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: For a shaft carrying a rotor, the natural whirling frequency can also be expressed in terms of the static deflection \( \delta \) the shaft would show under the rotor's own weight: \( \omega_n = \sqrt{\dfrac{g}{\delta}} \).
Step 2: This static deflection itself depends directly on both the supported mass (through the weight \( mg \)) and the shaft's stiffness \( k \), since \( \delta = \dfrac{mg}{k} \).
Step 3: Substituting this back:
\[ \omega_n = \sqrt{\dfrac{g}{mg/k}} = \sqrt{\dfrac{k}{m}} \]
confirming that critical speed depends on both mass \( m \) and stiffness \( k \).

Answer 2

A rotating shaft with a mounted rotor behaves dynamically much like a simple spring-mass oscillator, where the shaft acts as the spring (with stiffness \( k \)) and the rotor acts as the suspended mass \( m \). Just as changing either the spring or the mass changes a spring-mass system's natural frequency, changing either the shaft's stiffness or the rotor's mass changes the shaft's critical speed. Checking the options with this analogy:

1. Mass: In a spring-mass analogy, changing only the mass while keeping the spring the same does change the frequency, but changing only the spring (stiffness) while keeping the mass the same also changes it, so mass by itself cannot be the complete answer.
2. Stiffness: Similarly, stiffness alone cannot fully determine the frequency, since the same spring supporting different masses vibrates at different frequencies.
3. Mass and stiffness: Exactly as in the spring-mass analogy, both properties together set the natural frequency (and hence the critical, resonant speed) of the shaft-rotor system.
4. Stiffness and eccentricity: Eccentricity has no counterpart in the basic spring-mass natural-frequency relation at all; it affects the amplitude of the response once running near critical speed, not the resonant speed itself.

The spring-mass analogy confirms critical speed is governed jointly by mass and stiffness.

Therefore, the correct answer is mass and stiffness.