Rigid body translation

FIG. 2.6 Temperature dependence of the rigid-body translational mean-square amplitudes of quinolinic acid. The line represents the results from Eq. (2.51) with v = 44 cm. Source Takusagawa and Koetzle (1979). 

For a molecule with N nuclei, there are 3N degrees of freedom, with three ascribed to rigid-body translation, three to rotation, and 3N-6 to vibration, or the change of the reciprocal positions of the nuclei. The classical potential energy of such vibrations can be written as ... 

The constants in Eqns. (3.25a) and (3.25b) represent rigid-body translations in the X and y directions, respectively, and need not be considered in the calculation of... 

The constants in Eqns. (3.36) and (3.37) represent rigid-body translation and may be disregarded in considering deformation within the body. The functions fi(y) and fzix) may be examined through a consideration of the shearing strain yxy. From Eqns. (3.6), (3.7), and (3.24),... 

Fig. 10.5. HREM image of [001] tilt grain boundary in NiO. The = 5/(310) symmetrie tilt grain boundary shows two variants, separated by a small step. The sehematie CSL drawing (below) illustrates that the step introduees an effeetive rigid-body translation parallel to the grain boundary [10.30].

In its most general form, the Green s function matrix is of order 3N X 3N, where N is the number of atoms in the system. However, rigid body translation is usually not of interest, and its contribution to the Green s function matrix... 

In a formal treatment of such interfaces the different structures considered here can, in principle, be described by choosing different rigid-body translation vectors. However, such translations are not the small relaxations familiar in, for example, the 112 lateral twin boundary in Al, but are more closely related to stacking faults in fee metals. 

We caimot assume that all the ions have found their ideal sites in sintered material, e.g., some AP" ions in MgAl204 may be on tetrahedral sites. The structure predicted by computer modeling for the [112 lateral twin interface in NiO contains a rigid-body translation. Such a translation is not observed experimentally for the same type of interface in spinel, which has the same oxygen sublattice. It may be that the reason for this difference is that the translation-free configuration is what is present on a migrating GB and this becomes frozen in when the sample is cooled. The structure predicted by minimum energy calculations is a stationary structure. 

The natural vibration modes are shown in the accompanying figure. The first two modes, with zero natural frequencies, are rigid body translation and rigid body rotation, respectively. Modes 3 to 5 have the same vibration frequencies as the first three modes of the beam clamped at both ends, although the mode shapes are not identical. 

If the material is displaced relative to the coordinate system in a rigid-body translation, the displacement vectors are the same at any material point, u x) = const. This yields duijdxj = 0 and thus Sij = 0 as should be expected. This result is intuitively obvious, for a rigid-body translation does not cause strains. 

Newton-Euler equations for rigid body translation and rotation (9). For molecules possessing internal degrees of freedom as well, the equations become rapidly unmanageable when the molecular complexity increases. Also the solution of rigid body Newton-Euler equations involves special care because division by the sine of an Euler angle occurs which requires... 

All interfaces in spinel, even the S = 3, (111) twin boundary, can exist with at least two different structures. In a formal treatment of such interfaces the different structures considered here can, in principle, be described by choosing different rigid-body translation vectors. However, such translations are not the small relaxations familiar in, for example, the 112 lateral twin boundary in Al, but are more closely related to stacking faults in fee metals. The image illustrated in Figure 14.21 shows two parallel 111 twin boundaries (separated by a microtwin). The translations at the two twin boundaries are different as you can see in the insets. 

This leaves one point, the origin, unchanged, and thus rigid-body translations have been automatically eliminated by this approach. We adopt the Lagrangian approach to strain, in which the reference state is the undeformed state of the material, in contrast to the alternative Eulerian model, in which reference is made to the deformed state. We can rewrite Equation (3.1) in matrix form as... 

The rigid body orientation of a beam element is described by a rigid body translation and a sequence of rotations. The local coordinate systems, H and E, of the beam element are attached to the endpoint of that element. Coordinate system H differs from the inertial coodinate system, G, by a translation of Xj = [ Xg Yq Zg ] and coordinate system E is defined to have the same origin as H with its x-axis, colinear with the elastic axis of the beam element. The position vector, relative to local coordinate system, E, of any material point P, on an elastic beam element can be descibed as. 

All of the docking algorithms described above utilize GAs. Gehihaar et al. describe a rapid docking procedure that uses an EP and its application to HIV protease. An encoding of three rigid body translations, three rigid body rotations, and rotations around flexible bonds was used. The fitne.ss function was similar to that used by Verkhivker et al. and utilized simple pairwise atomic potentials. 

The projected frequencies can then be calculated from the projected Hessian in the usual fashion. If mass-weighted Cartesian coordinates are u.sed, rigid body translation and rotation are also projected out. In addition to the projected frequencies, one also needs the coupling matrix elements, B,... 

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