Thin Annular Plates Research Articles

BUCKLING DEFORMATION OF ANNULAR PLATES DESCRIBING NATURAL FORMS

It has been observed that undulating periodic patterns formed on an initial flat annular plate that model the leaves of plants are a physical response to the expansion of the surface under a lateral restraint. They are not visible at the beginning but become apparent only when the plants continue to grow. The behavior can be explained via the inhomogeneous deformation of gel materials that behave in a similar manner to hyperelastic materials in solid mechanics.This paper compares the stability of thin annular plates clamped along the inner edge and free along the outer periphery using numerical simulations of the swelling of thin gel annular plate held along the inner edge as well as analysis of a similar class of structures by solid mechanics concept via energy principle. The trends of results from both approaches compare favorably. The buckling patterns of annular plates with various values of inner radius to outer radius ratio illustrate the relationship between the geometry of the annular plate and the inhomogeneous deformation of gels or buckling patterns of solid mechanics materials. The undulating patterns on leaves such as those of flowering cabbage can thus be explained via the buckling behavior of annular plates, which can be regarded as thin soft materials adhered to a stiffer core. The study can also be extended to cover other stimuli under different environmental conditions and the outcome may bring further insights into the evolution of plants.

Non-axisymmetric bending of thin annular plates due to circumferentially distributed moments

The non-linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied. A von Kármán plate model is adopted to formulate the equations of motion. Free–free boundary conditions have been applied at both inner and outer edges. The matrix formulation of the Differential Quadrature Method is used to discretise and solve the governing equations. Linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome. However, the proposed non-linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non-axisymmetric cylindrical deformation. This is the case when at least one of two parameters reaches a critical value. These parameters are the non-dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius. Furthermore, it is noted that free-free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk. Hence, the principal axes of curvature in the deformed configuration do not have a preferred orientation. Therefore, the present work describes a shell possessing infinite identical equilibria, having different yet no favoured direction and, hence, links to previous researches on neutrally stable structures.

Elastic Stability of Annular Thin Plates with One Free Edge

The elastic stability of annular thin plates having one free edge and subjected to axisymmetric radial edge loads at the other edge is investigated. The supported edge is allowed to be either simply supported or clamped against axial (transverse) deflection. Both compression buckling and tension buckling (wrinkling) are investigated. To insure accuracy, two methods of solving the appropriate eigenvalue problems are used and found to yield essentially identical results. A selection of these results for both compression and tension buckling is presented graphically and used to illustrate interesting aspects of the solutions.

Buckling-resistant thin annular plates in tension

Motivated by the localized nature of elastic instabilities in radially stretched thin annular plates, we investigate the resistance to buckling of such configurations in the case when their mechanical properties are piecewise constant. By considering a plate consisting of two sub-annular regions perfectly bonded together and with different linear elastic properties, the neutral stability envelope corresponding to the case when radial constant displacement fields are applied on the inner and outer edges of the plate is investigated numerically in considerable detail. These results are complemented by an asymptotic reduction strategy that provides a greatly simplified eigenproblem capable of describing the original buckling problem in the limit of very thin plates.

Some applications of the WKB method to the wrinkling of bi-annular plates in tension

An application of WKB methods is proposed here for a stretched annular thin plate with piecewise-constant mechanical properties (also known as a bi-annular plate). Unlike the classical scenario involving only a simple annular such plate, in certain cases the neutral stability curve fails to be convex and the critical eigenmodes behave rather differently as the plate becomes progressively thinner (equivalent to \({\mu \to \infty}\) in our notations). On one side of this curve, the corresponding eigenmodes are localised near the inner rim of the annulus, while in the remaining part these functions are concentrated along the interface separating the two annular sub-regions. By using the asymptotic reduction technique proposed by Coman and Haughton in (Acta Mech 185:179–200, 2006), the original fourth-order three-point boundary-value problem is formally reduced to a pair of second-order differential equations coupled through a set of matching conditions at the interface. It is shown that for \({\mu \gg 1}\) the critical eigenvalues for both cases mentioned above can be approximated by solving a couple of simple transcendental equations and that the results predicted compare well with the direct numerical simulations of the original problem.

