## 2 Description of the Systems Under Investigation and Experimental Setups
### Random Vibration Testing
#### 4.2.1 Side Accelerometer  
Figure 11 depicts the random vibration testing results taken from the side accelerometer in the Y-direction. It follows from the plotted curves in Figure 9 that an obvious nonlinear trend was obtained in these tests, especially in the first and second peaks. For the central-cut cases, the first peak is near 100 Hz, which matches what is expected in the FEA and the impact tests shown in Figure 7. The mode shapes seen in Table 2 also verify that the response direction is the Y-direction, except for the no-cut case, which shows a higher response in the X-direction than in the Y-direction for the second mode of vibration. As a result, the data for mode 2 are not listed in Table 5, since it is difficult to distinguish using Figure 7d.  
Figure 11: Random vibration results from the Y-direction of the side accelerometer: (**a**) 0.5\({}^{\prime\prime}\) cut, (**b**) 0.25\({}^{\prime\prime}\) cut, (**c**) 0.1\({}^{\prime\prime}\) cut, and (**d**) no-cut BARC systems.  
By comparing the central-cut width results, trends should be easy to find. A percentage change is found by determining the change in frequency between the lowest excitation (\(1\times 10^{-7}\) V\({}^{2}\)/Hz) and the highest excitation (\(1\times 10^{-4}\) V\({}^{2}\)/Hz). For example, the system with the highest nonlinear softening is the 0.5\({}^{\prime\prime}\) cut width, particularly in mode 1, with a nonlinear softening of 4.41% between the fundamental frequencies of the lowest and highest excitations. Further, it should be noticed that as the central-cut width decreases, the nonlinear softening decreases as well. As stated, the 0.5\({}^{\prime\prime}\) cut has the highest nonlinear softening, but the no-cut case has the lowest nonlinear softening, with only a 1.01% decrease in the dominant frequency. This trend cannot be stated for the second mode, where the highest nonlinear softening is for the 0.25\({}^{\prime\prime}\) cut with a 1.12% decrease. Mode 2 for the 0.5\({}^{\prime\prime}\) cut and the 0.1\({}^{\prime\prime}\) cut has percentage decreases of 0.55% and 0.42%, respectively. These are almost linear, compared to the nonlinear softening behaviors seen in the first modes.  
Three important notes should be stated here. First, an additional frequency peak is present at around 200 Hz (may be a superharmonic of order 2 for the first mode) for the central case configurations, which can be due to the presence of quadratic nonlinearity in the system. This frequency peak's amplitude increases when the input excitation level is increased. Second, it should be mentioned that the sixth mode for the central-cut configurations shows the possible presence of nonlinear hardening. Indeed, the resonant frequency is increased by 0.91%, 1.2%, and 1.31% for 0.5\({}^{\prime\prime}\), 0.25\({}^{\prime\prime}\), and 0.1\({}^{\prime\prime}\), respectively, when the input excitation is increased from \(1\times 10^{-7}\) V\({}^{2}\)/Hz to \(1\times 10^{-4}\) V\({}^{2}\)/Hz. Third, inspecting the plots in Figure 11d and the dominant frequencies tabulated in Table 5, it can be stated that the no-cut configuration may have a nonlinear softening and then nonlinear hardening behavior. Indeed, when the input excitation level is increased, the dominant frequency decreases and then increases. As mentioned earlier, random vibration testing enables qualitative investigations for various modes of vibration, and hence, to precisely determine the type of nonlinearity (softening or/and hardening), quantitative investigations should be performed, as presented in the experimental harmonic testing section. In addition, it should be mentioned that, due to the coupling between the removable component and the box assembly, as well as the complexity of the BARC system, random vibration testing might result in the collection of low-quality data, which may result in a misinterpretation of the results.