Understanding Shaft Critical Speed

The question asks us to calculate the critical speed of a vertically mounted shaft. The critical speed, also known as the whirling speed, is the rotational speed at which resonance occurs, leading to large amplitude vibrations.

We are given the following information:

• Mass of the rotor, $m = 200 \text{ g}$
• Stiffness of the shaft at the rotor location, $k = 72000 \text{ N/m}$

The rotor is located at the mid-point of the shaft, but for calculating the critical speed using the given stiffness, we can model the system as a simple mass-spring system.

Critical Speed Calculation Formula

The formula for the critical speed ($\omega_c$) of a system with mass ($m$) and stiffness ($k$) is given by the natural angular frequency:

$$ \omega_c = \sqrt{\frac{k}{m}} $$

Where:

• $ \omega_c $ is the critical speed in radians per second (rad/s).
• $ k $ is the stiffness in Newtons per meter (N/m).
• $ m $ is the mass in kilograms (kg).

Step-by-Step Calculation

1. Convert Mass to Kilograms: The mass is given in grams, so we need to convert it to kilograms.
   $ m = 200 \text{ g} = \frac{200}{1000} \text{ kg} = 0.2 \text{ kg} $
2. Identify Stiffness: The stiffness is provided directly.
   $ k = 72000 \text{ N/m} $
3. Apply the Critical Speed Formula: Substitute the values of $k$ and $m$ into the formula.
   $ \omega_c = \sqrt{\frac{72000 \text{ N/m}}{0.2 \text{ kg}}} $
4. Calculate the Result:
   $ \omega_c = \sqrt{360000 \text{ s}^{-2}} $
5. Final Result:
   $ \omega_c = 600 \text{ rad/s} $

Conclusion

The calculated critical speed of the shaft is 600 rad/s. This corresponds to the speed where the shaft is most likely to experience significant vibrations due to resonance.