Question

how to derive the undamped vibration

Answer 1

vibration of a damped, single degree of freedom, linear spring mass system. ... Usually, if you start something vibrating, it will vibrate with a progressively decreasing amplitude and eventually stop moving. The reason our simple models predict the wrong behavior is that we neglected energy dissipation.

Answer 2

Undamped vibration : When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be undamped. The vibrations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position. An example of this type of vibration is the vehicular suspension... undampened by the shock absorber.

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