*3.3. Reduced Order Modelling on Pre-Processed Data for Friction Variations*

As described in Section 2.5, the friction variations were computed according to Equation (4), both for experimental and ROM predicted data. Similarly to Section 3.2, a prediction line was obtained to assess the accuracy of the results, obtained as the combination of the two (untextured and textured) ROM models. The results are shown in Figure 7.

**Figure 7.** 2-ROM-models prediction for combined untextured and textured friction data, according to Equation (4). The blue dots are obtained by evaluating the combined 2-ROM predictions with the combined experimental data (Equation (4)), the dashed black line is the linear regression that fits the data, while the solid red line represents the ideal prediction result, where the predicted values perfectly match the experimental ones.

From Figure 7 it is observed that the ROM prediction is highly accurate, since the regression line (dashed black) is highly comparable to the goal prediction (solid red).

Moreover, an error propagation was performed on the friction percent variation variable (Equation (4)), in order to assess the maximum error assumed by the predictive model. For this purpose, experimental deviations on friction measurements were not included, since these are not considered intrinsic properties of the prediction accuracy, but depend on the a-priori goodness of the dataset solely. The results obtained when taking into account both the error propagation on Δ*μ* and the standard deviation obtained for the 2-ROM prediction, i.e., *σ* = 1.4412 as shown in Figure 7, show values between 2% and 9%, with a mean value of 3% and 95% of data with a prediction deviation below 5%. Moreover, a study was performed to check the impact that each input parameter has on the measured friction, according to Equations (1) and (3), as shown in Figure 8 for textured ROM.

**Figure 8.** Textured data ROM's first term. The one-dimensional functions, for each variable are shown separately in the plots, according to Equation (1), labelled with letters from (**a**–**e**).

The one-dimensional functions (specified in Equation (1)) for the textured ROM, shown in Figure 8, represent the impact of each input on the friction variation. The textured ROM shows *α*<sup>1</sup> = 46.0479 for the first term (see Equations (1) and (3)), being its weighting factor one or two orders of magnitude higher than the remaining terms. The first term of the ROM series expansion can be, hence, considered a fair approximation of the system, where other terms are corrections to it, as described in Section 1.2. From Figure 8 one can see that bigger friction variations occur at extreme surface texturing parameters, e.g., smaller dimple depth and bigger diameter values.

## *3.4. Statistical Analysis Results of Real Dimple Dimensions*

As described in Section 2.6, PDFs were extrapolated for all texture parameters and were used to assess the measured differences from the desired nominal dimple values. The observed variations from the nominal dimple dimensions, described in Section 2.3, were introduced into the datasets in order to predict the corresponding friction variation using the previously computed ROM. The observed PDFs were normal distributions for both dimple depth and distance, while for diameter a skewed normal distribution was observed. dimple depth PDF showed a right shift of the mean equal to 3 μm and a standard deviation equal to 1.5 μm. Concerning the distance, the right shift of the mean was equal to 41 μm and 37 μm for corresponding nominal values of 200 μm and 300 μm, with a standard deviation of 13 μm and 9 μm, respectively. These findings led to significant differences in observed friction values, which means that deviations from nominal texture values, shown in Figures 9 and 10a,b, do actually affect friction as observed in Figure 11a,b.

**Figure 9.** Dimple depth specific PDF and corresponding nominal value.

A ROM model allows verifying that the introduction of a statistical noise on nominal surface texture values affects friction, as shown in Figure 11a,b. From these Figures it is possible to observe significant variations in friction PDFs distributions when depth or diameter and distances, respectively, vary from nominal values, and to compare the obtained friction distribution to a normal PDF with zero mean and *σ* = 0.01.

A two-tailed t-Student test was performed to check the compatibility of the two PDFs, i.e., the noisy nominal values friction distribution (orange) and the ideal friction distribution (green), shown in Figure 11a,b. The test result showed a statistically significant difference between the noisy and the ideal PDF in case of dimple diameter-distance variations, according to what can be expected from Figures 9 and 10a,b.

