13. Design of Flywheel free study note

1. What is Flywheel ?

A flywheel is a heavy rotating mechanical element that stores kinetic energy when the machine has excess energy and releases it when the machine requires additional energy. Its primary function is to reduce fluctuations in speed and ensure smooth operation of the machine.

Principle of Working

A flywheel works on the principle of conservation of angular momentum and kinetic energy storage.

• When the driving torque is greater than the resisting torque, the flywheel accelerates and stores energy.
• When the resisting torque is greater than the driving torque, the flywheel slows down and releases the stored energy.
• Thus, it maintains an almost constant rotational speed.

Applications:

• IC Engine
• Steam Engine
• Punch Press
• Rolling mills, compressors

2. Functions of a Flywheel

• Reduces fluctuation in speed
• Stores excess energy
• Supplies energy during peak load
• Ensures smooth operation

3. Energy Fluctuation in Machines

Energy fluctuation in machines is the variation in the energy stored in a rotating machine during one cycle of operation due to the difference between the energy supplied and the energy required.

In many machines such as internal combustion engines, punching machines, shearing machines, and reciprocating compressors, the torque produced by the driving source is not constant throughout the cycle. Similarly, the resisting torque may also vary.

As a result:

• When driving torque > resisting torque, the machine accelerates and gains kinetic energy.
• When driving torque < resisting torque, the machine decelerates and loses kinetic energy.

This continuous gain and loss of kinetic energy is called energy fluctuation.

Why Energy Fluctuation Occurs

• Energy fluctuation occurs because:
• Some operations (e.g., punching, forging) require a large amount of energy in a short time.
• Engine torque varies with crank angle.
• Load torque is not uniform.

4. Turning Moment Diagram

Relation with Work Done

The work done during a small crank movement $d\theta$ is:$$dW = T\,d\theta$$

and the total work done over a cycle equals the area under the turning moment diagram.

The fundamental relation is:

$W=\int T\,d\theta$

• Graph of torque vs crank angle
• Area between mean torque line and curve gives energy fluctuation

5. Maximum Fluctuation of Energy

Maximum Fluctuation of Energy is the maximum difference between the maximum and minimum kinetic energy of a flywheel during one complete cycle of operation.

It represents the largest amount of energy that the flywheel must store and release to maintain nearly uniform speed.

Mathematically,

$$\Delta E = E_{max} – E_{min}$$where:

• $E_{\min}$ = Minimum kinetic energy of the flywheel (J)
• $\Delta E$ = Maximum fluctuation of energy (J)
• $E_{\max}$​ = Maximum kinetic energy of the flywheel (J)
• Represents energy stored/released by flywheel

6. Coefficient of Fluctuation of Energy

The Coefficient of Fluctuation of Energy is the ratio of the maximum fluctuation of energy to the energy stored in the flywheel at its mean speed.

It indicates the percentage of the flywheel’s stored energy that is exchanged during one complete cycle.

Mathematically,

$$C_E = \frac{\Delta E}{E_{mean}}$$

where:

• $C_E$​ = Coefficient of fluctuation of energy (dimensionless)
• $\Delta E$ = Maximum fluctuation of energy (J)
• $E$ = Kinetic energy stored in the flywheel at mean speed (J)

7. Coefficient of Fluctuation of Speed

$$C_s = \frac{N_{max} – N_{min}}{N_{mean}}$$

• Indicates permissible speed variation

8. Energy Stored in Flywheel

$E = \frac{1}{2} I \omega^2$

Where:

• $I$ = mass moment of inertia
• $\omega$ = angular velocity

Energy Fluctuation Relation

$$\Delta E = I \omega^2 C_s$$

9. Design of Flywheel

(a) Determination of Mass Moment of Inertia

$$I = \frac{\Delta E}{\omega^2 C_s}$$

(b) Rim Design

• Most energy stored in rim
• Assume rim carries ~90% of total energy

(c) Stresses in Flywheel Rim

Centrifugal Stress

$$\sigma_c = \rho v^2$$Where:

• $\rho$ = density
• $v$ = peripheral velocity

Bending Stress

• Due to restraint of arms

(d) Design of Arms

• Subjected to bending
• Usually elliptical cross-section

(e) Design of Shaft and Hub

• Shaft designed for torque
• Hub fitted to shaft with key

10. Materials for Flywheel

• Cast Iron (most common)
• Steel (high-speed applications)

Properties required:

• High strength
• Good fatigue resistance
• Good castability

11. Advantages of Flywheel

• Smoothens speed fluctuations
• Improves efficiency
• Reduces vibration

12. Limitations

• Cannot control cyclic speed variations completely
• Adds weight and space
• Initial cost

13. Flywheel vs Governor

Feature	Flywheel	Governor
Function	Controls speed fluctuation	Controls mean speed
Energy	Stores energy	Does not store energy
Application	Cyclic variation	Load variation

14. Numerical Examples

Problem: A flywheel stores 5000 J of kinetic energy at its mean speed. During one cycle, the maximum fluctuation of energy is 250 J. Find the coefficient of fluctuation of energy.

Given:

• Energy stored, $E = 5000\ \text{J}$E=5000 J
• Maximum fluctuation of energy, $\Delta E = 250\ \text{J}$ΔE=250 J

Solution:$$C_E=\frac{\Delta E}{E}$$$$=\frac{250}{5000}$$$$\boxed{C_E=0.05}$$​

Answer: The coefficient of fluctuation of energy is 0.05, or 5%.

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