= Modal damping coupling
<Modal damping coupling> is the interaction between undamped modal coordinates produced by off-diagonal entries of the <modal damping matrix>. With mass-normalized modes and $D=\Phi^TC\Phi$, the equations are
$$
\ddot q_i+\omega_i^2q_i+\sum_jD_{ij}\dot q_j=g_i.
$$
Thus $D_{ij}\dot q_j$ contributes to mode $i$'s force when $j\ne i$. Distinct undamped <natural frequencies> fix the modal directions up to scaling and ordering; repeated-frequency subspaces admit additional basis rotations and must be treated before interpreting an off-diagonal entry as unavoidable <nonclassical damping>.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[§§2.1–2.2, modal damping and classical-decoupling criterion]{ref}
For an original example, take
$$
D=\begin{pmatrix}0.3&0.2\\0.2&0.3\end{pmatrix}.
$$
Its eigenvalues are $0.5$ and $0.1$, so it is positive definite. At modal velocities $\dot q=(1,-2)^T$, total dissipated <power> is
$$
\dot q^TD\dot q=0.3(1)^2+2(0.2)(1)(-2)+0.3(-2)^2=0.7.
$$
A diagonal-only approximation would give1.5 instead, despite leaving both diagonal coefficients unchanged.
The individual modal energies $E_i=\tfrac12(\dot q_i^2+\omega_i^2q_i^2)$ obey, with no external forcing,
$$
\dot E_i=-\dot q_i\sum_jD_{ij}\dot q_j.
$$
For these velocities, $\dot E_1=+0.1$ while $\dot E_2=-0.8$: one modal energy grows temporarily, even though the total falls at rate0.7. This <modal energy transfer> is compatible with a <passive mechanical network>; it is not evidence that an external energy source drives the growing coordinate.
For harmonic forcing, the complete response solves
$$
(\Omega^2-\omega^2I+i\omega D)Q=G.
$$
Off-diagonal effects therefore depend on forcing frequency and the other dynamic terms, not solely on the size of $D_{ij}$ relative to $D_{ii}$. Neglecting them is an approximation whose error can be enhanced near nearby resonances. Solving the coupled equations or the <damped quadratic eigenvalue problem> retains those interactions; assigning independent scalar <damping ratios> does not, by itself, justify dropping them.
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