Showing papers on "Axial symmetry published in 1971"

RMS Envelope Equations with Space Charge

Abstract: Envelope equations for a continuous beam with uniform charge density and elliptical cross-section were first derived by Kapchinsky and Vladimirsky (K-V). In fact, the K-V equations are not restricted to uniformly charged beams, but are equally valid for any charge distribution with elliptical symmetry, provided the beam boundary and emittance are defined by rms (root-meansquare) values. This results because (i) the second moments of any particle distribution depend only on the linear part of the force (determined by least squares method), while (ii) this linear part of the force in turn depends only on the second moments of the distribution. This is also true in practice for three-dimensional bunched beams with ellipsoidal symmetry, and allows the formulation of envelope equations that include the effect of space charge on bunch length and energy spread. The utility of this rms approach was first demonstrated by Lapostolle for stationary distributions. Subsequently, Gluckstern proved that the rms version of the K-V equations remain valid for all continuous beams with axial symmetry. In this report these results are extended to continuous beams with elliptical symmetry as well as to bunched beams with ellipsoidal form, and also to one-dimensional motion.

Apparatus for refining molten aluminum

Abstract: The apparatus of the invention is capable of injecting gas in the form of small discrete bubbles into a mass of molten metal. The apparatus comprises a rotatable shaft coupled to drive means at its upper end and a vaned rotor at its lower end. Gas under sufficient pressure to be injected into the melt is fed into a passageway extending axially through the device whereby upon rotation of the rotor the gas is injected into the molten metal and subdivided into discrete gas bubbles. The process of the invention utilizes the above described gas injection apparatus for refining molten aluminum by introducing an inert gas into the metal beneath the surface of the melt.

Stability of Nonlinear Oscillations of an Elastic Rod

Abstract: The stability of plane steady oscillations of a slender elastic rod subjected to lateral harmonic excitation is investigated. For certain values of excitation amplitudes and frequencies, the planar response is unstable, and nonplanar motions are parametrically excited. Steady nonplanar motions are characterized by each point on the rod centerline tracing an elliptical path perpendicular to the rod axis. The investigation is restricted to cantilevered rods with nearly equal principal moments of inertia of the cross section, and excitation is always in the direction of one of the principal axis of inertia. The effects of damping and of slight variations of the natural frequencies associated with motions in the two principal planes are investigated. Supporting experimental evidence is presented. Finally, the principal features of the transversely driven rod response are compared with the response for the same type rod excited axially.

Attitude stability of a symmetric satellite at the equilibrium points in the restricted three-body problem

Abstract: A problem of attitude motion of the smallest body for the restricted three-body problem is analyzed. Axial symmetry is assumed for the body, and attention is focused on the case in which the symmetry axis is normal to the orbit plane. For libration point satellites, results are similar to those for a satellite in orbit about a single body. However, for orbit equilibrium points lying on the line joining the two larger bodies, attitude stability results depart markedly from those for the two-body problem.

Axially symmetric explosion in a spheroid.

Abstract: The method of Laumbach and Probstein is applied to a point explosion in a spheroid with exponential density distribution. It is shown that the shock wave propagates strongly along the direction of symmetry axis and the envelope of the shock front elongates to the same direction. The rate of elongation of the shock envelope increases with the eccentricity of the spheroid and finally the blowout of the shock wave along the polar axis occurs when the eccentricity exceeds some critical value. It is suggested that such a blowout of the shock may be connected with some galactic outburst having axial symmetry.

Magnetic and Drift Surfaces using a New Stellarator Expansion

Abstract: A new stellarator expansion has been devised, wherein the depths of the helical and toroidal field modulations are comparable and the total rotational transform is finite. To accomplish this, we define λ = ρ/R ≪ 1 (R is the major radius, ρ is the minor radius) and choose p, the number of helical periods, to be of order λ−2/3. Detailed calculations of vacuum magnetic field properties have been carried out, including the magnetic surfaces, V″, shear, etc. For the particle surface calculations, the ratio of ion gyroradius to major radius of the torus, e = RL/R, is chosen to be of order λ3/2. Using the same methods as for the vacuum field calculations, analytic solutions are obtained for particle drift surfaces in stellarators with finite departures from axial symmetry. These surfaces, derived from perturbation techniques, are obtained for four particle classes that are, for the first time, precisely categorized according to trapping state. Transitions between classes are interpreted. Numerical calculation of...

