Short Answer:

Whirling speed of a shaft is the speed at which a rotating shaft begins to vibrate violently in the lateral direction due to resonance. It occurs when the natural frequency of the shaft coincides with the frequency of rotation. At this speed, the shaft experiences large bending and may even break if the vibration is not controlled.

The whirling speed mainly depends on the stiffness, mass, and distribution of the shaft. To avoid failure, machines are designed so that their working speed is always well below or above the whirling speed.

Detailed Explanation:

Whirling Speed of Shaft

Whirling speed of a shaft (also known as critical speed or resonant speed) is the rotational speed at which the shaft tends to bend or vibrate excessively in the lateral direction. At this speed, the centrifugal force acting on the unbalanced mass of the rotating shaft equals the restoring elastic force of the shaft, causing a condition of resonance.

When a shaft rotates, any small unbalance or mass eccentricity produces a centrifugal force. As the speed increases, this force increases rapidly, and the shaft starts bending. When the speed reaches a certain value, known as the whirling speed, the deflection becomes very large, and the shaft may start to whirl or rotate in a bent condition about a displaced axis.

Concept of Whirling or Critical Speed

Every rotating shaft has its own natural frequency of vibration, depending on its material, dimensions, and stiffness. When the speed of rotation matches this natural frequency, resonance occurs. During resonance, even a small unbalance can produce a large deflection, causing the shaft to bend continuously.

At whirling speed, the shaft does not rotate about its geometric center but about its mass center. This creates large stresses, vibrations, and noise. If this speed is sustained, it can lead to mechanical damage or complete failure of the shaft. Therefore, designers always make sure that the shaft operates either below or above the whirling speed range.

Mathematical Expression

For a simple uniform shaft carrying a single disc at its mid-span, the whirling speed (N) is given by:

Where:

•  = whirling speed in revolutions per second
•  = acceleration due to gravity (9.81 m/s²)
•  = static deflection of the shaft due to the weight of the disc

This formula shows that the whirling speed depends on the stiffness (through deflection) and the weight of the rotating parts. A stiffer shaft with less deflection will have a higher whirling speed.

Causes of Whirling

1. Unbalanced Mass:
   If the mass of the shaft or attached parts is not distributed uniformly, centrifugal forces are created when it rotates, causing bending.
2. Eccentric Mounting:
   If the center of gravity of the shaft does not coincide with its geometric axis, unbalanced forces appear, leading to whirling.
3. Flexible Shaft:
   Shafts with low stiffness can easily deflect under centrifugal forces and reach whirling speed even at low rotation.
4. Improper Alignment:
   Misalignment of couplings or bearings introduces bending stresses, increasing the chance of whirling.
5. Resonance Conditions:
   When the rotational speed equals the natural frequency, resonance causes large amplitude vibrations, leading to whirling.

Effects of Whirling Speed

1. Excessive Vibration:
   At whirling speed, vibration amplitude becomes very high, causing noise and instability.
2. Fatigue Failure:
   Continuous bending of the shaft due to vibration may lead to fatigue cracks and eventual breakage.
3. Bearing Damage:
   The lateral motion can overload and damage bearings, reducing their life span.
4. Loss of Efficiency:
   Energy is wasted in vibration and friction, lowering the overall efficiency of the machine.
5. Safety Risk:
   If not controlled, whirling vibration can cause catastrophic mechanical failure, especially in high-speed machines like turbines and rotors.

Methods to Control Whirling Speed

1. Balancing the Shaft:
   Proper balancing of the shaft and attached components ensures uniform mass distribution and reduces unbalanced centrifugal forces.
2. Increasing Shaft Stiffness:
   Using materials with higher modulus of elasticity or increasing the shaft diameter helps raise the whirling speed.
3. Reducing Shaft Length:
   Shorter shafts are less flexible and have higher critical speeds, reducing the chances of whirling.
4. Using Dampers:
   Damping systems help absorb the vibration energy and limit the amplitude during resonance.
5. Operating Below or Above Whirling Speed:
   Machines are designed to operate safely below or above the critical speed range to avoid resonance.

Practical Example

In engines, turbines, and generators, the rotating shafts must pass through their critical speed during acceleration. For example, when starting a turbine, the speed increases from zero to operating speed. It may briefly pass through its whirling speed range, but if the system has sufficient damping and stiffness, the vibration amplitude will remain small. Therefore, careful design and balancing are essential to prevent excessive vibration during operation.

Conclusion

The whirling speed of a shaft is the rotational speed at which it experiences resonance due to the equality of the natural frequency and the frequency of rotation. At this speed, the shaft undergoes large lateral vibrations, which can lead to failure if not controlled. The whirling speed depends mainly on the shaft’s stiffness, mass, and static deflection. Engineers avoid operating machines near this speed by ensuring proper design, balancing, damping, and support, thus maintaining safe and efficient operation of rotating systems.