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Qwestrum Engineering360 · Mechanical Engineering · Dynamics of Machines

Flywheel

A flywheel stores kinetic energy
E=12Iω2E=\tfrac{1}{2}I\omega^{2}
and smooths cyclic speed fluctuation. Required inertia is
I=ΔEω2CsI=\dfrac{\Delta E}{\omega^{2}C_{s}}
, where
Cs=ωmax⁡−ωmin⁡ωmeanC_{s}=\dfrac{\omega_{\max}-\omega_{\min}}{\omega_{\mathrm{mean}}}
is the coefficient of fluctuation of speed (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.

  • Kinetic energy: E=12Iω2=12I(πN/30)2E=\tfrac{1}{2}I\omega^{2}=\tfrac{1}{2}I(\pi N/30)^{2}.
  • Energy fluctuation (small Δω): ΔE=IωmeanΔω=Iωmean2Cs\Delta E=I\omega_{\mathrm{mean}}\Delta\omega=I\omega_{\mathrm{mean}}^{2}C_{s}.
  • Required inertia: I=ΔEω2CsI=\dfrac{\Delta E}{\omega^{2}C_{s}} (use ωmean\omega_{\mathrm{mean}}).
  • Coefficient of speed fluctuation: Cs=(ωmax⁡−ωmin⁡)/ωmeanC_{s}=(\omega_{\max}-\omega_{\min})/\omega_{\mathrm{mean}} — typically 0.010.01–0.050.05 for engines.
  • Flywheel stores energy during the power stroke and releases it during idle strokes.
  • Rim contributes ~90% of II for a conventional rim-type flywheel.

Topic details

Introduction

Flywheel design is a staple of Indian DOM and machine-design papers, usually posed with a turning-moment diagram of an engine or press. The flywheel absorbs energy when supply exceeds demand and releases it otherwise, limiting the speed swing.

Scope in B.Tech and GATE syllabus

SS Rattan defines the coefficient of fluctuation of energy (C_e, from the turning-moment diagram) and the coefficient of fluctuation of speed (C_s, the allowable speed swing). The maximum fluctuation of energy ΔE is the area between the mean torque line and the actual torque curve.

Why this topic matters in practice

Candidates must not confuse a flywheel (controls cyclic speed variation within a cycle) with a governor (controls mean speed between cycles). The rim carries most of the inertia, so hoop stress σ = ρv² limits the rim speed — a design check examiners like to include.

Key relations & formulas

E=12Iω2=12I(πN30)2E = \frac{1}{2} I \omega^{2} = \frac{1}{2} I (\frac{\pi N}{30})^{2}
(kinetic energy of flywheel)
ΔE=I⋅ωmean⋅Δω\Delta E = I\cdot \omega_{mean}\cdot \Delta\omega
(energy fluctuation, small speed variation)
I=E(ω2⋅Cs)I = \frac{E}{(\omega^{2}\cdot C_{s})}
(required moment of inertia, C_s = coefficient of speed fluctuation)
σmax=(ρ⋅ω2⋅r2)2\sigma_{max} = \frac{(\rho\cdot \omega^{2}\cdot r^{2})}{2}
(hoop stress in rim, thin rim approximation)

Notation and sign conventions

Relation 1 —
E=12Iω2=12IE = \frac{1}{2} I \omega^{2} = \frac{1}{2} I
E=12Iω2=12I(πN30)2E = \frac{1}{2} I \omega^{2} = \frac{1}{2} I (\frac{\pi N}{30})^{2}
(kinetic energy of flywheel)
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 —
ΔE=I⋅ωmean⋅Δω\Delta E = I\cdot \omega_{mean}\cdot \Delta\omega
ΔE=I⋅ωmean⋅Δω\Delta E = I\cdot \omega_{mean}\cdot \Delta\omega
(energy fluctuation, small speed variation)
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 —
I=E/I = E/
I=E(ω2⋅Cs)I = \frac{E}{(\omega^{2}\cdot C_{s})}
(required moment of inertia, C_s = coefficient of speed fluctuation)
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 —
σmax=\sigma_{max} =
σmax=(ρ⋅ω2⋅r2)2\sigma_{max} = \frac{(\rho\cdot \omega^{2}\cdot r^{2})}{2}
(hoop stress in rim, thin rim approximation)
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

Kinetic energy stored is E = ½Iω²; a small speed change Δω changes this by ΔE = I·ω_mean·Δω = I·ω_mean²·C_s, since Δω = ω_mean·C_s. Rearranged, the required inertia is I = ΔE/(ω_mean²·C_s).

Governing relations in practice

ΔE, the maximum fluctuation of energy, is read from the turning-moment (T–θ) diagram as the largest cumulative area above/below the mean torque. Smaller allowable C_s (steadier speed) demands a larger flywheel.

