= Nonclassical damping
<Nonclassical damping> occurs when a linear mechanical system's <damping matrix> cannot be diagonalized together with its <mass matrix> and <stiffness matrix> by an undamped <normal mode> transformation. Consequently the damping terms couple the modal equations even though their inertia and stiffness terms are separated. This can matter when a localized dissipative element is attached to a structure whose elastic and inertial responses are distributed differently.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[Introduction and §2.2, inadequacy of classical modal analysis]{ref}
For $M\ddot u+C\dot u+Ku=0$, take a complete mass-normalized undamped mode matrix $\Phi$ with $\Phi^TM\Phi=I$ and $\Phi^TK\Phi=\Omega^2$. Then $u=\Phi q$ gives
$$
\ddot q+D\dot q+\Omega^2q=0,\qquad D=\Phi^TC\Phi.
$$
With distinct undamped <natural frequencies>, a nonzero off-diagonal $D_{ij}$ shows <modal damping coupling>. If a frequency is repeated, first allow a rotation of the modes within its repeated-frequency subspace: an off-diagonal entry in one arbitrary basis is not by itself proof of <nonclassical damping>. For symmetric $M,K,C$ with $M$ positive definite, the basis-independent condition for classical decoupling is
$$
CM^{-1}K=KM^{-1}C.
$$
Failure of this condition establishes <nonclassical damping> under those assumptions.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[§2.2, equation(12), necessary and sufficient commutation condition]{ref}
For an original example in consistent normalized units, let
$$
M=I,\quad K=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\quad
C=\begin{pmatrix}0.4&0\\0&0\end{pmatrix}.
$$
The damping acts only on the first physical coordinate. The undamped mode vectors $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$ give
$$
D=\begin{pmatrix}0.2&0.2\\0.2&0.2\end{pmatrix}.
$$
Thus an initial modal state $q=(1,0)^T$, $\dot q=(1,0)^T$ has $\ddot q_2=-0.2$. Dropping the off-diagonal damping terms would incorrectly predict zero initial acceleration in the second modal coordinate. Nevertheless $\dot u^TC\dot u=0.4\dot u_1^2\ge0$: passive dissipation does not require modal decoupling.
The <damped quadratic eigenvalue problem> follows from $u(t)=r e^{\lambda t}$:
$$
(\lambda^2M+\lambda C+K)r=0.
$$
In oscillatory cases its complex vectors can encode different phases at different physical coordinates. One can solve the coupled equations directly or use an appropriate damped-mode method; assigning separate scalar decay equations to the undamped modes is an additional approximation, not a consequence of positive dissipation.
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