Principal axes of translation

The subscripts 1, 2, 3 refer to the principal axes of translation, which are three mutually perpendicular axes fixed to the body defined such that if the body translates without rotation parallel to one of them it will experience a force only in that direction. 

The two triads of orthonormal vectors lie parallel to the principal axes of translation and to the principal axes of rotation at O, respectively. 

If a body possesses three mutually perpendicular planes of reflection symmetry, its center of reaction lies at the point of intersection of these planes. The coupling dyadic is zero at this point, whereas the translation dyadic and rotation dyadic at R adopt the forms shown in Eqs. (44) and (45), in which the principal axes of translation and rotation (at R) coincide and lie normal to the three symmetry planes. An ellipsoid is an example of such a body [see Eqs. (58)-(60)]. 

Only six coefficients are required to characterize the coupling dyadic at the center of reaction. But then an additional three scalars are required to specify the location of this point, so that the total number of independent scalars required for a complete characterization is still nine. Similarly, three scalars suffice for the translation dyadic if we refer them to the principal axes of translation [see Eq. (44)], but then three additional scalars (e.g., an appropriate set of Eulerian angles) are required to specify the orientations of these axes. So it comes down to the same thing—namely, that six scalars are required. The same is true of the rotation dyadic at any point, and of the coupling dyadic at the center of reaction. 

The utility of (134a) is most simply seen by considering the highly symmetrical case where one of the principal axes of translation of the particle lies parallel to a principal axis of the wall dyadic. In this case (134a) is equivalent to the scalar relation (B16)... 

The r vectors are the principal axes of inertia determined by diagonalization of the matrix of inertia (eq. (12.14)). By forming the matrix product P FP, the translation and rotational directions are removed from the force constant matrix, and consequently the six (five) trivial vibrations become exactly zero (within the numerical accuracy of the machine). 

Figure 12.17 Upper left panel contours of constant response in two-dimensional factor space. Upper right panel a subset of the contours of constant response. Lower left panel canonical axes translated to stationary point of response surface. Lower right panel canonical axes rotated to coincide with principal axes of response surface.

The skew-symmetric part S 4 is equivalent to a vector (x t)/2 with components (/. t),/2 = (/.jtk — /.ktj)/2, involving correlations between a libration and a perpendicular translation. The components of S 4 can be reduced to zero, and S made symmetric, by a change of origin. It can be shown that the origin shift that symmetrizes S also minimizes the trace of T. In terms of the coordinate system based on the principal axes of L, the required origin shifts p, are... 

The resulting description of the average rigid-body motion is in terms of six independently distributed instantaneous motions—three screw librations about nonintersecting axes (with screw pitches given by S11/Lll, etc.) and three translations. The parameter set consists of three libration and three translation amplitudes six angles of orientation for the principal axes of L and T six coordinates of axis displacement and three screw pitches, one of which has to be chosen arbitrarily again, for a total of 20 variables. 

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

Each selected configuration is translated and rotated in such a way that all of the solvent coordinates can be referred to a reference system centred on the centre of mass of the solute with the coordinate axes parallel to the principal axes of inertia of the solute. 

To actually use these results, it is of course necessary to actually calculate the components of the resistance tensors. We have seen that it is necessary to solve only three problems for translation and three problems for rotation in the coordinate directions to specify all of the components of A, B, C, and D. It probably does not need to be said that the orientation of the coordinate axes should be chosen to take advantage of any geometric symmetries that can simplify the fluid mechanics problems that must be solved. For example, if we wish to determine the force and/or torque on an ellipsoid for motions of arbitrary magnitude and direction (with respect to the body geometry), we should specify the components of the resistance tensors with respect to axes that are coincident with the principal axes of the ellipsoid, as this choice will simplify the fluid mechanics problems that are necessary to determine these components. If arbitrary velocities U and ft are then specified with respect to these same coordinate axes, the Eqs. (7-22) will yield force and torque components in this coordinate system. 

The symmetric tensor K = [Kij] is called translational. It characterizes the drag of a body under translational motion and depends only on the size and shape of the body. In the principal axes, the translation tensor is reduced to the diagonal form... 

A symmetric tensor ft is called a rotational tensor. It depends both on the shape and size of the particle and on the choice of the origin. The rotational tensor characterizes the drag under rotation of the body and has the diagonal form with entries fii, Q2, 3 in the principal axes (the positions of the principal axes of the rotational and translational tensors in space are different). For axisymmetric bodies, one of the major axes (for instance, the first) is parallel to the symmetry axis, and in this case = O3. For a spherical particle, we have fii = fl2 = flj. 

If the velocity of a spherical particle in Stokes settling is always codirected with the gravity force, even for homogeneous axisymmetric particles the velocity is directed vertically if and only if the vertical coincides with one of the principal axes of the translational tensor K. If the angle between the symmetry axis and the vertical is [Pg.85]

The sliip s movement translates into loads on the three principal axes of the vessel. Saddles and lashings must be strong enough to resist these external forces without exceeding some allowable. stress point in the vessel. The point of application of the load is at the C.G. of the vessel. These loads affect the vessel in the same manner as seismic forces... 

The standard model for diffusive motion in polymers is Brownian diffusion, which occurs as a series of infinitesimal reorientational steps. This model is most appropriate for intermediate-to-large sized spin probes and spin-labeled macromolecules, where the macromolecule is much larger than any solvent molecules. Because of this broad applicability, the Brownian diffusion model is the most widely used. This type of rotational diffusion is completely analogous to the one-dimensional random walk used to describe translational diffusion in standard physical chanistry texts, with the difference that the steps are described in terms of a small rotational step 59 that can occur in either the positive or negative direction. In three dimensions, rotations about each of three principal axes of the nitroxide must be taken into account. A diffusion constant may be defined for each of these rotations motions, in a way that is completely analogous to the definition of translational diffusion constant for the one-dimensional random walk. 

CoMMA uses 13 descriptors of shape and electrostatic properties. The pure shape descriptors are the three principal moments of inertia, and the two pure electrostatic descriptors are the magnitude of the dipole moment and the magnitude of the principal quadrupole moment. To complement these, six additional descriptors that relate shape and charge are calculated the dipolar components as well as the magnitudes of the components of displacement between the center of mass and the center of dipole along the three principal axes of inertia. Last, quadrupolar components are calculated with respect to a translated initial reference frame whose origin coincides with the center of dipole. 

Liquid crystals are classified by symmetry. As it is well known, isotropic liquids with spherically symmetric molecules are invariant under rotational, 0(3), and translational, T(3), transformations. Thus, the group of symmetries of an isotropic liquid is 0(3)xT(3). However, by decreasing the temperature of these liquids, the translational symmetry T(3) is usually broken corresponding to the isotropic liquid-solid transition. In contrast, for a liquid formed by anisotropic molecules, by diminishing the temperature the rotational symmetry is broken 0(3) instead, which leads to the ap>p)earance of a liquid crystal. The mesophase for which only the rotational invariance has been broken is called nematic. The centers of mass of the molecules of a nematic have arbitrary positions whereas the principal axes of their molecules are spontaneously oriented along a preferred direction n, as shown in Fig. 1. If the temperature decreases even more, the symmetry T(3) is also partially broken. The mesophases exhibiting the translational symmetry T(2) are called smectics (see Fig. 1), and those having the symmetry T(l) are called columnar phases (not shown). 

The polar decomposition theorem expresses a general second order tensor as a product of a positive symmetric tensor with an orthogonal tensor. This is very useful in interpreting deformation processes in terms of a translation, a rigid rotation of the principal axes of strain, and stretching along these axes (Jaunzemis 1967). 

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