Rayleigh damping ($C=\alpha M+\beta K$)

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]