The fault feature extraction and classification of gear using principal component analysis and kernel principal component analysis based on the wavelet packet transform
Introduction
Gear transmission is the basic form of mechanical transmission, and it has the advantages of a compact structure, stable transmission ratio, large transmission power, high transmission efficiency long service life, etc. So, it is widely used on the transmission system of machine tools, vehicles, construction machinery and other machinery equipment. Its running state is directly related to the working effect of the entire mechanical equipment. However, due to reasons of the complex structure of the gear transmission system itself and poor working conditions, it is very prone to damage, break and other faults. According to statistics, 80% of the failures are caused by the gear faults while there are gears in the mechanical transmission system, and gear faults also account for about 10% in rotating machinery failures [1]. Therefore, the failure analysis and fault diagnosis of the gear transmission system and its various components are a very important research work, and it has always been hotspot and frontier of international and domestic academics [2], [3], [4].
Fault diagnosis generally includes three main components, namely characteristic signal detection, feature extraction and fault classification. The feature extraction of faults is generally divided into feature selection and feature extraction, and the theory and method of fault feature selection and extraction are the most active research area in fault diagnosis discipline. Feature extraction is an important step in fault identification. If the feature extraction is incorrect or incomplete and it will inevitably lead to erroneous classification and false positives. How to extract the effective fault characteristic information from the complex dynamic mechanical signals is the key to solve the problem of large-scale complex mechanical and electrical equipment fault diagnosis. Traditional method of signal processing can not take into account the signals’ localization characteristics and overview that show in the time domain and frequency domain analysis and it is powerless to analyze and process nonlinear and nonstationary of complex electromechanical device signals. Although modern signal processing methods provides the possibility for the analysis of non-linear, non-Gauss and non-stationary signals, it has its own unique advantages and disadvantages, such as short-time Fourier transform has no cross-term interference but the time–frequency resolution is low [5], [6]; Wigner distribution has a high resolution of time–frequency and it is always real function. Whether it is real signal or complex signal, but for the multi-component signal, Wigner distribution will be serious cross-interference and resulting in false frequency components [7].
Wavelet transform (WT) is a multi-resolution analysis method, which has a good time–frequency localization features. Through the wavelet of the “stretching” and “translation”, it can obtain good time domain resolution from high-frequency part which the duration is very short; it obtains good frequency domain resolution in the low-frequency component analysis, and it is a powerful tool in dealing with non-stationary signal [8], [9]. Wavelet transform has been widely used in signal and image processing, speech analysis, numerical computation, pattern recognition, quantum physics, fault diagnosis and other fields [10] and is considered to be a major breakthrough in the tools and methods. Hansang Kim utilize wavelet analysis to predict the location of structural damage identification [11]; Pinzheng Zhang and Huazhong Shu of Southeast University in China adopt a method that combines wavelet analysis and neural network to study face recognition [12]. The wavelet packet transform is a form of wavelet transform. The method for mechanical fault diagnosis based on redundant second generation wavelet package transform, neighborhood rough set and support vector machine is presented in Ref. [13], which was applied to vibration signals for achieving high classification performance in mechanical equipment fault diagnosis [13]. The length of decomposition results of traditional wavelet packet transform (WPT) will decrease by half in the next level for downsampling, then the length of sequences in the last level will become very short, and this is very inconvenient for further analysis of these sequences. One kind of WPT (convolution WPT) based on convolution definition is put forward to overcome the defect of traditional WPT [14]. And wavelet packet transform has also been received a wide range of applications.
Feature extraction based on spatial transform is to transform the original sample data into a new space to make the data between the different classes have maximum separation in the new space, or to make the data in new space have the best ability to describe the original data. Transform feature extraction technology is divided into linear and nonlinear, respectively, based on linear transformation and nonlinear transform. PCA (Principal Component Analysis) [15] method is optimal dimension compression technology whose feature is based on the minimum mean square error. Namely, in the same dimensions, the data transformed from the original using the method of principal component analysis will contain most of the original data information. This method is based on second-order statistics of the data (namely based on the corresponding covariance matrix) for analysis and extracts each irrelevant characteristic component. By solving the characteristic equation and choosing the eigenvector corresponding larger eigenvalue as transformation axis. This ensures the converted data with minimal loss relative to the original data. By introducing a kernel function, Kernel Principal Component Analysis (KPCA) [16] implicitly maps the input space into a nonlinear feature space, and in the feature space linear principal component analysis can be conducted. Since the mapping is non-linear, KPCA is a method of nonlinear principal component analysis. Compared with PCA, if the original data exists with complex non-linear relationship, KPCA is more suitable for the feature extraction and can achieve the purpose that reacts the original data in the greatest degree while compressing the dimensions of the original data [17]. Therefore, KPCA can simplify the steps and process of feature extraction and it can be better used in feature extraction and classification of the fault.
Section snippets
Wavelet packet transform and wavelet packet frequency band energy
Wavelet packet decomposition can provide a more meticulous analysis method to signal. Through dividing the frequency band to multi-layer, wavelet packet decomposition can further decompose the high frequency part which is no decomposed in multi-resolution analysis. Therefore, it has also a high resolution ratio in the high frequency part. According to the characteristics of the analyzed signal wavelet packet decomposition can adaptively select the appropriate frequency band to matches with the
Principal Component Analysis PCA
The basic principle of PCA algorithm: Given input data matrix Xm×n (usually m > n), it is constituted by the centralized sample data , where xi ∊ Rn, moreover:PCA converts the input vector to a new vector by the formula (9):where U is an orthogonal matrix of n × n, whose the i-th column ui is the i-th feature vector of the sample covariance matrix C.Namely, PCA firstly need to solve eigenvalues and eigenvectors of the sample covariance matrix C:
Establishment of gear system vibration test equipment and acquisition of test signal
Gear vibration testing experimental apparatus is shown in Fig. 1, the whole system is driven by motor, through the coupling, the power will been transferred to reducer gear, after the reducer outputting, through gear coupling, torque and speed sensor, the power will been transferred to the magnetic powder loader. Install the acceleration sensor, which sit on the vicinity of symmetry position on fault gear, on the testing gear. Drive gear is set to failure. In addition, the transmission rings,
Gear fault feature selection based on wavelet threshold de-noising
Firstly, the collected gear vibration acceleration signal is denoised by wavelet threshold to remove random noise from the original signal and improve signal-to-noise ratio (SNR). Then 4 layers wavelet packet decomposition is applied to the signal that has been denoised, and the 16 frequency bands of signal is obtained, which is: 0–160 Hz, 160–320 Hz, 320–480 Hz, 480–640 Hz, 640–800 Hz, 800–960 Hz, 960–1120 Hz, 1120–1280 Hz, 1280–1440 Hz, 1440–1600 Hz, 1600–1760 Hz, 1760–1920 Hz, 1920–2080 Hz, 2080–2240 Hz,
Gear fault extraction and classification based on PCA and KPCA
The methods of PCA and KPCA are respectively used in the process of fault feature extraction and classification for wavelet packet frequency band energy of gear vibration acceleration signal. Then the dimensionality of the data is reduced. Meanwhile second-order relationship and higher order nonlinear relationship among data are extracted. Using the extracted first and second principal components classifies and displays the different faults and the results of the two methods were compared.
Conclusions
- (1)
Utilizing wavelet threshold de-noising to preprocess the original signal, it greatly increases the signal to noise ratio of the original signal; according to engineering experience and experimental parameters, it shows which frequency band of the fault feature distributes in approximately. According to the principle that the different fault characteristic frequency are in different frequency bands, the principle determines the number of layers of wavelet packet decomposition. Utilizing wavelet
Acknowledgments
This research is sponsored by the Natural Science foundation of China (NSFC) (Grant No. 51275422), the fund of Chinese Aviation Science (Grant No. 01I53073), the Natural Science foundation of Shaanxi province in China (Grant No. 2004E219) and the graduate carve out seed fund of Northwestern Polytechnical University (Grant Nos. Z2013032, 51275422, 01I53073, 2004E219, Z2013032).
References (23)
- et al.
Gear fault identification and localization using analytic wavelet transform of vibration signal
Measurement
(2013) - et al.
Vibration model of rolling element bearings in a rotor–bearing system for fault diagnosis
J. Sound Vib.
(2013) - et al.
Time–frequency analysis of the first and the second heartbeat sounds
Appl. Math. Comput.
(2007) - et al.
On the usefulness of STFT phase spectrum in human listening tests
Speech Commun.
(2005) - et al.
Spectral decomposition of seismic data with reassigned smoothed pseudo Wigner–Ville distribution
J. Appl. Geophys.
(2009) - et al.
Monitoring gear vibrations through motor current signature analysis and wavelet transform
Mech. Syst. Signal Process.
(2006) - et al.
Haar wavelet for machine fault diagnosis
Mech. Syst. Signal Process.
(2007) - et al.
Damage detection of structures by wavelet analysis
Eng. Struct.
(2004) - et al.
Mechanical fault diagnosis based on redundant second generation wavelet packet transform, neighborhood rough set and support vector machine
Mech. Syst. Signal Process.
(2012) - et al.
Convolution wavelet packet transform and its applications to signal processing
Digit. Signal Process.
(2010)
Classification of fault location and performance degradation of a roller bearing
Measurement
Cited by (164)
A novel denoising method of the hydro-turbine runner for fault signal based on WT-EEMD
2023, Measurement: Journal of the International Measurement ConfederationA whole-building data-driven fault detection and diagnosis approach for public buildings in hot climate regions
2023, Energy and Built EnvironmentRobust kernel principal component analysis and its application in blockage detection at the turn of conveyor belt
2023, Measurement: Journal of the International Measurement ConfederationCriteria for optimizing kernel methods in fault monitoring process: A survey
2022, ISA TransactionsCitation Excerpt :Dong et al. [77] apply the RBF-KPCA to reduce the feature mutual correlation, while a Morlet-SVM classifier [78] is used to identify the bearing running state. On the other hand, Shao et al. [79] utilize the RBF-KPCA for the fault feature extraction on gear systems. Otherwise, Liu et al. [80] integrate KPCA and TWSVM for analyzing the health conditions of bearing and bevel gear faults.
Damage monitoring of pultruded GFRP composites using wavelet transform of vibration signals
2022, Measurement: Journal of the International Measurement Confederation