Lesson 5: Cam-Follower Systems and Motion Programming

Every mechanism so far was a linkage you were handed and then analysed. The cam inverts the problem: you decide the exact motion the follower must make, its rise, its dwell, its return, and then you design the curved surface that delivers it. The danger is in the curve. Choose a motion law whose acceleration jumps, and the follower receives an infinite jerk at that instant: a hammer blow that wears the contact, makes noise, and shakes the machine at high speed. The whole craft of cam design is shaping the motion so that nothing jumps. In this lesson you program the motion with SVAJ diagrams, lay out the cam profile by hand, and size the base circle from the pressure angle. #CamDesign #MotionProgramming #Cycloidal

Learning Objectives

By the end of this lesson, you will be able to:

1. Program a follower motion with the SVAJ diagrams (displacement, velocity, acceleration, jerk)
2. Compare motion laws and explain why cycloidal motion is smooth where harmonic motion is not
3. Lay out the cam profile graphically by the inversion method
4. Size the base circle from the pressure angle

Real-World System Problem: A Motion You Cannot Make from a Linkage

An engine valve must snap open, stay fully open while gas flows, then close, all in a fixed fraction of a crank revolution, and repeat without fail millions of times. A packaging machine must advance a film, pause precisely while a print head fires, then advance again. These are dwell motions: the output must hold still for part of the cycle and move on a programmed schedule for the rest. A linkage cannot dwell cleanly, but a cam can: its profile is a stored motion program that the follower reads off as the cam turns.

The cam-follower contact is the higher pair from the joint classification: line contact that both rolls and slides, removing only one degree of freedom. That single contact is where all the force passes, so the shape of the cam decides both the motion and whether the follower is driven smoothly or hammered.

The Cam Design Problem

Engineering Question: Given the motion the follower must make (its rises, dwells, and returns), what surface produces it, and is that motion smooth enough to run at speed?

Why Cam-Follower Systems Matter

Fundamental Theory: SVAJ and Motion Laws

The Displacement Diagram and the SVAJ Family

S, V, A, J

A cam motion is described by four diagrams against cam angle :

• S, displacement , the follower position;
• V, velocity (multiply by for time rate);
• A, acceleration (multiply by );
• J, jerk (multiply by ), the rate of change of acceleration.

The fundamental law of cam design: for a cam to run at speed, the displacement and its first two derivatives (, , ) must be continuous across the whole cycle, including the joins to the dwells. A discontinuity in acceleration means an infinite jerk and an impulsive contact force.

Comparing the Motion Laws

Four Motion Laws for a Rise of Height h over Angle β

Motion law	Peak acceleration	Acceleration at the ends	Suitability
Uniform (constant velocity)	zero (infinite at ends)	infinite velocity jump	only with rounded ends
Parabolic (constant accel)		jumps (finite, but step)	low speed
Simple harmonic (SHM)		jumps from zero at a dwell	moderate speed
Cycloidal		zero at both ends	high speed, smoothest

Cycloidal motion has the highest peak acceleration of the four, yet it is the best for high speed because its acceleration starts and ends at zero, so it joins a dwell with no jump and no infinite jerk. Simple harmonic motion looks smooth but its acceleration is a cosine that is at full value at the ends, so where a rise meets a dwell the acceleration steps from a maximum to zero, an infinite jerk.

Pressure Angle and the Base Circle

Pressure Angle

For a radial translating follower (no offset), the pressure angle between the contact normal and the follower motion is:

where is the base-circle radius (the smallest radius of the cam). A large pressure angle means most of the contact force pushes sideways on the follower stem rather than driving it, which causes jamming and wear. The standard guide is to keep for a translating follower. The denominator shows the fix: a larger base circle lowers the pressure angle, at the cost of a bigger cam.

Application 1: Program the Motion and Draw the SVAJ Diagram

This is the central worked example. We program a rise and draw its displacement, velocity, and acceleration to scale, the graphical heart of cam design.

Build it and explore

The cam-follower is best built and turned in CAD: the Cam and Follower Mechanism lesson in the Parametric Mechanical CAD course models exactly this system. For a mechanism that produces pure simple-harmonic follower motion, see the Scotch-Yoke Mechanism, the physical generator of the SHM law compared below.

Step 1: Draw the Displacement Diagram (S, V, A)

Plot the three curves to scale against cam angle. Choose a vertical scale for each (for example 1 cm = 5 mm of lift for ); the shapes are what matter.

