Question

The critical speed of the shaft is affected by the

• A. Diameter and the eccentricity of the shaft
• B. Span and the eccentricity of the shaft
• C. Diameter and the span of the shaft
• D. Frequency of vibrations

Hint:

Critical speed of a shaft depends entirely on its structural stiffness and mass distribution: - Higher diameter ($d \uparrow$) $\implies$ Stiffness ($I \uparrow$) $\implies$ Critical speed $\uparrow$ - Longer span ($L \uparrow$) $\implies$ Stiffness $\downarrow$ $\implies$ Critical speed $\downarrow$ - Eccentricity influences vibration amplitude, but does not alter the critical speed value itself.

Answer 1

The Correct Option is C

The question asks about the factors that affect the critical speed of a shaft. The critical speed of a shaft is defined as the speed at which the shaft starts to vibrate dangerously in resonance due to its natural frequency.

The correct option given is Diameter and the span of the shaft. Let's understand why this option is correct step-by-step.

1. Critical Speed Formula:

   The critical speed (N_c) for a shaft is usually calculated using the Rayleigh-Ritz or Dunkerley's method. One basic theoretical formula considering deflection is:

   N_c = (1 / 2π) √(g / δ)

   where g is the acceleration due to gravity, and δ is the static deflection of the shaft typically affected by material properties and geometry.

2. Effect of Diameter:

   The diameter of the shaft influences its stiffness. A larger diameter increases the moment of inertia I, which increases shaft stiffness. Thus, higher stiffness can increase the critical speed because the shaft will have a higher natural frequency.

3. Effect of Span:

   The span is the length between supports of the shaft. A longer span results in lower stiffness and hence a decrease in the critical speed because the natural frequency decreases as the span increases.

4. Exclusion of Other Options:
   • Eccentricity of the Shaft: Although eccentricity can contribute to unbalance forces causing vibrations, it primarily affects the amplitude of oscillations rather than directly influencing the critical speed itself.
   • Frequency of Vibrations: The natural frequency, rather than any arbitrary frequency of vibrations, is what determines the critical speed. The critical speed is reached when the shaft's natural frequency matches the rotational speed.

Thus, from the analysis above, it is evident that the primary factors affecting the critical speed of a shaft are the diameter and the span of the shaft.