*3.5. Vibration Damping Properties*

Examples of the frequency dependencies of the transfer damping function of the tested rubber composites measuring *t* = 10 mm in thickness with different carbon black particle sizes are shown in Figure 12. It is evident from this comparison that the size of the carbon black particles has a significant influence on the vibration damping properties. It can be concluded that the material´s ability to damp mechanical vibration generally increased with an increase in the carbon black particle size. This is caused by lower stiffness (or by higher damping) of the rubber composites, which were produced with larger carbon black particle sizes. These facts result in a higher transformation of input mechanical energy into heat during forced oscillations [37] and a decrease in the values of the damped and undamped natural frequencies [38]. Therefore, the first resonance frequency (*fR*1) value was shifted to the left (see Figure 12) with an increase in the carbon black particle size, i.e., from 1520 (N 110) to 968 Hz (N 990), as indicated in Table 4. These findings are in excellent agreemen<sup>t</sup> with the results that were experimentally determined by the abovementioned methods, namely the hardness, tensile, shear, and viscoelastic measurements. It was verified in these cases that the increasing carbon black particle size led to a decrease of the break stress, the Shore A hardness, and the storage moduli *E´* and *G´*. In contrast, the rebound resilience was higher for these particle sizes.


**Table 4.** The first resonance frequency (fR1) in Hz of the studied rubber composite materials, as induced by harmonic force vibration for different rubber thicknesses and inertial masses.

The vibration damping properties of the investigated harmonically loaded composite rubber samples are also influenced by their thickness *t*, the excitation frequency *f*, and the inertial mass *m*. The effect of the inertial mass on the vibration damping properties of sample N 330 is shown in Figure 13. It is visible that better damping properties were obtained with higher inertial mass *m*, which led to a decrease in the undamped natural frequency, and thus, the damped frequency. This is due to the fact that the natural frequency of an undamped system is proportional to the square root of the material stiffness for the applied inertial mass [22]. For this reason, the inertial mass has a positive influence on vibration damping, which is reflected by a shift of the first resonance frequency peak position to lower frequencies, i.e., by the decrease of the *fR*1 (see Table 4) from 1441 (*m* = 0 g) to 412 Hz (*m* = 500 g). The vibration isolation properties of the investigated rubber composites are also significantly influenced by their thickness *t*, as shown in Figure 14 for the N 330 sample with an inertial mass *m* of 90 g. It is evident that the higher material thickness led to lower values of the *fR*1, i.e., from 1138 (*t* = 10 mm) to 520 Hz (*t* = 20 mm), as indicated in Table 4. For this reason, the rubber thickness generally has a positive influence on vibration damping properties. It is also visible from Figures 12–14 that the material´s ability to damp mechanical vibration is significantly influenced by the excitation frequency *f*. It is evident that resonant mechanical vibration (*D* < 0) was achieved at low excitation frequencies, depending on the rubber sample type, the thickness *t*, and the inertial mass *m*. For example, for the N 110 sample type measuring *t* = 10 mm in thickness and without inertial mass (*m* = 0 g), the resonant mechanical vibration was observed at frequencies *f* < 2950 Hz (see Figure 12). For the N 990 sample type measuring *t* = 20 mm in thickness and with inertial mass *m* = 500 g, the resonant mechanical vibration was achieved at considerably lower excitation frequencies (at *f* < 330 Hz). In contrast, damped mechanical vibration (*D* > 0) was generally achieved at higher excitation frequencies (see Figures 12–14).

The material´s ability to damp mechanical vibration was also evaluated for the tested material samples, which were harmonically loaded by the compression force with an amplitude of 10 kN at an excitation frequency of 20 Hz. In the case of the N 990 sample type, it was not possible to perform this evaluation due to the low stiffness of this rubber sample compared to the other tested rubber sample types. The frequency dependencies of the transfer damping function of the investigated rubber samples (thickness *t* = 20 mm, inertial mass *m* = 90 g) after 750,000 loading cycles are demonstrated in Figure 15. Again, as in the case of the cyclically unloaded rubber samples (see Figure 12), the material´s ability to dampen mechanical vibration increased with an increase in the carbon black particle size. For this reason the rubber composites, which were produced with larger carbon black particle sizes, exhibited lower stiffness, resulting in a decrease of the first resonance frequency peak position to lower excitation frequencies. As shown in Table 5, similar results were obtained independently of the number of loading cycles.

**Figure 12.** Frequency dependence of the transfer damping function (D) for the tested rubber composite measuring t = 10 mm in thickness, without inertial mass (m = 0 g).

**Figure 13.** Frequency dependencies of the transfer damping function (D) for the tested N 330 rubber composite measuring t = 10 mm in thickness, loaded with different inertial masses.

**Figure 14.** Frequency dependencies of the transfer damping function (D) for the tested N 330 rubber composites of different thicknesses and loaded with inertial mass m = 90 g.


**Table 5.** The first resonance frequency (fR1) in Hz of the studied rubber composite materials measuring t = 20 mm in thickness, as induced by harmonic force vibration for different inertial masses and numbers of loading cycles.

The effect of the number of loading cycles on the vibration damping properties of the N 330 sample type is shown in Figure 16. It is evident that the vibration damping ability of the sample generally increased with an increase in the number of loading cycles, which led to a decrease of the first resonance frequency peak position to lower frequency values (see Table 5) with the increasing number of loading cycles. Therefore, the higher number of loading cycles led to a reduction in rubber sample stiffness, which was accompanied by better damping properties in this rubber sample. As shown in Table 5, similar findings were observed for the other tested rubber composites.

**Figure 15.** Frequency dependencies of the transfer damping function (D) for the tested rubber composites measuring t = 20 mm in thickness after 750,000 loading cycles, with inertial mass m = 90 g.

**Figure 16.** Effect of number of loading cycles on frequency dependencies of the transfer damping function (D) for the N 330 rubber composite measuring t = 20 mm in thickness, with inertial mass m = 500 g.
