Vibrodiagnostics of the Presence of Surface Symmetrical Cracks in Beams of Rectangular and Round Cross-Sections by the Characteristics of Their Nonlinear Longitudinal Vibrations

This work is devoted to the determination of vibrodiagnostic indicators of the presence of symmetric and annular breathing cracks in cantilever beams at their nonlinear longitudinal vibrations. The results of determining the vibrodiagnostic indicators of the presence of cracks, namely, the relative change in the fundamental frequency of longitudinal vibration and the ratio of the amplitudes of the dominant harmonics at the fundamental and superharmonic resonances for different depths and locations of cracks, are presented. To carry out computational studies, various computational models were developed, namely, with concentrated and distributed parameters, beam and volumetric finite element (FE) models, and methods for their research. The use of the model with concentrated parameters consists of the analytical determination of the nonlinearity parameter of the system, which takes into account the change in the stiffness of the system in the presence of damage and is determined by changing the potential deformation energy of the beam due to the opening of the crack during oscillations. The determination of vibrodiagnostic indicators of the presence of a crack using a model with distributed parameters is carried out by solving the differential equation of forced longitudinal vibrations of the second order, where the crack is represented as a section of the beam with a reduced cross-sectional area. The numerical analysis of vibrations of the objects of study with a breathing crack is carried out using the developed beam and volumetric FE models. In the beam FE model, the breathing crack is modeled by a finite element with variable stiffness, and in the volumetric model, it is represented as a mathematical section with the setting of contact interaction conditions for its banks. The main dependences of vibrodiagnostic indicators of the presence of cracks on their parameters are obtained analytically using a model with concentrated parameters and numerically using a beam FE model. To confirm the reliability of the obtained dependences, their individual values are compared with the data obtained for the volumetric FE model, which obviously describes the stress–strain state in the vicinity of the crack most accurately among the proposed models but at the same time requires more computational resources. The obtained numerical dependences of the ratio of the amplitudes of the dominant harmonics of displacements at the main resonance from the place of vibration regis