## 2 Aerodynamic Characteristics Analysis Model of a Cylindrical Projectile
### Dynamic Derivative Identification  
The forced vibration method [24] has high accuracy and computational efficiency, and is a widely used method for calculating the dynamic derivative of a projectile. In this paper, the forced pitch vibration method is employed to calculate the dynamic derivative of the projectile. When the projectile is subjected to a forced vibration with a small amplitude, the dynamic derivative can be rewritten according to the assumption of the linearized aerodynamic theory of small perturbations as  
\[C_{\rm m}(t)=C_{\rm m00}+C_{\rm m}^{\alpha}\Delta\alpha+C_{\rm m}^{\dot{\alpha }}\Delta\dot{\alpha}+C_{\rm m}^{\dot{\alpha}}\Delta q+C_{\rm m}^{\dot{q}} \Delta\dot{q}+\hat{\Delta}(\Delta\alpha,q) \tag{6}\]  
where \(C_{\rm m0}\) represents the static pitching moment of the cylindrical projectile at an angle of attack of \(\alpha_{0}\), \(C_{\rm m}^{\dot{\alpha}}\) is the zeroth-order static derivative of the pitching moment with respect to the angle of attack, \(C_{\rm m}^{\dot{\alpha}}\) is the first-order derivative of the pitching moment with respect to the angle of attack, \(q\) is the pitch angular velocity of the projectile vibration, \(C_{\rm m}^{q}\) is the rotational derivative of the pitching moment with respect to the angular velocity of the pitching motion or the derivative of the damping, \(C_{\rm m}^{\dot{q}}\) is the first-order derivative of the pitch moment with respect to the angular velocity of the pitching motion, and \(\hat{\Delta}(\Delta\alpha,q)\) represents the higher-order derivative term of \(\Delta\alpha\) and \(q\). \[\Delta\alpha=\alpha_{\rm m}\sin(\omega_{\rm q}t),\ \Delta\dot{\alpha}=\omega \alpha_{\rm m}\cos(\omega_{\rm q}t)=q,\ \Delta\ddot{\alpha}=-\omega^{2}\alpha_{\rm m}\sin(\omega_{\rm q}t)=\Delta \dot{q} \tag{7}\]  
In the numerical calculation, a vibration period is divided into \(N\) equal parts, and the aerodynamic force of the projectile at each time step can be calculated to obtain the discrete aerodynamic data of \(N\) groups of the projectile in the whole vibration cycle, and then the combined dynamic derivative can be obtained based on Equation (8), as follows  
\[ \begin{split} C_{\rm m}^{\dot{\alpha}}+C_{\rm m}^{q}& =\frac{2}{\omega_{\rm q}\alpha_{\rm m}T_{0}}\int_{-T_{0}/2}^{T_{0}/2}C_{\rm m }(t)\cos(\omega_{\rm q}t)\mathrm{d}t\\ &=\frac{2}{\omega_{\rm q}\alpha_{\rm m}T_{0}}\sum_{i=1}^{N-1} \big{(}\frac{C_{\rm m}(t(i))\cos(\omega_{\rm q}t(i))+C_{\rm m}(t(i+1))\cos( \omega_{\rm q}t(i+1))}{2}\big{)}\Delta t\end{split}  \tag{8}\]  
Based on the assumption of the linearized aerodynamic theory of small perturbations, the amplitude of the forced vibration should not be too large. In this paper, the amplitude of the forced vibration is set at \(1^{\circ}\). A new parameter, dimensionless reduction frequency \(k_{\rm d}\), is introduced, which is expressed as follows:  
\[k_{\rm d}=\frac{\omega_{\rm q}L_{\rm r}}{2V_{\infty}} \tag{9}\]  
where \(L_{\rm r}\) represents the characteristic length of the cylindrical projectile. Given that the cylindrical projectile is to be flown at a large angle of attack, the total length of the projectile body is taken as \(L_{\rm r}\), and \(V_{\infty}\) is the velocity of the free flow (refer to the standard value for the working condition setting). According to the test results of the design index, it is considered that the process of actively adjusting the 90\({}^{\circ}\) pitch attitude angle of the test specimen in 70 ms is its typical flight state. The typical average angular velocity can be selected as \(\overline{\omega_{\text{t}}}\) = 22.44 rad/s.