Lesson 3.2: Dynamics of Machines (Force Analysis, Flywheel, Governors)

Dynamics of Machines deals with forces, motion, and energy in mechanical systems. GATE and PSU exams often test force analysis, flywheel design, and governor operation.

🔹 1. Force Analysis of Mechanisms

• Purpose: Determine forces on links, joints, and shafts under moving loads.

• Types of Forces:

  1. Inertia Forces: Due to acceleration of masses ($F=ma$)

  2. Gravity Forces: Due to weight of components

  3. Frictional Forces: At bearings or sliding contacts

• Free Body Diagram (FBD): Essential tool to analyze forces in each link.

Applications: Slider-crank, connecting rods, engine mechanisms

🔹 2. Flywheel

• Definition: Rotating mass storing kinetic energy to smooth out fluctuations in speed.

• Applications: Engines, punching machines, presses

• Kinetic Energy of Flywheel:

KE=12Iω2KE = \frac{1}{2} I \omega^2KE=21​Iω2

Where:

• $I$ = Mass moment of inertia

• $\omega$ = Angular velocity

• Fluctuation of Speed:

δ=ωmax−ωminωmean×100%\delta = \frac{\omega_{\text{max}} – \omega_{\text{min}}}{\omega_{\text{mean}}} \times 100\%δ=ωmean​ωmax​−ωmin​​×100%

• Design Considerations:

  1. Maximum torque & energy variation

  2. Material strength & weight

  3. Diameter & rim thickness

Example:
Engine develops 1000 N·m torque, flywheel mass = 200 kg, radius = 0.5 m. Find maximum speed fluctuation.

• Use KE=1/2Iω2KE = 1/2 I \omega^2KE=1/2Iω2, I=mr2I = m r^2I=mr2 for solid rim flywheel

🔹 3. Governors

• Definition: Device to regulate engine speed automatically.

• Types:

  1. Centrifugal Governors:

     • Watt governor, Porter governor, Proell governor

     • Speed regulation depends on radius of rotation of balls

  2. Inertia Governors: Use motion of masses to control speed

• Parameters:

  • Sensitivity: Change in speed for given change in load

  • Stability: Resistance to hunting/oscillation

  • Control Force: Maintains balance between centrifugal & gravity forces

• Equation for Watt Governor:

ω2r=gtan⁡θ\omega^2 r = g \tan \thetaω2r=gtanθ

Where:

• $\omega$ = angular speed

• $r$ = radius of ball path

• $\theta$ = arm angle

Applications: Steam engines, turbines, pumps

🔹 4. Solved Examples (PYQ Style)

Example 1 (GATE ME 2016):
Flywheel with I = 500 kg·m², ω_mean = 50 rad/s, torque fluctuation ΔT = 2000 N·m. Find maximum speed variation.
👉 Solution: Use $\Delta KE=\Delta T\cdot \theta$, then δ=(ωmax−ωmin)/ωmean\delta = (\omega_{max}-\omega_{min})/\omega_{mean}δ=(ωmax​−ωmin​)/ωmean​

Example 2 (PSU Exam):
Centrifugal governor, ball radius 0.2 m, angle 30°, g = 9.81 m/s². Find angular speed for equilibrium.
ω=gtan⁡θ/r=9.81∗tan⁡30°/0.2≈12.3 rad/s\omega = \sqrt{g \tan \theta / r} = \sqrt{9.81 * \tan 30° / 0.2} ≈ 12.3 \text{ rad/s}ω=gtanθ/r​=9.81∗tan30°/0.2​≈12.3 rad/s

🔹 5. Practice Exercises

1. Draw FBD of a slider-crank mechanism and compute inertia forces on connecting rod.

2. A flywheel of radius 0.5 m stores 10 kJ of energy. Find mass if ω = 100 rad/s.

3. Watt governor: r = 0.15 m, θ = 25°, find speed of rotation.

4. Compute speed fluctuation of flywheel for given torque variation in a punching machine.

5. Compare sensitivity of Porter vs Watt governor for same engine speed.

🔹 6. Summary

• Force Analysis: Determines forces in links/joints → essential for design

• Flywheel: Stores energy, reduces speed fluctuation, KE = 1/2 Iω²

• Governors: Maintain engine speed → centrifugal, inertia types

• Key Equations:

  • F = ma (link forces)

  • ΔKE = Torque fluctuation × angular displacement

  • ω² r = g tan θ (governor)