The specification of the low rigidity parts machining process is consider by introducing suitable equations of constrains, which describe additional elastic deformations in one of equations describing the force controlling influence. This paper introduces general and detailed mathematical models of the DS of turning the longitudinal, low rigidity shafts. Research results and examples of their approximation were introduced. Some method of synthesis and examples structure of regulator P were shown for one detailed model of DS under received approximated operational transmittance of DS. The way of controlling the accuracy of shafts turning in the elastic deformable condition and controlling system was described.