Cams and Follower Systems

Introduction

A cam mechanism converts rotary motion into a prescribed reciprocating, oscillating, or translating motion. It consists of a cam, which is the driving member having a specially shaped profile, and a follower, which is constrained to remain in contact with the cam. Unlike linkages, whose motion is determined primarily by link geometry, a cam mechanism can generate an almost arbitrary motion law by suitable profile design.

Cams are used in valve trains, fuel-injection pumps, textile machinery, printing machines, indexing devices, packaging equipment, and automated production systems. Their principal advantages are compactness, accurate timing, and the ability to provide dwell periods. Their limitations include sliding or rolling contact, high contact stress, sensitivity to manufacturing errors, and dynamic forces at high speed.

Learning Objectives

After studying this chapter, the reader should be able to:

• classify cams and followers;
• interpret displacement, velocity, acceleration, and jerk diagrams;
• select and apply standard follower-motion laws;
• construct a cam profile from a displacement diagram;
• determine pressure angle, radius of curvature, and possible undercutting;
• estimate contact forces and assess high-speed performance; and
• make appropriate design decisions for practical cam mechanisms.

1. Fundamental Terminology

A cam is a rotating or translating machine element that imparts a specified motion to a follower. The follower is the output element that contacts the cam. The cam profile is the actual working contour of the cam. The pitch curve is the locus of the follower reference point as the cam rotates. For a roller follower, it is the locus of the roller centre; for a knife-edge follower, it coincides with the cam profile.

The prime circle is the smallest circle centred at the cam axis and tangent to the pitch curve. Its radius is the prime-circle radius, denoted by R_p. For a roller follower, the base circle is the smallest circle tangent to the cam profile, with radius R_b. If the roller radius is r_r, then, for an external cam and concentric radial follower, approximately R_p = R_b + r_r.

The lift or stroke, h, is the maximum follower displacement. A rise is follower motion away from the reference position, and a return is motion toward it. A dwell is an interval during which the follower remains stationary while the cam continues to rotate. The cam angle associated with a rise, return, or dwell is called its angle of action or dwell angle, depending on the motion.

The line of stroke is the path of a translating follower. If it passes through the cam centre, the follower is radial or in-line. If it is offset by a distance e, the follower is offset. An oscillating follower rotates about a fixed pivot through an angular displacement rather than translating.

The pressure angle, φ, is the angle between the follower direction of motion and the common normal at the contact point. It measures the severity of the force transmission. A large pressure angle produces a large transverse force and may cause jamming, wear, or excessive bearing load.

2. Classification of Cams and Followers

2.1 Cam classifications

Cams may be classified according to their geometry and motion:

A radial cam is the most common type in elementary design. A groove cam can positively control both rise and return, whereas a surface cam generally requires a spring, gravity, or another force to maintain contact.

2.2 Follower classifications

Followers are classified by the nature of contact, motion, and arrangement.

A knife-edge follower has theoretically point contact and is simple to analyze, but it experiences high wear and is rarely used for substantial loads. A roller follower replaces sliding contact with rolling contact, reducing friction, although it introduces roller-bearing stress and requires careful profile construction. A flat-faced follower is robust and suitable for high loads, but it usually experiences sliding contact and may require a larger cam size. A spherical-faced follower reduces edge contact when the follower face tilts.

A force-closed follower is held against the cam by a spring, gravity, or fluid pressure. A form-closed follower is constrained mechanically, as in a groove or conjugate cam. Force closure is simpler, but loss of contact can occur if the spring force is insufficient.

3. Kinematic Description of Follower Motion

Let the cam rotate through angle θ at angular speed ω, and let the follower displacement from its reference position be s(θ). If the cam rotates at constant speed,

The follower velocity is

and the follower acceleration is

The jerk, important in high-speed operation, is

These equations assume constant cam speed. If the cam speed varies, acceleration includes an additional term:

where α = dω/dt is cam angular acceleration. In most introductory cam design, α is neglected because the camshaft is treated as rotating uniformly.

