Approximate analytical method for determining the vibration-diagnostic parameter indicating the presence of a crack in a distributed-parameter elastic system at super- and subharmonic resonances

Abstract

620.178.5:620.179 and A. P. YakovlevAn approximate method for calculating the vibration-diagnostic parameter indicating the presence of a crack in an elastic distributed-parameter system at super-and subharmonic resonances is considered. The method involves the finding of the non-linearity characteristic of an elastic system based on the analysis of its forced vibrations in the undamaged state and the use of results of the approximate analytical determination of parameters of nonlinearity for vibrations of an elastic body with a closing crack, which is simulated by a single-degree-of-freedom system with an asymmetric bilinear characteristic of the restoring force at the above-mentioned resonances.Keywords: super-and subharmonic resonances, vibration-based diagnostics of fatigue damage, closing crack, nonlinear vibrations, distributed-parameter elastic system. Introduction.Interest in studying vibrations of elastic bodies in the presence of damages such as fatigue cracks is motivated by the necessity of evaluating a possible change in the vibrational state of structural elements during their operation on the one hand, and by the development of the efficient methods of vibration-based diagnostics of this kind of damages, on the other. Here, one of the most investigated lines of research is establishing the calculated relationships between the parameters of the closing Mode I crack and vibration process at super-and subharmonic resonances. This is due to the complexity of the analytical solution of the problem on forced vibrations of a substantially nonlinear system derived.In this field, a number of investigations are known. Thus, in [1], the problem of transverse oscillations of a prismatic beam with a closing crack at subharmonic 2-nd order resonance under the action of a harmonic transverse concentrated load was considered using the Ostrogradsky-Hamilton principle and Ritz method. The weakening of the bending rigidity of the girder with an open crack was interpreted as a local decrease in the effective cross section of the beam in the limited region approximated by a triangular prism having a right angle at the crack tip and a height equal to the crack depth. It was assumed that in the region of the resonance, the beam performs vibrations close to one of the natural modes, and the mathematical model of such vibrations was represented by one ordinary differential equation in which the sign of the term corresponding to the so-called reduced depth of the crack was governed by the sign of the quasi-normal coordinate. Next, using the averaging method, the set of the time-independent differential equations was set up for determining the amplitude and phase of the second harmonic of the vibration process.The problem on forced vibrations of a rectangular plate containing longitudinal defects of continuity, such as closing cracks, was considered in [2] for the case of superharmonic second-order resonance of the first mode of transverse vibration. To solve it, the Ostrogradsky-Hamilton principle and Ritz method were a...

Nonlinear Vibration Analysis of a Beam with a Breathing Crack

“…Over the last decades, many scholars have exploited the different nonlinear characteristics of beams with breathing cracks by using analytical methods [18][19][20], numerical methods [21,22] and experimental approaches [23,24]. Their studies showed that nonlinear vibration characteristics have a higher sensitivity to breathing cracks, especially the presence of sub-and super-harmonic resonances [25].…”

“…Their studies showed that nonlinear vibration characteristics have a higher sensitivity to breathing cracks, especially the presence of sub-and super-harmonic resonances [25]. For example, Matveev et al [18] presented an approximate analytical method to calculate the relative amplitude of the dominant harmonics of a rod-like structure with a closing crack at sub-and super-harmonic resonances, and they showed that the acceleration amplitude of the second harmonic is more sensitive than that of the fundamental harmonic. Bovsunovskii et al [19] developed the dynamic model of a beam with a closing crack which considered the damping change due to crack propagation.…”

Nonlinear Vibration Analysis of Beam and Plate with Closed Crack: A Review

“…In order to determine the relative amplitude of the dominant harmonics of a rod-like structure with a closing crack at subharmonic and superharmonic resonances, Matveev et al [122] presented an approximate analytical method. They demonstrated that the acceleration amplitude of the second harmonic is more sensitive than that of the fundamental harmonic.…”