Henry T. Brown's movement 219 places a crown wheel eccentrically on its shaft so the relative radius at the pinion changes through rotation. The calculator evaluates an entered instantaneous engagement radius and pinion pitch radius; it does not solve the eccentric geometry or a complete speed waveform.
Variable Circular Motion by Crown-wheel and Pinion Interactive Calculator
Enter an instantaneous crown engagement radius and pinion pitch radius. Results are ideal local ratio arithmetic—not a complete eccentric-crown motion, tooth-geometry, load, efficiency or design calculation.
Equation Used
- The pinion is treated as the input and the crown wheel as the output for the local ratio calculation.
- R and r are positive instantaneous pitch radii in the same arbitrary length unit.
- Ideal rolling pitch contact and lossless power transfer are assumed at the evaluated instant.
- The crown and pinion rotate about perpendicular axes and speed outputs are magnitudes.
- The animation uses an assumed eccentricity and treats the entered crown radius as a reference for its illustrative variation. Its moving ratio is separate from the numerical outputs calculated directly from the entered instantaneous radius.
- Full-cycle geometry, contact maintenance, tooth form, backlash, interference, sliding, force, torque capacity, shaft motion, inertia, stress, efficiency, wear, lubrication, operating limits and safety are omitted.
Brown Movement 219
Henry T. Brown identifies movement 219 as variable circular motion by crown wheel and pinion and states that the crown wheel is placed eccentrically to its shaft, so the relative radius changes. This is the source topology shown here.
The animation uses an illustrative eccentricity and varying local radius to show changing pinion speed. These assumed dimensions do not establish conjugate tooth profiles or continuous physical contact.
What This Calculator Represents
The tool evaluates one entered instantaneous engagement radius. It divides that radius by entered pinion radius for an ideal reduction ratio, inverts the ratio for crown revolutions per pinion revolution, and reports the same ratio as an ideal lossless torque factor.
Ideal Instantaneous Ratio Calculation
crown revolutions per pinion revolution = r / R
ideal lossless torque factor = R / r
fixed-reference output index = 100(r/R)/(1/8)
The final quantity is only a normalized comparison with a fixed 1/8-output reference. NASA gear references use pitch radius and equal pitch-line velocity for ideal gear-speed relations; real tooth contact and efficiency require additional geometry and load data.
Worked Instantaneous Example
At entered instantaneous crown radius 4 units and pinion radius 1 unit, the ideal reduction ratio is 4.00:1, crown motion is 0.250 revolution per pinion revolution, ideal lossless torque factor is 4.00 and the fixed-reference output index is 200%.
Model Boundary
- Documented and illustrated: crown wheel eccentric to its shaft with a perpendicular pinion.
- Calculated: one instantaneous ideal radius ratio and normalized output index.
- Not calculated: eccentricity, phase-dependent radius, full speed waveform, moving support geometry, tooth conjugacy, backlash, interference, load, efficiency, stress, life, manufacture or safety.
Frequently Asked Questions
Why does the engagement radius change?
The crown wheel is mounted eccentrically to its shaft, so its radius relative to the pinion varies during rotation.
Do the inputs determine eccentricity?
No. They describe only one instantaneous radius state.
Is torque factor actual output torque?
No. It is an ideal lossless ratio.
What is the output index?
Current ideal crown motion divided by a fixed reference of 1/8 revolution per pinion revolution.
Does the animation prove continuous tooth contact?
No.
Can this design or rate a crown-wheel drive?
No.
References
- Henry T. Brown, 507 Mechanical Movements, movement 219: eccentric crown wheel and pinion producing variable circular motion.
- E. & F. N. Spon, Spons' Dictionary of Engineering, figure 3215: eccentric crown-wheel and pinion description.
- NASA gear reference: pitch radius, gear speed and pitch-line velocity relations.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
