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 $t⩾0,$ where the vector of displacements $Xt∈Rn,$ the symmetric and non-negative matrix $C∈Rn×n$ 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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