# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 2 Methods
### Computational Methods
Replacing integrals with quadrature formulas for calculations \(u_{nm}=u_{nm}(t_{i})\), \(v_{nm}=v_{nm}(t_{i})\), and \(w_{nm}=w_{nm}(t_{i})\) results in the following system:\[\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}a_{klnm}u_{pmn}=\sum\limits_{n=1}^{N} \sum\limits_{m=1}^{M}a_{klnm}(u_{0 nm}+\overset{\bullet}{u}_{omn}t_{p})+\] \[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{jNM},\] \[\sum\limits_{s=0}^{j}B_{s}\phi_{1 kl}(t_{j},t_{s},u_{s11},\ldots,u_ {sNM},v_{s11},\ldots,v_{sNM},w_{s11},\ldots,w_{sNM})\quad\bigg{)}-\lambda_{ nmkl}^{2}u_{jkl}\bigg{\}},\] \[\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}b_{klnm}v_{pmn}=\sum \limits_{n=1}^{N}\sum\limits_{m=1}^{M}b_{klnm}(v_{0 nm}+\overset{\bullet}{v}_{ omn}t_{p})+\] \[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{jNM }, \tag{5}\] \[\sum\limits_{s=0}^{j}B_{s}\phi_{1 kl}(t_{j},t_{s},u_{s11},\ldots,u_ {sNM},v_{s11},\ldots,v_{sNM},w_{s11},\ldots,w_{sNM})\quad\bigg{)}-\chi_{ nmkl}^{2}v_{jkl}\bigg{\}},\] \[\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}c_{klnm}w_{pmn}=\sum \limits_{n=1}^{N}\sum\limits_{m=1}^{M}c_{klnm}(w_{0 nm}+\dot{w}_{omn}t_{p})+\] \[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{ jNM},\] \[\sum\limits_{s=0}^{j}B_{s}\psi_{1 kl}(t_{j},t_{s},u_{s11},\ldots,u_ {sNM},v_{s11},\ldots,v_{sNM},w_{s11},\ldots,w_{sNM})\quad\bigg{)}-\omega_{ nmkl}w_{jkl}\bigg{\}}\]  
The next stage of the numerical method is the regularization of nonlinear integrodifferential Equation (5) with weakly singular kernels. By changing variables  
\[t-\tau=z^{\frac{1}{2}},\,0\leq z\leq t^{\alpha}\,\,(0<\alpha<1) \tag{6}\]  
the integral at the Koltunov-Rzhanitsyn kernel with a singularity of the following form:  
\[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{jNM},\]0  
takes the form  
\[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{jNM},\]1  
After the change of variables, the integrand concerning z becomes regular. Direct replacement of the integrals in the system (5) by a sum over some quadrature formula yields a numerical solution to the numerical solution of system (5). The trapezium formula used is  
\[+\sum\limits_{j=0}^{p-1}A_{j}(t_{p}-t_{j})\bigg{\{}\chi_{kl}\bigg{(} \quad t_{j},u_{j11},\ldots,u_{jNM},v_{j11},\ldots,v_{jNM},w_{j11},\ldots,w_{jNM},\]2  
where the coefficients are  
\[ \begin{array}{c}B_{0}=\frac{1}{2}(\Delta t)^{\alpha};\,\,B_{i}=\frac{1}{2}( \Delta t)^{\alpha}(i^{\alpha}-(i-1)^{\alpha});\\ B_{k}=\frac{1}{2}(\Delta t)^{\alpha}(\left(k+1\right)^{\alpha}-\left(k-1 \right)^{\alpha}),\,\,k=\overline{1,i-1}.\end{array}  \tag{10}\]  
Thus, due to the double integration of the original system (4) over the time t and the use of the quadrature formula, system (5) is obtained to find deflections \(w_{nm}=w_{nm}(t_{i})\) and displacements \(u_{nm}=u_{nm}(t_{i})\),\(v_{nm}=v_{nm}(t_{i})\). The solutions of the system of Equation (5) are found by the Gauss method.