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Torque-induced

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Torque-induced precession (gyroscopic precession) is the phenomenon in which the axis of a spinning object (e.g., a part of a gyroscope) "wobbles" when a torque is applied to it. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. If the speed of the rotation and the magnitude of the torque are constant, the axis will describe a cone, its movement at any instant being at right angles to the direction of the torque. In the case of a toy top, if the axis is not perfectly vertical, the torque is applied by the force of gravity tending to tip it over.

The response of a rotating system to an applied torque. When the device swivels, and some roll is added, the wheel tends to pitch.

The device depicted on the right here is gimbal mounted. From inside to outside there are three axes of rotation: the hub of the wheel, the gimbal axis, and the vertical pivot.

To distinguish between the two horizontal axes, rotation around the wheel hub will be called 'spinning', and rotation around the gimbal axis will be called 'pitching.' Rotation around the vertical pivot axis is called 'rotation'.

First, imagine that the entire device is rotating around the (vertical) pivot axis. Then, spinning of the wheel (around the wheelhub) is added. Imagine the gimbal axis to be locked, so that the wheel cannot pitch. The gimbal axis has sensors, that measure whether there is a torque around the gimbal axis.

In the picture, a section of the wheel has been named dm₁. At the depicted moment in time, section dm₁ is at the perimeter of the rotating motion around the (vertical) pivot axis. Section dm₁, therefore, has a lot of angular rotating velocity with respect to the rotation around the pivot axis, and as dm₁ is forced closer to the pivot axis of the rotation (by the wheel spinning further), due to the Coriolis effect dm₁ tends to move in the direction of the top-left arrow in the diagram (shown at 45°) in the direction of rotation around the pivot axis. Section dm₂ of the wheel starts out at the vertical pivot axis, and thus initially has zero angular rotating velocity with respect to the rotation around the pivot axis, before the wheel spins further. A force (again, a Coriolis force) would be required to increase section dm₂'s velocity up to the angular rotating velocity at the perimeter of the rotating motion around the pivot axis. If that force is not provided, then section dm₂'s inertia will make it move in the direction of the top-right arrow. Note that both arrows point in the same direction.

The same reasoning applies for the bottom half of the wheel, but there the arrows point in the opposite direction to that of the top arrows. Combined over the entire wheel, there is a torque around the gimbal axis when some spinning is added to rotation around a vertical axis.

It is important to note that the torque around the gimbal axis arises without any delay; the response is instantaneous.

In the discussion above, the setup was kept unchanging by preventing pitching around the gimbal axis. In the case of a spinning toy top, when the spinning top starts tilting, gravity exerts a torque. However, instead of rolling over, the spinning top just pitches a little. This pitching motion reorients the spinning top with respect to the torque that is being exerted. The result is that the torque exerted by gravity - via the pitching motion - elicits gyroscopic precession (which in turn yields a counter torque against the gravity torque) rather than causing the spinning top to fall to its side.

Precession or gyroscopic considerations have an effect on bicycle performance at high speed. Precession is also the mechanism behind gyrocompasses.

Gyroscopic precession also plays a large role in the flight controls on helicopters. Since the driving force behind helicopters is the rotor disk (which rotates), gyroscopic precession comes into play. If the rotor disk is to be tilted forward (to gain forward velocity), its rotation requires that the downward net force on the blade be applied roughly 90 degrees (depending on blade configuration) before, or when the blade is to one side of the pilot and rotating forward.

To ensure the pilot's inputs are correct, the aircraft has corrective linkages that vary the blade pitch in advance of the blade's position relative to the swashplate. Although the swashplate moves in the intuitively correct direction, the blade pitch links are arranged to transmit the pitch in advance of the blade's position.

Classical (Newtonian)

The torque caused by the two opposing forces F_g and -F_g causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.

Precession is the result of the angular velocity of rotation and the angular velocity produced by the torque. It is an angular velocity about a line that makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque. The permanent axis must turn towards this line, since the body cannot continue to rotate about any line that is not a principal axis of maximum moment of inertia; that is, the permanent axis turns in a direction at right angles to that in which the torque might be expected to turn it. If the rotating body is symmetrical and its motion unconstrained, and, if the torque on the spin axis is at right angles to that axis, the axis of precession will be perpendicular to both the spin axis and torque axis.

Under these circumstances the angular velocity of precession is given by:

${\boldsymbol {\omega }}_{p}={\frac {\ mgr}{I_{s}{\boldsymbol {\omega }}_{s}}}$

In which I_s is the moment of inertia, ${\boldsymbol {\omega }}_{s}$ is the angular velocity of spin about the spin axis, and m*g*r are the force and radius that comes from the torque. The torque vector originates at the center of mass. Using ${\boldsymbol {\omega }}$ = ${\frac {2\pi }{T}}$, we find that the period of precession is given by:

$T_{p}={\frac {4\pi ^{2}I_{s}}{\ mgrT_{s}}}$

In which I_s is the moment of inertia, T_s is the period of spin about the spin axis, and ${\boldsymbol {\tau }}$ is the torque. In general, the problem is more complicated than this, however.