Cam profile generator

A disc cam turns at constant speed and pushes a follower along a straight line. The follower's motion over one turn is the cam's program, made of rises, where the follower moves out, dwells, where it stands still, and returns, where it moves back. The generator draws the cam from this program and the sizes of the follower. It also reports the pressure angle and the smallest radius of the pitch curve, the two values that decide whether the cam runs well.

The motion laws

Over a segment of angle β, the follower moves through the lift h along a motion law in u = θ/β, the fraction of the segment, with θ counted from the segment's start. The four laws differ in how the follower starts and stops.

• The cycloidal law, s = h [u − sin(2πu)/(2π)], starts and ends with zero velocity and zero acceleration, so the follower runs smoothly into and out of a dwell, even at speed.
• The simple harmonic law, s = h (1 − cos πu)/2, starts and ends with zero velocity, but its acceleration jumps where it meets a dwell.
• The 3-4-5 polynomial, s = h (10u³ − 15u⁴ + 6u⁵), also has zero velocity and acceleration at both ends, with a lower peak acceleration than the cycloidal law.
• Constant velocity, s = h u, has an infinite acceleration where the segment starts and ends. The follower knocks there, and the generator gives a note.

The pitch curve and the prime circle

The roller's center travels along the pitch curve. At the cam angle 0°, the follower stands on the base circle, the smallest circle of the cam, and the roller's center lies on the prime circle of radius R_p = R_b + r, the base circle radius plus the roller radius. With the follower's axis offset by e from the cam's center, the roller's center starts at d = √(R_p² − e²) along the axis and lies at d + s(θ) at the cam angle θ.

Pressure angle

The pressure angle φ is the angle between the follower's axis and the normal of the pitch curve, the direction in which the cam pushes the roller. For a translating follower, tan φ = (ds/dθ ∓ e)/(d + s), with the sign of e set by the direction of rotation. A large pressure angle presses the follower sideways into its guide, and the usual limit for a translating follower is 30°. A larger base circle or a longer rise angle lowers it, and an offset follower lowers it on the rise at the cost of the return. The generator reports the largest pressure angle and the cam angle where it occurs.

Undercut

The cam's outline lies inside the pitch curve, at the roller radius. Where the pitch curve bulges outward and bends tighter than the roller radius, the outline would cross itself, so the cam is undercut and the follower cannot follow the motion there. The generator computes the pitch curve's radius of curvature over the whole turn and reports its smallest value where the curve bulges outward, together with the outline's radius there, the pitch curve's radius minus the roller radius. Where the pitch curve is hollow, the outline lies further from the center of curvature and cannot be undercut. A roller radius of 0 is a knife-edge follower, whose outline is the pitch curve itself.

Bore and keyway

A bore diameter above 0 cuts a bore at the cam's center, and a keyway width and depth cut a keyway into its top. The depth t2 counts from the bore's edge, as keyway tables give it for the hub.

Units

Sizes and lifts are in millimeters and angles in degrees. The velocity and acceleration are taken per radian of cam rotation, in mm/rad and mm/rad². For a cam turning at ω in rad/s, multiplying them by ω and ω² gives the follower's velocity in mm/s and its acceleration in mm/s².

1. Load

   The page loads Python and planex2d into this tab. Your browser keeps both in its cache, so later visits start faster.

2. Draw

   planex2d draws the cam for the settings, again whenever one changes.

3. Save

   The cam is saved as DXF, STEP, PDF or CSV, made in the tab as well.

The outline

Every segment of the motion becomes one curve of the roller's center, an exact parametric curve of the cam angle, which planex2d fits as a spline to about 10⁻¹² of its size. The curves join into one closed loop, the pitch curve. The cam's outline is that loop offset inward by the roller radius, fitted within a millionth of the cam's size, far below any machining tolerance. The pressure angle and the radius of curvature follow from the motion's formulas and their derivatives, at every quarter degree of the turn. The tests check them against numeric derivatives of the drawn pitch curve, and check that the roller, centered on the pitch curve, touches the outline.

Supported files

Run it in Python

The generator's cam takes a few lines with the library. The example draws the default cam, a cycloidal rise of 12 mm over 120°, a dwell of 60°, the return over 120° and another dwell, for a base circle of 30 mm and a roller of 8 mm. Each segment is one expression curve for the roller's center at the distance r(t) from the cam's center, with the cam angle t in degrees. The rise is the cycloidal law above with u = t/120, and since the expressions take angles in degrees, sin(2πu) is written sin(3*t). The offset by the roller radius gives the outline. See Parametric curves and Offsets in the docs, and the disc cam practice drawing for a cam with its drawing.

$ pip install planex2d

import planex as px

rise = "38 + 12*(t/120 - sin(3*t)/(2*pi))"
fall = "50 - 12*((t - 180)/120 - sin(3*(t - 180))/(2*pi))"
pitch = px.QuickSketch((0, 38))
for r, angles in [(rise, (0, 120)), ("50", (120, 180)), (fall, (180, 300)), ("38", (300, 360))]:
    pitch.Curve((f"({r})*sin(t)", f"({r})*cos(t)"), angles)
cam = pitch.Wire().Offset(-8).Face()
(left, bottom), (right, top) = cam.bounding_box
print("top", round(top, 6), "bottom", round(bottom, 6))
px.WriteDXF(cam, "cam.dxf", unit=px.mm, scale=1)

It prints a top of 30 and a bottom of −42. The outline lies on the base circle at 0°, 30 above the center, and at the base circle plus the lift at 180°, 42 below it. The file cam.dxf holds the same outline as the generator's DXF file of the default cam.

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Limits

• The generator draws disc cams for translating roller and knife-edge followers. Flat-faced and oscillating followers and cylindrical cams are not included.
• The motion holds up to 12 segments. It starts on the base circle at 0°, and the returns must bring the follower back to it.
• The values are kinematic. Speeds, spring forces, contact stresses and the follower lifting off the cam at speed are not computed.
• The largest pressure angle and the smallest radius are found by checking every quarter degree of the turn, so the exact extremes may differ slightly.
• The STEP file holds a flat face, not a solid. The thickness appears in the cam data and on the PDF.
• A cam that fails or a result that looks wrong? Open an issue on GitHub and describe the settings.