Free Vibration Analysis Of Plates Research Articles

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

Analysis of free vibration in thin-walled plates using an enhanced polygonal plate element with selective interpolation approach

A highly efficient selective element-interior interpolation approach is proposed that impressively enhances the performance of a spatially isotropic polygonal element (SI-ARS-Poly) and provides outstanding solutions in the displacement and energy norms for the static analysis of thin-walled plate structures (Nguyen and Phan (2023) [22]). In this approach, the element-edge rotations and shear strains are respectively interpolated into interior of polygonal element using linear and quadratic approximations. In order to comprehensively explore the advantages of this element, we extend it to free vibration analysis. Numerical results demonstrate that the proposed approach is stable in the free vibration analysis of plates with smoothed mode shapes. Notably, its solutions for free vibration remain robust in the face of mesh distortion. In comparison with other existing polygonal elements, it is superior with the lowest frequencies closely approximate to the exact/reference ones.

Variable stiffness discrete Ritz method for free vibration analysis of plates in arbitrary geometries

Ritz method runs into difficulties when dealing with complex geometry domain due to the difficulty of constructing a global trial function to approximate structural deformation and the complexity of integration. In this paper, a novel numerical approach, discrete Ritz method (DRM), is proposed for the free vibration analysis of arbitrary-shaped plates with arbitrary cutouts. Legendre polynomials are adopted to generate the admissible function. DRM transforms the complex geometry domain into a system of rectangular domain with variable stiffness. Standard energy functionals and computation procedures are established for arbitrary-shaped plates. Then, in terms of Gauss-Legendre quadrature, integration over a complex geometry domain can be numerically simulated by discretizing the rectangular domain using Gauss points, and putting the thickness and stiffness of the Gauss points within the cutouts to be zero. With sufficient Gauss points, the innovative integration strategy can provide high precision integration results, and hence resolve the eigenvalue problem for free vibration of arbitrary-shaped plates with arbitrary cutouts. Numerical examples considering plates with various shapes, cutouts, aspect ratios, and boundary conditions are demonstrated to show the precision and stability of DRM, and results are compared with other available numerical, analytical, and experimental results existing in the literature.

Free Vibration Analysis of Plates with Arbitrarily Oriented Branched Stiffeners

Free Vibration Analysis of Plates with Arbitrarily Oriented Branched Stiffeners

Free vibration analysis of plates with steps and internal line supports by using a modified matched interface and boundary method

This paper deals with free vibration of plates with steps and internal line supports by using a modified matched interface and boundary (MMIB) method. Different kinds of interfaces caused by steps, rigid and elastic line supports and their combinations are taken into account. Detailed MMIB procedures for dealing with these different interfaces are presented. Various examples are chosen to illustrate the accuracy and convergence of MMIB method. Numerical results show that the proposed MMIB is a highly accurate and convergent approach for solving the title issue. This study will extend the application range of MMIB method.

Experimental and numerical study on free vibration characteristics of laminated composite plate with/without cut-out

The experimental and numerical study related to the free vibration characteristics of laminated composite plates with/without cut-out is presented in this paper. The natural frequencies are obtained from the experimental analysis using FFT analyzer. Similarly, these frequencies are also obtained from the finite element analysis employing nine-nodded isoparametric plate element. The effects of aspect ratio (0.5, 1.0. 1.5 and 2.0), number of layers (8, 12 and 16), different weight fraction of fibre and epoxy (70:30, 50:50 and 40:60) and different orientations of fibres, i.e. (0/90)2s, (45/−45)2s and (30/−30)2s, on the natural frequencies of the laminated composite plates are studied for various boundary conditions. The different parameters considered for the free vibration analysis of plate with cut-out are the size of cut-out, location of cut-out, i.e. concentric/eccentric cut-out, and thickness and the ply orientation of the laminated plate. It is observed that the natural frequencies obtained from both the experimental and finite element methods for these cases are in good agreement. Moreover, the above parameters greatly influence the natural frequencies of the laminated composite plate with/without cut-out. Further, this study will help in providing adequate knowledge and confidence to the designer for the design of laminated composite plates with sufficient accuracy.

A Coupled Improved Element Free Galerkin-Finite Strip (IEFG-FS) Method for Free Vibration Analysis of Plate

In this paper, a coupling of improved element-free Galerkin (IEFG) method and semi-analytical finite strip (FS) method is presented for free vibration analysis of thin plates. This method is very easy to implement and has advantages of both IEFG and FS methods, so that IEFG method is used in sub-domain with complex geometry, and FS method is used for the remaining domains. The use of the FS method considerably reduces the analysis time, and the essential boundary conditions are easily enforced in FS sub-domain. In the IEFG method, the shape function does not have the Kronecker delta function property. Therefore, Lagrange multipliers method is used to satisfy the boundary conditions. Finally, five examples are presented to show the effectiveness of this work.

