# Aerothermoelastic Analysis of Conical Shell in Supersonic Flow
## 2 Dynamic Model
### Dynamic Equation  
Based on the mode function of the functionally graded conical shell obtained using the Rayleigh-Ritz method, the displacement components of the conical shell are stated as below:  
\[\left\{ \begin{array}{l}u(x,\theta,t)=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{ n}U_{ij}(x,\theta)q_{ij}^{u}(t)=\mathbf{U}\mathbf{q}_{u}(t)\\ v(x,\theta,t)=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}V_{ij}(x,\theta)q_{ij}^ {v}(t)=\mathbf{V}\mathbf{q}_{v}(t)\\ w(x,\theta,t)=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n}W_{ij}(x,\theta)q_{ij}^ {w}(t)=\mathbf{W}\mathbf{q}_{w}(t)\end{array} \right., \tag{23}\]  
where \(m\) is the modal truncation number. \(\mathbf{U}\), \(\mathbf{V}\) and \(\mathbf{W}\) are the mode functions in meridional, circumferential and radial directions, respectively. \(\mathbf{q}_{u}\), \(\mathbf{q}_{v}\) and \(\mathbf{q}_{w}\) are the generalized coordinates. According to the linear piston theory with a curvature correction term, the aerodynamic pressure generated by supersonic flowing through the outer surface of the functionally graded conical shell is given by [2; 32]:  
\[\Delta p=\frac{\gamma p_{\infty}M_{\infty}^{2}}{\left(M_{\infty}^{2}-1\right)^ {1/2}}\left[\frac{M_{\infty}^{2}-2}{M_{\infty}^{2}-1}\frac{1}{\alpha_{\infty} M_{\infty}}\frac{\partial w}{\partial t}+\frac{\partial w}{\partial x}-\frac{w}{2 r(x)\left(M_{\infty}^{2}-1\right)^{1/2}}\right], \tag{24}\]  
where, \(p_{\infty}\), \(\gamma\), \(M_{\infty}\) and \(\alpha_{\infty}\) are the free-stream static pressure, air-specific heat ratio, Mach number and speed of sound, respectively. It should be noted that the linearized first-order potential theory is valid for \(\sqrt{2}<M_{\infty}<5\). Moreover, since the flow over a shell behind the attached shock wave is nonuniform, the constant Mach number is effective for a sufficiently short shell [19]. The virtual work caused by the aerodynamic load is written as  
\[\delta W=-\int_{0}^{L}\int_{0}^{2\pi}\left(\Delta p\delta w\right)\cdot r(x) \mathrm{d}x\mathrm{d}\theta=Q_{p}\delta q_{w}, \tag{25}\]  
where the expression of generalized force \(Q_{p}\) is  
\[ \begin{array}{ll}Q_{p}&=\frac{\gamma p_{\infty}M_{\infty}^{2}}{\left(M_{ \infty}^{2}-1\right)^{1/2}}\int_{0}^{L}\int_{0}^{2\pi}\left[\mathbf{W}\frac{ \mathbf{W}^{T}}{\partial x}r(x)-\frac{1}{2\left(M_{\infty}^{2}-1\right)^{1/2} }\mathbf{W}\mathbf{W}^{T}\right]\mathrm{d}x\mathrm{d}\theta\mathbf{q}_{w}\\ &+\int_{0}^{L}\int_{0}^{2\pi}\left[\frac{M_{\infty}^{2}-2}{M_{\infty}^{2}-1} \frac{r(x)}{\alpha_{\infty}M_{\infty}}\mathbf{W}\mathbf{W}^{T}\right]\mathrm{ d}x\mathrm{d}\theta\dot{\mathbf{q}}_{w}.\end{array}  \tag{26}\]  
Substituting expressions (9)-(11), (23) and (26) into Lagrange equations,  
\[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{q}_{p}} \right)-\frac{\partial L}{\partial q_{p}}=Q_{p}, \tag{27}\]  
where \(L=T-U_{\epsilon}-U_{\Delta T}\), \(p=u,\ v,\ w\). The dynamic equation of the functionally graded conical shell can be described by the following equation:  
\[\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}_{\Delta P}\dot{\mathbf{X}}+(\mathbf{K}_ {\epsilon}+\mathbf{K}_{\Delta T}+\mathbf{K}_{\Delta p})\mathbf{X}=0, \tag{28}\]  
where \(\mathbf{X}=\left[\mathbf{q}_{u}^{T},\ \mathbf{q}_{v}^{T},\ \mathbf{q}_{w}^{T}\right]^{T}\). \(\mathbf{K}_{\epsilon}\) and \(\mathbf{M}\) are, respectively, the stiffness matrix and mass matrix of the functionally graded conical shell. \(\mathbf{K}_{\Delta P}\) and \(\mathbf{C}_{\Delta P}\) are, respectively, the stiffness matrix and damping matrix related to aerodynamic load, and \(\mathbf{K}_{\Delta T}\) is the stiffness matrix related to thermal load.