Three-dimensional Free Vibration Research Articles

Three-Dimensional Free Vibration Analysis of Sandwich Plates with VSCL Face Sheets and Porous Functionally Graded Core

Three-Dimensional Free Vibration Analysis of Sandwich Plates with VSCL Face Sheets and Porous Functionally Graded Core

Levy-type analytical series solution for the three-dimensional free vibrations of functionally graded material rectangular plates with piezoelectric layers

Analytical solutions founded on three-dimensional theories play a crucial role in evaluating the credibility and precision of different plate theories and numerical methodologies. While Levy-type analytical solutions are widely recognized, they have been primarily confined to purely elastic plates. This study introduces a Levy-type analytical series solution for three-dimensional vibrations in a sandwich rectangular plate featuring a functionally graded material (FGM) core, along with piezoelectric material (PM) layers on the top and bottom surfaces. The behaviors of the FGM and PM layers were described using three-dimensional elasticity and piezoelasticity theories, respectively. In this study, the displacement functions and electric potential of each layer were expanded by Fourier series and polynomial auxiliary functions. An analytical series solution was then established by satisfying the governing equations of each layer, the mechanical and electric boundary conditions on the six faces of the plate, and the continuity conditions on the interfaces between the PM and FGM layers. To validate the proposed solutions, in-depth convergence studies were conducted for the vibration frequencies of the first six modes of sandwich square plates with various boundary conditions on the other pair of side faces. The well-converged results were then compared with published data based on various plate theories to verify the accuracy of these published data. Finally, accurate nondimensional frequencies were tabulated for the first six modes of sandwich rectangular plates with various aspect ratios, thickness-to-width ratios, PM-to-FGM layer thickness ratios, power law indices for the FGM layer, and six combinations of boundary conditions. These new numerical results when piezoelectric coupling is considered should be very useful to future analytical and numerical studies.

A Size-Dependent Finite Element Method for the 3D Free Vibration Analysis of Functionally Graded Graphene Platelets-Reinforced Composite Cylindrical Microshells Based on the Consistent Couple Stress Theory

Within a framework of the consistent couple stress theory (CCST), a size-dependent finite element method (FEM) is developed. The three-dimensional (3D) free vibration characteristics of simply-supported, functionally graded (FG) graphene platelets (GPLs)-reinforced composite (GPLRC) cylindrical microshells are analyzed. In the formulation, the microshells are artificially divided into numerous finite microlayers. Fourier functions and Hermitian C2 polynomials are used to interpolate the in-surface and out-of-surface variations in the displacement components induced in each microlayer. As a result, the second-order derivative continuity conditions for the displacement components at each nodal surface are satisfied. Five distribution patterns of GPLs varying in the thickness direction are considered, including uniform distribution (UD) and FG A-type, O-type, V-type, and X-type distributions. The accuracy and convergence of the CCST-based FEM are validated by comparing the solutions it produces with the exact and approximate 3D solutions for FG cylindrical macroshells reported in the literature, for which the material length scale parameter is set at zero. Numerical results show that by increasing the weight fraction of GPLs by 1%, the natural frequency of FG-GPLRC cylindrical microshells can be increased to more than twice that of the homogeneous cylindrical microshells. In addition, the effects of the material length scale parameter, the GPL distribution patterns, and the length-to-thickness ratio of GPLs on natural frequencies of the FG-GPLRC cylindrical microshells are significant.

Three-dimensional bending and free vibration analyses of laminated cylindrical panel with/without elastic foundation using two-dimensional discrete method

In this paper, a novel isogeometric scaled boundary finite element method (IGSBFEM) is developed to provide a comprehensive investigation into bending and free vibration of laminated cylindrical panels resting on Pasternak foundation. The significant novelty of this method lies in that the proposed formulation is derived based on the three-dimensional (3D) theory of elasticity, while only two-dimensional (2D) reference surface associated with the curvilinear coordinate system needs to be discretized by using isogeometric elements constructed with the non-uniform rational B-splines (NURBS) basis functions, which allows to reduce the computation cost and maintain the exact geometry of problem with desired precision flexibly. The SBFEM governing equation of each single lamina is reduced to a matrix form of first-order ordinary differential equations by introducing a vector of dual variables made of nodal displacements and forces, and the solution can be expressed analytically in terms of the radial coordinate. Therefore, shear locking phenomenon can be overcome naturally without any shear correction factors or reduced integration. The global solving equations of laminated cylindrical panels resting on Pasternak foundation are assembled with a layer-wise method. In addition, the high-order continuous features of isogeometric elements are insensitive to distorted mesh. Therefore, it is compatible with the research of complex geometrical structures with holes. The superior performances of the current approach are testified in the analyses of cylindrical panels with or without elastic foundation, solutions obtained by many other researchers are used for comparison purpose, and effects of several parameters including length and thickness, fiber orientation, stacking sequence and foundation stiffnesses coefficient on the bending or free vibration characteristics are also discussed.

