Annular Elements Research Articles

Vibration of Hemi-Ellipsoidal Axisymmetric Domes Submerged in Water

Ten thin-walled domes, of hemi-ellipsoidal form, were vibrated in air and while partially and fully submerged in water. The domes varied in shape from oblate ones of aspect ratio (AR) 0.25 to prolate ones of aspect ratio 4. The fundamental modal patterns for the oblate domes tended to be of axisymmetric form, while the fundamental modal patterns of the hemispherical and prolate domes tended to be of asymmetric or lobar form. The theoretical analysis was carried out by the finite element method, where the motion of the shell structure was represented by three different types of element and the motion of the water was represented by a solid annular element. Comparison between theory and experiment was found to be good.

Vibration of annular plates partially treated with unconstrained damping layer

Mass and stiffness matrices of an annular element consisting of base plate and unconstrained damping layer have been derived assuming a modal solution to the equation of motion of the plate. The complex eigen equations have been solved for frequencies and loss factors by an extension of the simultaneous iteration technique. Frequencies of unlayered plates and loss factors of fully layered plates have been compared with those available in literature. The effect of staggering of layers on the vibration characteristics of the plate has been investigated. It is found that when the plate is partially layered starting from its inner edge the loss factors will be higher for all boundary conditions and modes than when the plate is fully layered provided the mass of the damping layer is kept the same in both cases.

Nearfield calibration array utilizing piezoelectric polymer

The Trott nearfield calibration array (NFCA) concept allows the farfield acoustic properties of underwater transducers to be determined in measurement facilities that are too small for direct farfield measurements. As a projector, the NFCA produces a nearly uniform plane wave throughout a large adjacent volume. The farfield sensitivity of a receiver can then be measured directly by placing it in the “plane‐wave” volume. As a receiver, the NFCA measures only a single plane‐wave component of the sound radiated by a projector located in the “plane‐wave” volume. Existing NFCA's are primarily used as receivers and utilize small piezoceramic hydrophone elements arranged in planar or cylindrical configurations. The spacing between elements limits the upper frequency of operation of the NFCA; e.g., a 1‐in. spacing allows operation up to about 50 kHz. This paper describes the design and construction of a planar NFCA utilizing a mosaic of nonelectroded sheets of the piezoelectric polymer PVDF attached to full area electrodes that subdivide the array into independently controlled annular elements. The PVDF NFCA was tested successfully as both a projector and a receiver by calibration of a high‐frequency transducer. Although the available electronics limited the upper frequency of the test to 150 kHz, the PVDF NFCA should be useable, with suitable electronics, up to megahertz frequencies. [Work partially supported by ONR Code 220.]

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.

Effects of space charge and lens aberrations in the focusing of an electron beam by a solenoid lens

The effects of space charge and lens aberrations on a 5-keV, 80–200-mA electron beam focused by a solenoid lens are investigated theoretically and experimentally. A third-order radial trajectory equation is derived with space charge added in the form of discrete annular current elements. This equation is numerically solved for the trajectories of focused electrons in a short solenoid lens. Trajectories launched near the axis are ‘‘reflected’’ from the axis at the waist due to space charge, and trajectories launched farther from axis, cross the axis. The resulting discontinuity is best illustrated in a transverse phase-space plot where it may be described as a ‘‘tearing’’ of the electron distribution. Due to the combined action of the nonlinear lens forces and the space charge, the initially uniform radial current density distribution becomes markedly nonuniform (hollow or peaked depending on distance) downstream from the lens. Detailed experimental measurements of the density profiles at different positions are found to be in very good agreement with computations.

