Normal vibration frequencies of two- and three-dimensional arrays of alternating point masses |
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Authors: | S.M. Lee |
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Affiliation: | Department of Physics, Michigan Technological University, Houghton, Michigan 49931, U.S.A. |
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Abstract: | Normal vibration frequencies are calculated for alternating point-masses of mass M and m in two-dimensional rectangular and a three-dimensional parallelepiped arrays. The method of intermediate co-ordinate transformation is used in the analysis. In applying this method, the vibrating systems consisting of alternating masses are treated as the basic systems of linear chains of identical point-masses of mass m (the zero-order system) that are coupled together to form the two- and three-dimensional systems and then further modified by inclusion of an additional mass ?m at alternating sites, where M = (1 + ?)m. The zero-order solution for the linear array of identical particles of mass m is well known. The present calculation shows the advantage of the process of intermediate co-ordinate transformation in simplifying problems which are otherwise complicated and also illustrates how such a process can be applied in more than one successive step. |
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