Gyroscopic couple and its applications

Why it matters

Any fast-spinning rotor whose axis is turned, such as an aircraft engine during a turn, a ship's turbine when the ship pitches, the wheels of a motorcycle on a curve, or a drone's propellers during a manoeuvre, exerts a couple on its bearings and on the vehicle. The same physics drives gyrocompasses, ship stabilisers and the control-moment gyroscopes that point satellites. You need the magnitude to size bearings and the direction to know whether the nose lifts or the vehicle tends to overturn.

Key ideas

Angular momentum and precession. A rotor of polar moment of inertia I spinning at ω has angular momentum H = I·ω, a vector along the spin axis (right-hand rule: curl the fingers in the sense of rotation, the thumb gives the vector). Turning the spin axis itself at angular velocity Ω about a perpendicular axis is called precession. The angular momentum vector then changes direction at the rate dH/dt = Ω × H, magnitude I·ω·Ω, and a couple of this size must act on the rotor.

Active and reactive couples.

• Active gyroscopic couple: the couple that must be applied to the rotor (through its bearings) to make it precess. Vector C = Ω × H, i.e. along the axis perpendicular to both the spin and the precession axes.
• Reactive gyroscopic couple: the equal and opposite couple that the rotor exerts on its bearings and hence on the vehicle. This is the one that tilts the aircraft or ship. Spin axis, precession axis and couple axis are mutually perpendicular. If the spin axis makes an angle θ with the precession axis, the couple is I·ω·Ω·sin θ; when they are parallel there is no gyroscopic couple.

Direction rule in practice. Draw the spin vector, rotate it by the precession, and the change of the vector points along the active couple; reverse it for the reactive couple. Then read the reactive couple's effect with the right-hand rule. For a rotor turning clockwise when viewed from the rear (stern) of the vehicle:

• Aeroplane turning left, or ship steering to port: reactive couple raises the nose (bow) and dips the tail (stern). Turning right gives the opposite.
• Ship pitching, bow rising: reactive couple turns the ship towards starboard (right); bow falling turns it to port.
• Ship rolling: the spin axis is parallel to the roll axis, so there is no gyroscopic effect. Reversing the rotor's sense reverses every effect.

Vehicles on curves. Wheels spinning at ω_w = v / r_w precess at Ω = v / R when the vehicle follows a curve of radius R. In a car this gyroscopic couple (wheels and engine) adds to the centrifugal overturning couple, unloading the inner wheels. A motorcycle rider leans inward so that the weight couple balances both the centrifugal and the gyroscopic couples.

Applications. Gyrocompasses and rate gyroscopes for navigation; ship stabilisers; control-moment gyroscopes for spacecraft attitude; precession loads on helicopter and wind-turbine rotor bearings; drone propellers creating unwanted coupling between pitch and yaw.

Formulas

H = I·ω (kg·m²/s)

C = I·ω·Ω (gyroscopic couple, N·m; spin and precession axes perpendicular)

• I = polar moment of inertia of the rotor about its spin axis (kg·m²); I = m·k²
• ω = spin velocity (rad/s); Ω = precession velocity (rad/s)

C = I·ω·Ω·sin θ (θ = angle between spin and precession axes)

Ω = v / R (vehicle on a curve of radius R at speed v)

Ω_max = φ₀·(2π / T_p) (ship pitching in SHM with amplitude φ₀ in rad and period T_p in s)

Two-wheeler of mass m, CG height h, leaning at θ, wheel radius r_w, engine gear ratio G = ω_engine / ω_wheel:

• gyroscopic couple C_g = (2·I_w + G·I_e)·(v² / (R·r_w))·cos θ
• centrifugal couple C_c = (m·v² / R)·h·cos θ
• balance m·g·h·sin θ = C_g + C_c, so tan θ = [(2·I_w + G·I_e)·v²/(R·r_w) + m·v²·h/R] / (m·g·h)

Worked examples

Example 1 (standard): aeroplane in a turn. Given: engine and propeller rotating parts 400 kg, radius of gyration 0.3 m, 2400 rpm clockwise viewed from the rear. The aircraft turns left on a 50 m radius at 200 km/h.

1. I = m·k² = 400 × 0.3² = 36 kg·m²
2. ω = 2π × 2400 / 60 = 251.3 rad/s
3. v = 200 / 3.6 = 55.56 m/s → Ω = v / R = 55.56 / 50 = 1.111 rad/s
4. C = I·ω·Ω = 36 × 251.3 × 1.111 = 10 053 N·m
5. Direction: clockwise from the rear and a left turn → the reactive couple raises the nose and dips the tail. Answer: C ≈ 10.05 kN·m, nose up and tail down.

Example 2 (GATE level): leaning a motorcycle. Given: bike and rider 250 kg, CG 0.6 m above the road, each wheel I_w = 1.2 kg·m², wheel radius 0.3 m, engine rotating parts I_e = 0.2 kg·m² turning in the same sense as the wheels at G = 5 times wheel speed. Speed 72 km/h (20 m/s) on a 50 m radius curve. Find the lean angle.

1. Gyroscopic term (upright): (2 × 1.2 + 5 × 0.2) × 20² / (50 × 0.3) = 3.4 × 26.67 = 90.7 N·m
2. Centrifugal term (upright): (250 × 20² / 50) × 0.6 = 2000 × 0.6 = 1200 N·m
3. Balancing term coefficient: m·g·h = 250 × 9.81 × 0.6 = 1471.5 N·m
4. tan θ = (90.7 + 1200) / 1471.5 = 0.877 → θ = 41.3° Answer: lean ≈ 41° from the vertical; the gyroscopic part is small (about 7%) here but grows with lighter bikes and larger wheels.

Quick extra (ship pitching). A ship's turbine rotor (8 t, k = 0.6 m, 1800 rpm) while the ship pitches with amplitude 6° and period 20 s: Ω_max = (6π/180)(2π/20) = 0.0329 rad/s, C_max = 2880 × 188.5 × 0.0329 = 17.9 kN·m.

Common mistakes

• Using rpm or km/h directly; convert to rad/s and m/s.
• Using mass × radius² with the vehicle's turning radius; I belongs to the rotor about its own spin axis.
• Mixing up active and reactive couples, so getting the effect backwards.
• Expecting a gyroscopic effect when a ship rolls; the axes are parallel.
• Using the pitching amplitude in degrees in Ω_max = φ₀·2π/T_p.
• Forgetting the engine term (with its gear ratio and sense) in vehicle problems.

For GATE ME

Expect a numerical on the gyroscopic couple of an aircraft, ship or vehicle, usually paired with a direction choice (nose up or down, bow to port or starboard). Practise the vector rule until you can sketch spin, precession and couple vectors in seconds, and remember that ship rolling gives no effect.

Quick check

1. I = 5 kg·m², ω = 200 rad/s, Ω = 0.5 rad/s. Couple?
2. Which two axes must the couple axis be perpendicular to?
3. Ship rotor clockwise from stern, ship steers to starboard. Effect?
4. A car takes a curve: do the wheels' gyroscopic couple and the centrifugal couple add or oppose (wheels spinning forward)?
5. Why does a rolling ship feel no gyroscopic couple from its turbine?

Answers: 1. 500 N·m. 2. The spin axis and the precession axis. 3. Bow dips and stern rises. 4. They add, both tending to overturn the car outward. 5. The roll axis is parallel to the spin axis, so H does not change direction.