A Hermite-Family C1 Finite Layer Method for the Three-Dimensional Free Vibration Analysis of Exponentially Graded Piezoelectric Microplates Based on the Consistent Couple Stress Theory

Based on the consistent couple stress theory (CCST), we develop a Hermite-family [Formula: see text] finite layer method (FLM) for the three-dimensional (3D) free vibration analysis of a simply-supported, exponentially graded (EG) piezoelectric microplate under open- and closed-circuit surface conditions. In the formulation of the FLM, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric microplates is reduced to the Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric macroscale plates and functionally graded (FG) elastic microplates by assigning a value of zero to the material length scale parameter and by ignoring the piezoelectric and flexoelectric effects in the formulation, respectively. The accuracy and convergence rate of the FLM are assessed by comparing their solutions with the benchmark solutions of both the EG piezoelectric macroplates and the power-law-type FG elastic microplates that are available in the relevant literature. We examine and discuss some key effects on the free vibration characteristics of an EG piezoelectric microplate, including the impact of the material length scale parameter, the material-property gradient index, the length-to-thickness ratio, the piezoelectric effect, and the flexoelectric effect.

Three-dimensional free vibration analysis of functionally graded graphene platelets-reinforced composite toroidal shells

In this work, based on the weak form of three-dimensional (3D) theory of elasticity, we develop a mixed isoparametric finite element method (IFEM) for the analysis of the free vibration behavior of freely-supported, functionally graded (FG) graphene platelets-reinforced composite (GPLRC) toroidal shells. The functions of five distribution patterns of graphene platelets (GPLs) through the thickness direction are considered, for which the effective Young’s modulus is estimated using the Halpin-Tsai model, and the effective Poisson’s ratio and the effective mass density are estimated using the rule of mixtures. By incorporating Hamilton’s principle and Reissner’s strain energy functional, we derive the system equations of the mixed IFEM. Based on the mixed IFEM, the 3D solutions for the natural frequency parameters of the FG-GPLRC toroidal shells converge rapidly, and the convergent solutions are in excellent agreement with the accurate solutions for both homogeneous isotropic toroidal shells and FG transversely isotropic toroidal shells which are reported in the literature. Numerical results show that the impacts of the GPL distribution, the weight fraction of the GPL, and the radius-to-thickness ratio on the lowest natural frequencies of the FG-GPLRC toroidal shells is significant.

Free vibrations of spatial frame structures: Analytical modelling and solution

This study analytically handles the three-dimensional free vibrations of spatial frames using the initial values method, considering the axial and shear deformations alongside rotary inertias, namely, torsional, in-plane bending and out-of-plane bending. To handle the spatial geometry, the direction cosine matrices are used. Validation is performed with three cases in total, one case available in the literature alongside two numerical examples that are solved analytically and compared to the finite element models. Excellent agreements are found between the analytical results and the results in the literature, as well as those obtained from the finite element models.

A direct approach for three-dimensional elasto-static and elasto-dynamic solutions in curvilinear cylindrical coordinates with application to classical cylinder problems

This paper deals with introducing a unique representation of the three-dimensional Navier's equations of motion in cylindrical coordinate system in an exact simplified form without any approximation, aiming at facilitating solution procedure for different 3-D elasto-static and elasto-dynamic problems in the future. A novel form of the 3-D elasticity equations of motion including the body forces in cylindrical coordinate system is derived in an uncoupled form in terms of the longitudinal (axial) displacement component and the ‘r-θ’ in-plane anti-symmetric rotation function instead of introducing any additional auxiliary unknown potential function. The other displacement components (i.e., circumferential and radial displacement components) are shown to be obtained from two independent equations in terms of the determined axial displacement and the aforementioned rotation component. The correctness, validity and easy implementation of the introduced elasticity approach for obtaining exact elasticity solutions for various 3-D elasto-static and elasto-dynamic problems are demonstrated through solving a number of known elasticity problems. Three-dimensional static and free vibrations of finite-length solid cylinders as well as thick-walled hollow cylindrical shells are analytically solved. Numerical comparative results and discussion are conducted. Excellent agreement between the obtained results and those reported in the literature is observed in all cases, confirming the validity of the proposed new approach.

