Gyroscopes: Fundamentals, Dynamics, and Aircraft Applications, and Erection Systems

Fundamentals of Motion and Inertia

• Newton’s 1st Law of Motion (Inertia):
  • An object in motion will remain in motion and an object at rest will remain at rest unless acted on by an unbalanced force.
  • In a frictionless environment (e.g., space), a pushed object will continue at the same speed indefinitely.
  • On Earth, atmospheric friction and gravity provide the additional forces that oppose initial motion.
  • The core concept is that a moving mass continues in the same direction unless an external force intervenes.

Gyroscope Elements and Construction

• Rotor: A perfectly balanced mass mounted on a central shaft.
• Rotor Construction:
  • The unit must be perfectly circular and symmetrical about the spin axis to prevent imbalance during rotation.
  • Weight Distribution: To maximize momentum and stability, weight is concentrated on the rim of the rotor.
  • Rim Concentration Principle: Inertia depends on the square of the radius, so rotors are made as large as possible with mass at the outer edge.
  • Compromise: Increasing weight improves momentum but causes excessive bearing friction and drift. A balance must be maintained between momentum and friction.
• Gyroscopic Balance:
  • Rotors rotate at high speeds and must be both statically and dynamically balanced to minimize vibration.
  • Static Balance: The center of gravity must act exactly upon the spin axis.
  • Dynamic Balance: The plane of spin must act at right angles ($90^\circ$) to the axis of spin.
• Rigidity Factors: The heavier the rotor and the closer the weight is to the rim, the higher the gyro's rigidity.
• Inner Gimbal: Formed by adding a frame with bearings to the rotor shaft, creating the first axis of spin.

Fundamental Properties of Gyroscopes

• Gyroscopic Rigidity (Gyroscopic Inertia):
  • The ability of the mass inertia to keep the axis rigid or pointing in a constant direction in space.
  • It resists any force tending to change the plane of rotation.
• Gyroscopic Precession:
  • The effect of angular displacement when a force is applied to the gyro.
• Principle of Conservation of Angular Momentum: Both properties depend on this principle, stating angular momentum remains constant unless a force is applied.
• Mathematical Representation of Angular Momentum:
  • Angular momentum is the product of the moment of inertia ($I$) and angular velocity ($\omega$).
  • Formula: $\text{Angular Momentum} = I \times \omega$.
  • Values are referred to the center of gravity of the gyro.

Natural and Mechanical Examples

• The Earth: Acts as a high-speed rotating mass possessing rigidity and precession, despite lacking a mechanical gimbal system.
• Bicycle: Spinning the front wheel off the ground demonstrably shows rigidity (resisting turning) and precession (twisting the handlebars).
• Motor-car Engine Flywheel: The spin axis aligns with the motion of the car. When turning, the flywheel’s rigidity causes precession, tending to move the front of the car up or down.
• Aerospace Applications: Aircraft propellers, compressor assemblies, and turbine assemblies of jet engines all exhibit gyroscopic properties.

Gimbals and Degrees of Freedom

• Universal Mounting: A gyroscope is defined as a heavy wheel universally mounted to have three degrees of freedom:
  • Spinning Freedom: Rotation about the centre axis of spin ($XX_1$).
  • Tilting Freedom: Rotation about a horizontal axis at right angles to the spin axis ($YY_1$).
  • Veering Freedom: Rotation about a vertical axis perpendicular to both spin and tilt axes ($ZZ_1$).
• Gimbal System Structures:
  • Inner Gimbal: Provides tilting freedom.
  • Outer Gimbal: Provides veering freedom and is supported in the frame or case.
  • Configuration: The rotor is mounted in two concentrically pivoted rings. All axes are mutually at right angles and intersect at the rotor's center of gravity.
• Modern Terminology:
  • A gyro must spin to show properties but cannot measure along the spin axis.
  • Two-frame gyroscope: Possesses one degree of freedom for measurement.
  • Three-frame gyroscope: Possesses two degrees of freedom for measurement.

Increasing Gyroscopic Rigidity

• Rigidity is determined by high angular velocity and the kinetic energy of the rotor. It can be increased by:
  • Increasing the mass of the rotor.
  • Increasing the rotor speed.
  • Concentrating more mass near the rim (increasing the ‘radius of gyration’).

Detailed Dynamics of Precession

• Definition: Precession is the angular change in the direction of the plane of rotation under an applied force.
• 90-Degree Rule: The change occurs not in line with the force, but at a point $90^\circ$ away in the direction of rotation.
• Factors Influencing Precession Rate:
  • Strength and direction of the applied force.
  • Moment of inertia of the rotor (weight).
  • Angular velocity of the rotor (speed).
• Rate Relationships:
  • Greater force = Greater rate of precession.
  • Greater rigidity (Inertia/Speed) = Smaller rate of precession for a given force.
• Cessation of Precession: Precession continues until the plane of rotation aligns with the plane of the applied force and the directions are coincident.
• Input and Output Axes:
  • Input Axis: The axis about which torque (force) is applied.
  • Output Axis: The axis about which precession takes place.