Electromagnetic Wave Propagation in Radially and Axially Nonuniform Media: Geometrical-Optics Approximation

Abstract: This paper presents an investigation of the propagation of a cylindrically symmetrical beam in radially and axially nonuniform media, when the dielectric constant ∊ is given by ∊ = ∊0−∊2(z)r2. A differential equation for the dimensionless beam-width parameter has been set up in the geometrical-optics approximation and solved for some specific dependences of ∊2 on z. The relevance of this investigation to self-focusing fibers or rods is discussed.

The space-time metric inside a rotating cylinder

Abstract: It is shown that the space-time region inside an axially symmetric, infinite, rotating, cylindrical mass distribution is necessarily Minkowskian.

Studies in the Motion and Decay of Vortices

Abstract: Solutions of Navier-Stokes equations are constructed as an asymptotic expansion in terms of a small parameter related to the Reynolds number of the vortex. A general scheme is presented for the matching of the inner viscous core of the vortex to the outer inviscid solution. The singularities in the inviscid theory are removed and the condition of regularity in the flow field defines the velocity of the vortex line. This general scheme is applied to study vortices in two dimensional, axially symmetric and in three dimensional flow fields. Results for the outer solution which do or do not require the solution in the viscous core are obtained separately. Several examples are presented to show that the solution of this analysis can be identified with that of classical inviscid theory with the same initial vorticity distribution for the initial instant. They disagree afterwards because the inviscid theory ignores the diffusion of vorticity in the core. For a general three dimensional problem, the present scheme of analysis can be carried out provided the shape of the vortex line and the three dimensional flow field fulfill a constraint condition. This condition is automatically fulfilled for the two dimensional and axially symmetric problems. When the initial velocity of the vortex is prescribed to be different from that given by the analysis, the necessary modifications to the expansion scheme and the solution for the subsequent motion are presented.

Reversed Field Equilibria in Axially Symmetric Electron Current Layers

Abstract: Self‐consistent, nonrelativistic E‐layer equilibria in two dimensions are calculated by numerical methods. A rigid‐rotor model for the electron distribution, together with a neutralizing ion background, leads to a nonlinear equation for the magnetic flux function ψ. Solutions to this equation are displayed, and the existence of bifurcations or nonuniqueness is demonstrated.

Rotational motion of axially symmetric molecules in liquids

Abstract: After Langevin's method, orientational diffusion of a non-spherical particle is investigated. By use of this method the study of the molecular rotational motion is extended to that of axially symmetric molecules, in which the cross terms of angular velocity components are not neglected. The components of the rotational friction coefficient tensor are expressed quite generally. Angular autocorrelation functions are given by use of the distribution function. It is shown that the contribution of the precessional motion is never neglected at small time. The calculated results are compared with the experimental results of infrared absorption measurements.

Oscillations of a body filled with a viscous fluid

Abstract: The oscillations of a physical pendulum containing a spherical cavity filled with an incompressible viscous liquid were discussed in [1]. In this paper we consider the mote general problem of the motion of an axially symmetric solid with a spherical cavity filled with an incompressible viscous fluid and moving about a fixed point. It is assumed that the center of the cavity and the fixed point lie on the axis of symmetry of the body.

Principal- and slip-line methods of numerical analysis in plane and axially-symmetric deformations of rigid/plastic media

Abstract: Some new principal- and slip-line methods of numerical analysis are presented for plane and axiallysymmetric deformations of rigid/plastic media. These methods are applicable to the solution of problems of incompressible isotropic, or incompressible anisotropic, or (special) compressible isotropic media obeying generalized Tresca/von Mises constitutive equations. The essential idea in the numerical methods considered is an artifice of the reduction of elliptic (and similar) problems of systems of partial differential equations to hyperbolic ones by means of a procedure based upon the initial estimation of one of the field quantities followed by iterations to determine all field quantities. Solutions for the particular problem of combined extension and expansion of a thick-walled circular cylinder under axial force and internal pressure are discussed.

Influence of Molecular Geometry, Orientation, and Dynamics on Angular Correlation Patterns from Rotationally Labeled Macromolecules

Abstract: In order to determine the sensitivity of gamma‐ray angular correlation patterns from solute macromolecules labeled with rotational tracers such as 111mCd, a theoretical study was made of the behavior expected under certain conditions. A nucleus of spin 52, acted upon by an axially symmetric electric field gradient, and bound to a rodlike macromolecule, was considered. Under static conditions (no molecular rotation), the time‐dependent correlation pattern is quite sensitive to molecular orientation and, for oriented molecules, to the angle between the axis of the field‐gradient tensor and the molecular axis. A general equation and results for selected geometric configurations are given. When molecular rotation is allowed, a classical model is applicable if the rotation is sufficiently slow. This model is used to calculate relaxation curves for several geometrical configurations under the condition that the macromolecules rotate about their long axes. These curves are shown to have considerable diagnostic v...

