Vibration of M.D.O.F system

Equation of motion of multi degree of freedom system in matrix form --

[m]{x''} + [x]{x'} + [k]{x} = -xg(t)[m]{I}

where,

[m] = Mass Matrix.

[k] = Stiffness Matrix.

[c] = Damping Matrix

{I} = Unit Vector

x''g(t) = Ground Motion

The physical coordinates {x} may be related with normal or prinicipal coordinates {q} as,

{x} = [φ]{q}

where,[φ] = Modal Matrix.

Time Derivate of {x} are

{x'} = [φ]{q'}

{x''} = [φ]{q''}

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