Spherical Shell Segments Research Articles

Semi-analytical vibration modeling of complex axisymmetric shells using shifted Legendre series

This paper presents a semi-analytical approach based on the shifted Legendre series for vibration modeling of complex axisymmetric shells. This method divides the structure along its generator into complex axisymmetric shells comprised solely of conical and spherical shell segments. A global cylindrical coordinate system is established to unify the coordinate system of complex axisymmetric shells. Coupling between segments and the simulation of arbitrary boundary conditions are achieved using artificial spring technology. Based on the first-order shear deformation theory (FSDT), the energy functional of each segment is derived. The circumferential displacement is expanded using Fourier series, while the meridional displacement is expanded using the shifted Legendre series at shifted Gauss-Lobatto nodes. Utilizing the inner product matrix and differential matrix of the shifted Legendre series, the integration and differentiation processes in the vibration characteristic equation are simplified into a product form, significantly enhancing computational efficiency. Through convergence analysis and comparison of the computational results of the present method with numerical simulation results, experimental results and literature results demonstrate the advantages of high convergence, high accuracy and fast solution speed of this solution method. On this basis, the vibration characteristics of ring-stiffened double-layer hemispherical shells are investigated. The results demonstrate the great applicability of this method for vibration modeling of complex axisymmetric shells in engineering.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.

Geometrically nonlinear vibration analysis of eccentrically stiffened porous functionally graded annular spherical shell segments

This article investigates nonlinear free vibrations of porous functionally graded (FG) annular spherical shell segments surrounded by elastic medium and reinforced by circumferential stiffeners. Porous FG material contains distributed even and un-even porosities and is modeled based on refined power–law function. The governing equations of stiffened porous annular spherical shell segments have been derived according to thin shell theory with the geometrical nonlinear in von Karman–Donnell sense and the smeared stiffeners method. An analytical trend has been provided for solving the nonlinear governing equations. Obtained results demonstrate the significance of porosity distribution, geometric nonlinearity, foundation factors, stiffeners and curvature radius on vibration characteristics of porous FG annular spherical shell segments.

Scope of inextensible frame hypothesis in local action analysis of spherical reservoirs

Spherical reservoirs, as objects perfect with respect to their weight, are used in spacecrafts, where thin-walled elements are joined by frames into multifunction structures. The junctions are local, which results in origination of stress concentration regions and the corresponding rigidity problems. The thin-walled elements are reinforced by frame to decrease the stresses in them. To simplify the analysis of the mathematical model of common deformation of the shell (which is a mathematical idealization of the reservoir) and the frame, the assumption that the frame axial line is inextensible is used widely (in particular, in the manual literature). The unjustified use of this assumption significantly distorts the concept of the stress-strain state. In this paper, an example of a lens-shaped structure formed as two spherical shell segments connected by a frame of square profile is used to carry out a numerical comparative analysis of the solutions with and without the inextensible frame hypothesis taken into account. The scope of the hypothesis is shown depending on the structure geometric parameters and the load location degree. The obtained results can be used to determine the stress-strain state of the thin-walled structure with an a priori prescribed error, for example, in research and experimental design of aerospace systems.

A unified accurate solution for vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints

A unified accurate solution procedure for free vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints is presented. The material properties of the spherical shells are assumed to change continuously in the thickness direction and two different four-parameter power-law distributions are considered. The proposed method is formulated by the Ritz procedure on the basis of the first-order shear deformation shell theory. Each of admissible functions, regardless of boundary conditions, is composed of a standard Fourier cosine series and several auxiliary functions introduced to ensure and accelerate the convergence of series representations. The accuracy and reliability of the current solution are validated by comparing the results with existing results and those generated from the finite element analyses, and numerous new results for functionally graded spherical shells subjected to elastic restraints are presented, which can serve as the benchmark solutions for other computational techniques in the future research. The effects of the boundary conditions, power-law exponents, and shell segments on the free vibrations of the spherical shells are also investigated, and some interesting insights into the parameter effects on frequency behaviors are illustrated.

Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions

Based on the Reissner-Naghdi’s thin shell theory, this paper concentrates on the free vibration of a joined cylindrical-spherical shell with elastic-support boundary conditions by a domain decomposition method. The joined shell is first separated from the elastic-support boundary and then subdivided into some cylindrical and spherical shell segments along the axis of revolution. The elastic-support boundary is regarded as a combination of distributed linear springs and can be treated as a special interface as well as the interface between two adjacent shell segments. Through the variational operation of the whole energy functional, the discretized equations of motion are derived by the expansions of the displacement field for each shell segment with Fourier series and Chebyshev orthogonal polynomials in the circumferential and longitudinal direction, respectively. To analyze the numerical convergence and precision of the present method, a number of case studies have been conducted and the solutions are compared with those derived by ANSYS and those presented in the previous literature to confirm the reliability and accuracy.

