Cams and Cam-Follower Design: Acceleration Forces

Acceleration Forces

The fundamental equation for the force acting on a translating body under acceleration:

$$R = \frac{Wa}{g}$$

where:

• R = resultant force (excluding friction)
• W = effective weight
• a = acceleration
• g = gravitational constant (386 in/sec² or 9.81 m/s²)

The effective weight is not just the follower. It includes:

$$W = W_f + \frac{1}{3}W_s + W_e$$

where:

• W_f = follower weight
• W_s = return spring weight (⅓ of total, since a spring's mass doesn't move uniformly)
• W_e = external mechanism weight (pistons, pushers, etc.)

Spring Forces

The return spring must be strong enough to keep the follower on the cam at all times. At the point of maximum negative acceleration (where the cam is trying to throw the follower off), the spring force must exceed the inertia force:

$$F_s = R - W_f - F_e - F_f$$

where F_e = external force and F_f = friction force.

The spring should also have some preload — an initial compression — to account for spring "set" (loss of force over repeated use) and to prevent roller sliding at the start of movement.

The required spring constant:

$$K_s = \frac{F_s - \text{preload}}{y_a}$$

where y_a = rise of cam from R_min to the height where maximum negative acceleration occurs.

The spring force at any height y above R_min:

$$F_y = K_s \cdot y + \text{preload}$$

Pressure Angle and Friction Forces

The pressure angle creates a sideways force component that must be resisted by the follower guides. In a translating follower with two guide bushings:

• The normal force from the cam on the follower: F_n
• The force pushing the follower forward: F_n cos α
• The sideways force: F_n sin α
• Friction forces in the bushings: μF₁ and μF₂

The complete force equation for calculating the normal load F_n on the cam:

$$F_n = \frac{\frac{Wa}{g} + yK_s + \text{preload} + W_f + F_e}{\cos\alpha - \mu\sin\alpha \cdot \frac{l_2}{l_1 + l_2} - \mu_d}$$

where:

• μ = coefficient of friction in guide bushings
• μ_d = coefficient of friction at cam-roller contact
• l₁, l₂ = distances defining the bushing geometry

The Effect of Manufacturing Errors

the practitioner showed the practitioner the most sobering calculation in cam design.

Scenario: A cam running at 900 RPM has a tiny manufacturing error — a "bump" that rises 0.001 units in height over 1° of rotation, somewhere along the profile.

The change in acceleration caused by this error:

$$\Delta a = \pm 2e\left(\frac{6N}{\Delta\phi}\right)^2$$

For e = 0.001, Δφ = 1°, and N = 900 RPM:

$$\Delta a = +2 \times 0.001 \times \left(\frac{6 \times 900}{1°}\right)^2 = 58{,}320 \text{ length/sec}^2$$

This is 10 times the acceleration calculated for a perfect cam.

At high speed, even the tiniest imperfection — a chatter mark from machining, a poor blend between profile segments — can generate forces large enough to damage the cam surface. This is why precision in cam manufacturing is not optional. It is survival.

Cam Torque

The torque that the cam exerts on its shaft at any position:

$$T_o = (R_{min} + y) \cdot F_n \cdot \sin\alpha$$

This torque varies throughout the cycle and must be considered in selecting the cam shaft, bearings, and driving motor.

Contact Stresses and Material Selection: The Final Gatekeepers

After the forces are known, the next step is determining whether the cam and roller materials can withstand those forces without surface failure.

Hertzian Contact Stress Formula

When a roller follower is loaded against a cam, the maximum compressive stress at the contact surface is:

For a steel roller on a steel cam:

$$S_c = 2290\sqrt{\frac{F_n}{b}\left(\frac{1}{r_f} \pm \frac{1}{R_c}\right)}$$

For a steel roller on a cast iron cam, use 1850 instead of 2290.

