Whirling speed refers to the rotational speed at which a shaft exhibits a resonance phenomenon known as whirling or critical speed. At this speed, the natural frequency of the shaft's transverse vibration coincides with the frequency of the rotating shaft, resulting in excessive deflection and potential failure.

Here's a detailed explanation of why the whirling speed of a shaft coincides with the natural frequency of its transverse vibration:

1. Understanding Whirling Speed:
- Whirling speed is a critical phenomenon that occurs when the natural frequency of the shaft's transverse vibration matches the frequency of the rotating shaft.
- When this happens, the centrifugal forces acting on the rotating shaft can cause it to deflect excessively, leading to vibration and potential failure.

2. Shaft Vibration Modes:
- A shaft can vibrate in different modes: longitudinal, transverse, and torsional.
- Longitudinal vibration refers to the back-and-forth motion along the axis of the shaft.
- Transverse vibration refers to the up-and-down or side-to-side motion perpendicular to the axis of the shaft.
- Torsional vibration refers to the twisting motion of the shaft around its axis.
- Coupled bending torsional vibration involves a combination of bending and twisting motion.

3. Natural Frequencies:
- Every shaft has a set of natural frequencies at which it tends to vibrate.
- These natural frequencies depend on various factors such as the shaft's length, material properties, and boundary conditions.
- Each vibration mode (longitudinal, transverse, torsional) has its own set of natural frequencies.

4. Resonance Phenomenon:
- Resonance occurs when an external force or excitation matches the natural frequency of a vibrating system.
- In the case of a rotating shaft, the excitation is the rotational speed, and the natural frequency is the frequency of the transverse vibration mode.
- When the rotational speed coincides with the natural frequency of the transverse vibration, resonance occurs, resulting in excessive deflection and potential failure.

5. Whirling Speed and Transverse Vibration:
- Whirling speed specifically refers to the rotational speed at which the natural frequency of the shaft's transverse vibration is matched.
- At this speed, the transverse vibration mode of the shaft is most susceptible to resonance and excessive deflection.
- It is important to avoid operating a shaft at or near its whirling speed to prevent vibration-related failures.

In conclusion, the whirling speed of a shaft coincides with the natural frequency of its transverse vibration. This is because when the rotational speed matches the natural frequency of the transverse vibration mode, resonance occurs, leading to excessive deflection and potential failure.

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