Short Answer:

The whirling speed is the speed at which a rotating shaft starts to vibrate violently in a circular or whirling motion due to resonance. It occurs when the natural frequency of the shaft coincides with the frequency of rotation. At this speed, even a small unbalance in the shaft can cause large deflections, bending, and vibrations.

When the shaft rotates at its whirling speed, it becomes unstable and may lead to excessive noise, vibration, or mechanical failure. Hence, machines like turbines, rotors, and engines are designed to operate either below or above their whirling speed to avoid damage.

Detailed Explanation :

Whirling Speed

The whirling speed of a shaft is the speed at which the shaft tends to rotate with a large lateral (sideways) vibration due to resonance. It is also known as the critical speed or resonant speed. This speed occurs when the natural frequency of lateral vibration of the shaft equals the rotational speed of the shaft.

When a shaft is perfectly straight and balanced, it rotates smoothly around its axis. However, in real-world applications, no shaft is perfectly straight or uniform. Small manufacturing errors, uneven weight distribution, or material irregularities cause the shaft’s mass center to be slightly displaced from its geometric center. This displacement results in an unbalanced centrifugal force when the shaft rotates.

At certain speeds, this centrifugal force matches the shaft’s natural frequency of vibration, resulting in resonance. During resonance, the amplitude of vibration increases sharply, and the shaft begins to whirl, forming a conical or circular path instead of rotating straight. This speed is called the whirling speed of the shaft.

1. Basic Concept of Whirling

When a shaft rotates, the unbalanced mass causes centrifugal force acting outward from the axis of rotation. The shaft bends or deflects slightly to balance this force.
At low speeds, the deflection is very small and stable. But as the speed increases, the centrifugal force also increases proportionally to the square of the speed.

At a particular speed, the centrifugal force becomes equal to the restoring force due to the shaft’s stiffness. At this condition, the shaft begins to vibrate with large amplitude. This speed is called the whirling speed or critical speed. Beyond this speed, the amplitude again decreases, and the shaft becomes stable at higher speeds.

2. Mathematical Expression of Whirling Speed

Consider a shaft carrying a disc at its center.
Let,

•  = mass of the disc (kg)
•  = acceleration due to gravity (m/s²)
•  = static deflection of the shaft due to the weight of the disc (m)
•  = angular velocity of rotation (rad/s)

The natural frequency of vibration of the shaft is given by:

When the rotational speed equals the natural frequency, the whirling speed () occurs.

Thus,

The corresponding whirling speed in revolutions per second (N_c) is:

Or in revolutions per minute (r.p.m):

This expression shows that the whirling speed depends on the stiffness (through δ) and the mass distribution of the shaft. A stiffer shaft with smaller deflection will have a higher whirling speed.

3. Factors Affecting Whirling Speed

1. Shaft Stiffness:
   A stiffer shaft resists bending and thus has a higher whirling speed.
2. Shaft Length:
   A longer shaft is more flexible and has a lower whirling speed.
3. Shaft Diameter:
   Increasing the shaft diameter increases stiffness, thereby increasing the whirling speed.
4. Mass Distribution:
   Uneven distribution of mass or eccentricity causes unbalance, reducing the whirling speed.
5. Bearing Support:
   The position and type of bearing support affect the boundary conditions of the shaft, which in turn influence the whirling speed.
   • Shaft with both ends supported has higher whirling speed.
   • Shaft with one end fixed and one free has lower whirling speed.

4. Physical Behavior During Whirling

At the whirling speed, the shaft does not remain straight along its axis of rotation. Instead, it forms a circular or conical shape and rotates around its deflected center.
If the shaft is flexible and runs for a long time at or near its whirling speed, it can lead to:

• Excessive vibration and noise.
• Bending and misalignment of the shaft.
• Loosening of couplings and bearings.
• Mechanical failure due to fatigue stress.

Therefore, the design of rotating machinery must ensure that the operating speed does not coincide with the whirling speed. Generally, machines are designed to operate safely below 80% or above 120% of their whirling speed.

5. Methods to Avoid Whirling

1. Increase Shaft Stiffness:
   By using a larger diameter or high-rigidity materials to increase the natural frequency.
2. Improve Balancing:
   Dynamic balancing of the rotor minimizes unbalanced forces that cause vibration.
3. Add Damping Devices:
   Dampers absorb vibration energy and reduce the amplitude at resonance.
4. Proper Bearing Placement:
   Bearings should be positioned to provide optimum support and reduce bending.
5. Operate Away from Critical Speed:
   Design the system so that the working speed is far from the whirling speed.

6. Real-Life Example

In a steam turbine, as the rotor accelerates during startup, it passes through its whirling speed. During this time, vibrations are temporarily high, but the turbine quickly accelerates past this region. To avoid prolonged resonance, engineers use critical speed analysis and damping mechanisms to ensure safety and stability.

Similarly, in automobile drive shafts, the design ensures that the normal operating speed remains below the whirling speed to prevent excessive vibration.

Conclusion:

The whirling speed is the critical rotational speed at which a shaft experiences resonance, resulting in large amplitude lateral vibrations. It depends on shaft stiffness, mass distribution, and support conditions. Operating near this speed can cause severe vibration and mechanical damage. Hence, it is essential to design and operate rotating machinery either below or above the whirling speed to ensure safe, smooth, and reliable performance.