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

55 5

(°) + (°) 29.10 0.13

(s -1 ) 5.68

Table 1. Input values and calculated results of logarithmic decrement and damping ratio for two different rolling surfaces.

Rolling surface

n

δ

Without rubber tubing With rubber tubing

26 13

0.21 0.40

0.03 0.06

31.46 6.09 The damping ratio values were further used to generate the viscous damping curves directly on the response graphs. Alongside this curve, a linear line connecting the amplitudes and + is also added to the graphs. This line represents the damping caused by dry friction. The envelope of viscous damping is described by the following equation: = − , (3) where is the natural angular frequency of the absorber and t is time. The natural frequency of the absorber was determined as the average period length of the first five free oscillations; the corresponding values are listed in Table 1. The response graphs are shown in Fig. 3(a) and Fig. 3(b). Figure 3(a) depicts the response of the ball moving along the track without the tubing, while Fig. 3(b) shows the response of the track with the rubber tubing applied. 0.18

Fig. 3. Free oscillation response of the centre of the ball with added envelopes representing 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.

Experiments show that oscillations decay at slower rates when the spherical absorber moves over a surface without tubes. This conclusion is supported both by visual observation and by quantitative analysis using the logarithmic decrement. In this case, approximately 26 cycles are needed for the oscillations to fully decay. On the contrary, for motion along the surface with rubber tubing, the damping is significantly faster, with stabilisation occurring after only 13 cycles. The higher damping level is also confirmed by a higher value of the logarithmic decrease in this case. When analysing the amplitude decay graphs, it is evident that, for the surface without tubing, the response curve lies between the envelope characterising viscous damping and the straight line corresponding to friction damping. This suggests that the actual damping is a combination of both mechanisms. In contrast, for the absorber moving on the surface with tubing, the amplitudes clearly converge to the line representing the friction model. To evaluate which of the considered models provides the best representation of the real damping process, a quantitative analysis was performed in the form of a functional approximation (fitting) to the measured data. This procedure allowed a more accurate determination of the nature of the amplitude decay and which models best fit the experimental observations.