## 2 Dynamic Derivatives Identification Based on the Forced Oscillation Method  
At present, there are two commonly used methods to calculate the dynamic derivatives using CFD: the forced oscillation method and the free oscillation method. The forced oscillation method calculates the unsteady aerodynamic force of the aircraft's motion process by giving it a harmonic motion around the center of gravity, and finally calculates the dynamic derivatives. The free oscillation method is the disturbance motion of an aircraft with a given support, and it further identifies the dynamic derivatives. In comparison, the forced oscillation method has become the mainstream method for identifying dynamic derivatives due to its controllable motion conditions and easy implementation. Taking the pitch direction as an example, the forced oscillation method is generally given a simple harmonic oscillation, as follows:  
\[\alpha=\alpha_{1}+\theta=\alpha_{1}+\alpha_{\text{m}}\sin(2\pi ft) \tag{1}\]  
In the formula, \(\alpha_{1}\) is the initial angle of attack of oscillation; \(\alpha_{\text{m}}\) is the oscillation amplitude; \(f\) is the oscillation frequency. By numerically solving the time history of the aircraft undergoing forced pitch oscillation, the time history curve of the dynamic pitch moment coefficient \(C_{\text{m}}\) and the hysteresis loop \(C_{\text{m}}-\alpha\) can be obtained. By integrating the hysteresis loop mentioned above within one cycle, the dynamic derivatives at the angle of attack can be calculated. \[\left(\frac{\partial C_{\text{m}}}{\partial\theta}\right)0=\frac{1}{\alpha_{ \text{m}}\pi}\int\limits_{t-T}^{t+T}(C_{\text{m}}-C_{\text{m}0})\cos(2\pi ft)dt \tag{2}\]  
In the formula, \(C_{\text{m}0}\) is the pitch moment coefficient at the initial angle of attack; \(T\) is the oscillation period. The free oscillation method does not limit the rotational degrees of freedom of the aircraft, and the aircraft oscillates freely under its own aerodynamic moment. Taking the pitch direction as an example, the free pitch oscillation formula for a single degree of freedom without mechanical damping can be written as  
\[\overset{\bullet\bullet}{\theta}=\frac{1}{I}C_{\text{m}}=\frac{1}{I}\left(C_ {\text{m}0}+\left(\frac{\partial C_{\text{m}}}{\partial\theta}\right)0\theta+ \left(\frac{\partial C_{\text{m}}}{\partial\theta}\right)0\theta\right) \tag{3}\]  
In the formula, \(I\) is the dimensionless moment of inertia. After obtaining the time history curve of free pitch oscillation \(\theta=\theta(t)\) and \(C_{\text{m}}=C_{\text{m}}(t)\) through numerical calculation, the frequency of free pitch oscillation \(f=\frac{1}{I}\) can be determined. Then, by selecting two peak values of the same period, the dynamic derivatives at the equilibrium angle of attack can be obtained:  
\[\left(\frac{\partial C_{\text{m}}}{\partial\theta}\right)0=2 If\ln\frac{\theta _{2}}{\theta_{1}} \tag{4}\]  
The above methods take the pitch direction as an example to illustrate the calculation process of pitch dynamic derivatives. The yaw dynamic derivatives and roll dynamic derivatives can be calculated similarly. In addition, the above method obtains the combined derivatives, and it is necessary to add the simulation of translation motion to separate the combined items. Although the dynamic derivatives identification method based on the forced oscillation method is widely used, it has shortcomings. In general, it is necessary to simulate the steady-state flow field under computational conditions, obtain initial aerodynamic values, and then use dynamic grid technology to solve the unsteady flow field. Although the motion form is a periodic simple harmonic motion, it is affected by the initial inertial effect of the unsteady flow field, and it needs to calculate multiple periods to achieve the stability of the unsteady aerodynamic force, as shown in Figure 1. In the first cycle, peak data of unsteady aerodynamic force cannot be used due to initial effects.