## 2 Statement of the Problems
Note that the formulated problem of finding the PS field is a coefficient inverse problem [4], the solution of which is a nonlinear ill-posed problem [31] and requires the use of special methods. Additionally, to construct a scheme for solving inverse problems, it is necessary to develop a method for solving direct problems for calculating bending vibrations of plates with PS, the realization of which is possible using the numerical methods only, due to the nonlinearity of Equations (11). On the basis of the constructed method, it is possible to analyze the effect of PS on AFCs of plates, as well as on the deflection function. Solving the Direct Problems of Vibration Analysis of Plates: Analysis of Amplitude-Frequency Characteristics  
A technique based on the [PERSON] method [32; 33] was developed for solving the direct problems of calculating the vibrations of plates. The functions of the deflection and the angle of rotation of the normal were represented in the form of expansions in the following sets of orthogonal functions, satisfying the boundary conditions, in the following form:  
\[w(\xi,\kappa)=\sum_{m=1}^{N}a_{m}(\kappa)\varphi_{1m}(\xi),\ \theta(\xi, \kappa)=\sum_{m=1}^{N}b_{m}(\kappa)\varphi_{2m}(\xi). \tag{12}\]  
The basic functions for a solid plate are selected in the form:  
\[\varphi_{1m}(\xi)=(1-\xi^{2})^{2}\xi^{2(m-1)},\ \varphi_{2m}(\xi)=\sin(m\pi \xi). \tag{13}\]  
The basic functions for an annular plate are selected in the form:  
\[\varphi_{1m}(\xi)=(1-\xi^{2})^{2}(\xi-\xi_{0})^{2}\xi^{2(m-1)},\ \varphi_{2m}(\xi)=\sin \biggl{(}m\pi\frac{\xi-\xi_{0}}{1-\xi_{0}}\biggr{)}. \tag{14}\]  
The numerical Galerkin method for solving direct problems of calculating oscillations and amplitude-frequency characteristics was implemented in the Maple package. To achieve an accuracy of 10, it is enough to use 6-7 basis functions. The numerical scheme for solving direct problems was tested on model examples, including homogeneous plates without PS, for which the solution can be written in an analytical form, and showed sufficiently high accuracy. For each studied problem, a set of computational tests was carried out to calculate the frequency response and deflection functions in the neighborhood of the first dimensionless resonant frequency for different levels of PS. As an example, in the problem for a solid plate, a uniform PS field, corresponding to the solution of the problem of planar vibrations of a plate [29], was considered, and for an annular plate of inner radius \(\xi_{0}=0.2\), the solution of the corresponding Lame problem for a ring [34] loaded with latent internal pressure,the form of which is given below (19). The dimensionless level of PS in both cases was determined as a level of the dimensionless initial loading \(p_{0}\) and varied in the range of \(p_{0}=10^{-4},\ldots,10^{-3}\) corresponding to the most common cases in practice. The results of tests on calculating the AFC at the probing point \(\xi_{z}\) and deflection functions at the probing frequency \(\kappa_{z}\) for a solid plate are shown, respectively, in Figure 2a,b and for an annular plate in Figure 3a,b. In the figures, the problem solutions at \(p_{0}=0\) (no PS) are marked by a solid line, and the values of the dimensionless PS level \(p_{0}=5\cdot 10^{-4},10^{-3},5\cdot 10^{-3}\) are marked by dotted lines. It can be seen from the graphs that there are differences in the AFC, but they become significant near the resonance only. Accordingly, near resonance, the amplitude of vibrations depends most of all on the PS level. It should be noted that a similar situation arose in the study of other problems of modeling PS [2; 26; 27]. Based on the analysis performed,  
Figure 3: Influence of the PS level on the AFC (**a**) and the deflection function (**b**) of an annular plate in the vicinity of the first resonance. Figure 2: Influence of the PS level on the AFC (**a**) and the deflection function (**b**) of a solid plate in the vicinity of the first resonance.