Water-wheel Governor: Belt Speed and Illustrative Gate Command

Posted by Robbie Dickson on

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Historic water-wheel governors sensed shaft speed through belts/gearing and moved the water gate in the corrective direction. This calculator evaluates an entered belt ratio/slip and a declared proportional gate-command rule only.

Water Wheel Governor Interactive Calculator

Enter wheel speed, pulley ratio, assumed belt slip and target spindle speed. The gate value is a declared command: 50% minus 1.5 percentage points per percent speed error, clipped to 0–100%; it is not a governor simulation.

0°

Calculated belt-driven speed
--
Signed target-relative error
--
Illustrative proportional command
--
Slip-associated speed difference
--

Equation Used

ideal spindle=Nw R; spindle=Nw R(1-s/100); speed error=100(N-Nt)/Nt; illustrative gate command=clip(50-1.5 error,0,100); belt loss=Nw R-N.
Pulley ratio and slip are entered constants. The 50% bias and gain 1.5 are arbitrary scenario settings embedded in this tool. There is no time response, inertia, gate travel rate, water dynamics, stability or integral action.
  • Wheel RPM, ratio, slip and target are entered independently.
  • Slip is a constant fractional speed reduction.
  • The gate command uses a fixed 50% bias and proportional gain 1.5.
  • The command is clipped to 0–100% but no actuator or plant dynamics are modeled.
  • Torque, belt capacity, flyball geometry, backlash, flow, transient stability and load disturbances are omitted.
FIRGELLI Automations - Interactive Mechanism Calculators
Same mechanism and inputs as the interactive calculator.

Belt-Sensed Water-Wheel Governor Topology

U.S. Patent 432,105 describes a water-wheel governor driven from the main shaft through a belt/shafting/gearing connection. U.S. Patent 228,566 states the corrective sense: open the gate when speed falls and close it when speed rises. Those sources support topology, not this page's numerical control gain.

What This Calculator Represents

The tool converts wheel speed through an entered ratio and slip factor, compares the result with an entered target, and applies a declared clipped proportional command.

Belt-Speed and Command Calculation

N_i=N_w R
N=N_i(1-s/100)
e=100(N-N_t)/N_t
gate=clip(50-1.5e,0,100)

The bias and gain are assumed—not derived from a governor.

Worked Scenario

At 12 wheel rpm, ratio 10:1, zero entered slip and target 120 rpm, spindle speed equals target, error is zero, gate command is the declared 50% bias and slip-associated speed difference is zero.

Assumptions and Limits

  • Calculated: ratio/slip speed, signed error and clipped proportional command.
  • Illustrated: belt drive, flyball sensing cue and gate-command direction.
  • Not calculated: governor geometry, dynamics, stability, water flow, actuator response, torque or belt capacity.

Frequently Asked Questions

Is the 1.5 gain historical?

No; it is a declared calculator assumption.

Does 50% mean the real gate equilibrium?

No.

Does the model include flyball dynamics?

No.

Is belt slip predicted?

No.

Will the command stabilize a wheel?

That cannot be determined here.

Can this design a governor?

No.

References

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