{c}
{title2=$C=\alpha M+\beta K$}

<Rayleigh damping> approximates the <damping matrix> of a linear <structural dynamics> model by a combination of its <mass matrix> and <stiffness matrix>:
$$
M\ddot u+C\dot u+Ku=f(t),\qquad C=\alpha M+\beta K.
$$
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.<Rayleigh Damping Command in OpenSees Documentation>[§3.1.13.1, matrix formula and coefficient table]{ref}

To see the modal consequence, take an undamped <normal mode> $K\phi_i=\omega_i^2M\phi_i$ and its modal mass $m_i=\phi_i^TM\phi_i$. Then
$$
c_i=\phi_i^TC\phi_i=(\alpha+\beta\omega_i^2)m_i,
\qquad \zeta_i=\frac{c_i}{2m_i\omega_i}
=\frac12\left(\frac{\alpha}{\omega_i}+\beta\omega_i\right),\qquad \omega_i>0.
$$
Thus <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 $\omega_1=10\,\mathrm{rad/s}$ and $\omega_2=50\,\mathrm{rad/s}$ gives
$$
\beta=\frac{2(0.05)}{10+50}=\frac1{600}\,\mathrm s,
\qquad \alpha=\frac{2(0.05)(10)(50)}{10+50}=\frac56\,\mathrm{s^{-1}}.
$$
The resulting <damping ratio> is $0.0375$ at $20\,\mathrm{rad/s}$ and $0.0875$ at $100\,\mathrm{rad/s}$. 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> $\dot u^TC\dot u$ 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.<Rayleigh Damping Command in OpenSees Documentation>[§3.1.13.1, Note, warning about concentrated-plasticity nonlinear models]{ref}


 Rayleigh damping (source code)