Three-dimensional Free Vibration Analysis Research Articles

A C1 continuous hexahedral element for nonlinear vibration analysis of nano-plates with circular cutout based on 3D strain gradient theory

Abstract The main objective of the present study is to develop a hexahedral C 1 continuous element for the three-dimensional nonlinear free vibration analysis of rectangular nano-plates with circular cutout based on the strain gradient elasticity theory. In the proposed element, in addition to the values of displacement components, the corresponding first, mixed second and mixed third order derivatives are taken into account as nodal values. Since arbitrary-shaped hexahedral elements are considered, spatial derivatives are presented using isoparametric formulation. To derive the nonlinear governing equations, the matrix form of kinetic and strain energies are written based on the three-dimensional strain gradient elasticity theory which can be reduced to the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST). The accuracy and convergence of the numerical results are first investigated. In the parametric study, the nonlinear free vibration response of nano-plate is studied for different length scale parameters, geometrical factors and boundary conditions.

A study on the vibrational properties of weight-efficient plates made of material with functionally graded porosity

Functionally graded (FG) porous structures are a novel class of weight-efficient engineering materials and characterized by introducing graded non-uniform porosities. Due to the wide range of applications of porous materials in lightweight structures, biomedical systems and separation processes, in this paper, three-dimensional (3D) free vibration analyses of light-weight plates made of uniform and non-uniform graded porous materials resting on two-parameter elastic foundations are investigated. The mathematical formulation is based on linear and small strain elasticity assumptions. The plate with FG porosity is simply supported at the edges and is assumed to have three variations of porosities through the thickness. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. Validation and convergence studies are also done to demonstrate the accuracy of the results. The natural frequencies of the simply-supported plate made of material with functionally graded porosity are calculated and presented in both tabular and graphical forms. The effects of different porosity distributions, porosity parameter, elastic coefficients of foundation and geometrical parameters on the results are investigated. The results of the present analysis indicate the advantages of using a plate with graded porosity over a traditional plate with uniform porosity.

Three-dimensional free vibration analysis of variable stiffness laminated composite rectangular plates

The free vibration of variable stiffness laminated composite rectangular plates is studied for the first time on the basis of three-dimensional elasticity theory combined with the p-version of the finite element method. Each layer is modeled as one brick p-element with curvilinear fibers. The element stiffness and mass matrices are derived based on the principle of virtual displacements. Inter-element compatibility is achieved by matching the generalized displacements at vertices, edges, and faces shared by elements. Results are obtained for frequencies, modal displacements, and modal stresses of symmetric and anti-symmetric laminates with various combinations of free, simply supported, and clamped boundary conditions. The method is validated through convergence study and comparison with published three-dimensional frequencies for constant stiffness laminated composite plates. The frequencies predicted by the equivalent single-layer classical plate theory and first-order shear deformation theory show deviation from three-dimensional solutions. Phenomena such as the discontinuity of inter-laminar modal flexural stresses, change of sign of modal transverse shear stresses through the thickness, and modal cross-sectional warping are observed and explained. New three-dimensional frequencies for variable stiffness laminated composite plates are provided, which may serve as a benchmark for future studies.

Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell

The buckling and free vibration of initially stressed functionally graded cylindrical shell reinforced with non-uniformly distributed graphene platelets (GPLs) are investigated using the state-space formulation based on three-dimensional elasticity theory. The shell is under an axial initial stress and composed of multilayers with GPLs uniformly dispersed in each individual layer but its weight fraction changing layer-by-layer along the thickness direction. The modified Halpin-Tsai model and rule of mixtures are employed to evaluate the effective elastic properties of the GPL-reinforced shell. Analytical buckling and frequency solutions are obtained for simply supported shells. Numerical results are presented for functionally graded GPL-reinforced cylindrical shells with five GPL dispersion patterns (GPL-UD, GPL-V, GPL-A, GPL-X, and GPL-O). The effects of GPL weight fraction, dispersion pattern, geometry, and size as well as the influence of initial stress on the buckling and free vibration characteristics of the shell are discussed in detail. It is found that the addition of a small amount of GPLs significantly increases the critical buckling stress and natural frequencies. The GPL-X pattern outperforms other patterns for thin composite shells while the uniform pattern GPL-UD works better for thick composite shells.

