Boundary Effect on Forming the Natural Frequency Spectrum of Flexural Vibrations of Beams with Local Surface Defects

The authors present the results of calculation investigations on the determination of the mechanisms of the boundary effect for the beam with a constant rectangular cross section on the formation of the natural frequency spectrum of flexural vibrations in the presence of an open edge fatigue crack on the beam surface. For various boundary conditions of the beam fixation, variations of the first-mode flexural vibration natural frequencies at different locations of the above surface defect (modeled by a 1 mm-wide groove with variable depth) along the beam length are established. The analysis of the obtained dependencies revealed that the beam with both free edges contained a segment where the natural frequency of flexural vibrations of the damaged beam was no less than the intact beam frequency. Moreover, there is stress symmetry about the beam middle, which undergoes the maximum frequency variation due to damage. Meanwhile, in the beam with two rigidly fixed edges, the maximal variations of the natural flexural frequencies of the damaged beam occurred at some distance from these edges, namely at 0.025 and 0.975 relative lengths. When the defect/groove was located in the middle of the beam, the damaged beam’s natural frequency varied less (almost twice). Moreover, the data symmetry about the beam middle was observed in the dependencies (0.25 and 0.75), where the natural frequencies of the damaged and intact beams were the same. In the beam with one rigidly fixed end, the smallest value of the natural frequency of the damaged beam vibrations is observed near the fixed edge. With the groove shift from the edge, the vibration frequency increases, reaching or even exceeding that of the intact beam with increased relative length value. For the beam with one or two rigidly fixed edges, the identical dependencies in the direction of one or two axes were obtained. Therefore, for the beam boundary conditions implying one or two free edges, there were segments with the natural frequencies of vibrations of the damaged beam exceeding those of the intact beam. On the contrary, for the boundary conditions of the beam with fixed ends, the above values are equal only locally, i.e., within a very narrow beam segment.