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I have an 2D elastodynamics problem, that is a problem which is driven by the Cauchy equation: $$\rho\ddot u-\mathrm{div}\sigma=\rho f$$ where $u$ is the displacement, $\sigma$ the Cauchy stress tensor, $f$ the density of volumetric forces, and $\rho$ the mass density. The constituve law is Hook's law from linear elasticty (homogeneous and isotropic medium). To solve it numerically, I use the finite element method. After space discretization I will have the following equation: $$M\ddot u+Ku=F$$ where $M$ is the mass matrix, $K$ the stiffness matrix and $F$ the load vector. The solution of such an equation oscillate around an equilibrium position and my aim is to add a viscous damping term $C\dot u$ that will critically damp the system, by which I mean I would like to avoid the oscillatory behaviour as much as possible. The equation now reads: $$M\ddot u+C\dot u+Ku=F$$ The problem is how to choose the matrix $C$ that will achieve the desired effect?

In modal analysis of dynamic structures, the same term $C\dot u$ is added to the discretized Cauchy equation to take damped vibrations into account but for a different purpose: we search for proper modes of vibration. Here we already know the two proper modes of vibration: S-waves and P-waves. So taking into account this information, I would like to build a matrix $C$ that will reflect this physical fact and that contains only constants like $E$, $\mu$ (two classical elasticity modulus) and $\rho$, rather than assuming a Rayleigh damping term $C=\alpha M+\beta K$ and finding the relevant values of $\alpha$ and $\beta$ by hand (numerically).

Thank you very much for any idea!

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I'm a bit rusty on structural dynamics, but pages 87-92 of Meirovitch's "Fundamentals of Vibration" explain that a system

$$ m\ddot{x} + c\dot{x} + kx = 0 $$

is critically damped when $c=2m\omega$, where $\omega=\sqrt{\frac{k}{m}}$ is the natural frequency of vibration.

There is a natural generalization to the system of ODEs

$$ M \ddot{x} + C \dot{x} + Kx = 0. $$

For an eigenpair of the pencil $(K,M)$, say $(\omega^2,u)$, where $u$ is a unit vector, we have that

$$ K u = M u \omega^2, $$

and thus we could construct a $C$ which critically damps this mode of vibration if we set

$$ C := 2 \omega (Mu)u^T, $$

where the matrix $(Mu)u^T$ transforms $u \mapsto Mu$, so that $(Mu)u^T$ is analogous to the $m$ term in the equation $c = 2 \omega m$, and the $2$ and $\omega$ require no generalization.

For some set of modes $\{u_j\}_j$, with corresponding eigenvalues $\{\omega^2_j\}_j$, you could critically damp those modes by setting

$$ C := 2 \sum_j \omega_j (Mu_j)u_j^T. $$

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  • $\begingroup$ Thank you very much Jack, your answer gives me an idea that I will investigate soon. $\endgroup$ – Benjamin Apr 21 '12 at 6:24

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