KEVOS® MAKEOS · Mechanics
Modified Constant-Velocity Cam with Parabolic Matching (Rise Angles, Displacement Table, Acceleration)
My cam follower must rise partly at constant velocity, with constant acceleration before and constant deceleration after: what cam angles do the acceleration and deceleration parts need so the curves blend, what is the displacement at each step of cam rotation, and how large are the velocity and accelerations at my cam speed?
Modified constant-velocity cam by the handbook’s parabolic matching: a rise y1 at constant acceleration, y2 at constant velocity over the angle β2 and y3 at constant deceleration, with the parabolas tangent to the straight line so the velocity is continuous; the matching angles β1 = 2 y1 β2 ÷ y2 and β3 = 2 y3 β2 ÷ y2, the displacement y = y1 (θ ÷ β1)² in the first part, y1 + y2 (θ − β1) ÷ β2 in the middle and (y1 + y2 + y3) − y3 ((βt − θ) ÷ β3)² in the last, tabulated at your step of cam rotation, and at your cam speed the constant velocity y2 ω ÷ β2 and the accelerations 2 y1 ω² ÷ β1² and 2 y3 ω² ÷ β3².
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Rise at constant acceleration y1 | mm | Part of the rise made at constant acceleration from rest. |
| Rise at constant velocity y2 | mm | Part of the rise made at constant velocity. |
| Rise at constant deceleration y3 | mm | Part of the rise made at constant deceleration to rest. |
| Cam angle for the constant-velocity rise β2 | ° | Cam rotation during which the constant-velocity part of the rise takes place. |
| Step of cam rotation for the table | ° | Interval of cam rotation at which to tabulate the displacement. |
| Cam speed and limit | ||
| Cam speed | rpm | Rotational speed of the cam. |
| Largest follower acceleration you accept | m/s² | Largest follower acceleration you accept, from the follower mass and spring. |
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 at constant acceleration y1
- What it is
- Part of the rise made with constant acceleration from rest.
- 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 6.35 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the acceleration angle β1 and the acceleration.
- Get it right
- A small part, often a fifth or so of the whole rise.
What this means: Rise at constant velocity y2
- What it is
- Part of the rise made at constant follower velocity.
- 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 31.75 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the constant velocity together with β2.
- Get it right
- The working part of the stroke, where uniform speed is needed.
What this means: Rise at constant deceleration y3
- What it is
- Part of the rise made with constant deceleration to rest.
- 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
- Sets the deceleration angle β3 and the deceleration.
- Get it right
- Often larger than y1 to soften the stop.
What this means: Cam angle for the constant-velocity rise β2
- What it is
- Cam rotation for the constant-velocity part of the rise.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 359°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the matching angles and the constant velocity.
- Get it right
- Take it from the timing diagram of the machine.
What this means: Step of cam rotation for the table
- What it is
- Interval of cam rotation at which the displacement is tabulated.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets how many points the table lists.
- Get it right
- 5° is common; finer steps for large cams or accurate profiles.
Cam speed and limit
What this means: Cam speed
- What it is
- Rotational speed of the cam shaft.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Velocity rises with speed and acceleration with its square.
- Get it right
- Use the highest speed the machine runs at.
What this means: Largest follower acceleration you accept
- What it is
- Largest follower acceleration that your follower mass and spring can accept.
- Units and format
- Enter a number in m/s²: metres per second squared. The form accepts 0.000001 m/s² to 10000000 m/s². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 m/s² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Fails the cam when the acceleration is higher.
- Get it right
- Work it out from the spring force available and the moving mass.
Method and formulas
- β1 = 2 y1 β2 ÷ y2; β3 = 2 y3 β2 ÷ y2; βt = β1 + β2 + β3 (the tangent to a parabola bisects the abscissa to its vertex).
- 0 ≤ θ ≤ β1: y = y1 (θ ÷ β1)²; β1 ≤ θ ≤ β1 + β2: y = y1 + y2 (θ − β1) ÷ β2; last part: y = (y1 + y2 + y3) − y3 ((βt − θ) ÷ β3)².
- ω = 6 N °/s; velocity = y2 ω ÷ β2; acceleration 2 y1 ω² ÷ β1²; deceleration 2 y3 ω² ÷ β3².
- Example: the acceleration and deceleration take 20° and 40°, so the whole rise takes 110° of cam rotation.
Limitations
- The accelerations change suddenly at the ends of each parabola; for high-speed cams a cycloidal blend gives lower dynamic forces.
- The return is designed the same way; this tool covers the rise.
- Measure the displacements on radial lines from the cam centre when laying out the pitch curve.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Lay out a cam that moves a follower at constant velocity without impact at the ends
- Produce the displacement table for drawing or machining the cam
- Check the follower accelerations at the cam speed
Who it is for
- Machine designers
- Automation and packaging engineers
- Toolmakers
What you get
- Matching angles β1 and β3 and the total rise angle
- Displacement at each step of cam rotation
- Constant velocity of the follower
- Acceleration and deceleration of the follower
Interpreting the result
- Parabolic and cycloidal blends need the same angles because both reach the same velocity for the same rise and angle.
- A smaller acceleration part y1 shortens β1 but raises the acceleration as y1 ÷ β1².
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2288–2291. Topic reference; no source tables or recommended settings are reproduced.
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