## 4 Analysis and Modeling of Aircraft Parameters
### Analysis of Single Aircraft Parameter  
The aircraft parameters group studied in this paper includes lift coefficient, drag coefficient, inlet area, specific impulse, aircraft mass, and reference area. These six parameters can be divided into three aspects, as shown in Figure 8: aerodynamic shape (lift coefficient and drag coefficient), propulsion system (inlet area and specific impulse), and structural design (aircraft mass and reference area). The values of these six aircraft parameters are adjusted to explore their influence on the optimal cruise point denoted by (\(Ma_{opt}\), \(h_{opt}\)). The ratio between the value after adjustment and the original value is denoted by \(X_{L}\), \(X_{D}\), \(X_{A}\), \(X_{I}\), \(X_{m}\), and \(X_{S}\), respectively. Within the range of 0.8 to 1.5, the influence of a single factor in (\(X_{L}\), \(X_{D}\), \(X_{A}\), \(X_{I}\), \(X_{m}\), \(X_{S}\)) is investigated. Only one variable is changed at a time, and the rest remain at 1. After the aircraft parameters are adjusted, the corresponding optimal cruise points are obtained by the PSO algorithm in Figure 5.  
Figure 9 shows the influence of different factors. From Figure 9a, \(Ma_{opt}\) has a positive correlation with \(X_{A}\) and \(X_{I}\), and the effects of \(X_{A}\) and \(X_{I}\) on \(Ma_{opt}\) are very similar, which indicates that, if the magnitude of thrust determined by \(X_{A}\) and \(X_{I}\) is larger, \(Ma_{opt}\) will  
Figure 8: Considered parameters in this paper. Figure 7: The location of optimal cruise point in the contour of fuel consumption. increase as well. Further, \(Ma_{opt}\) has a negative correlation with \(X_{D}\) and \(X_{S}\), and the effects of \(X_{D}\) and \(X_{S}\) on \(Ma_{opt}\) are also similar, which indicates that, if the magnitude of drag determined by \(X_{D}\) and \(X_{S}\) is larger, \(Ma_{opt}\) is lower. Differently, \(X_{L}\) and \(X_{m}\) have little effect on \(Ma_{opt}\). It can be seen from Figure 9b that \(h_{opt}\) is sensitive to all the factors. In detail, \(h_{opt}\) is positively correlated with \(X_{L}\), \(X_{A}\), \(X_{l}\), and \(X_{S}\), and the effects of \(X_{A}\) and \(X_{I}\) on \(h_{opt}\) are very similar, which indicates that the magnitude of thrust determined by \(X_{A}\) and \(X_{I}\) also has a great impact on \(h_{opt}\), while \(h_{opt}\) has a negative correlation with \(X_{D}\) and \(X_{m}\). A multiple regression analysis is employed to quantify the influence of the six factors on the optimal cruise point, and the regression coefficient is normalized as an index of sensitivity. Figure 10 shows the sensitivity percentages of the different factors, where a negative sign indicates a negative correlation between the factor and the dependent variable. It can be seen that \(X_{A}\) and \(X_{I}\) have the largest impact on the optimal cruise Mach number, accounting for nearly 30%, followed by \(X_{D}\) and \(X_{S}\), which are negatively correlated, while the sensitivity of \(X_{L}\) and \(X_{m}\) is almost 0; \(X_{A}\), \(X_{I}\) and \(X_{m}\) have the greatest influence on the optimal cruise altitude, accounting for about 20%, where the influence of \(X_{m}\) is negative. Based on the conclusions above, the inlet area, specific impulse, and drag coefficient have the largest impact on the optimal steady-state cruise Mach number. If the optimal cruise Mach number needs to be enhanced, the inlet area and specific impulse should be larger or the drag of aircraft should be reduced. Due to the inlet area, the specific impulse and aircraft mass have the largest impact on the optimal cruise altitude. To improve the optimal cruise altitude, the most effective way is to increase the inlet area and specific impulse, or reduce the weight of the aircraft. Figure 10: Sensitivity of different factors on the optimal cruise point: (**a**) Sensitivity on \(Ma_{opt}\); (**b**) Sensitivity on \(h_{opt}\). Figure 9: Influence of different factors on the optimal cruise point: (**a**) Influence on \(Ma_{opt}\); (**b**) Influence on \(h_{opt}\).