Theory Of Machines And Design

Step 1: Understanding critical speed.
Critical speed of a shaft is the speed at which the shaft’s natural frequency of transverse vibration coincides with its rotational speed, leading to resonance and large deflections.
Step 2: Relation with natural frequency.
The critical speed of a shaft is directly related to its natural frequency , which for a single degree of freedom system is given by: where is the stiffness of the shaft and is the mass associated with the shaft.
Step 3: Identifying governing parameters.
From the expression, it is clear that the natural frequency—and hence the critical speed—depends on both the mass and the stiffness of the shaft.
Step 4: Conclusion.
Therefore, the critical speed of a shaft depends on its mass and stiffness.

Step 1: Formula for energy stored in a flywheel.
The kinetic energy stored in a flywheel is given by: For a rim type flywheel, moment of inertia .
Step 2: Comparing the two flywheels.
Let the radius of the first flywheel be . Then, radius of the second flywheel is .
Step 3: Ratio of energies.

Step 4: Conclusion.
The energy stored in the second flywheel is one-fourth of that stored in the first flywheel.