Prolate Spheroidal Shell Research Articles

The energy of a prolate spheroidal shell in a uniform magnetic field

The problem of the energy of a spheroidal magnetic shell, solved by methods of classical electrodynamics, arises, in particular, upon the study of thin-wall biocompatible microcapsules in connection with a pressing issue of targeted drug delivery. The drug inside a microcapsule should be released from the shell at a required instant of time by destroying the capsule’s shell. The placement inside a shell of magnetic nanoparticles sensitive to an external magnetic field theoretically makes it possible to solve both problems: to transport a capsule to the required place and to destroy its shell. In particular, the shell can be destroyed under the action of internal stress when the shape of a capsule is changed. In this paper, the analysis of the model of a magnetic microcapsule in the form of a prolate spheroidal shell is performed and formulas for the magnetostatic and magnetic free energy when the magnetic field is directed along the major axis of the spheroid are derived.

Coulomb energy of uniformly charged spheroidal shell systems.

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.

Dynamic response of a fish swim bladder to transient sound

High-level underwater sound exposures from impact pile driving activities and seismic air guns can cause physical damage to fishes. Damage to biological tissues depends on the strain rate induced by the initial rapid rise and fall times of the received sound pulse. So a time-domain analysis is needed to understand mechanisms of tissue damage resulting from exposure to sound from transient acoustic sources. To address this issue, the frequency domain mathematical model originally developed by Finneran and Hastings (JASA 108: 1308-1321, 2000) was modified to predict the response of a fish swim bladder and surrounding tissues to transient signals. Each swim bladder chamber was modeled as an elastic prolate spheroidal shell filled with gas and connected to other parts of the anatomy. Results of three case studies are presented showing correlation between the strain rate of the swim bladder wall and tissue damage reported in five different species of fish.

Potential Distribution Computation in Conducting Prolate Spheroidal Shells Using the Exodus Method

This paper presents the use of the Exodus method to compute the potential distribution in a conducting prolate spheroidal shell. The transition probabilities for the Exodus method simulation were obtained from the finite-difference scheme transformation of the prolate spheroid Laplace equation. This method is independent of the random number generation from the computing facility and gives more accurate results than the fixed random walk Monte Carlo method. The results obtained matched perfectly with those obtained by using an exact solution and explicit finite-difference solution method.

Centrifugal force induced by relativistically rotating spheroids and cylinders

Starting from the gravitational potential of a Newtonian spheroidal shell we discuss electrically charged rotating prolate spheroidal shells in the Maxwell theory. In particular we consider two confocal charged shells which rotate oppositely in such a way that there is no magnetic field outside the outer shell. In the Einstein theory we solve the Ernst equations in the region where the long prolate spheroids are almost cylindrical; in equatorial regions the exact Lewis ‘rotating cylindrical’ solution is so derived by a limiting procedure from a spatially bound system. In the second part we analyze two cylindrical shells rotating in opposite directions in such a way that the static Levi-Civita metric is produced outside and no angular momentum flux escapes to infinity. The rotation of the local inertial frames in flat space inside the inner cylinder is thus exhibited without any approximation or interpretational difficulties within this model. A test particle within the inner cylinder kept at rest with respect to axes that do not rotate as seen from infinity experiences a centrifugal force. Although in suitably chosen axes the spacetime there is exactly Minkowskian out to the inner cylinder, nevertheless, those inertial frame axes rotate with respect to infinity, so relative to the inertial frame inside the inner cylinder a test particle is traversing a circular orbit.

Analytical and numerical computation of prolate spheroidal shell capacitance

AbstractThis article presents analytical and numerical methods of computing the capacitance of a prolate spheroidal shell. Detailed and comprehensive equations for determining the exact results were developed. Also, an explicit finite difference (FD) numerical technique for solving Laplace's equation in prolate spheroidal coordinates systems for an axially symmetric geometry has been developed. Consequently, the FD results were used to compute the capacitance of the spheroidal shell in an easy to understand manner. This was achieved using the Dirichlet boundary conditions on the prolate spheroidal surfaces. The capacitance results obtained using FD did agree with those obtained using analytical method. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2361–2365, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24630

Frequency spectra of bilaminar prolate spheroidal shells.

The kinetic and strain energy densities were derived for the vibration of a bilaminar prolate spheroidal shell of constant thickness. The bilaminar shell is composed of two bonded concentric prolate spheroidal layers of different material properties and thicknesses. The elastic strain energy density has seven independent kinematic variables: three displacements, two thickness shears, and two thickness stretches. Continuity of displacements is enforced at the interfacial reference surface. The shell has constant thickness h=h1+h2, where h1 and h2 denote the thicknesses of the respective layers. The reference surface eccentricity is defined by 1/a, where a is its shape parameter. Using appropriate comparison functions in terms of Legendre polynomials that satisfy the boundary conditions for a closed spheroidal shell, the system is solved using the Galerkin method. Numerical results of the frequency spectra are presented for various ratios of h1/h2 and various material properties for the two layers. Initial results are presented for a=100 (a nearly spherical shape). [Work supported by the ONR/ASEE Summer Faculty Research Program.]

