Rotating flyballs · A rising sleeve

Centrifugal Governor

Why does the sleeve rise as the spindle speeds up? Between the travel stops, faster rotation opens the flyball arms farther. The equal upper and lower links pull the sliding sleeve upward as they open. Once the sleeve reaches a stop, its position stays fixed and the contact force changes with speed instead.

Drag a blue flyball or change the spindle speed. Watch the sleeve rise between two physical stops, then compare its low-speed, freely raised and travel-limited positions.

Arm spread $0$

Sleeve rise $0$

Inward force per ball $0$

One turn $0$

Upper-link tension $0$

Lower-link force $0$

Upward stop reaction $0$

Azimuth rate $0$

Grab a blue ball

Turn faster. Raise the sleeve.

Spindle speed

N

43\,\mathrm{rpm}

Link length

L

1\,\mathrm m

Gravity

g

9.81\,\frac{m}{s^2}

Mass of each bob

m

0.5\,\mathrm{kg}

Pivot offset / link length

a/L

0.08

Positions use metres, forces newtons and time seconds. Right, up and toward you are positive. Azimuth increases from the right toward you; its rate is positive in the calculations and CSV.

The upper and lower arms have the same pin-to-pin length and equal radial pivot offsets. The gold sleeve slides on the spindle. Its rise is measured from its lower stop. The stops hold the arms between fifteen and seventy-five degrees from vertical.

The motor prescribes the spindle speed. The page solves for a steady arm angle at each setting, including the force from a stop when the free position would lie beyond its travel. It does not animate the acceleration or settling between settings.

The blue line is a short motion trail. The separate curve shows the computed steady-orbit relation over the control range, with the current setting marked. CSV contains one complete calculated orbit; no measured acquisition is implied.

Upper-link force is positive in tension. Lower-link force is positive when it pulls toward the sleeve and negative in compression. The stop reaction is positive upward. Links and sleeve have no modeled mass; the added mass control refers to each flyball. No spring or external sleeve load is included.

This is the flyball-and-sleeve part of a governor. It does not close a feedback loop to an engine or predict hunting, impacts, power, wear or safe operating speed. Bob radii, joint thickness, clearances and the displayed trail width are schematic; pin-centre geometry uses the model lengths.

Two equal point masses rotate at a prescribed speed on equal, massless upper and lower links. Both pivot pairs have the same finite radial offset. The sleeve is massless and frictionless, with ideal lower and upper stops. Each setting shows a steady position. Engine feedback, hunting, friction, impact and structural limits are omitted.

Start with the essentials

Focus question: Does increasing spindle speed always raise the sleeve farther?
One-sentence intuition: Between the stops, greater speed opens the arms and raises the sleeve. At a stop, the contact force changes instead of the sleeve position.

Core mathematical model

Finite pivot offsets and sleeve travel

r=a+L\sin\theta,\quad y_s=H-2L\cos\theta

Both pivot pairs lie at radial distance a from the spindle. The upper and lower links each have length L.

The free steady position

\omega^2(a+L\sin\theta)=g\tan\theta

This relation balances the two force components when the sleeve is not touching a stop. Setting the offset to zero gives the ideal on-axis conical-pendulum limit.

The force from a stop

F_s=m\left[g-\omega^2r\cot\theta\right]

The upward reaction is positive. It vanishes in the free range; at the lower and upper stops it points up and down respectively. Mass refers to one of the two equal balls.

Common difficulties

A speed sensor is not a closed control loop

Typical misconceptionThe moving sleeve on this page proves that an engine will regulate its own speed stably.

Better mental modelThis page prescribes speed and calculates a steady position. Engine torque, load, valve feedback and transient dynamics are needed to study regulation and hunting.

New settings prepare a new steady motion

Typical misconceptionThe instantaneous change of angle is a prediction of how the real apparatus accelerates.

Better mental modelThe model compares exact steady configurations. It does not integrate the forces during preparation, so no settling time or transient power is reported.

Run the experiment

01
Grab the ball
Pause, then drag a blue ball outward or upward in the main scene.
What to observe: The linked sleeve rises as the arms open. The spindle speed updates to the corresponding steady setting.
02
Try both travel limits
Select Lower stop, Sleeve raised and Upper stop. Open More measurements.
What to observe: The sleeve moves only between its stops. Outside that interval the stop carries a nonzero vertical reaction.
03
Change gravity and mass
Try Moon gravity, then change Mass in the collapsed group.
What to observe: Gravity changes the speed needed to hold a given angle. At fixed speed and geometry, equal changes of ball mass scale the forces without changing the steady position.

Scientific sources

Richard Fitzpatrick · The conical pendulum · J. C. Maxwell · On Governors (1868) · TecQuipment · Flyballs and sleeve