This revision treats the pinion as the rotating member of a declared metric rack-and-pinion pair and keeps the calculator to nominal pitch-line geometry.
Pinion Pitch-Line Travel Explorer
Enter nominal pitch diameter, revolutions and tooth count. The worksheet reports ideal rack travel, module and circular pitch without sizing teeth or predicting load capacity.
Equation Used
- The pitch circle rolls without slip along the rack pitch line.
- D is the declared nominal pitch diameter and z is an integer tooth count.
- Nominal metric module is m=D/z and circular pitch is p=pi m.
- The animation uses schematic teeth and a fixed display scale.
- Pressure angle, involute profile, profile shift, undercut, backlash, tolerances, lubrication, load, strength, stiffness and efficiency are excluded.
Retains the correct nominal rack-travel and module identities while replacing unsupported case studies and disconnected artwork with exact pitch-line rolling.
A pinion converts rotation to rack translation
At the nominal pitch geometry, the pinion pitch circle rolls along the rack pitch line. An angular rotation theta therefore corresponds to rack displacement s=(D/2)theta. For a full revolution, theta=2 pi and s=pi D.
The visible tooth outlines are schematic. Real mating requires a compatible rack tooth system.
A geometry used in many engineered systems
Rack-and-pinion drives are used where rotary input must produce linear motion, but suitability depends on travel, speed, force, duty, accuracy, environment and the selected gear product. This page does not claim a universal machine configuration or service life.
Nominal metric pitch relationships
KHK's rack technical reference gives rack pitch p=pi m and the assembly distance as rack pitch-line height plus pinion pitch radius for a standard spur pinion. KHK's gear-dimension reference gives d=zm for standard spur gears.
These identities do not establish a complete tooth profile or an acceptable design.
A 60 mm pitch-diameter example
A 60 mm nominal pitch diameter produces 188.5 mm of ideal rack travel per revolution. With 12 teeth the declared module is 5 mm and the circular pitch is 15.71 mm.
Pitch geometry is only the first check
Actual selection must address compatible module and pressure angle, profile and undercut, face width, contact ratio, load, speed, lubrication, backlash, accuracy, stiffness, mounting alignment and safety.
The former CNC case study, brand comparisons, life predictions and universal accuracy or backlash claims are removed because they were not established by the cited references.
Pinion geometry questions
Is outside diameter the same as pitch diameter?
No. Use the reference or pitch diameter specified for the gear geometry.
Does pi D give exact machine travel?
It gives nominal pitch-line travel for ideal rolling; actual positioning also depends on backlash, compliance, errors and controls.
Does tooth count affect travel if pitch diameter is held fixed?
No; pi D sets travel per revolution, while tooth count changes the implied module D/z.
Pinion and rack references
- KHK, Racks Technical Information: reference pitch p=pi m and rack/pinion mounting-distance geometry.
- KHK, Calculation of Gear Dimensions: standard spur-gear reference diameter d=zm.
- ISO 21771-1:2024: concepts and parameters for cylindrical involute gears and compatible basic rack tooth profiles.
- Boston Gear Engineering Information: pitch-diameter and speed relationships for gears and sprockets.
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