12}} + M{\left( {\frac{L}{2} - \frac{L}{3}} \right)^2} + m \times {\left( {\frac{{2L}}{3}} \right)^2}\)\( = \frac{{M{L^2}}}{{12}} + \frac{{M{L^2}}}{{36}} + \frac{{4M{L^2}}}{9} = \frac{{2M{L^2}}}{9}\)BALANCING Torque about 0\(I\alpha = K \times \frac{{2L}}{3} \times \left( {\frac{{2L}}{3}\theta } \right) + K \times \frac{L}{3} \times \left( {\frac{L}{3}\theta } \right)\)\(\frac{{2M{L^2}}}{9}\frac{{{d^2}\theta }}{{d{t^2}}} = \frac{{5k}}{{2M}} = \OMEGA _n^2\theta \)\({\omega _n} = \sqrt {\frac{{5k}}{{2M}}} \)

"> 12}} + M{\left( {\frac{L}{2} - \frac{L}{3}} \right)^2} + m \times {\left( {\frac{{2L}}{3}} \right)^2}\)\( = \frac{{M{L^2}}}{{12}} + \frac{{M{L^2}}}{{36}} + \frac{{4M{L^2}}}{9} = \frac{{2M{L^2}}}{9}\)BALANCING Torque about 0\(I\alpha = K \times \frac{{2L}}{3} \times \left( {\frac{{2L}}{3}\theta } \right) + K \times \frac{L}{3} \times \left( {\frac{L}{3}\theta } \right)\)\(\frac{{2M{L^2}}}{9}\frac{{{d^2}\theta }}{{d{t^2}}} = \frac{{5k}}{{2M}} = \OMEGA _n^2\theta \)\({\omega _n} = \sqrt {\frac{{5k}}{{2M}}} \)

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A thin uniform rigid bar of length L and mass M is hinged at point O, located at a distance of L/3 from one of its ends. The bar is further supported using spring, each of stiffness k, located at the two ends. A particle of mass \(m = \frac{M}{4}\) is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about O, the natural frequency of the system is

Mechanical Vibrations Undamped Free Vibration in Mechanical Vibrations . 7 months ago

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Taking mass moment of inertia about 0\(I = \frac{{m{L^2}}}{{12}} + M{\left( {\frac{L}{2} - \frac{L}{3}} \right)^2} + m \times {\left( {\frac{{2L}}{3}} \right)^2}\)\( = \frac{{M{L^2}}}{{12}} + \frac{{M{L^2}}}{{36}} + \frac{{4M{L^2}}}{9} = \frac{{2M{L^2}}}{9}\)BALANCING Torque about 0\(I\alpha = K \times \frac{{2L}}{3} \times \left( {\frac{{2L}}{3}\theta } \right) + K \times \frac{L}{3} \times \left( {\frac{L}{3}\theta } \right)\)\(\frac{{2M{L^2}}}{9}\frac{{{d^2}\theta }}{{d{t^2}}} = \frac{{5k}}{{2M}} = \OMEGA _n^2\theta \)\({\omega _n} = \sqrt {\frac{{5k}}{{2M}}} \)

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Posted on 25 Oct 2024, this text provides information on Mechanical Vibrations related to Undamped Free Vibration in Mechanical Vibrations. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.

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