### Direct measurement

The most straightforward technique to obtain the interface forces is from force transducers mounted directly between the active and the passive side. This approach requires that the transducers at the interface are stiff enough in the frequency range of interest, such that the assumption of $$ \mathbf{u}\mathrm{_{2}^{B}} = \mathbf{u}\mathrm{_{2}^{A}} $$ is valid. The main drawback of this method is the inconvenience of mounting transducers between the active component and the receiving structure. Lack of space, distortion of the original mounting situation, and inability to measure all desired Degrees of Freedom at a connection point render this method generally impractical, especially for typical automotive applications. However, force transducers can be used effectively when built into a test bench setup to measure Blocked Forces.

### Mount stiffness method

In many automotive applications, the actively vibrating parts are attached to the vehicle body using resilient mounts to reduce vibration transmission. With proper tuning of the mount flexibility (stiffness) and absorption (damping) properties, a high level of vibration suppression can be achieved. The mount stiffness method takes advantage of the fact that the amount of vibration on either side of the mount will not be the same: $$ \mathbf{u}\mathrm{_{2}^{B}} – \mathbf{u}\mathrm{_{2}^{A}} $$ is no longer zero. By combining the measured displacement on either side of the mount (usually integrated from accelerometer data) with the stiffness of the mount $$ \mathbf{Z}\mathrm{^{mt}} $$, we can calculate the interface forces:

$$ \mathbf{g}\mathrm{_{2}^{B}} = \mathbf{Z}\mathrm{^{mt}} (\mathbf{u}\mathrm{_{2}^{A}} – \mathbf{u}\mathrm{_{2}^{B}}) $$

Although the mount stiffness method can be powerful and easy to conduct, it can be challenging to get an accurate characterization of the mount stiffness properties $$ \mathbf{Z}\mathrm{^{mt}} $$, particularly at the higher frequencies and amplitudes. The static stiffness of the mount provided from the supplier or measured on a test bench is likely only appropriate at the very low frequencies, above which a more sophisticated characterization is warranted. This can, for instance, be Inverse Substructuring using a Virtual Point FRF model of the mount with measurement adapters or a full decoupling approach.

### Matrix Inverse method

The perhaps most popular method for determining interface forces is the matrix inverse method. It was observed that responses at the passive side are obtained from the application of interface forces to the FRFs of the passive subsystem.

$$ \mathbf{u}\mathrm{_{3}^{B}} = \mathbf{Y}\mathrm{_{32}^{B}} \mathbf{g}\mathrm{_{2}^{B}} $$

This equation can also be inverted to determine the interface 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 $$ n $$ interface forces is properly observable, using $$ n > m $$ sensors. The indicator sensors are used to calculate interface forces as:

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

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 passive component $$ \mathbf{Y}\mathrm{_{42}^{B}} $$, so after dismounting of the active component.

Note that the second part requires dismounting of the active component(s) from the assembly. Nevertheless, the $$ \mathbf{Y}\mathrm{_{42}^{B}} $$ and $$ \mathbf{Y}\mathrm{_{32}^{B}} $$ 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 matrix inverse technique 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.