Lesson 5: Cam-Follower Systems and Motion Programming

Displacement of the cycloidal rise:

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Velocity and acceleration (per unit cam angle):

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Read the diagram. Plotted against , displacement runs smoothly from 0 to , velocity is a bell that starts and ends at zero, and acceleration is a full sine that also starts and ends at zero. Because the acceleration is zero at both ends, the rise joins the neighbouring dwells with no jump. ✅

Convert the rise angle to radians first. rad. Every factor formula divides by , so a rise angle left in degrees is the single most common way to lose these marks. ✅

Write down the factors for the cycloidal law: and . ✅

Peak velocity per unit cam angle, at mid-rise ():

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Peak acceleration per unit cam angle, at the quarter points:

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Convert to real units at the cam speed rad/s. The per-radian figures are geometry alone; the speed enters only here:

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Sanity-check the size of the answer. m/s² is about . A follower of mass kg therefore needs roughly N just to be accelerated, before any spring or load force. Doubling the cam speed would quadruple it, which is exactly why acceleration, not velocity, sets the limit on how fast a cam can run. ✅

Compare with simple harmonic motion. SHM over the same rise peaks at mm/rad², lower than cycloidal. Yet SHM acceleration is a cosine at full value at the start of the rise: where it meets the preceding dwell it jumps from 0 to 50, an infinite jerk. The cycloidal law trades a slightly higher peak acceleration for zero jumps, which is the right trade for a high-speed cam. ✅

Set it up, and set a speed. Open the simulator (siwit.co/CFM), choose a cycloidal rise of mm over with a knife-edge follower, and set the cam speed to 290 rpm, the nearest step to the rad/s used above. That gives rad/s. ✅

Expect real units, not per-radian ones. The readout is Peak velocity (at speed), in mm/s, and Peak acceleration in mm/s². The per-radian figures of Step 2 are pure geometry and never appear on screen; multiply them by and first: ✅

Change the speed slider and the panel numbers move while the per-radian ones do not. That is the whole reason the working is done per radian. ✅

Find where each peak sits. Go back to the normalised diagram of Step 1 and read the cam-angle axis: velocity peaks at mid-rise () and acceleration at the quarter points ( and ). Acceleration passes through zero at both ends of the rise, which is why the panel reports Fundamental law as obeyed. ✅

Break the law deliberately. Switch the rise to simple harmonic. Peak acceleration falls to about mm/rad², lower than cycloidal, yet the acceleration now steps at the dwell boundary and the panel flags the violation. Lower peak, worse cam: it is the discontinuity, not the magnitude, that breaks a high-speed follower. ✅

Draw the base circle of radius mm and mark the cam centre. Choose a length scale and mark it (for example 1 cm = 20 mm). ✅

Divide the cam angle into convenient steps (every , say), measured opposite to the cam’s rotation, because by inversion the follower travels backward around a fixed cam. ✅

Step off the displacement. At each angle, read from the displacement diagram of Application 1 and mark a point that distance radially beyond the base circle. During the rise the radius grows from to ; during the dwell it holds; during the fall it returns. ✅

Join the points with a smooth curve. This is the pitch curve, the path of the follower centre. For a roller follower, the actual cam surface is offset inward from the pitch curve by the roller radius. ✅

Pitch radius. The profile radius at any angle is , so it is mm through the low dwell and mm through the high dwell. ✅

The dwells are circular arcs. Where the follower holds still, the profile is a constant-radius arc (concentric with the base circle). The rise and fall are the shaped transitions between them. ✅

Roller offset. Drawing the pitch curve first and offsetting by the roller radius is the standard route, and it is exactly what the CAD model does when you sweep the roller around the program. ✅

The offset is not always possible. Offsetting inward only works while the roller is smaller than the profile’s tightest curve. Where the pitch curve is convex with radius of curvature , a roller of radius cuts into the flanks either side once approaches , and the cam can no longer deliver the motion you programmed. This is undercutting, and the usual working rule is or so. It is checked in Step 3, not assumed. ✅

Rebuild the same cam. Open the simulator (siwit.co/CFM), choose cycloidal for both rise and fall, and set the program of Application 1: lift mm, rise , dwell , fall , dwell , base radius mm. Tick Show pitch curve. ✅

Check the two radii you drew. The pitch curve should sit at a constant mm through the low dwell and a constant mm through the high dwell, with the rise and fall as the shaped transitions between the two arcs. Those are the same numbers as Step 2, read off a different construction. ✅

Watch the roller offset appear. Set the follower type to Roller and increase the roller radius from mm. The cam surface pulls inward from the pitch curve by exactly the roller radius, which is the offset you were told to apply by hand. ✅

Push it until it fails. Read Min radius of curvature: for this program it is about mm. Keep raising the roller radius and watch the ratio to close up, until the Undercut result reports the profile is no longer usable. A mm roller is comfortable here (), mm is still workable, and by mm you are into trouble. ✅

This is worth doing once deliberately, because undercutting is invisible on the pitch curve you drew. The hand layout will happily give you a profile that no roller of the chosen size can actually trace. ✅

Check the first guess by hand. Use the mid-rise form , with mm/rad from Application 1. At mm:

A full numerical sweep of the rise gives , so the hand estimate sits about a degree low, as expected. Either way it is well above the guide and the follower would jam. ✅

Size the base circle directly rather than guessing again. Rearranging for :

Because the hand form under-estimates by about a degree, round up rather than taking mm as final. ✅

Confirm the chosen size. At mm the hand form gives and the full sweep gives , just inside the guide with a little margin. This is why Application 2 used mm. ✅

Read the trade-off from the plot below: the larger base circle lowers the whole pressure-angle curve under the line, at the cost of a physically larger cam. Note the direction of the lever, the base circle buys pressure angle back cheaply at small and expensively at large , so there is little point oversizing far beyond the guide. ✅

Open the simulator (siwit.co/CFM) with the cycloidal rise and a knife-edge follower, set the base radius to mm, and read the pressure-angle chart: the peak is about , above the guide line. ✅

Enlarge the base circle. Raise the base radius to mm and the peak pressure angle falls to about , just under the guide, matching the hand iteration. ✅