PSI - Issue 73

Marek Kawulok et al. / Procedia Structural Integrity 73 (2025) 51–57 Marek Kawulok et al. / Structural Integrity Procedia 00 (20 2 5) 000 – 000

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In the first step of the fitting process, all peak amplitudes were extracted from the time response and then ordered according to their sequence (peak index). For viscous damping, equation (3) was modified to the following form (Al Hababi et al., 2020): , = − (4) where and are parameters adjusted during the fitting. The value represents the maximum amplitude at the start of the motion, and corresponds to the logarithmic decrement of damping. In the case of the linear dry friction model, the damping is described by a line: , = − (5) in which friction is the initial amplitude and is the damping coefficient indicating how much the amplitude decreases with each successive cycle. The quality of the fit was evaluated using the coefficient of determination (R 2 ) and the root mean square error (RMSE). The resulting values are presented in Table 2, while the graphical representation of the fits can be seen in Figure 5 for the ball moving on the untreated surface and in Figure 6 for the ball moving on the surface with applied rubber tubing.

Damping model (°) (°/cycle) R 2 Viscous 32.37 0.09 -

Table 2. Results of the fitting analysis.

Rolling surface

RMSE (°)

Without rubber tubing

0.97 0.98 0.96 0.99

1.37 1.18 1.92 0.98

Friction Viscous Friction

26.16 39.52 31.73

-

1.04

With rubber tubing

0.17

-

-

2.40

Fig. 4. Sorted amplitudes of free oscillations with fitted curves for viscous and frictional damping. Graph (a) shows the case without rubber tubing on the track, while graph (b) corresponds to the case with rubber tubing.

From the quantitative fitting analysis, it is evident that, for the ball moving on the surface without tubing, the damping cannot be attributed to a single model. Although both the viscous and friction models have high coefficients of determination (R 2 = 0.97 and 0.98), the difference in the RMSE values (1.37 vs. 1.18) indicates that the friction model better fits the experimental data. This is also confirmed by visual analysis of the graph, where the response amplitudes lie between the viscous and friction damping curves, suggesting that the actual damping may be a combination of both mechanisms. The initial amplitudes estimated by the models also differ, with the friction model predicting a lower initial amplitude (26.16) compared to the viscous model (32.37), possibly due to differences in the approximation of damping.