# Aerothermoelastic Analysis of Conical Shell in Supersonic Flow
## 2 Dynamic Model
### Constrained Mode
The stiffness coefficients \(Q_{ij}\) (\(i,j=1,\,2,\,6\)) are expressed as  
\[Q_{11}=Q_{22}=\frac{E(z,T)}{1-\mu^{2}(z,T)},\ Q_{12}=Q_{21}=\frac{\mu(z,T)E(z, T)}{1-\mu^{2}(z,T)},\ Q_{66}=\frac{E(z,T)}{2(1+\mu(z,T))}. \tag{8}\]  
The kinetic energy of the functionally graded conical shell is expressed as  
\[T_{s}=\frac{1}{2}\!\int_{-h/2}^{h/2}\int_{0}^{2\pi}\int_{0}^{L}\rho(\dot{u}^{2 }+\dot{v}^{2}+\dot{w}^{2})\cdot r(x)\mathrm{d}x\mathrm{d}\theta\mathrm{d}z. \tag{9}\]  
The strain energy of the stretching and bending of the functionally graded conical shell can be given by  
\[U_{\varepsilon}=\frac{1}{2}\!\int_{0}^{L}\int_{0}^{2\pi}\int_{-\frac{h}{2}}^{ \frac{h}{2}}(\sigma_{x}\varepsilon_{x}+\sigma_{\theta}\varepsilon_{\theta}+ \sigma_{x\theta}\varepsilon_{x\theta})r(x)\mathrm{d}x\mathrm{d}\theta\mathrm{d}z. \tag{10}\]  
The potential energy due to the thermal stresses is described as [22]  
\[U_{\Delta T}=\frac{1}{2}\!\int_{0}^{L}\int_{0}^{2\pi}N_{x}^{T}\left(\frac{ \partial w}{\partial x}\right)^{2}\!r(x)\mathrm{d}\theta\mathrm{d}\eta, \tag{11}\]  
where \(N_{x}^{T}\) is expressed as  
\[N_{x}^{T}=\int_{-\frac{h}{2}}^{\frac{h}{2}}\left[Q_{11}(z,T)+Q_{12}(z,T)\right] \beta(z,T)\Delta T(z)\mathrm{d}z. \tag{12}\]  
Considering that the temperature only changes along the thickness direction, the equation of heat conduction can be stated as below:  
\[-\frac{\mathrm{d}}{\mathrm{d}z}\left[k(z,T)\frac{\mathrm{d}T}{\mathrm{d}z} \right]=0, \tag{13}\]  
The thermal boundaries of the conical shell can be given by  
\[T=T_{m},\ z=h/2\ ;\ T=T_{c},\ z=-h/2. \tag{14}\]The solution of the equation, by means of polynomial series, can be written as [35]  
\[T(z)=T_{c}+\Delta T\cdot\eta(z),\ \ \Delta T=T_{m}-T_{c}. \tag{15}\]  
where  
\[ \begin{array}{c}\eta(z)=\frac{1}{c}\bigg{[}\left(\frac{2z+h}{2h}\right)-\frac{ k_{m}}{(p+1)k_{c}}\left(\frac{2z+h}{2h}\right)^{p+1}+\frac{k_{m}^{2}}{(2p+1)k _{c}^{2}}\left(\frac{2z+h}{2h}\right)^{2p+1}\\ -\frac{k_{m}^{3}}{(3p+1)k_{c}^{2}}\left(\frac{2z+h}{2h}\right)^{3p+1}+\frac{k_ {m}^{4}}{(4p+1)k_{c}^{2}}\left(\frac{2z+h}{2h}\right)^{4p+1}-\frac{k_{m}^{5}}{( 5p+1)k_{c}^{2}}\left(\frac{2z+h}{2h}\right)^{5p+1}\bigg{]},\\ C=1-\frac{k_{m}}{(p+1)k_{c}}+\frac{k_{m}^{2}}{(2p+1)k_{c}^{2}}-\frac{k_{m}^{3 }}{(3p+1)k_{c}^{2}}+\frac{k_{m}^{4}}{(4p+1)k_{c}^{2}}-\frac{k_{m}^{5}}{(5p+1)k _{c}^{2}},\end{array}  \tag{16}\]  
where \(k_{m}=k_{m}-k_{c}\). For thin conical shells, the temperature distribution can be considered linear:  
\[\eta(z)=\bigg{(}\frac{2z+h}{2h}\bigg{)}. \tag{17}\]  
The constrained modes of the composite conical shell are obtained using the Rayleigh-Ritz method. The displacement function of the structure is expressed as  
\[\left\{ \begin{array}{l}u(x,\theta,t)=U_{n}(x)\cos(n\theta)\sin\omega t=\sum \limits_{J=1}^{N}a_{Jn}\varphi_{u,J}(x)\cos(n\theta)\sin\omega t\\ v(x,\theta,t)=V_{n}(x)\sin(n\theta)\sin\omega t=\sum\limits_{J=1}^{N}b_{Jn} \varphi_{v,J}(x)\sin(n\theta)\sin\omega t\\ w(x,\theta,t)=W_{n}(x)\cos(n\theta)\sin\omega t=\sum\limits_{J=1}^{N}c_{Jn} \varphi_{w,J}(x)\cos(n\theta)\sin\omega t\end{array} ,\right. \tag{18}\]  
where \(n\) is circumferential wave number, and \(\omega\) is natural frequency. \(U_{n}(x)\), \(V_{n}(x)\) and \(W_{n}(x)\) are the meridional modal functions used to described the vibrational mode in the longitudinal direction, which can be approximated using the characteristic orthogonal polynomials satisfying the boundary conditions. \(N\) is the number of truncated terms used in practical calculation,\(a_{Jn}\), \(b_{Jn}\) and \(c_{Jn}\) are the unknown coefficients, and \(\varphi_{J,u}(x)\), \(\varphi_{J,v}(x)\) and \(\varphi_{J,w}(x)\) are the characteristic orthogonal polynomials which can be generated using the Gram-Schmidt procedure according to the boundary conditions. More details about the characteristic orthogonal polynomials are provided in the literature [36].