**Figure 10.** (**a**) Dimple diameter and (**b**) distance specific PDFs and respective nominal values.

As previously described in Section 2.5, two expanded datasets were generated introducing statistical noise to the textured data sample by varying the nominal dimple parameters (i.e., depth, diameter, and distance) according to the obtained PDFs. When depth was made to vary, only values equal to 20 μm could be used; in fact, nominal depth values of 10 μm and 30 μm would cause friction values to fall outside the ROM domain if a PDF was applied to generate data around these values. In this case the corresponding ranges for dimple diameter and distance were 200 μm and 200–300 μm, respectively. For the second dataset, given the ROM definition domain, feasible dimple diameter and distance ranges resulted in 100–200 μm and 200–300 μm, respectively, corresponding to depth values of 10 μm only. It is important to remark that, in this case, given the geometrical definition of the dimples, diameter and distance are inversely correlated and that the two PDFs are very different, being the first one a skewed distribution and the second a normal one (please refer to Section 2.5 for method details).

**Figure 11.** Friction coefficient distribution (orange) when (**a**) dimple depth or (**b**) diameter and distance are varried according to their specific PDFs (Figures 9 and 10a,b, respectively), centred and compared to a normal distribution (green) with zero mean and *σ* = 0.01.

Figure 12a,b show the findings for the friction prediction when the geometrical parameters of the seals deviate from nominal values.

**Figure 12.** (**a**) ROM prediction with statistical noise introduction on depth and (**b**) on diameter and distance. The blue dots are obtained by evaluating the ROM on the statistically expanded dataset, the solid red line represents the ideal ROM result and the error bars show the friction variations linked to (**a**) depth or (**b**) diameter and distance deviations from nominal values.

According to the results shown in Figure 12a,b, when the textured ROM was evaluated on the expanded noisy dataset, friction prediction turned out to be completely affected by the statistical nominal texture values deviations. Once again one can observe that when diameter and distance were made to vary, these variations produced big uncertainties on the predicted friction values and statistically significant differences on mean values, which was corrected when an ideal PDF was used for texture parameters. In order to prove this, friction prediction was repeated using the green distribution in Figure 11a,b, where the texture parameters are centred in their nominal value, with a standard deviation of *σ* = 0.01. The obtained results are shown in Figure 13a,b.

**Figure 13.** (**a**) ROM prediction with statistical noise introduction on depth and (**b**) on diameter and distance, with centred mean on nominal value and *σ* = 0.01. The blue dots are obtained by evaluating the ROM on the statistically expanded dataset, the solid red line represents the ideal ROM result and the error bars show the friction variations linked to (**a**) depth or (**b**) diameter and distance deviations from nominal values.

#### *3.5. Experimental Friction Measurement Results and ROM Friction Prediction Outcome*

The objective of the experimental testing procedure is to identify specific surface textures, which exhibit the lowest friction as a function of the relative velocity *vr* and the contact pressure *pc*. In Figures 14–16 the quasi-stationary friction coefficients *μ* of the 8 rubber specimens with different textures are shown as function of the relative velocity *vr*. Data points depicted in Figures 14–16 are based on 5 measurements, whose mean values are shown. Error bars are not provided in the figures due to better visibility, although sigma ranges for the whole pool of data varied between *σmin* = 0.004 and *σmax* = 0.196 with a mean value of *σmean* = 0.005.

**Figure 14.** (**a**) Friction coefficient *μ* as function of the relative velocity *vr* for the eight different rubber specimens and (**b**) the corresponding friction variation Δ*μ* in relation to the untextured rubber sample 8, for contact pressures of *pc*,*max* = 0.5 MPa.

**Figure 15.** (**a**) Friction coefficient *μ* as function of the relative velocity *vr* for the eight different rubber specimens and (**b**) the corresponding friction variation Δ*μ* in relation to the untextured rubber sample 8, for contact pressures of *pc*,*max* = 0.7 MPa.

Each figure visualises the measurement results for one of the three different contact pressures levels *pc* considered. In addition, the friction variations of the different rubber samples are presented in relation to the untextured reference rubber specimen 8. Under all operating conditions, friction-reducing, as well as friction-increasing textures, can be identified. Positive values of friction variation indicate a texture-related reduction of friction, while negative friction variation values describe a friction increase compared to the untextured reference sample 8, see also Equation (4). Besides a texture related vertical shift of the friction characteristic, a horizontal shift of the minimum can be observed as well. Hence, surface texturing modifies friction in all regimes, which will be discussed in another publication.

