## 4 Numerical Analysis Results and Data Analysis
### Interaction Analysis  
#### 4.3.1 Interaction Plot  
Interaction is an effect that occurs at a specific level combination of two or more design parameters. That is, the design parameters do not independently affect the response, but affect each other dependently. Previously, whether all two-factor interactions were statistically significant was determined through multi-way ANOVA. By plotting these multi-way ANOVA results, it is convenient to intuitively understand the interaction effect. The interaction plot according to the combinations of levels between the two factors is shown in Figure 6.  
In Figure 6a,c,d, it can be seen that there is a significant interaction effect because the slope changes according to the change in the factor level. On the other hand, in Figure 6b,e,f, the slope does not change even when the level of the factor changes. Therefore, the significant interaction factors are joint orientation \(\times\) aspect ratio, joint orientation \(\times\) vehicle speed and aspect ratio \(\times\) cover depth, which is consistent with the results of ANOVA.  
Figure 6a is a two-factor interaction plot of joint orientation \(\times\) aspect ratio. When the joint orientation was horizontal, the vibration increased rapidly as the aspect ratio level changed from 1.0 (semicircular shape) to 0.5 (half elliptical shape with the long axis being the vertical axis). When the lava tube had a complex joint, the vibration value showed linearity according to the aspect ratio level, and in the case of no joint, horizontal joint, and vertical joint, the slope increased when the aspect ratio level was 0.5.  
Figure 6c shows the interaction between the vehicle speed, which is an external influence factor, and the joint direction, which is an internal characteristic factor. The no joint, horizontal joint and compound joint had similar slopes according to the vehicle speed level change. However, in the case of the vertical joint, the value of the vibration was relatively large, but the slope of the graph according to the change in the vehicle speed was relatively small.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Factor** & \(\mathcal{S}\)**S** & \(\boldsymbol{\phi}\) & \(\boldsymbol{V}\) & \(\boldsymbol{F}\) & \(\boldsymbol{p}\)**-Value** \\ \hline Joint orientation & 4.835 & 3 & 1.612 & 395.416 & \(1.31\times 10^{-23}\) \\ Aspect ratio & 1.033 & 2 & 0.516 & 126.706 & \(4.65\times 10^{-15}\) \\ Cover depth & 1.185 & 1 & 1.185 & 290.73 & \(1.18\times 10^{-16}\) \\ Vehicle speed & 5.638 & 1 & 5.638 & 1383.32 & \(5.03\times 10^{-26}\) \\ Joint orientation \(\times\) Aspect ratio & 0.115 & 6 & 0.019 & 4.69 & \(1.90\times 10^{-13}\) \\ Joint orientation \(\times\) Vehicle speed & 0.183 & 3 & 0.061 & 14.967 & \(4.51\times 10^{-6}\) \\ Aspect ratio \(\times\) Cover depth & 0.345 & 2 & 0.173 & 42.351 & \(2.49\times 10^{-9}\) \\ e & 0.118 & 29 & 0.004 & & \\ T & 13.451 & 47 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 10: ANOVA table after pooling non-significant factors.  
Figure 6d is an interaction plot between aspect ratio and cover depth. When the aspect ratio was 0.5, the slope increased rapidly at a cover height of 2 m. Through this, it is clear that when a lava tube has a shallow cover depth, it is greatly affected by vibration depending on its shape ratio.  
Figure 6: Interaction plots between two factors. (**a**) Joint orientation and aspect ratio; (**b**) Joint orientation and cover depth; (**c**) Joint orientation and vehicle speed; (**d**) Aspect ratio and cover depth, (**e**) Aspect ratio and vehicle speed; (**f**) Cover depth and vehicle speed.