Rayleigh Damping

Rayleigh damping considers damping to be related to both the mass and stiffness

$$\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$$

For a critical damping ratio

$$\xi_{i} = \frac{1}{2\omega_{i}}\alpha + \frac{\omega_{i}}{2}\beta$$

For two distinct damping ratios

$$\begin{Bmatrix}\xi_{i} \\ \xi_{j} \\ \end{Bmatrix} = \frac{1}{2}\begin{bmatrix}\frac{1}{\omega_{i}} & \omega_{i} \\\frac{1}{\omega_{j}} & \omega_{j} \\\end{bmatrix}\begin{Bmatrix}\alpha \\\beta \\\end{Bmatrix}$$

This can be solved to determine the coefficients $\alpha$, $\beta$. In the case of the same damping at both frequencies this simplifies to

$$\beta = \frac{2\xi}{\omega_i+\omega_j}$$

$$\alpha = \omega_{i}\omega_{j}\beta$$