5.2 Gyroscopes

5.2.1 Introduction and Basic Principles A gyroscope is defined as any rotating body that exhibits two fundamental properties: gyroscopic inertia and precession. These properties are inherent in all rotating bodies, including the earth itself. The principle of the spinning rotor gyroscope was first demonstrated in 1852 by the physicist Jean Foucault as he was studying the earth's rotation.1

• Gyroscopic Inertia The rigidity in space of a gyroscope is a consequence of Newton's first law of motion, which states that a body tends to continue in its state of rest or uniform motion unless subject to outside forces. An example of this tendency is a rifle bullet that, because it spins on itself in flight, exhibits gyroscopic inertia, tending to maintain a straighter line of flight than it would if not rotating. Rigidity in space can also be demonstrated by a model gyroscope consisting of a flywheel supported in rings (see Figure 5.17) in such a way that the axle of the flywheel can assume any angle in space.

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Figure 5.17 Free gyroscope with two gimbals

• Precession When a force, or a torque, applied to a gyroscope tends to change the direction of the axis of rotation of this gyroscope, the axis of rotation will not move to the “expected” position, but to a direction perpendicular to that one. A simple example of precession can be seen in a bicycle wheel hanged at one of its extremities like the one shown on the following video:

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5.2.2 Categories of Gyro

Figure 5.18 Current gyro technology applications

5.2.3 Mathematical Consideration From Newton's second law, the angular momentum H of a body will remain unchanged unless a torque T acts upon the body:

T = dH / dt

where H = IωR (scalar relation) I: moment of inertia of the rotor about spin axis ωR: angular velocity of the rotor

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Figure 5.19 Gyroscopic precession A constant torque T applied on one axis of the gyro causes a drift of angular velocity Ω called the precession rate:

T = H dθ/dt = H Ω

The rule to remember which direction a gyro wheel moves when you try to turn it (with a torque T) is that its angular momentum H tends to be aligned with the torque T.

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m g Figure 5.20 Gyroscopic precession due to the gravity

The same figure can be used to explain the precession of the Earth.2

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5.2.4 Single Degree of Freedom/Single-axis Gyroscopes Two types of single axis gyro are considered:

- the rate gyro, which is an open-loop system - the rate integrating gyro, a closed-loop

version of the rate gyro.

Figure 5.21 Single degree of freedom gyroscope

• The Rate Gyro (open-loop system) The wheel is mounted in a gimbal that is attached to the instrument case by one or two torsion bars that limit the rotation of the gimbal.

Axis Definition Spinning axis : Defined by the wheel. Output axis : Defined by the gimbal. Input axis : Perpendicular to the two other axis.

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The gimbal turns about the output axis in response to a rotation about the input axis.

Figure 5.22 Rate gyroscope schematic This effect is a consequence of the property of precession. When the gyro is turned about the input axis of an angle ∆θ =Ω ∆t, the direction of the angular momentum is changed of the same angle. This variation of angular momentum, call it ∆H, will create a torque on the gyroscope wheel in accordance to Newton’s law: T= ∆H/∆t This torque will act on the gimbal that will rotate about the output axis of an angle proportional to the input velocity Ω.

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Damping Rate gyroscopes are often liquid filled to provide damping for the correct dynamic response, and to protect the gimbal against shock and vibration. The damping is a function of the fluid viscosity, which varies strongly with temperature. Note: Viscosity is the property that describes the magnitude of the resistance to shear forces in the liquid. As temperature increases, viscosity decreases.

Rather than control the temperature of the fluid, which acts as an modifying input, many gyros use mechanical damping compensators. The principle of damping compensation is based on the method of opposing inputs to correct the effect of temperature. Note on mechanical damping compensation: By making the parts of materials with different coefficients of expansion the gap between the gimbal and the case of the gyro can be arranged to close down as the viscosity reduces with increasing temperature. The gap opens up as the instrument gets cold.

Pick-off / Pick-up The pick-off used in gyroscopes is an electromagnetic system that can determined the position of the gimbal with respect to the case of the gyro. The system consists of a toothed rotor on the gimbal and a stator attached to the case (see Figure 5.23).

