Cams and followers: profiles and follower motions

Cam and follower types, cam nomenclature and pressure angle, the four standard follower motion laws with their peak velocity and acceleration, jump and undercutting.

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Why it matters

Every four-stroke engine opens and closes its valves with cams, and fuel-injection pumps, distributor points and many machine-tool feeds use them too. The cam profile sets the follower's displacement, and through its derivatives the follower's velocity and acceleration, which decide the valve-spring force, contact stress, noise and whether the follower stays in contact at high speed.

Key ideas

Cam and follower. A cam is a link with a curved surface (or groove) that gives a prescribed motion to a follower by direct contact, a higher pair. Rotary input gives a reciprocating or oscillating output, often with dwells that are hard to get from linkages.

Types of cam.

  • Radial (disc, plate) cam: the follower moves perpendicular to the camshaft axis. Most common, and used for engine valves.
  • Cylindrical (barrel) cam: a groove on a cylinder; the follower moves parallel to the axis.
  • Wedge (translating) cam, face cam, conjugate cam (two cams that drive the follower both ways, no spring).

Types of follower.

  • By contact surface: knife-edge (simple but wears fast, rarely used), roller (rolling contact, low friction and wear, common in diesel injection pumps and roller-tappet valve trains), flat-faced or mushroom (takes high side thrust, used in many overhead-cam engines; needs a convex cam profile everywhere), spherical-faced.
  • By motion: translating (reciprocating) or oscillating (rocker arms).
  • By line of motion: radial (passes through the cam centre) or offset.

Cam nomenclature.

  • Base circle: the smallest circle drawn from the cam centre to the cam profile. Its size sets the overall cam size.
  • Trace point: the knife edge, or the roller centre, whose motion represents the follower's.
  • Pitch curve: the path of the trace point relative to the cam. For a roller follower the actual cam profile is the inner envelope of roller circles drawn on the pitch curve.
  • Prime circle: the smallest circle from the cam centre to the pitch curve; prime circle radius = base circle radius + roller radius.
  • Pressure angle: the angle between the common normal at contact and the direction of follower motion. Large pressure angles produce side thrust that can jam a translating follower; keep it below about 30° for translating roller followers. A larger base circle reduces the pressure angle.

Follower motion programmes (rise h during cam angle β, at constant cam speed ω):

  • Uniform velocity: constant v but infinite acceleration at the start and end of the rise. Used only when modified with parabolic or circular arcs at the ends.
  • Simple harmonic motion (SHM): smooth displacement, finite but suddenly changing acceleration at the ends (infinite jerk at the transitions to dwell).
  • Uniform acceleration and retardation (parabolic): constant acceleration for the first half of the rise and equal constant retardation for the second half. Lowest peak acceleration for a given h and β, but acceleration jumps at the start, middle and end.
  • Cycloidal: acceleration is zero at both ends and varies smoothly, so jerk is finite. Best for high speeds despite a higher peak acceleration.

Contact and jump. A force-closed follower stays on the cam only while the spring force exceeds the inertia force m·a needed during the retarding part of the motion. If it does not, the follower leaves the cam (jump or float), which in an engine means valve float and possible valve-to-piston contact.

Undercutting. For a roller follower, if the radius of curvature of the pitch curve becomes smaller than the roller radius at a convex region, the required cam profile intersects itself and the follower cannot follow the intended motion. Cure: a larger base circle or a smaller roller.

Formulas

Notation for all motion laws: h = total lift (m); β = cam angle for the rise (rad); θ = cam angle from the start of the rise (rad); ω = cam angular speed (rad/s), ω = 2πN/60. Velocity v (m/s) and acceleration a (m/s²) of the follower. The same formulas apply to the return with β replaced by the return angle.

v = h·ω / β (uniform velocity)

s = (h/2)·(1 − cos(π·θ/β)), v_max = π·h·ω / (2β), a_max = π²·h·ω² / (2β²) (SHM; v_max at mid-rise, a_max at the ends)

v_max = 2·h·ω / β, a = 4·h·ω² / β² (uniform acceleration and retardation; a is constant in each half)

s = h·(θ/β − sin(2π·θ/β) / (2π)), v_max = 2·h·ω / β, a_max = 2π·h·ω² / β² (cycloidal; a_max at θ = β/4)

r_prime = r_base + r_roller

  • r_prime: prime circle radius (m); r_base: base circle radius (m); r_roller: roller radius (m).

