A trapezoidal Fourier p-element for membrane vibrations

A trapezoidal Fourier p-element for the analysis of membrane transverse vibrations is investigated. Trigonometric functions are used as shape functions instead of polynomials to avoid ill-conditioning problems. The element matrices are analytically integrated in closed form. With the enrichment degrees of freedom in Fourier series, the accuracy of natural frequencies obtained is increased in a stable manner. One element can predict many modes accurately. Since a triangle can be divided into three trapezoidal elements, the range of application is wider than the previously derived rectangular Fourier p-element. The natural modes of a square membrane consisting of two trapezoidal elements are computed as test cases and convergence is very fast with an increasing number of trigonometric terms. Comparison of natural modes calculated by the trapezoidal Fourier p-element and the conventional finite elements is carried out. The results show that the trapezoidal Fourier p-element produces higher accurate natural frequencies than the conventional finite elements with the same number of degrees of freedom.     

On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates

A comprehensive study on the use of a set of trigonometric functions, originally proposed by Beslin and Nicolas [Journal of Sound and Vibration 1997;202:633–55], as admissible solutions in the Ritz method for general vibration analysis of rectangular orthotropic Kirchhoff plates is presented. The approach is denoted here as Trigonometric Ritz method (TRM). Since its introduction, application of TRM was limited to a very few plate problems. The aim of this work is to extend the potential of the method on predicting natural flexural frequencies of plates with various complicating factors, including in-plane loads, elastically restrained edges, rigid/elastic concentrated masses, intermediate line and point supports or their combinations. Computational efficiency, stability, convergence and accuracy of the method are discussed and supported by extensive analysis. TRM-based solutions are compared with many reference cases available in the literature obtained with other methods or Ritz functions. Numerical results indicate that TRM exhibits good to excellent accuracy for all cases considered. New solutions are also presented for future comparison purpose.     

Free vibration of composite plates using the finite difference method

The finite difference method was used to solve differential equations of motion of free vibration of composite plates with different boundary conditions. The effects of shear deformation and rotary inertia on the natural frequencies of laminated composite plates are investigated in this paper. Four cases are studied: neglecting both shear deformation and rotary inertia, considering only rotary inertia, considering only shear deformation, and considering both. Solutions were obtained for symmetric and angle-ply laminated plates. The factors that affect natural frequencies of different composite plates, such as span-to-depth ratio, aspect ratio, angle-ply, and lamination sequence were also investigated. Results were found to agree well with exact and approximate solutions reported in literature. Shear deformation showed a considerable effect on the natural frequencies for composite plates, whereas the rotary inertia effect was found to be negligible.     

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R₁, the stiffness equation is first formulated in region [−R₁, R₁] with even number of nodes and then reduced to region [0, R₁] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.     

Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method

In the present study, the free vibration of laminated functionally graded carbon nanotube reinforced composite beams is analyzed. The laminated beam is made of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers. In each layer, single-walled carbon nanotubes are assumed to be uniformly distributed (UD) or functionally graded (FG) distributed along the thickness direction. Effective material properties of the two-phase composites, a mixture of carbon nanotubes (CNTs) and an isotropic polymer, are calculated using the extended rule of mixture. The first-order shear deformation theory is used to formulate a governing equation for predicting free vibration of laminated functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beams. The governing equation is solved by the finite element method with various boundary conditions. Several numerical tests are performed to investigate the influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, length-to-thickness ratios and the numbers of layers on the frequencies of the laminated FG-CNTRC beams. Moreover, a laminated composite beam combined by various distribution types of CNTs is also studied.     

A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution

This paper deals with the free vibration analysis of rotating laminated cylindrical shells. The analysis uses discrete singular convolution (DSC) technique to determine frequencies. Regularized Shannon's delta (RSD) kernel is selected as singular convolution to illustrate the present algorithm. The formulations are based on the Love's first approximation shell theory, and include the effects of initial hoop tension and centrifugal and coriolis accelerations due to rotation. The spatial derivatives in both the governing equations and the boundary conditions are discretized by the DSC method. Frequency parameters are obtained for different types of boundary conditions, rotating velocity and geometric parameters. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed. The analysis has been verified by comparing results with those in the literature and sufficient agreement is obtained.     

Crack analysis in orthotropic media using the extended finite element method

An extended finite element method has been proposed for modeling crack in orthotropic media. To achieve this aim a discontinuous function and two-dimensional asymptotic crack-tip displacement fields are used in a classical finite element approximation enriched with the framework of partition of unity. It allows modeling crack by standard finite element method without explicitly defining and re-meshing of surfaces of the crack. In this study, fracture properties of the models are defined by the mixed-mode stress intensity factors (SIFs), which are obtained by means of the domain form of the interaction integral (M-integral). Numerical simulations are performed to verify the approach, and the accuracy of the results is discussed by comparison with other numerical or (semi-) analytical methods.