For a flywheel, if $\omega_{Max}$ is the maximum speed and $\omega_{Min}$ is the minimum speed, then the coefficient of fluctuation of speed will be:
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Remember the basic concept: Fluctuation divided by Mean. Since the mean speed has a "2" in its denominator ($\frac{a+b}{2}$), it moves to the numerator when you divide, resulting in the factor of 2 at the top.
1. Definition and Mathematical Formula: The coefficient of fluctuation of speed is defined as the ratio of the maximum fluctuation of speed to the mean speed of the cycle.
• Maximum Fluctuation of Speed: This is the difference between the maximum and minimum speeds: $(\omega_{Max} - \omega_{Min})$.
• Mean Speed ($\omega_{mean}$): This is the average of the maximum and minimum speeds: $\omega_{mean} = \frac{\omega_{Max} + \omega_{Min}}{2}$.
2. Deriving the Coefficient ($C_s$): Substituting these into the ratio:
$$C_s = \frac{\omega_{Max} - \omega_{Min}}{\omega_{mean}}$$
$$C_s = \frac{\omega_{Max} - \omega_{Min}}{\frac{\omega_{Max} + \omega_{Min}}{2}}$$
$$C_s = \frac{2(\omega_{Max} - \omega_{Min})}{\omega_{Max} + \omega_{Min}}$$
3. Engineering Significance: A smaller value of $C_s$ indicates a more stable machine with less speed variation. For instance, in internal combustion engines, the flywheel must limit speed fluctuations to ensure smooth power delivery and prevent structural vibrations.