Examination of processes of a hydroelasticity of a ridge pipe ring a lateral view at action of vibration

Mathematics. Physics. Mechanics


Kondratov D. V.1*, Kalinina A. V.2**

1. Volga Management Institute named after P.А. Stolypin, 164, Moskovskaya str., Saratov, 410012, Russia
2. Russian Presidential Academy of National Economy and Public Administration, 23/25, Cathedral str. Saratov, 410031, Russia

*e-mail: kondratovdv@yandex.ru
**e-mail: kali1598@yandex.ru


One of the topical problems of modern machine industry and missile design is the development of engineering methods of computation of operating performances of the complex mechanical systems to secure their reliability and profitability. Machine elements are often subjected to significant vibrational loadings. For the machine elements being the complex mechanical systems various absorption systems are used. For instance, the magnetohydrodynamic dampers can be efficiently applied for the vibration absorption. The considered systems can be modeled by two cylindrical shells enclosed each other with fluid filling between them. This model is useful for explosion engines, flash float gears of navigation, liquid rocket engines, telescopic landing gears, actuating rams with a hollow plunger, and fuel systems of flight vehicles. It is necessary to score that the fluid between shells in such model is damping the eigenvibrations of shells as well as is cooling the shells. For the weight reduction thin-walled shells are used. The outer shell can be geometrically irregular, and the interior one can be modeled as absolutely rigid cylinder. Use of geometrically irregular outer shells with bulkheads not only allows to reduce the weight but also ensures the stability under various vibrational excitations.

The mechanical model of the system with the ribbed pipe of ring cross-section formed by two surfaces of the coaxial cylindrical shells interacting with a viscous incompressible fluid is considered. The exterior shell is geometrically irregular, and the interior one is the absolutely rigid cylinder (see Fig.1).

Fig. 1. Model of a mechanical system.

The mathematical model of this system consisting of partial differential equations of dynamics of the viscous incompressible fluids and the elastic ribbed shell with their boundary conditions is constructed.

The mathematical model is represented by the coupled equations: the non-linear Navier-Stokes partial differential equations, the continuity equation, the partial differential equations of dynamics of both the interior and exterior elastic cylindrical shells of Kirchhoff-Love type, and their boundary conditions. To derive the dynamical equations for the geometrically irregular shell the Hamilton’s variation principle was used. The irregular surfaces of the shell are described by means of generalized Heaviside functions.

The investigation techniques of hydroelasticity problem for ribbed pipes with ring cross-section under vibration loadings are analyzed. The coupled problem of hydroelasticity is solved by means of a perturbation technique and in the guess of the harmonic law of vibration; the dimensionless variables are used. The solution of the equations of hydromechanics is formulated as the monomial small-parameter decomposition where the parameter describes the relative width of the supporting stratum of the fluid, and the monomial small-parameter decomposition with the parameter describing the relative deflection for each shell.

To solve the dynamic equations for the outer shell the Bubnov-Galerkin method was used. The expressions for components of velocity of a fluid, hydrodynamic pressure, and displacements of the exterior elastic geometrically irregular cylindrical shell are obtained.

The proposed mathematical model and research techniques allow to explore the hydroelasticity of a ribbed pipe of a ring cross-section with elastic geometrically irregular outer shell and absolutely rigid interior one with the viscous incompressible fluid between them at presence of the vibration. The solutions of these problems allow one to find the causes of the cavity corrosion of machine elements and increase the strength and a reliability of mechanical systems.


hydroelasticity, tube with ring cross section, geometrically irregular shell, viscous incompressible liquid, vibration


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