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We propose in this paper a method to detect cracks in circular plates clamped at the circumference. It consists in analyzing and interpreting the frequency changes that occur due to a circular arc crack. We show that the effect of the crack depends on the strain energy loss, which actually is proportional to the curvature achieved by the affected region in the radial direction. Three types of cracks are considered herein to demonstrate this dependency, differing by their length. Modal analysis is performed involving a professional simulation software for the intact plate, and that with the generated cracks. We found a perfect fit when we compared the curvature of the plate in the radial direction obtained with the analytical relation with the frequency values for the crack displaced along the radius. Therefore, using the curvature of several vibration modes we can obtain damage patterns that can be used to locate circular cracks.

The paper presents a study dedicated to analyzing the sensitivity of structures with open contours to cracks. For this purpose, we analyze the particular case of the I-beam made by welding, for which we want to see how the frequencies of different modes of vibration change when a crack occurs in the weld joint between the web and the flange. The research is done using numerical simulations performed with the ANSYS program, from which the natural frequencies are obtained depending on the depth and extent of the crack. A large number of vibration modes and two types of boundary conditions are taken into account. We found that the frequency changes are small but still detectable and depend on the beam fixation mode. This makes it possible to detect cracks in the weld joint if the vibration modes where the decrease in frequency is significant are analyzed.

In this paper, we present a simple predictive model to estimate the natural frequencies of steel cantilever beams with branched cracks. The model considers the cracked beam as a beam with constant cross-section, on which acts an equivalent bending moment. This bending moment takes into account the modification of the rigidity on the beam segment on which the crack extends in the longitudinal direction and the additional rotation that occurs in the slices located at the ends of the damaged segment. Along with the theoretical background and the deduced mathematical model, we present several cases of deterioration defined by the depth of the transverse crack branch respectively the position and the extension of the longitudinal branch. These examples prove the accuracy of the predictions regarding the change of the natural frequencies due to the damage obtained with the proposed model.

The paper presents a mathematical relation to express the effects of a T-shaped crack on beam-like structures. The study is performed by using the finite element method (FEM) and contrived analytical formulas for proving the existing correlation between deflection, strain energy, and natural frequency changes. The damaged beam static and dynamic behaviors are analyzed by means of the FEM. The frequency shift for the first mode of vibration derived directly was compared with the frequency alteration calculated involving the change in the beam deflection for several scenarios of the T-shaped crack. The frequency shift is derived using the relations between deflection and natural frequency. As the final step in the research, we compared the outcomes for the same crack with the results achieved from a beam suffering a loss of stiffness.

The paper presents a study regarding the vibration behavior of Euler-Bernoulli beams which have cracks with complex shapes. The aim is to show that diverse orientations of the crack branches occurring in a given region of the structure produce different changes of the eigenfrequencies for the different vibration modes, dependent on the position and propagation angle of the crack. The research is conducted employing the finite element analyses, which are used to describe the dynamic response of a structure affected by a transversal open crack followed by two branches, each oriented at a different angle. These cracks are traditionally referred to as Y-shaped crack. We calculated the relative frequency shifts for the first six transverse vibration modes and extracted the damage signature for the considered branched crack location. The damage signatures can be used as patterns in damage detection procedures, transforming the procedure of detecting the crack position and severity from the eigenfrequency changes in an inverse problem.

Detecting damage by using vibration signals is popular because it permits evaluating the structural integrity without being necessary scanning of the whole structure. The effect of transverse cracks is presented in detail in the literature, but in reality, the cracks can shift the direction of propagation and even split, resulting in the so-called branched crack. The effect of this type of crack is less investigated due to its complexity. We herein propose a simple model to predict frequency changes that occur due to branched cracks. Initially, we present the effect of stiffness reduction along the damaged section on the structure’s natural frequency. Next, we show that the predicted frequency drop is smaller that happens in reality. This is caused by the sudden cross-section reduction in the slice on which the transverse crack branch is. The phenomenon is similar to the stress concentration for static loads. We propose for dynamic systems a factor that considers the energy stored at the delamination ends. Considering this factor and the stiffness reduction on the damaged segment, we obtain accurate frequency changes due to any type of crack that extends in the longitudinal direction. The model is implemented in Python and tested successfully against simulation with dedicated software.

We propose herein a mathematical relation to calculate the severity of transverse cracks which affect a prismatic beam. It is known that the damage severity depends on the depth of the crack located at the slice on which the biggest bending moment is achieved. The frequency drop due to this crack is proportional to the severity associated with it. For all other crack locations, the effect is diminished in relation to the curvature registered at the affected slice. It is essential to obtain a relationship to correctly express the severity with respect to the crack depth. To find it, we designed a model of a cantilever beam with cracks located near the fixed end. We observed that the deflection of the beam’s free end increases the closer the crack to the fixed end is, but after a certain limit, the deflection decreases. We concluded that the deflection decrease occurs because the deformation around the crack is restricted by the fixing condition. To find the true severity, we derived the pseudo-severity for six crack positions and estimate the severity using the linear and second-order polynomial regression curves. Next, we used twelve points for interpolation and found a very similar severity. Finally, we performed modal analysis for the beam with cracks at different positions and found the mathematical relation developed to predict frequency changes that involves the severity provides accurate results.

This paper presents the determining of natural frequencies of plates without and with damages using the finite element method of SolidWorks program. The first thirty natural frequencies obtained for thin rectangular rectangular plates clamped on contour without and with central damages a for different dimensions. The relative variation of natural frequency was determined and the obtained results by the finite element method (FEM) respectively relative variation of natural frequency, were graphically represented according to their vibration natural modes. Finally, the obtained results were compared.