Cams and followers: profiles and follower motions

Cam and follower types, cam terminology and pressure angle, and the uniform-velocity, SHM, parabolic and cycloidal follower motions with their peak velocity and acceleration, jump and undercutting.

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Why it matters

A cam turns steady shaft rotation into a precisely timed rise, dwell and return of a follower. Engine valve trains, packaging and printing machines, automatic lathes and textile looms rely on cams, and electronic cam profiles in servo drives use the same motion laws. The motion law you choose sets the follower's peak acceleration, and so its inertia force, noise, wear and the spring needed to keep contact.

Key ideas

Types of cams. Disc (radial) cam: profile on the edge of a plate, follower moves perpendicular to the shaft axis. Cylindrical (drum) cam: groove on a cylinder, follower moves parallel to the axis. Also face (grooved plate) cams and translating (wedge) cams.

Types of followers.

  • By contact: knife-edge (simple, high wear), roller (rolling contact, low wear, the most common), flat-faced or mushroom (low side thrust, compact, used for engine valves; it needs a convex cam profile), spherical-faced.
  • By motion: translating (reciprocating) or oscillating.
  • By line of motion: radial (in-line, passes through the cam centre) or offset.

Terminology (disc cam, roller follower).

  • Base circle: smallest circle tangent to the cam profile, drawn from the cam centre.
  • Trace point: the reference point on the follower (roller centre or knife-edge).
  • Pitch curve: path of the trace point relative to the cam. Prime circle: smallest circle tangent to the pitch curve; prime radius = base radius + roller radius.
  • Lift (stroke) h: maximum follower travel.
  • Rise, dwell, return: the cam angles over which the follower goes out, stays still, and comes back.
  • Pressure angle φ: angle between the common normal at contact (line of force) and the follower's direction of motion. Large φ raises side thrust and can jam a translating follower; keep φ below about 30° for translating roller followers.

Standard follower motions (displacement diagrams). θ is cam angle from the start of the rise, β is the cam angle for the full rise.

  • Uniform velocity: straight-line displacement. Velocity jumps at the start and end, so acceleration is theoretically infinite there. Used only with modified (rounded) ends.
  • Simple harmonic motion (SHM): the follower's displacement is the projection of a point moving uniformly on a semicircle of diameter h. Smooth velocity, but acceleration jumps at the ends of the rise (finite jerk spikes). Good for moderate speeds.
  • Uniform acceleration and retardation (parabolic): constant acceleration for the first half, constant deceleration for the second. It has the lowest peak acceleration of the common laws for a given h and β, but acceleration still jumps three times.
  • Cycloidal: the follower moves like a point on a rolling circle. Acceleration is zero at both ends and continuous throughout, so jerk stays finite. Best for high-speed cams, at the cost of a higher peak acceleration.

Dynamics. Follower inertia force = m·a. During deceleration near the top of the rise, the spring (force closure) must supply this force, or the follower leaves the cam (jump), then lands with impact. A stiffer spring, a lighter follower or a lower peak deceleration cures it; a grooved cam (form closure) avoids it.

Profile limits. A roller follower's radius must be less than the minimum radius of curvature of the convex pitch curve, or the profile is undercut (the follower cannot follow the designed motion). Increasing the base circle reduces both the pressure angle and the risk of undercutting.

Formulas

ω = cam angular velocity (rad/s), h = lift (m), β = rise angle (rad), θ = cam angle from start of rise (rad). Velocities in m/s, accelerations in m/s².

Uniform velocity: s = h·θ/β ; v = h·ω/β ; a = 0 except infinite at the ends.

SHM: s = (h/2)·(1 − cos(πθ/β)) v_max = π·h·ω / (2β) (at mid-rise) ; a_max = π²·h·ω² / (2β²) (at the start and end)

Uniform acceleration and retardation: s = 2h(θ/β)² for θ ≤ β/2 v_max = 2h·ω/β (at mid-rise) ; a = 4h·ω²/β² (constant magnitude)

Cycloidal: s = h·(θ/β − (1/2π)·sin(2πθ/β)) v_max = 2h·ω/β (at mid-rise) ; a_max = 2π·h·ω²/β² (at θ = β/4 and 3β/4)

Pressure angle, in-line translating roller follower: tan φ = (ds/dθ) / (R_p + s), R_p = prime circle radius (m).

Time for a rise: t = β/ω (s).

Worked examples

Example 1 (standard): SHM rise. Given: lift h = 40 mm, rise over 120° of cam rotation, cam speed 300 rpm, SHM.

  1. ω = 2π × 300 / 60 = 31.42 rad/s ; β = 120° = 2.094 rad.
  2. v_max = π·h·ω / (2β) = π × 0.04 × 31.42 / (2 × 2.094) = 0.943 m/s
  3. a_max = π²·h·ω² / (2β²) = 9.870 × 0.04 × 987.0 / (2 × 4.386) = 44.4 m/s²
  4. Rise time t = β/ω = 2.094 / 31.42 = 0.0667 s.
  5. Pressure angle at mid-rise with a prime circle of 50 mm: ds/dθ = πh/(2β) = 0.0300 m/rad, s = 0.020 m, tan φ = 0.0300 / (0.050 + 0.020) = 0.4286 → φ = 23.2°. Answer: v_max ≈ 0.94 m/s, a_max ≈ 44.4 m/s², rise time 66.7 ms, φ ≈ 23° at mid-rise.

Example 2 (GATE level): choosing the motion law and the spring. Same cam (h = 40 mm, β = 120°, 300 rpm). The follower and its parts have mass 2 kg. Compare peak accelerations, and find the minimum spring force needed during deceleration for the cycloidal law (ignore gravity).

