The critical speed of a rotating shaft is a crucial parameter in the design and operation of machinery, especially for components like the rotor shaft of an electric motor. It represents the rotational speed at which the shaft is likely to vibrate excessively due to resonance with its natural frequency. Understanding and calculating this speed is vital to prevent severe vibrations, damage to bearings, and ultimately, failure of the machinery.

Critical Speed and Static Deflection Relationship

The critical speed of a shaft is inversely proportional to the square root of its static deflection. This relationship is derived from the principle that the natural frequency of a system is related to its stiffness and mass, which, for a rotating shaft, can be linked to the deflection under its own weight.

For a shaft with uniform cross-section and supported at both ends (like knife edges, implying a simply supported beam), the critical speed (\(N_c\)) in revolutions per minute (rpm) can be determined using the following formula, which directly relates it to the static deflection (\(\delta\)) of the shaft under its own weight:

$$ N_c = \frac{60}{2\pi} \sqrt{\frac{g}{\delta}} $$

Where:

• \(N_c\) = Critical speed in revolutions per minute (rpm)
• \(g\) = Acceleration due to gravity (approximately 9.81 m/s²)
• \(\delta\) = Static deflection of the shaft in meters (m)

Rotor Shaft Deflection Data

From the question, we are given the static deflection of the rotor shaft in the middle:

• Static deflection, \(\delta\) = 1.8 mm

To use this value in the formula, we need to convert millimeters to meters:

$$ \delta = 1.8 \text{ mm} = 1.8 \times 10^{-3} \text{ m} = 0.0018 \text{ m} $$

Calculating Critical Speed of the Shaft

Now, we can substitute the given values into the formula for critical speed:

$$ N_c = \frac{60}{2\pi} \sqrt{\frac{g}{\delta}} $$

Substitute \(g = 9.81 \text{ m/s}^2\) and \(\delta = 0.0018 \text{ m}\):

$$ N_c = \frac{60}{2\pi} \sqrt{\frac{9.81}{0.0018}} $$

First, calculate the term under the square root:

$$ \frac{9.81}{0.0018} = 5450 $$

Now, take the square root:

$$ \sqrt{5450} \approx 73.8241 $$

Next, perform the multiplication and division:

$$ N_c = \frac{60}{2\pi} \times 73.8241 $$

Using \(\pi \approx 3.14159\):

$$ N_c = \frac{60}{2 \times 3.14159} \times 73.8241 $$

$$ N_c = \frac{60}{6.28318} \times 73.8241 $$

$$ N_c \approx 9.54929 \times 73.8241 $$

$$ N_c \approx 704.89 \text{ rpm} $$

Rounding this value to the nearest whole number gives 705 rpm.

Motor Shaft Resonance Prevention

It is important for the operating speed of the electric motor to be significantly different from this calculated critical speed (705 rpm) to avoid resonance. Operating an electric motor near its critical speed can lead to large amplitude vibrations, excessive stresses on the shaft and bearings, and potential fatigue failure. Engineers typically design operating speeds to be at least 20-30% away from the critical speed.