Annular Finite Element Research Articles

Analysis of a shell of revolution subjected to axisymmetric loading taking into account geometric nonlinearity on the basis of the mixed finite element method

An algorithm is presented to obtain the deformation matrix for the annular finite element with a quadrilateral cross-section. The displacement and strain increments are taken as nodal unknowns. For numerical realization of the algorithm, we take the functional obtained using the principle of equal works for the internal and external forces at a step of loading. A newly developed version of approximating the unknown quantities allowed us to solve the problem of taking into account the finite element displacement as a rigid body.

Axisymmetric dynamic instability of polar orthotropic sandwich annular plate with ER damping treatment

The axisymmetric dynamic instability of polar orthotropic sandwich annular plate combined with electrorheological (ER) fluid core layer and constraining layer are studied in this paper. And, the ER core layer and constraining layer are used to improve the stability of the annular plate system. The boundaries of instability regions for the polar orthotropic sandwich annular plate system are obtained by discrete layer annular finite element and the harmonic balance method. The rheological property of an electrorheological material, such as viscosity, plasticity, and elasticity can be controlled by applying different electric field strength. Thus, the damping characteristics of the sandwich system are more effective when the electric field is applied on the sandwich structure. Additionally, variations of the instability regions for the polar orthotropic sandwich annular plate with different applying electric field strength, thickness of ER layer and some designed parameters are investigated and discussed in this study.

Thermoelastic damping in rotating flexible micro-disk

A rotating flexible annular micro-disk under thermoelastic coupling is modeled to investigate the thermoelastic damping in the rotating micro-disk in Microdrives. Transverse deflection of the micro-disk is considered as a function of both disk radial and circumferential coordinates, and its thermoelastic temperature increment is modeled by simultaneously involving disk thickness, radial and circumferential coordinates. Heat convection boundaries are assumed at disk top and bottom surfaces. System governing equations with the centrifugal stresses of disk rotation and thermoelastic coupling terms are discretized along disk radial direction by an annular finite element, and a quadratic eigenvalue problem is yielded to determine the thermoelastic damping. Mode shapes of the transverse deflection and temperature increment are presented, effects of disk rotation, surface convection and ambient temperature on natural frequency, frequency shift and thermoelastic damping are investigated. The maximum thermoelastic damping is obtained by optimizing disk thickness, inner radius and radial width.

Parametric resonance of axisymmetric sandwich annular plate with ER core layer and constraining layer

The parametric resonance problems of axisymmetric sandwich annular plate with an electrorheological (ER) fluid core and constraining layer are investigated. The annular plate is covered an electrorheological fluid core layer and a constraining layer to improve the stability of the system. The discrete layer annular finite element and the harmonic balance method are adopted to calculate the boundary of instability regions for the sandwich annular plate system. Besides, the rheological property of an electrorheological material, such as viscosity, plasticity, and elasticity can be changed when applying an electric field. When the electric field is applied on the sandwich structure, the damping of the sandwich system is more effective. Thus, variations of the instability regions for the sandwich annular plate with different applying electric fields, thickness of ER layer, and some designed parameters are presented and discussed in this study. The ER fluid core is found to have a significant effect on the location of the boundaries of the instability regions.

Free vibration analysis of rotating polar orthotropic annular plate with ER damping treatment

The free vibration analysis of rotating polar orthotropic sandwich annular plate with electrorheological (ER) fluid core layer is studied in this study. By using discrete layer annular finite element method, the equation of motion for the rotating polar orthotropic sandwich system is derived. The ER fluid is utilized to control the vibration and damping characteristics of the rotating polar orthotropic sandwich annular plate system. Additionally, the extensional and shear modulus of ER fluid are described by complex quantities. And, the natural frequencies and loss factors of the system are obtained by solving the complex eigenvalues of the system. Natural frequency and modal loss of the rotating polar orthotropic sandwich annular plate with different designed parameters are calculated and discussed in this study.

Dynamics behavior of rotating bladed discs: A finite element formulation for the study of second and higher order harmonics

Annular finite elements for the computation of second and higher order harmonics modes of bladed rotating discs are developed. The elements take into account gyroscopic effect and stiffening due to centrifugal and thermal stresses (the latter not present in arrays of blades). The displacement field is expressed by a truncated Fourier series along the angle and by polynomial shape functions in the radial direction. This paper is the generalization of a previous study limited to zero- and first-order harmonics and deals only with second and higher order harmonics modes that are uncoupled from the modes involving the behavior of the rotor as a whole. Several cases have been studied to verify the accuracy of the disc and array of blades elements.

Axisymmetric vibration and damping analysis of rotating annular plates with constrained damping layer treatments

The axisymmetric vibration and damping of rotating annular plates with constrained damping layer treatments are analyzed in the present paper. The discrete layer annular finite element method is employed to derive the equations of motion for the rotating composite plate where the geometry stiffness matrix induced by rotation is evaluated from solutions of static problems. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The effects of stiffness and thickness of the viscoelastic and constraining layers on the natural frequencies and modal loss factors of rotating composite plates are presented. Also, the composite plates with various boundary conditions and partial treatments are discussed.

Vibration and damping analysis of annular plates with constrained damping layer treatments

The natural frequencies and modal loss factors of annular plates with fully and partially constrained damping treatments are considered. The equations of free vibration of the plate including the transverse shear effects are derived by a discrete layer annular finite element method. The extensional and shear moduli of the viscoelastic material layer are described by the complex quantities. Complex eigenvalues are then found numerically, and from these, both frequencies and loss factors are extracted. The effects of viscoelastic layer stiffness and thickness, constraining layer stiffness and thickness, and treatment size on natural frequencies and modal loss factors are presented. Numerical results also show that the longer constrained damping treatment in radial length does not always provide better damping than the shorter ones.

Axisymmetric dynamic stability of sandwich circular plates

The axisymmetric dynamic stability of sandwich circular plates with a constrained damping layer treatment subjected to a periodic uniform radial loading along the outer edge of the host plate is studied by the discrete layer annular finite element method. The viscoelastic material with frequency dependent properties in the damping layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. The regions of dynamic instability are determined by Bolotin’s method while two eigenvalue problem systems with frequency dependent parameters are formed. The modified complex eigensolution method is employed to solve the eigenvalue problems. Numerical results are shown that the constrained damping layer attached to circular plates tends to stabilize the circular plate system. The effects of various parameters, such as types of viscoelastic material response, treatment thickness and area, on dynamic stability are also investigated.

Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer

The natural frequencies and modal loss factors of the three-layered annular plate with a viscoelastic core layer and two polar orthotropic laminated face layers are considered. The discrete layer annular finite element is employed to derive the equations of motion for the three-layered annular plate. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by the complex quantities. Complex eigenvalued problems are then solved, and the frequencies and modal loss factors of the composite plate are extracted. The results of the symmetric and non-symmetric composite annular plates are both presented. The effects of material properties, radius to thickness ratio, stacking sequences and thickness of face layers, and thickness of the viscoelastic core layer are discussed.

NATURAL FREQUENCIES OF TRANSVERSE VIBRATIONS OF NON-UNIFORM CIRCULAR AND ANNULAR PLATES

This paper presents a method for the free transverse vibration analysis of thin, elastic, isotropic, uniform and non-uniform circular and annular plates. The circumferential mode numbers (n=0) and (n=1) are dealt with in this paper. The method is a hybrid of plate theory and finite element analysis. The plate is subdivided into one circular and many annular finite elements. Two new finite elements were developed, the first type being a circular plate and the second an annular plate, the displacement functions of the finite element model are the classical solution shape functions of plate theory. Mass and stiffness matrices are determined by precise analytical integration. The free vibrations of uniform circular and annular plates are studied by this method as well as non-uniform plates. The results obtained reveal that the frequencies calculated by this method are in good agreement with those obtained by other authors. This method combines the advantages of the standard finite element analysis and the high-accuracy formulation provided by the use of displacement functions derived from plate theory instead of the usual low-order polynomials. The present method is remarkable for the fact that it enables us to determine with equal precision both low and high natural frequencies.

Finite Element Analysis of the Lateral Vibration of Thin Annular and Circular Plates With Variable Thickness

The method of annular finite elements with variable thickness is applied for analyzing the lateral vibration of thin annular and circular plates. The material of the plates may be of isotropic or polar orthotropic and the plate thickness may vary arbitrarily with the radius. Natural frequencies and mode shapes of the axisymmetric and nonaxi-symmetric modes are obtained. The numerical convergence of the method has been tested and comparisons have been made with the results obtained in other studies. It has been proved that the convergence of this method is very rapid and obtained results are very accurate.