Three-Dimensional Free Vibration Analysis and Critical Speed of Pressurized Rotating Functionally Graded Cylindrical Shells

This paper presents an approximate solution for the free vibration analysis of rotating functionally graded cylindrical shells based on the three-dimensional theory, using layerwise theory. Equations of motion are derived by applying Hamilton’s principle. In order to accurately account for the thickness effects, the layerwise theory is used to discretize the equations of motion and the related boundary conditions through the thickness of the shells. The edges of the shell are restrained by simply supported boundary conditions. Material properties are assumed graded in the thickness direction according to a simple power-law distribution in terms of the volume fraction of the constituents. The results obtained include the relationship between frequency characteristics, different material properties, power-law index, rotating velocities and amplitude of internal pressure. In order to validate the present analysis, the comparison is made with the other published works for a cylindrical shell obtained from CLT to TSDT. In addition, another comparison is made with non-rotating isotropic cylindrical shell based on 3D theory. These comparisons confirm reliability of the present work as a measure to approximate solutions to the problem of rotating functionally graded cylindrical shells.

Three-dimensional free vibration analysis of annular sector plates with arbitrary boundary conditions

The objective of this investigation is to apply the spectro-geometric method to study the vibration behavior of the annular sector plates with arbitrary boundary restraints on the basis of the three-dimensional theory of elasticity. All the classical homogeneous boundary conditions can be universally considered as the special cases when the stiffness for each of restraining springs is set equal to either zero or infinity in this study. Under the current solution framework, each displacement of the annular sector plate is, regardless of boundary conditions, invariantly expanded as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series and all the series expanded coefficients are determined with the Rayleigh–Ritz procedure. The present solution method can be generally applied to a broad range of annular sector plates, regardless of their thicknesses and sector angles. The current results are compared with those previously published in literature and the finite element analysis data to validate the current formulation. More results for plates with different boundary restraints and geometrical parameters are presented, which may serve as benchmark solutions for others.

Three-dimensional free vibration analysis of levy-type laminated plates using multi-term extended Kantorovich method

Abstract A closed form three-dimensional (3D) free vibration solution for a rectangular laminated plate with Levy-type support conditions is presented first time using recently developed extended Kantorovich method. Hamiltons principle is extended to develop a mixed formulation in terms of displacements and stresses as primary variables. The present solution will serve as a benchmark for assessing two-dimensional (2D) theories and 3D numerical solutions. Natural frequencies are tabulated for composite and sandwich laminates (first time) for different boundary conditions. It is found that single term solution is sufficient enough for obtaining accurate natural frequencies. But stresses near the clamped edge are accurately predicted by multi-term solution ( n = 2). The effect of the span to thickness ratio and in-plane modulus ratios on the natural frequency is also studied.

Free vibration analysis of embedded single-layered nanoplates and graphene sheets by using the multiple time scale method

An asymptotic approach using the Eringen nonlocal elasticity theory and multiple time scale method is developed for the three-dimensional (3D) free vibration analysis of simply-supported, single-layered nanoplates and graphene sheets (GSs) embedded in an elastic medium. In the formulation, the small length scale effect is first introduced to the nonlocal constitutive equations by using a nonlocal parameter, then the mathematical processes of nondimensionalization, asymptotic expansion and successive integration are performed, and finally recurrent sets of motion equations for various order problems are obtained. The interactions between the nanoplates (or GSs) and their surrounding medium are modeled as a two-parameter Pasternak foundation. Nonlocal classical plate theory (CPT) is derived as a first-order approximation of the 3D nonlocal elasticity theory, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal CPT, although with different nonhomogeneous terms. Some 3D nonlocal elasticity solutions of the natural frequency parameters of nanoplates (or GSs) with and without being embedded in the elastic medium and their corresponding through-thickness distributions of modal field variables are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory.

