A new approach to the analysis of oscillations of one-dimensional spatially periodic structures

A new approach to the analysis of oscillations of spatially periodic structures, which is based on the method of direct separation of motions (MDSM), is proposed in the paper. As an example of the approach application the study of oscillations of a string with variable cross-section 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.It is noted that application of the multiple scales method and other classical asymptotic methods for solving the considered equations is rather cumbersome and leads to erroneous results, because an unknown parameter is present in them. The fundamental difference between the MDSM and these methods is revealed. In particular, it is shown that the MDSM may be employed for solving equations without small parameter. Thereby, the applicability range of this method turns out to be broader than the one of the classical asymptotic methods. To illustrate the validity of this statement a classical problem about the stability of a pendulum with vibrating suspension axis is considered.