# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 3 Results and Discussion
### Analytical Solution for an Orthotropic Viscoelastic Rectangular Plate of Variable Thickness
\begin{array}{c}L_{34}\big{(}u,v,w,C_{ij},D_{ij}\big{)}=2\frac{\partial^{2}w}{ \partial x^{2}}\Big{(}\frac{\partial D_{11}}{\partial x}+\frac{\partial D_{16}} {\partial y}\Big{)}+2\frac{\partial^{3}w}{\partial x^{3}}\Big{(}\frac{\partial D _{12}}{\partial x}+3\frac{\partial D_{26}}{\partial y}+2\frac{\partial D_{66} }{\partial x}\Big{)}+\\ +2\frac{\partial^{3}w}{\partial x^{2}}\Big{(}\frac{\partial D_{12}}{\partial y} +\frac{3\partial D_{16}}{\partial x}+2\frac{\partial D_{66}}{\partial y} \Big{)}+2\frac{\partial^{3}w}{\partial y^{3}}\Big{(}\frac{\partial D_{22}}{ \partial y}+\frac{\partial D_{26}}{\partial x}\Big{)}+\frac{\partial^{2}w}{ \partial x^{2}}\Big{(}\frac{\partial^{2}D_{11}}{\partial x^{2}}+\frac{\partial ^{2}D_{12}}{\partial y^{2}}+2\frac{\partial^{2}D_{16}}{\partial x\partial y} \Big{)}+\\ +\frac{\partial^{2}w}{\partial y^{2}}\Big{(}\frac{\partial^{2}D_{12}}{\partial x ^{2}}+\frac{\partial^{2}D_{26}}{\partial y^{2}}+2\frac{\partial^{2}D_{26}}{ \partial x\partial y}\Big{)}+2\frac{\partial^{2}w}{\partial x\partial y} \Big{(}\frac{\partial^{2}D_{16}}{\partial x^{2}}+\frac{\partial^{2}D_{26}}{ \partial y^{2}}+2\frac{\partial^{2}D_{26}}{\partial x\partial y}\Big{)}+\\ +\frac{\partial^{2}w}{\partial x^{2}}\Big{\{}C_{11}\Big{[}\frac{\partial u}{ \partial x}+\frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}-\frac{ 1}{2}\Big{(}\frac{\partial w_{0}}{\partial x}\Big{)}^{2}\Big{]}+C_{12}\Big{[} \frac{\partial w}{\partial y}+\frac{1}{2}\Big{(}\frac{\partial w}{\partial y }\Big{)}^{2}-\frac{1}{2}\Big{(}\frac{\partial w_{0}}{\partial y}\Big{)}^{2} \Big{]}+\\ +C_{16}\Big{[}\frac{\partial u}{\partial y}+\frac{\partial w}{\partial x} +\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\frac{\partial w_{ 0}}{\partial y}\Big{]}\Big{\}}+\\ +2\frac{\partial^{2}w}{\partial x\partial y}\Big{\{}C_{16}\Big{[}\frac{ \partial u}{\partial x}+\frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^ {2}-\frac{1}{2}\Big{(}\frac{\partial w_{0}}{\partial x}\Big{)}^{2}\Big{]}+C_ {26}\Big{[}\frac{\partial w}{\partial y}+\Big{[}\frac{\partial w}{\partial y }\Big{)}^{2}-\frac{1}{2}\Big{(}\frac{\partial w_{0}}{\partial y}\Big{)}^{2} \Big{]}+\\ +C_{66}\Big{[}\frac{\partial u}{\partial y}+\frac{\partial w}{\partial x} +\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}-\frac{\partial w_{ 0}}{\partial x}\frac{\partial w_{0}}{\partial y}\Big{]}\Big{\}}+\\ +\frac{\partial^{2}w}{\partial y^{2}}\Big{\{}C_{12}\Big{[}\frac{\partial u}{ \partial x}+\frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}- \frac{1}{2}\Big{(}\frac{\partial w_{0}}{\partial x}\Big{)}^{2}\Big{]}+C_{22} \Big{[}\frac{\partial w}{\partial y}+\Big{[}\frac{\partial w}{\partial y} \Big{)}^{2}-\frac{1}{2}\Big{(}\frac{\partial w_{0}}{\partial y}\Big{)}^{2} \Big{]}+\\ +C_{26}\Big{[}\frac{\partial u}{\partial y}+\frac{\partial w}{\partial x} +\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}-\frac{\partial w_{ 0}}{\partial x}\frac{\partial w_{0}}{\partial y}\Big{]}\Big{\}}+\\ +\frac{\partial w}{\partial x}[L_{11}(u)+L_{12}(v)+L_{14}(w)]+\frac{\partial w}{ \partial y}[L_{21}(u)+L_{22}(v)+L_{24}(w)]\end{array}  \tag{37}\]  
If the considered plate has orthotropic properties, then for the coefficients \(C_{16}=C_{26}=0\) and \(D_{16}=D_{26}=0\). Then, following the relations (11) and (12), the stiffness matrices take the following form:  
\[C=\left( \begin{array}{ccc}C_{11}&C_{12}&0\\ C_{12}&C_{22}&0\\ 0&0&C_{66}\end{array} \right) \tag{38}\]  
\[D=\left( \begin{array}{ccc}D_{11}&D_{12}&0\\ D_{12}&D_{22}&0\\ 0&0&D_{66}\end{array}