Annular Plates Research Articles

Three-Dimensional Free Vibration Analysis of Gears with Variable Thickness Using the Chebyshev–Ritz Method

To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue equations. We divided the gear model into three annular parts along the locations of the step variations, and the admissible function was a Ritz series that consisted of a Chebyshev polynomial multiplying boundary function. The convergence study demonstrated the high accuracy of the present method. We used a hammering method for a modal experiment to test two annular plates and one gear’s eigenfrequencies in a completely free condition. We also applied the finite element method to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, we found that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. The relative errors of four sets of data were greater than 4%, and the errors of the other 48 sets were less than 4%. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of gears with variable thickness.

Dynamic thermal buckling and postbuckling of clamped–clamped imperfect functionally graded annular plates

Dynamic thermal buckling and postbuckling of imperfect functionally graded material (FGM) annular plates are investigated based on the nonlinear plate theory. The transient temperature fields of the FGM annular plates under dynamic thermal loadings are obtained by Laplace transform in combination of power series method according to the theory of Fourier heat conduction. The nonlinear dynamic equations of large axisymmetric deformations are numerically solved by series expansions and Runge–Kutta method. The critical bucking and dynamic postbuckling responses are predicted by the maximum deflections of the FGM annular plates with positive or negative initial geometric imperfections. The effects of the loads, the material gradient and the initial geometric imperfections on the dynamic responses and the buckling critical temperatures of the FGM annular plates are analyzed in detail.

Over-critical load analysis for residual stresses and displacement around fastener-holes based on the decohesive failure mechanism and non-linear Mises yield criterion

Analytical calculations of residual stresses and displacement in the case of elevated loads applied to common engineering structures require particular attention. This is related to various types of singularities such as material and geometrical discontinuities or numerical errors, particularly, for elastic/perfectly-plastic material. In the present study, some important aspects leading to such singularities are discussed within the framework of small strains and plane-stress state. Structures such as double-shear bolted connections, shrink-fits, open fastener-holes and pressurized rigid containers are analyzed based on the classical problem of an annular plate with a central hole subject to uniformly distributed inner pressure. At the outer edge, the plate is either stress-free or embedded into a rigid container. Comparison of the results with related papers is provided.

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.

New analysis model for rotor-bearing systems based on plate theory

The purpose of this work is to develop a new analysis model for angular-contact, ball-bearing systems on the basis of plate theory instead of commonly known approaches that utilize spring elements. Axial and radial stiffness on an annular plate are developed based on plate, Timoshenko beam, and plasticity theories. The model is developed using theoretical and inductive methods and validated through a numerical simulation with the finite element method. The new analysis model is suitable for static and modal analyses of rotor-bearing systems. Numerical examples are presented to reveal the effectiveness and applicability of the proposed approach.

Aerodynamic Optimization of Squirrel-Cage Fan with Dual Inlet

In this paper, the rotor and the volute of the squirrel-cage fan with dual inlet were optimized to improve its aerodynamic performance. The blade inlet angle, blade exit angle and diameter ratio of the impeller were chosen as optimization variables using the response surface methodology (RSM) to improve the total pressure. Furthermore, another three optimization variables were adopted on the basis of previous optimum results, which are the width of impeller, the location of the impeller annular plate and the location of cutoff respectively. The simulation and experimental results show that the total pressure of the optimal model has been greatly improved without noise increase in comparison with the original model.

Chiral optical field generated by an annular subzone vortex phase plate

We proposed and experimentally demonstrated a novel method for generating a chiral beam with controllable intensity twist lobes and direction by using annular subzone (AS) vortex phase plates, which is composed of different ASs and different vortex phases. The phase distribution continuity between two adjacent ASs determines the intensity distribution of the light field. The rotated direction of the optical filed is determined by the topological charge sign. The number of intensity twist lobes is determined by the topological charge gradient between adjacent subzones. The experimental results show that this method is effective and practical, which offers broad potential applications in particle manipulation, chiral microstructure fabrication, and optical tweezers.

