Ellipsoidal axis ratio

For prolate spheroids, u,. is the ratio between the major and the minor axes of the elhpsoid. When applying Eq. 5.1 to cylinders of the length-to-diameter aspect ratio fl/j, one may take roughly. Empirical equivalent ellipsoidal axis ratios... 

Hadinata C, Zheng R, Kennedy PK, Zhu P, Edward G, Lee Wo D, Tanner RI (2008) The effects of additives on material properties and shrinkage characteristics. Keynote Lecture at 24th Annual Conference of the Polymer Processing Society, Salerno Hagerman EM (1973) Weld-line fracture in molded parts. Plast Eng 29 67-69 Hand GL (1962) A thecny of anisotropic fluids. J Fluid Mech 13 33-46 Harris JB, Pittma TFT (1975) Equivalent ellipsoidal axis ratios of slender tod-Uke particles. J Coll Interf Sci 50 280-282... 

Most standard ellipsoidal heads are manufactured with a major and minor axis ratio of 2 1. For this ratio, the following equation can be used to calculate the minimum thickness required ... 

The ASME design formula from UG-32 for ellipsoidal heads having a major-to-minor axis ratio of 2 1 and being subjected to pressure on the concave side is t = PD/ 2SE — 0.2P), where t is the minimum thickness of the head (in.), P is the MAWP (psi), D is the internal diameter of the major diameter of the ellipsoid (in.) (and equal to the inside diameter of the vessel), S is the allowable stress of the material, as listed in ASME Section II, and E is the weld joint efficiency. ... 

The ellipsoidal dished head with a major to minor axis ratio of 2 1 is popular for economic reasons, even though the theory for thin-walled vessels predicts that the head of this shape should have twice the thickness of a hemispherical head where the major and minor axes are equal. Such an ellipsoidal head used for vessels under internal pressure has the same thickness as the cylindrical shell if the same allowable stresses and joint efficiencies are applied to both parts. The 1962 ASME Code Section VIII, Division 1 gives the following equation for the thin-walled ellipsoidal dished heads with a 2 1 major to minor axis ratio ... 

At the extensions of the Maxwell Garnett theory for parallel-oriented ellipsoids, equation (13) [23, 24], all particles are ellipsoids with parallel the mean axis. Only one depolarization factor L is necessary. L describes the ratio between the axes of the ellipsoids, and values for L between 0spherical particles (L =i), equation (13) gives the same result as the Maxwell Garnett theory, equation (12). For particles embedded in one plane, the orientation of the ellipsoids in ratio to the substrate is parallel or perpendicular to the plane of the incident light. An orientation of the ellipsoidal particles diagonally to the substrate cannot be considered. 

Much greater insight into nuclear framework may be gained by making use of the Kerr effect. It is often possible to tell the polarization ellipsoid of the molecule by axis ratios and position relative to the nuclear frame-... 

Simha concerning the dissymetry of protein molecules. The axis ratio/ of an ellipsoid of revolution, which represents the particle, is calculated from diffusion measurements in the ultracentrifuge (column 2) and compared with values obtained from viscosity measurements using equation (103) on page 285. The agreement between the two methods is rather satisfactory. 

Figure 2. Pressure-volume curves for 4 ellipsoids (including sphere). The curve corresponding to undeformed major/minor axis ratio = 1-927 coincides with that for the sphere. (Reproduced from...

Volume Relationships. A cylindrical vessel closed at both ends with elliptical dished heads has a volume equal to the volume of the cylindrical section plus twice the volume contained in one of the heads. The volume contained in a head can be expressed in terms of a cylinder of e( uiva-lent volume having the same inside diameter as the cylindrical section of the heafl. Figure 5.2 is a cross section of an ellipsoidal head having a 2 1 major-to-minor-axis ratio. 

Tims the volume of two ellipsoidal heads having a major-to-minor-axis ratio of 2.0- is ... 

