Some Problems in the Theory of Engineering Systems (Geometric Approaches)

It has been proven that to any electric circuit there exists a corresponding two pairs of conjugate linear vector spaces. One of these pairs is generated by a homological group, while the other is cohomological. A new method of analysis of mechanical and electric circuits is proposed, which consists of representing the main variables and matrices of oscillatory circuits in terms of many-dimensional tensor objects. A solution is obtained for the problem of defining eigenvalues of...
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It has been proven that to any electric circuit there exists a corresponding two pairs of conjugate linear vector spaces. One of these pairs is generated by a homological group, while the other is cohomological. A new method of analysis of mechanical and electric circuits is proposed, which consists of representing the main variables and matrices of oscillatory circuits in terms of many-dimensional tensor objects. A solution is obtained for the problem of defining eigenvalues of pure-loop and pure-node circuits. A new method is developed for defining a full range of eigenvalues of linear oscillatory systems with a great number of degrees of freedom. The notion of three-dimensional generalized rotations has been introduced. Relations between the parameters of the spinor representation of a group of three-dimensional generalized rotations and the coordinates of the initial and terminal points of rotation have been obtained. The simple relations between the elements of a three-dimensional orthogonal matrix of the basic representation and the Euler angles, on the one hand, and the coordinates of the initial and terminal points of rotation, on the other hand were derived. The spinor method of solution of inverse kinematic problem for spatial mechanisms with spherical pairs has been developed and the corresponding algorithm has been proposed. The obtained results permitted to reduce the actual three-dimensional problem of spatial motion control to the one-dimensional problem; simple adaptive algorithms are suggested, by means of which various partial problems on the terminal control are solved under various terminal conditions. New algorithms of control of spatial rotations of manipulating robots are studied.