Toppling and Gimbal Lock

• Toppling (Tumbling/Spilling):
  • Occurs when deflective forces are too strong or applied too rapidly, causing the rotor to lose its orientation.
  • Dangers: Can jump off work benches if not bolted down; causes severe damage to gimbal/rotor bearings and aircraft mounting bases.
• Unavoidable Precession (Drifting): Small errors caused by aircraft maneuvering and internal gimbal friction.
• Caging Devices: Used in older gyros to lock gimbals in place during aerobatics to prevent damage. Caging knobs can erect or reset the gyro.
• Modern Gyro Limits: Modern gyros often do not tumble but stop reflecting accurate pitch beyond $85^\circ$ nose up/down. They feature self-erecting mechanisms.
• Gimbal Lock:
  • Occurs when the inner and outer gimbals become aligned.
  • Prevention: Mechanical stops limit the inner gimbal movement, usually set around $85^\circ$. Reaching these stops causes random precession and toppling.

Gyroscope Type Definitions

• Free (Space) Gyro: Has complete freedom in three planes at right angles. It has two gimbals (inner tilt and outer veer). Unreferenced to Earth or gravity.
• Tied Gyro: Has freedom in three planes but is controlled by an external source to point in a specific direction.
• Earth Gyro: A tied gyro controlled by gravity to maintain its position relative to Earth; the spin axis points to the center of the Earth.
• Rate Gyro: Has only one plane of freedom at right angles to rotation; utilizes one gimbal and springs to measure the rate of movement.

Apparent Drift and Earth Rate

• Apparent Drift: To an observer on Earth, the rotor appears to move over time. In reality, the rotor stays rigid in space while the Earth rotates underneath it.
• Earth Rate ($\omega_e$): The Earth rotates at $15^\circ$ per hour.
• Revolution Time: One full revolution occurs in $23 \text{ Hr } 56.4 \text{ Min}$.
• Latitudinal components: Apparent drift depends on the latitude and the orientation of spin/input axes.
• Calculation Formula: $\text{Apparent Drift} = 15 \times \sin(\lambda)$, where $\lambda$ is the angle of latitude.
• Compensation:
  • Achieved by precessing the gyro in the opposite direction of Earth's rotation.
  • Methods: Electrical torquing signals, unbalancing gimbals, or placing weights on the gyro rotor's spin axis.

Transport Wander and Real Drift

• Transport Wander: The tilting of the spin axis that appears to occur when the gyro is moved (transported) from one point on Earth to another.
  • Example: Transporting a gyro from the North Pole to the equator results in a $90^\circ$ apparent tilt.
  • Correction: Referencing the gyro to the center of Earth/gravity.
• Real Drift: Caused by manufacturing physical imperfections such as bearing friction and gimbal imbalances. Only minimized through precision engineering.

Rebalancing and Polar Regions

• Earth rate error is corrected by adjusting a balancing nut on the rotor housing to apply constant torque.
• Settings are effective up to $60^\circ$ of latitude.
• Polar Operations: Above $60^\circ$, a gimbal rebalance (readjusting the nut) is required due to increased error rates.

Gimbal Errors during Maneuvers

• Definition: Error induced when a directional gyro (DG) is displaced in pitch and roll simultaneously.
• Cause: Loss of the $90^\circ$ relationship between gimbals. The outer gimbal moves to maintain the rotor axis direction, which is detected by the heading synchro.
• Variables: Depends on angle of climb/descent/bank and the angle between rotor axis and the aircraft's longitudinal axis.
• Cardinal vs. Intercardinal Headings:
  • Cardinal (North, South, East, West): Simple bank or pitch does not usually produce gimbal error.
  • Intercardinal Headings: Errors occur during turns or combined pitch/roll maneuvers.
• Recovery: Errors are generally eliminated once the aircraft returns to straight and level flight.

Handling Precautions for Gyroscopes

• Delicacy: Handle like ‘rotten eggs’; minor mishandling destroys bearings.
• 30-Minute Rule: Do not move an aircraft or remove a vacuum gyro for 30 minutes after power-down to allow the rotor to stop spinning (inertia keeps it spinning).
• Electrical Gyros: Same 30-minute rule applies unless maintenance docs specify dynamic braking (which stops the rotor in <1 min).
• Storage: Never store a gyro on anything other than $100\text{ mm}$ of foam rubber.
• Testing Serviceability: Measure "run-down" time; longer times indicate healthier bearings. Noisy gyros indicate bearing deterioration.
• Contamination: Dirt and dust are critical issues; filters must be checked/changed regularly.