A displacement diagram plots s against cam angle θ. Its slope, multiplied by ω, gives velocity; its second derivative, multiplied by ω², gives acceleration. Consequently, a displacement diagram is not merely a geometric construction: it is the basis for evaluating dynamic suitability.

4. Standard Motion Laws

The motion law must satisfy the required lift, cam angle, speed, acceleration limits, and manufacturing constraints. Common laws are described below for a rise of h during a cam-angle interval β. Define the normalized angle

4.1 Uniform velocity motion

For constant velocity rise,

Thus velocity is constant within the rise and acceleration is zero except at the beginning and end, where idealized acceleration is infinite. This discontinuity causes impact and is unsuitable for high-speed operation. It may be acceptable in slow mechanisms with generous compliance.

4.2 Simple harmonic motion

For simple harmonic rise,

The corresponding velocity and acceleration are

Velocity is zero at the ends and maximum at mid-stroke. Acceleration is finite but changes sign abruptly at the ends if adjoining dwells are idealized. The maximum acceleration magnitude is π²hω²/(2β²).

4.3 Uniform acceleration and deceleration

The rise is divided into equal angular portions. During the first half, acceleration is constant and positive; during the second half, it is constant and negative. For the first half, 0 ≤ θ ≤ β/2,

For the second half, β/2 ≤ θ ≤ β,

The velocity is continuous, but acceleration changes abruptly at the midpoint and at the boundaries. This law is easy to construct graphically but produces jerk impulses at the transitions.

4.4 Cycloidal motion

Cycloidal motion is obtained from the rolling of a circle and is widely used where low jerk is required. For a rise,

The velocity and acceleration are

Acceleration is zero at both ends, and the idealized jerk is finite within the interval. Cycloidal motion generally provides better high-speed performance than simple harmonic motion, although its peak acceleration can be comparatively high.

4.5 Polynomial motion

Polynomial laws can be selected to impose specified endpoint conditions. A commonly used 3-4-5 polynomial is

It satisfies zero displacement and velocity at the beginning and unit displacement with zero velocity at the end. Its acceleration also becomes zero at both ends, making it useful for smooth transitions between dwells. Higher-order polynomials can impose zero jerk or other conditions, but excessive polynomial order may create undesirable internal peaks.

4.6 Comparison of motion laws

The best motion law is not selected solely by peak acceleration. Manufacturing accuracy, follower mass, spring characteristics, available cam angle, lubrication, and allowable pressure angle must also be considered.

5. Construction of Displacement Diagrams

A displacement diagram is constructed by dividing the relevant cam angle into equal parts and finding the corresponding displacement for each division.

For uniform velocity, equal angular divisions correspond to equal displacement divisions. For SHM, the displacement ordinates can be obtained from the cosine relation or by projecting points from a semicircle. For cycloidal motion, ordinates are calculated from the cycloidal equation or obtained using a tabulated construction. For polynomial motion, normalized values of s/h are calculated for selected values of u.

A complete diagram must show rise, dwell, return, and dwell intervals in their actual angular sequence. The return law need not be identical to the rise law. In practice, the return may be made faster if the follower does not need to perform useful work during that interval; however, reducing the return angle increases velocity and acceleration and may increase contact loss.

Worked Example 1: Cycloidal kinematics

A translating follower rises h = 40 mm in β = 120° while the cam rotates at N = 600 rpm. Determine the maximum velocity and acceleration for cycloidal motion.

First convert units:

β = 120π/180 = 2.094 rad

For cycloidal motion, maximum velocity occurs at θ = β/2:

Therefore,

v_max = 2(0.040)(62.83)/2.094 = 2.40 m/s.

The maximum acceleration is

Hence,

a_max = 2π(0.040)(62.83²)/(2.094²) ≈ 113.7 m/s².

