Question

Critical speed of a shaft depends upon its

• A. mass
• B. stiffness
• C. mass and stiffness
• D. stiffness and eccentricity

Hint:

Higher stiffness increases critical speed, while higher mass reduces it.

Answer 1

The Correct Option is C

Step 1: Understanding critical speed.
Critical speed of a shaft is the speed at which the shaft’s natural frequency of transverse vibration coincides with its rotational speed, leading to resonance and large deflections.
Step 2: Relation with natural frequency.
The critical speed \( \omega_c \) of a shaft is directly related to its natural frequency \( \omega_n \), which for a single degree of freedom system is given by: \[ \omega_n = \sqrt{\frac{k}{m}} \] where \( k \) is the stiffness of the shaft and \( m \) is the mass associated with the shaft.
Step 3: Identifying governing parameters.
From the expression, it is clear that the natural frequency—and hence the critical speed—depends on both the mass and the stiffness of the shaft.
Step 4: Conclusion.
Therefore, the critical speed of a shaft depends on its mass and stiffness.

Answer 2

Critical speed occurs when a rotating shaft's speed matches its natural transverse (whirling) frequency, causing resonance and large deflections. Since this natural frequency for a simple rotor-shaft system is governed by the same physics as any spring-mass system, \( \omega_n = \sqrt{k/m} \), let's check what determines it.

1. Mass: Mass alone is not sufficient, since two shafts with identical mass but very different stiffness (e.g. different shaft diameters or lengths) would have very different natural frequencies and hence different critical speeds; stiffness must also be accounted for.
2. Stiffness: Likewise, stiffness alone is not sufficient, since two shafts of the same stiffness but carrying different rotor masses would whirl at different speeds; the inertia of the rotating mass also matters.
3. Mass and stiffness: The natural frequency formula \( \omega_n = \sqrt{k/m} \) shows that both the shaft's stiffness \( k \) and the mass \( m \) it supports jointly determine the frequency at which resonance (critical speed) occurs; neither quantity alone is enough.
4. Stiffness and eccentricity: Eccentricity (the offset of the mass centre from the shaft axis) affects how large the resulting deflection or vibration amplitude becomes at a given speed, but it does not change the speed at which resonance itself occurs; that speed is set by mass and stiffness, not by how off-centre the mass is.

The natural frequency relation confirms that both mass and stiffness together determine the critical speed.

Therefore, the correct answer is mass and stiffness.