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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 3038179, 9 pages
https://doi.org/10.1155/2017/3038179
Research Article

Principal Component Analysis in the Nonlinear Dynamics of Beams: Purification of the Signal from Noise Induced by the Nonlinearity of Beam Vibrations

1Department of Applied Mathematics and Systems Analysis, Saratov State Technical University, 77 Politeknicheskaya Str., Saratov 41054, Russia
2Cybernetics Institute, National Research Tomsk Polytechnic University, 30 Lenin Avenue, Tomsk 634050, Russia
3Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland
4Department of Mathematics and Modelling, Saratov State Technical University, 77 Politeknicheskaya Str., Saratov 41054, Russia

Correspondence should be addressed to Jan Awrejcewicz

Received 10 October 2016; Accepted 28 November 2016; Published 16 January 2017

Academic Editor: Emmanuel Lorin

Copyright © 2017 A. V. Krysko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).