Shock and Vibration

Volume 2016, Article ID 3409897, 9 pages

http://dx.doi.org/10.1155/2016/3409897

## Detrended Fluctuation Analysis and Hough Transform Based Self-Adaptation Double-Scale Feature Extraction of Gear Vibration Signals

Hubei Province Key Lab of Machine Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, P.O. Box 222, Wuhan, Hubei 430081, China

Received 27 July 2015; Revised 2 December 2015; Accepted 7 December 2015

Academic Editor: Evgeny Petrov

Copyright © 2016 JiaQing Wang 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

This paper presents the analysis of the vibration time series of a gear system acquired by piezoelectric acceleration transducer using the detrended fluctuation analysis (DFA). The experimental results show that gear vibration signals behave as double-scale characteristics, which means that the signals exhibit the self-similarity characteristics in two different time scales. For further understanding, the simulation analysis is performed to investigate the reasons for double-scale of gear’s fault vibration signal. According to the analysis results, a DFA double logarithmic plot based feature vector combined with scale exponent and intercept of the small time scale is utilized to achieve a better performance of fault identification. Furthermore, to detect the crossover point of two time scales automatically, a new approach based on the Hough transform is proposed and validated by a group of experimental tests. The results indicate that, comparing with the traditional DFA, the faulty gear conditions can be identified better by analyzing the double-scale characteristics of DFA. In addition, the influence of trend order of DFA on recognition rate of fault gears is discussed.

#### 1. Introduction

Generally, the gear transmission systems are characterized with periodic behaviors. However, the defects of gears, bearings, or transmission shafts may cause the nonlinear vibration. The gearbox vibration signals captured by the sensors are complicated, nonlinear, and nonstationary [1, 2]. Many researchers verified that the vibration time series of the gear transmission systems exhibit nonlinearity and self-similarity [1, 3, 4]. Therefore, a lot of the nonlinear time series analysis methods and several nonlinear characteristic quantities such as fractal dimension [5], entropy [6], and the Lyapunov exponent [7] have been employed to detect the faults. Though these nonlinear based methods may be suitable to analyze the nonlinear characteristics of vibration signals, they are difficult to obtain the more accurate results without considering the real scale related features of the time series which are characterized with multiexponents or nonlinear parameters.

In the recent years, the fractal or multifractal time series have been observed in many fields, such as geophysics time series, medical time series, and technical time series [8]. The traditional approaches for the fractal analysis, such as Hurst’s rescaled-range analysis () [9] and fluctuation analysis (FA) [8], always assume the time series as the stationary data without considering the possible fluctuation caused by some reasons. The methods for the nonstationary time series include the wavelet analysis, the discrete wavelet transform (WT), and the detrended fluctuation analysis (DFA) [8]. The DFA which was first introduced by Peng et al. in 1992 is a new Hurst exponent calculation method [10] based on the random walk theory. Basically, it represents a detrending version of fluctuation analysis (FA), which is more reliable and suitable for analyzing the nonstationary signal compared to the or the FA analysis. This is because the DFA can remove the external polynomial trends of the differential orders in order to obtain the accurate intrinsic statistical characteristics from the time series. One advantage of the DFA is that it can detect the long-range correlations embedded in the seemingly nonstationary time series and also avoid the spurious detection of the apparent long-range correlations which are an artifact of nonstationarity. The DFA has been widely applied to various fields, such as meteorology [11], materials science [12], finance [13], biological signals [14], and hydrograph [15]. Several modified DFA methods have also been proposed [8].

The DFA was also used in equipment fault diagnosis. de Moura et al. [16] employed the DFA and the principal component analysis (PCA) to the cluster analysis of gear faults. Instead of using the long-range correlation or scale exponents of time series, the idea of Moura’s method is to use the fluctuation function as a mapping function from data space to characteristic space. Afterwards, De Moura et al. [17] used the DFA to analyze the bearing fault. Sridhar et al. [18] combined the EEMD with DFA to denoise the noise-corrupted signal. The DFA are used to determine the noise components in IMFs. Through the DFA, the crossover phenomenons are found in finance [19], meteorology [20], medical science [21], and equipment fault diagnosis. Lin and Chen [22] found the interesting crossover properties in vibration signals captured from gearboxes and rolling bearings. The scale exponents corresponding to different time scales in double logarithm plots were used as the feature parameters to describe the defective conditions of gears and rolling bearing. Liu [2] claimed that the DFA curves of bearing’s vibration signals can be quantified by two scale exponents and the exponents in a small time scale can be utilized to distinguish the faulty bearing conditions. The author’s previous research also showed that the gear vibration signals had crossover phenomenon. The scale exponents and intercepts of DFA curves were used for gear fault classification [23]. Jiang et al. [24] evaluated the optimal scaling intervals with Quasi-Monte Carlo algorithm and the least square support vector machine was used for multifault diagnosis of gearbox. Hough [25] combined the least squares method with sliding window to extract the scale exponent and the neural network algorithm was used for classification of gear fault.

