Development of numerical-analytical models for calculating the natural frequencies of vibrations of a shaft with a crack

Abstract

The problem of vibration diagnostics of damage to power equipment during operation is extremely relevant for increasing the reliability and safety of its operation. One of the most dangerous defects that occur in shafts and rotors of steam turbines is cracks, the development of which can lead to catastrophic failures. Therefore, the development of vibration diagnostics methods capable of detecting subcritical cracks is of important practical importance. The combination of modern computing capabilities with CAE systems allows for effective analysis of the dynamic behavior of structures under various operating conditions of thermal power equipment. The paper proposes a numerical-analytical approach to assess the impact of a crack on the change in shaft stiffness. The basis for building the model is the use of the provisions of fracture mechanics and the Castigliano theorem, which allows expressing the compliance of the section through the ratio between the rate of deformation energy release and stress intensity coefficients (SIF). To study the bending, longitudinal and torsional vibrations of the shaft, taking into account the presence of a crack, CAD models of a beam of circular cross-section with edge transverse and semi-elliptical cracks were built. The geometric parameters of the defect were considered, in particular the relative depth γ = a/D and the aspect ratio a/b, which determine the nature of the crack's influence on compliance. The obtained expressions for the change in the strain energy allowed us to form analytical dependencies for determining compliance under different types of loading: bending, tension and torsion. In the case of bending vibrations, the dependence of the SIF on the moment of resistance of the section and the normalized stress distribution were used; in the case of longitudinal vibrations, the influence of a semi-elliptical crack and the variation of the parameter a/b were taken into account; in the case of torsion, the combined effect of transverse-shear and longitudinal-shear deformations was taken into account, which required approximation of the coefficients of the FII and FIII functions. Thus, the proposed approach provides the possibility of taking into account the nonlinear behavior of the structure with a crack and creates a basis for further development of methods for vibration diagnostics of steam turbine shafts. The use of the obtained numerical-analytical models will contribute to increasing the reliability of equipment condition monitoring, extending its service life and preventing emergency situations at thermal power plants and nuclear power plants.