26 R. N. Coppolino where the matrix of eigenvalues, [. ω 2 n], is real and diagonal. In addition, the real modes, [Ф ] (when they are unit modal mass normalized) satisfy the following orthogonality conditions: .[Ф] T[M][Ф] =[I] , [Ф] T[K] [Ф] =⎡ω 2 n⎤. (3.6 ) Employing the real mode displacement transformation, . {u}=[Ф] {q}, (3.7 ) the system’s dynamic response may be described in terms of the uncoupled modal equations, . ¨qn +2ζnωn˙qn +ω 2 nqn =⎡Ф T nΓe⎤{Fe}, (3.8a ) where , .2ζnωn ={Фn} T[B] {Фn}. (3.8b ) An important additional (theoretical) decoupling transformation resulting from the Eq. 3.6 orthogonality property is . {¨q}=⎡Ф T M⎤{¨u}. (3.9 ) The above relationship forms the basis of the SMAC algorithm [24]; however, it does not satisfactorily decouple experimentally estimated modal responses when the measured system dynamics are more consistent with empirically estimated complex modes. 3.5.2 Complex Modes The matrix equations describing dynamics of a linearly behaving structural dynamic system (see Eq. 3.1) are now described in state-space form by introduction of the velocity array, . {v}={˙u}, (3.10 ) ultimately resulting in the (unsymmetric) state-space matrix equation set, . ⎡˙ v ˙u ⎤=⎡−M−1 B −M−1 K I 0 ⎤⎡v u ⎤+⎡M−1 Γe 0 ⎤{Fe}. (3.11 ) It should be noted here that various (two coefficient matrix) symmetric forms of the state-space equations are possible [3]; however, the above single coefficient matrix form suits the purposes of the present discussion. Complex state-space modes associated with the algebraic eigenvalue problem [25], . ⎡−M−1 B −M−1 K I 0 ⎤⎡ϕv ϕu ⎤+⎡ϕv ϕu ⎤[λ] ⇒[A] [Ф] =[Ф] [λ] , (3.12 ) are mutually orthogonal via definition of the left-hand eigenvectors (employing the full set of state-space eigenvectors or modes) , .[Ф]−1 =[ФL] =[ϕvL ϕuL] . (3.13 ) Therefore, the “left-hand” and “right-hand” eigenvectors satisfy the following orthogonality conditions:
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