## 2 Statement of the Problems  
Let us consider in a cylindrical coordinate system \((r,\varphi,z)\) the axisymmetric bending vibrations of a prestressed circular inhomogeneous plate having a radius \(R\) and a thickness \(h\) within the framework of the Timoshenko hypotheses. We consider cases of both solid and annular plates with an inner radius \(R_{0}\) (Figure 1). Due to the symmetry of the plate along the angular coordinate, the components of the displacement vector within the framework of the Timoshenko hypotheses [5] take the form:  
\[u_{r}=z\theta,\ \ \ u_{\varphi}=0,\ \ \ u_{z}=w, \tag{1}\]  
where \(w\) is the plate deflection function, and \(\theta\) is the angle of rotation of the normal along the axis of the radial coordinate. For modeling preliminary stresses, let us write down the general form of the Piola asymmetric stress tensor [2]:  
\[\underline{T}=\underline{\underline{\sigma}}^{0}\cdot\
abla\underline{u}+ \underline{\underline{\sigma}}, \tag{2}\]  
where \(\underline{u}\) is displacement vector, \(\underline{\underline{\sigma}}\) is symmetric Kirchhoff stress tensor, and \(\underline{\underline{\sigma}}^{0}\) is prestress tensor. This model makes it possible to describe inhomogeneous fields of \(\overline{\mathrm{PS}}\) of various natures, in particular, arising during unloading after the formation of localized plastic zones. Further constructions are common for solid and annular plates. Here, the most common case of a plane PS state [29] in plates will be considered, in which \(\sigma_{rr}^{0}(r)\
eq 0\) and \(\sigma_{\varphi\varphi}^{0}(r)\
eq 0\) are different from zero. In this case, the PS tensor will have the following non-zero components:  
\[ \begin{split}& T_{rr}=\sigma_{rr}+z\theta_{,r}\sigma_{rr}^{0}= \frac{Ez}{1-\sqrt{2}}\left(\theta_{,r}+\frac{\
u\theta}{r}\right)+z\theta_{,r} \sigma_{rr}^{0},\\ & T_{\varphi\varphi}=\sigma_{\varphi\varphi}+\frac{z\theta}{r} \sigma_{\varphi\varphi}^{0}=\frac{Ez}{1-\sqrt{2}}\left(\
u\theta_{,r}+\frac{ \theta}{r}\right)+\frac{z\theta}{r}\sigma_{\varphi\varphi}^{0},\\ & T_{rz}=\sigma_{rz}+w_{,r}\sigma_{rr}^{0}=\frac{E}{2(1+\
u)}( \theta+w_{,r})+w_{,r}\sigma_{rr}^{0},\\ & T_{zr}=\sigma_{rz}=\frac{E}{2(1+\
u)}(\theta+w_{,r}).\end{split}  \tag{3}\]  
Figure 1: Schematic representation of the considered solid and annular plates. Let us write out the relations for the bending moments and shearing forces of the Timoshenko plate:  
\[M_{r}=\int\limits_{-h/2}^{h/2}zT_{rr}dz= D\Big{(}\theta_{r}+\frac{\gamma\theta}{r}\Big{)}+\frac{\mu^{3}}{12} \partial_{r}\sigma_{rr}^{0},\] \[M_{\varphi}=\int\limits_{-h/2}^{h/2}zT_{\varphi\varphi}dz= D\Big{(}\
u\theta_{r}+\frac{\theta}{r}\Big{)}+\frac{\mu^{3}}{12}\frac{ \theta}{r}\sigma_{\varphi\varphi}^{0},\] \[M_{r\varphi}=0, \tag{4}\] \[Q_{r}=\int\limits_{-h/2}^{h/2}T_{rz}dz= \frac{E\mu}{2(1+\
u)}(\theta+w_{r})+hw_{r}\sigma_{rr}^{0},\] \[Q_{\varphi}=0.\]  
Here, \(D(r)=\frac{E(r)h^{3}}{12(1-\
u^{2})}\) is a function of the cylindrical rigidity. The equations of steady-state axisymmetric bending vibrations of a plate caused by a normal load \(\mathfrak{q}(r)\) in a cylindrical coordinate system within the framework of the Timoshenko model [25; 30] take the form:  
\[Q_{r}-\frac{1}{r}\Big{(}\frac{\partial(rM_{r})}{\partial r}-M_{ \varphi}\Big{)}=\frac{\rho\mu^{3}}{12}\omega^{2}\theta, \tag{5}\] \[\frac{1}{r}\frac{\partial(rQ_{r})}{\partial r}=\rho h\omega^{2}w- \mathfrak{q}.\]  
Here, \(\rho(r)\) is a density, and \(\omega\) is a vibration frequency.