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Identifying cracks in the incipient state is essential to prevent the failure of engineering structures. Detection methods relying on the analysis of the changes in modal parameters are widely used because of the advantages they present. In our previous research, we found that eigenfrequencies were capable of indicating the position and depth of damage when sufficient vibration modes were considered. The damage indicator we developed was based on the relative frequency shifts (RFS). To calculate the RFSs for various positions and depths of a crack, we established a mathematical relation that involved the squared modal curvatures in the healthy state and the deflection of the healthy and damaged beam under dead mass, respectively. In this study, we propose to calculate the RFS for beams with several cracks by applying the superposition principle. We demonstrate that this is possible if the cracks are far enough from each other. In fact, if the cracks are close to each other, the superposition method does not work and we distinguish two cases: (i) when the cracks affect the same beam face, the frequency drop is less than the sum of the individual frequency drops, and (ii) on the contrary, cracks on opposite sides cause a decrease in frequency, which is greater than the sum of the frequency drop due to individual damage. When the RFS curves are known, crack assessment becomes an optimization problem, the cost function being the distance between the measured RFSs and all possible RFSs for several vibration modes. Thus, the RFS constitutes a benchmark that characterizes damage using only the eigenfrequencies. We can accurately locate multiple cracks and estimate their severity through experiments and thus prove the reliability of the proposed method.

Nowadays, the damage severity evaluation in mechanical structures is mostly performed by analyzing the natural frequency shift. The non-isotropic materials, as the multi-layered ones, are widespread in industrial applications, due to their interesting physicmechanical properties. Thus, a deeper approach of multi-layered beams becomes an important request in the research domain. This paper introduces a damage severity estimator by expressing the crack evolution as a function of stored energy. It is well known that the energy stored in a beam without damage is greater than the energy of that damaged beam. As a consequence, the beam deflection can be related to the stored energy. In this regard, the possibility to split the damage localization and the damage severity assessment has been proven, and also the graphical evolution of the natural frequency shift has been achieved as a function of the crack depth. The results achieved by the finite element method (FEM) and experimental tests are given in tables and graphics. For the first five vibration modes, a comparison was made between frequencies accomplished by analytical, numerical and experimental analyses, in order to give more credibility to the accuracy of the research data presented in this paper.

The paper proposes an inertial system capable of generating a propulsive force from the rotation on an eccentric circular path of 8 steel balls placed between two rotating discs constructed with a radial slot system. In addition to a detailed description of the mechanical components of this proposed device, particular attention is paid to the theoretical treatment of the innovative principle upon which the device is based. The paper begins by detailing the equations of motion of the steel balls, from which the propulsion force of the system is computed. Furthermore, the influence of the ball radius on the propulsion force is investigated. The obtained findings support the capability of the proposed system to achieve a linear propulsive force.

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