Changes in Earth Stresses Around a Wellbore Caused by Radially Symmetrical Pressure and Temperature Gradients

Abstract Pressure and temperature gradients are created around wellbores during waterflooding or when fluids are injected in connection with any other secondary or tertiary recovery process. These gradients result in changes in earth stresses, which in turn cause hydraulic fracturing pressures to change. In this paper, analytical solutions have been used to determine the stresses resulting from radially symmetrical temperature and pressure changes around a wellbore. These stresses are required to predict the change in fracture extension pressure that is caused by the injection process. Exact, closed-form solutions are given for the stresses. These have been evaluated with a computer, and more convenient empirical formulas have been fitted to the calculated results. Solutions for discrete cylindrical or disk-shaped regions of changed temperature and pressure are shown. Also, the solutions can be adapted to annular elements of finite thickness that are convenient for incorporation to an r-z-type computer program. Such a program could then be used to compute the stresses resulting from temperature and pressure fields that vary gradually in the radial direction. This paper gives examples to illustrate the effect of injecting a large volume of liquid that is cooler than the insitu reservoir, as is common when waterflooding. The cooling can have a large effect on lateral earth stresses, and for some conditions vertical hydraulic fracturing pressures can be significantly reduced.

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.

The usefulness of static condensation in the finite element analysis of stiffened submersible cylindrical hulls

The usefulness of the static condensation technique in the finite element analysis of stiffened submersible. cylindrical hulls is examined in this paper. The finite element formulation used herein is essentially the same as outlined by the authors in an earlier paper wherein the stiffener is modeled rigorously using axisymmetric thin annular plate elements for the web and axisymmetric thin shell elements for the flange. The static condensation technique has been applied in this paper to reduce these stiffener finite elements so that their effect can be transferred to the shell node at the point of attachment of the stiffener with the shell. The advantage of such condensation of the stiffener elements is the smaller number of equations to be solved without the rigor of the stiffener modeling being lost in any way. The manner of incorporating the condensation in the computer program has been described. Examples of several stiffened submersible cylindrical hulls have been considered as an illustration of the use of the program.

Vibration of submerged hemi-ellipsoidal domes

Ten thin-walled hemi-ellipsoidal domes were vibrated in air and also underwater. The domes varied in shape from oblate hemi-ellipsoids to prolate ones, and the fundamental eigenmodes were either axisymmetric as asymmetric, depending on the profiles of the domes. A varying meridional curvature annular element was developed for the shell and a similar one for the fluid, except that the latter was of quadrilateral cross-section. For submerged vibration, the two finite elements were coupled together and solution carried out through simultaneous iteration. Theoretical analyses with these structural and fluid finite elements gave satisfactory results compared to the experiments.

Vibration of thermally stressed polar orthotropic annular plates

AbstractThis study sets out to analyse the effect of uniform temperature on the natural frequencies of transverse vibration of uniform polar orthotropic annular plates. The method employed is that of the annular finite element technique whereby the flexural and membrane components of the solution are separated. By close examination of the form of the resulting in‐plane stress distribution due to the thermal loading, it has been possible to obtain results for a wide selection of specimens by way of one equation and three graphs.

Determination of the conditions of formation of annular coatings with a linear dependence of thickness on rotation angle

The valuable characteristics of such elements are the case of tuning them from one spectral region to another, which is accomplished by means of a simple physical rotation, and the simplicity of calibration. The main optical parameters (width of the spectral working region, dispersion, resolving power, and so on) of such elements depend in a decisive way on the profile of the annular coatings, and in light of this knowledge of the conditions of formation of annular coatings with different profiles is equivalent to knowledge of the conditions of producing annular optical elements with different optical parameters.

Vibration and stability analysis of polar orthotropic uniform discs subjected to peripheral hydrostatic loading

An analysis based upon the annular finite element method is carried out of the transverse vibration and stability of uniform polar orthotropic discs subjected to peripheral hydrostatic loading. Both the axi- and anti-symmetrical modes are considered for both clamped and simply supported at the outer peripheral radius. By separation of the flexural and geometric (membrane) components, the effect on both with respect to the degree of orthotropy is plotted and subsequently both components can be easily combined to produce the natural frequencies of the loaded disc. A simple relationship is derived to determine whether the lowest critical buckling mode is that containing zero nodal diameters and circles or some higher mode.

