Vibration Analysis Of Mindlin Plates Research Articles

Forced vibration analysis of Mindlin plates resting on Winkler foundation

The purpose of this paper is to study shear locking-free parametric earthquake analysis of thick and thin plates resting on Winkler foundation using Mindlin’s theory, to determine the effects of the thickness/span ratio, the aspect ratio and the boundary conditions on the linear responses of thick and thin plates subjected to earthquake excitations. In the analysis, finite element method is used for spatial integration and the Newmark-β method is used for the time integration. Finite element formulation of the equations of the thick plate theory is derived by using higher order displacement shape functions. A computer program using finite element method is coded in C++ to analyze the plates clamped or simply supported along all four edges. In the analysis, 8-noded finite element is used. Graphs are presented that should help engineers in the design of thick plates subjected to earthquake excitations. It is concluded that 8-noded finite element can be effectively used in the earthquake analysis of thick plates. It is also concluded that, in general, the changes in the thickness/span ratio are more effective on the maximum responses considered in this study than the changes in the aspect ratio.

A combined boundary-finite element procedure for dynamic analysis of plates with fluid and foundation interaction considering free surface effect

This study combines a mixed finite element formulation and a boundary element procedure for the free vibration analysis of Mindlin plates resting on Pasternak foundation and interacting with a quiescent fluid with a free surface on the other side. The plate-foundation system is represented by a two field mixed finite element formulation, based on the Hellinger-Reissner variational principle, and the fluid-structure interaction is incorporated into the analysis through a boundary element solution. The proposed formulation provides the added mass matrix in terms of the plate deflection, which is appended into the plate equation of motion. The method is applied to the free vibration problem of circular and elliptical bottom plates of rigid fluid storage tanks supported by elastic foundation. Effects of system parameters, such as thickness to width and fluid depth ratios, foundation parameters and plate ellipticity, are extensively studied.

An Extended Cell-Based Smoothed Three-Node Mindlin Plate Element (XCS-MIN3) for Free Vibration Analysis of Cracked FGM Plates

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) was recently proposed and proven to be robust for static and free vibration analyses of Mindlin plates. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, it is very flexible to apply for arbitrary complicated geometric domains due to using only three-node triangular elements which can be easily generated automatically. However so far, the CS-MIN3 has been only developed for isotropic material and for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-MIN3 by integrating itself with functionally graded material (FGM) and enriched functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed three-node Mindlin plate (XCS-MIN3) for free vibration analysis of cracked FGM plates. Three numerical examples with different conditions are solved and compared with previous published results to illustrate the accuracy and reliability of the XCS-MIN3 for free vibration analysis of cracked FGM plates.

Free damped vibration analysis of Mindlin plates with hybrid material-foundation viscoelasticity

Suppressed vibration of shear deformable plates with hybrid viscoelastic damping of material and foundation is investigated. Viscoelasticity of foundation is established by adopting Kelvin-Voigt model while viscoelasticity of materials is expressed by Boltzmann superposition integral, which uses dynamic mechanical analysis (DMA) results in terms of Prony series. Application of Laplace transformation and Mindlin plate theory with Hamilton principle led to a system of coupled integro-PDE of motions. Weighted residual method, numerical iterative algorithm and Fourier transform are implemented to achieve complex frequencies and transient responses of plates. Assessment of accuracy is carried out in the both elastic and viscoelastic domains. In the earlier, the frequencies are compared with counterparts obtained by p-version Ritz method and in the latter, dynamic characteristics of Mindlin and classic plates are examined that represent acceptable correlation. The influences of hybrid material-foundation damping, edge conditions, foundation characteristics and geometrical parameters on dynamic characteristics of plates are studied via a set of parametric study.

Static and free vibration analyses of stiffened folded plates using a cell-based smoothed discrete shear gap method (CS-FEM-DSG3)

Recently, a cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) was proposed for static and free vibration analyses of Mindlin plates. The CS-FEM-DSG3 uses three-node triangular elements that can be easily generated automatically for arbitrary complicated geometric domains. In this paper, the CS-FEM-DSG3 is further extended for static and free vibration analyses of stiffened folded plates, by combining the original CS-FEM-DSG3 with an Allman's plane stress triangular element and with a stiffener modeled by Timoshenko beam element. The model of a stiffened folded plate is formed by (1) regarding the plate and the stiffener separately in which the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, (3) superimposing the strain energy of plate and stiffener, and (4) using a transformation of degrees of freedom from a local system to a global system. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical methods.

A cell-based smoothed three-node Mindlin plate element (CS-FEM-MIN3) based on the C0-type higher-order shear deformation for geometrically nonlinear analysis of laminated composite plates

A cell-based smoothed three-node Mindlin plate element (CS-FEM-MIN3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and free vibration analyses of Mindlin plates. In this paper, the CS-FEM-MIN3 is extended to geometrically nonlinear analysis of laminated composite plates. The nonlinear formulation is based on the C0-type high-order shear deformation plate theory (C0-HSDT) and the von Kármán strains, which deal with small strains and moderate rotations. In the process of formulating the system stiffness matrix of the CS-FEM-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the MIN3 is used to compute the strains. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The accuracy and reliability of the proposed method is verified by comparing its numerical solutions with those of available other numerical results.

