FUNDAMENTALS OF MECHANICAL VIBRATION (2nd Ed) Graham Kelly How did it reach to this answer 5.47 Derive the stiffness matrix for the system in the figure using the indicated generalized coordinates and stiffness influence coefficients

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A: For solution refer below images.

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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.