Complex frequency characteristic and Green’s function for cantilever beam under longitudinal oscillations

Abstract

Relevant objective of this article is elaboration of methods of efficient determination of the dynamic characteristics of the materials both within the framework of well-known and classical models as well as within the advanced ones in the context of interaction with the environment under the impact of external dynamic loads. In the present work free and forced oscillations are studied with a help of spectral methods what allows to compare theoretical results with experimental data, obtained by other authors. The key point of the work is theoretical determination of the proper functions of the oscillatory system, its frequency characteristic using Green’s function. We examine longitudinal oscillations, caused by the substitute load, which provokes oscillations of a sound spectrum. Univariate character of the task is determined by the linearity organization of the longitudinal oscillations of a beam. The oscillations are modeled with a help of the substitute load, which is a subject to Dirac impulse function, which belongs to a class of generalized functions. The obtained solution makes it possible to use the proposed approach in order to determine frequency characteristics in the acoustic spectrum, taking into account that the oscillations are included in a full frequency spectrum and can be used for an adequate valuation of the possibility of the appearance of resonance phenomena.