Common Axis Of Symmetry Research Articles

Calculations for spray concrete support systems of underground structures with estimation of the rock bed consolidated with concrete

For a more detailed analysis of the deformation process of sprayed concrete support, this paper describes a calculation method based on the solution of a two-dimensional contact problem of the elasticity theory for a two-layer arbitrary ring with a common symmetry axis in a weighty deformable medium modeling bed. The deformational characteristics of ring layers and the medium are different. The calculation scheme of the contact problem is illustrated. The method described here could also be used to estimate the characteristics of a monolithic support system in capital workings.

Singularities on the flow between two rotating surfaces

Two coaxial surfaces of revolution rotate about the common axis of symmetry with almost the same angular velocity. At large Reynolds numbers, three singular surfaces are found. One over each surface of revolution and a cylindrical shear layer touching the outer surface. Outside the cylinder the velocity is determined by the velocity distribution over the two boundary layers, while inside the cylinder the fluid is at rest.

A simultaneous concentration of a vortex and a magnetic field in magnetohydrodynamic flows

A three-dimensional magnetodydrodynamic flow with a combined effect of convection and diffusion is reduced to a two-dimensional problem by a variable transformation. The original field is composed of a unidirectional vorticity and an aligned magnetic field superimposed on an axisymmetric straining flow in an incompressible, viscous, electrically conducting fluid. By restricting consideration to the axisymmetric vorticity and magnetic fields, an exact solution is presented, which shows time evolution from general initial profiles. In a particular case of constant straining, the solution simultaneously approaches the state of a concentrated cylindrical vortex and a magnetic tube of Gaussian profile sharing a common symmetry axis.

Structure of the flow of gas escaping from a gap in a pipeline

A study is made of the two-dimensional steady flow of gas escaping from a circular gap formed when the ends of pipes move apart along their common symmetry axis. A study is made of the rearrangement of the flow from the shockless flow in the neighborhood of the symmetry axis to the shocked flow. A numerical investigation gave the value H* of the width of the gap at which the transition takes place from a shockless flow structure to one with a shock. A study is made of the influence on H* of the pressure ratio ɛa (ɛa = Pa/P0, where Pa is the pressure in the ambient space and p0 is the stagnation pressure) and the specific—heat ratio. Godunov's scheme [1] was used for the numerical realization.

Axisymmetric contact problem in the presence of a wedge

The integral equation of the axisymmetric contact problem, obtained in a new form, is used to investigate the case when thin smooth rigid axisymmetric wedges (posts) are inserted into the bodies making contact along their common axis of symmetry. Singularities are established in the solution, particularly the possibility of the origination of negative drawing together of the bodies and the annular pressure domain under the action of a flat stamp of circular planform on the half-space.

Damper saturation effects on nutation

Saturation of a nutation damper carried by one of an ensemble of bodies (stack) in a relative angular motion around a common symmetry axis is investigated analytically. Formulae for the nutation angle rate are found for weak and strong saturation, applicable both to “axial” and “tangential” dampers. One of the results is that, under certain conditions, a saturated—i.e. a shorter and thus lighter—damper is more effective than an unsaturated damper.

Properties and performance of grazing-incidence mirror systems

We investigate the performance of, and the origin of aberrations in, beam lines based on simple and the recently proposed CARSA mirror elements. New results include the identification of the sums of off-grazing angles and their squares as figures of merit for total reflectance and scattering losses in near-grazing-incidence systems, and the discovery that the usual image distortions and aberrations previously associated with simple elements can essentially be eliminated with pairs of elliptical mirrors having a common rotational symmetry axis. Slit throughput efficiencies and sensitivities to system stability are calculated for both horizontal and vertical dispersion for several representative beam lines using a ray-tracing program developed for CARSA systems. We find that high-resolution operation is possible with no entrance slit for CARSA combinations.

On the distribution of the field of electromagnetic waves emitted by a dipole in a homogeneous magnetoactive plasma

This paper deals with the emission of an electric dipole in a homogeneous magneto-active multicomponent plasma in all resonance frequency ranges ω which are lower than the Langmuir electron frequency ω 0. We show that the structure of the field represents two cones, one inserted into the other with the common symmetry axis along the vector of the magnetic field H ̄ 0 . The surface of one of them corresponds to the plasma resonance region. The other is an analogue of Storey's cone in the whistler mode range. In the vicinity of the surfaces of these cones and near the symmetry axis, in a narrow range of angle, a considerable amplification of the field can take place. In some cases the maximum value of the field strength exceeds the field inside the cones, and the field of a similar dipole located in vacuum, by approximately (10 1–10 5) times. This effect is considerable and must play a prominent part in various processes taking place in a plasma. The emission energy is redistributed with distance from the source: initially the maximal field strength is near the surface of the resonance cone, but starting from near the symmetry axis i.e. at sufficiently large distances, the field energy is concentrated along the magnetic line of force passing through the source. Asymptotic formulae are obtained for all the regions inside and outside the cones. The results of the numerical calculations characterizing the spatial field pattern and its change as a function of frequency are given for the whistler mode.

