Rayleigh Damping Alpha⚠ unverified
Mechanical / Vibration Analysis · Compute the mass-proportional Rayleigh damping coefficient alpha
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn1 | wn1 | rad/s | 1.0 | First reference natural frequency |
| wn2 | wn2 | rad/s | 1.0 | Second reference natural frequency |
| zeta | ζ | — | 1.0 | Target damping ratio at both reference frequencies (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | α | 1/s | Mass-proportional Rayleigh damping coefficient alpha, in inverse seconds (1/s) |
The science & history
Understanding the Parameters
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Two reference frequencies $\omega_1$, $\omega_2$ — Rayleigh damping can only match the target $\zeta$ exactly at two frequencies; between them the damping dips slightly below $\zeta$, outside them it rises. They are usually chosen to bracket the modes that matter for the analysis.
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Target damping $\zeta$ — the modal damping you want in that band (e.g. 2–5 % for steel structures).
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Why $\alpha$ damps low frequencies — the mass-proportional term contributes a modal damping $\zeta_i^{(\alpha)} = \alpha/(2\omega_i)$, which decreases with frequency. So $\alpha$ dominates the low-frequency (long-period) modes; the stiffness term $\beta$ (Rayleigh Damping Beta) handles the high ones.
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Physical picture — mass-proportional damping is like each node moving through a viscous medium (drag proportional to velocity and mass), appropriate for distributed/external damping.
Derivation (Approaching a Proof)
With $[C] = \alpha[M] + \beta[K]$, project onto mode $i$ (mass-normalised, $\phi_i^T M\phi_i = 1$, $\phi_i^T K\phi_i = \omega_i^2$). The modal damping ratio is
$$\zeta_i = \frac{\phi_i^T C \phi_i}{2\omega_i} = \frac{\alpha + \beta\omega_i^2}{2\omega_i} = \frac{\alpha}{2\omega_i} + \frac{\beta\omega_i}{2}.$$
Requiring $\zeta_1 = \zeta_2 = \zeta$ at two frequencies gives two linear equations in $\alpha$ and $\beta$. Solving them (subtract to eliminate, then back-substitute):
$$\alpha = 2\zeta\,\frac{\omega_1\omega_2}{\omega_1+\omega_2}, \qquad \beta = \frac{2\zeta}{\omega_1+\omega_2}.$$
The $\alpha$ expression is the harmonic-mean-like combination of the two frequencies scaled by $2\zeta$, reflecting that mass-proportional damping is anchored by the low-frequency behaviour.
Dimensional check. $\alpha = 2\zeta\dfrac{\omega_1\omega_2}{\omega_1+\omega_2} = \dfrac{(\text{rad/s})^2}{\text{rad/s}} = \text{rad/s} = 1/\text{s}$ — inverse time, as required ($\zeta$ dimensionless).
History and Development
Lord Rayleigh introduced proportional damping in The Theory of Sound (1877); its enormous practical value emerged with finite-element dynamics in the mid-20th century. A general damping matrix does not diagonalise in the undamped modes, forcing an expensive coupled solution; Rayleigh's $\alpha[M]+\beta[K]$ form does diagonalise, so structural-dynamics and earthquake-engineering FE codes use it universally to assign realistic modal damping with just two parameters. See Rayleigh Damping Beta for the companion coefficient.
Related Concepts: Rayleigh Damping Beta, Damping Ratio, Natural Frequency Mass Spring, Critical Damping, Quality Factor Q, Response Spectrum Value
Notes: Mass-proportional part of $[C]=\alpha[M]+\beta[K]$; damps low frequencies ($\zeta^{(\alpha)}= \alpha/2\omega$). Matches $\zeta$ exactly at $\omega_1,\omega_2$ only (dips between). Preserves modal orthogonality → mode-by-mode solution. Pair with Rayleigh Damping Beta.