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.

中文翻译：

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用Levenberg-Marquardt优化算法求解横裂纹削弱的杆的反谱问题

考虑了由横向裂纹削弱的杆的纵向振动。假定裂缝是开放的，并通过平移弹簧进行建模。弹簧的刚度对应于裂缝的大小。已经开发出一种方法，用于根据与杆端的两种状态对应的两个光谱来识别横向裂纹的数量和位置以及相应弹簧的刚度：自由-自由和固定-自由。所开发的方法基于最小化目标函数的特征，该目标函数表征了给定（测量）的固有频率与算法实现期间计算出的固有频率之间的差异。使用Levenberg-Marquardt算法将目标函数最小化。考虑了数值示例。