Advances in Mathematical Physics

Volume 2017, Article ID 3038179, 9 pages

https://doi.org/10.1155/2017/3038179

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

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

Correspondence should be addressed to Jan Awrejcewicz; lp.zdol.p@wecjerwa

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).

#### 1. Introduction

Numerous methods can be employed to clean a signal from noise. For instance, wavelet transform [1], Fourier transform [2], Hilbert-Huang transform [3], or empirical mode decomposition [4] is often applied for this purpose. In all of the methods, the goal is to decompose the signal to a variety of fundamental constituents and enable analysing each of them separately. As a result, various features of the signal can be detected and the signal can be properly analysed. In this paper, the attention is paid to the method called the principal component analysis (PCA), which is widely used to amplify variation in the spectral data and reveal strong patterns in a dataset.

PCA [5] is an approach usually utilised to simplify investigation and visualisation of data. In particular, it is commonly used in pattern recognition and compression of various types of data, especially images, having high correlation between their components. For instance, it is implemented in frontal face databases, where the goal is to process images in such a way that noise is removed from a neighbourhood of a block of pixels. In order to accomplish it, it is necessary to present this block as a set of points in a multidimensional space. Then, the PCA can be applied, as a result of which only the first conversion components, assumed to contain the most useful information, are left. The remaining components are said to comprise unwanted noise. Thus, if an inverse transformation is applied to the first conversion components, a denoised image is obtained as output. One of advantages of this method is that no specified accuracy is required.

To assess the number of principal components in the appropriate proportion of variance, an objective approach can be always applied. However, when the* signal* and* noise* are not separated, no predetermined accuracy is of particular meaning. Thus, if one considers projections of the components on different planes, the components of the* signal* are tightly packed, keeping the amplitude relatively large. On the contrary,* noise* components associated with a relatively small amplitude occupy larger space (are scattered more strongly). Thus, in the above-described case, the principal component analysis plays a role of a filter. Namely, the signal is mainly contained in the projection of the first principal components, whereas other components exhibit much higher noise content.

In this paper, the principal component analysis is employed to analyse the signal coming from vibrations of beams. For this purpose, it is necessary to supplement this method with a mathematical model of a vibrating beam. As it is well known, there are several theories associated with the beam model. Particularly, Bernoulli-Euler [6] and Timoshenko model [7] are widely employed in many mechanical problems. In addition, a generalisation of the Timoshenko model, invented in 1964 by two Ukrainian scientists, Sheremetev and Pelekh [8], is often considered. It should be mentioned that this model was rediscovered 27 years later by Levinson [9] and Reddy [10]; thus in English literature, it is typically called the Levinson-Reddy model.

In this paper, a comparative analysis of the amplitude-frequency characteristics for all the above-mentioned, that is, Euler-Bernoulli, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy, beam models is conducted. Comparison of the amplitude-frequency characteristics is based on the signal analysis of solutions of systems of linear and nonlinear differential equations.

#### 2. Problem Formulation

The study is conducted on a single-layer beam occupying a two-dimensional region of space with the Cartesian coordinate system introduced in the following way: the axis is directed from the left to the right along the beam midline, and the axis is directed downwards, perpendicularly to the axis (see Figure 1). In the introduced coordinate system, the two-dimensional domain Ω of the beam is defined as follows: , . Here and below we use the following notation: denotes the beam height and stands for the beam length. The beam is subjected to the load acting in the direction normal to the midline of the beam. In addition, the beam lies on an elastic Winkler foundation.