Design cam profiles with follower displacement diagrams. Choose dwell-rise-dwell sequences and generate displacement, velocity, acceleration curves.
This tool builds the follower displacement diagram of a cam mechanism: you split the cam's rotation into dwells and rises, assign a motion law to each rise, and it returns displacement, velocity, and acceleration curves plotted against cam angle. The curves come from the closed-form equations of classical cam motion - constant-velocity, simple-harmonic, and cycloidal rises, among others - evaluated across the whole cycle. The recurring surprise is that the displacement curve is a poor guide to dynamics: follower forces follow the acceleration curve, and acceleration scales with the square of cam speed, so a rise that looks gentle on the displacement plot can be violent at speed.
A concrete input and expected output from the current implementation.
Input
Rise 10 mm over 60 deg of cam rotation, cycloidal motion, cam speed 60 rpm
Expected output
Peak follower velocity 120 mm/s, reached at the rise midpoint (30 deg). Peak acceleration 2262 mm/s^2 (about 0.23 g), reached at 15 deg and 45 deg. Displacement at 15 deg is 0.91 mm, at 30 deg is 5.00 mm, at 45 deg is 9.09 mm. Velocity and acceleration are zero at both ends of the rise.
Cycloidal motion follows s = h(u - sin(2*pi*u)/(2*pi)), with u the fraction of the rise completed, so peak velocity is 2h*omega/B = 120 mm/s and peak acceleration is 2*pi*h*omega^2/B^2 = 2262 mm/s^2 for h = 10 mm, B = 60 deg, omega = 2*pi rad/s. Both derivatives vanish at the segment ends, which is why the dwell-to-rise transitions carry no jerk spike.
The follower's cycle is split into angular segments: dwells, where displacement stays constant, and rises or falls, where displacement follows the chosen law as a function of the cam angle within the segment. The tool evaluates these closed-form functions across the full rotation and plots them against cam angle. Velocity is the displacement curve's first derivative scaled by cam speed; acceleration is the second derivative scaled by cam speed squared.
Each law trades peak values against endpoint behavior. Constant velocity keeps acceleration near zero through the rise but demands infinite acceleration at both ends. Simple harmonic motion (SHM) has zero endpoint velocity, but its acceleration starts and ends at its maximum, so the value jumps at each transition - an infinite jerk spike. Cycloidal motion brings both velocity and acceleration to zero at the endpoints, making it the usual choice for higher-speed cams, at the cost of a peak acceleration about 27 percent above SHM.
An SHM rise is a half cosine displacement wave, and its second derivative is also a cosine - maximum at the endpoints, not zero. The instant the rise begins, acceleration leaps from zero to its peak value, and that instantaneous jump is an infinite jerk spike, felt as shock and vibration. Cycloidal motion exists precisely to eliminate it: both velocity and acceleration are zero at the endpoints, at the cost of a slightly higher peak acceleration mid-rise.
At any cam angle, the follower's inertia force is the moving mass times the acceleration at that angle, added to spring preload and friction. The curve's peak value gives the worst-case load for sizing the spring, bearings, and drive. Mind the scaling: double the cam speed and every acceleration value quadruples, so a profile that feels smooth at low speed can lose follower contact at high speed.
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