= Modal energy transfer
<Modal energy transfer> describes redistribution of energy among the modal coordinates of a mechanical model. In mass-normalized undamped coordinates with $\Omega^2=\operatorname{diag}(\omega_i^2)$, define
$$
E_i=\frac12\left(\dot q_i^2+\omega_i^2q_i^2\right).
$$
For $\ddot q+D\dot q+\Omega^2q=g$, differentiation yields
$$
\dot E_i=\dot q_i g_i-\dot q_i\sum_jD_{ij}\dot q_j.
$$
Off-diagonal terms of the <modal damping matrix> can therefore make an individual modal energy grow even when there is no external forcing. Their presence is <modal damping coupling>.<The Decoupling of Damped Linear Systems in Oscillatory Free Vibration by Fai Ma, Ali Imam and Matthias Morzfeld>[§§2.1–2.2, modal-coordinate equations and damping coupling]{ref}
For an original example, let
$$
D=\begin{pmatrix}0.4&0.3\\0.3&0.4\end{pmatrix},\qquad
\dot q=\begin{pmatrix}1\\-2\end{pmatrix},\qquad g=0.
$$
This damping matrix has positive eigenvalues0.7 and0.1. The energy rates are $\dot E_1=+0.2$ and $\dot E_2=-1.0$, whereas
$$
\frac{d}{dt}(E_1+E_2)=-\dot q^TD\dot q=-0.8.
$$
Mode1 gains energy while the combined system loses energy. No negative total dissipation or external power source is needed.
The cross terms deserve care: $-D_{12}\dot q_1\dot q_2$ and $-D_{21}\dot q_2\dot q_1$ are not generally equal and opposite. They contribute to the total dissipation as well as to each coordinate's budget. Thus a damping-based redistribution should not automatically be interpreted as a unique lossless exchange between two modal reservoirs. A different permitted choice of basis, especially within a repeated-frequency subspace, can change individual modal energies while preserving total physical energy.
Summing the budgets gives the physically decisive balance $\dot E=\dot q^Tg-\dot q^TD\dot q$. With no forcing and positive-semidefinite $D$, total mechanical energy cannot increase, in agreement with the <mechanical dissipation inequality>. This supplies a stronger interpretation of an observed growing modal amplitude than assuming that every separately plotted mode must decay monotonically.
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