Second moment of inertia

Fig. 4. In the top two panels, the experimental alignment (i) as a function of rotational frequency for bands in 185 186Au [LAR85], 184Pt [CAR85], and 185Pt [WAD85]. Reference parameters of JLC= 22 li2/MeV and =1101i4/MeV3 are used. In the bottom panel, the second moment of inertia is plotted versus frequency for selected bands.

The VD polyhedra model also allows one to introduce two important parameters describing distortion of the atomic coordination [16, 17], The first of these is the Da vector that originates in the A atom and ends in the centroid of the VD polyhedron. According to [18], length of the Da vector is proportional to the gradient of the local electric field created by the atoms surrounding the A atom in the structure. The second parameter is the second moment of inertia, Gs, which describes the deviation of the VD polyhedron from an ideal sphere and characterizes uniformity of distribution of the X and Z atoms around A. For an ideal sphere, G3 = 0.077, whereas for an ideal AXg octahedron (which correspond to a cubic VD polyhedron), G3 = 0.0833 [17]. 

CN - coordination number Nf- average number of faces of the VDPs N t, - average number of non-bonding contacts per one U-O bond PVdp -volume of the VD polyhedron Svdp - total area of faces of the VDP Rsd- radius of the sphere with volume equal to that of the VDP Da -vector that originates in the U atom and ends in the centroid of the VD polyhedron G3-the second moment of inertia, which describes deviation of the VD polyhedron from ideal sphere A - difference between the shortest and the longest bonds in the coordination polyhedron p - total number of faces. Standard deviations are given in parentheses. 

The variables A and I are the area and second moment of inertia of the cross section of the beam. The coefficient y is given by ... 

In all materials, elementary mechanical theory demonstrates that some shapes resist deformation from external loads or residual stresses in processing better than do others. This phenomenon stems from the basic physical fact that deformation in beam and sheet sections depends upon the product of modulus (E) and the second moment of inertia (/), commonly expressed as El. The physical part performance can be changed by varying the moment of inertia or the modulus or both (see Chapter 3). 

Combination differences of the observed wavenumbers of the lines in the absorption bands of symmetric, spherical, and asymmetric rotors can be obtained. However, the rotational constant B to be calculated will depend on the type of vibration the band represents. In considering a symmetric rotor, the rotational constant for one of the molecule s moments of inertia is obtained from a parallel vibration, while a second moment of inertia is determined from the perpendicular vibration. Thus, the spacing of lines in the bands of a symmetric rotor is dependent upon the type of vibration. Moreover the interlinear spaces are not equal, as they are for linear or symmetric triatomic molecules. The perturbations of the fine-line structure of bands that have been discussed in other sections must also be taken into account before a combination difference is formed and a rotational constant calculated. 

In the undamped case, tmlike in the case of longitudinal motion, the phase and group speeds related to flexural waves depend on the cross-section properties, given by the area and the second moment of inertia. The group speed is twice the phase speed and both depend on frequency and, hence, are dispersive. It should be noted that, with the Euler-Bemoulli assumptions, the wave speeds tend to infinity as the frequency increases. 

Since density is changing as a fimction of position inside the foam, the concept of apparent flexural modulus is used here. Using the definition of flexural stiflhess S, which is the product between modulus E and the second moment of inertia ly, one can write [6,9] ... 

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