# Rayleigh damping in structural mechanics

In structural mechanics, we often approximate a partial differential equation by a system of ordinary differential equations in matrix form

$$\boldsymbol{M}\ddot{\boldsymbol{q}}(t)+\boldsymbol{B}\dot{\boldsymbol{q}}(t)+\boldsymbol{K}{\boldsymbol{q}}(t)=\boldsymbol{f}(t),$$

in which $\boldsymbol{M}$ is the mass matrix, $\boldsymbol{B}$ is the damping matrix and $\boldsymbol{K}$ is the stiffness matrix. The vector $\boldsymbol{q}(t)$ is the displacement vector which contains the displacements at specific positions and the force vector $\boldsymbol{f}(t)$ does specify the forces applied at the same positions at which the displacements are specified.

It is possible to deduce the mass matrix $\boldsymbol{M}$ and the stiffness matrix $\boldsymbol{K}$ by introducing finite elements. The damping matrix $\boldsymbol{B}$ is not as easy to determine.

A possible model is Rayleigh’s damping model. It is given by

$$\boldsymbol{B}=\alpha \boldsymbol{M} +\beta \boldsymbol{K}.$$

Now what I do not understand is the problem of choosing $\alpha$ and $\beta$. Are they obtained by system identification, while testing measured data with simulated data?

• Perhaps you should rename your question because it's generally about Rayleigh damping and not specific to beams. The current title "Structural mechanics of a beam" has nothing to do with it. – user1318499 May 9 '18 at 2:43

If you know the damping ratios ($\zeta$) at two frequencies, those 4 values determine $\alpha$ and $\beta$ by

$$\zeta = {1 \over 2}{({\alpha \over \omega} + \beta \omega)}$$

In this graph, the blue curve is $\zeta$, and it's the sum of the other two curves which correspond to the two terms in brackets in the formula. If you know two points on the blue curve, that uniquely defines its parameters $\alpha$ and $\beta$.

In the special case where damping ratio is proportional to frequency, $\alpha = 0$, the mass matrix $\boldsymbol M$ is ignored, and the formula is exact. Otherwise, it's usually an approximation that's only correct at the two frequencies used to fit it. • Interesting. This seems like a correct answer but it’s been awhile since I’ve seen this type of problem - any chance you could explain further, I’m curious. – OnStrike May 9 '18 at 1:07
• +1: So one does use $\boldsymbol{B}$ to determine the damping ratio $\zeta$? How do I get the values of the damping ratio, if the system is not decoupled from the beginning? – MrYouMath May 9 '18 at 7:05
• You're trying to determine B, so you can't use it to obtain anything until you have alpha and beta. What information do you have about the damping? It's a function of both structure and material so it's not something that's easily listed in tables accurately and probably requires measurements. There are also lots of different damping models/variables and everybody seems to use a different one that can't be reliably converted. – user1318499 May 9 '18 at 7:21
• @user1318499: Ok, that is what I meant by system identification :). Thank you a lot for your help. – MrYouMath May 9 '18 at 11:25