Understanding Whirling and Critical Speeds of a Rotating Shaft

Whirling is a phenomenon observed in rotating shafts where the shaft axis rotates about the axis of rotation. This occurs due to various factors, including mass unbalance. At certain rotational speeds, known as critical speeds, the shaft can experience large lateral deflections, which can lead to failure. These critical speeds correspond to the natural frequencies of lateral vibration of the shaft.

Critical Speeds and Modes of Vibration

A simply supported long slender rotating shaft behaves similarly to a simply supported beam in lateral vibration. It has different natural frequencies corresponding to different mode shapes of vibration.

• The first critical speed ($\omega_{c1}$) corresponds to the first mode of vibration (n=1). In this mode, the shaft deflects into a single curve with the maximum deflection at the center. For a simply supported shaft, there are nodes only at the supports.
• The second critical speed ($\omega_{c2}$) corresponds to the second mode of vibration (n=2). This mode has one internal node (a point along the shaft's length that does not move) between the supports, in addition to the nodes at the supports.
• The third critical speed ($\omega_{c3}$) corresponds to the third mode of vibration (n=3). This mode has two internal nodes between the supports.
• In general, the n-th mode of vibration for a simply supported shaft has (n-1) internal nodes.

Relating Critical Speeds to Mode Number

For a simply supported shaft, the critical speeds are proportional to the square of the mode number. The relationship between the n-th critical speed ($\omega_{cn}$) and the first critical speed ($\omega_{c1}$) is given by:

$$ \omega_{cn} = n^2 \omega_{c1} $$

This means the second critical speed is $2^2 = 4$ times the first critical speed, the third critical speed is $3^2 = 9$ times the first critical speed, and so on.

Solving the Problem

The problem states that two nodes are noticed during whirling at a frequency of 1800 rpm. Since the shaft is simply supported, the nodes at the supports are always present. The "two nodes noticed" refers to the internal nodes. Therefore, the mode of vibration has 2 internal nodes.

• Number of internal nodes = (n-1)
• Given internal nodes = 2
• So, (n-1) = 2, which means n = 3.

This indicates that the whirling observed at 1800 rpm corresponds to the third critical speed ($\omega_{c3}$).

We are given $\omega_{c3} = 1800$ rpm.

Using the relationship $\omega_{cn} = n^2 \omega_{c1}$, we can find the first critical speed ($\omega_{c1}$) using n=3:

$$ \omega_{c3} = 3^2 \omega_{c1} $$

Substitute the given value of $\omega_{c3}$:

$$ 1800 \text{ rpm} = 9 \times \omega_{c1} $$

Solve for $\omega_{c1}$:

$$ \omega_{c1} = \frac{1800}{9} \text{ rpm} $$

$$ \omega_{c1} = 200 \text{ rpm} $$

Thus, the first critical speed of the simply supported long slender rotating shaft is 200 rpm.

Summary of Critical Speeds for this Shaft

Based on the first critical speed of 200 rpm:

• First critical speed ($\omega_{c1}$, n=1, 0 internal nodes): 200 rpm
• Second critical speed ($\omega_{c2}$, n=2, 1 internal node): $2^2 \times 200 = 4 \times 200 = 800$ rpm
• Third critical speed ($\omega_{c3}$, n=3, 2 internal nodes): $3^2 \times 200 = 9 \times 200 = 1800$ rpm

The observation of two internal nodes at 1800 rpm aligns with the third critical speed calculation derived from a first critical speed of 200 rpm.

The final answer is 200 rpm.