Smoothing Technique Based Beta FEM (βFEM) for Static and Free Vibration Analyses of Reissner–Mindlin Plates

Recently, the edge-based and node-based smoothed finite element method (ES-FEM and NS-FEM) has been proposed for Reissner–Mindlin plate problems. In this work, in order to utilize the numerical advantages of both ES-FEM and NS-FEM for static and vibration analysis, a hybrid smoothing technique based beta FEM ([Formula: see text]FEM) is presented for Reissner–Mindlin plate problems. A tunable parameter [Formula: see text] is introduced to tune the proportion of smoothing domains calculated by ES-FEM or NS-FEM, which controlled the accuracy of the results. Numerical illustrations in both static and free vibration analysis are conducted. The shear locking free property, converge property and dynamic stability are carefully examined via several well-known benchmark examples. Moreover, an experimental test is carefully designed and conducted for validations, in which the mode values and shape of a rectangular steel plate is tested. Numerical examples demonstrate the advantages of [Formula: see text]FEM, in comparison with the standard FEM, ES-FEM and NS-FEM using the same meshes. The numerical and experimental results are in good agreement with each other and the [Formula: see text]FEM achieves the best accuracy among all the methods for the static or free vibration analysis of plates.

Detailed Numerical Study of Free Vibration Analysis of Plates

Detailed Numerical Study of Free Vibration Analysis of Plates

A new nine-node element for analysing plates with varying thickness using basic displacement functions

The capability of the Finite Element Method in producing accurate and efficient results largely depends on the shape functions adopted to frame the displacement field inside the element. In this paper, a new nine-node Lagrangian element was developed to analyse thin plates with varying cross-sections using the shape functions obtained for non-prismatic straight beams with minimum number of elements. The formulated shape functions, which represent vertical displacements and rotations throughout elements, are rooted from a purely mechanical functions called Basic Displacement Functions (BDFs). These functions are obtained by implementing the force method in Euler–Bernoulli beam theory, which ensures that equilibrium equation is satisfied in all interior points of elements. To verify the competency of the proposed element, solutions for the static analysis of isotropic rectangular plates under various loading conditions, together with free vibration analysis of plates with linear thickness variation were obtained and compared with the previous literature. Results showed that the proposed nine-node Lagrangian element was computationally more cost-effective compared to other competing methods when small number of elements is employed.

Static and Dynamic Analyses of Mindlin Plates Resting on Viscoelastic Foundation by Using Moving Element Method

Presented herein is a novel computational approach using the moving element method (MEM) for simulating the dynamic response of Mindlin plate resting on a viscoelastic foundation and subjected to moving loads. The governing equations and the element mass, damping and stiffness matrices are formulated in a convected coordinate system in which the origin is attached to the point of the moving applied load. Thus, the method simply treats moving loads as ‘stationary’ at the nodes of the plate to avoid updating the locations of moving loads due to the change of the contact points on the plate. To verify the accuracy of the proposed computational approach, static and free vibration analyses of plates are investigated first. Next, the dynamic response of plate resting on a viscoelastic foundation subjected to a moving load is examined. A parametric study is performed to determine the effects of the load’s velocity, foundation damping and foundation stiffness on the dynamic response of a plate. Finally, the comparisons of the dynamic response of plates resting on viscoelastic foundation and subjected to moving vehicles with three models of load (single-wheel, single-axle and tandem-axle) are discussed.

Free vibration analysis of plates taking into account rotary inertia and shear deformation via three alternative theories: a Lévy-type solution

This paper deals with the exact calculation of natural frequencies of a plate with two opposite edges simply supported considering three versions of the Uflyand–Mindlin plate theory: the original Uflyand–Mindlin plate theory, the truncated version of this model as suggested by Elishakoff, and the recently proposed Uflyand–Mindlin plate theory based on slope inertia. The comparison between the frequencies using the different models and with those found in the literature using numerical methods shows the efficiency of the models and the methods presented hereinafter.

Multi-scale bending, buckling and vibration analyses of carbon fiber/carbon nanotube-reinforced polymer nanocomposite plates with various shapes

Using a finite element-based multi-scale modeling approach, the bending, buckling and free vibration of hybrid polymer matrix composites reinforced by carbon fibers and carbon nanotubes (CF/CNT-RP) are analyzed herein. Thick composite plates with rectangular, circular, annular and elliptical shapes are considered. First, the equivalent material properties of CF/CNT-RP are calculated for different volume fractions of CF and CNT. To accomplish this aim, a two-step procedure is presented through which the coupled effects of nano- and micro-scale are taken into account. In the first step, modeling of dispersion of CNTs into the polymer matrix is done with considering interphase formed by their chemical interaction with the matrix, and the equivalent properties of resulting composite material are determined accordingly. CFs are then dispersed into CNT-RP which is considered a homogenous material in this step. Both distributions of CNTs and CFs are assumed to be random. After computing the equivalent properties of CF/CNT-RP for different volume fractions of its constituents, the bending, buckling and free vibration analyses of plates with different shapes are performed. It is shown that the reinforcement of the polymer matrix with both CF and CNT significantly affects the bending, buckling and free vibration characteristics of plates.

