Sector Plates Research Articles

Thermoelastic Coupling Vibration and Stability Analysis of Rotating Annular Sector Plates

This study investigates the thermoelastic coupling vibration and stability of rotating annular sector plates. Based on Hamilton’s principle and thermal conduction equation with deformation effect, the differential equation of transverse vibration for a rotating annular sector plate is established. The differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Then, the thermoelastic coupling transverse vibrations under three different boundary conditions are calculated. The change curve of the first three order dimensionless complex frequencies of the rotating annular sector plate with the dimensionless angular speed are analyzed in the case of the thermoelastic coupling and uncoupling. The effects of the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient on transverse vibration and stability of the annular sector plate are discussed. Finally, we obtained the type of instability and corresponding critical speed of the rotating annular sector plate in the case of the thermoelastic coupling and uncoupling.

A unified procedure for free transverse vibration of rectangular and annular sectorial plates

A unified solution procedure applicable for analyzing the free transverse vibration of both rectangular and annular sectorial plates is presented in this study. For the annular sectorial plate, the basic theory is simplified by a variable transformation in the radial direction. The analogies of coordinate system, geometry and potential energy between the two different shapes are drawn and then unified in one framework by introducing the shape parameter. A generalized solving procedure for the two shapes becomes feasible under the unified framework. The solution adopts the spectro-geometric form that has the advantage of describing the geometry of structure by mathematical or design parameters. The assumed displacement field and its derivatives are continuous and smooth in the entire domain, thereby accelerating the convergence. In this study, the admissible functions are formulated in simple trigonometric forms of the mass and stiffness matrices for both rectangular and annular sectorial plates can be obtained, thereby making the method computationally effective, especially for analyzing annular sectorial plates. The generality, accuracy and efficiency of the unified approach for both shapes are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.

Free vibration analysis of annular sector sandwich plates with FG-CNT reinforced composite face-sheets based on the Carrera’s Unified Formulation

This paper presents an investigation of the free vibration of annular sector sandwich plates with carbon nanotube reinforced face-sheets and various edges boundary conditions. It is assumed that carbon nanotubes are radially aligned and distributed uniformly (UD) or functionally graded (FG) in the thickness direction. The effective material properties of CNT reinforced face-sheets are estimated using the extended rule of mixture, which contains efficiency parameters to consider the size-dependent material properties. A variable kinematic model which has been utilized for FGM cases is extended to FG-CNTs to describe the continuous variation of properties through the thickness. In this paper, the Carrera’s Unified Formulation (CUF) is developed for the analysis of asymmetric sector plates for the first time. The governing equations and associated boundary conditions are obtained employing the Principle of Virtual Displacements (PVDs) based on the CUF and solved using the generalized differential quadrature (GDQ) method. Numerical results for some special cases are presented and validated by comparison with available results in the literature. Some numerical results are tabulated to show the effects of the volume fraction of carbon nanotubes, boundary conditions and geometrical parameters on the free vibration behavior of the annular sector sandwich structures with CNTRC face-sheets.

Vibration analysis of FG annular sector in moderately thick plates with two piezoelectric layers

Free vibration of functionally graded (FG) annular sector plates embedded with two piezoelectric layers is studied with a generalized differential quadrature (GDQ) method. Based on the first-order shear deformation (FSD) plate theory and Hamilton’s principle with parameters satisfying Maxwell’s electrostatics equation in the piezoelectric layers, governing equations of motion are developed. Both open and closed circuit (shortly connected) boundary conditions on the piezoelectric surfaces, which are respective conditions for sensors and actuators, are accounted for. It is observed that the open circuit condition gives higher natural frequencies than a shortly connected condition. For the simulation of the potential electric function in piezoelectric layers, a sinusoidal function in the transverse direction is considered. It is assumed that properties of the FG material (FGM) change continuously through the thickness according to a power distribution law. The fast rate convergence and accuracy of the GDQ method with a small number of grid points are demonstrated through some numerical examples. With various combinations of free, clamped, and simply supported boundary conditions, the effects of the thicknesses of piezoelectric layers and host plate, power law index of FGMs, and plate geometrical parameters (e.g., angle and radii of annular sector) on the in-plane and out-of-plane natural frequencies for different FG and piezoelectric materials are also studied. Results can be used to predict the behaviors of FG and piezoelectric materials in mechanical systems.

A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment

A new method has been presented for static analysis of macro- and nano sector plates with variable thickness. The thickness variations are assumed linearly, functionally, and also irregularly. The governing equations have been derived based on the nonlocal modified higher-order shear deformation theory and have been solved applying the semi-analytical polynomial method. The obtained results have been compared with the results of the other relevant articles. The effective parameters on the analysis have been investigated such as the thickness, the size of the plate under uniform and nonuniform loadings, and the boundary conditions including free edges.Communicated by Marco Amabili.

