The natural frequencies of the bending vibrations of the pipeline are investigated. The pipe sags over the obstacle and is under the action of tensile force. Outside the sagging area, the pipe rests on elastic supports. The fluid transported through the pipeline is under pressure. The direct problem was solved earlier, in this article, the inverse problem of identifying the speed and density of the transported fluid by the known natural frequencies of bending vibrations is solved. The equation of bending vibrations of a pipeline is described by the Kirchhoff model. The characteristic equation is solved using Ferrari formulas. The general decision is determined. We substitute the general solution into the boundary conditions and obtain a system of equations. This system gives a frequency equation, which is solved numerically on a developed program in the Maple package. The method of successive approximations is applied, after the third iteration, the accuracy of calculating the parameters of the velocity and density of the liquid is approximately 10−3. Thus, it was found that with an increase in the oscillation frequency, the density of the liquid inside the pipe decreases. It is determined that with increasing natural frequencies of pipe bending vibrations, the fluid velocity parameter increases. It is shown that the two lower frequencies of bending vibrations of the pipeline can be used to determine the parameters of the velocity and density of the liquid. The dependence of the mass flow rate of the liquid on the first natural frequency of the pipe oscillations is given. It is shown that with increasing frequency, the mass flow rate decreases. The research results will help the development of acoustic diagnostic methods and non-destructive testing methods and will find technical application for monitoring and diagnosing the state of pipeline systems.
The paper presents solutions to the direct and inverse problems on longitudinal vibrations of a rod with a variable cross-sectional area. The law of variation of the cross-sectional area is modeled as an exponential function of a polynomial of degree n . The method for reconstructing this function is based on representing the fundamental system of solutions of the direct problem in the form of a Maclaurin series in the variables x and λ. Examples of solutions for various section functions and various boundary conditions are given. It is shown that to recover n unknown coefficients of a polynomial, n eigenvalues are required, and the solution is dual. An unambiguous solution was obtained only for the case of elastic fixation at one of the rod’s ends. The numerical estimation of the method error was made using input data noise. It is shown that the error in finding the variable crosssectional area is less than 1% with the error in the eigenvalues of longitudinal vibrations not exceeding 0.0001.
In the work investigated the flexural vibrations of the pipeline. Parts of the pipeline on both sides of the sagging section have elastic supports. It is assumed that a constant longitudinal force acts along the neutral line. An incompressible fluid flows through the pipe at a constant average speed. The influence of internal pressure in the pipe on these oscillations is taken into account. The direct problem of determining the eigenfrequencies of flexural vibrations of the pipeline by the Kirchhoff model using Ferrari formulas is solved. The frequency spectrum is determined depending on the fluid pressure, the elasticity of the supports, the velocity of the fluid through the pipe. Particular and limiting cases are considered, for example, when the stiffness of the supports is very large and when they are very small. Graphs of the dependence of the first and second eigenfrequencies on the velocity of the transported liquid at different values of the liquid density parameter are constructed. It is shown that with the growth of the velocity parameter there is a decrease in the natural frequencies of flexural vibrations of the pipeline, and the faster the higher the density parameter of the liquid. It is determined that with an increase in the mass of the liquid per unit length of the pipeline there is a decrease in the natural frequencies of bending vibrations of the pipe. It is found that with the increase in the mass flow through the pipe, the natural frequencies of bending oscillations also decrease. It is confirmed that the frequencies of flexural vibrations of the pipeline are the same for the cases of pipe fastening “rigid fixing — rigid fixing” and “free end — free end”. The results of the study will contribute to the development of methods of acoustic diagnostics and non-destructive testing and can find technical application for monitoring and diagnostics of pipeline systems.
Consideration is given to the direct and inverse problems for pipeline bending both by gravity and transported fluid. The effect of internal pressure drop and the velocity of a fluid are taken into account. The influence of point fixing of “pipeline-capacity” constructions for the deflection is also taken into account. The inverse problem is to determine the relative stiffness of distributed support under the instrument determining pipeline deflection or deformation of its outer fibers. The method of loading pipeline by the concentrated power and determination of appropriate instrument deflection or deformation is applied. In particular, loading and corresponding measurements are carried out at the midpoint of the pipeline span.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.