User:Jeremiahpalmer

The FastFRS (Fast Frequency Response Solver) software is used in computer-aided engineering to provide the modal frequency response analysis of systems created by finite element softwares.

The Frequency Response Problem
In FE (finite element) space, the frequency response problem for fluid-structure interaction is $$ \left[ \begin{array}{cc} -\omega^2 M + i\omega B + (1+i \gamma)K + i K_s & i \omega A \\ i \omega A^T & -(-\omega^2 E + i \omega C + H)/\rho \end{array} \right] \left\{ \begin{array}{c} X_s(\omega) \\ X_f(\omega) \end{array} \right\} = \left\{ \begin{array}{c} F_s(\omega) \\ F_f(\omega) \end{array} \right\} $$

where $$M$$, $$B$$, $$K$$, and $$K_s$$ represent the FE structural mass, viscous damping, stiffness, and structural damping matrices, respectively, $$\gamma$$ is the structural damping factor, $$A$$ is the area (or coupling) matrix, which converts fluid pressure to structural loading, and $$E$$, $$C$$, and $$H$$ represent the FE fluid mass, damping, and stiffness matrices, respectively. Also, $$i = \sqrt{-1}$$ and $$\rho$$ is the fluid mass density.

$$ \left[ \begin{array}{cc} -\omega^2 I + i\omega \bar B + (1+i \gamma) \Lambda_s+ i \bar{K_s} & i \omega \bar A \\ i \omega \bar A^T & -\omega^2 \bar E + i \omega \bar C + \bar H \end{array} \right] \left\{ \begin{array}{c} Z_s(\omega) \\ Z_f(\omega) \end{array} \right\} = \left\{ \begin{array}{c} \bar F_s(\omega) \\ \bar F_f(\omega) \end{array} \right\} $$