Asymptotic modeling of technical systems with technological deviations
A new asymptotic modeling method of complex technical systems, such as thin-walled shells, focused on numerical calculation of their parameters in the presence of technological deviations from the perfect structure was developed. A scheme of the method was presented.Initially, a reduction of the boundary problem for a system of partial differential equations to a problem for ordinary differential equations is performed by expanding all quantities in Fourier series. Then, the system is transformed to the normal form, and its modification is made by expanding all the components in the multivariate Maclaurin series by degrees of the independent variable and the desired functions. An artificial parameter is introduced in equations, boundary conditions and form of the boundary under a special scheme, the solution is found in the form of asymptotic series by the degrees of the parameter. The scheme provides a solution in the form of a Taylor series by the independent variable. Stability of the coefficients of the models at increasing the number of iterations and the convergence of approximations to the exact solution in its area of the meromorphy was proven.Improved convergence of the asymptotic expansions in mathematical models of technical systems that are built under the developed method due to the generalized meromorphic summation based on two-dimensional Pade transform was shown. Special scheme of transforms, providing their existence, uniqueness, and convergence to the exact problem solution was selected.Numerical verification of the developed modeling method on classes of singular and non-linear problems was carried out. Theadvantages of the method, such as a high rate of convergence in the area of the meromorphy of exact solution, calculation stability of the coefficients of the model at increasing the number of iterations, computation of the initial approximation and all subsequent differential equations in accordance with the form of the system were demonstrated. The mathematical models of stability of smooth thin-walled cylindrical shell under uniform external pressure and free oscillations of a stringer cylindrical shell were constructed.The proposed asymptotic modeling method can be used to solve the problem of reliable prediction of the state of thin-walled shells with imperfections.