José Abell's research blog

NTS-02. On Rayleigh damping coefficients for FE analysis

Note to self. How to compute Rayleigh damping coefficients for given damping ratios $\xi_1$ and $\xi_2$ at frequencies $f_1$ and $f_2$.

This is textbook content, I just need to remind myself too often how this is done and end up re-deriving the equations.

Given the second-order system of differential equations representing the FE model

$$ M \ddot{u} + C \dot{u} + K u = F(t) $$

The damping matrix $C$ can be written as a Rayleigh damping matrix:

$$ C = a_0 M + a_1 K $$

$a_0$ and $a_1$ are Rayleigh damping coefficients found by solving

$$ \left[ \begin{array}{cc} \dfrac{1}{2\pi f_1} & 2 \pi f_1 \ \dfrac{1}{2\pi f_2} & 2 \pi f_2 \end{array} \right] \left[ \begin{array}{c} a_0 \ a_1 \end{array} \right] = \left[ \begin{array}{c} \xi_1 \ \xi_2 \end{array} \right] $$

Which I do in the following code:

NTS (Note-To-Self): Creating dynamically linked libraries

A nice extensive tutorial can be found here.

In a nutshell:

gcc -Wall -fPIC -c \*.c  
gcc -shared -Wl,-soname, -o \*.o  
mv /opt/lib  
  • -fPIC: Compiler directive to output position independent code, a characteristic required by shared libraries. Also see “-fpic”.
  • -shared: Produce a shared object which can then be linked with other objects …