Design of Flywheels

Why it matters

Flywheels are crucial in mechanical systems for storing rotational energy and regulating speed fluctuations. They are widely used in engines, turbines, and other machinery to ensure smooth operation and energy efficiency.

Key ideas

• Function of Flywheels: Flywheels store energy in the form of rotational kinetic energy. They help in limiting cyclic speed variation by compensating for fluctuations in energy supply or demand.
• Energy Storage: The energy stored in a flywheel is proportional to the square of its rotational speed and its moment of inertia.
• Design Considerations: Key factors include material selection, size, shape, and the maximum allowable stress. The design must ensure that the flywheel can withstand the stresses during operation without failure.
• Applications: Commonly used in automotive engines, power plants, and industrial machines to smooth out the delivery of power.

Formulas

• E = 0.5 · I · ω²
  • E: Energy stored in the flywheel (Joules)
  • I: Moment of inertia of the flywheel (kg·m²)
  • ω: Angular velocity (rad/s)
• I = 0.5 · m · r² for a solid disc
  • m: Mass of the flywheel (kg)
  • r: Radius of the flywheel (m)

Worked example

Given: A solid disc flywheel with a mass of 50 kg and a radius of 0.5 m rotates at 3000 rpm.

1. Convert angular velocity from rpm to rad/s:
   • ω = 3000 · (2π / 60) = 314.16 rad/s
2. Calculate the moment of inertia:
   • I = 0.5 · m · r² = 0.5 · 50 · (0.5)² = 6.25 kg·m²
3. Calculate the energy stored:
   • E = 0.5 · I · ω² = 0.5 · 6.25 · (314.16)² = 308,425.14 J

Final Answer: 308,425.14 Joules

For speed limits ω_max and ω_min, the usable cyclic energy swing is ΔE = I(ω_max² - ω_min²)/2. With ω_mean = (ω_max+ω_min)/2 and C_s = (ω_max-ω_min)/ω_mean, ΔE = I ω_mean² C_s. Determine ΔE from the turning-moment diagram; it is not the entire stored energy. The solid-disc inertia assumes uniform density and rotation about its central axis. A flywheel smooths speed within a cycle; a governor adjusts input to control mean speed when load changes.

Common mistakes

• Unit Conversion Errors: Failing to convert rpm to rad/s correctly.
• Neglecting Material Limits: Ignoring the material's stress limits can lead to design failure.
• Incorrect Moment of Inertia Calculation: Using the wrong formula for different shapes of flywheels.

For GATE ME

Questions often involve calculating the energy stored in a flywheel, determining the moment of inertia, or analyzing the effects of speed changes. Practice problems on unit conversions and understanding the relationship between mass, radius, and energy storage.

Quick check

1. What is the primary function of a flywheel?
2. How is the energy stored in a flywheel related to its speed?
3. What is the formula for the moment of inertia of a solid disc?

Answers: 1. To store rotational energy and regulate speed. 2. Proportional to the square of its speed. 3. I = 0.5 · m · r².