Where:

• S_c = maximum compressive stress
• F_n = normal load on the cam
• b = width of cam (contact length)
• r_f = radius of roller follower
• R_c = radius of curvature of cam surface
• Plus sign (+) when roller contacts the convex portion of the cam
• Minus sign (−) when roller contacts the concave portion
• When roller contacts a flat portion, R_c = ∞ and 1/R_c = 0

In practice, the greatest compressive stress occurs where the roller contacts the convex section with the smallest radius of curvature.

Worked Example: Contact Stress Calculation

Given:

• Roller radius r_f = 0.25 units
• Convex cam radius R_c = 2.01 units (= ρ − r_f = 2.26 − 0.25)
• Contact width b = 0.3 units
• Normal load F_n = 110 force units

$$S_c = 2290\sqrt{\frac{110}{0.3}\left(\frac{1}{0.25} + \frac{1}{2.01}\right)} = 93{,}000 \text{ stress units}$$

This calculated stress must be compared against the allowable stress for the selected material.

Cam Material Selection Guide

No single material is "best" for every cam application. The choice depends on the duty cycle, speed, loads, and whether cost or longevity is the priority. However, since fatigue is the most common failure mode, the surface endurance limit is the key property.

The following table shows maximum allowable compressive stresses for various cam materials in contact with a hardened steel roller, based on 100,000,000 stress cycles for pure rolling:

(Stress values are in psi; adapt to local stress units as appropriate.)

Important notes:

• These values assume pure rolling. Where sliding is present, use more conservative values.
• For stress repetitions significantly greater than 100,000,000 cycles, reduce allowable stress.
• Misalignment also reduces the allowable stress.

The Main Factors Influencing Cam Design

the practitioner summarized the two categories of factors that drive cam engineering decisions:

Factors Influencing Cam Forces:

1. Displacement and cam speed (acceleration forces)
2. Dynamic forces due to backlash and flexibility
3. Linkage dimensions (weight and weight distribution)
4. Pressure angle and friction forces
5. Spring forces

Factors Influencing Cam Stresses:

1. Radius of curvature (cam and roller)
2. Materials (hardness, surface endurance limit)

When Design Changes Are Needed

If the calculated maximum stress exceeds the allowable stress for the selected material, the design must be modified. Available options include:

1. Increase cam size — decreases pressure angle and increases radius of curvature
2. Switch to an offset or swinging follower — reduces pressure angle
3. Reduce cam speed — reduces inertia forces
4. Increase the cam rise angle β — more degrees of rotation for the same rise h
5. Increase cam width — reduces contact stress (if deflections remain acceptable)
6. Use a more suitable cam curve — or modify the curve at critical points

Cylinder Cams: When the Cam Wraps Around a Drum

Not all cams are flat plates. Cylinder cams (also called barrel cams or drum cams) carry their profile as a groove or ridge on the surface of a rotating cylinder. The follower rides in the groove and translates parallel to the cylinder's axis.

Layout of a Cylinder Cam

To construct a cylinder cam profile, you must develop (unroll) the cylindrical surface into a flat plane.

For a uniformly accelerated motion cam on a cylinder:

1. Divide the base circle of the cylinder into equal parts (12 is typical)
2. Set off these equal parts along a horizontal line (the developed circumference)
3. Divide the total rise h into segments proportional to the acceleration pattern — for uniformly accelerated motion, the division proportions are 1 : 3 : 5 : 5 : 3 : 1 (for a half-cycle)
4. Draw horizontal lines from these division points and vertical lines from the circumferential divisions
5. The intersections define points on the developed cam curve
6. Project these points back onto the cylindrical surface

Shape of Rolls for Cylinder Cams

The rolls (followers) for cylinder cams that run in grooves require special attention. Three shapes are possible:

Straight (cylindrical) roll: The worst choice. Because the groove has different surface speeds at the top and bottom (different radii from the cylinder center), a straight roll cannot rotate freely without excessive friction and sliding.

Curved (barrel-shaped) roll: Better for matching the groove curvature, but the small contact area wears quickly, pressing grooves into the cam surface and creating backlash.

Conical roll: The correct choice for most applications. A conical roll permits true rolling action in the groove because the cone apex can be designed to coincide with the cam cylinder axis.