Magnetic flow of viscous gas between rotating surfaces of revolution

In this work the steady laminar magnetic flow of viscous gas is considered in a narrow space (slot) between two surfaces of revolution rotating with constant angular velocities around a common axis of symmetry. The linearised equations of magnetic motion of the viscous gas flow for axial symmetry in the intrinsic curvilinear orthogonal coordinate system x, φ, y are used. The obtained solutions of the equations of motion have been illustrated by examples of gas flow through the slot of constant thickness between rotating and fixed conical surfaces, and between rotating and fixed spherical surfaces.

The Game of Passing-Through of a Pair of Vortex Rings

When two similar vortex rings travel along the common axis of symmetry in the same direction, the game of passage of one vortex ring through the other was successfully realized both in air and in water. The phenomenon depends on the Reynolds number, Re , based on the initial traveling speed of vortex rings and the diameter of the ejecting orifice, and also depends on the time interval between two puffs of fluid by which the two vortex rings are produced. The region in the Re - T plane in which the game of passing-through is observed was experimentally determined both for air and for water, where T denotes a non-dimensional time interval. It was found that the region for water was smaller than that for air.

Correction to the retarding torque experienced by two rotating concentric spheres

A general expression is determined for the first correction to the retarding torque experienced by two concentric spheres which slowly and steadily rotate about a common axis of symmetry and which bound an incompressible viscous fluid.

Interaction of Two Vortex Rings Moving along a Common Axis of Symmetry

Interaction of two vortex rings is examined by experiments and numerical analysis. Experimental observations are carried out in a water tank using the electrolysis method and it is found that the interaction patterns are classified into three types. Numerical simulations are performed by integrating the Navier-Stokes equation and the results of the experiments and calculations are found in good agreement. The game of the passing through each other does not happen in this region of low Reynolds number.

Collisional ionization of helium by slow electrons

For single ionization of helium by electron impact the energies and scattering angles of the scattered and ejected electrons have been measured in a coincident experiment at primary energies E0 between 30 and 80 eV. Two types of collision are found, those in which the momentum transfer to the ion produced is small (binary encounter) and those in which the momentum transfer is large (recoil encounter). The maximum of the coincidence rate measured due to collisions of the binary encounter type is located in forward direction. Its angular position is varying with the collision parameters, whereas its angular half width is approximately 60 degrees for all data obtained. The recoil collisions cause a maximum of the coincidence cross section at about 160 degrees with a half width larger than 90 degrees . The relative intensity of the binary encounter peak and the recoil peak as well as the total number of ionizing transitions depend strongly on the energies and angles of the two outgoing electrons. Binary peak and recoil peak do not have a common symmetry axis, probably due to the interaction between the two ionization electrons.

Flows Between Stationary Surfaces of Revolution, Having Similarity Solutions

Laminar flow of an incompressible, Newtonian fluid is considered, in a narrow space between two stationary surfaces of revolution having a common axis of symmetry. The method of free parameters is used to investigate the existence of similarity solutions. It is found that there are no surface shapes for which similarity solutions exist, when the full Navier-Stokes equations are used to describe the flow. After order-of-magnitude arguments are employed to reduce the equations surface shapes are found for which similarity solutions exist; the shapes are delineated and the similarity problems are formulated. Finally, a method for solving the similarity problem is discussed and the solution is tabulated from the results of calculations conducted on a digital computer.

Electron spin rotation and beam focusing

It is shown that it is possible to produce superimposed electric and magnetic fields having a common axis of cylindrical symmetry and a common plane of mirror symmetry, which act in the following way. Electron beam focusing takes place both in the plane of mirror symmetry and in the direction perpendicular to it, in first order approximation image formation is stigmatic. The positions of cardinal points are independent of electron energy and, if the beam is polarized, the rotation of the direction of polarization between source and image is also independent of electron energy. The fields defined by the components E r = E 0( R ▪ ) 2, Eθ = E z = 0, B r = B θ = 0, B z = B 0( R ▪ ) R = const in the z = 0 plane of a cylindrical coordinate system represent a simple example.