However, the detail reasons for multiscales of fault vibration signals were not discussed in abovementioned description. Furthermore, as far as the authors know, the influence of detrend order of DFA on fault recognition was not discussed in previous literatures, and only limited methods for evaluating the crossover points of DFA were developed. In addition, the scale exponents of different time scale intervals were used as the characteristic parameter in previous researches. However, the intercept of double logarithmic plot of the DFA was not utilized. Acutally, the intercept that is used in our research involves a lot of information of vibration signal.

In this paper, the detrended fluctuation analysis (DFA) is employed to analyze the gear vibration signals. According to the double logarithmic plot of the DFA, it is verified that the gear vibration signals exhibit self-similarity in two ranges of time scales. The reason for the double-scale characteristic is discussed through the simulation analysis. Furthermore, the scale exponents and intercepts corresponding to different scale intervals are extracted as the feature vectors to describe the fault condition of gears. It is found that more pieces of information about the gear faults are involved in the small time scale interval. In order to detect the crossover point of two time scales and extract the parameters (scale exponents and intercepts) automatically, a new approach based on the Hough transform is proposed. The experiments were performed with the proposed parameters to classify the gear faults. Combining the Gaussian mixture model (GMM) and Bayesian maximum likelihood classifiers, the classification of gear vibration signals achieved successfully.

The remainder of the paper is organized as follows. Sections 2 and 3 overview the detrended fluctuation analysis (DFA) and the Hough transform (HT). By applying several gear fault simulated signals, the analysis and discussion about the DFA are presented in Section 4. In Section 5, a self-adaptive feature extraction and classification method for the vibration signals based on DFA and HT is introduced and verified by the experiments. Finally, Section 6 contains the conclusions.

#### 2. Detrended Fluctuation Analysis

Considering is a time series of length .

*Step 1. *Map to time series by integration:where is the mean of the time series :

*Step 2. *Divide into which are sub-time series with equal length *.* The length of a time series is usually not a multiple of the length , and redundant data of the time series may be left. Although the redundant data can be deleted in the following analysis, we suggest repeating the same process from the opposite end of the same time series. For each sub-time series, compute the corresponding least squares order fits:where is the trend of the th sub-time series. It is the fitting polynomial in this sub-time series. Linear, quadratic, cubic, or higher-order polynomials can be used in the fitting procedure (usually called DFA1, DFA2, DFA3, etc.). is the coefficient of th order.

*Step 3. *For each sub-time series, compute the fluctuation function: where

*Step 4. *Repeat Steps through 3 for a broad range of sub-time series (i.e., box) with length . If the time series are long-range power-law correlated, the relationship between and can be described as follows:where is the scale exponent. It can be calculated by taking the logarithm of both sides of (6),and subsequently plotting versus to obtain scale exponent and intercept by linear regression.

The scale exponent characterizes the long-range power-law correlation properties of the time series. It has a close relationship with the self-correlation function. If , 1, and 1.5, the characteristics of the time series correspond to the independent random process (white noise), process, and Brownian motion, respectively. If , the correlations in the signal are antipersistent (negative correlations). If , correlations in the signal are persistent (positive correlations).

#### 3. Hough Transform

The Hough transform [25] is an automatic image analysis technique which can be used to detect regular curves such as straight lines, circles, and ellipses within an image. The plotting of versus of DFA can be seen as an image. The linear relationship between and means a series of straight lines in the plotting. The Hough transform for detecting straight lines is introduced as follows.

Generally, in a Cartesian coordinate plane , a straight line can be described as , where parameters and are the slope and intercept, respectively. Only when the values of and are known, can we describe this line accurately. The point on this line can be written as and it can be changed as , which indicate a straight line in the coordinate plane . That means a point in plane corresponds to a line in plane and vice versa. If every point on a line in plane is mapped to plane , the lines will cross at one point and the line in plane will be identified. If there are several crossed points in plane , several straight lines will be identified in plane . However, vertical lines in the plane described as will give rise to unbounded values of the slope parameter . Thus, Duda and Hart proposed the use of a different pair of parameters used in polar coordinates, which are referred to as the Hough parameter space, to replace the pair of parameters used in Cartesian coordinates (Figure 1). Consider