An alternative pseudoperipheral node finder for resequencing schemes

AbstractMany resequencing algorithms for reducing the bandwidth, profile and wavefront of sparse symmetric matrices have been published. In finite element applications, the sparsity of a matrix is related to the nodal ordering of the finite element mesh. Some of the most successful algorithms, which are based on graph theory, require a pair of starting pseudoperipheral nodes. These nodes, located at nearly maximal distance apart, are determined using heuristic schemes. This paper presents an alternative pseadoperipheral node finder, which is based on the algorithm developed by Gibbs, Poole and Stockmeyer. This modified scheme is suitable for nodal reordering of finite meshes and provides more consistency in the effective selection of the starting nodes in problems where the selection becomes arbitrary due to the number of candidates for these starting nodes. This case arises, in particular, for square meshes. The modified scheme was implemented in Gibbs‐Poole‐Stockmeyer, Gibbs‐King and Sloan algorithms. Test problems of these modified algorithms include: (1) Everstine's 30 benchmark problems; (2) sets of square, rectangular and annular (cylindrical) finite element meshes with quadrilateral and triangular elements; and (3) additional examples originating from mesh refinement schemes. The results demonstrate that the modifications to the original algorithms contribute to the improvement of the reliability of all the resequencing algorithms tested herein for the nodal reordering of finite element meshes.

Axisymmetric parametric stability of composite annular plates

The problem of axisymmetric parametric stability of composite polar orthotropic circular and annular plates is analyzed, by using annular finite elements. The instability regions are found out by adopting Bolotin's procedure. Results of a detailed parametric study are reported for bilayered plates of clamped and simply supported boundaries, which show the effects of radii ratio, layer disposition etc., on the dynamic stability.

Axisymmetric Nonlinear Dynamic Response of Bimodulus Annular Plates

In the present study, the geometrically nonlinear dynamic response of bimodulus annular and circular plates is obtained. The governing equations of the problem are formulated using the energy method and are solved by using annular finite elements spacewise. The integration in the time domain is accomplished by the Wilson θ method. Numerical work has been done for different hole sizes under various edge conditions and loadings.

Dynamic Response of Sandwich Shells of Revolution

Development of an annular finite element, for the linear elastic analysis of sandwich shells, is reported here. The derivation of stiffness and mass matrices is based on improved shell theory which takes into account the effects of rotary inertia and transverse shear deformation. Flexural rigidity of the faces is included in the formulation. The core of the sandwich shell is assumed to be incompressible in the radial direction. Numerical examples of spherical sandwich shells with two types of boundary conditions—(i) fixed and (ii) pinned along the outer periphery—have been presented. The results are generated for displacements and frequencies.

Axisymmetric buckling and post-buckling of bimodulus annular plates

The buckling and post-buckling analysis of bimodulus circular and annular plates is presented. Annular finite elements are used to solve the problem. In the case of bimodulus materials, the elastic constants in tension and compression are different. The constitutive matrix based on the ‘Jones model’ is used for the analysis. The post-buckling problem is solved as an eigen-value problem with deflexion at a point, as an independent variable. Numerical results are compared for particular cases.

Finite element analysis of thick annular plates under internal forces

The axisymmetric vibration and stability of thick annular plates under internal forces are studied by the finite element method. The internal forces are taken to be the combination of a uniform compressive stress and a pure bending stress. An annular finite element based on a high-order plate theory is used. It has three nodes and 21 degrees of freedom. Accurate estimates of natural frequencies and buckling loads are obtained. The influences of some parameters are studied.

Thermal gradient effects upon the vibrations of certain composite circular plates, Part I: Plane orthotropic

The effect of a radial parabolic temperature distribution upon the natural frequencies of small free transverse vibration (axi-and non-axisymmetric) of polar orthotropic circular plates is considered. Although the problem is solved primarily by means of the annular finite element method, close examination of the exact form of the thermal stress distribution, and utilization of the Lamb and Southwell approximation, enables one to present the results in a very efficient form.

Efficiency of annular finite elements for flexural vibrations of thick disks

The efficiency is evaluated of various semianalytical annular finite elements in predicting the vibrational modes for disks where the effect of transverse shear and rotary inertia becomes important, with reference both to the minimum number of degrees of freedom and to the minimum number of operations required to achieve a certain approximation for the exact natural frequency value. From this point of view, a new element, in which normal displacements are interpolated by cubic polynomials and rotations by parabolic polynomials along the radius, is compared with other elements described in the literature.