Governors: Watt, Porter, Proell and Hartnell

Why it matters

A flywheel smooths speed within a cycle, but when the load on an engine changes, the mean speed drifts unless the fuel supply changes too. A governor senses speed and moves the fuel control: centrifugal governors still regulate diesel fuel-injection pumps, generator sets and tractor engines. The same analysis teaches equilibrium of rotating systems and the trade-off between sensitivity and stability that appears in every control system.

Key ideas

Governor versus flywheel. The flywheel limits the cyclic fluctuation of speed within one cycle at a given load; the governor controls the mean speed over many cycles when the load changes, by changing the fuel or steam supply. A governor has no influence on cyclic fluctuation.

Centrifugal governors. Balls (flyweights) rotate with a spindle driven from the engine. As speed rises, the centrifugal effect m·ω²·r moves the balls outward, which lifts a sleeve connected to the fuel control and reduces the fuel supply. The force that pulls the balls back in is the controlling force: gravity on loads (dead-weight governors) or a spring (spring-loaded governors).

Watt governor. Two balls on arms pivoted on the spindle, no central load. The height h of the cone (from the plane of the balls to the point where the arms meet the axis) depends only on speed: h = g / ω². At practical engine speeds h becomes very small, so the Watt governor suits only slow engines.

Porter governor. A Watt governor with a heavy central load M on the sleeve. The load raises the equilibrium speed for a given height and greatly increases the governor's effort and power, so it can work the fuel control against friction. With friction F at the sleeve the governor has a range of speeds (insensitiveness) at each position.

Proell governor. Like the Porter governor, but each ball is fixed to an upward extension of the lower arm, not at the joint of the upper and lower arms. For the same dimensions it runs at a lower speed than a Porter governor, so it needs smaller balls; it is also more sensitive.

Hartnell governor. Balls on the vertical arms of bell-crank levers pivoted on a frame that turns with the spindle; the horizontal arms bear on the sleeve, which is pressed down by a helical spring. Spring control gives large controlling force in a compact unit, allows high speeds and works in any orientation. Initial spring compression sets the speed range; spring stiffness sets the sensitivity.

Performance terms.

• Sensitiveness: (N₂ − N₁) / N_mean over the full sleeve travel. A small value means a sensitive governor.
• Stability: for each speed there is exactly one ball radius; the radius increases with speed.
• Isochronism: the equilibrium speed is the same at all radii (zero range). An isochronous governor is infinitely sensitive but in practice hunts. A Porter governor cannot be isochronous; a Hartnell governor can, with the right spring.
• Hunting: the governor over-corrects and the speed oscillates continuously above and below the mean. It happens when the governor is too sensitive.
• Effort: the mean force at the sleeve for a given fractional change of speed. Power: effort × sleeve lift.
• Controlling force curve: controlling force F against radius r. For stability F = a·r − b (the line, when extended, cuts the force axis below the origin); isochronous if it passes through the origin; unstable if it cuts above.

Formulas

h = g / ω² ≈ 895 / N²

• h: governor height (m); ω: spindle speed (rad/s); N: speed (rpm); g = 9.81 m/s². Watt governor; ball mass does not appear.

ω² = (m + M) · g / (m · h) (Porter, equal arms and links meeting on the axis, no friction)

• m: mass of each ball (kg); M: central (sleeve) load (kg).

N² = [m + (M/2)·(1 + q)] / m · (895 / h), q = tan β / tan α (Porter, general)

• α, β: angles of the upper arm and the lower link with the vertical. With friction at the sleeve, replace M·g by (M·g ± F): + while rising, − while falling.

