# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 3 Results and Discussion
### Proposed Numerical Solution  
This section presents the obtention of the proposed numerical solution for the dynamic stability of orthotropic viscoelastic rectangular plate of variable thickness Bubnov-Galerkin procedure. The solution of the resulting system of integrodifferential equations that satisfies the boundary conditions of the problem is sought concerning the in-plane displacements \(u\) and \(v\), as well as the normal deflection \(w\) in the form  
\[u(x,y,t)=\sum_{n=1}^{N}\sum_{m=1}^{M}u_{nm}(t)\phi_{nm}(x,y) \tag{64}\]  
\[v(x,y,t)=\sum_{n=1}^{N}\sum_{m=1}^{M}v_{nm}(t)\phi_{nm}(x,y) \tag{65}\]  
\[w(x,y,t)=\sum_{n=1}^{N}\sum_{m=1}^{M}w_{nm}(t)\phi_{nm}(x,y) \tag{66}\]  
Expressions (57)-(59) are substituted into the system of Equations (49)-(56). Then, performing the Bubnov-Galerkin procedure, the following dimensionless system of basic resolving nonlinear integrodifferential equations is obtained:\[ \begin{split}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}a_{klmn}\ddot{u}_ {nm}-\eta_{1}\bigg{\{}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}\big{\{}\big{[} (1-\Gamma_{11}^{*})d_{1 klmn}+(1-\Gamma^{*})d_{2 klmn}\big{]}u_{nm}+\ \big{[}(1-\Gamma_{12}^{*})d_{3 klmn}+(1-\Gamma^{*})d_{4 klmn}\big{]}v_{nm}\big{\}} +\\ +\sum\limits_{n,j=1}^{N}\sum\limits_{m,j=1}^{M}\Big{[}(1-\Gamma_{ 11}^{*})d_{7 klmnij}+(1-\Gamma_{12}^{*})d_{8 klmnij}+(1-\Gamma^{*})d_{9 klmnij} \Big{]}\Big{(}w_{nm}w_{ij}-w_{0 nm}w_{0 ij}\Big{)}\bigg{\}}=0\\ \sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}b_{klmn}\ddot{v}_{nm}- \eta_{2}\bigg{\{}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}\big{\{}\big{[}(1- \Gamma_{21}^{*})e_{1 klmn}+(1-\Gamma^{*})e_{2 klmn}\big{]}u_{nm}+\ \big{[}(1-\Gamma_{22}^{*})e_{3 klmn}+(1-\Gamma^{*})e_{4 klmn}\big{]}v_{nm}\big{\}} +\\ +\sum\limits_{n,j=1}^{N}\sum\limits_{m,j=1}^{M}\Big{[}(1-\Gamma_ {22}^{*})e_{7 klmnij}+(1-\Gamma_{21}^{*})e_{8 klmnij}+(1-\Gamma^{*})e_{9 klmnij} \Big{]}\Big{(}w_{nm}w_{ij}-w_{0 nm}w_{0 ij}\Big{)}\bigg{\}}=0\\ \sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}c_{klnm}\ddot{w}_{nm}+ \eta_{3}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}p_{klmn}^{2}(1-2\mu_{klmn} \cos\Theta t)w_{nm}-\\ -\eta_{3}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}\big{\{}\big{[} \Gamma_{11}^{*}f_{5 klmn}+\Gamma_{12}^{*}f_{6 klmn}+\Gamma_{22}^{*}f_{7 klmn}\ +\Gamma_{21}^{*}f_{8 klmn}+\ \Gamma^{*}f_{9 klmn}\big{]}w_{0 nm}\big{\}}-\\ -\eta_{3}\left\{\sum\limits_{n,i=1}^{N}\sum\limits_{m,j=1}^{M}w_ {nm}\Big{\{}\big{[}\big{(}1-\Gamma_{11}^{*}\big{)}\xi_{1 klmnij}+\big{(}1- \Gamma_{21}^{*}\big{)}\xi_{2 klmnij}+ \tag{69}\] \[+(1-\Gamma^{*})\xi_{3 klmnij}\big{]}u_{ij}+\Big{[}(1-\Gamma_{22}^{*}) \xi_{4 klmnij}+\big{(}1-\Gamma_{12}^{*}\big{)}\xi_{5 klmnij}+(1-\Gamma^{*})\xi_ {6 klmnij}\Big{]}v_{ij}\Big{\}}+\\ +\sum\limits_{n,j,r=1}^{N}\sum\limits_{m,j=1}^{M}w_{nm}\Big{\{} \big{(}1-\Gamma_{11}^{*}\big{)}g_{8 klmnijrs}+\big{(}1-\Gamma_{12}^{*}\big{)}g_{8 klmnijrs}+(1-\Gamma_{22}^{*})g_{7 klmnijrs}+\\ +\big{(}1-\Gamma_{21}^{*})g_{8 klmnijrs}+(1-\Gamma^{*})g_{8 klmnijrs }\Big{\}}\big{(}w_{ij}w_{rs}-w_{0 ij}w_{0 rs}\big{)}\Big{\}}=0\\ u_{nm}(0)=u_{0 nm},\ \dot{u}_{nm}(0)=\dot{u}_{0 nm},\ v_{nm}(0)=v_{0 mn},\ \dot{v}_{nm}(0)=\dot{v}_{0 nm},\ w_{nm}(0)=w_{0 nm},\ \dot{w}_{nm}(0)=\dot{w}_{0 nm}.\end{split}