Determining the cone angle:

The amount of taper depends on the spiral angle of the cam groove. Since this angle typically varies throughout the cycle, the roll and groove should be designed for the section where the heaviest duty is performed.

The method:

1. Let b = circumferential distance on the cam surface for the section requiring correct rolling action
2. Let a = throw (rise) of the cam for this distance
3. OU = development of the roll movement at the top of the groove
4. OV = development at the bottom of the groove
5. The top width and bottom width of the groove are made proportional to OU and OV

Cam Milling: Manufacturing the Profile

Milling Plate Cams on a Universal Milling Machine

Plate cams with a constant rise (common in automatic screw machines) can be cut using a universal milling machine with the spiral head set at a calculated angle.

The principle: When the spiral head is vertical, the "lead" of the cam equals the lead for which the machine is geared. By tilting the spiral head to angle α, any cam lead less than the machine's geared lead can be achieved.

The formula:

$$\sin\alpha = \frac{360° \times r}{\phi \times L}$$

where:

• α = angle to set the index head from horizontal
• r = rise of cam in a given portion of the circumference
• L = spiral lead for which the milling machine is geared
• φ = angle (in degrees) over which the rise is required

Example: A cam requires a rise of 0.125 units over 300°, and the machine is geared for a lead of 0.670 units:

$$\sin\alpha = \frac{360° \times 0.125}{300° \times 0.670} = 0.2239$$

$$\alpha = 12° 56'$$

Multi-lobe cams: When a cam has several lobes with different leads, gear the machine for a lead slightly larger than the greatest lobe lead. All lobes can then be milled without changing gears — just adjust the spiral head angle for each lobe.

Practical tip: Whenever possible, mill on the underside of the cam. This keeps chips clear and makes layout lines visible.

Cutting Uniform Motion Cams: A Simple, Accurate Method

For cams that must advance a precise number of thousandths per revolution:

1. Set the milling machine index to cut the desired number of divisions (e.g., 200 divisions for a full revolution)
2. Divide the total throw by half the number of divisions to get the increment per cut
3. Place the cam on an arbor between centers
4. Using a convex cutter at the correct distance from center, make the first cut
5. Lower the knee by the calculated increment and turn the index pin the required holes
6. Repeat around the entire cam

Example: For a heart cam with a throw of 1.1 units, using 200 divisions:

• 1.1 ÷ 100 = 0.011 units per cut
• Each cut: lower the knee 0.011 units and index one position

This method is simple, accurate, and repeatable — no layout or filing required.

Elastohydrodynamic Lubrication: Why Cams Survive at All

There is a remarkable phenomenon that explains why cam-and-follower systems can operate under pressures that would, according to classical lubrication theory, cause immediate metal-to-metal contact and failure.

The Grubin-Crook Discovery

In 1949, A.N. Grubin — and independently in 1958, A.W. Crook — explained why cams and gear teeth can operate under extreme contact pressures without seizing.

The mechanism: Most mineral oils experience an enormous increase in viscosity under high pressure. At pressures around 700 × 10⁶ N/m² (roughly 100,000 psi), the viscosity can increase 10,000-fold. Oil trapped between the cam and follower surfaces at these pressures behaves virtually like a solid separating layer.

This is elastohydrodynamic lubrication (EHL).

The Conditions for EHL to Work

EHL becomes effective when the oil film thickness is less than about 0.25 to 1 μm (microns). For it to succeed:

• Surfaces must be very smooth (surface roughness must be small relative to the film thickness)
• Surfaces must be very carefully aligned (misalignment disrupts the film)
• Mineral oils generally have good pressure-viscosity characteristics for EHL
• Synthetic oils may not have satisfactory pressure-viscosity characteristics

The Friction Coefficient Under EHL

The coefficient of friction under EHL conditions depends on load, contact geometry, and speed, but generally falls between:

• μ ≈ 0.01 at the lightest pressures
• μ ≈ 0.1 at the highest pressures

Film Thickness Estimation

For line contact (gears, cam-tappet systems), the EHL film thickness can be estimated from established formulas involving:

• α = pressure-viscosity coefficient (typical for mineral oil: 1.8 × 10⁻⁸ m²/N)
• ν = viscosity at atmospheric pressure (Ns/m²)
• U = entraining surface velocity = (U_A + U_B)/2 (m/s)

The Unsolved Problem

EHL theory brilliantly explains why lubrication works under extreme conditions. But it also creates a paradox: if squeezing the oil harder makes it harder to extrude, why do lubricants ever fail?