Comparisons of tropical cyclone simulations with and without the assumption of circular symmetry

Abstract: Results from a three-layer asymmetric hurricane model previously described by the authors are compared with results from an axially symmetric analog to investigate the effect of the symmetry assumption on the internal dynamics of model cyclones. The symmetric model storm initially develops more rapidly than the asymmetric storm. The differences in intensity during the first 100 hr are related to differences in horizontal resolution produced by the staggered grid used with the symmetric model. The symmetric model, on the other hand, does not produce the second period of intensification that starts at 120 hr in the asymmetric model. This fact supports the conclusion reached in the earlier paper that the development of large-scale asymmetries at 100 hr is closely related to the subsequent intensification. Although the life cycles of the two storms are different, the azimuthally averaged structure of the asymmetric storm at maximum intensity is similar to the corresponding structure of the symmetric ...

On the dead-zone formation in plastic axially-symmetric converging flow

Abstract: A n upper-bound analysis is used to examine dead-zone formation in rigid/plastic axially-symmetric converging flow. A quasi-stable internal shear is found to occur under certain conditions. Bounds on the geometry for quasi-stable behavior and bounds on the required power are presented. The analysis agrees favorably with experiments.

Approximate Theory of Torsional Wave Propagation in Elastic Circular Composite Cylinders

Abstract: An approximate one‐dimensional theory is developed for axially symmetric torsional deformation of elastic circular composite cylinders. The cylinders are composed of a core perfectly bonded to an outer casing. The equations of motion accommodate the first two frequency branches of the dispersion relation for symmetric torsional waves propagating along the axis of the cylinder. Important parameters are the ratios of densities, shear moduli, and radii of the core and casing. Dispersion relations obtained from this approximate theory are studied for ranges of values of these parameters, and a comparison made with the results of the “exact” theory, i.e., three‐dimensional elasticity. In order to match closely the results from the exact analysis, two correction factors are introduced, and values are tabulated for typical material and geometric properties. Sufficient conditions at the ends of finite cylinders are obtained to provide a unique solution.

Principal-stress differences in transverse planes of symmetry: In this analysis, the fact that the value of Poisson's ratio for most of the photoelastic materials at critical temperatures equals 0.5 is utilized to determine the displacement function in the transverse plane of symmetry of an axisymmetric body

Abstract: The value of Poisson's ratio for most of the photoelastic materials at critical temperature is very nearly equal to 0.5. This fact can be utilized to determine the values of the displacement function u(r) and the principal-stress differences in the transverse plane of symmetry of an axially symmetric body. This requires a single integrated-retardation pattern. The solutions are applied to the problem of a sphere under diametral compression and the results compared with those obtained from shear-difference method.

Axially Symmetric Hydromagnetic Equilibria with Spherical Plasma‐Vacuum Interfaces

Abstract: Solutions representing axially symmetric hydromagnetic equilibria with spherical plasma‐vacuum interfaces are extended from a low‐pressure paramagnetic regime to a high‐pressure diamagnetic regime.

Axially Symmetric Torsional Waves in Circular Composite Cylinders

Abstract: A method is developed to obtain the dispersion diagram for axially symmetric torsional waves in any type of elastic circular cylinder composed of a core perfectly bonded to an outer casing. The method is based on the application of two sets of bounds that are easily obtained. One set represents solutions of a solid cylinder with geometric and material properties identical with the core; the other set of bounds represents solutions of a hollow cylinder with its properties identical with the casing. Certain intersections of the two sets of bounds correspond to solutions of the frequency equation of the composite cylinder. Therefore the frequency branches weave through these intersections, but otherwise stay between the bounds. The dispersion diagram of the composite cylinder is thus obtained. Both real and imaginary wavenumbers occur. Other complex wavenumbers are shown not to exist.

Radial oscillations of a thick-walled spherical highly stressed nonhomogeneous shell☆

Abstract: Radial oscillations of a thick-walled spherical shell are investigated assuming the material of the shell to be elastic, isotropic, incompressible and continuously nonhomogeneous in the radial direction. The shell is first subjected to a static axially symmetric deformation and then set into infinitesimal motions. General equations are specialized to the neo-Hookean material and an explicit equation for the frequency of oscillations is derived. In the limit case of a homogeneous thin shell and of an infinite medium with a spherical cavity the results agree with the known results. Graphs are given displaying the frequency variations for various degrees of nonhomogeneity, wall-thickness and initial stretch.