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Rayleigh Damping

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

C=αM+βK\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}

For a critical damping ratio

ξi=12ωiα+ωi2β\xi_{i} = \frac{1}{2\omega_{i}}\alpha + \frac{\omega_{i}}{2}\beta

For two distinct damping ratios

{ξiξj}=12[1ωiωi1ωjωj]{αβ}\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 α\alpha, β\beta. In the case of the same damping at both frequencies this simplifies to

β=2ξωi+ωj\beta = \frac{2\xi}{\omega_i+\omega_j}
α=ωiωjβ\alpha = \omega_{i}\omega_{j}\beta