the amount of up elevator required to fly level i.e. reduced a significant amount. For an aircraft weight of 16 oz. this amounts to the equivalent of moving the c.g. forward by 6.4/16.0=0.4 inches without changing the amount of up elevator required to fly.

A second consideration is that with propellers with 1 or two blades the gyro torque oscillates. If the torque computed by the methods above was 10 inch-ounces then the torque on the aircraft would average this amount but oscillate between 0 and 20 inch-ounces 2 times per revolution. At 12000 RPM (200Hz) this is a 400 Hz oscillation in torque - getting close to middle C. With 3 or more blades the Inertia about any axis normal to the shaft is the same and there is no oscillation in torque.

In summary, propellers with three or more blades should be used if vibration is a problem and high pitch or yaw rates will be demanded. Control line aircraft need less up elevator to fly upright and more down elevator to fly inverted because of gyro torque. To reduce the amount of rudder and/or elevator required in sharp pull up maneuvers a low weight (carbon fiber) propeller should be considered.