Free vibration and buckling analyses of functionally graded annular thin sector plate in-plane loads using GDQM

In the present study, buckling and free vibration analyses of annular thin sector plate made of functionally graded materials (FGMs) resting on visco-elastic Pasternak foundation, subjected to external radial, circumferential and shear in-plane loads is investigated. Material properties are assumed to vary along the thickness according to an power law with Poisson\'s ratio held constant. First, based on the classical plate theory (CPT), the governing equation of motion is derived using Hamilton\'s principle and then is solved using the generalized differential quadrature method (GDQM). Numerical results are compared to those available in the literature to validate the convergence and accuracy of the present approach. Finally, the effects of power-law exponent, ratio of radii, thickness of the plate, sector angle, and coefficients of foundation on the fundamental and higher natural frequencies of transverse vibration and critical buckling loads are considered for various boundary conditions. Also, vibration and buckling mode shapes of functionally graded (FG) sector plate have been shown in this research. One of the important obtained results from this work show that ratio of the frequency of FG annular sector plate to the corresponding values of homogeneous plate are independent from boundary conditions and frequency number.

Nonlinear free and forced vibration analysis of FG‐CNTRC annular sector plates

This article deals with the large amplitude free and forced vibration analysis of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) annular sector plates based on a numerical approach. The modified rule of mixture is used to estimate the material properties. The equations of motion are developed based on the first‐order shear deformation theory (FSDT) and the von Kármán geometric nonlinearity. First, the discretized form of energy functional of structure is given with the aid of variational differential quadrature method. Then, a time periodic discretization is performed and the frequency response of the nanocomposite plate is determined using the pseudo‐arc length continuation method. After verifying the correctness of the proposed approach, a comprehensive parametric study is presented to investigate the effects of important factors on the nonlinear vibration characteristics of the FG‐CNTRC annular sector plates. The results imply that the volume fraction and distribution type of nanotubes have considerable effects on the fundamental frequency as well as nonlinear frequency response curves. POLYM. COMPOS., 40:E1364–E1377, 2019. © 2018 Society of Plastics Engineers

Three-dimensional exact solution for the free vibration of thick functionally graded annular sector plates with arbitrary boundary conditions

A new three-dimensional exact solution for the free vibrations of arbitrary thick functionally graded annular sector plates with arbitrary boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. According to a power law distribution of the volume of the constituents, the material properties change continuously through the thickness of the functionally graded annular sector plates. Each of displacements of the annular sector plates, regardless of boundary conditions, is expanded as a three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Ritz procedure by the energy functions of plate. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature, and numerous new results for thick FG annular sector plates with elastic boundary conditions are presented. The effects of gradient indexes are also illustrated.

A modified Fourier solution for sound-vibration analysis for composite laminated thin sector plate-cavity coupled system

This paper applied the modified Fourier series method to investigate the sound-vibration characteristics by establishing a composite laminated thin sector plate-cavity coupled model for the first time based on the classical plate theory (CPT) and Rayleigh-Ritz energy technique. The coupled system consists of an annular sector or circular sector plate backed by an acoustic cavity filled with air or water. Ignoring the influence of boundary conditions, displacements admissible functions of laminated sector plate and sound pressure admissible functions of cavity can be set up as a Fourier series superposition, whose composition are the superposition of Fourier cosine series and supplementary functions. The addition of these supplementary polynomials can effectively eliminate the discontinuity or jump phenomenon on the boundary. The correctness of the established analytical model has been validated by being compared with the results achieved by the finite element method (FEM). On this basis, the coupling mechanism of the weakly coupled system and the strongly coupled system are discussed in detail. In addition, some new results and discussions are given, including the cavity depth, plate thickness, anisotropic degree, varying boundary conditions and so on, which could provide reference for future research.

Vibration of FG-CNTRC annular sector plates resting on the Winkler-Pasternak elastic foundation under a periodic radial compressive load

The vibration and stability of functionally graded carbon nanotube-reinforced (FG-CNTRC) composite annular sector plates resting on Winkler-Pasternak elastic foundation subjected to a periodic radial compressive load are investigated for various boundary conditions. To this end, a shear deformable plate model is established according to a parabolic theory which can interpret the shear deformation and rotary inertia effects without using any shear correction factor. The modified micromechanical technique is employed to compute the effective material properties of the FG-CNTRCs. The discretized form of higher-order governing equations is directly accessed by employing Hamilton’s principle and the variational differential quadrature (VDQ) method. In addition, by considering the applied radial compressive force as a periodic function, the discretized equations are expressed as the Mathieu–Hill equations, and then the instability regions are determined via the Bolotin’s scheme. In numerical results, the effects of the CNT distribution pattern and volume fraction, geometry parameters, sector angle, elastic foundation and static load factor on the stability of FG-CNTRC annular sector plates subject to arbitrary edge conditions are thoroughly discussed.

