Modal energy transfer

Modal energy transfer describes redistribution of energy among the modal coordinates of a mechanical model. In mass-normalized undamped coordinates with ${\Omega }^2=\mathrm{diag}({\omega }_i^2)$, defineFor $\ddot{q}+D\dot{q}+{\Omega }^{2}q=g$, differentiation yieldsOff-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.^[ref]

For an original example, letThis damping matrix has positive eigenvalues0.7 and0.1. The energy rates are ${\dot{E}}_1=+0.2$ and ${\dot{E}}_2=-1.0$, whereasMode1 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 $\dot{E}={\dot{q}}^{T}g-{\dot{q}}^{T}D\dot{q}$. 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.