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Torsion Bar For high sensitivity the torsion bar stiffness, Ktb, should be small. The weaker the torsion bar, the more it twists and the larger the output signal. But at the same time the input axis must stay as close as possible from the reference position to prevent the gyro to sense rotation about an axis normal to the desired input axis. This effect of sensitivity to axis other than the input axis is called cross-coupling error. Therefore, stiff torsion bars are usually preferred as they also increase the bandwidth (meaning that the gyro can measure higher frequency input signal), and protect the gyro against shock and vibration.

Figure 5.23 The Northrop GRG5 rate gyro

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Pick-off stator

and rotor

• The Rate-Integrating Gyro (closed-loop system) We can overcome the cross-axis sensitivity (or remove the cross-coupling error) by operating the rate gyro in closed-loop. Torsion bars are removed and replaced by bearings. The gimbal is brought back to its equilibrium position by a torquer.

Figure 5.24 Rate integrating gyroscope schematic The pick-off output now drives a servo amplifier that supplies the current to the torquer. The gyro output is not the pick-off angle but the amount of current supplied to the torquer. The Torquer

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The torquer produces a torque along the output axis that is proportional to the current coming from the servo. The Output Axis Bearing For the gyro to have a low threshold, the gimbal must respond to very small torques. A typical gyro wheel of about 25-mm diameter, spinning at 24,000 rpm, will have an angular momentum of about 105 g.cm2/s. Example: Let an input rate (Ω) be 1 deg/h, the same as 1 arc-sec/s. The gyroscopic torque is T=HΩ =0.5 dyn.cm =5.10-8 Nm. This is the torque generated by a weight of 0.5 mg at a radius of 1 cm. Half of a milligram is the weight of a piece of aluminum oven foil about 3 mm square. For the gimbal to respond to such a small torque it must be supported in very low friction output axis bearings.

Figure 5.25 The Northrop GIG6 rate-integrating gyro

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5.2.5 Two-Degree of Freedom Gyroscopes The dynamically tuned gyroscope is the most common two-degree of freedom gyro. It is a widely used modern compact gyroscope covering a wide performance spectrum from 0.01°/hour to 30°/hour rate uncertainty (error on the rate measurement), typical size is bout 40mm diameter x 40mm height. a

Figure 5.26 Dynamically tuned gyroscope

The wheel is coupled to the spin motor drive shaft by a flexible joint, which provides a two-axis gimbal system.

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Torquer : The details of the torquer system are shown on Figure 5.19. The magnets of the torquer are fixed on the gyro

heel and the coils are mounted on the gyro case. w

Figure 5.27 Gyroscope torquer schematic

The two electromagnetic pick-offs detect the displacement of the wheel from the horizontal plane.

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References:

http://science.howstuffworks.com/gyroscope.htm

1: Foucault pendulum The inventor of the gyroscope, Jean Bernard Leon Foucault, demonstrated during the 1851 World's Fair that a pendulum could track the rotation of the Earth. A scientific tour de force, Foucault's demonstration forever attached his name both to the effect itself (the Foucault effect) and to the universal joint pendulum that freely swings and rotates at the same time (the Foucault pendulum). A basic Foucault pendulum is simply a weight on a wire. Practically any pocket watch has the potential to act as a pendulum, exhibiting up to a 10 to 15 degree rotation per hour around its hinge point. To an observer in a windowless room, the rotation that accompanies the swing is a kind of optical illusion: the pendulum is not turning, instead the Earth is actually rotating under the pendulum. Foucault's dramatic proof at the World's Fair is considered to be the first non-astronomical proof of the Earth's rotation. With rotating hinges raised to heights in excess of 90 feet, Foucault Pendulums are now massive display pieces in the lobbies of more than 60 museums and entrance halls around the world, including the United Nations Building in New York and at the Smithsonian Museum in Washington.

2: Precession of the Earth Axis The moon's gravity, primarily, and to a lesser degree the sun's gravity, acting on Earth’s oblateness tries to move Earth’s axis perpendicular to the plane of Earth’s orbit. However, due to gyroscopic action, Earth’s poles do not “right themselves” to a position perpendicular to the orbital plane. Instead, they precess at 90 degrees to the force applied. This precession causes the axis of Earth to describe a circle having a 23.5 degree radius relative to a fixed point in space over about 26,000 years.

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