Two externally tangent cylinders roll together with opposite rotation directions. Their pitch radii determine the ideal speed ratio, follower rotation and surface travel when the contact does not slip.
Ideal External Rolling-Contact Calculator
Enter two declared pitch radii and a driver rotation. The worksheet reports the ideal no-slip ratio, follower rotation magnitude, common pitch-surface travel and external center distance. It does not predict whether a real contact can transmit the requested load without slip.
Equation Used
- Two rigid circular pitch cylinders are externally tangent with fixed parallel axes.
- The model imposes zero relative tangential velocity at the ideal contact point.
- Radii are declared pitch radii and deformation is neglected.
- Traction, preload, torque, stress, lubrication, wear, heat and efficiency are outside this kinematic model.
The calculation relates pitch radii and rotation under an ideal no-slip constraint. Transmitting a load also requires adequate contact traction, which is not calculated here.
Equal tangent velocity is the no-slip constraint
At the common tangent, ideal rolling without slip requires the contacting surface points to have equal instantaneous velocity. For fixed external cylinders this gives opposite rotation senses and |omega1|r1=|omega2|r2.
The green point in the animation is the geometric tangent location; the colored radius marks are material orientation marks and rotate with their respective cylinders.
A kinematic reference for pitch cylinders
This ideal relation is useful for understanding pitch-circle ratios and friction-wheel geometry. A real friction drive additionally needs material-specific traction, normal force, torque, deformation, temperature and wear analysis. This page does not supply those design values.
Signed angular constraint and common arc travel
Ideal ratio example
With r1=80 mm, r2=120 mm and one driver revolution, the follower turns 0.667 revolution in the opposite sense, the pitch-surface arc-travel magnitudes are both 502.7 mm, and the centers are 200 mm apart.
What the no-slip model does not establish
The ideal ratio describes motion while the contact rolls without slip. Whether a real drive maintains that motion depends on its contact force, transmitted torque, materials and operating conditions; the ratio alone does not establish traction capacity or service life.
Rolling-contact questions
Is the follower direction included?
Yes. External contact makes theta2 negative when theta1 is positive; the result tile shows magnitude.
Does no slip mean the contact point is stationary?
It means the two contacting material points have equal instantaneous velocity. That shared velocity need not be zero in every reference frame.
Does this calculate torque capacity?
No.
Rolling-contact references
- MIT OpenCourseWare, Classical Mechanics, Week 12: rolling kinematics and the contact point of a wheel rolling without slipping.
- MIT OpenCourseWare 8.01L, angular motion and rolling without slipping: s=R theta and v=R omega.
- NASA Technical Paper 3629, A Rolling Analysis: defines roll without slip as no relative linear velocity between contacting bodies and distinguishes slipping contact.
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