# 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
 \tag{62}\]\[L_{34}(w)=-12\frac{\partial w}{\partial x}\Big{\{}h\Big{\{}\Delta \big{(}1-\Gamma_{11}^{*}\big{)}\Big{(}\lambda\delta\frac{\partial^{2}u}{ \partial x^{2}}+\frac{\partial w}{\partial x}\frac{\partial^{2}w}{\partial x^{2} }\Big{)}+\] \[+\big{[}\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}+(1-\mu_{1} \mu_{2})g(1-\Gamma^{*})\big{]}\Big{(}\lambda^{2}\delta\frac{\partial^{2}v}{ \partial x^{2}}+\lambda^{2}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial x^{2}}\Big{)}+\] \[+(1-\mu_{1}\mu_{2})g(1-\Gamma^{*})\Big{(}\lambda^{3}\delta\frac{ \partial^{2}u}{\partial y^{2}}+\lambda^{2}\frac{\partial w}{\partial x}\frac{ \partial^{2}v}{\partial y^{2}}\Big{)}\Big{\}}+\frac{\partial h}{\Delta\Big{(} 1-\Gamma_{11}^{*}\Big{)}\Big{[}\lambda\delta\frac{\partial u}{\partial x}+ \frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}-\frac{1}{2}\Big{(} \frac{\partial u_{0}}{\partial x}\Big{)}^{2}\Big{]}+\] \[+\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}\Big{[}\lambda^{2} \delta\frac{\partial v}{\partial y}+\frac{\lambda^{2}}{2}\Big{(}\Big{(}\frac{ \partial w}{\partial y}\Big{)}^{2}-\Big{(}\frac{\partial w_{0}}{\partial y} \Big{)}^{2}\Big{)}\Big{]}\Big{\}}+\] \[+\frac{\partial h}{\partial y}(1-\mu_{1}\mu_{2})g(1-\Gamma^{*}) \Big{(}\lambda^{3}\delta\frac{\partial u}{\partial y}+\lambda^{2}\delta\frac{ \partial v}{\partial x}+\lambda^{2}\Big{[}\frac{\partial w}{\partial x}\frac{ \partial w}{\partial y}-\frac{\partial w_{0}}{\partial x}\frac{\partial w_{0}} {\partial y}\Big{]}\Big{)}\Big{\}}-\] \[-12\frac{\partial^{2}w}{\partial x^{2}}h\Big{\{}\Delta\big{(}1- \Gamma_{11}^{*}\big{)}\Big{[}\lambda\delta\frac{\partial u}{\partial x}+ \frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}\Big{]}+\mu_{2} \Delta\big{(}1-\Gamma_{12}^{*}\big{)}\Big{[}\lambda^{2}\delta\frac{\partial v }{\partial y}+\frac{\lambda^{2}}{2}\Big{(}\frac{\partial w}{\partial y}\Big{)} ^{2}\Big{]}\Big{\}}-\] \[-12\frac{\partial^{2}w}{\partial x^{2}}h\Big{\{}\Delta\big{(}1- \Gamma_{11}^{*}\big{)}\Big{[}\lambda\delta\frac{\partial u}{\partial x}+ \frac{1}{2}\Big{(}\frac{\partial w}{\partial x}\Big{)}^{2}-\Big{(}\frac{ \partial w_{0}}{\partial x}\Big{)}^{2}\Big{]}+\] \[+\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}\Big{[}\lambda^{2} \delta\frac{\partial v}{\partial y}+\lambda^{2}\Big{(}\Big{(}\frac{\partial w }{\partial y}\Big{)}^{2}-\Big{(}\frac{\partial w_{0}}{\partial y}\Big{)}^{2} \Big{)}\Big{]}\Big{\}}-\] \[-12\frac{\partial w}{\partial y}\Big{\{}h\Big{\{}\frac{1}{\Delta \big{(}1-\Gamma_{22}^{*}\big{)}\Big{(}\lambda^{4}\delta\frac{\partial^{2}v}{ \partial y^{2}}+\lambda^{4}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial y^{2}}\Big{)}+\big{[}\frac{\mu_{1}}{\Delta}\big{(}1-\Gamma_{21}^{*} \big{)}+\big{(}1-\mu_{1}\mu_{2}\big{)}g(1-\Gamma^{*})\big{]}\times\] \[\times\Big{(}\lambda^{3}\delta\frac{\partial^{2}u}{\partial x}+ \lambda^{2}\frac{\partial w}{\partial x}\frac{\partial^{2}w}{\partial x}\Big{)} +(1-\mu_{1}\mu_{2})g(1-\Gamma^{*})\Big{(}\lambda^{2}\delta\frac{\partial^{2}v }{\partial x}+\lambda^{2}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial x^{2}}\Big{)}\Big{\}}+\] \[+\big{[}\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}+(1-\mu_{1} \mu_{2})g(1-\Gamma^{*})\big{]}\Big{(}\lambda^{2}\delta\frac{\partial^{2}v}{ \partial x^{2}}+\lambda^{2}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial x^{2}}\Big{)}+\]0 \[+\big{[}\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}+(1-\mu_{1} \mu_{2})g(1-\Gamma^{*})\big{]}\Big{(}\lambda^{2}\delta\frac{\partial^{2}v}{ \partial x^{2}}+\lambda^{2}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial x^{2}}\Big{)}+\]1 \[+\big{[}\mu_{2}\Delta\big{(}1-\Gamma_{12}^{*}\big{)}+(1-\mu_{1} \mu_{2})g(1-\Gamma^{*})\big{]}\Big{(}\lambda^{2}\delta\frac{\partial^{2}v}{ \partial x^{2}}+\lambda^{2}\frac{\partial w}{\partial y}\frac{\partial^{2}w}{ \partial x^{2}}\Big{)}+\]2  
The resulting system of Equations (56)-(63) with the corresponding boundary and initial conditions describes the motion of an orthotropic viscoelastic rectangular plate of variable thickness.