Analysis of Oscillations of One-Dimensional Spatially Periodic Structures. an Unconventional Approach and Some New Effects

Abstract eng:
The present study is concerned with the analysis of oscillations of one-dimensional spatially periodic structures. There are many approaches for wave examination in such structures, in particular, methods, based on the utilization of the Floquet theory [1]. However, in the framework of this theory it is problematic to incorporate the (external) boundary conditions. The averaging procedure for processes in periodic systems based on the multiple scales method [2] combined with the averaging method [3] was proposed in the monograph [4]. In the present paper a new approach for the analysis of oscillations of one-dimensional spatially periodic structures, which is based on the method of direct separation of motions (MDSM) [5,6], is proposed. The approach is introduced in order to detect new effects in such structures’ behavior, which may be employed to produce systems with advanced properties. As an example of the approach application the study of oscillations of a string with variable crosssection is conducted. As the result, analytical expressions for the eigenmodes and the eigenfrequencies of the system are obtained. It is shown that modulation of the string cross-section leads to a change of the eigenfrequencies as compared with their non-modulated values, and to the emergence of a spectrum of additional high eigenfrequencies, which correspond to large wave lengths. A simple physical explanation of the latter effect, which is noted, apparently for the first time, is proposed. It is shown that character of string oscillations may be controlled by modulation of its cross-section, in particular, for given initial conditions the effect of high-frequency components suppression may be achieved.

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