Cams and followers

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

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Why it matters

Cams give a machine an exactly timed motion that would need a complicated linkage otherwise: engine valves, automatic lathe tool slides, packaging and printing machines, and indexing drives. The follower motion chosen for the cam decides the peak velocity, the inertia force, the spring needed to keep contact and the noise and wear at speed.

Key ideas

Parts and classification.

  • Cams: radial (disc/plate) cam, cylindrical (drum) cam, wedge or translating cam, globoidal cam.
  • Followers by contact surface: knife-edge (simple, but high contact stress and wear), roller (rolling instead of sliding, less wear), flat-faced/mushroom (compact, small side thrust, needs a convex cam), spherical-faced.
  • Followers by motion: translating (reciprocating) or oscillating. A translating follower is radial when its axis passes through the cam centre and offset otherwise.
  • Contact is kept either by a spring/gravity (force-closed) or by a groove or conjugate cam (form-closed).

Cam nomenclature.

  • Base circle: smallest circle drawn from the cam centre that touches the cam profile.
  • Trace point: the follower point used to generate the pitch curve (knife edge, or roller centre).
  • Pitch curve: path of the trace point relative to the cam; prime circle: smallest circle from the cam centre touching the pitch curve (base circle radius + roller radius for a roller follower).
  • Lift (stroke) h: maximum follower travel. Angle of ascent, dwell, descent: cam angles for rise, rest and return.
  • Pressure angle φ: angle between the direction of follower motion and the common normal at contact. A large pressure angle increases side thrust and can jam a translating follower; designers usually keep it below about 30° by using a larger base circle or an offset.

Displacement diagrams (follower motions). For a rise h over cam angle β with the cam at constant ω:

  • Uniform velocity: simplest, but velocity jumps at the start and end, so acceleration is theoretically infinite there. Used only with modified ends.
  • Simple harmonic motion (SHM): s = (h/2)(1 − cos(πθ/β)). Finite acceleration within the rise, but acceleration jumps from zero to its peak at the start, so jerk is infinite there (if there is a dwell).
  • Uniform acceleration and retardation (parabolic): constant acceleration for the first half of the rise and equal retardation in the second half. It gives the smallest peak acceleration for a given h, β and ω, but acceleration still changes abruptly.
  • Cycloidal: s = h(θ/β − sin(2πθ/β)/(2π)). Acceleration is zero at both ends and continuous, so jerk stays finite. Preferred for high-speed cams.

Contact with the follower. When the inertia force of the follower and attached parts during retardation exceeds the spring force, the follower leaves the cam (jump). A stiffer or more preloaded spring, a lower peak retardation (choice of motion, larger β) or a positive-drive cam prevents it.

Undercutting. For a roller follower, if the radius of curvature of the pitch curve at a convex point is smaller than the roller radius, the profile loops and the cam cannot be cut correctly. A flat-faced follower needs a cam profile that is convex everywhere.

Formulas

ω = 2πN/60; time for the rise t = β/ω (β in rad).

SHM: v_max = π·h·ω/(2β), a_max = π²·h·ω²/(2β²) (at the ends of the stroke).

Uniform acceleration and retardation: v_max = 2·h·ω/β, a = 4·h·ω²/β² (constant magnitude).

Cycloidal: v_max = 2·h·ω/β, a_max = 2π·h·ω²/β² (at quarter and three-quarter points).

Uniform velocity: v = h·ω/β.

Eccentric circular cam with flat-faced radial follower: s = e(1 − cos θ), v = ω·e·sin θ, a = ω²·e·cos θ, an exact SHM.

Symbols: h lift (m), β cam angle for the rise (rad), ω cam angular velocity (rad/s), θ cam angle turned (rad), e eccentricity (m), v follower velocity (m/s), a follower acceleration (m/s²). All assume constant cam speed.

Worked examples

Example 1 (standard). A cam turning at 300 rev/min raises a knife-edge follower 40 mm with SHM during 120° of cam rotation. Find the time of rise, the maximum velocity and the maximum acceleration of the follower.

  1. ω = 2π × 300/60 = 31.42 rad/s; β = 120° = 2.094 rad; h = 0.04 m.
  2. Time of rise: t = β/ω = 2.094/31.42 = 0.0667 s.
  3. v_max = π·h·ω/(2β) = π × 0.04 × 31.42/(2 × 2.094) = 0.942 m/s.
  4. a_max = π²·h·ω²/(2β²) = 9.870 × 0.04 × 987.0/(2 × 4.386) = 44.4 m/s².
  5. t = 0.0667 s, v_max = 0.942 m/s, a_max = 44.4 m/s²

Example 2 (GATE level). A follower must rise 30 mm while the cam turns 90° at 600 rev/min. Compare the peak acceleration with SHM, uniform acceleration and retardation, and cycloidal motion.

  1. ω = 2π × 600/60 = 62.83 rad/s, ω² = 3948 rad²/s²; β = π/2 = 1.571 rad, β² = 2.467; h = 0.03 m.
  2. SHM: a_max = π²·h·ω²/(2β²) = 9.870 × 0.03 × 3948/(2 × 2.467) = 236.9 m/s².
  3. Uniform acceleration and retardation: a = 4·h·ω²/β² = 4 × 0.03 × 3948/2.467 = 192.0 m/s².
  4. Cycloidal: a_max = 2π·h·ω²/β² = 6.283 × 0.03 × 3948/2.467 = 301.6 m/s².
  5. Peak velocity: SHM 1.885 m/s; parabolic and cycloidal both 2hω/β = 2.4 m/s.
  6. Peak accelerations ≈ 237, 192 and 302 m/s². The parabolic motion has the lowest peak, but cycloidal is preferred at high speed because its acceleration starts and ends at zero, which avoids the jerk that excites spring and valve-train vibration.