Simplified description of out-of-plane waves in thin annular elastic plates

Dispersion relations are derived for the out-of-plane wave propagation in planar elastic plates with constant curvature using the classical Kirchhoff thin plate theory. The dispersion diagrams and the mode shapes are compared with their counterparts for a straight plate strip and the role of curvature is assessed for plates with unconstrained edges. Elementary Bernoulli–Euler theory for a beam of rectangular cross-section with the circular shape of its axis is also employed to analyze the wave guide properties of this structure in its out-of-plane deformation. The applicability range of the elementary beam theory is validated. The wave finite element method in the formulation of the three-dimensional elasticity theory is used to ensure that the comparison of dispersion diagrams is performed in the frequency range, where the classical thin plate theory is valid. Thus, the paper summarizes the effects brought to the propagation of out-of-plane waves in thin elastic plates by their constant curvature and the models of these plates.

The limitations on applying classical thin plate theory to thin annular plates clamped on the inner boundary

The experimentally measured resonance frequencies of a thin annular plate with a small ratio of inner to outer radii and clamped on the inner boundary are compared to the predictions of classical thin-plate (CTP) theory and a finite-element (FE) model. The results indicate that, contrary to the conclusions presented in a number of publications, CTP theory does not accurately predict the frequencies of a relatively small number of resonant modes at lower frequencies. It is shown that these inaccuracies are attributable to shear deformations, which are thought to be negligible in thin plates and are neglected in CTP theory. Of particular interest is the failure of CTP theory to accurately predict the resonance frequency of the lowest vibrational mode, which was shifted approximately 30% by shear motion at the inner boundary.

Finite element corotational formulation for geometric nonlinear analysis of thin shells with large rotation and small strain

Based on the consistent symmetrizable equilibrated (CSE) corotational formulation, a linear triangular flat thin shell element with 3 nodes and 18° of freedom, constructed by combination of the optimal membrane element and discrete Kirchhoff triangle (DKT) bending plate element, was extended to the geometric nonlinear analysis of thin shells with large rotation and small strain. Through derivation of the consistent tangent stiffness matrix and internal force vector, the corotational nonlinear finite element equations were established. The nonlinear equations were solved by using the Newton-Raphson iteration algorithm combined with an automatic load controlled technology. Three typical case studies, i.e., the slit annular thin plate, top opened hemispherical shell and cylindrical shell, validated the accuracy of the formulation established in this paper.

On an interaction of two elastic bodies: Analysis and algorithms

The paper deals with existence and uniqueness results and with the numerical solution of the nonsmooth variational problem describing a deflection of a thin annular plate with Neumann boundary conditions. Various types of the subsoil and the obstacle which influence the plate deformation are considered. Numerical experiments compare two different algorithms.

Effects of Misalignment of High Speed Flexible Coupling on the Fighter Aircraft Transmission Characteristics

The Fighter aircraft transmission system consists of a light weight, High Speed Flexible Coupling (HSFC) known as Power Take-Off shaft (PTO) for connecting Engine gearbox (EGB) with Accessory Gear Box (AGB). The HSFC transmits the power through series of specially contoured metallic annular thin flexible plates whose planes are normal to the torque axis. The HSFC operates at high speed ranging from 10,000 to 18,000 rpm. The HSFC is also catered for accommodating larger lateral and axial misalignment resulting from differential thermal expansion of the aircraft engine and mounting arrangement. The contoured titanium alloy flexible plates are designed with a thin cross sectional profile to accommodate axial and parallel misalignment by the elastic material flexure. This paper investigates the effect of misalignment on the transmission characteristics of the HSFC couplings. A mathematical model for the HSFC coupling with misalignment has been developed for analyzing the torque transmission and force interaction characteristics. An extensive testing has been conducted for validating characteristics of the designed coupling under various misalignment conditions. With this the suitability of the model adapted for the design iteration of HSFC development is validated. This method will reduce the design iteration cycles of HSFC and can be extended for the similar development of flexible couplings.