Free vibration analysis of cracked Mindlin plate using an extended cell-based smoothed discrete shear gap method (XCS-DSG3)

A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements was recently proposed for static and free vibration analyses of Mindlin plates. In this paper, the CS-DSG3 is extended for free vibration analysis of cracked Mindlin plates by integrating the original CS-DSG3 with discontinuous and crack-tip singular enriched functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-DSG3). Three numerical examples are considered to illustrate the reliability and accuracy of the XCS-DSG3 for the free vibration analysis of cracked Mindlin plates.

Free vibration analysis of Mindlin plates partially resting on Pasternak foundation

In this paper, the generalized differential quadrature (GDQ) method is used to study free vibration of moderately thick rectangular plate partially resting on Pasternak foundation. The foundation is considered to support the plate either completely or partially. The governing equations which consist of a system of partial differential equations (PDEs) are obtained based on the first-order shear deformation theory. Various combinations of simply supported, clamped and free boundary conditions are considered. Application of the GDQ method to the governing PDEs resulted in a system of algebraic equations. Solution of this system with accordance to the considered boundary conditions leads to an eigenvalue problem to obtain natural frequencies of the plate. Results of this study are validated with available results in the literature which reveal accuracy and fast convergence rate of the method. Effects of different parameters such as foundation stiffness, foundation geometry, boundary conditions and geometrical parameters on the natural frequencies of the plate are presented.

Vibration Analysis of Mindlin Plates with Cracks by MLSElement Method

This paper presents an investigation on the vibration behaviour of rectangular Mindlin plates with one or more preexisting through cracks. The moving least square element (MLS-element) method recently developed by the authors is further extended to carry out this study. The shape functions of an element for the displacement fields can be derived based on the MLS interpolation technique. The MLS-element method possesses the flexibility of finite element method and the advantages of MLS technique which can improve the accuracy of the analysis by increasing the number of MLS points within the elements or the number of terms of the basis functions in the MLS interpolation. Convergence and comparison studied for plates with cracks are carried out to verify the correctness and accuracy of the MLS-element method. Vibration frequencies for Mindlin plates with multiple cracks are obtained. The influence of the crack locations and lengths on the vibration behaviour of plates is presented and discussed.

Ｓｐｌｉｎｅ‐Ｒｉｔｚ法を用いたＭｉｎｄｌｉｎ板の高次振動解析

To avoid mechanical resonance and fatigue damage in engineering designs, it is crucial to predict and understand the responses of structure due to high frequency vibration and noise. This paper presents high frequency vibration analysis of Mindlin plates using spline Ritz method. To demonstrate an accuracy and convergence of the present method, some numerical examples are solved and the results are compared with those obtained by analytical method and other numerical methods. We also demonstrate the present method for high frequency vibration analysis by providing accurate frequencies for Mindlin plates vibrating in the first 5000 modes.

DSC‐Ritz method for the free vibration analysis of Mindlin plates

AbstractThis paper introduces a novel method for the free vibration analysis of Mindlin plates. The proposed method takes the advantage of both the local bases of the discrete singular convolution (DSC) algorithm and the pb‐2 Ritz boundary functions to arrive at a new approach, called DSC‐Ritz method. Two basis functions are constructed by using DSC delta sequence kernels of the positive type. The energy functional of the Mindlin plate is represented by the newly constructed basis functions and is minimized under the Ritz variational principle. Extensive numerical experiments are considered by different combinations of boundary conditions of Mindlin plates of rectangular and triangular shapes. The performance of the proposed method is carefully validated by convergence analysis. The frequency parameters agree very well with those in the literature. Numerical experiments indicate that the proposed DSC‐Ritz method is a very promising new method for vibration analysis of Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.

Vibration Analysis of Discontinuous Mindlin Plates by Differential Quadrature Element Method

A new numerical technique, the differential quadrature element method (DQEM), has been developed for solving the free vibration of the discontinuous Mindlin plate in this paper. By the DQEM, the complex plate domain is decomposed into small simple continuous subdomains (elements) and the differential quadrature method (DQM) is applied to each continuous subdomain to solve the problems. The detailed formulations for the DQEM and the connection conditions between each element are presented. Several numerical examples are analyzed to demonstrate the accuracy and applicability of this new method to the free vibration analysis of the Mindlin plate with various discontinuities which are not solvable directly using the differential quadrature method.

Free vibration analysis of mindlin plates with parabolically varying thickness

The free transverse vibration characteristics of rectangular plates by including the effect of transverse shear deformation and rotary inertia is presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions are presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.

Free vibration analysis of mindlin plates with parabolically varying thickness

The free transverse vibration characteristics of rectangular plates, including the effect of transverse shear deformation and rotary inertia, are presented. A method based on the variational procedure in conjunction with the finite difference technique is used to determine the natural frequencies and mode shapes. The convergence characteristics of the present method are studied by varying the thickness parameter and running a comparison with the classical thin plate solutions. The effects of the thickness parameter and taper thickness ratio on the vibration characteristics are studied. The predictions are presented for the simply supported and clamped plates and they are compared with existing solutions based on the thin plate theory.

Free vibration analysis of Mindlin plates with linearly varying thickness

A method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia. The validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness. Natural frequencies and mode shapes of Mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant.