Disc Cam, Four Segments

What is modelled

A cam turns at constant speed and a roller follower is driven up and down by its profile. The designer chooses the lift and how many degrees the rise, the dwell at the top, the return and the dwell at the bottom each take, and then the shape of the motion in each moving segment. A shape that starts and stops gently keeps the acceleration, and so the force, low.

Equations

• Rise s = h f(u) with u = θ/β. Harmonic f = (1 − cos πu)/2; cycloidal f = u − sin(2πu)/2π; 3-4-5 f = 10u³ − 15u⁴ + 6u⁵; constant acceleration f = 2u² then 1 − 2(1−u)².
• Velocity v = Cv h ω/β, acceleration a = Ca h (ω/β)² with Cv = 2, π/2, 2, 1.875 and Ca = 4, π²/2, 2π, 5.77 for the four laws.
• Pressure angle tan φ = (ds/dθ)/(Rp + s); inertia force m a.

Checks run in the page

• Peak velocity from numerical differentiation against the table value.
• Peak acceleration from the second derivative against the table value.
• The velocity integrated over the rise against the lift.
• The work of the inertia force around one revolution, which must vanish.

Honest simplifications

The follower is a rigid, frictionless roller on the pitch curve with no spring or bounce, the cam turns at constant speed, and the pressure angle is computed on the pitch curve rather than on the true cam surface, so the roller radius is ignored. The simple-harmonic and constant-acceleration laws have a jump in acceleration at the segment ends, which real machines avoid because it hammers the mechanism.

• Compare Simple harmonic with Cycloidal: the peak acceleration rises from 4.9 to 6.3 in the table, but the cycloidal law has no jump in acceleration.
• Press Fast rise, small cam and watch the pressure angle exceed the 30 degree limit.
• Raise the speed and see the acceleration, and with it the inertia force, grow with the square.