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Vibrations of Shells and Plates, Third Edition

With more and more refined buildings fascinated by glossy engineering, wisdom of the complicated vibration habit of plates, shells, curved membranes, earrings, and different advanced constructions is key for today’s engineering scholars, because the habit is essentially varied than that of easy buildings equivalent to rods and beams.

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The classification of normal vibrations by irreducible representation of a factor-group is very important at the analysis of the Raman spectra and infrared absorption spectra, because it permits to predict the activity and polarization of the vibration lines [10,49]. At the group theoretical analysis of the optical vibrational spectra of molecular crystal the method of site symmetry is widely adopted; it was first suggested by Halford [55] and Hornig [56] and is in detail described in many papers [10,49,57-59].

The calculations were carried out by Kitaigorodskii's method of AAP/2 in the approximation of rigid molecules by numerical differentiation of crystal potential energy. The calculations of frequencies and the external vibrations of both naphthalene and anthracene crystals were carried out. For example, the following results were obtained for librational vibrations of naphthalene (in cm-1): Experimental. 125 109 74 71 51 46 I 126 134 94 7O 61 55 II 129 139 92 77 62 49 where column I contains the results of calculations carried out with the assumption of the libration only with respect to inertial axes of molecules, column II contains the results of the complete calculation which showed that the effective librational axis forms an angle of 30 ~ with respect to the inertial axis.

The calculation is in detail described in Ref. [45] for the case of molecular crystal where the calculation of the external vibrations of the orientationaUy disordered nitrochlorobenzene crystal is carried out. We note that the expressions obtained in Sect. A for the force constants are also applied to the nonideal crystals. B. Symmetrical Properties of Force Constants and Dyn~mlcal Matrix The potential energy of a crystal and, consequently, the force constants are invariant to the symmetry transformation under which the interatomic distances are remained unchanged [46].

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