## 3 Materials and Methods  
Plate is a body constrained by two parallel planes and lateral surface. Upper and bottom planes are bases of the plate. Plane equidistant from the bases is called middle plane. Line of the cross-section middle plane with lateral surface is plate contour. Further we will be designate plate \(P\), surface \(S\) and contour \(C\) and we take into account the assumptions proposed by [PERSON]. Let us consider thin isotropic symmetrical plate with at least two symmetry axes which has thickness \(h\) and surface \(S\). Plate is referred to right-handed Cartesian coordinate system \(Ox_{1}x_{2}\) origin at its geometrical center and constrained by contour \(C\) (Figure 1a). We assume that it is balanced with outer load \(q(x_{1},x_{2})\) applied to surface \(S\) and complied with the boundary conditions. Symmetry of the plate geometry, mechanical properties, boundary conditions and outer load are taken into account in this paper. Plate is defined by coordinates of its vertices \(V_{i}(\xi_{i},\eta_{i}),\quad i=1,\dots,n\). We choose the solution of Equation (1) in the form of sum of general solution \(w_{0}\) of uniform Equation (2) and particular solution \(w_{*}\) of Equation (1). \[w=w_{0}+w_{*}. \tag{6}\]  
General solution of Equation (2) we take in the following form [73]. \[w_{0}=f_{kps}(x_{s})\cdot T_{kp(3-s)}(x_{3-s}). \tag{7}\]  
Figure 1: Scheme of the symmetrical plate. [PERSON] summation rule is used here. According to this convention indices repeated twice in a single term imply the summation is to be done. Summation indices in formula (7) are \(k\), \(p\) and \(s\). As particular cases, we obtain Levy \((p=1,s=2)\) and Timoshenko \((p=1,s=1,2)\) forms of representation. In the above \(p=1,2;\ s=1,2;\ k=1,\ldots,K\), where \(K\) is several approximations of the solution and determines its accuracy. The \(K\) is greater the accuracy of the solution is better. \(T_{kps}(x_{s})\) are given trigonometric functions  
\[T_{k1s}(x_{s})=\cos\left(\gamma_{kps}x_{s}\right);\quad T_{k2s}(x_{s})=\cos \left(\delta_{ks}x_{s}\right). \tag{8}\]  
Here  
\[\gamma_{ks}=\frac{k\pi}{a_{s}};\quad\delta_{ks}=\frac{(2k-1)\pi}{2a_{s}}. \tag{9}\]  
Parameters \(a_{s}\ (s=1,2)\) are model constants. They must be given a priori. In suggested model they are taken as sizes \(2a_{s}\ (s=1,2)\) of rectangular contour \(L\) described on plate contour \(C\) (Figure 1b). Unknown functions \(f_{kps}(x_{s})\) we take in the form  
\[f_{kps}(x_{s})=R_{kps}E_{kps}(x_{s}). \tag{10}\]  
Because indices \(k\), \(p\), \(s\) appear at the both sides of this formula, summation according to these indices does not perform. In the above \(R_{kps}\) are unknown coefficients which are determined from boundary conditions at the plate contour, \(E_{kps}(x_{s})\) are unknown functions. Substituting expressions (7)-(10) into Equation (2) we come to system of two unbounded forth order homogeneous differential equations with respect to these functions [70]. \[ \begin{split}& E_{kp1}^{(IV)}(x_{1})-2\kappa_{kp2}^{2}E_{kp1}^{ \prime\prime}(x_{1})+\kappa_{kp2}^{4}E_{kp1}(x_{1})=0;\\ & E_{kp2}^{(IV)}(x_{2})-2\kappa_{kp1}^{2}E_{kp2}^{\prime\prime}( x_{2})+\kappa_{kp1}^{4}E_{kp2}(x_{2})=0;\end{split}  \tag{11}\]  
where  
\[\kappa_{kps}= \begin{cases}\gamma_{ks},&p=1,\\ \delta_{ks},&p=2.\end{cases}  \tag{12}\]  
Functions \(E_{kps}(x_{s})\) are taken as the following  
\[E_{kps}(x_{s})=\exp\left(\lambda_{kps}x_{s}\right). \tag{13}\]  
Substituting expression (13) into Equation (11) we come to system of characteristic equations  
\[ \begin{split}&\lambda_{kp1}^{4}-2\kappa_{kp2}^{2}\lambda_{kp1}^{ 2}+\kappa_{kp2}^{4}=0,\\ &\lambda_{kp2}^{4}-2\kappa_{kp1}^{2}\lambda_{kp1}^{2}+\kappa_{kp1 }^{4}=0.\end{split}  \tag{14}\]  
Let \(\lambda_{kps\
u}\), where \(\
u=1,2\) be their roots.