## 4 Scheme for Solving Inverse Problems of PS Identification  
To solve inverse problems, we use a special projection method based on the representation of all unknown functions in the form of expansions in systems of linearly independent functions. To use it, let us formulate a weak statement of problem (11) taking into account boundary conditions (7) and (8) or (9). We scalarly multiply the first equation by the function of the possible field of the angle of rotation \(\theta_{1}(\xi_{\kappa})\) of the normal and scalarly multiply the second equation by the function of the possible deflection field \(w_{1}(\xi_{\kappa})\). Using the formula for integration by parts, boundary conditions and carrying out transformations, a weak statement of the problem was obtained in the form:  
\[\int\limits_{0}^{1}\Big{(}\gamma\xi\theta_{1}\Big{(}D\tfrac{(1- \
u)}{2}+s_{rr}^{0}\Big{)}w^{\prime}+\bigg{(}\Big{(}\gamma\xi D\tfrac{(1-\
u)} {2}+\tfrac{D+\xi\alpha_{rr}^{0}+\alpha_{rr}^{0}}{\xi_{\kappa}}-\kappa^{2}\varrho \xi\Big{)}\theta_{1}-D\
u\theta_{1}{}^{\prime}\Big{)}\theta-\] \[-\big{(}\xi\big{(}D+s_{rr}^{0}\big{)}\theta_{1}{}^{\prime}-D\
u \theta_{1}\big{)}\theta^{\prime}\big{)}\xi d\xi=0, \tag{15}\] \[\int\limits_{0}^{1}\Big{(}\xi\Big{(}D\tfrac{(1-\
u)}{2}+s_{rr}^{0 }\Big{)}w_{1}{}^{\prime}w^{\prime}+D\tfrac{(1-\
u)}{2}\xi w_{1}{}^{\prime} \theta-\kappa^{2}\varrho\xi w_{1}w\Big{)}\xi d\xi=-\int\limits_{0}^{1}\text{q} \xi^{2}w_{1}d\xi.\]  
A detailed description of the used projection technique is presented in [29]. Let us give its basic scheme for the problems under consideration. The required function \(s_{rr}^{0}(\xi)\) can be represented in the form of an expansion in terms of a system of orthogonal functions \(\psi_{j}(\xi)\):  
\[s_{rr}^{0}(\xi)=\sum\limits_{j=1}^{M}C_{j}\psi_{j}(\xi). \tag{16}\]  
Using the linearity of Equation (15) with respect to the functions \(w\), \(\theta\), \(\theta_{1}\), and \(w_{1}\), substituting expansions (12) and (16) into relation (15), and sequentially choosing the basic functions \(w_{1}=\varphi_{n}\), \(\theta_{1}=\gamma_{n}\), \(n=1\dots N\) as possible functions, a system of \(2N\) linear equations with respect to the coefficient functions \(a_{i}(\kappa)\) and \(b_{i}(\kappa)\) can be obtained from Equation (15). Having determined the coefficients \(a_{i}(\kappa)\), \(b_{i}(\kappa)\) in the form of expressions with respect to \(C_{j}\) and \(\kappa\), they are substituted into the expansion (12). Thus, it is possible to obtain an expression for the deflection function \(w(\xi,\kappa,C_{j})\), which must satisfy the additional condition (10) of inverse problems. Choosing frequencies \(\kappa_{m}(m=1\dots M)\), we obtain a system of \(M\) nonlinear equations of order \(2M\) with respect to a set of \(M\) coefficients \(C_{j},j=1\dots M\) in a form (17), from the solution of which we can determine the expansion coefficients (16) of the desired function \(s_{rr}^{0}(\xi)\).  
\[w(\xi_{z},\kappa_{m},C_{j})=f(\kappa_{m}),\ j=1\dots M,\ m=1\dots M. \tag{17}\]  
The developed projection approach was implemented in the Maple package. Its realization combined the solution of direct problems using the Galerkin method, as well as numerical integration and solution of systems of linear and nonlinear equations using built-in solvers. For the functions of the deflection and the angle of rotation of the normal, 6-7 basis functions were used, and the number and type of basis functions for determining the prestress were selected for each test separately and are given in their descriptions.