Three-Dimensional Free Vibration Analyses of Preloaded Cracked Plates of Functionally Graded Materials via the MLS-Ritz Method

In this study, the moving least squares (MLS)-Ritz method, which involves combining the Ritz method with admissible functions established using the MLS approach, was used to predict the vibration frequencies of cracked functionally graded material (FGM) plates under static loading on the basis of the three-dimensional elasticity theory. Sets of crack functions are proposed to enrich a set of polynomial functions for constructing admissible functions that represent displacement and slope discontinuities across a crack and appropriate stress singularity behaviors near a crack front. These crack functions enhance the Ritz method in terms of its ability to identify a crack in a plate. Convergence studies of frequencies and comparisons with published results were conducted to demonstrate the correctness and accuracy of the proposed solutions. The proposed approach was also employed for accurately determining the frequencies of cantilevered and simply supported side-cracked rectangular FGM plates and cantilevered internally cracked skewed rhombic FGM plates under uniaxial normal traction. Moreover, the effects of the volume fractions of the FGM constituents, crack configurations, and traction magnitudes on the vibration frequencies of cracked FGM plates were investigated.

Three-dimensional free bending vibrations of variable radius functionally graded circular column immersed in infinite fluid

The three-dimensional free bending vibrations of variable radius functionally graded circular column fully or partially immersed in infinite fluid are investigated for the first time using a coupled hierarchical finite-infinite element method based on curved elements. The fluid is incompressible, irrotational, and inviscid. The material properties vary continuously along the radial direction according to a simple power law function. The solid and fluid behaviors are described by the three-dimensional elasticity theory and Laplace’s equation, respectively. Coupling occurs on the interface via the boundary restraints imposed on it. Curved edges are described exactly by the blending functions. Examples of clamped-free and clamped-guided functionally graded circular columns with quadratic and cubic radius variations are used to illustrate the method. Convergence test and comparison study are conducted. New results for the frequencies of the lowest two bending modes are provided, which may serve as benchmark solutions. Furthermore, a parametric study is conducted to show the effects of various geometrical and mechanical parameters on the frequencies, modal deflections, and modal hydrodynamic pressures.

A new meshfree approach for three-dimensional free vibration analysis of thick laminated doubly-shell of revolution

In this paper, a new meshfree approach is proposed for the three-dimensional free vibration analysis of laminated composite elliptical and paraboloidal shells of revolution. The three-dimensional theory of elasticity has been applied to formulations for free vibration analysis of laminated composite elliptical and paraboloidal shells of revolution. The field function is approximated by a new Tchebychev-point interpolation method (TPIM) shape function that uses the Tchebychev polynomial as the basis for the point interpolation method (PIM) shape function. The governing equation and boundary conditions for laminated elliptical and paraboloidal shells are obtained by combining the governing equation and boundary conditions for individual layers using a continuous condition. The artificial spring technique is introduced to generalize the boundary and continuous conditions, and the type of boundary conditions is selected according to the stiffness of the spring. The accuracy and reliability of the proposed method is verified through comparison with the results of literature and ABAQUS software. Free vibration characteristics such as natural frequency and mode shape of laminated elliptical and paraboloidal shells of revolution under various boundary conditions are presented through numerical examples.

Three-dimensional free vibration analysis of thick laminated combination shell using a meshfree approach

In this paper, a novel meshfree approach is proposed for the three-dimensional free vibration analysis of laminated composite combination shells. The three-dimensional theory of elasticity is applied to the formulation for free vibration analysis of thick laminated elliptical–cylindrical and conical–cylindrical combination shells, and the field functions are approximated by a novel meshfree Tchebychev-point interpolation method shape function using Tchebychev polynomials as the basis. The governing equation and boundary conditions for the individual layers of the substructures of the laminated combination shell are derived, and the equations of the whole system are obtained by combining them using a continuous condition. The boundary and continuous conditions are generalized by the introduction of an artificial spring technique, and the type of boundary conditions is selected according to the spring stiffness. The accuracy and reliability of the proposed method are verified by comparing the results of the proposed method with those of literature and finite element program ABAQUS. The free vibration characteristics including natural frequencies and mode shapes of thick combination shells with various geometries and boundary conditions are presented through numerical examples.