The calculation assumes constant cam speed, an ideal rigid follower, and exact cycloidal motion. Actual acceleration may be higher because of flexibility, manufacturing error, and dynamic interaction with the spring and follower mass.

6. Construction of a Radial Cam Profile

The profile is commonly constructed from the displacement diagram by a graphical or numerical procedure.

1. Draw the base circle with centre at the camshaft axis.
2. Draw the follower reference line. For an in-line follower, it passes through the cam centre; for an offset follower, it is displaced by e.
3. Divide the cam rotation into angular intervals corresponding to the displacement-diagram divisions.
4. Rotate the follower reference line through each cam-angle division in the direction opposite to the actual cam rotation. This reversal is required because the construction observes the cam in a stationary frame while the follower line is effectively indexed relative to it.
5. On each indexed follower line, mark the required distance from the cam centre. For a knife-edge follower, this point lies on the profile. For a roller follower, it locates the roller centre on the pitch curve.
6. Draw a smooth curve through the pitch points. For a roller follower, draw the inner envelope of circles of radius r_r centred at the pitch points to obtain the actual cam profile.

The profile must be checked for cusps, undercutting, minimum radius, pressure angle, and adequate contact. Numerical construction using CAD is preferred for production, but the same geometric principles apply.

7. Pressure Angle and Force Transmission

The pressure angle is a central design criterion. For an in-line translating knife-edge follower with pitch-curve radius r(θ), the pressure angle can be related to the slope of the pitch curve. Since the radial distance changes by dr/dθ,

For a roller follower, r refers to the pitch-curve radius, not necessarily the cam-profile radius. For an offset follower, the geometry includes the offset and must be derived from the actual coordinates; the simple radial expression is not directly applicable.

The follower force can be resolved into a useful component and a transverse component. If the required follower force along the line of motion is F_f, the normal contact force is approximately

and the transverse component is

These relations assume a frictionless contact and define F_f as the net force required along the follower line. With friction coefficient μ, the actual contact force and torque depend on the direction of sliding and detailed contact geometry. A common preliminary design limit is a pressure angle below approximately 30° for translating followers, although the acceptable value depends on speed, load, offset, and bearing capacity.

The net follower force may be estimated as

where F_L is the external load, m_f is equivalent follower mass, a is follower acceleration, F_s is the spring force at the position considered, and F₀ represents other resisting or assisting forces according to the chosen sign convention. The equation must be applied consistently; spring preload and spring force should not be counted twice.

8. Curvature, Undercutting, and Profile Feasibility

The cam profile must have sufficient curvature to prevent sharp points and loss of a valid roller envelope. If the radius of curvature of the pitch curve is ρ_p and the roller radius is r_r, then a basic condition for an external roller cam is

at locations where the roller is on the concave side of the pitch curve. If ρ_p ≤ r_r, the offset envelope can develop a cusp or loop, and the physical profile becomes undercut or geometrically impossible. The exact sign and condition depend on whether contact is on the concave or convex side; therefore, profile curvature must be evaluated using the signed curvature of the actual pitch curve.

Undercutting may also occur when the base circle is too small relative to the lift and motion law. Increasing the base-circle radius generally reduces pressure angle and improves curvature, but it increases cam size and peripheral speed. Increasing the roller radius may reduce contact stress but can worsen undercutting in concave regions.

For a plane curve described in polar coordinates by r = r(θ), the radius of curvature is

The equation assumes a sufficiently smooth differentiable curve and uses a signed convention. In numerical design, derivatives should be evaluated carefully because noise in the displacement data can produce large curvature errors.

9. Contact Stress, Friction, and Wear

Cam contacts are often subjected to high local stress. A roller follower experiences rolling contact, but sliding may still occur because the roller and cam have different instantaneous velocities. A flat-faced follower experiences substantial sliding and requires effective lubrication and suitable surface finish.