N² = (FM / BM) · [m + (M/2)·(1 + q)] / m · (895 / h) (Proell)

• FM, BM: dimensions of the extended lower arm (m), from the governor's geometry; FM/BM < 1 for the usual layout.

m·ω²·r·x = (1/2)·(M·g + S)·y (Hartnell, ball weight and arm obliquity neglected)

• r: ball radius (m); x: length of the ball arm (m); y: length of the sleeve arm (m); S: spring force (N).

h_sleeve = (r₂ − r₁)·(y / x) and s = (S₂ − S₁) / h_sleeve

• h_sleeve: sleeve lift (m); s: spring stiffness (N/m).

sensitiveness = (N₂ − N₁) / N_mean

Worked examples

Example 1 (standard: Porter governor). All arms and links are 250 mm long and pivoted on the axis. Each ball is 5 kg, the central load is 25 kg. Find the speeds at ball radii of 150 mm and 200 mm, neglecting friction.

1. At r = 150 mm: h = √(0.25² − 0.15²) = √(0.0625 − 0.0225) = 0.200 m.
2. ω² = (m + M)·g / (m·h) = (30 × 9.81) / (5 × 0.200) = 294.3 rad²/s², so ω = 17.16 rad/s, N = 163.8 rpm.
3. At r = 200 mm: h = √(0.0625 − 0.04) = 0.150 m.
4. ω² = (30 × 9.81) / (5 × 0.150) = 392.4 rad²/s², ω = 19.81 rad/s, N = 189.2 rpm.
5. Range of speed = 25.4 rpm. A Watt governor (M = 0) of the same geometry would run at only 66.9 rpm and 77.2 rpm: the central load multiplies the speed by √6 here.

Example 2 (GATE level: Hartnell governor spring). A Hartnell governor has balls of 2 kg each, ball arms x = 100 mm and sleeve arms y = 60 mm. The ball radius is 100 mm at 300 rpm (minimum) and 130 mm at 320 rpm (maximum). Neglecting the ball weight, sleeve mass and arm obliquity, find the spring forces, the sleeve lift and the spring stiffness.

1. ω₁ = 2π × 300 / 60 = 31.42 rad/s; ω₂ = 2π × 320 / 60 = 33.51 rad/s.
2. Centrifugal forces: F₁ = 2 × 31.42² × 0.100 = 197.4 N; F₂ = 2 × 33.51² × 0.130 = 292.0 N.
3. Moments about the bell-crank pivot (M = 0): S = 2·F·x / y. S₁ = 2 × 197.4 × 0.1 / 0.06 = 658 N; S₂ = 2 × 292.0 × 0.1 / 0.06 = 973 N.
4. Sleeve lift: h = (0.130 − 0.100) × 60 / 100 = 0.018 m = 18 mm.
5. Stiffness: s = (973 − 658) / 0.018 = 17 500 N/m = 17.5 N/mm.
6. Initial compression of the spring at minimum speed = 658 / 17 500 = 37.6 mm.

Common mistakes

• Taking the governor height as arm length minus radius. Use h = √(l² − r²) for arms pivoted on the axis.
• Thinking ball mass changes the Watt governor's speed. It cancels; only h matters.
• Using rpm in m·ω²·r.
• In a Porter governor with friction, adding friction on both the rising and falling sides with the same sign.
• In a Hartnell governor, writing the moment balance with the sleeve force acting on one lever but forgetting that each lever carries half the spring force.
• Confusing hunting (too sensitive) with insensitiveness (friction).

For GATE ME

Questions ask for the equilibrium speed of a Watt or Porter governor from its geometry, the speed range or sensitiveness between two radii, the spring stiffness and initial compression of a Hartnell governor, and concepts such as isochronism, hunting, stability and the controlling force curve. Practise the Porter formula with friction and the Hartnell moment balance until you can set them up without a figure.

Quick check

1. Height of a Watt governor at 60 rpm?
2. Does doubling the ball mass change a Watt governor's speed?
3. What is an isochronous governor, and why does it hunt?
4. In a Porter governor, what does the central load do to the speed for a given height?
5. Controlling force line F = 1000r + 50 (N, r in m). Stable?

Answers: 1. 895 / 3600 = 0.249 m. 2. No. 3. One whose equilibrium speed is the same at every radius; any small speed change sends the sleeve to an extreme, so it over-corrects. 4. It increases it by √((m + M)/m) for equal arms and links. 5. No, the line cuts the force axis above the origin, so it is unstable.