## 2 Statement of the Problems
Substituting (4) into (5) and carrying out transformations, one can write down the problem as regards the dimensionless functions of the deflection and the angle of rotation of the normal in the form:  
\[\gamma\xi\Big{[}D\frac{(1-\
u)}{2}(\theta+w^{\prime})+s_{rr}^{0}w^ {\prime}\Big{]}-\Big{(}\xi\Big{[}D\Big{(}\theta^{\prime}+\frac{\
u\theta}{ \xi}\Big{)}+s_{rr}^{0}\theta^{\prime}\Big{]}\Big{)}^{\prime}+D\Big{(}\
u\theta ^{\prime}+\frac{\theta}{\xi}\Big{)}+s_{\varphi\varphi}^{0}\frac{\theta}{\xi}- \kappa^{2}\rho\xi\theta=0,\] \[\Big{(}\xi\Big{[}D\frac{(1-\
u)}{2}(\theta+w^{\prime})+s_{rr}^{0 }w^{\prime}\Big{]}\Big{)}^{\prime}-\kappa^{2}\rho\xi w+\mathfrak{q}\xi=0. \tag{6}\]  
The following dimensionless parameters and variables are introduced in (6): \(r=R\xi\), \(w=R\widetilde{w}\), \(D=D_{0}\widetilde{D}\), \(R_{0}=R\xi_{0}\), \(D_{0}=D(R)\), \(\sigma_{rr}^{0}=\frac{12D_{0}}{R^{3}}s_{rr}^{0}\)\(\sigma_{\varphi\varphi}^{0}=\frac{12D_{0}}{R^{3}}s_{\varphi\varphi}^{0}\), \(\rho=\rho_{0}(R)\widetilde{\rho}\), \(\kappa^{2}=\frac{\mu^{3}}{12D_{0}}R^{2}\omega^{2}\), \(\mathfrak{q}=\frac{12D_{0}}{R^{2}}\widetilde{\mathfrak{q}}(\xi)\), \(\gamma=12\frac{R^{2}}{R^{3}}\). The tilde signs \"\(\sim\)\" in (6) are omitted, and the derivative with respect to the dimensionless coordinate \(\xi\), is indicated by the prime \(\omega^{\prime}\)>>. It can be assumed that the plates are rigidly clamped along the contour, which is expressed by the following boundary conditions:  
\[M_{r}=\int\limits_{-h/2}^{h/2}zT_{rr}dz= D\Big{(}\theta_{r}+\frac{\gamma\theta}{r}\Big{)}+\frac{\mu^{3}}{12} \partial_{r}\sigma_{rr}^{0},\]0  
For a solid plate, add conditions to zero (due to the symmetry of the plate):  
\[M_{r}=\int\limits_{-h/2}^{h/2}zT_{rr}dz= D\Big{(}\theta_{r}+\frac{\gamma\theta}{r}\Big{)}+\frac{\mu^{3}}{12} \partial_{r}\sigma_{rr}^{0},\]1  
For an annular plate, the condition of rigid restraint is also considered on the inner radius:  
\[M_{r}=\int\limits_{-h/2}^{h/2}zT_{rr}dz= D\Big{(}\theta_{r}+\frac{\gamma\theta}{r}\Big{)}+\frac{\mu^{3}}{12} \partial_{r}\sigma_{rr}^{0},\]2  
The Poisson ratio \(\
u\) can be assumed to be constant, which is the most common situation in practice. The inverse problems for both plates consist of determining a set of dimensionless functions of prestress \(s_{rr}^{0}(\xi)\), \(s_{\varphi\varphi}^{0}(\xi)\), as well as the corresponding functions of the deflection \(w(\xi)\) and the angle of rotation of the normal \(\theta(\xi)\) using additional information about the amplitude frequency characteristic (AFC) measured at the point \(\xi=\xi_{z}\) in a certain frequency range:  
\[M_{r}=\int\limits_{-h/2}^{h/2}zT_{rr}dz= D\Big{(}\theta_{r}+\frac{\gamma\theta}{r}\Big{)}+\frac{\mu^{3}}{12} \partial_{r}\sigma_{rr}^{0},\]3  
Additionally, let us take into account that the functions of PS must satisfy the Cauchy equilibrium equation [29], which in the considered axisymmetric case, is written in the form \(\sigma_{rr}^{0}+\frac{\sigma_{rr}^{0}-\sigma_{0\omega}^{0}}{\xi_{z}}=0\), from which \(\sigma_{\varphi\,\varphi}^{0}=\xi{\sigma_{rr}^{0}}^{\prime}+\sigma_{rr}^{0}\). Then, the function \(s_{\varphi\,\varphi}^{0}(\xi)\) can be excluded from the oscillation equations by expressing it in terms of \(s_{rr}^{0}(\xi)\), and we can write (6) in the form:  
\[ \begin{split}&\gamma\xi\Big{[}D\frac{(1-\
u)}{2}(\theta+w^{ \prime})+s_{rr}^{0}w^{\prime}\Big{]}-\Big{(}\xi\Big{[}D\Big{(}\theta^{\prime}+ \frac{\
u\theta}{\xi}\Big{)}+s_{rr}^{0}\theta^{\prime}\Big{]}\Big{)}^{\prime }+D\Big{(}\
u\theta^{\prime}+\frac{\theta}{\xi}\Big{)}+\frac{\big{(}\xi{\sigma _{rr}^{0}}^{\prime}+s_{rr}^{0}\big{)}\theta}{\xi}-\kappa^{2}\varrho\xi\theta=0,\\ &\Big{(}\xi\Big{[}D\frac{(1-\
u)}{2}(\theta+w^{\prime})+s_{rr}^{0 }w^{\prime}\Big{]}\Big{)}^{\prime}-\kappa^{2}\varrho\xi w+\text{q}\xi=0.\end{split}  \tag{11}\]  
Thus, in inverse problems, it is also sufficient to define only a function \(s_{rr}^{0}(\xi)\).