A novel meshfree method for three-dimensional natural frequency analysis of thick laminated conical, cylindrical shells and annular plates

In this paper, a novel meshfree approach for three-dimensional free vibration analysis of thick laminated composite conical, cylindrical shells and annular plates is presented. The theoretical model for free vibration analysis of thick shells and plates is formulated by applying the three-dimensional theory of elasticity, and all displacement components are approximated by a novel meshfree Tchebychev-point interpolation method (TPIM) shape function using Tchebychev polynomials as a basis. After deriving the governing equations and boundary conditions for individual layers of the laminated shell, the governing equations and boundary conditions of the entire system are derived by combining them according to the order of layers. The boundary and combination conditions are generalized by introducing the artificial spring technique, and the type of conditions is selected by the spring stiffness values. The accuracy and reliability of the proposed method are verified by comparison with the results of literatures and finite element software ABAQUS. The free vibration characteristics such as the natural frequency and mode shape of laminated conical, cylindrical shells and annular plates under different boundary conditions are presented through some numerical examples.

Three-dimensional free flexural vibrations of fluid-filled functionally graded circular cylindrical shell with curvilinear radius variation

The three-dimensional free flexural vibrations of functionally graded circular cylindrical shell with curvilinear radius variation fully or partially filled with incompressible and inviscid fluid are investigated for the first time using a coupled axisymmetric solid-fluid hierarchical finite element method based on curved elements. Curved edges are described exactly by means of blending functions. The method is used in conjunction with the three-dimensional elasticity theory and Laplace’s equation to obtain the governing equations. The material properties follow the power law distribution in the thickness direction. The effectiveness of the method is illustrated using examples of clamped-free and clamped-guided functionally graded circular cylindrical shells with quadratic and cubic radius variations. The method is validated through convergence test and comparison study. New results for the cycle frequencies of the lowest two flexural modes are presented, which may serve as benchmarks for future studies. Moreover, the effects of several geometrical and mechanical parameters on the cycle frequencies, modal deflections, and modal hydrodynamic pressures are explored.

Three-dimensional free vibration analysis of rotating sandwich functionally graded truncated conical shells under various boundary conditions

In this work, the authors develop a Reissner’s mixed variational theorem (RMVT)-based weak formulation for the three-dimensional (3 D) free vibration analysis of sandwich functionally graded (FG) truncated conical shells under various boundary conditions rotating with a constant angular velocity with respect to the central axis of the shells. The material properties of the FG layers constituting the truncated conical shell are assumed to obey a power-law distribution of the volume fractions of the constituents through their thickness direction, for which the effective material properties are estimated using the rule of mixtures. Based on the weak formulation, a semi-analytical finite annular prism (FAP) method is developed for the current issue, where the effects of the initial hoop stress due to rotation and the centrifugal and Coriolis accelerations are considered. Implementation of the current FAP method indicates that its solutions converge rapidly, and the convergent solutions closely agree with accurate solutions available in the literature. The obtained 3 D frequency parameter solutions for rotating and non-rotating sandwich FG truncated conical shells can provide a standard by which to assess those obtained using assorted two-dimensional classical, advanced, and refined shell theories.

Analysis of three–dimensional free vibrations of isotropic visco-thermoelastic solid cylinder with two relaxation time parameters

We explore the analysis of exact three-dimensional free vibrations of visco-thermoelastic solid cylinder, which has been considered at a uniform temperature and undeformed initially in the framework of generalized thermoelasticity. The formulation has been implemented with two relaxation time parameters, i.e., mechanical and thermal relaxation parameters. The series solution has been implemented to solve the differential equations along with the evaluation of stresses, displacements, and temperature by applying stress-free, rigidly fixed, thermally insulated and isothermal boundary conditions. The relations of frequency equations for the existence of different modes of visco-thermoelastic solid cylinder have been developed in compact form using assumed boundary conditions. Thermoelastic damping and frequency shift are presented graphically from the data generated by computations and simulations of numerical results. Also, the natural frequencies are represented in the form of tables which show that with an increase in the ratio of length to the radius and axial wave-number, the variation of vibrations keeps increasing.

Effect of three-phase-lag model on the analysis of three-dimensional free vibrations of viscothermoelastic solid cylinder

Herein we explore the effects of three-phase-lag (TPL) model of generalized thermoelasticity on the exact analysis of three-dimensional free vibrations of viscothermoelastic solid cylinder, which is assumed undeformed initially and at uniform temperature. The formulation has been put into practice with three different phase-lag parameters i.e., Phase-lag of the heat flux, Phase-lag of the temperature gradient, and Phase-lag of the thermal displacement gradient. The solution of matrix Frobenius method of continued series solution has been implemented to solve the differential equations for the evaluation of field functions i.e. stresses, displacements, and temperature by applying stress-free/rigidly-fixed thermal boundary conditions (i.e. thermally insulated and isothermal). The obtained frequency equations for the existence of different modes of TPL viscothermoelastic solid cylinder have been developed using considered stress free/rigidly fixed boundary conditions. The free vibration field functions i.e. frequency shift and thermoelastic damping have been presented graphically from the data generated by simulations and computations of numerical results. Also, the natural frequencies are presented in the form of tables which shows that with an increase in the axial wave-number and length to radius ratio, the behavior of vibrations goes on increasing. The behavior of vibrations of field functions in case of TPL model has larger in contrast to other models of generalized thermoelasticity.

