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Qwestrum Engineering360 · Mechanical Engineering · Theory of Machines

Cams and Followers

The follower's motion is derived from the displacement diagram: velocity v = (dS/dθ)·ω and acceleration a = (d²S/dθ²)·ω². The pressure angle φ = arctan((dS/dθ − e)/(S + s₀ + e)) must stay small to avoid jamming, per SS Rattan.
Exam tip: use ω in rad/s for dynamics formulas; N in rpm needs ω = 2πN/60. Do not confuse flywheel C_s with governor speed control.

Key formulas & points

Skim these first — then read the full notes below.

  • SHM, uniform acceleration, and cycloidal motion laws compared
  • Undercutting occurs when base circle too small for roller follower
  • Prime circle = smallest circle centred on cam axis touching pitch curve

Topic details

Introduction

Cam design questions in Indian TOM papers ask the student to draw the displacement, velocity, acceleration and jerk (SVAJ) diagrams for a stated motion law, then lay out the cam profile. The motion laws — uniform velocity, SHM, uniform acceleration (parabolic), and cycloidal — differ chiefly in their acceleration and jerk characteristics.

Scope in B.Tech and GATE syllabus

SS Rattan stresses that uniform velocity gives infinite acceleration at start/stop (impact), SHM gives finite acceleration but a jerk discontinuity, and cycloidal motion gives zero acceleration and jerk at the ends — hence its use for high-speed cams.

Why this topic matters in practice

The pressure angle governs side-thrust on the follower; if it exceeds about 30° the follower jams. Increasing the base circle reduces the pressure angle, while too small a base circle causes undercutting for a roller follower — the two competing design limits examiners test.

Key relations & formulas

v=(dSdθ)⋅ωv = (\frac{dS}{d\theta})\cdot \omega
(follower velocity, S = displacement diagram)
a=(d2Sdθ2)⋅ω2a = (d^{2}\frac{S}{d\theta^{2}})\cdot \omega^{2}
(follower acceleration)

Formulas (Indian textbook notation)

  • Pressureangleϕ=arctan((dS/dθ−e)(S+sf+e))Pressure angle \phi = arctan(\frac{(dS/d\theta - e)}{(S + s_{f} + e)})
rc=(d2Sdθ2+S+sf)32/∣d3Sdθ3∣r_{c} = (d^{2}\frac{S}{d\theta^{2}} + S + s_{f})\frac{^{3}}{^{2}} / |d^{3}\frac{S}{d\theta^{3}}|
(radius of curvature)

Notation and sign conventions

Relation 1 —
v=v =
v=(dSdθ)⋅ωv = (\frac{dS}{d\theta})\cdot \omega
(follower velocity, S = displacement diagram)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
a=a =
a=(d2Sdθ2)⋅ω2a = (d^{2}\frac{S}{d\theta^{2}})\cdot \omega^{2}
(follower acceleration)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
Pressureangleϕ=arctanPressure angle \phi = arctan

Formulas (Indian textbook notation)

  • Pressureangleϕ=arctan((dS/dθ−e)(S+sf+e))Pressure angle \phi = arctan(\frac{(dS/d\theta - e)}{(S + s_{f} + e)})
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
rc=r_{c} =
rc=(d2Sdθ2+S+sf)32/∣d3Sdθ3∣r_{c} = (d^{2}\frac{S}{d\theta^{2}} + S + s_{f})\frac{^{3}}{^{2}} / |d^{3}\frac{S}{d\theta^{3}}|
(radius of curvature)
Write this relation with symbols exactly as in SS Rattan — Theory of Machines before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The cam converts rotation θ into follower lift S(θ) defined by the displacement diagram. Differentiating with respect to time via the chain rule gives v = (dS/dθ)ω and a = (d²S/dθ²)ω², so the shape of the lift curve dictates the dynamics.

Governing relations in practice

Simple harmonic motion gives S = (h/2)(1 − cos(πθ/β)), producing a cosine acceleration that is finite but non-zero at the ends — acceptable at moderate speed. Cycloidal motion, S = h(θ/β − sin(2πθ/β)/2π), zeroes acceleration at both ends, eliminating jerk, so high-speed automotive cams use it.