  1. Uniform acc. and ret.: a = 4h·ω²/β² = 4 × 0.04 × 987.0 / 4.386 = 36.0 m/s²
  2. SHM: 44.4 m/s² (Example 1).
  3. Cycloidal: a_max = 2π·h·ω²/β² = 2π × 0.04 × 987.0 / 4.386 = 56.5 m/s²
  4. Peak deceleration force (cycloidal), which the spring must supply: F = m·a_max = 2 × 56.5 = 113 N Answer: 36.0, 44.4 and 56.5 m/s² for parabolic, SHM and cycloidal; spring force ≥ 113 N at the point of peak deceleration. The cycloidal law has the highest peak but no acceleration jumps, so it is still preferred at high speed.

Common mistakes

  • Using degrees for β in v and a formulas; β must be in radians.
  • Writing SHM displacement as h·sin θ; it is (h/2)(1 − cos(πθ/β)).
  • Calling cycloidal motion the one with lowest peak acceleration; that is the parabolic law.
  • Taking stroke as the cam's largest radius; lift = maximum radius of the pitch curve − prime radius.
  • Forgetting that a flat-faced follower cannot follow a concave profile.
  • Ignoring the roller radius when checking undercutting.

For GATE ME

Expect questions on: maximum velocity or acceleration of the follower for SHM, parabolic or cycloidal motion; identifying the motion law from a displacement, velocity or acceleration diagram; pressure angle; which follower or motion suits high speed. Practise converting rpm and degrees and memorise the four v_max and a_max results.

Quick check

  1. Which motion law has zero acceleration at the start and end of the rise?
  2. For SHM, where along the rise is the velocity maximum?
  3. A cam turns at 600 rpm with a rise over 90°. How long does the rise take?
  4. What is the prime circle radius for a 30 mm base circle and a 10 mm roller?
  5. What happens if the spring force is less than m·a during deceleration?

Answers: 1. Cycloidal. 2. At mid-rise. 3. (90/360) × 0.1 s = 25 ms. 4. 40 mm. 5. The follower jumps off the cam.

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 and follower mechanism is a type of mechanical system used to convert rotary motion into linear motion. The cam is a rotating or sliding piece in a mechanical linkage, while the follower is a component that moves in response to the cam's motion. This mechanism is commonly used in engines to operate valves.

  2. 2.What are the main types of cams?Concept

    A disc (radial or plate) cam has its profile on the edge of a plate and drives a follower perpendicular to the shaft axis; it is the most common. A cylindrical (drum) cam has a groove cut around a cylinder and drives the follower parallel to the axis, which suits long strokes. Face (grooved plate) cams carry a groove in a disc face and give positive drive in both directions, and translating (wedge) cams move in a straight line instead of rotating. The choice depends on the required follower direction, stroke and whether positive (form-closed) return is needed.

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

    The four standard laws are uniform velocity, simple harmonic motion, uniform acceleration and retardation (parabolic) and cycloidal. Uniform velocity has a straight-line displacement but infinite acceleration at the ends, so it is only used with rounded ends. SHM has smooth velocity but acceleration jumps at the ends; the parabolic law has the lowest peak acceleration, 4hω²/β², but acceleration jumps three times. Cycloidal motion has zero acceleration at both ends and finite jerk, so it is preferred for high-speed cams despite a higher peak acceleration, 2πhω²/β².

  4. 4.Why is a cycloidal motion profile often preferred in cam design?Application

    Cycloidal motion profiles are often preferred in cam design because they provide smooth acceleration and deceleration, reducing the risk of mechanical shock and wear. This type of motion ensures that the follower moves smoothly, which is particularly important in high-speed applications where abrupt changes in motion could lead to mechanical failure.

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

    Typical faults are acceleration jumps that cause shock, noise and vibration; a pressure angle that is too large, giving high side thrust and possible jamming of a translating follower; undercutting when the roller radius exceeds the convex pitch curve's minimum radius of curvature; and follower jump when the spring cannot supply the inertia force during deceleration. The usual fixes are a smoother motion law such as cycloidal, a larger base circle, a smaller roller and a stiffer spring or a grooved cam.

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

    A knife-edge follower can follow any profile but wears fast because of very high contact stress. A roller follower turns sliding into rolling, so friction and wear are low and it suits most industrial cams, but its radius must be smaller than the profile's minimum radius of curvature. A flat-faced (mushroom) follower gives low side thrust and a compact valve train, but it slides on the cam and needs a convex profile. Offsetting or oscillating followers are used to reduce the pressure angle or fit the layout.

  7. 7.Explain the significance of dwell in cam design.Concept

    Dwell in cam design refers to a period during the cam's rotation where the follower remains stationary. This is significant because it allows for specific operations to occur without movement, such as holding a valve open in an engine. Properly designing the dwell period is crucial for ensuring the correct timing and operation of the mechanism.

  8. 8.A follower rises 20 mm with simple harmonic motion over 120° of cam rotation. Find its displacement after the cam has turned 60° from the start of the rise.Numerical

    For SHM, s = (h/2)(1 − cos(πθ/β)), with h = 20 mm, β = 120° and θ = 60°. Then πθ/β = π/2, so s = 10(1 − cos 90°) = 10 mm, half the lift at half the rise angle, as expected by symmetry. A common mistake is to use s = h·sin θ, which ignores the rise angle β.

  9. 9.What are the advantages of using a roller follower over a flat-faced follower?Application

    A roller follower offers several advantages over a flat-faced follower, including reduced friction and wear due to rolling contact instead of sliding contact. This leads to increased efficiency and a longer lifespan for the cam mechanism. Roller followers are particularly beneficial in high-speed applications where minimizing friction is crucial.

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