Vibration Analysis Of Orthotropic Plates Research Articles

New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates

• Reviewed various analytical methods applicable for plates with general boundary conditions. • Applied finite integral transform to vibration of rotationally-restrained orthotropic plates . • Proposed a new formulation of finite integral transform. • Discussed numerical issues related to application of finite integral transform method. The exact series solutions of plates with general boundary conditions have been derived by using various methods such as Fourier series expansion, improved Fourier series method, improved superposition method and finite integral transform method. Although the procedures of the methods are different, they are all Fourier-series based analytical methods. In present study, the foregoing analytical methods are reviewed first. Then, an exact series solution of vibration of orthotropic thin plate with rotationally restrained edges is obtained by applying the method of finite integral transform. Although the method of finite integral transform has been applied for vibration analysis of orthotropic plates, the existing formulation requires of solving a highly non-linear equation and the accuracy of the corresponding numerical results can be questionable. For that reason, an alternative formulation was proposed to resolve the issue. The accuracy and convergence of the proposed method were studied by comparing the results with other exact solutions as well as approximate solutions. Discussions were made for the application of the method of finite integral transform for vibration analysis of orthotropic thin plates.

A unified formulation for free transverse vibration analysis of orthotropic plates of revolution with general boundary conditions

ABSTRACTIn this investigation, a general formulation for free transverse vibration analysis of orthotropic plates of revolution subjected to arbitrary boundary supports is presented by using the so-called spectro-geometric method (SGM) and the Rayleigh–Ritz technique. Under the current solution framework, the geometry of a structure can be accurately described in terms of mathematical or design parameters, rather than a computational grid or mesh. The unknown expansion coefficients are treated as the generalized coordinates and are determined using the Rayleigh–Ritz method. The accuracy and versatility of the current approach is fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions.

Free vibration analysis of orthotropic plates by differential transform and Taylor collocation methods based on a refined plate theory

Based on the two-variable refined plate theory, free vibration of orthotropic plates is analyzed using the differential transform method (DTM) and the Taylor collocation method (TCM). The refined plate theory outperforms the classical plate theory, and its formulation is simpler than those of other higher-order theories. Without the need for any shear correction factor, the theory performs reliably. The plates considered have two opposite edges simply supported (Levy plates). The first part of the analysis considers three combinations of clamped, simply supported and free edge conditions for the other two edges, keeping one of them simply supported. Detailed formulations of DTM and TCM for the free vibration analysis are given and, consequently, used to predict the frequency parameters and the effect of various factors ranging from geometric to material parameters. Next, the paper presents analysis of some cases, the multi-span plates and plates with stepped thickness and end rotational springs, whose analytical solutions are not readily available, particularly based on the two-variable refined plate theory. In order to verify the results, formulations of three more plate theories, namely the classical or Kirchhoff plate theory, the first-order shear deformation theory of Mindlin and the high-order shear deformation theory of Sayyad and Gugal, were implemented and solved using the proposed methods.

Buckling and free vibration analysis of orthotropic plates by using exponential shear deformation theory

In the present paper, an exponential shear deformation theory is used to determine the natural frequencies and critical buckling loads of orthotropic plates. The theory accounts for a parabolic distribution of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The in-plane displacement field uses an exponential function in terms of thickness coordinate to include the effect of shear deformation and rotary inertia. Governing equations and boundary conditions are derived from the dynamic version of principle of virtual work. The Navier type solution is employed for solving the governing equations of simply supported square orthotropic plates. The results obtained using present higher order shear deformation theory are found to be agree well with those obtained by other several existing higher order theories for analyzing the buckling and free vibration behaviour of orthotropic plates.

Free vibration analysis of orthotropic plates with variable thickness resting on non-uniform elastic foundation by element free Galerkin method

This study intends to investigate the vibration behavior of a thin square orthotropic plate resting on non-uniform elastic foundation and its thickness varying in one or two directions. By using the classical plate theory and employing element free Galerkin method, it is shown that the fundamental frequency coefficients obtained are in good agreement with available results in the literature. The effects of thickness variation, foundation parameter and boundary conditions on frequency are investigated. The results show that the method converges very fast regardless of parameters involved.

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.

The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape

Free flexural vibrations of homogeneous, thin, orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the R-function method. The proposed method is based on the use of the R-function theory and variational methods. In contrast to the widely used methods of the network type (finite differences, finite element, and boundary element methods), in the R-function method all the geometric information given in the boundary value problem statement is represented in an analytical form. This allows one to seek a solution in a form of some formulas called a solution structure. These solution structures contain some indefinite functional components that can be determined by using any variational method. A method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described. Numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits.

A Discrete Method for Bending and Free Vibration Analysis of Orthotropic Plates with Non-uniform Thickness

A Discrete Method for Bending and Free Vibration Analysis of Orthotropic Plates with Non-uniform Thickness

Free vibration analysis of orthotropic plates resting on Pasternak foundation by mixed finite element formulation

In mixed finite element formulation three different methods: (1) Hellinger-Reissner principle (HR); (2) Hu–Washizu principle (HW); (3) Gâteaux Differential Method (GDM), are widely used. In this study using the GDM, a functional and a plate element capable of modeling the Kirchhoff type orthotropic plate resting on Winkler/Pasternak (isotropic/orthotropic) elastic foundation are given and numerical results of a free vibration analysis is performed. The GDM is successfully applied to various structural problems such as space bars, plates, shells by Omurtag and Aköz. The PLTEOR4 element has four nodes with 4×4 DOF. Natural angular frequency results of the orthotropic plate are justified by the analytical expressions present in the literature and some new problems for orthotropic plate on elastic foundation (Winkler and Pasternak type foundation) are solved. Pasternak foundation, as a special case, converges to Winkler type foundation if shear layer is neglected. Results are quite satisfactory.

Free vibration analysis of orthotropic plates by the initial value method

A modified method of the classical initial value technique is presented for the free vibration analysis of rectangular orthotropic plates. The method employs finite difference operators in one direction and uses integration in the other direction. The solutions are superimposed to satisfy the boundary conditions. A detailed discussion of the method is given. A comparative study is presented of the initial value method and other numerical methods. Three examples that show the accuracy of the method are presented.