Fatigue Crack Growth Arrestor for High Speed Flexible Coupling of Fighter Aircraft Transmission Systems

A lightweight High Speed Flexible Coupling (HSFC) is used to connect the engine gear box (EGB) with an accessory gear box (AGB) for fighter aircrafts. The HSFC transmits the power at high speeds ranging from 10000 to 18000 rpm from the EGB to AGB. The HSFC also accommodates larger angular and axial misalignments resulting from differential thermal expansion of the aircraft engine and mounting arrangement. The HSFC utilizes series of metallic contoured annular thin flexible plates. These flexible plates are susceptible to fatigue failures in the presence of minuscule surface defects. The present work describes the analysis carried out on a typical HSFC with incipient flaw for various kinds of imposed misalignments. Based on the sensitivity study of the flaw on the stress levels, a Fatigue Crack Growth Arrestor (FCGA) is designed. In order to contain the crack propagation, drilling a hole at the tip of the crack on the flexible plate as a FCGA is adapted. The fatigue life of the flexible plates is improved by finding appropriate size and location of FCGA through finite element analysis and simulations.

Non-resonant structure design of catenary transformer in ultrasonic gear honing

To solve the design problem of transformer composed of non-resonant structure in ultrasonic gear-honing, force coupling conditions for moderately thick annular plate (MTAP) and catenary horn are proposed, and the frequency equations of the transformer, which consist of an MTAP and a catenary horn, are derived based on Mindlin’s theory. The design parameters of the transformer were obtained by solving the frequency equations with the help of MATLAB, and its mode and frequency were deduced by the modal analysis of finite element method (FEM), which are consistent with the theoretical design demands. The transformer design can be extended from system with thin annular plate to that with MTAP as the theoretical method by the simulation analysis of various ratios of the thickness to radius.

Determination of residual stress in deep submicron thick films by the critical buckling technique of annular thin plates

Critical buckling of annular plates has been proposed to measure residual stress in thin films with submicron thickness and succeeds in measuring stress ranging from decades of MPa to hundreds of MPa. In this paper, critical buckling of annular plates was further researched in order to enhance their sensitivity, and improved structures were proposed to measure ultra small residual stress in scale of several MPa. Specimens of improved annular structures with thickness down to 200 nm were prepared on silicon-on-insulator wafer, and released in hydrofluoric HF solution to get the critical etching length under an on line monitoring system. The critical etching length was inputted into the ANSYS software to predict the residual stress by the eigenvalue buckling analysis. Small compressive stress of 3.07 MPa within 100 nm film was resolved successfully with this improved annular structure, but the micro rotating structures fabricated in the same die cannot generate a distinguishable deformation.

Exact analytical solution for free vibration of functionally graded thin annular sector plates resting on elastic foundation

This article introduces an exact analytical method for free vibration analysis of functionally graded (FG) thin annular sector plates resting on Winkler and Pasternak elastic foundations. The annular sector plate has simply supported radial edges and arbitrary boundary conditions along the circular edges. Based on the displacement field of Kirchhoff plate theory, the governing equations of motion are obtained considering the in-plane displacements and rotary inertia. Using a set of functions, the three coupled governing equations of motion are converted into two decoupled equations. By applying the boundary conditions at inner and outer radii, an eigenvalue problem for finding the natural frequencies is obtained. The nine distinct cases are considered involve all possible combinations of boundary conditions along the circular edges. Accurate non-dimensional frequency is presented for over a wide range of sector angles, some inner to outer radii (aspect ratio) and different powers of functionally graded material. Accurate natural frequencies of FG annular sector plates resting on elastic foundations are presented for the first time and can be used as reference values for numerical analyses.

The Higher Frequency of Free Vibrations of Thin Annular Plates Made of Functionally Graded Material

AbstractSubject of this paper is a thin plate with a characteristic geometry: periodic in one direction and smoothly varying along the other. The aim of the contribution is to formulate and apply the averaged model which can describe higher frequencies of free vibrations. Problem of finding frequencies of free vibrations is very important. It could be applied to many engineering problems such as resonance phenomena, wave propagation, absorption of vibrations and many others. When the considered plate is made of an isotropic material, we can find the first, the second and the others frequencies in simply way. But if we consider plate made of a functionally graded material [1], which have varying properties and which is made of two components, this problem is more complicated. In this cause, apart from family of the base frequencies (first, second, etc), which depend on macroscopic properties of the plate, we have the higher frequency, which depends on a microscopic structure. We can find many papers describing the base frequencies of free vibrations for many different types of structures (for example [2] for considered type of plate). In this paper, we find the higher frequency. For solving this problem, we use the tolerance averaging technique described in [3]. This theory allows to take into account the microstructure size and to find the higher frequency of free vibrations. The equations have smooth coefficients. They can be solved numerically with help of the finite difference method, in polar coordinates for an annular plate. Next, we use special procedure for selecting of the higher frequencies of free vibrations, which depend on the microstructure size, from the list of all frequencies. After that we analyze an influence of ratios of material properties and the microstructure size on the higher frequency of free vibrations. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Natural vibration of circular and annular thin plates by Hamiltonian approach

