# Effects of Central Cut Width on the Dynamical Characteristics of Box Assembly Structure
## 3 Effect of the Central Cut on the Box Assembly Characteristics: Computational Modeling
#### 4.3.2 Harmonic Vibration Testing Investigating Results from the X-Direction  
When comparing the Y-direction harmonic testing results to the X-direction counterpart, it is shown how difficult it is to observe the present nonlinearities in each system due to the dominance of the Y oscillation for the first modes of vibration. This can be seen in the frequency response curves in Figure 12. Each system is seen to have mode coupling as well with the poor quality of data shown. It can also be shown how low the amplitudes of oscillation will be when gathering data from this direction. Previously, the Y-direction showed an amplitude nearing almost 60 g's. Here, in the X-direction, the amplitude is almost reaching only 6 g's. \begin{table}
\begin{tabular}{c|c c c c c c c} \hline  & \multicolumn{2}{c}{**0.5\({}^{\prime\prime}\) Cut (g/V)**} & \multicolumn{2}{c}{**0.25\({}^{\prime\prime}\) Cut (g/V)**} & \multicolumn{2}{c}{**0.1\({}^{\prime\prime}\) Cut (g/V)**} & \multicolumn{2}{c}{**No Cut (g/V)**} \\ \cline{2-7}  & **L Peak** & **R Peak** & **L Peak** & **R Peak** & **L Peak** & **R Peak** & **Peak** \\ \hline
50 mV & 179.20 & 685.33 & 151.53 & 706.63 & 104.27 & 492.21 & 158.22 \\ \hline
100 mV & 175.41 & 531.31 & 120.06 & 444.04 & 107.73 & 414.06 & 104.00 \\ \hline
125 mV & 173.51 & 482.87 & 119.72 & 451.21 & 111.22 & 381.48 & 92.976 \\ \hline
150 mV & 168.59 & 447.98 & 125.33 & 474.42 & 106.68 & 338.04 & 85.813 \\ \hline
**\% Change** & 5.92\% & 34.63\% & 17.24\% & 32.86\% & -2.31\% & 31.32\% & 45.76\% \\ \hline \end{tabular}
\end{table} 
Table 10: Amplitude found using harmonic vibration testing from the output/input plots of the side accelerometer. Figure 11: Time history of the 0.5\({}^{\prime\prime}\) cut case near resonance from the side accelerometer at 150 mV excitation, showing **(a)** 114 Hz (resonance), **(b)** 114.5 Hz, and **(c)** 115 Hz. Figure 12: FRF harmonic testing results in X-direction from (**a**) 0.5” cut top, (**b**) 0.5” cut side, (**c**) 0.25” cut top, (**d**) 0.25” cut side, (**e**) 0.1” cut top, (**f**) 0.1” cut side, (**g**) no cut top, and (**h**) no cut side showing backbone trends among each test. The tabulated dominant frequencies can be seen in Table 11. The top accelerometer is chosen for this analysis into the first mode of vibration. A nonlinear softening trend can be observed in all the test results. Again, notice how a softening trend occurs when comparing peaks between one another. As the cut width decreases, so do the dominant frequencies. This trend is true for both the left and right peaks. The system with the highest nonlinear softening is the 0.1\" peak, primarily in the right peak with a percent change value of 1.09%. This value is very similar to what is seen in Table 9.  
3.3 Discussion of the Impacts of the Central Cut Width on the Dynamics of the Box Assembly Structure Using Harmonic Vibration Testing  
To further examine how the central cut width (1) impacts the existence and strength of nonlinearities and (2) influences the overall dynamics of the system in terms of the broadband regions, dominant frequencies, and amplitudes of oscillation, a deep investigation was performed for determining the quality factor of each box assembly configuration and a comparative study was carried out. First, the quality factor for each accelerometer in the Y-direction was calculated and is depicted in Figure 13. Finding the quality factor allows for a better understanding of the nonlinear damping present in the system, particularly since the resonant frequency does not vary much in terms of percentage (less than 1.5%) for all the box assembly configurations, as shown in Table 11. As demonstrated earlier, the first modes 1(a) and 1(b) are mostly oscillating in the Y-direction. Further, the X-direction plots have a lot of variability due to the low levels of output amplitudes and the existence of several harmonics in the time history, so the main focus in this section is on the Y-direction results.