## 3 Sensitivity Analysis of Aerodynamic Characteristics
The first-order sensitivity \(\hat{S}_{i}\) and total sensitivity \(\hat{S}_{i}^{\mathrm{T}}\) of the input \(d_{i}\) are solved by the following method [21; 27]. \[\hat{S}_{i}=\frac{\hat{D}_{i}}{\hat{D}}\ \,\ \ \ \ \ \ \  
center of mass is positioned at a distance of 0.12 m from the head of the model. The basic settings are as follows: the structural mesh is used to divide the small area around the model. A component mesh is generated, and then the structural mesh is also used to divide the entire flow field calculation domain and generate a background mesh. Finally, the two sets of meshes are merged together by overlapping mesh technology, as shown in Figure 3. The overall calculation is based on the FVM, the convection terms of the governing equations are discretized according to the second order upwind scheme, and the time discretization adopts the first-order implicit format to ensure the convergence and stability of the calculation. Due to the large background mesh range, the influence of the projectile on the far-field boundary is negligible, so the far-field boundary of the computational domain is set to the pressure far-field boundary condition. This boundary can only be used for compressible gases, and the density of compressible gases must be obtained from the ideal gas equation of state. The same far-field flow velocity and flow direction are set at these boundaries in order to simulate the initial flight speed and angle of attack of the projectile. The surface of the projectile body is set to a non-slip solid wall boundary condition. The projectile is assumed to move synchronously at the same rotational speed following the rotating area. The interface between the non-rotating and rotating regions is set to the overset interface boundary condition. The initial parameters of the flow around the flow field used in the numerical simulation are presented in Table 1.  
The selection of the number of meshes and the physical time step of the model has a significant impact on the simulation results, particularly in the context of unsteady calculation. Therefore, it is essential to select a reasonable time step and the number of meshes in the simulation. In order to simulate the forced vibration of the cylindrical projectile model in the dynamic derivative simulation, the component mesh can be made to vibrate with the wall of the model by overlapping mesh technology. A vibration period is divided into 200 steps, with each time step being \(4.9\times 10^{-5}\) s. Table 2 shows the calculation results of the drag coefficient \(C_{d}\) and the lift coefficient \(C_{1}\) by using different grid numbers at inlet flow field velocity of 0.3 Ma and initial attack angle of \(30^{\circ}\). It can be observed that, as the mesh is progressively encrypted, the aerodynamic characteristics parameters converge. Considering that the discrepancy in the calculation results for the number of grids in the three groups is minimal, the subsequent calculations are based on 110,000 grids. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Mach (Ma)** & **Pressure (Pa)** & **Temperature (K)** & **Reduction** & **Amplitude** & **Roll Angle** \\  & & & & **Frequency \(k_{d}\)** & \(\alpha_{m}\)/deg & \(\phi\)/deg \\ \hline Value & 0.3 & 100 & 288 & 3 & 1 & 90 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the flow field around the model under initial certain conditions. Figure 3: Meshing of the external flow field zone and surface of the projectile. On this basis, the numerical simulation method established in Section 2 is employed to compute the flow field and aerodynamic parameters of the cylinder projectile model in the range of 0\({}^{\circ}\)to 90\({}^{\circ}\)angles of attack. Figures 4 and 5 illustrate the velocity distributions of the circumferential flow field in the forced vibration state of the projectile. The initial angles of attack of the model are \(\pm 1^{\circ}\) at 0\({}^{\circ}\), 10\({}^{\circ}\), 30\({}^{\circ}\), 60\({}^{\circ}\), and 90\({}^{\circ}\), respectively.