Two hypotheses:

1. High temperature flashes may locally degrade the lubricant
2. High shear rates may actually fracture the lubricant film — since under extreme pressure, the oil is momentarily more like a wax than a liquid

The Problem

The original cam used constant velocity motion on a machine running at 900 RPM. The displacement diagram was a straight line — theoretically infinite acceleration at the start and end of every stroke.

The Fix

the practitioner redesigned the cam using cycloidal motion for the primary rise and return, with the following specifications:

1. Motion curve: Cycloidal — zero jerk discontinuities, smooth acceleration throughout
2. Pressure angle: Verified to stay below 30° at all points using Formula (9a)
3. Minimum radius of curvature: Calculated using Formula (13d), verified ρ_min > r_f with adequate margin
4. Contact stress: Calculated using the Hertzian formula, confirmed below the endurance limit of SAE 4340 induction-hardened steel (226,000 psi allowable)
5. Spring constant: Recalculated to hold the follower against the cam at maximum negative acceleration with a 1.05 correction factor
6. Manufacturing tolerance: Specified surface finish and profile tolerance tight enough that manufacturing errors would produce acceleration perturbations less than 10% of the designed maximum

The Result

The machine ran continuously for 14 months without a single cam-related stoppage. Noise levels dropped by 18 dB. Maintenance costs fell to near zero.

the practitioner smiled. "Now you understand why we don't just draw shapes. We design motion."

Complete Cam Design Workflow: Your Step-by-Step Reference

Whether you're designing your first cam or your hundredth, this is the process:

Phase 1: Define the Motion Requirements

• What must the follower do? (Rise, dwell, return, dwell — specify angles and distances)
• What is the operating speed (RPM)?
• What are the external loads?
• What are the space constraints?

Phase 2: Select the Displacement Curve

• Low speed, no precision needed → Constant velocity (modified)
• Moderate speed → Parabolic or simple harmonic
• High speed → Cycloidal (almost always the right choice)
• Custom requirements → Displacement diagram synthesis (composite curves)

Phase 3: Draw the Displacement Diagram

• Plot follower displacement vs. cam angle for the full 360°
• Calculate velocity and acceleration at all critical points
• Apply the dynamic correction factor (1.05 for cycloidal, ≥2.0 for parabolic)

Phase 4: Determine Cam Size and Follower Type

• Choose follower type (radial, offset, swinging)
• Specify maximum pressure angles (≤30° translating, ≤45° swinging)
• Use graphical or analytical methods to determine R_min
• Choose between open-track and closed-track

Phase 5: Verify Radius of Curvature

• Calculate ρ_min at critical points
• Ensure ρ_min > r_f (no undercutting)
• Ensure R_c > r_cutter (can be manufactured)

Phase 6: Calculate Forces and Stresses

• Determine acceleration forces (F = Wa/g)
• Size the return spring (if open-track)
• Calculate normal force F_n including friction
• Calculate Hertzian contact stress
• Select materials from endurance limit data

Phase 7: Construct the Cam Profile

• Use the inversion technique
• Plot the pitch curve from displacement diagram data
• Draw roller circles and find the cam surface envelope
• Verify the profile at closely spaced intervals

Phase 8: Manufacture and Verify

• Specify surface finish and profile tolerance
• Select manufacturing method (CNC milling, grinding, EDM)
• Inspect the finished cam against the theoretical profile
• Test at operating speed with vibration monitoring

Key Formulas Reference Card

Displacement Curves

Critical Design Limits

Dynamic Correction Factors

The Lesson That Lasts a Lifetime

the practitioner learned something that every engineer, designer, and builder eventually discovers:

The shape of a curve is not just geometry. It is a decision about forces, stresses, noise, vibration, wear, and the lifespan of every component in the system.

