The subject matter of the article is the processes of virtual localization of near shape parts during adaptive machining. The aim is to develop an effective method for finding the starting location of a CAD model of a part with virtual localization inside a point cloud obtained by laser scanning of a workpiece. The task is to formalize the procedure for starting positioning of the part model as the first stage of the virtual localization process. The second stage for final localization proposed to use iterative algorithms with the objective function which is sensitive to the intersection of the surface parts and the workpiece. In solving the problem the starting position used tools available in today's CAD packages and 3D scanning tools. The methods used are the methods of matrix algebra, in particular, the methods for finding the main central moments of inertia of three-dimensional objects based on the tensor of inertia. The following results were obtained. When calculating the inertia tensor components is proposed to use three-dimensional scanning data of workpiece and geometrical data of part obtained from the CAD system. The result is an algorithm starting location of CAD model in the virtual localization, which in the case of blanks with oversize close to uniform can provide enough current location parts for adaptive machining tasks. It is shown that to minimize computational errors and to ensure satisfactory accuracy of localization proposed algorithm can require several iterations of the shift vector search model. Conclusions. The scientific novelty of the results obtained is as follows: in contrast to the previously used approaches, when solving the problem of virtual localization for the starting position, using the condition of coincidence of the centers of the weight of thin shells coinciding with the surfaces of the workpiece and the part, it was proposed to additionally ensure the alignment of the main central axes of inertia of these shells, which, in the case of near shape blanks, provides a positioning accuracy that may not require additional iterative procedures.
Spectral methods have indisputable advantages over numerical methods in solving various problems of mathematical physics. The advantages are the high convergence and accuracy of approximate solutions, which is most relevant for calculating the strength of aerospace machinery. The problem of choosing basic functions inevitably arises when applying spectral methods. The problem is that, in addition to providing exponential convergence, the system of basic functions has to satisfy some other requirements: stability of approximate solutions and procedures for their obtaining, reduction of calculations, convenience, and some more. This paper compares eleven basic systems: the systems constructed in the form of linear combinations of Legendre polynomials that satisfy either only the main boundary conditions or the main and natural ones, similar systems constructed using Chebyshev polynomials, and the functions proposed by Khalilov S. A., systems of Lagrange – Lobatto interpolation polynomials using the Legendre and Chebyshev interpolation points, system of trigonometric functions, exponentiation, and system of finite functions of the finite element method. The convergence speed of the approximate solution to the exact one, the error in the equations of boundary value problems, and the condition numbers of matrices of linear algebraic equation systems, which arise when using variational, projection and collocation methods, were compared. The study performed on three test problems modeling beam bending: classic beam bending under unevenly distributed load, bending of additionally stretched beam on the elastic basis and geometrically nonlinear bending. The impact of the Gibbs effect on the approximate solution convergence is investigated. Among the considered basic systems, the system of basis functions in the form of linear combinations of Legendre polynomials has proved to be the best, as they satisfy all boundary conditions. This basis leads to the highest speed at which approximate solution approaches the exact one, the error in the equations approaches to zero, and also it has the smallest increase in the condition number with the increase in the order of SLE matrices, which appear due to variational and projection methods. The finite functions of the finite element method have proved to be the worst in terms of accuracy and convergence.
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