The paper considers the problem of determining the local inhomogeneity of the medium from the natural frequencies of string oscillation. The inhomogeneity is modeled in three sections: in the first and third sections medium is homogeneous, and on average section the elastic characteristics are modeled by a quadratic function. This model is implemented using the conjugation conditions at boundary between media. It is shown that to identify the center of an inhomogeneity and determine its size, two natural frequencies are enough, and in the case of rigid fixing of both ends of the string, the solution of the problem is dual. The problem is solved by expanding the fundamental system of solutions into a power series in the variables x and λ. The estimates of the error of the method are given.
There are situations when there is a breakthrough of a pipeline with oil products under water. As a result, oil spills on the surface, polluting the environment. Underwater currents and wind can carry an oil stain away from the point of leakage. Therefore, it is not always possible to visually determine the location of a pipeline break by a spot of oil on the surface.To solve such problems, it is proposed to install strain sensors along the pipeline, which take the values of the displacement derivative ∂u(x,t)/∂x (deformations) at different instants of time, and use the simplest model of a pipeline based on the equations of longitudinal oscillations of a homogeneous rod. Formulas for determining the moment and location of the pipeline rupture were obtained from the strain sensors data and a scheme of interaction with GLONASS was proposed that allows instantly detecting leaks and damages of pipelines laid under water and timely eliminating the consequences of the accident. The application of the proposed scheme minimizes the consequences of the accident for the environment and financial costs.
Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.
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