Consider the arrangement shown in the figure below where J is the combined polar mass moment of inertia of the disc and the shafts. $K_{1}, K_{2}, K_{3}$ are the torsional stiffness of the respective shafts. The natural frequency of torsional oscillation of the disc is given by

\sqrt{\frac{K_{1} + K_{2} + K_{3}}{J}}

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\sqrt{\frac{K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1}}{J (K_{1} + K_{2})}}

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\sqrt{\frac{K_{1} + K_{2} + K_{3}}{J (K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1})}}

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\sqrt{\frac{K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1}}{J (K_{2} + K_{3})}}

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Solution

The correct option is B $\sqrt{\frac{K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1}}{J (K_{1} + K_{2})}}$
Equivalent stiffness,

$K_{e q} = \frac{K_{1} K_{2}}{K_{1} + K_{2}} + K_{3} = \frac{K_{1} K_{2} + K_{1} K_{3} + K_{2} K_{3}}{K_{1} + K_{2}}$

Now natural frequency

$= \sqrt{\frac{K_{e q}}{J}} = \sqrt{\frac{K_{1} K_{2} + K_{2} K_{3} + K_{3} K_{1}}{J (K_{1} + K_{2})}}$

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