Acoustic backscattering from fluid‐filled submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The fluid‐filled shell is insonified by a plane wave with an arbitrary angle of incidence. The external and internal fluid loading impedances were computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far‐field backscattered acoustic pressure spectra are presented as a function of the angle of incidence for several shell thickness‐to‐half‐length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). A comparison of the backscattering from fluid‐filled and empty shells is presented at selected plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

AXISYMMETRIC FREE VIBRATION OF THIN PROLATE SPHEROIDAL SHELLS

In this paper a detailed study of the theory of free axisymmetric vibration of thin isotropic prolate spheroidal shells is presented. The analysis is performed according to Rayleigh – Ritz method. This method as well as an approximate modeling technique were attempted to estimate the natural frequencies for the shell. This technique is based on considering the prolate spheroidal as a continuous system constructed from two spherical shell elements matched at the continuous boundaries. Through out the obtained results it is found that this method predicted fairly well the natural frequencies of a prolate spheroidal shell for all values of eccentricities.

Nonaxisymmetric acoustic scattering from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamiltons principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The shell is insonified by a plane wave with arbitrary incident angle. The external (heavy) fluid-loading impedance was computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far-field backscattered acoustic pressure spectra are presented as a function of incident angle for several shell thickness-to-half-length ratios ranging from .005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic scattering is presented at selected frequencies and plane wave incident angles. [Work supported by the ONR/ASEE Summer Faculty Research Program and the NAVSEA Newport ILIR Program.]

Solution of the 3-D equations of elasticity in prolate spheroidal coordinates

The 3-D equations of elasticity in prolate spheroidal coordinates are solved using a Helmholtz decomposition for the displacement vector. The displacement vector is written in terms of a scalar potential (dependent on the dilatational wave number) and a vector potential (dependent on the shear wave number). These wave potentials are solutions, respectively, of the scalar and vector wave equations cast in prolate spheroidal coordinates. An eigenfunction expansion of spheroidal wave functions is employed to solve the scalar wave equation. The solution of the vector wave equation is facilitated using expansions of prolate spheroidal vector wave functions. These vector wave functions are constructed by applying certain vector differential operators to the scalar wave functions. The canonical problem of free vibration of a traction-free confocal elastic layer (shell) is considered. The shape of the prolate spheroid is defined by the major-axis length L and minor-axis length D. Since the sphere is a limiting form of the prolate spheroid, the analytical approach is validated by comparing the frequency spectra of a prolate spheroidal shell with L/D∼1 to those of a spherical shell using a 3-D solution in spherical coordinates. [Work supported by the NAVSEA Newport ILIR Program.]

Axisymmetric acoustic scattering from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The shell is insonified by a plane wave incident along the major axis. The external (heavy) fluid loading impedance was computed using an eigenfunction expansion of prolate spheroidal wavefunctions. Far-field scattered acoustic pressure spectra are presented for several shell thickness-to-half-length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic scattering is presented at selected frequencies. [Work supported by the ONR/ASEE Summer Faculty Research Program.]

Axisymmetric acoustic radiation from submerged prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamiltons principle [S. I. Hayek and J. E. Boisvert, J. Acoust. Soc. Am. 114, 2799–2811 (2003)]. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. For axisymmetric vibration of a submerged shell, the external (heavy) fluid loading impedance was computed using expansions of prolate spheroidal wavefunctions. The shell was excited by axisymmetric normal surface forces, including a point load at the shell apex and ring load at other locations. Far-field radiated pressure spectra are presented for several shell thickness-to-half-length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroidal shell (a=1.01) to a spherical shell (a∼100). The far-field directivity of acoustic radiation is presented at selected frequencies. [Work supported by the NUWC ILIR Program and the ONR/ASEE Summer Faculty Research Program.]

Vibration of prolate spheroidal shells with shear deformation and rotatory inertia: axisymmetric case.

This paper presents the derivation of the equations for nonaxisymmetric motion of prolate spheroidal shells of constant thickness. The equations include the effect of distributed mechanical surface forces and moments. The shell theory used in this derivation includes three displacements and two thickness shear rotations. Thus, the effects of membrane, bending, shear deformation, and rotatory inertia are included in this theory. The resulting five coupled partial differential equations are self-adjoint and positive definite. The frequency-wave-number spectrum has five branches, two acoustic and three optical branches representing flexural, extensional, torsional, and two thickness shear. For the case of axisymmetric motion, these were computed for various spheroidal shell eccentricities and thickness-to-length ratios for a large number of modes. The axisymmetric dynamic response for damped shells of various eccentricities and thicknesses under point and ring surface forces are presented.

