A rotor has a mass of 12 kg and is mounted midway on a horizontal shaft which is supported at the ends by two bearings (assumed as simply supported). The bearings are 1 m apart. What will be the critical (whirling) speed of the shaft? [EI = 4 kN - m², g = 10 m/s²]

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BHEL Engineer Trainee Mechanical 24 Aug 2023 Official Paper

Answer 1

Concept:

For a simply supported shaft with a rotor (concentrated mass) at the center, the critical (whirling) speed is determined using the relation:

\( \omega = \sqrt{\frac{g}{\delta}} \)

Where, \( \delta \) is the static deflection due to the weight of the rotor.

Static deflection at center of simply supported shaft due to point load is given by:

\( \delta = \frac{W L^3}{48 EI} \)

Calculation:

Given:

Mass of rotor, \(m = 12~kg, so ~weight, W = mg = 12 \times 10 = 120~N\)

Span of shaft, \(L = 1~m, ~EI = 4~kN\cdot m^2 = 4000~N\cdot m^2\)

Substitute in deflection formula:

\( \delta = \frac{120 \times 1^3}{48 \times 4000} = \frac{120}{192000} = \frac{1}{1600}~m \)

Now, critical speed:

\( \omega = \sqrt{\frac{g}{\delta}} = \sqrt{10 \times 1600} = \sqrt{16000} = 40\sqrt{10}~rad/s \)