Clockwork Goats Cam

What is modelled

A disc cam turns at constant speed and pushes a follower up and down. The lift here is cycloidal: the follower speeds up and slows down smoothly, with no jump in acceleration at the ends, then dwells at the top (where the goats meet) and returns. A spring keeps the follower on the cam; if the inertia force is larger than the spring force the follower leaves the profile. A steep cam also pushes the follower sideways: the angle between the push and the follower motion is the pressure angle.

Equations

• Rise s = h (u − sin(2πu) / 2π) with u = θ / β.
• Speed v = ω h (1 − cos 2πu) / β, peak 2 h ω / β; acceleration peak 2π h ω2 / β2.
• Contact force N = k (δ + s) + m a; contact is lost where N < 0.
• Pressure angle tan φ = (ds/dθ) / (Rb + s), kept below 30 degrees.

Checks run in the page

• Finite-difference slope and curvature of the lift against the analytic formulas.
• The peak speed and acceleration from a scan of the whole turn.
• The smallest base circle for a 30 degree pressure angle found by bisection.
• The spring energy gained over the rise against the integrated force times lift.

Honest simplifications

The follower is a rigid in-line roller with no friction, no gravity and no cam flexibility, the spring is linear, the cam turns at constant speed, and the goats are only a drawing of the follower travel. Use it for the trends, not for designing a mechanism.

• Press Follower jumps off and watch the contact force go below zero.
• Press Small cam, steep flank and see the pressure angle exceed 30 degrees.
• Raise the rise angle to soften the acceleration.