F_spring ≥ m·a_ret (to avoid jump, ignoring friction and gravity)

  • F_spring: spring force at that instant (N); m: equivalent mass of follower train (kg); a_ret: follower retardation (m/s²).

Worked examples

Example 1 (standard: SHM). A cam rotating at 300 rpm lifts a follower 40 mm with SHM during 120° of cam rotation. Find the maximum velocity and acceleration of the follower during the rise.

  1. ω = 2π × 300 / 60 = 31.42 rad/s; β = 120° = 2.094 rad; h = 0.04 m.
  2. v_max = π·h·ω / (2β) = π × 0.04 × 31.42 / (2 × 2.094) = 3.948 / 4.189 = 0.943 m/s.
  3. a_max = π²·h·ω² / (2β²) = 9.870 × 0.04 × 987.0 / (2 × 4.386) = 389.6 / 8.773 = 44.4 m/s².

Example 2 (GATE level: comparing motion laws). For the same cam (h = 40 mm, β = 120°, 300 rpm), find v_max and a_max for uniform acceleration and retardation and for cycloidal motion. The follower train has an effective mass of 0.8 kg; what minimum spring force is needed at the point of peak retardation for cycloidal motion?

  1. Uniform acceleration and retardation: v_max = 2hω / β = 2 × 0.04 × 31.42 / 2.094 = 1.20 m/s; a = 4hω² / β² = 4 × 0.04 × 987.0 / 4.386 = 36.0 m/s².
  2. Cycloidal: v_max = 2hω / β = 1.20 m/s; a_max = 2π·h·ω² / β² = 6.283 × 0.04 × 987.0 / 4.386 = 56.5 m/s².
  3. Comparison: parabolic 36.0 < SHM 44.4 < cycloidal 56.5 m/s², but only cycloidal has zero acceleration at the ends and finite jerk.
  4. Peak retardation (cycloidal, at θ = 3β/4) is 56.5 m/s². Minimum spring force = m·a = 0.8 × 56.5 = 45.2 N at that position.

Common mistakes

  • Using β in degrees inside the formulas. β must be in radians when ω is in rad/s.
  • Treating the base circle and prime circle as the same for a roller follower. They differ by the roller radius.
  • Saying SHM has smooth acceleration everywhere: it jumps at the start and end of the rise when there are dwells.
  • Assuming the largest peak acceleration is always worst. Cycloidal has the highest peak but the gentlest start, which matters more at high speed.
  • Thinking a bigger base circle changes the lift or the motion law. It reduces the pressure angle and curvature problems; lift is set by the displacement diagram.
  • Checking jump only at the highest speed but at the wrong point: the critical point is where retardation is largest and the spring is least compressed.

For GATE ME

Questions ask for maximum velocity or acceleration of a follower for a given motion law, identification of which law has zero end acceleration or minimum peak acceleration, cam nomenclature (pressure angle, prime circle, trace point), and follower suitability (flat-faced needs a convex profile; roller for high load and speed). Memorise the four motion laws in terms of h, β and ω, and practise converting rpm and degrees correctly.

Quick check

  1. Which follower motion has zero acceleration at the start and end of the rise?
  2. Prime circle radius for a 30 mm base circle and 8 mm roller?
  3. SHM rise of 20 mm over 90° at 600 rpm: v_max?
  4. Which motion law gives the lowest peak acceleration for a given lift and rise angle?
  5. What is the effect of increasing the base circle radius on the pressure angle?

Answers: 1. Cycloidal. 2. 38 mm. 3. π × 0.02 × 62.83 / (2 × 1.571) = 1.26 m/s. 4. Uniform acceleration and retardation. 5. It decreases.

Interview questions

All Theory of Machines and Vibrations interview questions

Try answering each one aloud before you open it.

  1. 1.What is a cam and follower mechanism?Concept

    A cam is a link with a shaped surface or groove that drives a follower through direct point or line contact, a higher pair. Rotation of the cam gives the follower a prescribed reciprocating or oscillating motion, including dwells, that is set entirely by the cam profile. The follower is kept in contact by a spring or gravity (force closure) or by a groove (form closure). The best-known example is the camshaft opening and closing an engine's valves.