Evaluation of exact electro-elastic static and free vibration solutions of multilayered plates for benchmarking: Piezoelectric composite laminates and soft core sandwich plates

This work provides a study on three-dimensional exact electro-elastic static and free vibration solutions of multilayered plates, focused on a comprehensive evaluation of well-known benchmarks for piezoelectric and/or composite laminates as well as soft core sandwich plates, adding much to thus far available in the literature. The exact solution method for simply supported multilayered plates is fully described in line with earlier leading works, compiled in a single study in a consistent form throughout. The layers involved can be either piezoelectric layers poled through-thickness (i.e. extension mode) or purely elastic layers, including composite layers. For each layer, the form of the through-thickness exact solution depends strongly on its material properties, thus each case arising from an orthotropic, transversely isotropic or simply isotropic layer is considered specifically. Within the free vibration solution, in agreement with an in-plane mode (nx, ny), the so-called special modes for either nx=0 or ny=0, though often overlooked, are purposely addressed, along with thickness modes for each pair (nx, ny). The benchmarks cover purely elastic solutions, as in composite laminates and soft core sandwich plates, as well as electro-elastic solutions, namely piezoelectric composite laminates, involving not only but especially thick plates. For each multilayered plate, the static solution considers either an applied load or an applied electric potential, providing a detailed through-thickness evaluation of displacements and stresses, and when present, the electric potential and electric displacements. The respective free vibration solution reveals the first twenty natural frequencies and associated modes, including all together special modes and thickness modes.

Three-dimensional free vibration analyses of functionally graded laminated shells under thermal environment by a hierarchical quadrature element method

This paper carries out three-dimensional (3D) free vibration analyses of laminated shells made of functionally graded materials (FGMs) in both thermal and non-thermal environments by a hierarchical quadrature element method (HQEM). Two configurations of FGM laminated shells were considered, the first with FGM face sheets and homogeneous core, while the second with homogeneous face sheets and FGM core. Effective material properties of the FGMs were estimated in terms of two micromechanical models called Voigt’s rule of mixture (ROM) and Mori-Tanaka (MT) scheme. For shells in thermal environment, a nonlinear temperature distribution along thickness direction was considered while the elasticity properties were assumed to be temperature-dependent. Natural frequencies obtained by the presented formulation for FGM laminated shells under non-thermal conditions but varied geometric and boundary conditions were compared with those obtained by two-dimensional (2D) finite element method (FEM) and semi-analytical approaches in literatures first to assess the correctness and accuracy. Then the influence of volume fraction index, core thickness, and temperature gradient on natural frequencies of FGM laminated shells under thermal environment were investigated. The three-dimensional formulations of this work were shown to need similar computational cost as 2D ones but with the accuracy of 3D theory.

Three-Dimensional free vibration analysis of composite FGM rectangular plates with in-plane heterogeneity: An EKM solution

A three-dimensional elasticity based approximate analytical solution is proposed for free vibration analysis of composite functionally graded Levy-type rectangular plates having an in-plane gradation of material properties. The density and elastic properties of the composite plates are considered to vary linearly along the x-direction. Modified Hamilton’s principle has been applied to derive the weak-form of governing equations in which all stresses and displacements consider as primary variables to ensure the exact point-wise satisfaction of all interlaminar continuity and edge support conditions. The extended Kantorovich method, in conjunction with the Fourier series and power series approach, has been used to obtain the approximate solution in analytical form. New benchmark numerical results are presented for a single-layered and multi-layered in-plane functionally graded rectangular plates. The influence of the in-plane heterogeneity, thickness ratio, and support conditions on the flexural frequencies and mode shapes of the plate are presented in detail. The present study shows that the free vibration response of the rectangular plate is affected significantly by the in-plane gradation of material properties, but the extent of influence largely depends on edge support conditions of the plate as well as on configuration. The present benchmark solution can be used to validate other 3D numerical approaches and two-dimensional (2D) plate theories.