Solution of the Inverse Spectral Problem for a Rod Weakened by Transverse Cracks by the Levenberg—Marquardt Optimization Algorithm

Abstract

The longitudinal vibrations of a rod weakened by transverse cracks are considered. Cracks are assumed to be open and modeled by translational springs. The stiffness of the springs corresponds to the size of the cracks. A method has been developed for identifying the number and position of transverse cracks, as well as the stiffness of the corresponding springs according to two spectra that correspond to two types of conditions at the ends of the rod: free—free and fixed—free. The developed method is based on minimizing the objective function that characterizes the difference between the given (measured) natural frequencies and natural frequencies calculated during the implementation of the algorithm. The objective function is minimized using the Levenberg—Marquardt algorithm. Numerical examples are considered. The stability of the obtained results with respect to noise in the initial data is investigated.