In the machining of shafts with low rigidity it is difficult to achieve the required parameters of accuracy of form and dimensions and of surface quality. Under specific conditions, the low inherent rigidity and the relatively low shaft rigidity, compared to the rigid assemblies of the machine tool, cause the appearance of vibrations. The paper presents a 7-level hierarchical mathematical model of the system of longitudinal turning and a mathematical model of the dynamic system of machining of shafts with low rigidity in the elastic-deformable state, taking into account factors interfering with and destabilising the process of shaft machining. For the purpose of validation of the models, a study was conducted on the conformance of the dynamic characteristics of the mathematical model with the actual object. The study was realized on specially designed experimental workstation for the study of characteristics of the dynamic system of turning of elastic-deformable shafts with low rigidity. The experiment demonstrated a high conformance of the dynamic characteristics of the mathematical model with the actual object.