Flywheels, Governors, Balancing and Gyroscope
Turning moment diagrams and flywheel size, coefficient of fluctuation, governors (Watt, Porter, Hartnell) and their sensitivity, balancing of rotating and reciprocating masses, and gyroscopic couple.
These topics deal with controlling speed and forces in running machines: a flywheel smooths speed within each cycle, a governor controls mean speed as load changes, balancing removes shaking forces, and the gyroscopic couple acts on turning rotors.
Flywheels
The engine torque varies over a cycle (the turning moment diagram), while the load torque is roughly constant. The flywheel stores the extra energy and gives it back.
- Maximum fluctuation of energy ΔE = E_max − E_min (from the areas of the turning moment diagram).
- Coefficient of fluctuation of speed Cs = (ω_max − ω_min)/ω_mean.
- ΔE = I ω² Cs (ω = mean speed). So the required I = ΔE/(ω²Cs).
- For a rim-type flywheel, I ≈ mR².
Governors
A governor changes the fuel supply as speed changes.
- Watt governor: height h = g/ω² (h in m, ω in rad/s), about 895/N² m with N in rpm.
- Porter governor: a Watt governor with a central load; more sensitive at higher speeds.
- Hartnell governor: a spring-controlled centrifugal governor.
- Sensitiveness = (N₂ − N₁)/N_mean. Isochronous: equilibrium speed is the same at all radii (infinitely sensitive, but it hunts). Stable: radius increases with speed. Hunting: the governor keeps overshooting.
Balancing
- Rotating masses in one plane: the vector sum Σmr must be zero. A single balancing mass m_b r_b = −Σmr.
- In several planes: both Σmr = 0 (forces) and Σmrl = 0 (couples). Two balancing masses in two chosen planes are always enough.
- Reciprocating masses (single cylinder): primary force mω²r cos θ, secondary force mω²r cos 2θ/n. Usually only a fraction (2/3 to 3/4) of the primary force is balanced by a rotating mass, because a rotating mass adds an unbalanced vertical force.
Gyroscopic couple
A rotor with moment of inertia I spinning at ω, whose axis turns (precesses) at ω_p, needs a couple
C = I ω ω_p.
This acts on ships turning or pitching, aircraft turning, and vehicles going round curves.
Worked example
A flywheel must keep speed within ±1% of 300 rpm when ΔE = 6 kJ. Find I.
- ω = 2π × 300/60 = 31.42 rad/s. Cs = 0.02.
- I = 6000/(31.42² × 0.02) = 304 kg·m².
Practice
What is the height (in mm) of a Watt governor running at 60 rpm? Use h = g/ω² with g = 9.81 m/s². (Round to the nearest mm.)
Show answer and explanation
Answer: 248ω = 2π rad/s, h = 9.81/(4π²) = 0.2485 m ≈ 248 mm.
A flywheel has I = 100 kg·m² and mean speed 20 rad/s with Cs = 0.05. What is the maximum fluctuation of energy in J?
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Answer: 2000ΔE = Iω²Cs = 100 × 400 × 0.05 = 2000 J.
A rotor with I = 0.5 kg·m² spins at 200 rad/s while its axis precesses at 0.2 rad/s. What is the gyroscopic couple in N·m?
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Answer: 20C = Iωω_p = 0.5 × 200 × 0.2 = 20 N·m.
A governor whose equilibrium speed is the same at every radius of rotation is called:
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Answer: BAn isochronous governor has one equilibrium speed for all radii; in practice it tends to hunt.