Free vibration analysis of laminated and FGM composite annular sector plates

In this article, the authors used two different numerical approaches for frequency response of annular sector and sector plates with functionally graded materials and laminated composite cases. First-order shear deformation (FSDT) and Love's conical shell theories are used for obtaining the annular sector plate equations via some suitable angles and geometric parameters. The method of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) have been used for numerical solution of the resulting governing equations for modal analysis. Simple power-law and four-parameter power law distributions in terms of the volume fractions of constituents have been used for FGM composites. Comparison and convergence study for the present numeral methods have been made via existing results available in the literature for sector and annular sector plates. Frequencies values for annular sector/sector plates have been obtained for different material and geometric parameters, boundary conditions and sector angles.

Dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints

In this paper, the dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints is performed in a unified form for the first time. The overall theoretical model is based on the first order shear deformation theory. The kinetic energy and potential energy function of the plates are unified representation of five kinds of displacement admissible function. Then, each of displacement admissible function is expanded as a modified Fourier series to obtain general elastic restraints. Lastly, the solutions are obtained by using the variational operation. The convergence and accuracy of the present modeling are validated by comparing its results with those available in the literature and FEM results. Based on that, a series of innovative results are also highlighted in the text, which may be as the basic data for other algorithm research in the future.

Vibration of carbon nanotube reinforced composite (CNTRC) annular sector plates by discrete singular convolution method

Modeling and numerical solution of free vibration problem of annular and annular sector plates with carbon nanotube reinforced (CNTR) composites have been presented. First-order shear deformation theory has been used for modeling of laminated composite and CNTR composite materials. The method of discrete singular convolution (DSC) is used for solution of equations of motion for shells and plate. Verification of the accuracy of the present DSC results is verified by appropriate convergence study and checked with the results available in the open literature for isotropic, laminated and CNTR cases. The influence of volume fraction index, boundary conditions, types of CNTR and geometrical parameters on results have been investigated in details.

Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded nanocomposite sandwich sectorial plates are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes are assumed to be graded in the thickness direction. To determine the effect of carbon nanotube agglomeration on the elastic properties of carbon nanotube-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is considered to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight carbon nanotubes. The two-dimensional generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed sectorial plates are simply supported at radial edges, while all possible combinations of free, simply supported, and clamped boundary conditions are applied to the other two circular edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of agglomeration, geometrical, and material parameters together with the boundary conditions on the frequency parameters of the sandwich functionally graded nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of carbon nanotubes agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analyze the sandwich sectorial plates.

Stability Analysis of Composite Perforated Annular Sector Plates Under Thermomechanical Loading by Finite Element Method

This paper presents the stability analysis of a perforated plate with sector geometry made of composite materials. The sector of concern is a compound of graphite-epoxy and glass-epoxy with identical ply thickness but different fiber angles for each layer. The mechanical load conditions considered include uniform axial, circumferential, and biaxial pressure, while the thermal loading is specified to be uniform temperature increase over the whole sector. The existence of one or two circular holes has increased the complexity of analysis. To obtain solutions of high accuracy, the three-dimensional elasticity theory relations have been employed. Using the finite element method along with the stability condition of Trefftz, the buckling equation of the structure is derived. Green nonlinear strain-displacement relations are used to form the geometrical stiffness matrix. Unlike the finite element method used by other researches, a novel curved 3D B-Splined element is used to more accurately trace the displacement and stress variations of the structure. This element can be used in solution domains with geometric discontinuities, such as perforated plates and also meshed in the thickness direction. Moreover, instead of using the common von Karman assumptions, the most general form of the strain tensors in the curvilinear coordinates is adopted. The buckling load is obtained by extremizing the second variations of the total potential energy. The finite element formulation is coded in the MATLAB software. The effects of various parameters such as sector dimensions, dimensions of the hole, mechanical load directions, and fiber angles of each layer on the thermomechanical buckling is investigated.

