Short Answer:

Whirling speed of a shaft is the speed at which the shaft starts to vibrate violently in a circular manner due to resonance. At this speed, the natural frequency of the shaft becomes equal to the frequency of rotation. When this condition occurs, even a small unbalance in the shaft can produce large vibrations and bending. The whirling speed is also known as the critical speed of the shaft.

In simple terms, when a rotating shaft’s speed coincides with its natural frequency, it starts whirling like a skipping rope. This phenomenon is dangerous because it can lead to bending, fatigue failure, or even complete breakdown of the machine if not controlled. Therefore, engineers design machines to operate safely below or above the whirling speed.

Detailed Explanation:

Whirling Speed of Shaft

The whirling speed of a shaft is the rotational speed at which the shaft tends to vibrate violently in a circular or transverse direction due to resonance. It is also called the critical speed or resonant speed of the shaft. The phenomenon of whirling occurs when the frequency of rotation of the shaft becomes equal to its natural frequency of lateral vibration.

Every rotating shaft has a natural frequency that depends on its stiffness, mass, and boundary conditions. When the shaft rotates at a speed that matches this natural frequency, even a small disturbance or unbalanced mass causes excessive vibrations. These vibrations make the shaft appear as if it is “whirling” or rotating about a deflected axis instead of its original centerline.

Whirling is one of the most important aspects to consider in mechanical design because it directly affects the stability, performance, and safety of rotating machinery such as turbines, engines, and compressors.

Concept of Whirling

When a shaft rotates, it experiences centrifugal forces due to unbalanced masses. If the center of gravity of the shaft is not exactly on the axis of rotation, an eccentricity is created. This eccentric mass causes a periodic force that bends the shaft.

At low speeds, the deflection caused by this force is small and stable. However, as the speed increases, the deflection also increases. When the speed reaches a certain critical value — the whirling speed — the deflection becomes very large because the rotational frequency equals the natural frequency of the shaft.

If the speed is increased further beyond this point, the amplitude decreases again, and the shaft becomes stable once more. This shows that the whirling speed is a dangerous zone and must be avoided during machine operation.

Mathematical Expression for Whirling Speed

Consider a simply supported shaft carrying a small disk of mass   at its center and having eccentricity  .

At equilibrium, the centrifugal force due to the rotating mass is balanced by the restoring force due to shaft stiffness:

Where:

•  = mass of the rotating disk,
•  = angular speed in rad/s,
•  = eccentricity of the mass center,
•  = stiffness of the shaft,
•  = lateral deflection.

From this, the whirling speed   occurs when:

This equation shows that the whirling speed depends on the stiffness and mass of the system.

In terms of natural frequency,

Factors Affecting Whirling Speed

1. Shaft Stiffness:
   A stiffer shaft has a higher natural frequency, which increases the whirling speed.
2. Mass of the Shaft or Attached Disk:
   Heavier shafts have lower whirling speeds because the natural frequency decreases with increased mass.
3. Length of the Shaft:
   Longer shafts are more flexible, leading to lower stiffness and therefore a lower whirling speed.
4. Type of Supports:
   The end conditions (such as fixed, simply supported, or free) affect stiffness and hence alter the whirling speed.
5. Eccentricity of Mass:
   Greater eccentricity increases unbalanced forces, making the shaft reach large vibrations near the critical speed.

Significance of Whirling Speed

Whirling speed plays a crucial role in the design of rotating machinery. Engineers must ensure that the operating speed of the shaft is sufficiently away from its whirling speed. Generally, machines are designed to work below the first critical speed or above the second critical speed.

In practical applications:

• Turbines, rotors, and engines are checked for critical speeds during design and testing.
• Flexible shafts are carefully balanced to minimize eccentricity.
• Damping mechanisms are added to control vibrations near critical speed.

If a machine passes through its whirling speed during start-up or shutdown, the speed must be changed quickly to avoid prolonged resonance.

Effects of Whirling Speed

1. Excessive Vibrations: Causes large lateral deflection of the shaft.
2. Noise and Instability: The system becomes noisy and unstable during resonance.
3. Fatigue and Failure: Repeated bending can cause fatigue cracks and eventual shaft failure.
4. Bearing Damage: High vibration loads can damage bearings and couplings.
5. Reduced Machine Life: Continuous operation near whirling speed shortens component lifespan.

Methods to Control Whirling

1. Balancing of Rotors:
   Proper balancing ensures the center of mass coincides with the center of rotation.
2. Increasing Shaft Stiffness:
   Using larger diameters or stronger materials raises the natural frequency.
3. Reducing Eccentricity:
   Accurate machining and alignment help minimize unbalance.
4. Use of Damping Devices:
   Dampers absorb vibration energy, reducing amplitude.
5. Operating Speed Selection:
   Machines are designed to operate well below or above the whirling speed to ensure safety.

Conclusion:

Whirling speed of a shaft is the rotational speed at which the natural frequency and the rotational frequency become equal, causing resonance and large vibrations. It is a critical condition that must be avoided in mechanical systems to prevent failure and ensure smooth operation. By controlling stiffness, mass, balance, and damping, engineers can design machines that operate safely away from their whirling speed, ensuring better performance and durability.