Rotation of axes

FIGURE 10.3 Data from the 2D pulse sequence in Fig. 10.2. (a) Spectra resulting from Fourier transform of S(t2) at various values of f,. (f>) Rotation of axes in (a) to show modulation in signal amplitude as a function of. (c) Signal after second Fourier transform with respect to showing a single peak at coordinates (Q,t/2TT, D./27T). (d) Contour plot, showing the peak from (c). Courtesy of Ad Bax (National Institutes of Health). 

Rotation of axes with an equation chosen to make y = 0 ... 

The measurement of the diffracted intensities, which is necessary for a complete study of a film s texture, is done by recording the variations of the intensity diffracted at the angle 20 that corresponds to this family of planes during the rotation of axes and %. Therefore, most of the time, a grid along ( ) and x is defined, and the intensity distribution is measured incrementally, with each increment equal to a fraction of a degree. 

For finite strain in isotropic media, only states of homogeneous pure strain will be considered, i.e. states of uniform strain in the medium, with all shear components zero. This is not as restrictive as it might first appear to be, because for small strains a shear strain is exactly equivalent to equal compressive and extensional strains applied at 90° to each other and at 45° to the original axes along which the shear was applied (see problem 6.1). Thus a shear is transformed into a state of homogeneous pure strain simply by a rotation of axes by 45°. A similar transformation can be made for finite strains, but the rotation is then not 45°. All states of homogeneous strain can thus be regarded as pure if suitable axes are chosen. 

Figure 2.20 Two-dimensional rotation of axes around Xj, and associated angles a..

Any symmetric second-order tensor can be transformed by a rotation of axes to a form in which the tensor only contains the three diagonal components, the olf-diagonal components being zero. For this state, the diagonal components are called the principal values and the particular set of axes are called the principal axes. 

As with stresses, transformations of strain elements from one coordinate-axis system into another obtained by rotations of axes obey the same transformation laws as those of stresses, utilizing the same direction cosines and making the symmetrical strain matrix also a tensor (McClintock and Argon 1966). 

I his change of coordinates is more general than a mere rotation of axes, which is a special case. The inverse of (6) may be written as... 

In the most general case of rotation of axes the expansion will involve 81 terms. Here, however, the z axis is perpendicular to the y axis, so that ay2 will be zero. 

No transformation is needed for which is always real. The notation reflects the fact that 2/fc transforms like cos k(j) and like sin k(j). The factors of /T/2 ensure that a rotation of axes induces an orthogonal transformation of the moments. The first few of these moments coincide precisely with the Cartesian charge and dipole moment, and later ones describe the quadrupole, octopole and so on. A complete list is given in Table 1 for moments up to hexadecapole. 

The first rotation of axes will be to rotate the (1, 2, 3) axes by an angle 4> in the 1-2 plane as shown in Fig. 1.6a. We label these new axes as (1,2,3) and the transformation matrix, by analogy with the two-dimensional system. 

Figure 1.6 The specific rotations of axes that define the Euler angles 9, 4>, and Y.

In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required first to simplify terms in Hq, second to simplify terms in He, and third to suspend nonautonomous terms in The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. 

Rotation of axes. The cross product terms in (3.22b) may be eliminated by... 

The values of the coefficients will depend on the geometry of the system. However, there are certain relations involving the which are independent of the geometry. Equation (IIIB-5) is equi valent to a rotation of axes from the coordinate system (i = 1. . . iV) to the coordinate system j. (if = 1. . . Af) therefore, the are components of a unitary matrix (Eyring, Walter,... 

Isotropic Materials which have the same mechanical properties in all directions at an arbitrary point or materials whose properties are invariant upon rotation of axes at a point. Amorphous materials are isotropic. 

Equation (87) is already a reduced standard form coming from a rotation of axes and a suitable scaling of E of any more general form. E should have also a non-vanishing quadratic approximation for all other variables y. Now we look at the 2D germ of form (87) and treat different cases of the constants a and b. We look for straight lines... 

Other mles are based on numerical evaluation of the eigenvalues it is assumed that in the case of perfect independence among variables, the PC will be the same as the original variables (PCA represents an invariant rotation of axes) and will account for unitary variance in case of autoscaled data, thus a PC with an eigenvalue less than 1 contains less information of one original variable and could be discarded (this rule sometimes is also taken as eigenvalues... 

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