Approximate Analytical Determination of Vibrodiagnostic Parameters of the Presence of a Closing Crack in Bar Elements Under Subharmonic Resonance

Introduction. Detection of damages such as fatigue cracks is of prime importance for ensuring reliability of machine parts and structures operating under dynamic loading conditions. Vibration tests represent one of the most efficient and viable methods of nondestructive evaluation. This has motivated an increased interest in analyzing the influence of cracks on vibrations of structural components, which serves as a basis for the elaboration of feasible methods for vibrodiagnostics of the presence of these damages and for the determination of their parameters. Among the best known and simplest methods are those based on the determination of variation of natural frequencies of a structure. They are focused mostly on beam-type structures with fatigue opening-mode edge cracks. However, for the most important cases of detection of small cracks these methods have usually a poor sensitivity and require certification of natural frequencies and vibrodiagnostic conditions for initial (intact) structural components. Also, in operation of components under high-cycle deformation prior to a crack initiation they can undergo a change of their frequency and dissipative characteristics. For example, it is mentioned in [1] that the available frequency-based methods provide a reliable assessment of medium-sized cracks (equal to about 20% of the depth of beam). Nevertheless, researcher keep on improving these methods, in order to raise their resolution in the determination of crack size and location both for a single crack as well as for several ones [2–8]. It was emphasized in [1] that the methods based on the assessment of natural modes were more sensitive to detection of damage. The development of research along this direction in combination with frequency-based methods was addressed in [1, 6, 9–12]. Also, the use of wavelet transform methods was noted to be quite efficient [1, 6, 13]. Generally, during the vibration analysis of beam-type structural components the researchers consider the presence of open cracks or notches, which permits applying the linear approaches, and less often they investigate the closing or breathing cracks [9, 14]. The closing or breathing cracks are responsible for a characteristic nonlinearity of the vibrating system, thus allowing the use of new vibrodiagnostic methods. Even the earliest of the known research efforts [15–17] demonstrated a high sensitivity of the proposed superand subharmonic methods for the detection of fatigue cracks, which is more than ten times higher than that of the conventional frequency-resonance approaches. Analysis of various nonlinear effects in vibration of beam-type structural components with breathing cracks still attracts researcher’s attention [18–27]. They study the vibration nonlinearity-induced distortion of harmonicity of the strain signal in the section close to the crack location [20], variation of coefficients of harmonic constituents of