Vibration characteristics of a spherical–cylindrical–spherical shell by a domain decomposition method

A domain decomposition method is used to analyze the free and forced vibration characteristics of a spherical–cylindrical–spherical shell, based on Reissner–Naghdi's thin shell theory. The joined shell is divided into some cylindrical and spherical shell segments along the meridional (longitudinal) direction. Double mixed series, i.e., Fourier series and Chebyshev polynomials, are employed as the admissible displacement functions to obtain the discretized equation of motion for the joined shell. Numerical comparisons with the results derived by FEM and those available in the previous literature are made to validate the present method. Moreover, the effects of length-to-radius and radius-to-thickness ratios on the natural frequencies are also investigated.

A Domain Decomposition Method for Forced Vibration Analysis of Joined Conical-Cylindrical-Spherical Shell

Based upon the Reissner-Naghdi-Berry shell theory, a semi-analytical domain decomposition method is presented to analyze the forced vibration of a joined conical-cylindrical-spherical shell with general boundary conditions. The joined shell was divided into some conical, cylindrical and spherical shell segments along the axis of revolution. The constraint equations derived from interface continuity conditions between two adjacent shell segments were introduced into the energy functional of the joined shell. Displacement variables of each shell segment are expressed as a mixed double series in the forms of Fourier series in the circumferential direction and Chebyshev orthogonal polynomial in the longitudinal direction. The forced vibration response of the joined shells subjected to various harmonic excitations and boundary conditions was calculated and compared with those FEM results obtained by finite element software ANSYS to confirm the reliability and accuracy of this analytical solution.

Modeling the Near-Surface Shear Layer: Diffusion Schemes Studied With CSS

As we approach solar convection simulations that seek to model the interaction of small-scale granulation and supergranulation and even larger scales of convection within the near-surface shear layer (NSSL), the treatment of the boundary conditions and minimization of sub-grid scale diffusive processes become increasingly crucial. We here assess changes in the dynamics and the energy flux balance of the flows established in rotating spherical shell segments that capture much of the NSSL with the Curved Spherical Segment (CSS) code using two different diffusion schemes. The CSS code is a new massively parallel modeling tool capable of simulating 3-D compressible MHD convection with a realistic solar stratification in rotating spherical shell segments.

Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness

Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness

TURBULENT DYNAMOS IN SPHERICAL SHELL SEGMENTS OF VARYING GEOMETRICAL EXTENT

We use three-dimensional direct numerical simulations of the helically forced magnetohydrodynamic equations in spherical shell segments in order to study the effects of changes in the geometrical shape and size of the domain on the growth and saturation of large-scale magnetic fields. We inject kinetic energy along with kinetic helicity in spherical domains via helical forcing using Chandrasekhar-Kendall functions. We take perfect conductor boundary conditions for the magnetic field to ensure that no magnetic helicity escapes the domain boundaries. We find dynamo action giving rise to magnetic fields at scales larger than the characteristic scale of the forcing. The magnetic energy exceeds the kinetic energy over dissipative time scales, similar to that seen earlier in Cartesian simulations in periodic boxes. As we increase the size of the domain in the azimuthal direction we find that the nonlinearly saturated magnetic field organizes itself in long-lived cellular structures with aspect ratios close to unity. These structures tile the domain along the azimuthal direction, thus resulting in very small longitudinally averaged magnetic fields for large domain sizes. The scales of these structures are determined by the smallest scales of the domain, which in our simulations is usually the radial scale. We also find that increasing the meridional extent of the domains produces little qualitative change, except a marginal increase in the large-scale field. We obtain qualitatively similar results in Cartesian domains with similar aspect ratios.