A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation

This paper presents a free vibration analysis of plates made of functionally graded materials and resting on twolayer elastic foundations by proposing a non-polynomial four variable refined plate theory. Undetermined integral terms are introduced in the proposed displacement field and unlike the conventional higher shear deformation theory (HSDT), the present one contains only four unknowns. Equations of motion are derived via the Hamilton\'s principles and solved using Navier\'s procedure. Accuracy of the present theory is demonstrated by comparing the results of numerical examples with the ones available in literature.

Eigenfrequencies of simply supported taper plates with cut-outs

Free vibration analysis of plates is necessary for the field of structural engineering because of its wide applications in practical life. Free vibration of plates is largely dependent on its thickness, aspect ratios, and boundary conditions. Here we investigate the natural frequencies of simply supported tapered isotropic rectangular plates with internal cutouts using a nine node isoparametric element. The effect of rotary inertia on Eigenfrequencies was demonstrated by calculating with- and without rotary inertia. We found that rotary inertia has a significant effect on thick plates, while rotary inertia term can be ignored in thin plates. Based on comparison with literature data, we propose that the present formulation is capable of yielding highly accurate results. Internal cutouts at various positions in tapered rectangular simply supported plates were also studied. Novel data are also reported for skew taper plates.

Three-dimensional stress and free vibration analyses of functionally graded plates with circular holes by the use of the graded finite element method

Static and free vibration analyses of plates with circular holes are performed based on the three-dimensional theory of elasticity. The plates are made of a functionally graded material (FGM), and the volume fractions of the constituent materials vary continuously across the plate. The effective properties of the FGM plate are estimated by using the Mori–Tanaka homogenization method. A graded finite element method based on the Rayleigh–Ritz energy formulation is used to solve the problem. Effects of different volume fractions of the materials and hole sizes on the behavior of FGM plates under uniaxial tension are investigated. Natural frequencies of a fully clamped FGM plate with a circular cutout are derived. The results obtained are compared with available experimental data.

Free Vibrations of Plates and Shells with an Isogeometric RM Shell Element

Free vibration analysis of plates and shells is carried out by using isogeometric approach. For this purpose, an isogeometric shell element based on Reissner-Mindlin (RM) shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and it is also used to derive all terms required in the isogeometric element formulation. New anchor positions are proposed to calculate the shell normal vector. Gauss integration rule is used for the formation of stiffness and mass matrices. The proposed shell element is then used to examine vibrational behaviours of plate and shell structures. From numerical results, it is found to be that reliable natural frequencies and associated mode shapes can be predicted by the present isogeometric RM shell element.

Convergence analysis of the exponential matrix method for the solution of 3D equilibrium equations for free vibration analysis of plates and shells

Abstract The three-dimensional equilibrium dynamic equations written in general curvilinear orthogonal coordinates allow the free vibration analysis of one-layered and multilayered plates and shells. The system of second order differential equations is transformed into a system of first order differential equations. Such a system is exactly solved using the exponential matrix method which is calculated by means of an expansion with a very fast convergence ratio. In the case of plate geometries, the differential equations have constant coefficients. The differential equations have variable coefficients in the case of shell geometries because of the curvature terms which depend on the thickness coordinate z. In shell cases, several mathematical layers must be introduced to approximate the curvature terms and to obtain differential equations with constant coefficients. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the solution of shell equations. Both N and M values are analyzed for different geometries, thickness ratios, materials, lamination sequences, imposed half-wave numbers, frequency orders and vibration modes.

Formulating the Stochastic Finite Elements for Free Vibration Analysis of Plates with Variable Elastic Modulus

Authors : Mojtaba Aghamiri Esfahani, Mohammad Karkon, Seyed Majid Hosseini Nezhad, Reza Hosseini-Ara Abstract : In this study, the effect of uncertainty in elastic modulus of a plate on free vibration response is investigated. For this purpose, the elastic modulus of the plate is modeled as stochastic variable with normal distribution. Moreover, the distance autocorrelation function is used for stochastic field. Then, by applying the finite element method and Monte Carlo simulation, stochastic finite element relations are extracted. Finally, with a numerical test, the effect of uncertainty in the elastic modulus on free vibration response of a plate is studied. The results show that the effect of uncertainty in elastic modulus of the plate cannot play an important role on the free vibration response.

An efficient plate element for the vibration of composite plates

An element is presented for the free vibration analysis of composite plates. The element is capable of representing high orders of displacement using very few degrees of freedom. Hierarchy in the orders of deflection, rotation, moment and shear has been preserved in the element through scrupulous choice of field variables. The effect of rotary inertia has been considered through an innovative scheme for inclusion into mass lumping procedures. Performance of the element has been tested in a wide range of problems which indicate that free vibration analysis of plates with the help of the present element can reach a high level of accuracy at reduced computation cost, compared to standard elements in literature.