Residual stress measurements in an autofrettaged cylinder through the Initial Strain Distribution method

An experimental activity aimed at determining the complete residual stress field in an autofrettaged cylinder is reported and the obtained results are presented and discussed.A set of strain gauges were located on annular plates extracted from the cylinder in order to measure the relaxed strains produced by the gradual extension of a radial cut. The relaxed strains were related to the initial strain distribution induced by the autofrettage process by solving a set of integral equations with dedicated influence functions.In order to estimate the global errors and to assess the reliability of the obtained residual stress distributions, two nominally identical annular plates were subjected to similar tests.Experimental arrangements, including the strain gauges positioning and the machining set up of the cuts along with the post-processing of the relaxed strains are presented and the obtained results of the two tests are compared and discussed. The whole procedure appears suitable for an accurate evaluation of the residual stress distribution in the autofrettage component.

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.

Sensitivity of Composite Structure with Directional Properties of Annular Three-Layered Plate Mechanically and Thermally Loaded

Abstract Paper presents the sensitivity of the three-layered plate structure on the acting of mechanical and thermal loads. The cases of the annular plates, whose individual layers: facings and core have homogeneous building and/or heterogeneous one expressed by the variable material properties in radial direction have been examined. Numerical investigations have been carried out modelling the select examples of plate structure with the use of the finite element method. Plate is loaded in the plane of facings or is subjected to the flat temperature field. The evaluation of the structure sensitivity has been carried out analysing the values of critical loads or critical temperatures and corresponding with them buckling modes. Numerous results presented in diagrams create the image of plate behaviours, show responses of plate structure and indicate on the means of structure designs, which can fulfil the expected conditions of plate work.

Refined Model of the Dynamic Behavior of Flexible Reinforced Shallow Shells Made from Nonlinear Elastic Materials

An initial-boundary-value problem is formulated for the dynamic deformation of flexible reinforced shallow shells made of composite materials whose components are nonlinearly elastic. The geometric nonlinearity is taken into account in the Karman approximation. The resulting equations allow one, with different degrees of accuracy, to determine the strain state of such thin-wall structures with account of their weakened resistance to transverse shears. The relatios of the traditional nonclassical Reddy theory follow from these equations as a special case. Numerical integration of the problem stated is carried out on the basis of the method of steps in time by using an explicit “cross”-type scheme. Thin shallow spherical shells and flexible annular plates are considered. They are reinforced in the radial and circumferential directions, contain an absolutely rigid insert, and are clamped along the outer edge. The dynamic behavior of such structures is studied in relation to the level of pressure caused by the blast of air waves and to the form (convex or concave of the face) to which this pressure is applied. It is revealed that, at time intervals exceeding 0.1 s, mechanical behavior of the reinforced shallow shells and plates calculated by the Reddy theory significantly differs from the dynamic response determined according to the refined theory proposed.

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.

Electromechanical properties and actuation capability of an extension mode piezoelectric fiber composite actuator with cylindrically periodic microstructure

An extension mode piezoelectric fiber composite (PFC) actuator with cylindrically periodic microstructure is presented in this work especially for directional actuation of plane structures of revolution. The PFC actuator is designed in the form of a thin annular disk where the continuous piezoelectric fibers are periodically distributed along the circumferential direction to yield the directional actuation along the radial direction in the cylindrical principal material coordinate system. This kind of microstructure of the annular PFC actuator yields its radially varying electromechanical properties that are determined through the asymptotic segmentation of its (PFC) volume into a large number of microvolumes of different fiber volume fractions. The closed-form expressions for the effective electromechanical coefficients of the microvolumes are derived, and the corresponding verification is carried out through the numerical homogenization using finite element procedure. The results reveal the indicative magnitude of an effective piezoelectric coefficient ( $$e_{31})$$ that quantifies the directional actuation along the radial direction. But, the magnitude of this coefficient ( $$e_{31})$$ decreases indicatively with the increasing radius, and thus the annular PFC actuator is redesigned in a special manner for the improved magnitude of the coefficient ( $$e_{31})$$ at any radius. With these improved properties of the annular PFC actuator, its indicative actuation capability in control of vibration of an annular plate is observed, and thus this annular PFC actuator may be recommended for active control of plane structures of revolution specifically where the actuation along the radial direction is the major requirement.

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.

Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory

The present work is concerned with the application of the variational differential quadrature (VDQ) method (Faghih Shojaei and Ansari, 2017), in the area of computational mechanics, to the nonlinear large deformation analysis of shell-type structures. To this end, based on the six-parameter shell model, the functional of energy in quadratic form is derived based on Hamilton’s principle which is then directly discretized by the VDQ technique. The formulation of article is presented in a general form so that it can be readily used for different structures such as beams, annular plates, cylindrical shells and hemispherical shells under various loading conditions. In order to reveal the accuracy of developed solution strategy, it is tested in several popular benchmark problems for the geometric nonlinear analysis of shells. The results show that the present numerical method is capable of yielding highly accurate solution in the nonlinear large deformation analysis of shells. It is also easy to implement due to its compact and explicit matrix formulation.

