 Condensed mechanical load vector

The condensed mechanical load vector is the effective forcing on retained coordinates after mechanical dynamic condensation eliminates internal coordinates. It includes the effect of forces applied to the eliminated coordinates; it is not simply the retained subset of the original load vector. With partitioned mechanical dynamic-stiffness matrix $\mathbf{D}$ and invertible internal block ${\mathbf{D}}_{ii}$,and the retained equation is ${\hat{\mathbf{D}}}_{rr}{\mathbf{X}}_r={\hat{\mathbf{F}}}_r$. This follows by solving the internal block equation and substituting it into the retained equation, using the same Schur complement that constructs the reduced dynamic stiffness. Coupled force/response models require their boundary conditions to be stated.^[ref]

For an illustrative two-coordinate system at one frequency, let diagonal entries of $\mathbf{D}$ be175N/m and off-diagonal entries be $-100N/m$. Eliminating coordinate2 givesIf $F_1=0$ and $F_2=1\mathrm{N}$, then ${\hat{F}}_1=4/7\mathrm{N}$ and $X_1=100/20625\mathrm{m}$. Discarding the eliminated-coordinate force would incorrectly remove this response.

Conversely, choose simultaneous real complex amplitudes $F_1=1\mathrm{N}$ and $F_2=-1.75\mathrm{N}$. The condensed load vanishes and $X_1=0$, even though neither applied force is zero. The recovered internal displacement is $X_2=-0.010\mathrm{m}$, which satisfies both original equations. This is a frequency-specific response cancellation, rather than a vanishing total physical force: $F_1+F_2=-0.75\mathrm{N}$.

Because the elimination uses ${\mathbf{D}}_{ii}^{-1}(s)$, the effective load can depend on frequency and phase even when the original force amplitudes are simple. An arbitrary change of load or frequency therefore requires recomputing that transformation; a reduced matrix alone does not preserve every forced-response problem.