# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 2 Methods
### Analytical and Numerical Method  
Consider an anisotropic viscoelastic rectangular plate of an arbitrarily varying thickness \(h(x,y)\) with sides \(a\) and \(b\) under the action of axial dynamic loads along the side \(a\) (Figure 1). In the general case, the law of change in thickness \(h\) can be specified analytically as the function of the \(x\) and \(y\) coordinates. This function \(h(x,y)\) must have continuous second derivatives in each of the \(x\) and \(y\) coordinates. Further, a certain form of dependence (106) is used in numerical examples, but it is not the only possible one.  
The mathematical model of the problem in a geometrically nonlinear formulation follows the classical Kirchhoff-Love theory. The system of equations of motion within the framework of the chosen theory has the following form [46;47]:  
\[\frac{\partial N_{x}}{\partial x}+\frac{\partial N_{xy}}{\partial y}+p_{x}- \rho h\frac{\partial^{2}u}{\partial t^{2}}=0 \tag{1}\]  
\[\frac{\partial N_{xy}}{\partial x}+\frac{\partial N_{y}}{\partial y}+p_{y}- \rho h\frac{\partial^{2}v}{\partial t^{2}}=0 \tag{2}\]  
\[\begin{array}{l}\frac{\partial^{2}M_{x}}{\partial x^{2}}+\frac{\partial^{2} M_{y}}{\partial y^{2}}+2\frac{\partial^{2}H}{\partial x\partial y}+\frac{ \partial}{\partial x}\left(N_{x}\frac{\partial uv}{\partial x}+N_{xy}\frac{ \partial uv}{\partial y}\right)+\\ +\frac{\partial}{\partial y}\left(N_{xy}\frac{\partial uv}{\partial x}+N_{y} \frac{\partial uv}{\partial y}\right)+P_{x}(t)\frac{\partial^{2}v}{\partial x ^{2}}+q-\rho h\frac{\partial^{2}w}{\partial t^{2}}=0.\end{array} \tag{3}\]  
Here, \(N_{x}\) and \(N_{y}\) are the normal forces per unit length; \(N_{xy}\) is the shear force; \(M_{x}\) and \(M_{y}\) are the bending moments, \(H\) is the torque moment; \(p_{x}\) and \(p_{y}\) are the in-plane loads per unit area; \(q\) is the normal distributed load per unit area; \(P_{x}(t)\) is the compressive force per unit length applied to the plate edge; \(\rho\) is the density of the material; and \(u\), \(v\), and \(w\) are the components of the displacement vector \(\{U\}\) in the directions of the axes \(Ox\), \(Oy\), and \(Oz\), respectively. The compressive force \(P_{x}(t)\) is applied to the side of dimension \(b\) and is a function of time according to the law \(P_{x}=P_{0}+P_{t}\cos(\theta\ t)\), where \(P_{0}\) is the constant part of the load, \(P_{t}\) is the amplitude of the variable part, and \(\theta\) is the frequency of the load pulsations.  
The system of Equations (1)-(3) is supplemented by the corresponding boundary conditions [46], which will be used to solve the problems:  
1. All edges are simply supported: at \(x=0,a\): \(u=0\), \(v=0\), \(w=0\); \(M_{x}=0\); at \(y=0\), \(b\): \(u=0\), \(v=0\), \(w=0\); \(M_{y}=0\).
2. All edges are clamped: at \(x=0,a\): \(u=0\), \(v=0\), \(w=0\); \(\frac{\partial w}{\partial x}=0\); at \(y=0\), \(b\): \(u=0\), \(v=0\), \(w=0\); \(\frac{\partial w}{\partial y}=0\).
3. Two opposite edges are simply supported, the other two edges are clamped: at \(x=0,a\): \(u=0\), \(v=0\), \(w=0\); \(\frac{\partial w}{\partial x}=0\); at \(y=0\), \(b\): \(u=0\), \(v=0\), \(w=0\); \(M_{y}=0\). The initial conditions at \(t=0\)[46] are as follows: \(u(x,y,0)=u_{0}(x,y)\), \(\dot{u}_{0}(x,y,0)=\dot{u}_{0}(x,y)\), \(v(x,y,0)=\dot{v}_{0}(x,y)\), \(w(x,y,0)=w_{0}(x,y)\), and \(\dot{w}(x,y,0)=\dot{w}_{0}(x,y)\).  
Here, \(u_{0}(x,y)\), \(v_{0}(x,y)\), \(w_{0}(x,y)\), \(u_{0}(x,y)\), \(\dot{v}_{0}(x,y)\), and \(\dot{w}_{0}(x,y)\) are given functions.  
Figure 1: An orthotropic viscoelastic rectangular plate of variable thickness under the action of periodic in-plane loads.