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''}