A cam that looks right can fail spectacularly. A cam that is mathematically correct can run for decades. The difference is not talent or luck — it is understanding the physics hiding inside the mathematics.

Cams are one of the oldest mechanical elements in existence. They powered the first automated looms, the first internal combustion engines, and the first manufacturing lines. They will power machines that haven't been invented yet.

The principles in this guide will not expire. Newton's second law does not have a shelf life. The Hertzian contact stress equation does not become obsolete. The cycloidal curve will still produce smoother motion a century from now.

Master these principles, and you don't just design cams. You design motion itself.

Your Next Step

Pick one machine you work with — or one machine you're designing — and answer these three questions:

1. What displacement curve is the cam using? (If you don't know, that's your first problem.)
2. What is the maximum pressure angle, and where does it occur?
3. What is the minimum radius of curvature, and is it safely above the roller radius?

If you can answer all three with confidence, you understand cams. If you can't, now you know exactly where to start.

This guide was developed from authoritative references in cam design and mechanism engineering. All formulas, material data, and design principles are based on established engineering practice validated across decades of industrial application.

The Complete Guide to Cams and Cam Design: From First Principles to Factory-Floor Mastery

A Machine Stopped. A Career Almost Ended. Then the Cam Profile Changed Everything.

The packaging line at Richter & Holtz ran twenty-two hours a day, six days a week. It fed, folded, sealed, and stacked corrugated boxes at a rate that made accounting smile and the maintenance crew sweat.

Then, on a Thursday evening during second shift, the main indexing cam on Unit 7 shattered.

Not cracked. Shattered.

Metal fragments punched through the guard cover. The follower arm twisted sideways. The entire downstream conveyor ground to a halt. Twelve minutes of silence cost the company over six figures in lost output.

When mechanical engineer the practitioner Engström pulled the wreckage apart the next morning, the root cause was obvious to anyone who understood cam dynamics—but invisible to anyone who didn't.

The original cam had been designed with a constant-velocity displacement curve. At the operating speed of 900 RPM, the theoretical acceleration at the transition points was infinite. Every revolution hammered the cam surface with forces no material could withstand indefinitely.

the practitioner didn't just replace the cam. She redesigned the entire follower system from the displacement diagram up—selecting a cycloidal motion curve, recalculating the minimum base circle radius, verifying the pressure angle, and choosing a material that could handle the surface stresses.

Unit 7 ran for the next four years without a single cam-related failure.

This guide teaches you everything the practitioner knew—and more.

Whether you are designing your first cam mechanism or troubleshooting a high-speed production line, the principles in this guide are universal, timeless, and mathematically precise. No fluff. No shortcuts. Just the engineering that separates machines that run from machines that break.

What Exactly Is a Cam—And Why Should You Care?

A cam is a mechanical component that converts rotary motion into a prescribed linear or oscillating motion through direct contact with a follower. That single sentence contains an entire universe of engineering.

Every automatic screw machine, every internal combustion engine valve train, every textile loom, every packaging machine, and every automated assembly station depends on cams to orchestrate precise, repeatable, timed motion. If gears are the muscles of a machine, cams are the choreographers.

The beauty of a cam lies in its profile—the carefully shaped surface that dictates exactly how, when, and how fast the follower moves. Change the profile, and you change the machine's behavior entirely.

Here is the fundamental relationship you must internalize:

The shape of the cam determines the displacement of the follower. The first derivative of that displacement gives you velocity. The second derivative gives you acceleration. And acceleration is where forces live—forces that will either serve you or destroy your machine.

Classes of Cams: The Two Families You Must Know

Every cam you will ever encounter falls into one of two broad classes. Understanding the difference is not academic—it determines whether your mechanism runs smoothly or beats itself to death.

Uniform Motion Cams

A uniform motion cam moves the follower at a constant rate of speed from the beginning to the end of the stroke. The displacement diagram is a straight line.

This sounds ideal. Constant speed. Predictable movement. Simple to manufacture.

