= Mechanical dynamic condensation
<Mechanical dynamic condensation> eliminates selected coordinates from a frequency-dependent mechanical response while retaining their effect on the remaining coordinates. It is exact algebraic elimination at the chosen transform variable when the eliminated block is invertible. It is not equivalent to clamping those coordinates. Starting from the partitioned <mechanical dynamic-stiffness matrix>,
$$
\begin{pmatrix}\mathbf D_{rr}&\mathbf D_{ri}\\\mathbf D_{ir}&\mathbf D_{ii}\end{pmatrix}
\begin{pmatrix}\mathbf X_r\\\mathbf X_i\end{pmatrix}
=\begin{pmatrix}\mathbf F_r\\\mathbf F_i\end{pmatrix},
$$
solve the internal equation as $\mathbf X_i=\mathbf D_{ii}^{-1}(\mathbf F_i-\mathbf D_{ir}\mathbf X_r)$. Substitution gives
$$
\underbrace{(\mathbf D_{rr}-\mathbf D_{ri}\mathbf D_{ii}^{-1}\mathbf D_{ir})}_{\widehat{\mathbf D}_{rr}}\mathbf X_r
=\underbrace{\mathbf F_r-\mathbf D_{ri}\mathbf D_{ii}^{-1}\mathbf F_i}_{\widehat{\mathbf F}_r}.
$$
The effective matrix is a <Schur complement>; the right side is the <condensed mechanical load vector>. Both matter when the internal coordinate is externally forced. The coupled dynamic equations and measurement boundary conditions are the starting model, rather than independent scalar stiffness reciprocals.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[Printed p.11, coupled analytical mechanical models; pp.14–16, force/response descriptions]{ref}
For the two-mass example at $\omega=5\,\mathrm{rad/s}$, take $D_{11}=D_{22}=175\,\mathrm{N/m}$ and $D_{12}=D_{21}=-100\,\mathrm{N/m}$. With $F_1=0$ and $F_2=1\,\mathrm N$, eliminating coordinate2 gives effective stiffness $117.8571\,\mathrm{N/m}$ and effective load $100/175\,\mathrm N$. Hence $X_1=100/20625\,\mathrm m\approx4.848\,\mathrm{mm}$. Omitting the load correction would incorrectly predict no response at coordinate1.
For constant original <mass matrix>, <damping> and <mechanical stiffness matrix>, the condensed matrix is generally a rational function of frequency, not simply a smaller constant mass–damping–stiffness triple. A low-frequency approximation requires its own justification. The elimination can also become singular at an internal-block frequency even when the complete system matrix remains invertible. For example, $\begin{pmatrix}0&-k_c\\-k_c&0\end{pmatrix}$ has invertible full matrix for $k_c\ne0$, but its lower-right scalar block cannot be inverted. One must then retain that coordinate or choose a different elimination.
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