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Free Vibrations of the Aboveground Section of the Main Gas Pipeline after the Movement of the Cleaning or Diagnostic Piston inside It

The authors solve the boundary-value problem regarding the identification of the free vibrations of the above-ground section of the gas pipeline which are caused by the passage of the cleaning (diagnostic) piston. The boundary conditions in this problem correspond to the pinching of the ends of the pipeline section. The initial conditions are derived from the solution of the problem of forced vibrations of the same aboveground section of the gas pipeline, which are caused by the movement of the cleaning (diagnostic) piston inside the pipeline. This problem was solved earlier when using the Laplace integral transform, taking into account the initial deflection of the pipeline section under its own weight. The obtained solution of this problem is the sum of double integrals and several simple summands. The desired solution to the problem of free vibrations of the aboveground section of the gas pipeline is presented as the product of two functions. The former is a function of the pipeline coordinates only, and the later is a function of time. The first function is the sum of the products of unknown coefficients which have been identified using the known boundary conditions of the problem and Krylov functions which include the roots of the characteristic equation. It is a fundamental function which characterizes the mode of free vibrations of the pipeline section. There is a unlimited number of such functions, since there is an infinite number of roots of the frequency equation. In the second function, the cosines and sines have unknown coefficients, which are calculated by using the given initial conditions of the problem. The calculation of these coefficients is based on identifying the integrals of the product of the functions of the initial problem situations and fundamental functions. Since the functions of the initial conditions of the problem are complex and represent the sum of double integrals and some simple functions, Lagrange interpolation polynomials are used to facilitate the calculation of the mentioned coefficients. At the pipeline section which is 100 m long the numerical values ​​of the interpolation polynomials coincide with the functions of the initial conditions at 12 points (including the extreme points 0 and 100 m).