Three-Dimensional Free Vibration Analysis of Thick Laminated Composite Circular Plates with Simply-Supported Boundary Conditions

This paper presents a numerical analysis of free vibration of thick laminated circular plates, with simply supported boundary conditions at the edges of plates. A finite element analysis is made for obtaining the first five natural frequencies using three-dimensional ‘SOLID185’ of ANSYS. The effects of different fibre orientation angle, thickness to radii ratios, on the free vibration responses are discussed in detail. The accuracy and numerical reliability have been verified by appropriate convergence studies and comparisons with the existing results from literature. These results should be a valuable alternative for validating new computational techniques in future, due to the accuracy, simplicity and versatility of the present analysis.

Three-dimensional free vibration analysis of thick laminated circular plates

In this communication, a numerical analysis regarding free vibration of thick laminated circular plates, having free, clamped as well as simply-supported boundary conditions at outer edges of plates is presented. The employment of finite element is made in this communication. The finite element methodology operates on the basis of three-dimensional theory of elasticity and was employed to assess the natural frequencies for laminated circular plates of various thickness-to-outer radius ratios. The first five natural modes of flexural vibrations for different boundary conditions are presented in pictorial forms. Verification of the accuracy of the results was made using the necessary convergence analysis and checked using literature results.Keywords: Laminated Composite Circular Plate; Natural Frequency; Mode Shape; Finite Element Method

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.

Three-dimensional vibration analysis of layered and functionally graded plates through sampling surfaces formulation

The paper deals with a recently developed method of sampling surfaces (SaS) and its implementation for the three-dimensional (3D) free vibration analysis of layered plates with embedded functionally graded (FG) layers. The SaS method is based on choosing inside the nth layer In not equally spaced SaS parallel to the middle surface of the plate in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree In-1 in the assumed distributions of the displacements and mechanical properties through the thickness of the layer leads to the robust FG plate formulation. The SaS are located inside each layer at Chebyshev polynomial nodes that makes it possible to minimize uniformly the error due to the Lagrange interpolation. As a result, the SaS formulation can be applied efficiently to obtaining the analytical solutions for layered and FG plates, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity.

Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.

Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method

The three-dimensional free vibration analysis of a multi-directional functionally graded piezoelectric (FGP) annular plate resting on two parameter (Pasternak) elastic foundations is investigated under different boundary conditions. The material properties are assumed to vary continuously along the radial and thickness directions and have exponent-law distribution. A semi-analytical approach named the state space based differential quadrature method (SSDQM) is used to provide an analytical solution along the thickness using the state space method (SSM) and an approximate solution along the radial direction using the one-dimensional differential quadrature method (DQM). The influence of the Winkler and shear stiffness of the foundation, the material property graded variations, and the circumferential wave number on the non-dimensional natural frequency of multi-directional FGP annular plates is studied.

Three-dimensional free vibration analysis of sandwich FGM cylinders with combinations of simply-supported and clamped edges and using the multiple time scale and meshless methods

Three-dimensional free vibration analysis of sandwich FGM cylinders with combinations of simply-supported and clamped edges and using the multiple time scale and meshless methods

Three-dimensional free and transient vibration analysis of composite laminated and sandwich rectangular parallelepipeds: Beams, plates and solids

This paper presents an efficient method for predicting the free and transient vibrations of multilayered composite structures with parallelepiped shapes including beams, plates and solids. The exact three-dimensional elasticity theory combined with a multilevel partitioning hierarchy, viz., multilayered parallelepiped, individual layer and layer segment, is employed in the analysis. The continuity constraints on common interfaces of adjacent layer segments are imposed by a modified variational principle, and the displacement components of each layer segment are assumed in the form of orthogonal polynomials and/or trigonometric functions. Numerical studies are given for free vibrations of composite laminated and sandwich beams, plates, and solids. Some in-plane shear vibration modes missed in previous elasticity solutions for multilayered plates are examined. The natural frequencies derived from Reddy’s high-order shear deformation theory and layerwise theory for soft-core sandwich plates show significant deviation from elasticity solutions. Transient displacement and stress responses for angle-ply laminated and sandwich plates under four types of impulsive loads (including rectangular, triangular, half-sine and exponential pulses) are obtained by the Newmark integration procedure. The present solutions may serve as benchmark data for assessing the accuracy of advanced structural theories and new developments in numerical methods.