Common mistakes

  • Using degrees for β and θ in the velocity and acceleration formulas; they must be in radians.
  • Thinking cycloidal motion has the lowest peak acceleration; it has the highest of the three, but no acceleration jumps.
  • Confusing the base circle with the prime circle for a roller follower.
  • Applying SHM formulas to the descent with the rise angle; use the angle of descent.
  • Calling the uniform-velocity cam smooth: its velocity jumps, so ideal acceleration is infinite at the ends.

For GATE PI

Expect follower velocity or acceleration for SHM, parabolic or cycloidal rise at a given speed, displacement of an eccentric cam with a flat follower, identification of the motion with no jerk, and the effect of base circle size and offset on pressure angle. Practise converting rev/min to rad/s and degrees to radians quickly.

Quick check

  1. Which standard follower motion has zero acceleration at both ends of the rise?
  2. An eccentric cam has e = 10 mm and turns at 20 rad/s with a flat-faced follower. What is the peak follower velocity?
  3. How do you reduce the pressure angle of a cam without changing the lift?
  4. A cam dwells for 30° at 600 rev/min. How long is the dwell?

Answers: 1. cycloidal; 2. 0.2 m/s; 3. increase the base circle (or use a suitable offset); 4. 8.33 ms.

Interview questions

All Theory of Machines and Machine Design interview questions

Try answering each one aloud before you open it.

  1. 1.What is a cam and how does it function in a mechanical system?Concept

    A cam is a rotating or sliding piece in a mechanical linkage used especially in transforming rotary motion into linear motion. It is typically a part of a camshaft and works by pushing a follower along a specific path. The shape of the cam determines the motion of the follower, which can be used to open and close valves in engines, among other applications.

  2. 2.Explain the different types of followers used in cam mechanisms.Concept

    Followers in cam mechanisms can be classified based on their shape and motion. Common types include knife-edge followers, roller followers, flat-faced followers, and spherical followers. Knife-edge followers have a sharp edge that contacts the cam, while roller followers use a rolling element to reduce friction. Flat-faced followers have a flat surface in contact with the cam, and spherical followers have a curved surface. Each type has its own advantages and is chosen based on the specific application requirements.

  3. 3.Why are cams used in internal combustion engines?Application

    Cams are used in internal combustion engines to control the timing and movement of the engine's valves. They convert the rotary motion of the camshaft into the linear motion needed to open and close the intake and exhaust valves at the correct times during the engine cycle. This precise control is crucial for the engine's efficiency, performance, and emissions.

  4. 4.What happens if the cam profile is not designed correctly?Application

    A profile with abrupt changes of velocity or acceleration gives very large or infinite acceleration and jerk, which produces shock, noise, vibration of the follower spring and wear. If the peak retardation is too high for the spring force, the follower jumps off the cam. Too small a base circle gives a large pressure angle, so a translating follower can jam, and with a roller follower a pitch-curve radius smaller than the roller radius causes undercutting, so the intended motion cannot be produced.

  5. 5.How does the choice of follower type affect the performance of a cam mechanism?Application

    A knife-edge follower can follow any profile but has very high contact stress and wear, so it is rare. A roller follower replaces sliding with rolling, reducing wear, and is common in stationary engines and machine tools, but its profile can undercut. A flat-faced follower is compact, has little side thrust because the pressure angle is zero for a radial follower, and is used in automotive valve trains, but it needs a cam profile that is convex everywhere.

  6. 6.Explain the term 'dwell' in the context of cam mechanisms.Concept

    In cam mechanisms, 'dwell' refers to a period during the cam's rotation where the follower remains stationary. This occurs when the cam profile is designed to have a flat or constant-radius section, allowing the follower to pause in its motion. Dwell is used in applications where a pause in motion is necessary, such as holding a valve open for a specific duration in an engine.

  7. 7.What is the significance of the base circle in cam design?Concept

    The base circle is the smallest circle, centred on the cam axis, that touches the cam profile; the follower sits on it during the lowest dwell and lift is measured from it. Its size fixes the cam size and largely sets the pressure angle: a larger base circle lowers the pressure angle and side thrust and reduces the risk of undercutting, at the cost of a bigger, heavier cam. For a roller follower the related prime circle has radius equal to base circle radius plus roller radius.

  8. 8.Calculate the lift of a follower if the cam profile has a maximum radius of 50 mm and a base circle radius of 30 mm.Numerical

    The lift of the follower is the difference between the maximum radius of the cam profile and the radius of the base circle. Lift = Maximum radius - Base circle radius = 50 mm - 30 mm = 20 mm. Therefore, the lift of the follower is 20 mm.

  9. 9.What are the potential consequences of excessive wear in cam and follower systems?Application

    Excessive wear in cam and follower systems can lead to increased friction, loss of motion accuracy, and eventual failure of the mechanism. This wear can cause changes in the cam profile, leading to improper timing and movement of the follower. In engines, this can result in reduced efficiency, increased emissions, and potential engine damage. Regular maintenance and proper material selection can help mitigate these issues.

  10. 10.A cam rotates at 600 RPM and has a dwell period of 30 degrees. How long does the follower remain stationary during one rotation?Numerical

    First, calculate the time for one rotation: Time per rotation = 60 seconds / 600 RPM = 0.1 seconds. The dwell period is 30 degrees out of 360 degrees, so the fraction of time the follower remains stationary is 30/360. Time stationary = 0.1 seconds * (30/360) = 0.00833 seconds. Therefore, the follower remains stationary for approximately 0.00833 seconds during one rotation.

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