 Modal damping matrix

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]