{title2=$D=\Phi^TC\Phi$}

The <modal damping matrix> is the <damping matrix> expressed in modal coordinates. Starting from $M\ddot u+C\dot u+Ku=f$ and writing $u=\Phi q$ gives
$$
M_q\ddot q+D\dot q+K_q q=\Phi^Tf,
\qquad M_q=\Phi^TM\Phi,\quad D=\Phi^TC\Phi,\quad K_q=\Phi^TK\Phi.
$$
When the columns of $\Phi$ are mass-normalized undamped <normal modes>, $M_q=I$ and $K_q=\operatorname{diag}(\omega_i^2)$. Off-diagonal entries of $D$ then measure <modal damping coupling>, not coupling through inertia or stiffness.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[§§2.1–2.2, modal transformation and diagonal versus coupled modal damping]{ref}

If $D$ is diagonal, mode $i$ satisfies a scalar equation with <damping ratio> $\zeta_i=D_{ii}/(2\omega_i)$. Without mass normalization the corresponding formula is $\zeta_i=D_{ii}/(2m_i\omega_i)$, where $m_i=\phi_i^TM\phi_i$. Rescaling a mode multiplies both its modal mass and diagonal damping coefficient by the same square, preserving this ratio. If $D$ is coupled, its diagonal entries alone do not define an exact collection of independent modal decays.

For a numerical example in consistent units, take
$$
M=\begin{pmatrix}2&0\\0&1\end{pmatrix},\quad
K=\begin{pmatrix}8&0\\0&9\end{pmatrix},\quad
\Phi=\begin{pmatrix}1/\sqrt2&0\\0&1\end{pmatrix}.
$$
The mass-normalized <natural frequencies> are $2$ and $3$. A physical <damping matrix>
$$
C=\begin{pmatrix}0.8&0.2\\0.2&0.6\end{pmatrix}
$$
becomes
$$
D=\begin{pmatrix}0.4&0.2/\sqrt2\\0.2/\sqrt2&0.6\end{pmatrix}.
$$
Both diagonal-only <damping ratios> would be $0.10$, but the off-diagonal coefficient $0.2/\sqrt2\approx0.1414$ still couples their velocities. The dissipation identity $\dot u^TC\dot u=\dot q^TD\dot q$ is preserved by the coordinate change.

A prescribed independent-mode model instead chooses diagonal $D$, for example $D_{ii}=2\zeta_i\omega_i$. Such a prescription is a modeling choice about physical dissipation. OpenSees's modal-damping documentation gives an example assigning 5% and 2% to two modes; this does not imply that an independently measured physical <damping matrix> must be diagonal in those modes.<Modal Damping Command in OpenSees Documentation>[§3.1.13.2, two-mode damping-ratio example]{ref}


 Modal damping matrix (source code)