Arbitrary orientation

A two-dimensional slice may be taken either parallel to one of the principal co-ordinate planes (X-Y, X-Z and Y-Z) selected from a menu, or in any arbitrary orientation defined on screen by the user. Once a slice through the data has been taken, and displayed on the screen, a number of tools are available to assist the operator with making measurements of indications. These tools allow measurement of distance between two points, calculation of 6dB or maximum amplitude length of a flaw, plotting of a 6dB contour, and textual aimotation of the view. Figure 11 shows 6dB sizing and annotation applied to a lack of fusion example. 

If the first plane is rotated through a full circle, the first radius of curvature will go through a minimum, and its value at this minimum is called the principal radius of curvature. The second principal radius of curvature is then that in the second plane, kept at right angles to the first. Because Fig. II-3 and Eq. II-7 are obtained by quite arbitrary orientation of the first plane, the radii R and R2 are not necessarily the principal radii of curvature. The pressure difference AP, cannot depend upon the manner in which and R2 are chosen, however, and it follows that the sum ( /R + l/f 2) is independent of how the first plane is oriented (although, of course, the second plane is always at right angles to it). 

Figure 18.12 The electron-density map is interpreted by fitting into it pieces of a polypeptide chain with known stereochemistry such as peptide groups and phenyl rings. The electron density (blue) is displayed on a graphics screen in combination with a part of the polypeptide chain (red) in an arbitrary orientation (a). The units of the polypeptide chain can then be rotated and translated relative to the electron density until a good fit is obtained (b). Notice that individual atoms are not resolved in such electron densities, there are instead lumps of density corresponding to groups of atoms. [Adapted from A. Jones Methods Enzym. (eds. H.W. Wyckoff, C.H. Hirs, and S.N. Timasheff) 115B 162, New York Academic Press, 1985.]...

Now that the basic stiffnesses and strengths have been defined for the principal material coordinates, we can proceed to determine how an orthotropic lamina behaves under biaxial stress states in Section 2.9. There, we must combine the information in principal material coordinates in order to define the stiffness and strength of a lamina at arbitrary orientations under arbitrary biaxial stress states. 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as... 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

Invariant stiffness concepts as developed by Tsai and Pagano [7-16 and 7-17] can be used as an aid to understanding the stiffnesses of laminates of arbitrary orientation and how those stiffnesses can be varied. The concepts and their use are discussed in Sections 7.7.3.1 through 7.7.3.3. 

The invariant stiffness concepts for a iamina will now be extended to a laminate. All results in this and succeeding subsections on invariant laminate stiffnesses were obtained by Tsai and Pagano [7-16 and 7-17]. The laminate is composed of orthotropic laminae with arbitrary orientations and thicknesses. The stiffnesses of the laminate in the x-y plane can be written in the usual manner as... 

Now we consider the two principal challenges presented by the systems under study. The first involves determination of the ground state for a lattice system of static dipole moments and implies Ho minimization over all possible orientations of the vectors eR,. Molecules are assumed to have uniform dipole moments, = n with arbitrary orientations, eR> in the absence of dipole-dipole interactions. On... 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

It should be noted that for geometrically equivalent nuclei a complicated ENDOR spectrum may be observed for arbitrary orientations of B0, if the hfs tensors are nearly... 

For an arbitrary orientation of B0, Beff will no longer be parallel or antiparallel to B0. The intensity ratio of transitions induced by l.h. and r.h. rotating fields is then not only determined by the anisotropic enhancement factor but also by the noncoincidence of Beff and B0. For proton hfs with Af1 < 15 MHz the residual lines induced by a r.h. rotating ... 

Fig. 23a, b. ENDOR with circularly polarized rf fields (CP-ENDOR). Single crystal ENDOR spectra of Cu(bipyam)2 (C104)2 diluted into Zn(bipyam)2(C104)2 arbitrary orientation, temperature 20 K. a) Conventional ENDOR spectrum (linearly polarized rf field), b) CP-ENDOR spectrum applied rf field right hand rotating. The spectrum is dominated by the eight nitrogen transitions with msAN > 0. (From Ref. 104)... 

We see that the requirement discussed in last section is satisfied we can associate one of the three values of E with each of the existing combinations nm. (Incidentally Pi = 0, in every case.) This shows that our system of polar coordinates with an arbitrary orientation of the axis is an adequate system for the quantization or, in other words, the atomic axis may have any direction. 

Additionally, one can consider the thermal expansivity in an arbitrary orientation direction. For films, the linear coefficient of thermal expansion at angle (p to the orientation axis is determined by... 

We can also use link polynomials to prove that certain unoriented links are topologically chiral. For example, let L denote the (4,2)-torus link which is illustrated on the left in Figure 12. This is called a torus link because it can be embedded on a torus (i.e. the surface of a doughnut) without any self-intersections. It is a (4,2)-torus link, because, when it lies on the torus, it twists four times around the torus in one direction, while wrapping two times around the torus the other way. Let L denote the oriented link that we get by putting an arbitrary orientation on each component of the (4,2)-torus link, for example, as we have done in Figure 12. Now the P-polynomial of L is P(L ) = r5m l - r3m x + ml 5 -m3r + 3m r3. 

In the principal coordinates, of course, there are only three nonzero components of the stress and strain-rate tensors. Upon rotation, all nine (six independent) tensor components must be determined. The nine tensor components are comprised of three vector components on each of three orthogonal planes that pass through a common point. Consider that the element represented by Fig. 2.16 has been shrunk to infinitesimal dimensions and that the stress state is to be represented in some arbitrary orientation (z, r, 6), rather than one aligned with the principal-coordinate direction (Z, R, 0). We seek to find the tensor components, resolved into the (z, r, 6) coordinate directions. 

The asymmetric-top eigenfunctions are linear combinations of symmetric-top eigenfunctions, which allows the asymmetric-top selection rules to be found. Since the dipole moment can have an arbitrary orientation, we get the three kinds of selection rules in (5.89) details are omitted. 

The kinetic equation (4.27) with the energy function (4.52), first studied by Brown [47], has since been investigated extensively [48,54-59]. However, due to mathematical difficulties, the case of arbitrary orientation of the external and anisotropy fields (i.e., vectors h and n) has been addressed only relatively recently. The numerical solution of the relaxation problem for h and n crossed under an arbitrary angle for the first time was given in Ref. 60. 

Fig. 1.5. Direction of the dipole moment d of an optical transition in a diatomic molecule (a), (c) parallel transition (b), (d) perpendicular transition (e) arbitrary orientation of the angular momentum J.

The powdered material sample to be tested is generally further ground in order to get a very fine powder, where the crystalline grains have arbitrary orientations. With the help of these random grain... 

Fig. 6. Bond Order Effects in the Type A Zwitterion Rearrangement. Note Basis orbitals are shown with arbitrary orientation, hence plus-minus overlaps do not imply antibonding between any particular orbital pair. However, with an odd number of such overlaps, the system is Mobius...

In this section the maximum stress concentration developed by an elliptic hole at arbitrary orientation in an infinite elastic sheet subjected to general biaxial stress loading is considered. In Figure 1, consider an elliptic hole of major axis a, minor axis b, and orientation angle fl with respect to an axis of applied uniaxial tension Si. If the sheet is isotropic, elastic, and infinite, then the major principal stress ([Pg.42]

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