Let us now consider the case of a polarized-distorted sphere such as the one in Figure 8. We would expect that associated with the polarization process there is a distortion process because dipoles are associated with some real molecular group that has a shape. In other words, when the dipoles orient in the electric field, there is an average projection in the field direction that is different from the average when the field is turned off. At this point it is necessary to postulate a molecular mechanism for the distorted sphere. Consider the sphere to consist of N ellipsoids with a major (2a) to minor (a) axis ratio of 2. The dipole moment = 4D units, which was computed from Eq. (12), , = 1) 7 = room... 

In the above discussion and in reports published up to now, it is always considered that the appearance of a voxel resembles a spinning ellipsoid with axis lengths a=bgeneral linearly-polarized laser beam is employed. Shown in Fig. 30a is the lateral SEM image of a voxel formed using 1.4 NA optics when the Ti Sapphire laser output was utilized as it was. The voxel axis lengths are 2a=325 nm (x direction) and 2b = 295 nm (y direction), giving rise to a lateral axis ratio jUjj=a/b=l.l. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

A similar acoustic technique was applied by Pickles and Bittleston (1983) to investigate blast produced by an elongated, or cigar-shaped, cloud. The cloud was modeled as an ellipsoid with an aspect ratio of 10. The explosion was simulated by a continuous distribution of volume sources along the main axis with a strength proportional to the local cross-sectional area of the ellipsoid. The blast produced by such a vapor cloud explosion was shown to be highly directional along the main axis. 

The insight from AFM images may be greatly boosted by sophisticated image analysis. Fritzsche and Henderson [30,31] have extracted cross-sections of nucleosomes at half-maximum height and have fitted them to virtual ellipsoids. These ellipsoids had relatively smooth perimeter and an aspect ratio of 1.2 1.4 moreover, the orientation of the ellipsoids was correlated with the direction of the fiber axis, with more than 50% of nucleosomes aligned with the axis. While this orientation effect may result from surface interactions, as discussed by the authors themselves, it may also represent an actual, and structurally important, feature of fiber structure. Ellipsoid-shaped nucleosomes have been reported in electron EM studies [32,33], and have been predicted in models of chromatin... 

Adding LCP and RCP components of different amplitudes, yields elliptically polarized light. The major axis of the ellipsoid is the sum of amplitudes AR and AL, and the minor axis is their difference, as shown in Fig. 5. The ellipticity 0 is defined as the arctangent of the ratio of the minor axis to the major axis ... 

Corpuscular particles which deviate from spherical shape can often be treated theoretically as ellipsoids of revolution. Many proteins approximate this shape. An ellipsoid of revolution is characterised by its axial ratio, which is the ratio of the single half-axis a to the radius of revolution b. The axial ratio is greater than unity for a prolate (rugby-football-shaped) ellipsoid, and less than unity for an oblate (discus-shaped) ellipsoid. 

Fiber motion — Jeffery orbits. The motion of ellipsoids in uniform, viscous shear flow of a Newtonian fluid was analyzed by Jeffery [32, 33] in 1922. For a prolate spheroid of aspect ratio a (defined as the ratio between the major axis and the minor axis) in simple shear flow, u°° = (zj), the angular motion of the spheroid is described... 

The sedimentation of isometric particles, such as cubes and octa-hedra, deviates only slightly from the Stokes equation. Significantly anisometric particles that can be approximated by ellipsoids of revolution are amenable to rigorous sizing by centrifugation if the axial ratios q are known. If the major semi-axis is a and the minor semi-axis is c, the sedimentation velocity dx/dt can be written as... 

Various hydrodynamic studies of proteins, for example viscosimetry and diffusion, have shown that proteins behave in solution as fairly compact and symmetric particles. In time, equations were developed to describe the overall size and shape of proteins in terms of ellipsoids of revolution. Two basic types of ellipsoids of revolution, the prolate and oblate, are used to depict the approximate three-dimensional shapes of protein molecules. These are shown in Figure 4.12. Each ellipsoid of revolution may also be described by an axial ratio, that is, the ratio of the short axis to the long axis. A spherical protein molecule has an axial ratio of 1 1, whereas an asymmetric molecule, such as fibrinogen, may have an axial ratio of 1 10. 

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