Short Answer:

The critical speed of a shaft is the speed at which the shaft begins to vibrate violently due to resonance. It occurs when the natural frequency of the shaft coincides with the frequency of rotation. At this speed, even small unbalances in the shaft can produce large amplitude vibrations.

When the shaft runs at its critical speed, it experiences excessive deflection, noise, and stress, which can lead to mechanical failure. Therefore, it is very important to design and operate rotating machines such as turbines, rotors, and engines below or above their critical speed to ensure smooth and safe operation.

Detailed Explanation :

Critical Speed of Shaft

The critical speed of a shaft is the speed at which the rotational frequency of the shaft matches its natural frequency, causing resonance and large amplitude vibrations. This phenomenon occurs mainly in rotating systems such as turbines, compressors, electric motors, and crankshafts. When resonance occurs, the shaft bends excessively, and this condition can lead to severe mechanical failure if not controlled properly.

In other words, the critical speed is the speed at which the shaft tends to rotate with a large whirling motion. The deflection of the shaft increases drastically because the external exciting frequency (due to unbalanced mass or other factors) equals the system’s natural frequency.

1. Concept of Whirling or Critical Speed

When a shaft rotates, it is usually not perfectly balanced because of slight manufacturing defects or uneven mass distribution. This creates a centrifugal force that acts outward and causes the shaft to deflect slightly from its original axis.

At low speeds, this deflection is small and stable. However, as the speed increases, the centrifugal force increases proportionally to the square of the speed. When the rotational speed becomes equal to the shaft’s natural frequency, the deflection becomes very large and the shaft starts to whirl. This speed is called the critical speed or whirling speed of the shaft.

The critical speed depends on factors like the shaft stiffness, mass distribution, and support conditions.

2. Mathematical Expression

Consider a simple shaft carrying a disk at its center. The shaft acts like a spring that provides restoring force when bent.

Let,

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

The natural frequency of the shaft is given by:

Now, the critical speed (N_c) in revolutions per second is:

Or, in revolutions per minute (r.p.m):

This equation shows that the critical speed depends on the deflection (δ) of the shaft under its own weight and the gravitational acceleration (g). A smaller deflection (stiffer shaft) results in a higher critical speed.

3. Factors Affecting Critical Speed

1. Mass Distribution:
   Uneven mass distribution along the shaft causes imbalance, which can lower the critical speed.
2. Shaft Stiffness:
   A stiffer shaft has higher natural frequency, and hence, a higher critical speed. Flexible shafts have lower critical speeds.
3. Bearing Support:
   The type of bearing and its position significantly affect the critical speed. A shaft supported at both ends has a higher critical speed compared to one supported at a single end.
4. Length and Diameter of Shaft:
   A longer shaft or smaller diameter reduces stiffness and thus lowers the critical speed.
5. Type of Loading:
   The critical speed also depends on whether the load on the shaft is distributed or concentrated at a single point.

4. Types of Critical Speeds

There can be more than one critical speed depending on the mode of vibration of the shaft.

• First Critical Speed: The lowest speed at which resonance occurs.
• Higher Critical Speeds: These occur at multiples of the first critical speed but are generally less harmful because the amplitude of vibration is smaller.

Designers focus mainly on the first critical speed to ensure it is well separated from the normal operating speed.

5. Effects of Operating Near Critical Speed

When the shaft runs close to its critical speed, the following problems may occur:

• Excessive vibration and noise.
• Increased bending stress in the shaft.
• Misalignment of bearings and couplings.
• Fatigue failure due to repeated stress cycles.

Therefore, machines are generally designed to operate below 80% or above 120% of the critical speed to avoid resonance and ensure safety.

6. Methods to Increase Critical Speed

1. Increase Shaft Stiffness:
   Use larger diameter or stiffer materials to raise natural frequency.
2. Decrease Shaft Length:
   Shorter shafts are less flexible and have higher critical speeds.
3. Use Proper Balancing:
   Dynamic balancing of rotors reduces unbalanced forces and helps to minimize vibration.
4. Improve Bearing Support:
   Using additional bearings or altering bearing locations can raise the critical speed.
5. Add Dampers:
   Damping devices absorb vibration energy and prevent excessive oscillation near the critical speed.

7. Practical Example

Consider a rotor shaft in a steam turbine. As the turbine accelerates, it passes through different speeds, including the critical speed region. If the shaft is not designed correctly, it may vibrate excessively at this point. To prevent damage, the shaft is designed with suitable stiffness, balancing, and damping so that it can safely pass through the critical speed range during startup and shutdown.

Conclusion:

The critical speed of a shaft is the rotational speed at which resonance occurs due to the matching of the shaft’s natural frequency with its rotational frequency. It is a vital concept in mechanical design as excessive vibration at this speed can lead to damage or failure. Proper design, balancing, and damping are essential to ensure that machinery operates safely either below or above the critical speed range.