Rigid body model

A considerable simplification is achieved when molecules can be treated as rigid bodies, as was done for naphthalene and anthracene (Fig. 2.2), the frequency spectra of which were derived using atom-atom potential functions. The mean-square displacements due to the internal modes can be calculated from the experimental infrared and Raman force constants, and added to the values obtained with Eq. (2.58). The rigid-body model for thermal vibrations is further discussed in section 2.3.3. 

The next refinement of the model takes into account that the shape of most molecular species differs from being rod-like typical nematogenic molecules are given in Table 4.6-1. The resulting behaviour of such a bi-axial molecule is often associated with hindered rotation, however it can also be understood from a rigid-body model where different moments of inertia lead to oscillations of different angular amplitudes in spite of identical (thermal) excitation and identical repulsive forces (Korte, 1983). This can be summarized by order parameters defined as above but referring to one of the two shorter. 

The experimental and hypothetical crystal structures are superimposed in Figure 6.9 using the COMPACK algorithm. The RMS deviation between these structures is fairly small. The W99 potential with the multipole description of the electrostatics, after a minimization of the structures using a partial rigid body model and the DREIDING potential, is a potential good model for this systan. 

Fig. 1. A simple rigid body model of a lumbar motion segment unit coupled by springs...

We have at our disposal various scales at which we can look at details of the molecule under study. In the crudest approach we may treat the molecule as a point mass, which contributes to the gas pressure. Next we might become interested in the shape of the molecule, and we may approximate it first as a rigid rotator and get an estimation of rotational levels we can expect. Then we may leave the rigid body model and allow the atoms of the molecule to vibrate about their equilibrium positions. In such a case we need to know the corresponding force constants. This requires either choosing the structural formula (chemical bond pattern) of the molecule... 

If the ADPs from a standard least-squares are consistent with the rigid-body model, bond lengths can... 

Owing to the fact that absolute values of correlation times are usually not available, interproton distances cannot be directly calculated. Distances are instead obtained by calibration of the cross peak intensities against an internal distance standard, usually the distance between diastereotopic geminal protons (178 pm) or aromatic protons of Tyr (242 pm). Assuming isotropic tumbling and rigid-body model for all parts of the molecule. Equation [9] is then used to calculate all interproton distances ... 

In the present pqrer, we present a simple rigid body model of a spline coupling and use it to determine normal and tangential displacements in the cont region r en misalignment is imposed. We also consider the overall equilibrium and stability of the coiqrling. These results are then compared to those from a boundary element model Mdiich includes elasticity and stick-slip friction in the contact Wear depth predictions are also made. A locked spline, in which the shaft axial location is controlled by a nut and a shoulder is also examined. 

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