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Gyroscopic Principles
All spinning objects have gyroscopic properties. The main properties that an object can experience in any gyroscopic motion are Rigidity in Space and Precession.

Rigidity in Space
Rigidity in space describes the principle that a gyroscope remains in the fixed position on the plane in which it is spinning, unaffected by the Earth's rotation. For example, a bike wheel.

Source

Precession
A simple case of precession, also known as steady precession, can be described by the following relation to Moment:

$$\sum M_x = -I\phi\prime^2 sin\theta cos\theta +I_z\phi\prime sin\theta(\phi\prime cos\theta + \psi\prime ) $$

where $$\phi\prime$$ represents precession,  $$\psi\prime$$ is represented by spin, $$\theta$$ is the nutation angle, and $$I$$ represents inertia along its respective axis. This relation is only valid with the Moment along the Y and Z axises are equal to 0.

The equation can be further reduced noting that the angular velocity along the z-axis is equal to the sum of the Precession and the Spin: $$\omega_z = \phi\prime cos \theta + \psi\prime$$, Where $$\omega_z $$ represents the angular velocity along the z axis.

$$\sum M_x = -I\psi\prime^2 sin \theta cos \theta + I_z \psi \prime(sin\theta)\omega_z $$

or

$$\sum M_x = \psi\prime sin \theta (I_z\omega_z-I\psi\prime cos \theta)$$

Gyroscopic Precession is torque induced. Described as the rate of change of the angular momentum and angular velocity that was produced by the same applied torque. This physical phenomenon results in the seemingly impossible dynamic occurrences. For example, a spinning top. This gyroscopic process is taken advantage of in many aerospace circumstances, such as airplanes and helicopters to help guide them into a desired orientation.

Bike Wheel Demonstration
A demonstration was done by Walter Lewin during one of his lectures at MIT involving a motor, a vertically fastened rope, and a bike wheel. This presentation is often recreated to display the properties of this gyroscopic behavior. The bike wheel is spun using the motor when the wheel is attached to the rope, the wheel maintains a perpendicular angle in respect to the rope with a continuous rotation in the direction of the wheel's spin. As the bike wheel begins to slow down, the bike wheel's perpendicular angle is lost, and its rotational velocity decreases until the wheel falls at rest, with the bike wheel's rod in line with the attached rope.

External Links: Lecture Lecture Notes

Helicopters
The main rotor of a helicopter acts like a gyroscope. Its motion is influenced by the principle of gyroscopic precession which is the concept that a force applied to a spinning object will have a maximum reaction approximately 90 degrees later. The reaction may differ from 90 degrees when other stronger forces are in play. To change direction, helicopters must adjust the pitch angle and the angle of attack.

Gyro Bowl
The Gyro Bowl is a commercial product for toddlers that utilizes gyroscopic principles to keep the center bowl (where food is) upright even while the child tosses and turns the device.

Gyro X
A prototype vehicle created by Alex Tremulis and Thomas Summers in 1967. The car utilizes gyroscopic precession to drive on two wheels. An assembly consisting of a flywheel mounted in a gimbal housing under the hood of the vehicle acted as a large gyroscope. The flywheel was rotated by hydraulic pumps creating a gyroscopic effect on the vehicle. A precessional ram was responsible for rotating the gyroscope to change the direction of the precessional force to counteract any forces causing the vehicle imbalance. The one-of-a-kind prototype is now at the Lane Motor Museum in Nashville, Tennessee.