Cam Profile Calculator
This cam profile calculator finds how far a follower has lifted at a given cam angle, using the simple harmonic motion law, one of the standard cam-design motion programs. A cam is a shaped disk that pushes a follower up and down as it rotates, turning rotary motion into a precise reciprocating stroke for valves, pumps and machinery. During the rise, the simple harmonic motion law moves the follower along half a cosine wave: the displacement starts at zero, accelerates smoothly, and reaches full lift exactly when the cam has turned through its rise angle. The displacement equals half the lift times one minus the cosine of pi times the cam angle over the rise angle. This smooth law avoids the abrupt acceleration jumps of simpler programs, which is why it features in automotive cam design, including the valvetrain analysis behind vehicle safety work at the US National Highway Traffic Safety Administration. Enter the total lift, the rise angle, and the current cam angle, and the calculator returns the follower displacement and the fraction of lift completed. Every figure is computed deterministically from the simple harmonic motion formula shown in full below, with a worked example that reconciles exactly to the calculator.
Simple harmonic motion moves the follower along half a cosine: a 20 mm lift over a 90 degree rise, at a cam angle of 45 degrees (halfway), gives a displacement of 10.00 mm, exactly half the lift at the midpoint.
Simple harmonic motion cam formula
y = ( h / 2 ) x ( 1 - cos( pi x theta / beta ) )
h = total lift, beta = rise angle
theta = current cam angle within the rise
y = follower displacement at that angle
The follower follows half a cosine wave during the rise. At theta equals zero the cosine is one, so the displacement is zero. At theta equals beta the cosine is minus one, so the displacement is the full lift. At the midpoint the cosine is zero, giving exactly half the lift, with the smoothest possible motion at the ends.
Worked example
Find the displacement of a follower with a 20 mm lift over a 90 degree rise, at a cam angle of 45 degrees.
- angle ratio = theta / beta = 45 / 90 = 0.5000
- cos(pi x 0.5) = cos(90 deg) = 0.0000
- 1 - cos = 1 - 0 = 1.0000
- y = (20 / 2) x 1.0000 = 10.00 mm
The follower displacement is 10.00 mm, exactly half the total lift at the midpoint of the rise. These are the calculator's default inputs, so the result above matches the widget exactly.
Displacement through the rise
For a 20 mm lift over a 90 degree rise, the follower moves along half a cosine wave.
| Cam angle (deg) | Displacement (mm) |
|---|---|
| 0 | 0.00 |
| 22.5 | 2.93 |
| 45 | 10.00 |
| 67.5 | 17.07 |
| 90 | 20.00 |
Engine valvetrain and transportation safety context: US National Highway Traffic Safety Administration (NHTSA).
Cam Profile Calculator: frequently asked questions
What is simple harmonic motion in cam design?
It is a motion program where the follower displacement traces half a cosine wave during the rise, giving zero velocity at both ends and a smooth, sinusoidal acceleration. It is one of the basic cam laws, smoother than uniform or parabolic motion because the acceleration starts and ends at finite values rather than jumping abruptly.
Why is the midpoint exactly half the lift?
At the midpoint of the rise the cam angle equals half the rise angle, so the cosine term is the cosine of 90 degrees, which is zero. The displacement formula then gives half the lift times one minus zero, which is exactly half the lift. The motion is symmetric about that midpoint.
What does the rise angle mean?
The rise angle, often written beta, is the angle the cam rotates through while lifting the follower from zero to full lift. A larger rise angle spreads the lift over more rotation, lowering the follower's velocity and acceleration for the same lift, which reduces wear and noise at high speed.
Can the cam angle exceed the rise angle?
The rise formula applies only within the rise, for cam angles from zero up to the rise angle. Beyond that the follower is in the dwell or return phase, which uses a different segment of the cam profile. Keep the entered cam angle within the rise to get a meaningful displacement from this tool.
Is simple harmonic motion the best cam law?
It is smooth and easy to compute but its acceleration is non-zero at the start and end of the rise, which can cause a small shock if the adjacent dwell has zero acceleration. Cycloidal motion removes that mismatch and is often preferred for high-speed cams, at the cost of higher peak acceleration. Simple harmonic motion remains a common, well-behaved choice.
Official sources
- Vehicle powertrain and mechanical safety reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.