Modal damping matrix ($D={\Phi }^{T}C\Phi$)

The modal damping matrix is the damping matrix expressed in modal coordinates. Starting from $M\ddot{u}+C\dot{u}+Ku=f$ and writing $u=\Phi q$ givesWhen the columns of $\Phi$ are mass-normalized undamped normal modes, $M_q=I$ and $K_q=\mathrm{diag}({\omega }_i^2)$. Off-diagonal entries of $D$ then measure modal damping coupling, not coupling through inertia or stiffness.^[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 $m_i={\varphi }_i^{T}M{\varphi }_i$. 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, takeThe mass-normalized natural frequencies are $2$ and $3$. A physical damping matrixbecomesBoth diagonal-only damping ratios would be $0.10$, but the off-diagonal coefficient $0.2/\sqrt{2}\approx 0.1414$ still couples their velocities. The dissipation identity ${\dot{u}}^{T}C\dot{u}={\dot{q}}^{T}D\dot{q}$ 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.^[ref]

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 $D={\Phi }^{T}C\Phi$, the equations areThus $D_{ij}{\dot{q}}_j$ 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.^[ref]

For an original example, takeIts eigenvalues are $0.5$ and $0.1$, so it is positive definite. At modal velocities $\dot{q}=(1,-2{)}^T$, total dissipated power isA diagonal-only approximation would give1.5 instead, despite leaving both diagonal coefficients unchanged.

The individual modal energies $E_i=\frac{1}{2}({\dot{q}}_i^2+{\omega }_i^2{q}_i^2)$ obey, with no external forcing,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 solvesOff-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.

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.