THE ABERRATIONS OF CYLINDRICAL OPTICAL SYSTEMS

本文首先由柱面系统中描写光线所须的参数和对称性,确定了同轴柱面系统的独立象差数是八个,而对无限远物体而言则只有六个。而后由矢量代数方法证明空间光线在柱面系统中的折射光路与发生折射率改变后的子午截面内光路全同,由此得出柱面系统有别于球面系统的二种象差与色差相当。从而导出了校正柱面象差的条件并对柱面望远镜象差作了估值。结果表明,校正柱象差一般而言是很困难的,另一方面虽不加以校正影响也不算严重。最后讨论了柱面望远镜的高斯光学,结果表明,望远镜使一方向余弦按定比例缩小,而另一方向余弦不变时,它所成出的直线的象是弯曲的。

5. The axisymmetric case in hydromagnetics

Recent work at the Yerkes Observatory has been concerned with the study of configurations in which the magnetic and velocity fields possess a common axis of symmetry. In those cases where the density ρ may be assumed constant, it has proved advantageous to employ a representation suggested by Lüst and Schlüter[1]: in cylindrical co-ordinates (ω, ϕ, z) let and where is a unit vector and T, P, V, and U are independent of the azimuthal angle Φ. The hydrodynamic equation may then be replaced by the pair of equations (cf. Ghandrasekhar [2]) and where Δ5 is the Laplacian operator in 5 dimensions. The equation for the magnetic field, may similarly be replaced by the pair of equations and

On Surfaces of Stationary Area Bounded by Two Circles, or Convex Curves, in Parallel Planes

pendicular to the straight line joining their centers. The question presents itself whether these are the only minimal surfaces bounded by the two circles. The answer is in the affirmative, and we therefore have explicit examples of curves in space bounding more than one minimal surface and all of them known. More generally, we shall also consider the case when the two bounding circles lie in parallel planes but do not necessarily have a common axis of symmetry. It will be shown that any minimal surface bounded by them has the property that its intersection by a plane parallel to the planes of the two bounding circles is again a circle. This property allows the minimal surfaces bounded by the two circles to be explicitly determined and indeed this determination has already been made by B. Riemann under this assumption [6, pp. 305-352, especially pp. 341-347]. In precise terms, let S be a minimal surface in space bounded by two plane curves IN, r2 lying in parallel planes. The following theorems will be proved: THEOREM 1. If Fi, F2 are circles, then the intersection of S by a plane parallel to the planes of F1, F2 is again a circle. THEOREM 2. If ri, r2 are convex curves, the the intersection of S by a plane parallel to the planes of rF, r2 is again a convex curve. With reference to Theorem 1, it easily follows that when the two circles have a common axis of symmetry, then all the minimal surfaces bounded by ri, r2 are rotationally symmetric and are therefore catenoids. This completes a classical example, and furnishes an example of boundary curves bounding more than one minimal surface all of which are known. In this connection, mention should be made of the question posed by T. Rado in the Princeton Bicentennial Conference in 1946: to estimate the number of minimal surfaces bounded by a given curve or curves. 2. Let S be a minimal surface given in terms of conformal parameters (u, v). It is given by the equations x = x(u, v), y = y(u, v), z = z(u, v), where the functions x(u, v), y(u, v), z(u, v) are harmonic functions of (u, v) and x? + yu + zu = xv + y? + zV, +YYv + zuZv = O.

On the Axisymmetric Problem of the Theory of Elasticity for an Infinite Region Containing Two Spherical Cavities

Abstract This paper contains a solution in series form for the stress distribution in an infinite elastic medium which possesses two spherical cavities of the same size. The loading consists of tractions applied to the cavities, as well as of a uniform field of tractions at infinity, and both are assumed to be symmetric with respect to the common axis of symmetry of the cavities and with respect to the plane of geometric symmetry perpendicular to this axis. The loading is otherwise unrestricted. The solution is based upon the Boussinesq stress-function approach and apparently constitutes the first application of spherical dipolar co-ordinates in the theory of elasticity. Numerical evaluations are given for the case in which the surfaces of the cavities are free from tractions and the stress field at infinity is hydrostatic. The results illustrate the interference of two sources of stress concentration in a three-dimensional problem. The approach used here may be extended to cope with the general equilibrium problem for a region bounded by two nonconcentric spheres.

High Intensity X-Ray Monochromator

A monochromator for concentrating radiation of wave-length 0.71A has been made by disposing about a common symmetry axis a group of eight rocksalt crystals each plastically deformed and ground to the spindle surface curvature required for focusing radiation from a point source to a point image. The maximum monochromatic intensity at the focus exceeds the intensity of the total radiation from the molybdenum-target source when received directly at the same location.