# X-DoF

Calculating Blocked Forces is easy in SOURCE. However, selecting the right number of forces and the right DoFs to use can be challenging. How many forces do I actually need to describe the system? Should I include the rotational DoFs? These are common questions you might ask yourself when calculating Blocked Forces.

To answer them, VIBES has developed X-DoF, a new patent-pending algorithm!

## How many DoFs to describe the source?

With Blocked Forces, you model an excitation source independently of the receiver. You might think that the more degrees of freedom you have, the better you can describe your system. However, when calculating Blocked Forces using the in-situ method (which includes a matrix inversion), you can run into the problem of modeling measurement noise and not only the source dynamics. So,

With too many DoFs you will model measurement noise and not only the system dynamics.

### Existing approaches to improve the matrix inversion

To avoid overfitting and modeling noise with Blocked Forces, you need to find a way to understand which are the essential DoFs for your case and reduce the calculation to these.

To do that, you could follow these well-known approaches:

• Performing a Matrix regularization. By looking at the CMIF matrix, you can understand how to regularize the matrix. In particular, you can:
1. Truncate: remove a fixed number of Singular Values (SVs).
2. Set a relative threshold: remove SVs based on the ratio largest/smallest.
3. Set an absolute threshold: remove SVs based on a fixed ratio.
• Manually select DoFs. Manually select and deselect DoFs from the analysis inputs.

However, it might be the case that some DoFs are needed in a certain range but not in another. With regularization and manual DoF selection, you can just decide whether to include the DoF for the entire frequency range or remove it.

In addition, these methods produce processes that are not robust and not repeatable, because they are subjective. To prove that, I provided the same measurements to two engineers and asked them to improve the quality of Blocked Forces. They both performed a matrix regularization to exclude some SVs. One of them decided to manually set the number of SVs to include while the other decided to exclude some by setting a relative threshold. The results are shown below and have some differences.

Non-robust and non-repeatable processes (such as matrix regularization and manual DoF selection) are not the best solution.

## X-DoF

Not repeatable and subjective processes are not a good solution. X-DoF solves this problem by automatically selecting a set of blocked forces that accurately represent the indicator data without overfitting, at each frequency block. This way, it provides robust and objective results and makes the whole process more efficient.

The advantages of using X-DoFs when calculating Blocked Forces:

• It reduces the risk of overfitting.
• It gives objective results.
• It is repeatable.
• It reduces the impact of indicator noise on the calculation.
• It removes the task of selecting DoFs manually and lets engineers focus on their main objective.

In the picture above you can see how Blocked Forces of a component change when applying X-DoF (second picture). X-DoF provides the necessary Blocked Forces only.

X-DoF automatically selects a set of Blocked Forces that accurately represents the indicator data without overfitting.

## Example case: mechatronic component

In this example, you will see how using X-DoF improves the quality of the Blocked Forces and avoids including noise in the results.

We have calculated Blocked Forces of a mechatronic component on two test benches: a stiff test-bench (bench A) and a compliant one (bench B), using the in-situ method.

We have then performed a transfer validation (link for theory), applying Blocked Forces of bench A to B and vice-versa and comparing measured data with the obtained predictions.

### Step 1: Predictions with matrix inversion

We started by calculating Blocked Forces of the component on the two benches, using the in-situ method and performed a transfer validation. We applied the Blocked Forces obtained in bench A to bench B and vice-versa. In the following pictures you can see the predictions at the target sensors after performing a transfer validation. The black line corresponds to the measured data and the colored one to the TPA syntheses. In both cases, there is an important overprediction at the target sensors.

These overpredictions can be caused by overfitting. For this reason, we also applied regularization and X-DoF separately.

### Step 2: Improving results with matrix regularization

To improve the predictions, we decided to perform some matrix regularization. In particular, we have tried to:

1. Add a relative threshold to the matrix regularization.
2. Select a certain amount of Singular Values.

From the picture, it can be noticed that when applying the regularization, the results improve. However, there is still overprediction at the target sensors and according to the strategy used for the matrix regularization, the results are quite different. These results are improved but not objective.

### Step 3: Improving results with X-DoF

The best option is to use X-DoF to calculate Blocked Forces, to avoid including noise in the calculation and to have a robust and objective process.

With X-DoF, the match between measured and predicted results is highly improved. At higher frequencies, it can be noticed that X-DoF underpredicts the results. This is the case because, in that frequency range, the signal presents noise at the indicators. For this reason, since it is not possible to estimate a force from a noisy signal, X-DoF calculate conservative results.

When using X-DoF, you can speed up the calculation by analyzing only a segment of the operational data or by limiting the frequency range.