A semi-analytical solution for free vibration of thick orthotropic annular sector plates with general boundary conditions, internal radial line and circumferential arc supports

The aim of this paper is to put forward a semi-analytical solution to conduct the free vibration analysis of thick orthotropic annular sector plates with general boundary conditions, internal radial line and circumferential arc supports for the first time. Based on the Mindlin plate theory, stationary energy principle and Rayleigh-Ritz method, the governing eigenvalue equation is derived for thick orthotropic annular sector plates of different radii, sector angles, thickness ratio, line/arc supports and various boundary conditions along the radial and circumference edges. The main innovation point of this paper lies in breaking the classical boundary barrier by using an improved Fourier series to replace the traditional Fourier series. By comparing with those results in open reported literature, the present method proves to be of excellent accuracy and reliability. In addition, in order to enrich the research, some results involving thick orthotropic annular sector plates with various boundary conditions are presented firstly, which can be acted as benchmark data for the future research.

Differential Quadrature Free Vibration Analysis of Functionally Graded Thin Annular Sector Plates in Thermal Environments

In this paper, the free vibration behavior of functionally graded (FG) thin annular sector plates in thermal environment is studied using the differential quadrature method (DQM). The material properties of the FG plate are assumed to be temperature dependent and vary continuously through the thickness, according to the power-law distribution of the volume fraction of the constituents. The nonlinear temperature distribution along the thickness direction of the plate is considered. Based on the classical plate theory, the governing differential equations of motion of the plate are derived and solved numerically using DQM. The natural frequencies of thin FG annular sector plates in thermal environment under various combinations of clamped, free, and simply supported boundary conditions (BCs) are presented for the first time. To ensure the accuracy of the method, the natural frequencies of a pure metallic plate are calculated and compared with those existing in the literature for the homogeneous plate where the results are in good agreement. The effects of temperature field, BCs, volume fraction exponent, radius ratio, and the sector angle on the free vibrations of the FG-plate are examined.

Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates

The intent of this paper is to firstly perform the vibration analysis of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates with arbitrary boundary conditions by means of the semi-analytical method which is proposed by the author’s team. In the material model, the distribution of the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures and four types of the CNTs distribution are studied in this paper. The refined rule of mixtures approach containing the efficiency parameters is adopted to determine the properties of the composite media. The admissible displacement functions of the FG-CNTRC circular, annular and sector plates are uniformly expanded as the modified Fourier series which embodies a standard cosine Fourier series and several auxiliary functions which are introduced to eliminate the limit of the boundary conditions. On this foundation, the first-order shear deformation elasticity theory is employed to construct the energy expression of the FG-CNTRC circular, annular and sector plates. Then the Ritz-variational energy method is used to decide the natural frequencies and the associated mode shapes. To examine the convergence, accuracy, stability and efficiency of the computational model, the comprehensive studies are conducted. Based on that, some crucial parametric studies covering the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs and boundary conditions are also investigated in detail.

A semi-analytical method for transverse vibration of sector-like thin plate with simply supported radial edges

• The concept of sector-like plate is proposed due to the lack of researches on these kinds of structures. • A semi-analytical method is used to analyze the vibration of sector-like plates. • The method converges quickly and the results are accurate by comparing the results with the reference results. • The method is applicable for the vibration problems of sector-like plates. In this paper, a semi-analytical method is presented to study the transverse vibration of sector-like thin plate with various boundary conditions. A sector-like plate is generated from a sector plate whose circular edge is replaced by edges of other geometries. The expressions of boundary conditions for arbitrary point on the arc edge are given in a local rectangular coordinate system and transformed into the global polar coordinate system. Fourier expansion is performed on the expressions of the boundary conditions so that the boundary conditions are satisfied on the whole edge. Several examples are presented and the results are calculated numerically and compared with those in literature and obtained by FEM. The good agreement between the results validates the accuracy and applicability of this method for sector-like plates.

Free Vibration Of Functionally Graded Carbon Nanotube Reinforced Composite Annular Sector Plate With General Boundary Supports

Abstract In this paper, an efficient and unified approach for free vibration analysis of the moderately thick functionally graded carbon nanotube reinforced composite annular sector plate with general boundary supports is presented by using the Ritz method and the first-order shear deformation theory. For the distribution of the carbon nanotubes in thickness direction, it may be uniform or functionally graded. Properties of the composite media are based on a refined rule of the mixture approach which contains the efficiency parameters. A modified Fourier series is chosen as the basic function of the admissible function to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges. To establish the general boundary supports of the annular sector plate, the artificial spring boundary technique is implemented at all edges. The desired solutions are obtained through the Ritz-variational energy method. Some numerical examples are considered to check the accuracy, convergence and reliability of the present method. In addition, the parameter studies of the functionally graded carbon nanotube reinforced composite annular sector plate are carried out as well.