In this article, we analyze the effect of transverse cracks on the natural frequencies of a Euler-Bernoulli functional gradient beam. The studied beam was discretized into finite elements and the global matrices of the motion equation are determined by applying the Lagrange equation to the beam kinetic and deformation energies. The material properties are considered to vary in the directions of the beam thickness, the gradation is described by the power-law distribution, the stiffness of the cracked element is determined based on the reduction of the beam cross-section. The numerical results obtained are compared with those available in the previous study. Finally, case studies were carried out to analyses the influence of the power law index, the depth and the opposition of the crack on the natural frequencies of the beam for different boundary conditions; these studies demonstrate the advantage of the FGM beam over the purely metal beam.
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.
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