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Abstract.The article considers the definition of the stability range of a dynamic system for turning nonrigid shafts with different technological set-ups: standard and developed ones; they are improved as a result of this research. The topicality of the study is due to the fact that processing such parts is associated with significant difficulties caused by deformation of the workpiece under the cutting force as well as occurrence of vibration of the part during processing, they are so intense and in practice they force to significantly reduce the cutting regime, recur to multiple-pass operation, lead to premature deterioration of the cutter, as a result, reduce the productivity of machining shafts on metal-cutting machines. In this connection, the purpose of the present research is to determine the boundaries of the stability regions with intensive turning of non-rigid shafts. In the article the basic theoretical principles of construction of a mathematical system focused on the process of non-free cutting of a dynamic machine are justified. By means of the developed mathematical model interrelations are established and legitimacies of influence of various technological set-ups on stability of the dynamic system of the machine-tool-device-tool-blank are revealed. The conducted researches allow to more objectively represent difficult processes that occur in a closed dynamic system of a machine.

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The article deals with the construction of an equivalent mechanical model of the elastic system of the machine for turning non-rigid shafts with various technological equipment, including machining in centers and machining in a chuck with the rear rotating center being clamped without using additional support and using a vibration-damper lunette. The relevance of this study due to the fact that in view of low stiffness processed non-rigid shaft technology system machine-tool-tool-workpiece is extremely sensitive to the deformations caused by the action of cutting forces. As a result, significant processing errors are generated. High sensitivity to deformations serves as a source of intensive vibrations of the workpiece, which leads to deterioration of the quality of the machined surface of the part, a decrease in the durability and durability of the cutting tool, accelerated wear and loss of guaranteed machine accuracy parameters. This makes it essential to significantly reduce the cutting conditions, to resort to multi-pass processing, which greatly increases the laboriousness and cost of manufacturing products. In this context, the aim of this study – the creation of an equivalent mechanical model of the machine, allowing the process of studies to determine the boundaries of sustainable non-rigid turning shafts considering the different ways of mounting the workpiece on the machine. The materials of the article can be useful for engineers and scientists in the field of engineering.