[Efficient Engineer FEM Video](https://www.youtube.com/watch?v=GHjopp47vvQ) [COMSOL FEM Article](https://www.comsol.com/multiphysics/finite-element-method) All linear structures can be fully characterized by the following equation: $ M\ddot{x}(t) + C\dot{x}(t) + K x(t) = f(t) $ $M =$ mass matrix $C =$ damping matrix $K =$ stiffness matrix $x(t) =$ displacement shape $f(t) =$ forcing function Taking the fourier transform reduces this to $ \underbrace{ (-\omega^{2} M +i\omega C + K) }_{ D }~x(\omega) = f(\omega) $ Defining the dynamical matrix $D$ as the combined matrix on the left and the frequency response function matrix $H$ as it's inverse $H = D^{-1}$, we have $h_{jk} = \frac{x_{j}(\omega)}{f_{k}(\omega)}$ Each entry in $H$ corresponds to the FRF between the displacement of the $j$th degree of freedom and the $k$th force. - $H$ tends to be tall because there are more DoFs than force application points - Each entry is an array of complex numbers corresponding to each discrete frequency in the spectrum The deformation state of a structure can be expressed by discretizing it into nodes and storing the deformation of each degree of freedom for each node in the column vector $x$. $ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} = \begin{bmatrix} & & \\ & \phi & \\ & & \end{bmatrix}\begin{bmatrix} \underline{X}_{1} \\ \underline{X}_{2} \\ \underline{X}_{3} \end{bmatrix} $ Physical coordinates $= X$ Mode shape Matrix $=[\phi]$ Modal Coordinates $= [\underline{X}]$