The helicity constraint in spherical shell dynamos

AbstractThe motivation for considering distributed large scale dynamos in the solar context is reviewed in connection with the magnetic helicity constraint. Preliminary accounts of 3‐dimensional direct numerical simulations (in spherical shell segments) and simulations of 2‐dimensional mean field models (in spherical shells) are presented. Interesting similarities as well as some differences are noted. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A 14-MeV Intense Neutron Source Based on Muon-Catalyzed Fusion - III: Thermohydraulic Regime of the Synthesizer

Design calculations of thermohydraulic parameters of the secondary target of the intense neutron source (INS) based on muon-catalyzed fusion (μCF) (the μCF-INS) are presented for a liquid deuterium-tritium (D-T) mixture. The synthesizer is connected to an external cooler by input and output pipelines. The optimal mixture composition, synthesizer layout, and dimensions are determined. The possibility of creating a D-T mixture flow with a quasi-uniform velocity distribution is demonstrated. Possible vortexes were found to be eliminated by installation of corresponding hydraulic resistance in the shape of a spherical shell segment. At the μCF-INS operation with its design parameters [neutron flux as high as 1014 n/(cm2·s)], the total heat deposit in the D-T mixture due to fusion and charged-particle ionization losses is estimated at ~117 kW. However, even at such conditions, with the appropriate synthesizer geometry and mass flow rate, the mixture temperature does not exceed the boiling point in any part of the synthesizer.

Ideal stretch forming for minimum weight axisymmetric shell structures

A companion paper “Minimum Weight Design of Axisymmetric Shell Structures” by Richmond and Azarkhin describes the design of axisymmetric thin-walled structures and parts that can support specified loads with minimum material. The result is an optimum shape and thickness distribution in the final part when the strength of the material is assumed to be uniform. Here, we demonstrate the application of ideal forming theory to design sheet stretching processes that can produce the optimum shapes and thickness distributions from flat sheets of uniform thickness. Specific designs are achieved for producing minimum weight shell structures that will support a specified uniform pressure assuming both the Mises and the Tresca yield criteria along with the rigid-perfectly plastic flow condition. In the case of the Tresca yield condition, the optimum structure is a spherical shell segment with uniform thickness, and an associated ideal stretching process is hydraulic bulging. Because the effects of strain hardening have been neglected in the structural optimization theory, it has been possible here to design the minimum weight structure and its forming process sequentially. In subsequent work, we plan to include the effects of strain hardening on the shell strength, which will then require coupled design of the structure and its forming process. Also extension of these methods to three-dimensional geometries will be considered.

Three-dimensional vibrations of thick spherical shell segments with variable thickness

A three-dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of spherical shell segments with variable thickness. Displacement components u φ , u z , and u θ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the φ and z directions. Potential (strain) and kinetic energies of the spherical shell segment are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the spherical shell segments. The method is applicable to thin spherical shell segments, as well as thick and very thick ones.

Three-Dimensional Vibration Analysis of Thick Shells of Revolution

A 3D method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow bodies of revolution (i.e., thick shells), not limited to straight-line generators or constant thickness. The middle surface of the shell may have arbitrary curvatures, and the wall thickness may vary arbitrarily. Displacement components u;gf, uz, and u;gu in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the entire body are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials are increased, frequencies converge to the exact values. Novel numerical results are presented for two types of thick conical shells and thick spherical shell segments having linear thickness variations and completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of both types of shells. The method is applicable to thin shells, as well as thick and very thick ones.

Axisymmetric spherical shell with variable wall thickness

This paper is engaged in research of the problem of axisymmetric spherical shell with variable wall thickness. The solutions for the problem are given for the spherical shell segment which does not contain the pole of sphere and the point of zero wall thickness.

Spherical Foundation Shell with a Ring Beam

ABSTRACT Elastic shell bending theory is applied to the problem of an axisymmetric foundation that consists of two spherical shell segments and a ring beam. The loading consists of a uniform vertical line load applied axisymmetrically on the ring beam. A two parameter foundation model is assumed and shell-beam as well as shell-soil stress and deformation interaction is considered. Theoretical results are computed and compared with results from two approximate analyses.

A theoretical elastic analysis of spherical thin shell segments in pressure vessels with a circular hole under transverse load [formula omitted] and Moment [formula omitted

In this paper, according to the theory of thin shells, the uniformly valid asymptotic homogeneous solution and the exact particular solution are derived for spherical thin shell segments with a circular hole under transverse load P 0 and moment M 0 .

Multiple beam lens transducer for sonar systems

A compact apparatus for transmitting and receiving multiple sonar beams utilizes an acoustic lens to direct plane waves incident in desired directions to electroacoustic transducers positioned on spherical shell segments centered in the focal regions of the lens associated with the incident beams. The electroacoustic transducers transmit spherical waves that are transformed by the acoustic lens to plane waves emergent in the desired directions.