= Proportional damping
{title2=$D=\Phi^TC\Phi$}
<Proportional damping> chooses a <damping matrix> compatible with the undamped <normal modes>, making a linear <structural dynamics> calculation separate into scalar modal equations. Its common two-term realization is <Rayleigh damping>, $C=\alpha M+\beta K$, with <mass matrix> $M$ and <stiffness matrix> $K$. The important computational property is modal decoupling; the two-term formula is a sufficient construction, rather than the only matrix that can possess that property.<Rayleigh Damping Command in OpenSees Documentation>[§3.1.13.1, mass- and stiffness-proportional matrix formula]{ref}
For constant symmetric $M,K$, with $M$ positive definite, choose a complete set of mass-normalized <normal modes> as the columns of $\Phi$:
$$
\Phi^TM\Phi=I,\qquad
\Phi^TK\Phi=\Omega^2=\operatorname{diag}(\omega_1^2,\ldots,\omega_n^2).
$$
Substitute $u=\Phi q$ into $M\ddot u+C\dot u+Ku=f$ and multiply by $\Phi^T$. The result is
$$
\ddot q+D\dot q+\Omega^2q=\Phi^Tf,
\qquad D=\Phi^TC\Phi.
$$
This <modal damping matrix> is diagonal precisely when the chosen modes uncouple the <damping>. For <Rayleigh damping>, $D=\alpha I+\beta\Omega^2$ is automatically diagonal, and $D_{ii}=2\zeta_i\omega_i$, where $\zeta_i$ is the modal <damping ratio>. More generally, prescribed diagonal $D$ can be realized by $C=M\Phi D\Phi^TM$; it need not fit a two-parameter <Rayleigh damping> curve. OpenSees's separate modal-damping command, for example, accepts individual <damping ratios> rather than requiring one common pair of mass/stiffness coefficients.<Modal Damping Command in OpenSees Documentation>[§3.1.13.2, example assigning 5% to the first mode and 2% to the second]{ref}
For a two-coordinate illustration in consistent normalized units, take
$$
M=I,\qquad K=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\qquad
C=\begin{pmatrix}0.3&-0.1\\-0.1&0.3\end{pmatrix}.
$$
The undamped mode vectors $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$ have <natural frequencies> $1$ and $\sqrt3$. The transformed <damping matrix> is $D=\operatorname{diag}(0.2,0.4)$, giving <damping ratios> $0.10$ and $0.2/\sqrt3\approx0.1155$. Here $C=0.1M+0.1K$ explicitly verifies the proportional construction.
In contrast, replacing $C$ by $\operatorname{diag}(0.2,0.4)$ in the original physical coordinates gives
$$
D=\begin{pmatrix}0.3&-0.1\\-0.1&0.3\end{pmatrix}.
$$
The damping is still dissipative, but the modal equations are coupled: positive dissipated <power> alone does not imply modal decoupling. Such <nonclassical damping> can arise when dissipation is distributed differently from the inertial and elastic responses. Dropping its off-diagonal modal terms is an additional approximation that needs justification for the particular system.
Back to article page