For preliminary analysis, Hertzian contact theory may be used to estimate local contact stress. The exact expression depends on whether the contact is point, line, or elliptical and on the equivalent elastic modulus. In all cases, contact stress increases with normal load and decreases as the effective contact radius and material stiffness increase. Production design should use the appropriate Hertzian equations for the contact geometry rather than a generic allowable pressure.

Important design variables include:

• material hardness and through-hardening or case-hardening depth;
• surface finish and conformity;
• lubricant viscosity and supply;
• roller-bearing rating;
• contact temperature;
• contamination and sealing; and
• fatigue life under repeated cycles.

Wear is accelerated by high sliding speed, inadequate lubrication, surface roughness, misalignment, and edge loading. A cam can be kinematically correct yet fail prematurely if tribological conditions are neglected.

10. High-Speed Dynamic Performance

At high speed, the follower does not necessarily follow the theoretical displacement curve. The follower, spring, cam, and supporting structure form a dynamic system. The effective follower mass and spring stiffness can produce oscillation, bounce, and loss of contact.

A simplified undamped model is

where y is actual follower displacement, s is prescribed cam displacement, m_eq is equivalent moving mass, and k is spring stiffness. A more realistic model includes damping, friction, spring nonlinearity, camshaft torsional flexibility, and contact constraints.

The natural circular frequency of the simple mass–spring model is

The ratio ω/ω_n indicates the likelihood of dynamic amplification. Operation near resonance should be avoided. The spring must maintain contact during the most unfavorable acceleration and load condition, but excessive spring force increases contact stress, bearing load, and power loss.

A useful high-speed design strategy is to select a motion law with continuous acceleration and preferably controlled jerk, enlarge the base circle where practical, minimize moving mass, provide adequate damping, and verify the design using time-domain dynamic simulation or experimental measurement.

11. Applied Case: High-Speed Camshaft for a Small Four-Stroke Engine

Background

Consider an illustrative design case involving a single-cylinder, four-stroke engine operating at 3,600 rpm. The camshaft must open an inlet valve over a specified crank-angle duration, provide a dwell-like closed period, and return the valve without loss of contact.

Problem

An initial design uses a small base circle and uniform acceleration–deceleration motion. During testing, the engine exhibits valve-train noise at high speed and inconsistent closing. The design team must identify likely causes and propose improvements.

Analysis

The camshaft rotates at half crankshaft speed, so its speed is 1,800 rpm. The valve train has a spring, rocker arm, valve mass, and small clearances. Uniform acceleration–deceleration introduces abrupt changes in acceleration at the midpoint and endpoints. These changes produce high jerk and excite structural modes. The small base circle also increases pressure angle and may create a small radius of curvature near the nose. At closing, the spring may not provide enough force to maintain contact against inertia.

Approach and solution

The team replaces the basic motion law with a polynomial or spline-based law having zero velocity and acceleration at the transitions and controlled jerk. The base-circle radius is increased, subject to packaging limits. The roller and cam materials are surface-hardened, and lubrication is improved. A multibody dynamic model is used to estimate valve bounce, and spring preload is adjusted so that contact is maintained without excessive seating impact. The design is then checked for pressure angle, curvature, contact stress, and fatigue life.

Outcome

The revised design reduces acceleration discontinuities, pressure angle, and dynamic separation. Noise is reduced and valve closing becomes repeatable. The larger cam increases envelope size, but the improvement in reliability justifies the packaging change.

Learning points

Kinematic smoothness alone does not guarantee satisfactory performance. Cam design is an integrated problem involving motion law, geometry, force transmission, contact mechanics, dynamics, materials, lubrication, and manufacturing capability.

