This calculator defines a selected sector input sweep and an output-to-input angular ratio. It constructs externally tangent pitch circles with fixed centers and reports the corresponding opposite-direction pinion sweep.
Sector Gear Pitch-Circle Kinematics
Explore exact fixed-center pitch-circle motion for a finite sector and pinion.
Equation Used
- The pitch circles are externally tangent with fixed centers.
- Ideal no-slip rolling holds at the pitch point.
- The selected toothed arc covers the entire input sweep.
- Tooth geometry, backlash, interference, loads and strength are excluded.
The selected ratio is output-angle magnitude divided by input-angle magnitude.
Fixed-center pitch-circle reference
The sector and pinion pitch circles remain externally tangent at one fixed pitch point. With no slip at that reference point, Rsθs + Rpθp = 0.
The finite toothed arc rotates with the sector and contains the pitch point over the selected input sweep.
What the calculator determines
Select sector sweep Θ and magnitude ratio k = |θp/θs| = Rs/Rp. The calculator reports opposite-direction pinion sweep kΘ, blank sector arc 360°−Θ, pitch-radius ratio and toothed-circle share.
KHK's technical reference defines pitch cylinders tangent at the pitch point and relates angular velocity ratio to the reciprocal tooth-count ratio.
Ideal rolling kinematics
At the selected pitch circles, Rsθs + Rpθp = 0. Therefore θp = −(Rs/Rp)θs = −kθs. Output sweep magnitude is kΘ.
Selected-input arithmetic
For a 90° sector sweep and k = 1.5, the pinion sweeps 135° in the opposite direction. The blank sector arc is 270°, and 25% of the full sector pitch circle is designated as toothed.
Model boundary
This is ideal pitch-circle kinematics, not a tooth design. Module or diametral pitch, integer tooth counts, pressure angle, involute profile, backlash, contact ratio, interference, strength, lubrication and manufacturing tolerances are not calculated.
Questions
What does the ratio mean?
It is output angular magnitude divided by input angular magnitude, equal to sector pitch radius divided by pinion pitch radius.
Why does the pinion reverse?
External pitch circles have opposite angular directions at their shared tangent point.
Is the blank arc engaged?
No.
Can this specify teeth?
No.
References & Further Reading
- Wikipedia contributors. Gear. Wikipedia
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