In the study on limiting possibilities of a casting process in terms of resource saving it is considered which solidstate constructions can optimally fill the space of the mold. The possibility of designing patterns for the Lost Foam Casting process in accordance with mathematical concepts is shown. The examples of patterns of frame cellular castings are described. The examples of cellular castings, designed with the help of computer technologies, are shown that often inherit structures of nature. Such castings allow expanding of the existing range of properties of metal products.

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions results in the impossibility to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameter together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different character take place, are defined. Dependence of the vibration frequencies on physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allow one to control the frequency spectrum, in which excitation of vibrations is possible.Mathematics Subject Classification 2000: 93C20, 74H10, 74H15, 74H45.

-This paper describes designs of metal castings, produced with the help of 3D technologies. Methods for materials 3D processing are related to the additive method of manufacturing, which is associated with the third industrial revolution, characterized by the resource saving ecological production. The examples of frame cellular castings are shown, which inherit structures of nature with the optimal combination of materials consumption, strength and attractive appearance. The described 3D technologies expand the existing range of metal products. Use of 3D printers reduces the technogenic impact on the environment, saving up to 90 % of the starting material, in contrast to the current "subtractive production". Among the new foundry processes, at the Physico-Technological Institute of Metals and Alloys (Kyiv, Ukraine) there have been patented 3D technologies of sand products molding by means of the deformation of loose materials as well as obtaining sand shell molds for one-time patterns.

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Dynamics of the hybrid systems of aerial cable-ways is investigated. The eigenvalue problems are considered for such hybrid systems with different assumptions. An overview of different methods for eigenvalue problems is given. In the research, the method of the normal fundamental systems is applied, which turns out to be very effective for the considered problems. Changes of dynamical characteristics of the systems depending on the controlled parameter are studied.Mathematics Subject Classification 2000: 37N35, 35B37, 93C20.