### Direct Blocked Forces

Consider the following scenario. The operating source is rigidly clamped on its interface so that the interface vibration $$ \mathbf{u}\mathrm{_{2}} $$ is zero, see figure above. The reaction forces in the clamped support (Blocked Forces) $$ \mathbf{g}\mathrm{_{2}^{B}} $$ ensure that:

$$ \begin{bmatrix} \mathbf{u}\mathrm{_{1}}\\

\mathbf{u}\mathrm{_{2}^{A} = 0}

\end{bmatrix} = \begin{bmatrix}

\mathbf{Y}\mathrm{_{11}^{A}} & \mathbf{Y}\mathrm{_{12}^{A}}\\

\mathbf{Y}\mathrm{_{21}^{A}} & \mathbf{Y}\mathrm{_{22}^{A}}

\end{bmatrix} \begin{bmatrix}

\mathbf{f}\mathrm{_{1}}\\

\mathbf{g}\mathrm{_{2}^{A}} = – \mathbf{g}\mathrm{_{2}^{B}}

\end{bmatrix} $$

This yields:

$$ \mathbf{g}\mathrm{_{2}^{B}} = (\mathbf{Y}\mathrm{_{22}^{A}})^{-1} \mathbf{Y}\mathrm{_{21}^{A}} \mathbf{f}\mathrm{_{1}} $$

$$ \mathbf{f}\mathrm{_{2}^{bl}} = \mathbf{g}\mathrm{_{2}^{B}} $$

The direct Blocked Forces method assumes the boundary to be infinitely stiff in all directions, which is rarely the case in practice. Hence the accuracy of the Blocked Forces is subject to the stiffness of the boundary relative to the component at hand. An additional difficulty is the measurement of rotational moments, as most commonly used sensors cannot measure collocated 6-DoF interface loads. Consequently, the direct Blocked Force method is expected to perform best at the low-frequency end for which the rigid boundary assumption is most valid and rotational effects are, in practice, least prominent.

### Free velocity method

The direct counterpart of the direct Blocked Forces method is the free velocity method, as depicted above. In this case, the component’s interfaces are left free, such that all vibrations are seen as “free displacements” $$ \mathbf{u}\mathrm{_{2}^{free}} $$ at the interface DoFs. From here on, equivalent forces can be calculated by inverting the admittance measured at the free subsystem’s interfaces, which can be understood by comparing the free displacements with the Blocked Force definition.

$$ \begin{bmatrix}

\mathbf{u}\mathrm{_{1}}\\

\mathbf{u}\mathrm{_{2}^{A} = \mathbf{u}\mathrm{_{2}^{free}} }

\end{bmatrix} = \begin{bmatrix}

\mathbf{Y}\mathrm{_{11}^{A}} & \mathbf{Y}\mathrm{_{12}^{A}}\\

\mathbf{Y}\mathrm{_{21}^{A}} & \mathbf{Y}\mathrm{_{22}^{A}}

\end{bmatrix} \begin{bmatrix}

\mathbf{f}\mathrm{_{1}}\\

\mathbf{g}\mathrm{_{2}^{A}} = \mathrm{0}

\end{bmatrix} $$

This yields:

$$ \mathbf{u}\mathrm{_{2}^{free}} = \mathbf{Y}\mathrm{_{21}^{A}} \mathbf{f}\mathrm{_{1}} $$

$$ \mathbf{f}\mathrm{_{2}^{bl}} = (\mathbf{Y}\mathrm{_{22}^{A}})^{+} \mathbf{u}\mathrm{_{2}^{free}} $$

In practice, it is almost impossible to measure under these conditions. For one, we must settle for a “quasi-free” condition above the decoupling frequencies. Secondly, it is often challenging to run the active component in operation in a free boundary condition.

### In-situ characterization

As direct Blocked Force and free velocity methods both pose their practical limitations, another more versatile method is often used. In practice, the active component will often be measured on a test rig whose dynamics fall somewhere between the completely rigid boundary and free conditions.

It was observed here that responses at the passive side are obtained from the application of Blocked Forces to the FRFs of the assembled system AB.

$$ \mathbf{u}\mathrm{_{3}^{AB}} = \mathbf{Y}\mathrm{_{32}^{AB}} \mathbf{f}\mathrm{_{2}^{bl}} $$

This equation can also be inverted to determine the Blocked Forces. To perform this inversion, the FRFs must contain sufficient independent responses to describe all interface forces and moments. The set of receiver responses $$ \mathbf{u}\mathrm{_{3}} $$ is typically too small in number and too distant from the interfaces to be suitable for inversion. In practice, to improve the conditioning of this FRF matrix, we equip the passive structure with additional indicator sensors $$ \mathbf{u}\mathrm{_{4}} $$, as shown in the figure above. The $$ n $$ indicator sensors are positioned close to the interfaces such that the full set of $$ m $$ interface forces is properly observable, using $$ n > m $$ sensors. The indicator sensors are used to calculate Blocked Forces as:

$$ \mathbf{f}\mathrm{_{2}^{bl}} = (\mathbf{Y}\mathrm{_{42}^{AB}})^{+} \mathbf{u}\mathrm{_{4}^{AB}} $$

Two sets of measurements are now required to execute:

- Operational indicator responses $$ \mathbf{u}\mathrm{_{4}^{AB}} $$, obtained from test cycles on the full assembly;
- An FRF test of the assembled system $$ \mathbf{Y}\mathrm{_{42}^{AB}} $$

$$ \mathbf{Y}\mathrm{_{42}^{AB}} $$ and $$ \mathbf{Y}\mathrm{_{32}^{AB}} $$ FRFs (the latter is needed to calculate responses at the target locations) can be obtained from the same FRF measurement campaign, as it only involves mounting of additional sensors.

With proper selection of indicator sensors, the in-situ characterization is usually quite accurate and relatively straightforward to execute. Like with any method where a matrix inversion is solved, proper conditioning of the FRF matrix and the operational data is crucial. Using Virtual Points for the description of the interface DoFs already helps achieve better conditioning, as one has more control over the actual DoFs that get inverted. For example, the XYZ force vectors are naturally orthogonal using Virtual Points and can be put nicely in the center of the connection points, which can be troublesome otherwise.

As the Blocked Forces are a property of the active source alone, these forces can be calculated in the target assembly AB or in an assembly of the source system on a test rig R (see figure above).