Transformation from Principal Coordinates

In this equation the velocity components align with the (z, r, 6) coordinates, and not the principal coordinates. 

With the coordinate transformation in general form as discussed above, it is very difficult to see the cancellations and simplifications that render the result independent of the particular principal directions. In this section we work out all the details from one of the principal axes. 

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. 

As an illustration, we determine the stress components on the z face illustrated in Fig. 2.18. The cosines of the three angles between each of the principal coordinates 

There are three such angles for the r axis and three more for the G axis. Thus, in all, there are nine angles and nine directions cosines that completely define the rotation from (Z, R, 0) to (z, r, G). 

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Just as with a transformation from cartesian coordinates to polar coordinates, we do not necessarily use the principal value of the arctangent function, but must obtain an angle in the proper quadrant, with 0 ranging from 0 to 27t. 

It is possible to make a transformation from Cartesian coordinates to principal axes so that these expressions take the simpler forms ... 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

The principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

The bond polarization model gives the chemical shift of an atom a as the sum over the Na bonds of this bond. The bond contributions are formed of a component for the unpolarized bond (which also includes the inner shell contributions to the magnetic shielding) and a polarization term. The bond contributions are represented by a tensor with its principal axes along the basis vectors of the bond coordinate system. The transformation from the bond coordinate system into a common cartesian system is given by the transformation matrix Z). ... 

The Wigner rotations describe the coordinate transformations from the principal axis frame (P ) in which the tensor describing the interaction X is diagonal, via a molecule-fixed frame (C) and the rotor-fixed frame (R) to the laboratory frame (L) as illustrated in an ORTEP representation in Fig. 1. 

Fig. 3.14. Schematic representation of the coordinate transformation from the principal axis frame (Cp) through the molecular frame (Cm) and the reference frame (Cr) to the laboratory frame (Ci ).

It is sometimes convenient to transform to principal directions of/in steps, by going first to symmetry coordinate 5,. With the notation of Chapter 2, the displacement vector from the symmetric reference structure is... 

The three Euler angles used to characterize the coordinate transformation from the PAS to the FRAG are principally determined by the molecular electronic structure. Figure 3 shows the CS tensor directions with respect to the local molecular frame of 60CB and the FILAG frame. For C(ar)-C(ar) such as at carbon sites 2, 5, 6, and 9 in 60CB, the 5n component in the C CS tensor is parallel to the C-C bond and the 33 component is perpendicular to the molecular plane. For C(ar)-FI such as at sites 3, 4, 7, and 8, both 5n and 22 components lie on the molecular plane and the 5n component is parallel to the C-FI bond. For C=N such as at site 1, the 33 component is parallel to the C=N bond and the 5n component is perpendicular to the molecular plane of the benzene ring. Therefore, the Euler... 

The coefficients of the transformation from arbitrary to principal axes may then be determined by a procedure analogous to that employed in the transformation to normal coordinates as described in Chap. 2. 

Let us now deviate from the principal coordinate system by turning it by 45°. Such a transformation allows one to subdivide two additive parts of the tensor ... 

In the above analysis, and Y are expressed in the principal coordinate system, but in general, it is necessary to transform these vector functions from the principal coordinate system to the particle coordinate system through a rotation. The vector quasi-spherical wave functions can also be defined for biaxial media (e 7 y z) by considering the expansion of the tangential vector function T>c (3,a)Va + T>is j3, a)vjj in terms of vector spherical harmonics. 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

As an alternative to the foregoing procedure, we can express the strains in terms of the stresses in body coordinates by either (1) inversion of the stress-strain relations in Equation (2.84) or (2) transformation of the strain-stress relations in principal material coordinates from Equation (2.61),... 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

For each of the failure criteria, we will generate biaxial stresses by off-axis loading of a unidirectionally reinforced lamina. That is, the uniaxial off-axis stress at 0 to the fibers is transformed into biaxial stresses in the principal material coordinates as shown in Figure 2-35. From the stress-transformation equations in Figure 2-35, a uniaxial loading obviously cannot produce a state of mixed tension and compression in principal material coordinates. Thus, some other loading state must be applied to test any failure criterion against a condition of mixed tension and compression. 

As With the shear strength, the maximum shear strain is unaffected by the sign of the shear stress. The strains in principal material coordinates, 1- yi2 be found from the strains in body coordinates by transformation before the criterion can be applied. 

PCA finds a smaller number of factors that describe the majority of the variability or spread in the dataset. Using these factors often called principal components, the row space is transformed or mapped into a new coordinate system in which the principal components, which can combine the information from several variables, redefine the axes based upon the factors, and these new axes describe the degree of variation or spread in the dataset. 

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