Annular Sector Plates Research Articles

A simplified plate theory for vibration analysis of composite laminated sector, annular and circular plate

A unified analysis model for vibration characteristics of composite laminated annular sector plate, circular sector plate, annular plate and circular plate with various elastic boundary conditions is established based on the simplified plate theory (SPT) and two-dimensional (2D) improved Fourier-Ritz method. It has significant advantages in the study of free and forced vibration of thin-to-moderately thick rotary composite laminated plates. Under the current framework, the displacement admissible functions of the rotary plate are generally expressed as superposition of simple periodic functions, which includes the multiplication of two cosine functions and two complementary polynomials. The introduction of these polynomials can effectively eliminate jumping or discontinuity at the boundaries. The vibration characteristics of the rotary plate can be obtained by Rayleigh-Ritz energy technique. The present method shows fast convergence and good accuracy. The parameterization study on geometric parameters, material parameters and boundary conditions has been carried out to systematically reveal the vibration characteristics of rotary composite laminated plates, which can be the benchmark for the future research.

Nonlinear periodic response of bimodular laminated composite annular sector plates

The steady state non-linear response of bimodular composite laminated annular sector plates is investigated using the field consistent eight-noded finite element based on the first-order shear deformation theory and Bert's constitutive model. The periodic forced response is obtained using the modified shooting method and arc-length/pseudo arc-length continuation techniques. Within a shooting cycle, the solution of the governing equation is obtained using Newmark's time integration coupled with Newton Raphson method. The effects of bimodularity, geometric nonlinearity, boundary conditions, load amplitude, lamination scheme and sector angle on the response characteristics are presented. Significantly large difference in the peak amplitudes is predicted with and without geometric nonlinearity. The higher harmonic contributions in the steady state displacement/stresses are demonstrated using frequency spectra and phase plane plots. Through the strain energy plots, the participation of even and odd order harmonics is correlated to the quadratic and cubic restoring forces in a cycle. The total number of degrees of freedom for converged results is 805 for CCCC and 865 for CSCS/SCSC plates resulting in a system seldom treated in the literature on nonlinear steady state periodic response and the results presented may serve as reference for validation of approximate solutions.

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.

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.

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.

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.

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.

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.

Geometrically nonlinear resonance of higher-order shear deformable functionally graded carbon-nanotube-reinforced composite annular sector plates excited by harmonic transverse loading

This article presents an attempt to study the nonlinear resonance of functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) annular sector plates excited by a uniformly distributed harmonic transverse load. To this purpose, first, the extended rule of mixture including the efficiency parameters is employed to approximately obtain the effective material properties of FG-CNTRC annular sector plates. Then, the focus is on presenting the weak form of discretized mathematical formulation of governing equations based on the variational differential quadrature (VDQ) method and Hamilton’s principle. The geometric nonlinearity and shear deformation effects are considered based on the von Karman assumptions and Reddy’s third-order shear deformation plate theory, respectively. The discretization process is performed via the generalized differential quadrature (GDQ) method together with numerical differential and integral operators. Then, an efficient multi-step numerical scheme is used to obtain the nonlinear dynamic behavior of the FG-CNTRC annular sector plates near their primary resonance as the frequency-response curve. The accuracy of the present results is first verified and then a parametric study is presented to show the impacts of CNT volume fraction, CNT distribution pattern, geometry of annular sector plate and sector angle on the nonlinear frequency-response curve of FG-CNTRC annular sector plates with different edge supports.

3D thermomechanical buckling analysis of perforated annular sector plates with multiaxial material heterogeneities based on curved B-spline elements

In the present paper, thermomechanical buckling of perforated functionally graded annular sector plates under uniform temperature rises and radial, circumferential or biaxial mechanical loads are investigated. Furthermore, effects of the circular cutouts on the buckling strength and deformation pattern are studied. Influence of the heterogeneity of material of the FGM sector in radial, circumferential, and transverse directions on the thermomechanical buckling is investigated for plates without or with one or two holes. Governing equations are derived based on the 3D energy-based theory of elasticity and buckling occurrence is detected by Treftz instability criterion. A novel curved 3D B-splined C2-continuous element is proposed to trace the inter-element variations of the displacements and stresses and model the geometric discontinuities (cutouts). A proper algorithm is also proposed to relate events of the original cylindrical coordinates to those of the natural coordinates and vice versa. Instead of using the common von Karman assumptions, the most general form of the strain tensor in curvilinear coordinates is used. Finally, effects of the sector dimensions, size and orientation of the cutout, heterogeneity direction, direction of the mechanical loads and the combination of the thermal and mechanical loads on the buckling loads and mode shapes are investigated.

Numerical approaches for vibration response of annular and circular composite plates

In the present investigation, by using the two numerical methods, free vibration analysis of laminated annular and annular sector plates have been studied. In order to obtain the main equations two different shell theories such as Love\'s shell theory and first-order shear deformation theory (FSDT) have been used for modeling. After obtaining the fundamental equations in briefly, the methods of harmonic differential quadrature (HDQ) and discrete singular convolution (DSC) are used to solve the equation of motion. Accuracy, convergence and reliability of the present HDQ and DSC methods were tested by comparing the existing results obtained by different methods in the literature. The effects of some geometric and material properties of the plates are investigated via these two methods. The advantages and accuracy of the HDQ and DSC methods have also been examined with different grid numbers and shell theory. Some results for laminated annular plates and laminated circular plates were also been supplied.

3-D Vibration analysis of FG-MWCNTs/Phenolic sandwich sectorial plates

In this study, based on the three-dimensional theory of elasticity, free vibration characteristics of sandwich sectorial plates with multiwalled carbon nanotube-(MWCNT)-reinforced composite core are considered. Modified Halpin-Tsai equation is used to evaluate the Young's modulus of the MWCNT/epoxy composite samples by the incorporation of an orientation as well as an exponential shape factor in the equation. The exponential shape factor modifies the Halpin-Tsai equation from expressing a straight line to a nonlinear one in the MWCNTs wt% range considered. In this paper, free vibration of thick functionally graded sandwich annular sectorial plates with simply supported radial edges and different circular edge conditions including simply supported-clamped, clamped-clamped, and free-clamped is investigated. A semi-analytical approach composed of twodimensional differential quadrature method and series solution are adopted to solve the equations of motion. The material properties change continuously through the core thickness of the plate, which can vary according to a power-law, exponentially, or any other formulations in this direction. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analysis of laminated sectorial plates.