## 2 Description of the Systems Under Investigation and Experimental Setups
### Harmonic Vibration Testing
#### 4.3.3 Harmonic Vibration Testing: Discussion on the Influence of the Central-Cut Width  
The harmonic vibration testing results show an overall nonlinear softening behavior throughout the 0.5\({}^{\prime\prime}\) and 0.25\({}^{\prime\prime}\) central-cut width systems, with a maximum nonlinear softening seen in the 0.25\({}^{\prime\prime}\) cut system, having a percentage change of 0.60% in the Y-direction of both the RC and side accelerometers. Interestingly, nonlinear hardening is strongly observed in the no-cut systems. After investigating all these accelerometers near the fundamental mode of vibration, a comparison between them takes place in Figure 20. It follows from the plotted frequency response functions in this figure that the RC accelerometer in the Y-direction has the highest amplitude for any accelerometer analyzed, followed by the side accelerometer in the Y-direction, with the RC accelerometer in the Z-direction having the lowest oscillating amplitudes. This shows that there is a majority of movement in the Y-direction for the first mode, as already examined in both the FEA and the impact data. Figure 19: A comparison of central-cut width configurations using the RC accelerometer in the Y-direction for the 50 mV and 150 mV input levels of excitation. Figure 20: A comparison using the Y-direction data from the side and RC accelerometers and Z-direction data from the RC accelerometer using the (**a**) 50 mV and (**b**) 150 mV harmonic excitation data. Figure 21 shows the results of harmonic testing from the side and RC accelerometers in the Y-direction and the RC accelerometer in the Z-direction for the 0.5\({}^{\prime\prime}\) cut system, using data from the 150 mV excitation test. Notice the various amplitudes at each excitation frequency. At resonance (\(\sim\)102.4 Hz), the amplitude is highest for the system. The amplitudes before and after resonance (\(\sim\)102 and \(\sim\)102.8, respectively) are lower than what was seen at resonance. Additionally, it should be noted that there is a difference in amplitude at the same frequency between accelerometers. The RC accelerometer experiences much higher oscillations in the Y-direction than the side accelerometer, and the RC accelerometer in the Z-direction shows the lowest amplitudes, matching what was discussed when comparing each of these in Figure 20.  
From the harmonic testing results carried out in this investigation, it should be noted that the resonant amplitude of the output/input is varied when the input excitation is increased, which indicates the presence of nonlinear damping in these BARC structures. Indeed, for a linear system, the output/input or transmissibility should be constant. The graphs of output/input are omitted for the sake of brevity. On the other hand, a quantitative analysis to find the quality factors for the systems under investigation was conducted. The quality factor provides a comprehensive understanding of the nonlinear damping in each system. The results are displayed in Figure 22 and depict the quality factors  
Figure 21: Time histories found around resonance from harmonic testing for the 0.5\({}^{\prime\prime}\) cut configuration with 150 mV excitation for the (**a**) side accelerometer and (**b**) RC accelerometer in the Y-direction and (**c**) RC accelerometer in the Z-direction. found from the first mode using the RC accelerometer in the Y-direction. The results presented in Figure 22 show that the Q-factor has an overall decreasing trend as the input excitation level is increased. This result indicates that nonlinear damping is present in each system, especially the no-cut system, with a monotonic increase in damping (a decrease in the Q-factor). It should be mentioned that the resonant frequency is only varied by a small percentage, as indicated in Table 9, and hence, the effective damping in the system should have a similar trend to the inverse of the quality factor.