Method description

4.4.3.1 The global spectral analysis description

The feature vector, of dimension m, will be formed with the absolute amplitude of spectral lines at the ball pass frequencies, their multiple harmonics and the side-bands related to their modulations by other frequencies including mainly the ro-tational frequency and the cage frequency. These spectral lines are automatically extracted from the spectrum of the envelope vibration signal corresponding to each fault condition, by selecting the dominant frequency components with respect to a specified threshold. The formed vector contributes, using the Fast Fourier Trans-form (FFT) technique, to construct an amplitude matrix X carrying inTrans-formation from the different fault conditions of interest. To fill the matrix X, there is no need to have a huge amount of data corresponding to each fault condition. Besides, each

envelope signal is decomposed into several data segments using a sliding window of samples. The FFT is computed on the different segments to get the amplitudes of retained spectral features. Afterwards, Principal component Analysis (PCA) is applied on the spectral matrix of information to get a lower dimensional principal subspace highlighting the differences and the similarities contained into X. This subspace represents the output space for classification and diagnosis.

Fig.4.13 summarizes the global approach. It comprises of four main steps:

Figure 4.13: The global spectral analysis description

• Step 1: Measurements space. It contains vibration signals corresponding to all conditions of interest: the healthy and all faulty conditions of the bearings.

Signals acquired from several operating points of the machine can be included in order to get a general behaviour space for diagnosis.

• Step 2: Data pre-processing. It consists mainly in the envelope analysis. The vibration signal corresponding to each fault condition is first band-pass fil-tered around the region of natural frequencies. The spectral kurtosis, the kurtogram or its recent improvements [64, 65] can be used to select the bear-ing resonant frequency bands. However, the difference in dB-spectra compared

to the kurtogram has been shown effective enough to identify the frequency region of interest [66, 46]. It represents the difference between the dB-spectra of a healthy vibration signal and a faulty one, and it just requires historical data to be available. The envelope signal is then given as the modulus of the analytic signal obtained using Hilbert Transform. FFT is applied to each envelope signal in order to extract the spectral features from its spectrum.

• Step 3: Spectral features selection. The line spectrum of each envelope signal is scanned to pick out only the frequencies of highest amplitude. The set of dom-inant frequencies associated with each fault condition is supposed to include the specific fault frequencies, i.e. the ball pass frequency, several harmonics and sidebands related to their modulations by the rotational frequencies. The rotational frequencies and their multiple harmonics can be selected during the extraction. However, all the common frequencies are withdrawn from the to-tal set of features in order to keep only the distinguishing components. The scanned frequency range can be reduced, since for normal speeds the charac-teristic fault frequencies are usually less than 500 Hz.

Figure 4.14: Matrix design for analysis

• Step 4: Fault diagnosis space. The X matrix is formed as shown in Fig.4.14 with m retained frequencies fi, i ={1, ..., m}, considered as variables or

fea-tures. It contains several (k in Fig.4.14) sub-matrices each referring to an operating point of the machine, i.e. a particular fault type with a particular severity level. Each envelope signal is decomposed into several data segments using a sliding window of samples in order to construct the associated sub-matrix in X. At each position of the window, the FFT is computed on the corresponding data segment and the absolute amplitudes of the retained vari-ables are acquired and arranged into X as a row sample. The size of the window is chosen with reference to a trade-off between the frequency resolu-tion and the number of samples in the sub-matrices. PCA is then applied to the covariance matrix of X to get the output PCA space for classification.

4.4.3.2 Discrimination of faults in the bearing balls using LDA

PCA and LDA are linear transformations that can be used to perform dimensional-ity reduction. However, theoretically, LDA leads to better data classification than PCA: PCA performs dimensionality reduction while preserving as much of the total variance in the high dimensional space as possible whereas LDA performs dimen-sionality reduction while preserving as much of the class discriminatory information as possible. This analysis technique considers maximising the following objective, also called Fisher criterion [67]:

J(u) = u^TSBu

u^TSwu (4.7)

where SB is the between classes scatter matrix and Sw is the within classes scatter matrix. If the number of classes to be discriminated is c, the LDA projects the space of the original variables onto a (c− 1) - dimensional space which axes u are obtained by maximising (4.7). Intuitively, this operation is a compromise between maximis-ing the distance between the projected centers of classes, and the minimisation of their variances, thereby facilitating classification. By using Lagrange multipliers, the solution to this problem yields u as the eigenvector of S_w⁻¹SB associated with the largest eigenvalue. The eigenvectors associated with the largest eigenvalues define discriminating axes which span the LDA space for classification. For any c - class problem we would always have c− 1 non-zero eigenvalues.

In the global spectral analysis method, LDA will substitute PCA in order to improve the discrimination among different sizes of ball faults [68]. So, LDA will be applied to the data set excluding data of races’s faults. As shown in Fig.4.15, only the samples corresponding to the healthy and the ball faults (BF) conditions (including all operating points) are considered to build the LDA transformed space.

The original space is still formed with the specific frequencies fi for i ={1, ..., m},

Figure 4.15: Data set design for LDA

that were extracted according to the diagram given in Fig.4.13.

Mathematically formulating the transformation, consider N samples{x¹, x2, ..., xN} of dimension m where Ni of the samples belongs to class Ci, i = 1, 2, ..., c. Let µi

be the mean of class Ci and µ be the mean of entire data given by:

µi = 1 Ni

X

xn∈Ci

x_n, µ = 1 N

XN n=1

x_n. (4.8)

The between classes scatter matrix can be estimated as:

SB=

Xc i=1

(µi− µ)^T(µi− µ) (4.9)

The within classes scatter matrix is estimated as:

Sw =

Xc i=1

Si where Si = ^X

xn∈Ci

(xn− µⁱ)^T(xn− µⁱ) (4.10)

The eigenvector decomposition of S_w⁻¹SB yields the transformed LDA space as spanned by the eigenvectors corresponding to non-zero eigenvalues. Projecting the original data set{x¹, x2, ..., xN} onto this space allows the classes separability.