Ring Laser Gyros (RLG) and the Sagnac Effect

• Strapdown System: Unlike conventional Inertial Navigation Units (INU), RLGs are ‘strapped down’ and do not require a stabilized platform.
• Technology: LASER (Light Amplification by Stimulated Emission of Radiation).
• Operation: Uses a mix of Helium and Neon gas at high voltage ($3000\text{ V}$) to produce monochromatic radiation.
• The Sagnac Effect: Two laser beams (clockwise and counter-clockwise) travel around an enclosed loop (triangle). If the loop rotates, one beam travels a shorter distance, and the other a longer distance.
• Frequency Shift: Motion causes a compressed wavelength (higher frequency) in one beam and an expanded wavelength (lower frequency) in the other.

Fibre Optic Gyros (FOG)

• Principle: Uses the Sagnac effect via windings of fibre optic lines.
• Advantages over RLG:
  • Smaller and more compact.
  • Longer light paths (100m to over $5\text{ km}$) lead to higher accuracy.
• Reliability: MTBF is measured in excess of 50,000 hours (over 6 years of continuous operation).
• Accuracy: Approximately $15\text{ metres}$ or less worldwide.
• Example (Airbus A380): Northrop Grumman unit combining GPS, inertial reference, and air data. Weight < $8\text{ kg}$; power consumption < $36\text{ W}$.

Gyroscope Erection Systems

• Vacuum/Air Driven Systems:
  • Wedge Plate: Deflects air across a wedge. Unbalanced deflection (tilting) creates differential flow to precess the gyro to vertical.
  • Pendulous Vane: Four ports at $90^\circ$. Gravity adjusts vanes during tilts to direct air differentially and precess the rotor back to balance.
• Mechanical Erection Systems:
  • Ball Cage: 5 to 8 steel balls on a radiused disc. Captured by hooks when tilted to apply precession force.
  • Rolling Ball: Slotted disc rotating at $\text{approx. } 30 \text{ RPM}$. Tilt causes the ball to provide torque based on its travel in the slot.
• Electrical Erection Systems:
  • Torque Motors: Squirrel-cage motors with reference and control windings.
  • Mercury Switches (Levelling): Glass tubes with 3 electrodes and mercury in inert gas. Tilt causes mercury to complete a circuit to the torque motor control winding.

Fast Erection Systems

• Required if the gyro is toppled or out of vertical by >$10^\circ$.
• Electrical Switch Method: Supplying torque motors with full line voltage ($115\text{ V}$) instead of normal ($20\text{ V}$). Increases rate from $5^\circ/\text{min}$ to $120^\circ$–$180^\circ/\text{min}$.
• Precaution: Limit use to $15 \text{ seconds}$ to avoid coil overheating.
• Electromagnetic Method (DC coils): Magnet pulls an umbrella-shaped armature into line. Operates for $\text{approx. } 20 \text{ seconds}$ during start-up.

Erection System Errors and Compensation

• Acceleration/Deceleration Error:
  • Acceleration induces a false ascent indication.
  • Deceleration induces a false descent indication.
• Turning Errors: Centrifugal force acts on pendulous vanes/mercury switches, creating a false vertical.
• Compensation Methods:
  • Inclination of rotor axis: Fixed tilt of $1.6^\circ$ or $2.5^\circ$ set during calibration.
  • Erection Cut-out: Disconnects the erection system when bank exceeds a threshold (e.g., $10^\circ$ or $0.18g$) using commutators or liquid level switches.

Power Supplies for Gyros

• Vacuum Systems:
  • Venturi Tube: Tapered tube on the fuselage using high-velocity air in flight to create suction ($2\text{"}$ or $4\text{"}$ of mercury). Inefficient; susceptible to icing.
  • Wet Vacuum Pumps: Steel vane pumps lubricated by engine oil. Oil is later separated or discharged overboard.
  • Dry Vacuum Pumps: Modern standard using Teflon or carbon components. Features a weak-link shear drive to protect the engine if the pump fails.
• Electrical Systems:
  • Direct Current (DC): Rotor is the armature, stator is a permanent magnet. Speed approx. $4,200 \text{ RPM}$. Rugged and low cost but has brush wear.
  • Alternating Current (AC): Uses 3-phase $115\text{V } 400\text{Hz}$ power. Squirrel-cage induction motor redesigned so the rotor is outside the stator (increasing mass/inertia). Speed approx. $22,500 \text{ RPM}$. High accuracy but higher cost.