In the following, the textures with the highest and lowest friction are discussed for each contact pressure level *pc*,*max*. For *pc*,*max* = 0.5 MPa rubber specimens 2 and 4 exhibit the lowest friction when considering the entire velocity range, while sample 1 reveals the highest friction level. Samples 2 and 4 have the highest area densities of 39% and 55%, respectively, while sample 1 has the lowest area density of 9% within all textures examined. Therefore, a clear trend is evident for *pc*,*max* = 0.5 MPa, where the largest area densities

analysed lead to the lowest friction levels, while the lowest area density leads to the highest friction level. The aspect ratio does not appear to have a major influence in this pressure condition as it fluctuates for samples 1, 2, and 4 between 0.11, 0.05, and 0.10, so that no clear trend is apparent.

**Figure 16.** (**a**) Friction coefficient *μ* as function of the relative velocity *vr* for the eight different rubber specimens and (**b**) the corresponding friction variation Δ*μ* in relation to the untextured rubber sample 8, for contact pressures of *pc*,*max* = 0.9 MPa.

For *pc*,*max* = 0.7 MPa and *pc*,*max* = 0.9 MPa rubber sample 1 and 3 exhibit the lowest friction level over the entire range of velocities. In contrast, specimen 7 shows the highest friction. Despite this agreement for both pressure levels, no clear trend can be derived in terms of dimple dimensions, area density, or aspect ratio. For example, sample 3 and 7 have similar area density values of 28% and 25%, respectively, as well as aspect ratios of 0.04 and 0.06. However, as mentioned above, the measured friction results of both textures differ greatly from each other.

Since there is no clear trend for dimple texture, it is concluded that the texture needs to be individually adapted to the existing operating conditions to minimize friction. For this purpose, ROM is an extremely efficient tool. In order to demonstrate the benefits of the ROM in vivid examples, four exemplary use cases were defined. The first theoretical use case is a friction contact intended to operate at the lowest contact pressure level *pc*,*max* = 0.5 MPa and the highest velocity *vr* = 251 mm/s within the operational conditions experimentally analysed. During the friction measurement procedure, a maximum friction reduction of 37% was already achieved with sample 4. However, the theoretical customer requires a further enhanced friction reduction of at least 60%. By using the ROM, a friction reduction of 63% can be predicted with the texture parameters given in row 1 of Table 5.

**Table 5.** Dimple dimensions predicted by the ROM, which further reduce friction based on the use cases.


The second use case is an application that operates at a contact pressure *pc*,*max* = 0.7 MPa and a velocity of *vr* = 31 mm/s. Friction measurements reveal a maximum friction reduction of 63% for sample 1. Again, the friction can be further reduced by a suitable texture, which is predicted by the ROM. The ROM indicates a friction reduction of 81% compared to reference seal 8 for the values specified in row 2 of Table 5.

The third use case is an imaginary technical requirement for a seal, which operates under the highest contact pressure level of *pc*,*max* = 0.9 MPa and the lowest velocity of *vr* = 6 mm/s within the considered conditions. The experimental friction results show a maximum friction reduction of 48% for seal sample 1. With the use of the ROM, a reduced friction of 72% can be predicted in relation to the untextured case with the values given in row 3 of Table 5.

Based on the ROM results, one additional data point, indicated by a triangle, is added to each of the three plots of Figures 14–16.

In addition to friction predictions, considering the experimentally analysed operating conditions, one of the major advantage of the ROM is the ability to freely interpolate within the parameter space of the operating conditions. This is why the fourth use case shows the ROM prediction for experimentally untested working conditions, i.e., *pc* = 0.6 MPa and velocity of *vr* = 100 mm/s, where the friction reduction is about 79%, for the texture parameters shown in row 4 of Table 5.

The resulting aspect ratio for all four use case textures is equal to 0.04, while the textured area ranges from 30% to 42%. Compared to the experimentally analysed dimple textures given in Table 3, the dimple diameters are rather at the upper edge of the investigated texture dimensions, while the dimple distances and the depths are found in the lower range of the dimple dimensions. However, the dimple values are not similar. Thus, again it is obvious that a surface texture has always to be determined as a function of the operating conditions in order to achieve a maximum friction reduction. For each distinct application and individual technical requirement, textures have to be identified which result in an optimum friction with respect to a defined reference. These ROM results are not experimentally confirmed within the scope of this paper, as it would be necessary to re-produce rubber samples with appropriate textures, which was not part of MouldTex project [39].