12. Design Procedure and Best Practices

A practical cam-design workflow is as follows:

1. Specify the required follower events: lift, rise, return, dwell, and timing angles.
2. Determine speed, load, allowable pressure angle, packaging limits, and environmental conditions.
3. Select follower type and whether the mechanism will be force-closed or form-closed.
4. Choose a motion law based on speed, allowable acceleration, jerk, and manufacturing requirements.
5. Construct the displacement, velocity, acceleration, and, where necessary, jerk diagrams.
6. Select a preliminary base-circle or prime-circle radius.
7. Construct the pitch curve and actual profile.
8. Check pressure angle, curvature, undercutting, contact stress, and minimum feature size.
9. Calculate follower and spring forces, bearing loads, and camshaft torque.
10. Perform dynamic simulation and tolerance analysis.
11. Specify materials, heat treatment, surface finish, lubrication, and inspection requirements.
12. Validate the prototype using displacement, vibration, temperature, noise, and wear measurements.

Computer-aided design and numerical optimization can efficiently vary base-circle radius, offset, roller radius, and motion-law coefficients. However, software output must be checked against definitions, sign conventions, units, and physical constraints. Artificial intelligence can assist with parameter search or fault detection, but it does not replace kinematic verification, contact mechanics, safety assessment, or engineering judgment.

Challenges and Limitations

Ideal cam theory assumes rigid members, exact geometry, constant speed, perfect contact, and negligible friction. Real mechanisms violate all of these assumptions. Manufacturing tolerances alter the displacement and pressure angle. Shaft torsion changes timing. Bearing clearance and structural flexibility create vibration. Temperature affects lubricant viscosity and material dimensions. Spring surge may cause contact loss even when the static force calculation appears satisfactory.

The designer must therefore distinguish between a theoretical profile and a manufacturable, durable mechanism. Safety-critical applications require documented analysis, traceable material specifications, validated calculations, and appropriate industry standards. In engine and industrial applications, guarding is necessary because exposed rotating cams and pinch points can cause injury.

Chapter Summary

Cams provide precise programmed motion through the geometry of a driving profile. They may be radial, cylindrical, wedge, face, groove, or conjugate types, while followers may be knife-edge, roller, flat-faced, translating, oscillating, radial, offset, force-closed, or form-closed. The displacement function determines velocity, acceleration, and jerk through differentiation with respect to cam angle and time.

Uniform velocity is simple but dynamically harsh. Simple harmonic, cycloidal, uniform acceleration–deceleration, and polynomial laws offer different compromises among smoothness, peak acceleration, jerk, and construction effort. A cam profile is generated by indexing follower positions, constructing the pitch curve, and forming the roller envelope where applicable.

Pressure angle governs force transmission; curvature governs profile feasibility and undercutting; contact stress, friction, lubrication, and material properties govern durability. At high speed, dynamic effects such as resonance, spring surge, and follower separation become decisive. Reliable cam design therefore combines kinematics, geometry, dynamics, tribology, materials, manufacturing, and experimental validation.

Key Terms

Cam: A machine element that prescribes follower motion through its profile.

Follower: The output member that remains in contact with a cam.

Cam profile: The actual working contour of a cam.

Pitch curve: Locus of the follower reference point.

Base circle: Smallest circle tangent to the cam profile.

Prime circle: Smallest circle tangent to the pitch curve.

Lift: Maximum follower displacement.

Rise: Follower motion away from its reference position.

Return: Follower motion toward its reference position.

Dwell: Interval of zero follower displacement change.

Pressure angle: Angle between follower motion direction and common contact normal.

Offset follower: Translating follower whose line of stroke does not pass through the cam centre.

Force-closed cam: Cam mechanism using a spring, gravity, or another force to maintain contact.

Form-closed cam: Mechanism that mechanically constrains the follower through a groove or conjugate profile.

Undercutting: Removal or geometric loss of the intended cam profile caused by insufficient curvature.

Jerk: Time rate of change of acceleration.

Pitch radius: Distance from cam centre to a point on the pitch curve.

Cycloidal motion: Motion law based on a cycloidal displacement function.