Vibration of Thick Circular Plate--Determination of Elastic Properties Through the Resonant Behaviour

The authors examine, from two complementary points of view, the main problem deriving from the necessity of deducing elastic properties of materials by considering the resonant condition of transversely vibrating discs, that is the determination of the efficiency at high frequencies of finite elements formulated with the assumptions of the thick plate theory. The first approach consists, having standardized the basic relations for various thick annular semi-analytical finite elements, in testing convergence and correspondence to known analytical solutions. The second consists in the experimental evaluation of the influence of thickness in deducing the Young’s modulus of a series of polycarbonate resin discs at frequencies corresponding to modes with up to eight nodal circles.

Finite Element Analysis of the Buckling of Variable Thickness Discs

This paper demonstrates the application of the annular finite element method to predicting the critical buckling loads of variable thickness discs. Provision is made in the analysis to accommodate discs made of materials exhibiting polar orthogonility. Results are compared with those obtained using altenative techniques for various cases of uniform isotropic and orthotropic specimens and variable thickness isotropic specimens.

Natural frequencies of transverse vibration of polar orthotropic variable thickness annular plates

The method of annular finite elements is applied to the case of varying thickness annular discs constructed of materials exhibiting polar orthotropic characteristics. The numerical convergence of the method is tested and comparisons are made with the corresponding results obtained in other studies. The method is applied specifically to polar orthotropic annular discs of linearly varying thickness profile for various transverse boundary conditions, aspect ratios and values of the parameter describing the thickness profile. The results are plotted and conclusions drawn.

Experimental study of residual stresses in thin-walled rotational shells made of composite materials by means of the annular element removal method

Experimental study of residual stresses in thin-walled rotational shells made of composite materials by means of the annular element removal method

A simplified design method and wind-tunnel study of horizontal-axis windmills

The paper presents a method of desingning blade geometry of a horizontal-axis medium-speed windmill using blade element theory and an inversion technique which is simplified by defining a local speed ratio at annular blade elements. This approach eliminates iterative computation, and a unique relationship between the local speed ratio and the inflow angle of resultant fluid velocity is established for optimum power coefficient of any blade element. The theoretical design method and the experimental results obtained in a wind tunnel with windmill models of 1 m diameter having different number of blades, twist angle of blade, angle of incidence, tip speed ratio and blade profiles are presented and discussed. Two simple methods of construction of the windmill blades are described briefly and the performance of these blades is presented.

Vibration analysis of polar orthotropic discs using the transfer matrix method

The well known transfer matrix scheme of Ehrich for calculating the vibration characteristics of uniform and varying thickness discs is extended to include polar orthotropic material behaviour. The exact deflection equations for an annular element of material are used in the solution. Comparisons with alternative solutions in the literature show excellent agreement for both annular and complete discs of uniform and varying thickness profile.

Stability and vibration of an annular plate with concentrated edge load

Buckling and free vibration of an annular plate with clamped inner boundary and a concentrated, in-plane edge load at the outer boundary are studied using the semi-analytical Finite Element Method. The annular plate elements developed by Pardoen are used for this purpose. Exact in-plane stress field is obtained for various harmonics. Geometric stiffness matrices are developed which are general and can handle any in-plane stress field once its Fourier expansion is known. Results indicate that theoretical buckling loads and frequencies of free vibration are considerably lower than the values reported in earlier analyses.

Bending of an annular plate in the presence of a concentrated, in-plane edge load

Bending behaviour of an annular plate with clamped inner boundary and a concentrated edge load at the outer boundary is studied using the semi-analytical finite element method. The annular plate elements developed by Pardoen are used for this purpose. Exact in-plane stress field is obtained for various harmonics. Geometric stiffness matrices are developed which are general and can handle any in-plane stress field once its Fourier expansion is known. Results are presented for two radii ratios and for chosen values of the in-plane load.