The mathematical model of the mechanical system, which is a section of the pipeline with built-in motor pump system and damper for vibration damping, has been developed. The mathematical model is represented by a system of partial differential equations of hydrodynamics for moving fluid in the pipeline and a system of two ordinary differential equations for the damping device. Solution the system of equations is performed using analytical and numerical methods. The numerical solution of the

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... tions of hydrodynamics carried out using the finite element method, implemented in the programming environment of mathematical package freeFEM ++.