In 1, the authors consider n-degree-of-freedom non-classically damped linear systems. They write the system representation in the modal coordinates as  
X¨t+2CX˙t+Λ2Xt=0,X0=X0,X˙0=X˙0
(1)
for all t0, where the vector of displacements XtRn, the symmetric and non-negative matrix CRn×n corresponds to the modal damping matrix, and the diagonal matrix  
Λ2=diag[ω12,ω22,,ωn2]Rn×n
(2)
has the square of the undamped natural frequencies of the system, ω12,ω22,,ωn2, on its diagonal.

The authors...

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