Efficiency of the method of spectral vibrodiagnostics for fatigue damage of structural elements. Part 3. Analytical and numerical determination of natural frequencies of longitudinal and bending vibrations of beams with transverse cracks. solution

摘要

Based on the concepts of linear fracture mechanics,we derive analytical expressions for the determination of naturalfrequencies of longitudinal and bending vibrations of beams whichare rectangular in cross section, are fixed in different ways, havevariable ratios of the section height to the beam length, and havetransverse cracks of various types. The results of' the analyticalsolution are compared with those obtained by the finite-elementmethod as well as with experimental data obtained by the authorsand other scientists. The analytical solution under consideration isshown to be quite simple and provide a fairly good accuracy of theresults obtained. Using a cantilever beam with one or twosymmetrical edge cracks or a central through crack as an example,we consider the possible dependence of the relative change innatural frequencies of vibration on the relative crack length, cracklocation, and the vibration mode of a beam. We discuss thepossible methods of evaluating the crack size and location fromthe results of experimental determination of the change in naturalvibration frequencies of a cracked beam.