Prediction of the magnetic field beyond a rectangular array of sensors

A mathematical method is developed for the reconstruction of the magnetic field of a source with sensors arranged in a rectangular array on a plane. A prolate spheroidal shell encloses the source. The magnetic field of the source is expressed in prolate spheroidal coordinates and the prolate spheroidal expansion coefficients are determined by the application of the derived selective functions to the x component of the measured magnetic field. In a simulation with 192 sensors, the magnetic field is accurately reconstructed for sample sources in the presence of sensor noise, sensor position, and orientation error. The reconstruction error is the least for sources with significant dipolar component.

Vibration of submerged prolate spheroidal shells with internal fluid loading

The equations of motion for nonaxisymmetric vibration of submerged prolate spheroidal shells of constant thickness with internal fluid loading were derived using Hamiltons principle. The shell theory used in this derivation includes shear deformations and rotatory inertias. The shell displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. The external and internal fluid loading impedances were computed using expansions of prolate spheroidal wave functions in each domain. The shell was excited by axisymmetric normal surface forces, including a point load at the shell apex and ring load at other locations. Numerical results were obtained for the driving and transfer mobilities for several shell thickness-to-half-length ratios ranging from 0.005 to 0.1, and for various shape parameters, a, ranging from an elongated spheroid shell (a=1.01) to a spherical shell (a=100). Results are presented for various combinations of external and internal fluid loading, and comparisons are made to the in vacuo shell vibration. [Work supported by ONR and the Navy/ASEE Summer Faculty Program.]

Density of Axisymmetric Modes of Closed Prolate Spheriodal Shells

The frequencies of resonance of prolate spheriodal shells may be estimated either by numerical methods or analytically via variational methods. With either approach, the computational effort increases as the order of the mode increases, making it difficult to obtain accurate estimates at frequencies where the modal density is high. However, at high frequencies, it is often necessary to obtain estimates only for the modal density since modal overlap will reduce the effects of individual modes on the response of the shell. In this paper, approximations for the modal density of the axisymmetric modes of closed prolate spheroidal shells are derived based on results obtained using analytic variational methods.

Vibration of in vacuo hemiprolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of hemiprolate spheroidal shells of constant thickness were derived using Hamilton’s principle. The thin shell theory used in this derivation includes shear deformations and rotatory inertias. The shell is clamped at the equator and is excited by mechanical surface force fields. The displacements and rotations were expanded in infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate and circular functions in the azimuthal angle coordinate. Five-branched frequency spectra were computed for several shell thickness-to-length ratios ranging from 0.005 to 0.1, and for various diameter-to-length ratios, including the limiting case of a spherical shell. Numerical results were obtained for the frequency spectra of the driving and transfer mobilities of these shells due to surface force excitations. Some comparisons of the dynamic response of hemiprolate and full prolate spheroidal shells are presented. [Work supported by ONR and the Navy/ASEE Summer Faculty Program.]

Dynamic response of in vacuo prolate spheroidal shells excited by surface force fields

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle. The thin shell theory used in this derivation includes three displacements and two changes of curvature. The effects of membrane, bending, shear deformations, and rotatory inertias are included in this theory. The resulting five partial differential equations are self-adjoint and positive definite. The shells are excited by axisymmetric line forces. The axisymmetric modal solutions are expanded in an infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate. Numerical results for the frequency response were obtained for several shell thickness-to-length ratios ranging from 0.005 to 0.1, and for various diameter-to-length ratios, including the limiting case of a spherical shell. Sample mode shapes were obtained at selected resonant frequencies. [Work supported by ONR and the Navy/ASEE Summer Faculty Program.]

Natural frequencies of nonaxisymmetric vibration of prolate spheroidal shells

The equations of motion for nonaxisymmetric vibration of prolate spheroidal shells of constant thickness were derived using Hamilton’s principle. The thin shell theory used in this derivation includes three displacements and two changes of curvature. The effects of membrane, bending, shear deformations, and rotatory inertias are included in this theory. The resulting five partial differential equations are self-adjoint and positive definite. The nonaxisymmetric modal solutions are expanded in a doubly infinite series of comparison functions. These include associated Legendre functions in terms of the prolate spheroidal angular coordinate, and circular functions of the circumferential coordinate. The natural frequencies and the mode shapes were obtained by the Galerkin method for each circumferential mode. Numerical results were obtained for several shell thickness-to-length ratios ranging from 0.005 to 0.1, and for various diameter-to-length ratios, including the limiting case of a spherical shell. [Work supported by Office of Naval Research and the Navy/ASEE Summer Faculty Program.]