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A damping matrix $C$ maps generalized velocity to the velocity-dependent resisting-force term in a linear structural dynamics model:Here $M$ is the mass matrix and $K$ is the stiffness matrix. The sign convention places $C\dot{u}$ on the left, so the actual resisting force is $-C\dot{u}$. For pure translational coordinates, entries have units $\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$; mixed rotational and translational coordinates require their corresponding generalized units.

Consider an original two-coordinate model with three viscous dampers: one from each coordinate to a fixed support, and a third between them. With positive coefficients $c_1,c_2,c_{12}$,For $\dot{u}=(v_1,v_2{)}^T$, the removed mechanical power isThis makes $C$ a positive semidefinite matrix. In particular, the connecting damper dissipates nothing when both coordinates move at the same speed. Choosing $c_1=c_2=10$ and $c_{12}=5$ in newton-seconds per metre gives $C=\left(\begin{array}{cc}15& -5\\ -5& 15\end{array}\right)$. Speeds $(1,0)\mathrm{m}{\mathrm{s}}^{-1}$ dissipate15watts, while $(1,1)\mathrm{m}{\mathrm{s}}^{-1}$ dissipate20watts.

For symmetric constant $M,K$, multiplying the equation by ${\dot{u}}^T$ gives an energy balance:The matrix therefore encodes where dissipation occurs and which relative motions activate it.

Rayleigh damping approximates $C$ by a weighted combination of the mass and stiffness matrices. OpenSees exposes separate coefficients for mass and different stiffness states, and cautions that this model can give incorrect results in some nonlinear concentrated-plasticity analyses.^[ref] A convenient matrix prescription must still represent the intended dissipation mechanism.

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.^[ref]

For constant symmetric $M,K$, with $M$ positive definite, choose a complete set of mass-normalized normal modes as the columns of $\Phi$:Substitute $u=\Phi q$ into $M\ddot{u}+C\dot{u}+Ku=f$ and multiply by ${\Phi }^T$. The result isThis 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 $C=M\Phi D{\Phi }^{T}M$; 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.^[ref]

For a two-coordinate illustration in consistent normalized units, takeThe undamped mode vectors $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$ have natural frequencies $1$ and $\sqrt{3}$. The transformed damping matrix is $D=\mathrm{diag}(0.2,0.4)$, giving damping ratios $0.10$ and $0.2/\sqrt{3}\approx 0.1155$. Here $C=0.1M+0.1K$ explicitly verifies the proportional construction.

In contrast, replacing $C$ by $\mathrm{diag}(0.2,0.4)$ in the original physical coordinates givesThe 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.

Rayleigh damping approximates the damping matrix of a linear structural dynamics model by a combination of its mass matrix and stiffness matrix:Here $\alpha$ has units ${\mathrm{s}}^{-1}$ and $\beta$ has units $\mathrm{s}$. This is a convenient form of proportional damping: for constant symmetric $M,K$ and $M$ positive definite, the same normal modes that separate the undamped system also separate this damping matrix. The OpenSees implementation additionally permits coefficients multiplying current, initial and last-committed stiffness matrices; these are distinct choices in a nonlinear calculation.^[ref]

To see the modal consequence, take an undamped normal mode $K{\varphi }_i={\omega }_i^{2}M{\varphi }_i$ and its modal mass $m_i={\varphi }_i^{T}M{\varphi }_i$. ThenThus Rayleigh damping specifies a frequency-dependent damping ratio, rather than the same damping ratio for every normal mode. The mass-proportional term dominates at low nonzero angular frequency, and the stiffness-proportional term dominates at high angular frequency.

For example, fitting a damping ratio of $0.05$ at both ${\omega }_1=10rad/s$ and ${\omega }_2=50rad/s$ givesThe resulting damping ratio is $0.0375$ at $20rad/s$ and $0.0875$ at $100rad/s$. Agreement at two fitted frequencies therefore does not validate damping outside that band. In particular, a zero-frequency rigid motion has no modal damping ratio of this form, yet the term $\alpha M\dot{u}$ can still resist that motion.

For nonnegative $\alpha ,\beta$ and positive semidefinite $M,K$, the dissipated power ${\dot{u}}^{T}C\dot{u}$ is nonnegative. This supplies an immediate physical check on a fit. It does not establish that the chosen damping matrix represents a particular material or joint: an apparatus would need measured modal decay or another measured dissipative response to justify its coefficients. OpenSees specifically cautions that Rayleigh damping can give incorrect results in nonlinear time-history analysis with concentrated plasticity.^[ref]