Contact stress: Local stress developed at the cam–follower contact.

Valve bounce: Unwanted separation and repeated impact in a high-speed valve train.

Review Questions

Short Answer Questions

1. What is the difference between a cam profile and a pitch curve?
2. Define pressure angle and explain why it is important.
3. State two disadvantages of a knife-edge follower.
4. Why is jerk relevant in high-speed cam design?
5. What is meant by undercutting?

Long Answer Questions

1. Classify cams and followers, and explain the practical applications of each major type.
2. Derive the velocity and acceleration expressions for simple harmonic follower motion.
3. Explain the graphical construction of a radial cam profile for a roller follower.
4. Compare uniform velocity, cycloidal, simple harmonic, and polynomial motion laws.
5. Discuss the effects of spring stiffness, moving mass, damping, and cam speed on high-speed follower performance.

Analytical / Application Questions

1. A follower rises 30 mm through 90° of cam rotation at 900 rpm using cycloidal motion. Determine the maximum velocity and acceleration, stating all unit conversions and assumptions.
2. A proposed roller-cam profile has a minimum signed pitch-curve radius of 8 mm in a concave region and a roller radius of 10 mm. Assess the feasibility of the profile and recommend two design changes.
3. A mechanism shows follower separation only during rapid return. Identify the likely kinematic and dynamic causes, and propose a systematic redesign and validation plan.

Multiple Choice Questions with Answers

Question 1

Which component is the output member that remains in contact with the cam?

A. Camshaft

B. Follower

C. Base circle

D. Pitch circle

Correct Answer: B. Follower

Question 2

For constant cam speed, follower velocity is given by which relation?

A. v = ω ds/dθ

B. v = ω² d²s/dθ²

C. v = dθ/ds

D. v = s/ω

Correct Answer: A. v = ω ds/dθ

Question 3

Which motion law has zero acceleration at both ends of the motion interval?

A. Uniform velocity

B. Uniform acceleration

C. Cycloidal motion

D. Constant displacement

Correct Answer: C. Cycloidal motion

Question 4

The pressure angle primarily indicates which design characteristic?

A. Cam material hardness

B. Severity of force transmission and side thrust

C. Lubricant viscosity

D. Shaft diameter only

Correct Answer: B. Severity of force transmission and side thrust

Question 5

A force-closed follower is maintained in contact with the cam by which means?

A. A spring or gravity force

B. A larger base circle only

C. A pitch circle

D. A displacement diagram

Correct Answer: A. A spring or gravity force

Question 6

For a roller follower, the actual cam profile is obtained from the pitch curve by constructing an envelope of circles having which radius?

A. Base-circle radius

B. Prime-circle radius

C. Roller radius

D. Camshaft radius

Correct Answer: C. Roller radius

Question 7

If a cycloidal follower rises 40 mm in 120° at 600 rpm, its maximum velocity is approximately:

A. 0.40 m/s

B. 1.20 m/s

C. 2.40 m/s

D. 4.80 m/s

Correct Answer: C. 2.40 m/s

Question 8

What condition indicates likely undercutting in a concave region of an external roller cam?

A. Pitch-curve radius is much larger than roller radius

B. Roller radius exceeds the local pitch-curve radius

C. Pressure angle is zero

D. Cam speed is constant

Correct Answer: B. Roller radius exceeds the local pitch-curve radius

Question 9

For the simple mass–spring model, the natural circular frequency is:

A. ω_n = k/m_eq

B. ω_n = √(k/m_eq)

C. ω_n = m_eq/k

D. ω_n = k m_eq

Correct Answer: B. ω_n = √(k/m_eq)

Question 10

Which change generally reduces pressure angle, subject to packaging constraints?

A. Reducing the base-circle radius

B. Increasing the base-circle radius

C. Increasing follower mass only

D. Introducing abrupt acceleration changes

Correct Answer: B. Increasing the base-circle radius