Shaft Critical Speed Calculation for Multiple Rotors

When a shaft supports multiple rotors, its vibration behavior becomes more complex than that of a shaft with a single rotor. The interaction between the rotors affects the shaft's whirling motion, leading to coupled critical speeds. Resonance occurs when the operating speed matches a critical speed, potentially causing large amplitude vibrations.

The question provides the natural frequencies ($f_1$ and $f_2$) of the system considering each rotor separately. For a system with two rotors, the lowest critical speed ($f_{c1}$) can be estimated using the following relationship, derived from considering the system's flexibility coefficients:

$$ \frac{1}{f_{c1}^2} = \frac{1}{f_1^2} + \frac{1}{f_2^2} $$

Where:

• $f_{c1}$ is the lowest critical speed (in Hz).
• $f_1$ is the natural frequency considering only the first rotor (100 Hz).
• $f_2$ is the natural frequency considering only the second rotor (200 Hz).

Calculating the Lowest Critical Speed

Let's substitute the given values into the formula:

$$ f_1 = 100 \text{ Hz} $$ $$ f_2 = 200 \text{ Hz} $$

Now, calculate $f_{c1}$:

$$ \frac{1}{f_{c1}^2} = \frac{1}{(100 \text{ Hz})^2} + \frac{1}{(200 \text{ Hz})^2} $$

$$ \frac{1}{f_{c1}^2} = \frac{1}{10000 \text{ Hz}^2} + \frac{1}{40000 \text{ Hz}^2} $$

To add these fractions, find a common denominator:

$$ \frac{1}{f_{c1}^2} = \frac{4}{40000 \text{ Hz}^2} + \frac{1}{40000 \text{ Hz}^2} $$

$$ \frac{1}{f_{c1}^2} = \frac{5}{40000 \text{ Hz}^2} $$

Now, solve for $f_{c1}^2$:

$$ f_{c1}^2 = \frac{40000 \text{ Hz}^2}{5} $$ $$ f_{c1}^2 = 8000 \text{ Hz}^2 $$

Take the square root to find $f_{c1}$:

$$ f_{c1} = \sqrt{8000 \text{ Hz}^2} $$ $$ f_{c1} \approx 89.44 \text{ Hz} $$

Converting Critical Speed to RPM

Critical speeds are often expressed in revolutions per minute (RPM). To convert the frequency from Hz to RPM, multiply by 60 (since 1 Hz = 1 cycle/second and there are 60 seconds in a minute):

$$ \text{Critical Speed (RPM)} = f_{c1} \times 60 $$

$$ \text{Critical Speed (RPM)} \approx 89.44 \text{ Hz} \times 60 \frac{\text{seconds}}{\text{minute}} $$ $$ \text{Critical Speed (RPM)} \approx 5366.4 \text{ RPM} $$

Rounding this value gives 5367 RPM.

Result Comparison

Comparing the calculated value with the given options:

The calculated lowest critical speed is approximately 5367 RPM, which matches Option 2.