Here is the trap: The movement starts from zero velocity and reaches full speed instantaneously. It also stops instantaneously. This means the acceleration at both transition points is theoretically infinite.

In practice, infinite acceleration means infinite force—which translates to shock, impact, noise, vibration, and rapid wear. At anything beyond very slow operating speeds, pure uniform motion cams are unacceptable.

Accelerated Motion Cams

Accelerated motion cams use mathematically shaped profiles to control how the follower speeds up, cruises, and slows down. The goal is to eliminate infinite accelerations and manage the forces acting on the cam-follower system.

There are several varieties, and choosing the right one depends on your operating speed, load requirements, and tolerance for vibration. The four most important displacement curves are covered in depth in the next section.

The key insight: In machinery working at high speeds, cams must be constructed so that sudden shocks are avoided when starting motion or reversing the direction of the follower. The cam profile is not just a shape—it is a force management strategy.

Cam Follower Systems: The Three Configurations That Drive Industry

Before you can design a cam profile, you must choose your follower configuration. The three most widely used systems are:

. Radial Translating Roller Follower

The roller follower sits directly above the cam shaft center. As the cam rotates, the roller moves straight up and down along a line that passes through the cam's axis of rotation.

Best for: Simple mechanisms where the follower path must be purely vertical, moderate loads, and applications where the pressure angle can be kept within acceptable limits.

         [Roller]
            |
            |  (Translating motion)
            |
       __|__
      /     |     \     ← Cam Profile
     /      ●      \       (● = Cam Shaft Center)
     \             /
      \__/

. Offset Translating Roller Follower

The follower's line of motion is displaced (offset) from the cam shaft center by a distance e. This seemingly small change has a powerful effect: it allows the designer to reduce the pressure angle during the rise stroke, which reduces side thrust and friction.

Best for: High-speed mechanisms, applications requiring smaller cam size, and systems where unequal pressure angles on rise and return are acceptable.

     [Roller]
        |
        |  ← Offset distance "e"
        |
   __|__
  /     |         \     ← Cam Profile
 /      ●          \       (● = Cam Shaft Center)
 \                 /
  \__/

. Swinging Roller Follower

The roller is mounted at the end of a pivoting arm. Instead of translating, the follower swings through an arc. This configuration is common in valve trains and high-speed indexing mechanisms.

Best for: Applications requiring compact design, high-speed operation, situations where translating guides would introduce excessive friction, and mechanisms with longer strokes.

    M (Pivot)
     \
      \  Lf (Arm Length)
       \
       [Roller]
          |
     __|__
    /     |     \     ← Cam Profile
   /      ●      \       (● = Cam Shaft Center)
   \             /
    \__/

Open-Track vs. Closed-Track Cams

The three configurations above are typically shown as open-track cams—the roller rides on the outside surface of the cam and a spring holds it in contact.

Closed-track cams (also called groove cams or positive-action cams) force the roller to move inside a machined groove or track. The roller is captured, so no spring is needed.

When to use closed-track: Whenever a broken spring would cause serious damage to the machine or product, closed-track cams provide the positive drive assurance you need. The trade-off is increased size, weight, and manufacturing cost.

Displacement Diagrams: Where Every Cam Design Begins

Every successful cam design starts not with a profile, but with a displacement diagram—a graph that plots follower position (vertical axis) against cam rotation angle or time (horizontal axis).

One complete revolution of the cam equals one cycle, which equals 360 degrees. The displacement diagram defines what happens during each degree of that revolution.

Reading a Displacement Diagram

A typical cycle consists of four phases:

  h ─────────── B ──────── C
  |           /              \
  |         /                  \
  |       /                      \
  |     /                          \
  |   /                              \
  0 A ─────────────────────────────── D ──── E
    |   Rise   | Dwell |   Return   | Dwell |
    0°  100°    120°     300°          360°

Critical insight: The shape of the rise and return curves is everything. A straight line (constant velocity) creates infinite acceleration at the transitions. A properly shaped curve manages acceleration, velocity, and displacement simultaneously.