Identification of a longitudinal notch of a rod by natural vibration frequencies

Objectives. To study the direct and inverse problem of vibrations of a rectangular rod having a longitudinal notch, to analyze regularities of the behavior of natural frequencies and natural forms of longitudinal vibrations when changing the location and size of the notch, and to develop a method for uniquely identifying the parameters of the longitudinal notch using the natural frequencies of longitudinal vibrations of the rod.Methods. The rod with a longitudinal notch is modeled as two rods, where the first one does not have a notch, while the second one does. For connection, conjugation conditions are used, in which longitudinal vibrations and deformations are equated. The solution of the inverse problem is based on the construction of a frequency equation under the assumption that the desired parameters are included in the equation. Substituting natural frequencies into this equation, the nonlinear system with respect to unknown parameters is derived. The solution of the latter is the desired notch parameters.Results. Tables of eigenfrequencies and graphs of eigenforms are given for different notch parameters. The results for different boundary conditions are obtained and analyzed. A method for identifying notch parameters by a finite number of eigenfrequencies is presented. The inverse problem is shown to have two solutions, which are symmetrical about the center of the rod. The unambiguous solution requires eigenfrequencies of the same problem with different boundary conditions at the right end. By adding additional conditions at the ends of the rod, the inverse problem can be solved with new boundary conditions to construct the exact solution and develop an algorithm for checking the uniqueness of the solution.Conclusions. The developed method can be used to solve the problem of identification of geometric parameters of various parts and structures modeled by rods.