# Free Vibrations and Flutter Analysis of Composite Plates Reinforced with Carbon Nanotubes
## 2 Materials and Methods
### Equations of Motion
Assuming that the pair of edges in the \(y\)-direction is simply supported, the solution of Equation (13) is looking for the out-of-plane deflection \(w\) in the form of a single Fourier series  
\[w(y)=B_{n}sin\Big{(}\frac{n\pi y}{b}\Big{)},\quad n=1,\ 2,\ \cdots, \tag{14}\]  
where \(n\) is the wave number in \(y\)-direction and \(B_{n}\) are constants. The solution is assumed to be in the form  
\[w=e^{rx}, \tag{15}\]  
where \(r\) denotes the roots of the characteristic equation. Thus, the solution is a typical eigenvalue problem. Using the above relations and eliminating the damping, Equation (13) is reduced to a one-dimensional problem with an unknown parameter \(r\) in the form  
\[D_{11}\frac{r^{4}}{a^{4}}-2(D_{12}+2D_{66})\frac{r^{2}}{a^{2}}\big{(}\frac{n \pi}{b}\big{)}^{2}+D_{22}\big{(}\frac{n\pi}{b}\big{)}^{4}-N_{x}^{0}\frac{r^{2} }{a^{2}}+N_{y}^{0}\big{(}\frac{n\pi}{b}\big{)}^{2}-\rho h\omega^{2}+ \tag{16}\]  
where \(\omega\) denotes frequency. The solution of the flutter problem (Equation (16)) is related to the applied boundary conditions along the \(x\)-axis. The boundary conditions considered here are:  
* Simply supported: \(w=w^{\prime\prime}=0\) on \(x=0\) and \(x=a\);
* Clamped supported: \(w=w^{\prime}=0\) on \(x=0\) and \(x=a\). The divergence instability is present for \(\omega=0\), and the flutter instability occurs for \(\omega\
eq 0\)[53]. The characteristic Equation (16) for the CPT plate can be written in a more convenient way according to [PERSON] [56]  
\[\Big{(}r^{2}-\pi^{2}k\Big{)}^{2}+\beta r-\lambda=0, \tag{17}\]  
where  
\[ \begin{array}{l}k=\frac{a^{2}}{m^{2}D_{11}}\Big{[}\big{(}D_{12}+2D_{66}\big{)} \big{(}\frac{n}{b}\big{)}^{2}+\frac{N_{y}^{0}}{2\pi^{2}}\Big{]},\beta=\frac{a ^{3}}{m^{2}D_{11}}\Lambda,\\ \lambda=\frac{a^{4}}{m^{4}D_{11}}\Big{[}\rho h\omega^{2}-D_{22}\big{(}\frac{n \pi}{b}\big{)}^{4}-N_{y}^{0}\big{(}\frac{n\pi}{b}\big{)}^{2}\Big{]}-4\pi^{4} k^{2},\end{array}  \tag{18}\]  
and \(m\) is the wave number in the \(x\)-direction. The solution of the characteristic Equation (17) is a function of only two variables and is written as  
\[r_{1,2}=\mu\pm iv,\qquad r_{3,4}=-\mu\pm\sqrt{\
u^{2}-4\mu^{2}},\ i=\sqrt{-1}. \tag{19}\]  
Moreover, the parameters \(\beta\) and \(\lambda\) can be expressed using \(\mu\) and \(\
u\) as  
\[\beta=4\mu\Big{(}\
u^{2}-\mu^{2}+k\pi^{2}\Big{)},\qquad\quad\lambda=k^{2}\pi^ {4}+\Big{(}\mu_{b}^{2}+\
u^{2}\Big{)}\Big{(}\
u^{2}-3\mu^{2}+2k\pi\tau^{2} \Big{)}. \tag{20}\]  
Considering boundary conditions, [PERSON] [56] gave the explicit form of the determinant in terms of the variables \(\mu\) and \(\
u\)  
* For a simply supported plate (SSSS) \[ \begin{array}{l}\Delta(k,\lambda,\beta)=\mu^{2}\Big{[}cosh2\mu-cosh\sqrt{ \
u^{2}-2\mu^{2}+2k\pi^{2}}cos\
u\Big{]}+\\ \frac{\big{(}\
u^{2}-\mu^{2}-k\pi^{2}\big{)}^{2}+2\mu^{2}\big{(}\mu^{2}-k\pi^{2 }\big{)}^{2}}{\sqrt{\
u^{2}-2\mu^{2}+2k\pi^{2}}}\frac{sin\
u}{2\
u}sinh\sqrt{ \
u^{2}-2\mu^{2}+2k\pi^{2}}=0.\end{array} \] (21)
* For a clamped plate (SCSC)\[ \begin{array}{c}\Delta(k,\lambda,\beta)=cosh2\mu-cosh\sqrt{v^{2}-2\mu^{2}+2k\pi^{2 }}cosv+\\ \frac{kr^{2}-3\mu^{2}}{\sqrt{v^{2}-2\mu^{2}+2k\pi^{2}}}\frac{sim}{v}sinh\sqrt{v^ {2}-2\mu^{2}+2k\pi^{2}}=0.\end{array}  \tag{22}\]  
The eigenvalue curve can be drawn assuming the arbitrary value of constant \(k\) in the coordinates (\(\mu,\
u\)).