= Mechanical dynamic-stiffness matrix
{title2=$\mathbf D(s)=s^2\mathbf M+s\mathbf C+\mathbf K$}
A <mechanical dynamic-stiffness matrix> maps displacement coordinates to applied <forces> in a linear <structural dynamics> model at a specified transform variable or frequency. Starting with constant <mass matrix> $\mathbf M$, <damping> matrix $\mathbf C$ and <mechanical stiffness matrix> $\mathbf K$,
$$
\mathbf M\ddot{\mathbf x}+\mathbf C\dot{\mathbf x}+\mathbf K\mathbf x=\mathbf f,
$$
zero <initial conditions> give $\mathbf F(s)=\mathbf D(s)\mathbf X(s)$ with $\mathbf D=s^2\mathbf M+s\mathbf C+\mathbf K$. For the $e^{i\omega t}$ convention, $\mathbf D(i\omega)=\mathbf K-\omega^2\mathbf M+i\omega\mathbf C$. Its <matrix inverse>, when it exists, is the complete <mechanical receptance> matrix. Models and measured responses must use matching coordinates and boundary conditions.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[Printed p.11, coupled analytical dynamic models and boundary conditions; pp.14–16, complex force/displacement response functions]{ref}
Consider two $1\,\mathrm{kg}$ masses, each grounded by a $100\,\mathrm{N/m}$ <spring> and joined by another $100\,\mathrm{N/m}$ <spring>, with no <damping>. At <angular frequency> $5\,\mathrm{rad/s}$,
$$
\mathbf D=\begin{pmatrix}175&-100\\-100&175\end{pmatrix}\,\mathrm{N/m}.
$$
A <force> amplitude $F_1=1\,\mathrm N$ with $F_2=0$ gives $X_1=175/20625\,\mathrm m\approx8.485\,\mathrm{mm}$ and $X_2=100/20625\,\mathrm m\approx4.848\,\mathrm{mm}$. Clamping coordinate2 instead gives $X_1=1/175\,\mathrm m\approx5.714\,\mathrm{mm}$, because the clamp supplies a reaction <force>.
Partitioning retained and internal coordinates gives <mechanical dynamic condensation>. With $\mathbf F_i=0$ and invertible $\mathbf D_{ii}$,
$$
\mathbf F_r=(\mathbf D_{rr}-\mathbf D_{ri}\mathbf D_{ii}^{-1}\mathbf D_{ir})\mathbf X_r.
$$
The effective matrix is a <Schur complement>, generally frequency dependent even when the original <mass matrix>, <damping> and <mechanical stiffness matrix> are constant. In the example its scalar value is $175-100^2/175\approx117.857\,\mathrm{N/m}$, rather than175. A reduced force-free internal coordinate and a clamped coordinate describe different experiments. At a singular block, that particular elimination is unavailable; it does not establish that the full mechanical system lacks a valid solution.
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