Rayleigh Damping Beta⚠ unverified
Mechanical / Vibration Analysis · Compute the stiffness-proportional Rayleigh damping coefficient beta
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 | β | s | Stiffness-proportional Rayleigh damping coefficient beta, in seconds (s) |
The science & history
Understanding the Parameters
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Reference frequencies $\omega_1$, $\omega_2$ — the two frequencies where the modal damping equals the target $\zeta$ exactly. $\beta$ (with $\alpha$) is solved to enforce this.
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Why $\beta$ damps high frequencies — the stiffness-proportional term contributes a modal damping $\zeta_i^{(\beta)} = \beta\omega_i/2$, which increases linearly with frequency. So $\beta$ dominates the high-frequency modes, whereas $\alpha$ dominates the low ones — together they shape the damping curve across the band.
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Numerical-damping use — because $\beta$ heavily damps high frequencies, it is often used deliberately to suppress spurious high-frequency numerical noise in explicit time-integration, at the cost of some over-damping of real high modes.
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Units of time — multiplying the stiffness matrix (which relates force to displacement), $\beta$ scales damping with the rate of deformation, giving it units of seconds.
Derivation (Approaching a Proof)
From the modal projection of $[C] = \alpha[M] + \beta[K]$, the modal damping ratio of mode $i$ is
$$\zeta_i = \frac{\alpha}{2\omega_i} + \frac{\beta\omega_i}{2}.$$
Enforcing $\zeta_1 = \zeta_2 = \zeta$ at frequencies $\omega_1$ and $\omega_2$ gives two equations. Writing them out and eliminating $\alpha$ (subtract the two, factor):
$$\frac{\alpha}{2}\left(\frac{1}{\omega_1}-\frac{1}{\omega_2}\right) + \frac{\beta}{2}(\omega_1-\omega_2) = 0,$$
together with $\zeta = \alpha/2\omega_1 + \beta\omega_1/2$, solves to
$$\beta = \frac{2\zeta}{\omega_1+\omega_2}, \qquad \alpha = 2\zeta\frac{\omega_1\omega_2}{\omega_1+\omega_2}.$$
The compact $\beta = 2\zeta/(\omega_1+\omega_2)$ reflects that stiffness-proportional damping is anchored by the high-frequency end of the matched band.
Dimensional check. $\beta = \dfrac{2\zeta}{\omega_1+\omega_2} = \dfrac{\text{–}}{\text{rad/s}} = \text{s}$ — time, as required ($\zeta$ dimensionless).
History and Development
Stiffness-proportional damping is the second half of Rayleigh's proportional-damping model (The Theory of Sound, 1877), indispensable in finite-element structural dynamics and earthquake engineering. The $\alpha$–$\beta$ pair lets an analyst prescribe realistic modal damping across a frequency band with two numbers while preserving modal orthogonality — avoiding the prohibitive cost of a full non-proportional damping matrix. $\beta$'s strong high-frequency damping is also exploited to control numerical high-frequency content in time-domain solvers. See Rayleigh Damping Alpha for the companion coefficient.
Related Concepts: Rayleigh Damping Alpha, Damping Ratio, Natural Frequency Mass Spring, Critical Damping, Quality Factor Q, Response Spectrum Value
Notes: Stiffness-proportional part of $[C]=\alpha[M]+\beta[K]$; damps high frequencies ($\zeta^{(\beta)} =\beta\omega/2$). Matches $\zeta$ exactly at $\omega_1,\omega_2$ only. Also used to suppress numerical high-frequency noise. Pair with Rayleigh Damping Alpha.