Whirling of Rotating Shafts: Understanding Critical Speeds

The question asks us to determine the first critical speed of a simply supported long slender rotating shaft when two nodes are observed at a whirling frequency of 1800 rpm.

Critical Speed Definition

Critical speed of a rotating shaft is the speed at which the shaft's rotational frequency matches one of its natural frequencies of transverse vibration. When the shaft rotates at a critical speed, it can experience large amplitudes of vibration, known as whirling, which can lead to excessive stresses and potential failure.

Understanding Nodes in Shaft Vibration

When a shaft vibrates, certain points along its length may remain stationary or have minimal displacement. These points are called nodes. The number of nodes observed corresponds to the mode shape of vibration.

• For the first mode of vibration (fundamental mode) of a simply supported shaft, there are no nodes between the supports.
• For the second mode, there is one node between the supports.
• For the third mode, there are two nodes between the supports.

In general, for the \(n\)-th mode of vibration of a simply supported beam or shaft, there are \((n-1)\) nodes between the supports.

Relationship Between Critical Speeds and Modes

For a simply supported long slender rotating shaft, the critical speeds are proportional to the square of the mode number. This relationship can be expressed as:

$$\omega_n = n^2 \omega_1$$

Where:

• \(\omega_n\) is the \(n\)-th critical speed.
• \(n\) is the mode number.
• \(\omega_1\) is the first critical speed.

Calculating the First Critical Speed

Given in the problem:

• Number of nodes observed = 2.
• Observed whirling frequency = 1800 rpm.

Step 1: Determine the Mode Number (\(n\))

Since 2 nodes are observed during whirling, this corresponds to the third mode of vibration for a simply supported shaft.

Number of nodes \(= n - 1\)

\(2 = n - 1\)

\(n = 2 + 1\)

\(n = 3\)

So, the observed speed of 1800 rpm is the third critical speed (\(\omega_3\)).

Step 2: Apply the Critical Speed Formula

We use the relationship:

$$\omega_n = n^2 \omega_1$$

Substitute the known values:

$$\omega_3 = 3^2 \omega_1$$

$$1800 \text{ rpm} = 9 \omega_1$$

Step 3: Solve for the First Critical Speed (\(\omega_1\))

$$\omega_1 = \frac{1800}{9}$$

$$\omega_1 = 200 \text{ rpm}$$

Therefore, the first critical speed of the shaft is 200 rpm.

The final answer is 200 rpm.