Machine kinematics · Roller envelope

Cam Motion Designer

Can a follower stop while the cam keeps turning? Yes. Where the profile is a circular arc centered on the shaft, the roller stays the same distance from the shaft, so the follower rests while the cam keeps turning. The rest of the profile sets how it rises and falls. For the same lift, a shorter rise makes the follower move faster and accelerate harder, and doubling the shaft speed makes the acceleration four times larger.

Turn a cam, drawn with its true machined edge, under a roller that slides in a straight guide. Design the rise, dwell and return, compare cycloidal and quintic motion, and see why a shorter rise raises both the acceleration and the pressure angle.

Follower lift $0$

Signed output velocity $0$

Output acceleration $0$

One shaft turn $0$

Signed pressure angle $0$

Peak absolute pressure angle $0$

Minimum normal-offset margin $0$

Contact / linkage residual $0$

Turn & compare

Shape the cam edge to set the motion.

Shaft phase

\theta

65.00\,{}^\circ

Prescribed shaft rate

\omega

60.00\,{}^\circ/\mathrm s

Base-circle radius

R_b

1.05\,\mathrm m

Follower lift

h

0.45\,\mathrm m

Rise and return angle each

\beta

120.00\,{}^\circ

Roller radius

a

0.15\,\mathrm m

In the plane of motion, right and up are positive, and depth points toward you. The shaft angle is measured counterclockwise from the crank pointing right. The controls use degrees; the calculations and the CSV export use radians, metres and seconds.

The follower rises out of the low dwell and rests at the top. Its return starts at half a turn and ends in a low dwell. The rise and the return take the same cam angle, and each dwell fills the rest of its half-turn. The presets choose cycloidal or quintic motion; the other controls keep the chosen law.

The pitch curve is the path of the roller centre, seen from the turning cam. The gold profile is that curve moved inward along its normal by one roller radius. The pressure angle runs from the follower's upward motion to the contact normal and is positive when the normal leans right; the peak readout uses its largest absolute value. A positive offset margin means the profile has no local cusp; it is not a strength rating.

Playback runs through one cycle and stops. All curves and exports are exact ideal kinematics sampled over one turn; they are not measured data. Positions, velocities and accelerations come from the same model. Mass, friction, backlash, elastic deformation, real contact loads and drive safety are not modeled.

An ideal external constraint keeps the roller on the cam; no return spring or preload is simulated, and the roller's spin is not tracked. Both motion laws keep displacement, velocity and acceleration continuous where they meet the dwells, but the jerk can jump. Smooth motion on screen does not prove that a real follower would never lose contact.

The cam and roller are ideal planar parts, and the roller moves in a straight line. The roller is assumed to stay on the cam, and the shaft turns at a fixed speed. Spring sizing, contact force, friction, roller spin, elastic deformation and what happens if the roller lifts off are not modeled.

Start with the essentials

Focus question: Can a follower stop while the cam keeps turning?
One-sentence intuition: Where the cam edge is a circular arc around the shaft, the follower stays still; the rest of the edge is shaped to follow the chosen displacement law.

Core mathematical model

Cycloidal rise

s=h\left(u-\frac{\sin(2\pi u)}{2\pi}\right),\quad u=\theta/\beta

This holds during the rise. The return runs the same curve backward, and the dwells hold the displacement constant.

Quintic rise

s=h(10u^3-15u^4+6u^5)

Both laws meet each dwell with zero velocity and zero acceleration, but the jerk can still jump.

Pitch curve of the roller center and the true edge

\rho=R_b+a+s,\quad\mathbf p=(\rho\sin\theta,\rho\cos\theta),\quad\mathbf c=\mathbf p-a\mathbf n

To build the profile, the cam frame is turned opposite to the input. The normal points outward from the pitch curve. Turning the edge forward again puts the contact point on the roller surface.

Pressure angle and a cusp-free offset edge

\tan\varphi=-\frac{s'}\rho,\quad\kappa=\frac{\rho^2+2(s')^2-\rho s''}{(\rho^2+(s')^2)^{3/2}},\quad m=1-a\kappa

Primes mean derivatives with respect to shaft angle in radians. The minimum-margin readout scans one full turn. Separate geometric tests check contact and a cusp-free offset edge across the whole control range.

Common difficulties

A kinematic model is not a force model

Typical misconceptionIf the animation runs smoothly, a real machine can run at that speed.

Better mental modelSmooth motion only shows that the geometry works. Contact loads, inertia, friction, elastic deformation and the drive itself need their own models, and the part thicknesses are schematic.

The roller center is not the cam edge

Typical misconceptionYou can machine the cam edge straight along the pitch curve, the path of the roller center.

Better mental modelThe cam edge sits one roller radius inside the pitch curve, measured along the curve's normal. Subtracting the roller radius along the radial line is generally wrong.

Run the experiment

01
Run one turn
Choose the motion-plane view and run or drag the coral crank.
What to observe: During each dwell the cam keeps turning, but the lift holds steady and the velocity stays at zero.
02
Make the rise shorter
Keep the lift, shorten the rise angle, then enlarge the base circle.
What to observe: A shorter rise raises the speed and acceleration. A larger base circle lowers the pressure angle and leaves the lift curve unchanged.
03
Compare motion laws
Compare cycloidal and quintic presets at the same dimensions, then double shaft rate.
What to observe: Both laws reach the same lift and dwell. Doubling shaft rate doubles velocity and quadruples acceleration.

Geometry & apparatus: CMU · Cams · NITK Virtual Labs · Crank and Slotted Mechanism