Damping

Rayleigh damping: a damping matrix built as a linear combination of the mass and stiffness matrices,

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

🔧 Grasshopper component

DampingAlpaca4d ▸ 07_Analysis

Inputs

AlphaMNumber0.0Factor on the mass matrix.

Damps the low frequencies.

BetaKCurrNumber0.0Factor on the current stiffness matrix.

BetaKInitNumber0.0Factor on the initial stiffness matrix.

BetaKCommNumber(none)Factor on the committed stiffness matrix.

Use exactly one of the three stiffness terms. They damp the high frequencies.

Outputs

DampingDampingFor Analysis Settings.

💡 Picking the coefficients

Target a damping ratio $\xi$ at two frequencies ${\omega }_i$ and ${\omega }_j$ (in rad/s):

$${\alpha }_M=\xi \frac{2{\omega }_i{\omega }_j}{{\omega }_i+{\omega }_j}{\beta }_K=\xi \frac{2}{{\omega }_i+{\omega }_j}$$

Get the frequencies from Natural Vibration first — usually the first mode and whichever higher mode still carries meaningful mass. Between the two the damping ratio dips slightly below $\xi$; outside them it rises, steeply at the high end.

With both frequencies the same, ${\alpha }_M=\xi \omega$ and ${\beta }_K=\xi /\omega$ give exactly $\xi$ at $\omega$. The Response Spectrum from Sticks example does that for thirty models, one a branch, each damped 5% at its own frequency.

Which stiffness matrix to use:

Term	Behaviour
BetaKInit	Damping stays tied to the elastic stiffness. Predictable, but can over-damp a structure that has softened a lot.
BetaKCurr	Damping follows the tangent stiffness, and falls away as the structure yields.
BetaKComm	Uses the last committed stiffness — a middle course, and the usual recommendation for non-linear work.

📈 When to use it

Use it when you are running a Transient analysis. Without damping a dynamic model rings indefinitely, and the peak response is overestimated.

Do not use it when the analysis is static — the damping matrix is never formed, and the input is ignored.

🔗 Relation to OpenSees

rayleigh $alphaM $betaKcurr $betaKinit $betaKcomm