## 5 Computational Tests for Identification of PS
** Reconstruction of the PS distribution function \(s_{rr}^{0}(\xi)=\frac{5.10^{-3}}{\xi_{0}^{-2}-1}\left(1-\frac{1}{\xi^{2}}\right)\) corresponding to the case of elastic deformation in the presence of the initial internal pressure (19) in the annular plate of the inner radius \(\xi_{0}=0.2\). The frequency range \([8.1,\ 8.5]\) is selected for identification. The renewal was carried out in the class of quadratic functions (\(M=3\),\(\psi_{j}(\xi)=\xi^{j-1}\)). In this case, the relative error of reconstruction does not exceed 5% throughout the entire section, except for the vicinity of the inner radius (rigid restraint), where it reaches 20%. Figure 4a shows the results of the reconstruction; the solid line denotes the exact function, and the dots denote the reconstructed function. Note that the developed approach can be applied in two stages: at the first stage, the PS is renewed as an arbitrary function; then, if the form of the renewed function is similar to the solution in the case of elastic or plastic deformations [26; 35], it is possible to carry out repeated renewal in order to determine the parameters of the solution, the form of which is known (as in tests 1 and 2), and significantly improve the identification accuracy. The result can be evaluated by the coincidence of the AFC. This was carried out in this test; the relative error did not exceed 1%. The result of the reconstruction is shown in Figure 4b. **Test 4. ** Model example. Reconstruction of the function \(s_{rr}^{0}(\xi)=0.2\cdot 10^{-3}\left(\xi+0.5\right)^{-2}\), similar to a distribution function of the PS, corresponding to the case of elastic deformation of an annular plate under radial loading. The case of a solid plate was considered, which can be obtained from a prestressed annular plate using an insert. The frequency range \([7.8,\ 8.1]\) is selected for identification. The identification was carried out in the class of quadratic functions (\(M=3\), \(\psi_{j}(\xi)=\xi^{j-1}\)). The relative error of reconstruction does not  
Figure 4: (**a**,**b**) Two-stage scheme for reconstructing the PS distribution function for an annular plate in the case of elastic deformation. First stage—(**a**), second stage—(**b**). exceed 9%. Figure 5 shows the results of the reconstruction; the solid line denotes the exact function, the dots denote the reconstructed function. **Test 5. ** Model example. Reconstruction of the function \(s_{rr}^{0}(\xi)=10^{-3}\cdot\left(0.5-\xi^{2}\right)\) describing the PS field, which is compressive for a part of an annular plate of radius \(\xi_{0}=0.2\), and stretching for another part, that is, it changes sign. The frequency range \([8.2,\ 8.5]\) is selected for identification. The identification was carried out in the class of linear functions (\(M=2\),\(\psi_{j}(\xi)=\xi^{j-1}\)). The relative error of reconstruction does not exceed 8%. Figure 6 shows the results of the reconstruction; the solid line denotes the exact function, and the dots denote the reconstructed function. Note that in practice, the problems of identification of a level and character of monotonicity of the laws of PS variation in plates, which are most often monotonic functions arising as a result of elastic or plastic deformation of the plates, are of the greatest importance. In view of this, renewal in the class of linear functions, which is the most efficient in terms of speed in action, can be considered the most promising. Figure 5: Reconstruction of the PS decreasing distribution function for a solid plate.