A rigid block of mass ? is mounted on four elastic supports, as shown in the figure. A mass ? drops from a height ? and adheres to the rigid block without rebounding. If the spring constant of each elastic support is ?, derive the equation of motion and find the natural frequency of vibration of the system (a) without the mass ?, and (b) with the mass ?. Also, find the resulting motion of the system in case (b)

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Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Generally, there are three types of vibration: longitudinal vibration, transverse vibration, and torsional vibration.