Mixed Navier-layerwise differential quadrature three-dimensional static and free vibration analysis of functionally graded carbon nanotube reinforced composite laminated plates

The three-dimensional free vibration and static analysis of the laminated plates with functionally graded (FG) carbon nanotube reinforced composite (CNTRC) layers is presented using a semi-analytical approach. The individual layers are assumed to be made from a mixture of aligned and straight single-walled carbon nanotubes (CNTs) with volume fractions graded in the thickness direction, and an isotropic matrix. The effective material properties of the resulting FG-CNTRC layers are estimated through a micromechanical model. The through-the-thickness variations of the displacement components are accurately modeled using a layerwise-differential quadrature method, and their in-plane variations are approximated using the trigonometric series. After demonstrating the convergence and accuracy of the method, the effects of geometrical parameters, type of CNTs distribution and volume fractions, and also lamination scheme on the natural frequencies, displacement and stress components of the FG-CNTRC layered plates are investigated.

Three-dimensional static and free vibration analysis of laminated cylindrical panel with viscoelastic interfaces

In this paper, three-dimensional theory of elasticity is employed to investigate bending and free vibration behavior of simply supported cylindrical panel with special emphasis on incorporating viscoelastic interfacial imperfection. Analysis is carried out by using Fourier series along the axial and circumferential directions and state space technique along the radial direction. Time-dependent behavior is investigated by solving first-order differential equation of sliding displacement at the viscoelastic interfaces. Numerical results depict good agreement between the present numerical results and the available published results. In addition, the effects of solid, elastic and viscous interfaces, time, aspect ratio and mid radius to thickness ratio on the bending and vibration behavior of laminated cylindrical panel are discussed.

Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations

Abstract A unified method based on the three-dimensional theory of elasticity is developed for the free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations. Each shell displacements, regardless of boundary conditions, is expanded as a standard Fourier cosine series supplemented with four auxiliary functions introduced to eliminate any possible discontinuities of the original displacement and its derivatives throughout the entire shell space including the boundaries and then to effectively enhance the convergence of the results. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Since the displacement field is constructed adequately smooth throughout the whole solution domain, an accurate solution can obtained by using Rayleigh–Ritz procedure based on the energy functions of the shell. The current method can be universally apply to a variety of end conditions including all the classical cases and their combinations and arbitrary elastic foundations. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. Effects of the boundary restraining parameters and foundation coefficients on frequency parameters are investigated as well. New results for thick cylindrical shells with various end conditions and resting on elastic foundations are presented, which may serve as benchmark solutions for validating new computational techniques in future.

Three-dimensional free vibration analysis of functionally graded carbon nanotube-reinforced composite plates with various boundary conditions

A unified formulation of finite prism methods (FPMs) based on Reissner's mixed variational theorem is developed for the three-dimensional (3D) free vibration analysis of functionally graded (FG) carbon nanotube-reinforced composite (CNTRC) plates and laminated fiber-reinforced composite (FRC) plates, the edge conditions of which are considered such that one pair of opposite edges is simply supported and the other pair may be combinations of free, clamped or simply supported edges. The single-walled carbon nanotubes (CNTs) and polymer are regarded as the reinforcements and matrices, respectively, to produce the CNTRC plate. Four different distributions of CNTs varying in the thickness direction are considered (i.e. the uniformly distributed, and FG V-, rhombus-, and X-type variations), and the through-thickness distributions of effective material properties of the CNTRC plate are determined using the rule of mixtures. In the formulation, the CNTRC/FRC plate is divided into a number of finite prisms in the domain, in which the trigonometric functions and Lagrange polynomials are used to interpolate the y-direction and plane variations for the primary variables of each individual prism, respectively, and the related orders used for expansion of assorted primary variables in the thickness coordinate can be freely chosen. It is shown that the eight- and nine-node quadratic FPM solutions of frequency parameters of simply supported, CNTRC plates and laminated FRC plates are in excellent agreement with the exact 3D solutions available in the literature, and with the ones obtained using the ANSYS software for those plates with various boundary conditions.