Fluid-structure interactions of internal pressure pipeline using the hierarchical finite element method

We study the influence of the fluid with the structure in vibration between fluid and structure of a cylinder of circular section granted by the phenomenon of the interaction fluid structure of a conditioned flow of laminar nature and incompressible in the form of the macrostructure. These two phenomena by the mechanical relations of stresses according to displacements, modelled by a cylinder. The analysis of the vibrations of cylinders filled with fluid is studied with limiting conditions of fluid and the solid with the coupling conditioned by its limits of action-reaction in forces. The problem of the cylindrical pipe is formulated by deriving the deformation and the kinetic energies of the vibrating cylinder with and its fluid to have different natural frequencies, we use the principle of Hamilton change the problem in the expression of the equation cylindrical differential which gives three displacement functions in a system of partial differential equations of the cylindrical coordinate of circular section which meet the limiting conditions imposed at both ends. Let us apply the Navier-Stocks equation in cylindrical coordinates, with the fluid continuity equation, for the solid equation of mechanical behaviour of stresses in terms of displacement by strain. To obtain the results of natural frequencies we use the Galerkin method for solid and for Galerkin-time fluid. Where the liquid influences the inner surface of the circular cylinder, depending on the condition of the coupling that the stresses of the solid are equal to the stresses fluid. The modelling is done by a computer language (MATLAB), the hierarchical finite element method is presented by a Legendre polynomial with double integral of Rodrigues, to arrive at the final formula of the mass-rigidity matrix, which dissects on three parts (fluid, coupling and structure). Based on a comparison with experimental results. We continue to study some geometrical and physical parameters which influence the natural frequencies, in a proportional or inversely proportional way.

An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.