## 4 The Aeroelastic Coupling Effects in Maneuvering Flight of Slender Body
### The Forced Pitching Motion of Aircraft Considering Elastic Effect  
In the wind tunnel test, when the model is forced in the longitudinal pitch degree of freedom, it needs to be subjected to an external driving force. According to the D'Alembert's principle, the force (moment) acting on the model is equal to the sum of its own inertial force, driving force, and aerodynamic force, and can be expressed as follows:  
\[\ddot{I\theta}=M_{aero}+M_{d} \tag{30}\]  
where \(\theta\) is the change in pitch angle, \(I\) is the moment of inertia of the model, \(\ddot{I\theta}\) is the moment of inertia, \(M_{aero}\) is the aerodynamic moment, and \(M_{d}\) is the motor driving moment.  
Compared with rigid-body models, elastic models exhibit altered aerodynamic moment \(M_{aero}\) due to deformation-induced changes in surface pressure distribution, leading to discrepancies in response characteristics. To investigate the aeroelastic effects on aircraft longitudinal pitch dynamics, this chapter analyzes the forced pitching motion characteristics of the missile. In the calculation process, the degrees of freedom of the yaw and roll directions of the missile are fixed, and its longitudinal degree of freedom is released, giving the law of the forced pitch motion of the missile:  
\[\alpha(t)=\alpha_{0}+\alpha_{m}sin(wt) \tag{31}\]  
where the initial angle of attack is \(\alpha_{0}=0\), the amplitude is \(\alpha_{m}=5^{\circ}\), and the natural angle frequency is \(w=10\).  
Figure 13: The computational model and surface grid.  
In order to explore the influence of stiffness on the stability of the missile in detail, the calculation results of four states are selected for comparative analysis, namely the rigid state, the initial structural stiffness state, the one-half initial structural stiffness state and the one-quarter initial structural stiffness state; the initial angle of attack is \(\alpha_{0}=0\) and the computational Mach number is 1.8.  
Figure 14 demonstrates the temporal evolution of first-order generalized displacement under varying stiffness conditions. The results reveal an inverse correlation between structural stiffness and displacement amplitude: lower stiffness values yield greater generalized displacements, indicating enhanced missile body deformation. Furthermore, systems with reduced stiffness exhibit prolonged transient phases before achieving a stable sinusoidal displacement pattern.  
The hysteresis loop change in the pitching moment coefficient with the pitching angle of the rigid model and the model under different stiffness levels is presented in Figure 15; the hysteresis loop of the rigid body model and the initial structural stiffness state is clockwise, and as the stiffness continues to decrease, the direction of the hysteresis loop changes, indicating that the dynamic stability of the calculated model changes, that is, the stability changes from dynamic stability to dynamic instability, which indicates that if the aeroelastic effect of the aircraft is considered, the stability analysis method based on the rigid body form loses its accuracy.  
In order to show the deformation of the model more intuitively, Figure 16 shows the deformation of the missile at a certain point in time, where red represents the rigid body model and green represents the elastomeric model; it can be seen that the deformation of the missile is mainly first-order bending.  
The results of the comparison of the flow field are shown in Figure 17, which presents both the pressure contour of the rigid model and the elastic model at different motion moments, due to the deformation, the pressure distribution of the rigid and elastic aircraft changes, and it can be seen that the shock area of the elastic model is smaller than that of the rigid model in the lower part of the tail.  
Figure 14: First-order generalized displacements with different stiffness.