This calculator applies ideal standard-module pitch geometry and no-slip speed ratio to a simple internal ring-and-pinion pair with both gear centers fixed. It is not a planetary-stage model or a gear-capacity calculation.
Internal Spur Gear and Pinion Interactive Calculator
Explore ideal pitch diameters, internal-mesh center distance, same-direction speed ratio and a geometric center-distance comparison.
Equation Used
- Standard metric module is common to both gears and tooth counts are positive with Nr>Np.
- Rigid fixed parallel axes and ideal involute pitch-circle rolling without slip.
- Pinion rpm is a nonnegative speed magnitude. In this fixed-axis internal pair, the ring rotates in the same direction as the pinion.
- Distance comparison uses Cexternal=(Dr+dp)/2 for the same pitch diameters only.
- No pressure angle/profile shift, addendum/dedendum, internal interference, undercut, tooth thickness/backlash, contact ratio, face width, torque, load distribution, stress, lubrication, efficiency, noise, strength, fatigue, tolerances, bearings or safety model.
The distance-saved percentage compares ideal center distances for internal versus external meshes using the same two pitch diameters. It is not package-volume, mass or stiffness savings. Internal-tooth interference and manufacturability require full tooth-system data and are not checked.
Ideal internal pitch-circle mesh
For common module m, pitch diameters are m times tooth count. An internal mesh places the centers Dr/2−dp/2 apart. Equal pitch-line velocity gives omega_r=omega_p Np/Nr, with the two members rotating in the same direction.
The animation makes the pitch circles internally tangent using the entered tooth-count ratio. Tooth marks are symbolic, and playback speed is independent of the entered rpm; the speed tiles give the calculated values.
Use as a first-principles geometry check
The outputs can verify module, tooth-ratio, center-distance and ideal speed arithmetic for a simple fixed-axis pair. They cannot analyze a planetary set, select teeth or establish torque capacity and life.
Internal versus external center distance
Cinternal=m(Nr−Np)/2, while an external mesh using the same pitch diameters would have Cexternal=m(Nr+Np)/2. The displayed reduction is only the relative difference between those center distances.
Actual outer envelope also depends on rims, hubs, shafts, bearings, housing and adjacent components.
Worked example
For module 1.5, Nr=72 and Np=24, pitch diameters are 108 and 36 mm, ideal internal center distance is 36 mm and ratio is 3:1. At 1,200 rpm pinion speed, ideal ring speed is 400 rpm in the same direction. Center-distance reduction versus an external mesh is 50%.
What pitch geometry cannot establish
A physical internal mesh needs a compatible tooth system, pressure angle, clearances and manufacturing method. Internal interference can rule out a pair even when the pitch circles fit. Torque capacity and life also depend on face width, materials, loads, alignment and lubrication; those are not calculated here.
Questions
Do internal gears rotate in the same direction?
For a simple fixed-axis internal mesh, yes.
Is center-distance reduction equal to package savings?
No. It compares only ideal shaft-center separation.
Does tooth count alone prove the internal mesh can be manufactured?
No. Internal interference and tool constraints require full tooth geometry.
References & Further Reading
- Wikipedia contributors. Epicyclic gearing. Wikipedia
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