KEVOS® MAKEOS · Mechanics
Cam Size for a Maximum Pressure Angle and Radius of Curvature (Smallest Base Radius, Undercut, Cutter Check)
For a disc cam with a radial translating roller follower, how small can the cam be (the least radius to the pitch curve) without exceeding my maximum pressure angle, where does that angle occur, what is the smallest radius of curvature of the pitch curve, and will my roller undercut the cam or my cutter be too large for the concave parts?
Cam proportions by the handbook formulas for uniform velocity, parabolic, simple harmonic and cycloidal rises with a radial translating roller follower: the pressure angle tan φ = (dr ÷ dθ) ÷ r along the pitch curve, the smallest radius Rmin to the pitch curve that keeps the largest pressure angle within your limit (closed form Rmin = 180 h ÷ (π β tan φmax) for uniform velocity and 360 h ÷ (π β tan φmax) − h ÷ 2 for parabolic motion, and the same condition solved for simple harmonic and cycloidal motion), the angle and radius where the largest pressure angle occurs, the radius of curvature ρ = (r² + r′²)^1.5 ÷ (r² + 2 r′² − r r″) along the rise, the cam surface radius ρ − rf on convex parts and ρ + rf on concave parts, and the checks against undercutting by the roller and against the radius of the cutter.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Rise motion | choice | Motion of the follower during the rise. |
| Rise h | mm | Total rise of the follower. |
| Cam angle for the rise β | ° | Cam rotation during which the rise takes place. |
| Largest pressure angle you accept φmax | ° | Largest pressure angle you accept between the follower’s line of motion and the normal to the cam; 30° or less is usual. |
| Least radius to the pitch curve to use (0 = smallest for φmax) | mm | Least radius from the cam centre to the pitch curve that you will use; 0 uses the smallest that meets the pressure angle limit. |
| Roller and cutter | ||
| Roller radius rf | mm | Radius of the follower roller. |
| Radius of the cutter or wheel for concave parts (0 = none) | mm | Radius of the milling cutter or grinding wheel that forms concave parts of the cam profile; 0 to skip the check. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Rise motion
- What it is
- Motion law the follower follows during the rise.
- Units and format
- Choose one of 4 options from the list: Uniform velocity; Cycloidal; Simple harmonic; Parabolic (constant acceleration).
- Example
- The form opens with “Cycloidal” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the accelerations, pressure angles and curvature.
- Get it right
- Uniform velocity is included for comparison; it needs blending at the ends.
What this means: Rise h
- What it is
- Total rise of the follower from its lowest to its highest position.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Larger rises raise the pressure angle and the accelerations.
- Get it right
- Take it from the motion the machine needs.
What this means: Cam angle for the rise β
- What it is
- Cam rotation during which the follower makes its rise.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Accelerations fall with the square of the rise angle.
- Get it right
- A longer rise angle lowers the accelerations and the pressure angle.
What this means: Largest pressure angle you accept φmax
- What it is
- Largest angle you accept between the follower motion and the normal to the cam.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 1° to 80°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A smaller limit needs a larger cam.
- Get it right
- 30° or less is usual for translating followers.
What this means: Least radius to the pitch curve to use (0 = sma…
- What it is
- Least radius from the cam centre to the pitch curve that you will use.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger cam lowers the pressure angle and raises the curvature radius.
- Get it right
- Enter 0 to use the smallest that meets the pressure angle; enter a larger value if the hub needs it.
Roller and cutter
What this means: Roller radius rf
- What it is
- Radius of the roller on the follower.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12.7 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger roller reduces the cam surface radius and can undercut it.
- Get it right
- Choose it smaller than the least convex radius of curvature.
What this means: Radius of the cutter or wheel for concave parts…
- What it is
- Radius of the milling cutter or grinding wheel used for concave parts of the profile.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Fails the check when the concave cam radius is smaller than the cutter.
- Get it right
- Enter 0 to skip the check.
Method and formulas
- r = Rmin + y(θ); r′ = dr ÷ dθ and r″ = d²r ÷ dθ² from the motion law (θ in radians); tan φ = r′ ÷ r.
- Uniform velocity: Rmin = 180 h ÷ (π β tan φmax); parabolic: Rmin = 360 h ÷ (π β tan φmax) − h ÷ 2; simple harmonic and cycloidal: the least Rmin whose largest φ equals φmax, found by successive trial along the rise.
- ρ = (r² + r′²)^1.5 ÷ (r² + 2 r′² − r r″); positive on convex parts, negative on concave parts.
- Convex: cam surface radius ρ − rf (undercut when ρ ≤ rf); concave: |ρ| + rf, which must exceed the cutter radius.
- Example: the least radius to the pitch curve is 38.999 mm and the smallest convex radius of curvature 34.341 mm.
Limitations
- The formulas are exact for radial translating roller followers and approximate for offset and swinging followers.
- Only the rise is examined; check the return in the same way.
- Keep the least radius large enough for the cam bore and hub as well.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the smallest cam that keeps the pressure angle within a limit
- Check that the roller will not undercut the convex parts of the cam
- Check that a milling cutter or grinding wheel can form the concave parts
Who it is for
- Machine designers
- Automation and packaging engineers
- Toolmakers
What you get
- Smallest least radius Rmin for the pressure angle limit
- Largest pressure angle at the radius used, and where it occurs
- Smallest radius of curvature of the pitch curve on convex and concave parts
- Cam surface radius and the undercut check
- Concave radius against the cutter
Interpreting the result
- A smaller cam raises the pressure angle and lowers the radius of curvature together.
- Undercutting changes the motion the roller actually follows; enlarge the cam or use a smaller roller.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2297–2301. Topic reference; no source tables or recommended settings are reproduced.
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