  2. 2.Explain the different types of cams.Concept

    A radial (disc or plate) cam has its profile on the edge of a disc and moves the follower perpendicular to the camshaft axis; it is the most common type and drives engine valves. A cylindrical (barrel) cam has a groove around a cylinder and moves the follower parallel to the axis, as in some machine-tool feeds. A wedge or translating cam reciprocates instead of rotating, and a conjugate cam pair drives the follower in both directions without a spring.

  3. 3.What are the different types of follower motions?Concept

    The standard motion laws are uniform velocity, simple harmonic, uniform acceleration and retardation (parabolic) and cycloidal. Uniform velocity gives infinite acceleration at the ends, so it is used only with modified ends. SHM has finite acceleration that jumps at the start and end of the rise. Parabolic motion gives the lowest peak acceleration (4hω²/β²) but jumps in acceleration at the start, middle and end. Cycloidal motion has zero acceleration at both ends and finite jerk, which makes it best for high speeds.

  4. 4.Why is a cycloidal motion profile preferred in cam and follower mechanisms?Application

    Cycloidal motion starts and ends the rise with zero velocity and zero acceleration, and its acceleration varies smoothly, so the jerk is finite everywhere. That avoids the sudden force changes that excite vibration of the follower and its spring, cause noise and wear, and promote follower jump at high speed. Its peak acceleration, 2πhω²/β², is higher than for SHM or parabolic motion, but at high speed the smoothness matters more than the peak.

  5. 5.What happens if a cam profile is not properly designed?Application

    A poorly designed profile can give a pressure angle that is too large, so the follower jams or wears its guide from side thrust; it can undercut, where a roller cannot follow a sharp convex region and the intended motion is lost; and it can have acceleration discontinuities that cause shock, noise and follower jump. In an engine, jump means valve float, lost power and possible valve-to-piston contact. The cures are the right motion law, a large enough base circle, a suitable roller size and an adequate spring.

  6. 6.How does the choice of follower type affect the cam design?Application

    A knife-edge follower can follow any profile but wears quickly, so it is rarely used. A roller follower has rolling contact with low friction and wear, but the pitch curve's radius of curvature must exceed the roller radius in convex regions to avoid undercutting, and its prime circle is the base circle plus roller radius. A flat-faced follower tolerates large side thrust and has zero pressure angle for a radial follower, but the cam profile must be convex everywhere. So the follower choice sets limits on base circle size and profile curvature.

  7. 7.A cam rotating at 120 rpm lifts a follower 10 mm with simple harmonic motion during 90° of cam rotation. Find the follower displacement after 45° of rotation and the maximum velocity during the rise.Numerical

    For SHM, s = (h/2)(1 − cos(πθ/β)). At θ = 45° and β = 90°, s = 5 × (1 − cos 90°) = 5 mm. The maximum velocity is v_max = πhω/(2β) with ω = 2π × 120/60 = 12.57 rad/s and β = π/2 rad: v_max = π × 10 × 12.57 / π = 125.7 mm/s, at mid-rise. SHM is defined in terms of cam angle, not as sin(ωt) of the cam speed.

  8. 8.What is the effect of increasing the base circle diameter of a cam on the follower motion?Application

    A larger base circle does not change the lift or the motion law, which are set by the displacement diagram. It makes the profile less steep, so the pressure angle and side thrust on the follower fall, and it increases the profile's radius of curvature, which reduces contact stress and the risk of undercutting with a roller follower. The cost is a bigger, heavier cam with higher sliding velocity at the contact.

  9. 9.Explain the significance of dwell in cam and follower mechanisms.Concept

    A dwell is a period of cam rotation during which the follower stays still; on the cam it is a circular arc concentric with the cam centre. In an engine the valve dwells closed while the follower rides on the base circle, which keeps the valve seated for compression and power strokes. Dwells are a main reason to use cams, because a linkage cannot easily give an exact stationary period, and the transition into and out of a dwell is where the choice of motion law matters most.

  10. 10.A cam rotating at 1500 rpm lifts a follower 15 mm with uniform acceleration and retardation during 180° of cam rotation. Calculate the maximum velocity of the follower.Numerical

    ω = 2π × 1500 / 60 = 50π = 157.1 rad/s and β = π rad. For uniform acceleration and retardation the maximum velocity, at mid-rise, is v_max = 2hω/β = 2 × 15 × 157.1 / π = 1500 mm/s = 1.5 m/s. The acceleration is 4hω²/β² = 4 × 0.015 × 24 674 / 9.87 = 150 m/s² in each half.

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