Design and analysis considerations

The pressure angle φ is the angle between the follower motion and the normal to the cam profile; large φ means much of the contact force pushes sideways, bending the follower stem. Keeping φ < 30° is the rule of thumb.

Advanced theory and extensions

Undercutting occurs when the base (or prime) circle is too small relative to the roller radius, so the desired profile cannot physically exist. The designer enlarges the base circle to cure both large pressure angle and undercutting.

Assumptions and validity limits

State assumptions explicitly before using any relation for cams and followers — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Theory of Machines viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Theory of Machines papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to cams and followers.
4. Use equation 1:
v=v =
.
5. Use equation 2:
a=a =
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Cams and Followers appears in linkages, cams, gear trains, and governors. In Indian mechanical curricula this topic is tested because it connects theory to kinematics and kinetics of mechanisms.
GATE and semester exams often combine cams and followers with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use cams and followers?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Forgetting the ω and ω² factors when converting S(θ) to velocity and acceleration
• Choosing uniform-velocity motion for high speed, ignoring its infinite end acceleration
• Measuring the pressure angle from the wrong reference (tangent instead of common normal)
• Using too small a base circle, causing undercutting with a roller follower

Quick revision checklist

Before attempting cams and followers problems, confirm you can:
1. SHM, uniform acceleration, and cycloidal motion laws compared
2. Undercutting occurs when base circle too small for roller follower
3. Prime circle = smallest circle centred on cam axis touching pitch curve
Revise the solved examples in SS Rattan — Theory of Machines and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Peak follower acceleration (SHM)

Problem

A cam gives the follower SHM with total lift h = 40 mm during outstroke over cam angle β = 90° (π/2 rad). Camshaft speed ω = 20 rad/s. Find the maximum acceleration.

Solution

a_max = (h/2)(πω/β)² = (0.040/2)(π×20/(π/2))² = 0.020×(40)² = 0.020×1600 = 32 m/s².

Conceptual check — Cams and Followers

Problem

In a Theory of Machines semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of cams and followers." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Cams and Followers, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    The follower's motion is derived from the displacement diagram: velocity v = (dS/dθ)·ω and acceleration a = (d²S/dθ²)·ω². The pressure angle φ = arctan((dS/dθ − e)/(S + s₀ + e)) must stay small to avoid jamming, per SS Rattan.
  2. 2
    State the relation v = and name each symbol.

    Model answer

    The governing relation is v=v =. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation a = and name each symbol.

    Model answer

    The governing relation is a=a =. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation Pressure angle φ = arctan and name each symbol.

    Model answer

    The governing relation is Pressureangleϕ=arctanPressure angle \phi = arctan. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation r_c = and name each symbol.

    Model answer

    The governing relation is rc=r_{c} =. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: SHM, uniform acceleration, and cycloidal motion laws compared

    Model answer

    SHM, uniform acceleration, and cycloidal motion laws compared — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Undercutting occurs when base circle too small for roller follower

    Model answer

    Undercutting occurs when base circle too small for roller follower — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Prime circle = smallest circle centred on cam axis touching pitch curve

    Model answer

    Prime circle = smallest circle centred on cam axis touching pitch curve — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Forgetting the ω and ω² factors when converting S(θ) to velocity and acceleration?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Choosing uniform-velocity motion for high speed, ignoring its infinite end acceleration?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Measuring the pressure angle from the wrong reference (tangent instead of common normal)?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Using too small a base circle, causing undercutting with a roller follower?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Cycloidal cam gives zero jerk at ends — preferred for high-speed cams.
  • 2
    Avoid: Forgetting the ω and ω² factors when converting S(θ) to velocity and acceleration
  • 3
    Avoid: Choosing uniform-velocity motion for high speed, ignoring its infinite end acceleration
  • 4
    Avoid: Measuring the pressure angle from the wrong reference (tangent instead of common normal)

📖 Standard books (India)

  • Theory of Machines — SS Rattan

    Read: Syllabus unit

    Kinematics, cams, governors, and balancing

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