The present paper deals with the natural vibration of thin circular and annular plates using Hamiltonian approach. It is based on the conservation principle of mixed energy and is constructed in a new symplectic space. A set of Hamiltonian dual equations with derivatives with respect to the radial coordinate on one side of the equations and to the angular coordinate on the other side are obtained by using the variational principle of mixed energy. The separation of variables is employed to solve Hamiltonian dual equations of eigenvalue problem. Analytical frequency equations are obtained based on different cases of boundary conditions. The natural frequencies are the roots of the frequency equations and corresponding mode functions are in terms of the dual variables q 1( r, θ). Three basic edge-constraint cases for circular plates and nine edge-constraint cases for annular plates are calculated and the results are compared well with existing ones.

Free vibration analysis of functionally graded thin annular sector plates using the differential quadrature method

In this article, the differential quadrature method (DQM) is used to study the free vibration of functionally graded (FG) thin annular sector plates. The material properties of the FG-plate are assumed to vary continuously through the thickness, according to the power-law distribution. The governing differential equations of motion are derived based on the classical plate theory and solved numerically using DQM. The natural frequencies of thin FG annular sector plates under various combinations of clamped, free, and simply supported boundary conditions 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. In this case, the result shows very good agreement. For the FG-plates, the effects of boundary conditions, volume fraction exponent, and variation of Poisson's ratio on the free vibrational behaviour of the plate are studied.

Global asymptotic approximations for wrinkling of polar orthotropic annular plates in tension

The role of polar orthotropy is investigated in relation to an elastic wrinkling problem for annular thin plates in tension. An appealing feature of this generalised constitutive framework is the possibility of allowing the radial and circumferential directions to have different bending stiffnesses. By suitably rescaling the buckling equation, three main parameters are identified as being responsible for the behaviour of the linear wrinkling mechanism. We conduct a number of singular-perturbation analyses in order to capture the dependence of the critical loads and mode numbers on the various physical parameters that enter in the description of the problem.

Free vibrations of thin annular plates made from functionally graded material

AbstractSubject of the consideration is thin annular plate made of a two‐phase functionally graded composte. The plate has periodically inhomogeneous microstructure slowly varying in space: the λ‐periodic structure along circular coordinate, but smoothly graded apparent (averaged) properties in the perpendicular, radial direction. The aim of the contribution is to derive and apply a deterministic macroscopic model describing the free vibrations of this plate. Modeling procedure is based on tolerance averaging technique. We received, equations system with smooth coefficients. We made numerical solution of this problem, using finite difference method, and analyze influence of material proportion and microstructure size on first frequency of free vibrations. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the strain-stress state of the reconstructed middle ear after inserting a malleus-incus prosthesis

The reconstructed middle ear (RME) consists of a thin annular plate made from cartilage, the intact stapes and a T-like prosthesis replacing the malleus-incus chain and interposed between the stapes and the reconstructed eardrum. Inserting the prosthesis leads inevitably to an initial pretension in the reconstructed ear. A high pretension can lead to a stiffening of the ligaments and a significant shift of the eigenfrequency spectrum of the middle-ear system. Only a weak pretension is required for satisfactory sound transmission. This paper presents a static model of the RME permitting one to estimate the initial strain-stress state of the sound-conducting system after insertion of a prosthesis. Several positions of the prosthesis with respect to the eardrum are considered. The anisotropic stiffness of the stapes annular ligament is represented by a fully occupied stiffness matrix. The equilibrium of the modelled system is described by four vector equations. In the case of small eccentricity of attachment of the prosthesis plate to the eardrum, the perturbation method and the Fourier series are used for solving the plate equation. Otherwise, the finite-difference method is applied. The influence of the geometrical dimensions of the inclined prosthesis and its position on the stresses in the prosthesis-stapes coupling as well as on the pretension of the annular ligament are analysed. Analysis of the calculations performed allows one to conclude that in order to avoid high pretensions and exclude possible luxation of the stapes, the prosthesis should be positioned as close to the centre of the reconstructed eardrum as possible.