Click to reveal the cycloidal motion equations and diagram

1. Displacement of the cycloidal rise:

   ✅

2. Velocity and acceleration (per unit cam angle):

   ✅

3. Read the diagram. Plotted against , displacement runs smoothly from 0 to , velocity is a bell that starts and ends at zero, and acceleration is a full sine that also starts and ends at zero. Because the acceleration is zero at both ends, the rise joins the neighbouring dwells with no jump. ✅

Step 2: Find the Peak Velocity and Acceleration

Click to reveal the peaks and the comparison with SHM

1. Peak velocity at mid-rise ():

   ✅

2. Peak acceleration at the quarter points:

   ✅

3. Compare with simple harmonic motion. SHM over the same rise peaks at mm/rad², lower than cycloidal. Yet SHM acceleration is a cosine at full value at the start of the rise: where it meets the preceding dwell it jumps from 0 to 50, an infinite jerk. The cycloidal law trades a slightly higher peak acceleration for zero jumps, which is the right trade for a high-speed cam. ✅

Application 2: Lay Out the Cam Profile

With the motion programmed, the cam profile is laid out graphically by inversion: imagine the cam fixed and the follower walking around it. At each cam angle the follower sits a distance beyond the base circle, and the locus of those points is the cam profile.

Build it and explore

Lay the profile out on paper as below, then reproduce it in CAD with the Cam and Follower Mechanism lesson, which generates the same profile from the same motion program.

Step 1: Lay Out the Pitch Curve by Inversion

Click to reveal the inversion construction

1. Draw the base circle of radius mm and mark the cam centre. Choose a length scale and mark it (for example 1 cm = 20 mm). ✅

2. Divide the cam angle into convenient steps (every , say), measured opposite to the cam’s rotation, because by inversion the follower travels backward around a fixed cam. ✅

3. Step off the displacement. At each angle, read from the displacement diagram of Application 1 and mark a point that distance radially beyond the base circle. During the rise the radius grows from to ; during the dwell it holds; during the fall it returns. ✅

4. Join the points with a smooth curve. This is the pitch curve, the path of the follower centre. For a roller follower, the actual cam surface is offset inward from the pitch curve by the roller radius. ✅

Step 2: Read the Profile

Click to reveal what the profile shows

1. Pitch radius. The profile radius at any angle is , so it is mm through the low dwell and mm through the high dwell. ✅

2. The dwells are circular arcs. Where the follower holds still, the profile is a constant-radius arc (concentric with the base circle). The rise and fall are the shaped transitions between them. ✅

3. Roller offset. Drawing the pitch curve first and offsetting by the roller radius is the standard route, and it is exactly what the CAD model does when you sweep the roller around the program. ✅

Application 3: Size the Base Circle from the Pressure Angle

The base circle is not free: too small and the pressure angle grows until the follower jams. This is where the cam’s size is actually decided.

Step 1: Plot the Pressure Angle and Iterate

Click to reveal the pressure-angle check

1. Compute across the rise. With a first guess mm, the pressure angle peaks at , well above the guide. The follower would jam. ✅

2. Increase the base circle. Raising enlarges the denominator. At mm the peak falls to , just inside the guide. ✅

3. Read the trade-off from the plot below: the larger base circle lowers the whole pressure-angle curve under the line, at the cost of a physically larger cam. This is why Application 2 used mm. ✅

Design Guidelines for Cam Design

Summary and Next Steps

Key Concepts Mastered

1. SVAJ diagrams: a cam motion is its displacement and the first three derivatives; for high speed, , , and must be continuous.
2. Motion laws: cycloidal motion is smoothest because its acceleration starts and ends at zero, unlike simple harmonic motion which steps at the dwells.
3. Cam profile by inversion: the profile radius is , the displacement diagram wrapped around the base circle.
4. Pressure angle: sizes the base circle; a larger base circle lowers the pressure angle.

Cam Design at a Glance

Quantity	Relation
Cycloidal displacement
Peak velocity / acceleration	/
Pressure angle
Pitch-curve radius

A Note on Tools

The SVAJ diagrams, cam profile, and pressure-angle curves here were drawn from the motion equations and reproduced with a few lines of Python (NumPy). To turn the profile into a part, build it in the Cam and Follower Mechanism CAD lesson, which sweeps the same program into a solid cam.

Next, Force Analysis and Mechanism Synthesis returns to the linkages and closes the course: free-body diagrams and force polygons give the joint reactions, the transmission angle measures force quality, and synthesis runs the whole process backward to design a mechanism that meets a force and motion specification.

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