Design and analysis considerations

Most inertia should sit at large radius because I = mr²; hence the rim-heavy design where the rim contributes ~90 % of I. The rim's own rotation creates hoop (centrifugal) stress σ = ρv² = ρω²R², capping the safe rim velocity.

Advanced theory and extensions

The flywheel therefore trades mass and size for speed steadiness. In presses it lets a small motor deliver a large intermittent punching energy; in engines it carries the crankshaft smoothly through the non-power strokes — the applied context worth stating.

Assumptions and validity limits

State assumptions explicitly before using any relation for flywheel — 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 Dynamics 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 Dynamics 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 flywheel.
4. Use equation 1:
E=12Iω2=12IE = \frac{1}{2} I \omega^{2} = \frac{1}{2} I
.
5. Use equation 2:
ΔE=I⋅ωmean⋅Δω\Delta E = I\cdot \omega_{mean}\cdot \Delta\omega
.
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

Flywheel appears in engines, flywheels, and high-speed shafts. In Indian mechanical curricula this topic is tested because it connects theory to balancing, vibration, and rotational dynamics.
GATE and semester exams often combine flywheel with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use flywheel?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Confusing the flywheel (cyclic fluctuation) with the governor (mean-speed regulation)
• Mixing coefficient of fluctuation of energy C_e with that of speed C_s
• Using ω_max or ω_min instead of ω_mean in I = ΔE/(ω²C_s)
• Ignoring rim hoop stress σ = ρv² when checking the maximum rim speed

Quick revision checklist

Before attempting flywheel problems, confirm you can:
1. Flywheel stores energy during power stroke, releases during idle stroke
2. Rim contributes ~90% of I for conventional flywheel design
3.
Cs=(ωmax−ωmin)ωmeanC_{s} = \frac{(\omega_{max} - \omega_{min})}{\omega_{mean}}
— typically 0.01–0.05 for engines
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.

Flywheel moment of inertia

Problem

An engine flywheel must limit speed fluctuation to C_s = 0.02 at mean speed ω = 25 rad/s, absorbing a maximum energy fluctuation ΔE = 2000 J. Find the required moment of inertia.

Solution

I = ΔE/(ω²·C_s) = 2000/(25² × 0.02) = 2000/(625 × 0.02) = 2000/12.5 = 160 kg·m².

Conceptual check — Flywheel

Problem

In a Dynamics of Machines semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of flywheel." 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 Flywheel, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    A flywheel stores kinetic energy E = ½Iω² and smooths speed fluctuation. Its required inertia is I = E_fluctuation/(ω²·C_s), where C_s = (ω_max − ω_min)/ω_mean is the coefficient of fluctuation of speed, following SS Rattan.
  2. 2
    State the relation E = ½ I ω² = ½ I and name each symbol.

    Model answer

    The governing relation is E=12Iω2=12IE = \frac{1}{2} I \omega^{2} = \frac{1}{2} I. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation ΔE = I·ω_mean·Δω and name each symbol.

    Model answer

    The governing relation is ΔE=I⋅ωmean⋅Δω\Delta E = I\cdot \omega_{mean}\cdot \Delta\omega. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation I = E/ and name each symbol.

    Model answer

    The governing relation is I=E/I = E/. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation σ_max = and name each symbol.

    Model answer

    The governing relation is σmax=\sigma_{max} =. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Flywheel stores energy during power stroke, releases during idle stroke

    Model answer

    Flywheel stores energy during power stroke, releases during idle stroke — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Rim contributes ~90% of I for conventional flywheel design

    Model answer

    Rim contributes ~90% of I for conventional flywheel design — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: C_s = (ω_max − ω_min)/ω_mean — typically 0.01–0.05 for engines

    Model answer

    Cs=(ωmax−ωmin)ωmeanC_{s} = \frac{(\omega_{max} - \omega_{min})}{\omega_{mean}} — typically 0.01–0.05 for engines — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Confusing the flywheel (cyclic fluctuation) with the governor (mean-speed regulation)?

    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: Mixing coefficient of fluctuation of energy C_e with that of speed C_s?

    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: Using ω_max or ω_min instead of ω_mean in I = ΔE/(ω²C_s)?

    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: Ignoring rim hoop stress σ = ρv² when checking the maximum rim speed?

    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
    Distinguish flywheel (energy) from governor (speed regulation).
  • 2
    Avoid: Confusing the flywheel (cyclic fluctuation) with the governor (mean-speed regulation)
  • 3
    Avoid: Mixing coefficient of fluctuation of energy C_e with that of speed C_s
  • 4
    Avoid: Using ω_max or ω_min instead of ω_mean in I = ΔE/(ω²C_s)

📖 Standard books (India)

  • Theory of Machines — SS Rattan

    Read: Syllabus unit

    Kinematics, cams, governors, and balancing

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