CONTINUOUS ELEMENT FORMULATIONS FOR COMPOSITE RING-STIFFENED CYLINDRICAL SHELLS

This research studies the free vibration of composite ring-stiffened cylindrical shells by the continuous element method (CEM). The dynamic stiffness matrix (DSM) of the investigated structure has been constructed based on the analytical solutions of the governing equations of motion for composite cylindrical shells and annular plates. By applying the powerful assembly procedure of continuous elements method, natural frequencies and harmonic responses of composite ring-stiffened cylindrical shells have been obtained. In addition, the proposed model allows extracting exactly ring-stiffener vibration modes by choosing appropriate points of response. Numerical examples have confirmed many advantages of the developed model.

Mechanical buckling analyses of sandwich annular plates with functionally graded carbon nanotube-reinforced composite face sheets resting on elastic foundation based on the higher-order shear deformation plate theory

The main objective of this article is to analyze the buckling of sandwich annular plates with carbon nanotube-reinforced face sheets subjected to in-plane mechanical loading resting on the elastic foundation. It is assumed that the sandwich plate is composed of the homogeneous core layer and two functionally graded carbon nanotube-reinforced composite face sheets. The effective material properties of the functionally graded carbon nanotube-reinforced composite face sheets are estimated using the modified rule of mixture method. The higher-order shear deformation theory along with the variational differential quadrature method is employed to derive the governing equations. To this end, the quadratic form of energy functional of the structure is derived based on higher-order shear deformation theory which is directly discretized using numerical differential and integral operators. The validity of the proposed numerical approach is first shown and the effects of various parameters are then investigated on the buckling of sandwich annular plates. It was found that the elastic foundation coefficients, type of distribution of carbon nanotubes, inner-to-outer radius ratio and core-to-face sheet thickness ratio play important roles in the stability of the structure. Furthermore, the numerical results of the higher- and first-order shear deformation theories are compared.

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.

Vibro-acoustic analysis of the annular segment flexible plate coupled with an impedance walled enclosure

This paper proposes a structural-acoustic coupling model which is composed of an annular segment thin plate with various elastic boundary constraints and an acoustic enclosure with rigid-wall or various impedance-wall boundary conditions. The admissible displacement and sound pressure of the coupling system are written as superposition of simple periodic functions on the basis of an improved Fourier series method. The introduction of supplementary polynomials can effectively eliminate the discontinuous or jumping phenomenon in the boundaries. The coupling system is described by constructing energy formulation. Therefore, all the unknown series expansion coefficients can be solved by Rayleigh-Ritz energy technique. The fast convergence of the present method has been given by selecting smaller truncation values. In addition, the accuracy of the present method has been proved by being compared with the finite element method (FEM). The effect of various geometric parameters and boundary conditions on the vibro-acoustic characteristics of coupling system is investigated. Besides, the coupling responses are given by applying the point force on the annular segment plate or point sound source inside the enclosure.

Vibration of FG-GPLs eccentric annular plates embedded in piezoelectric layers using a transformed differential quadrature method

In this paper, a transformed differential quadrature method (TDQM) for the free vibration analysis of functionally graded (FG) multilayer nanocomposite eccentric annular plates reinforced by graphene nanoplatelets (GPLs) and embedded in piezoelectric layers is developed. In this regards, a suitable conformal mapping is utilized to calculate the transformed weighting coefficients of the differential quadrature method (DQM) in physical domain which enables one to directly discretize the partial differential equations in the reference domain. The modified Halpin–Tsai model is employed to estimate the effective mechanical properties of the nanocomposites. The governing equations are derived based on the first-order shear deformation theory (FSDT) using Hamilton’s principle and Maxwell’s equation. The numerical results show that the TDQM has a very good convergence rate. Furthermore, the accuracy and reliability of the proposed method are verified by comparing the results with those obtained by simulating some problems via ABAQUS software and also, in the limit cases with those available in the literature. The effects of GPLs weight fractions, number of GPLs reinforced layers, distribution types of GPLs, different boundary conditions and various plate geometrical parameters such as the plate thickness, piezoelectric layers thickness, external electrical voltage and off-set parameter on the natural frequency parameters are investigated.