**2. Theoretical foundations of the processing method**

The most popular diagnostic methods bearings can be divided into two broad categories: spectral methods and time methods.

In the general case, the analysis algorithm based on the spectral method is as follows: using the properties of the Fourier transform, the signal spectrum is calculated:

*Principles of Diagnosing: The Technical Condition of the Bearings of the Gas Turbine… DOI: http://dx.doi.org/10.5772/intechopen.108400*

$$F(\boldsymbol{\alpha}) = \mathcal{S}\_0(\boldsymbol{\alpha}) \sum\_{n=0}^{N} \exp\left(-j\alpha \tau\_n\right),\tag{1}$$

where *<sup>S</sup>*0ð Þ¼ *<sup>ω</sup>* <sup>Ð</sup> ∞ �∞ *S t*ð Þ exp ð Þ �*jωt dt*—bearing signal spectral density (*S t*ð Þ). The

calculation *F*ð Þ *ω* is based on statistical properties *τ<sup>n</sup>* . The parameter *τ<sup>n</sup>* is a random value that characterizes the quasi-periodicity of the process under study, and the index *n* shows the pulse number. As a rule, when analyzing the bearing signals by the method (1), it is assumed that process is stationary, so its properties, determined in short time intervals, do not change from the interval to the interval. Thus, *τ<sup>n</sup>* ¼ *nT* which is the period of the impulses. Using spectral methods, we can confidently detect defects that are characterized by the occurrence of a periodic component. This is the wear of the rolling bearings, the weakening of mechanical joints in the engine, and much more. Spectral methods include methods based on the analysis of the broadband energy spectrum [6], the frequency component [7], the spectrum of the envelope signal [8], and the window conversion of Fourier [9].

Another direction in technical diagnosis is temporary analysis. Due to this, it can find some characteristics that are difficult to identify using spectral analysis. For example, according to the parameters of a temporary signal, you can detect the number of shock pulses, which in the spectrum may not be distinguishable due to the presence of interference and noise [10].

The disadvantages of classical methods of spectral and temporary processing include that they give only the general condition of the system under study. In addition, classical methods significantly reduce their effectiveness in the analysis of nonstationary processes, which complicates the assessment of dynamic parameters (quasi-periodicity), which means they cannot effectively give a forecast about how the system will behave in the future.

A method is proposed that occupies an intermediate position between spectral and temporary methods and is based:

1.On the main properties of the transformation of the Hermit;

2.On the main properties of the wavelet transformation;

3.On the classical theory of optimal filtration.

Thanks to the use of the basic properties of the wavelet transformation (scaling and localization), the method allows us to adapt to the local features of the bearing signal. As basic functions, the method uses the functions of the Gauss-Hermite (FGH), which are defined in the transformation of the Hermit [11]. The undoubted advantage of the FGH is that they are orthogonal in both the temporary and frequency area, which allows them to adapt them not only to the form of the signal in time, but also to its frequency spectrum. It is also worth noting that the method works throughout the sample of the signal under study, which increases its effectiveness but affects the calculation speed. The stability of the method for noise increases due to the fact that it is based on the principles of optimal filtration. At the output of the method, we have a cross-correlation function, according to which the dynamics of the pulse is evaluated (assessment of quasi-periodicity or *τn*) by measuring the distance between

the peaks. For clarity and information content, this procedure can be replaced with an assessment of the rhythmogram of the process under study. The rhythmogram is an apparatus for evaluating quasi-revisionism adopted in the medical field. The rhythmogram reflects the variability of the heart rhythm. Thus, the use of a rhythmogram to assess the state of the bearing is a new field in the diagnosis of technical systems.

#### **2.1 Mathematical description of the method**

The mathematical procedure for processing the signal of the bearing can be described by the following scheme:

The input digital signal *Sin*ð Þ *tk* (signal after sampling and quantization), obtained from the vibration sensors, changes with a reference signal ~ *Sptrn*ð Þ *tk*, *m* . A reference signal is a fragment of a record *Sin*ð Þ *tk* , the dynamics of which we want to trace. A reference signal can be interpreted as a defect or a useful signal, the choice depends on the objectives of the study. A reference signal can be built in the following expression:

$$\tilde{S}\_{\rm ptm}(t\_k, m) = \frac{1}{\sqrt{2^m}} \sum\_{q=0}^{Q} W(q, q\_{\rm cf}) A\_q(m) \Psi\_q(t\_k, m) \tag{2}$$

The main computing element (2) is the calculation, which is based on the transformation of the Hermit or FGH and the discrete wavelet transformation:

$$A\_q(m) = \left(q! 2^q \sqrt{\pi}\right)^{-1/2} \sum\_{k=-K}^{K} \mathbb{S}\_{ptm}(t\_k) \Psi\_q(t\_k, m), \tag{3}$$

where *Sptrn*ð Þ *tk* is the vector of samples of the selected fragment (or the standard is a fragment from *Sin*ð Þ *tk* ), *tk* ¼ Δ*t* � *k*, Δ*t* is the sampling step, *K* ¼ *Nptrn=*2, *Nptrn* is the duration of the selected fragment in samples, *k* ¼ �*K*,*:* � 1,0,1,*::K*, borders on Δ*t* ¼ 3*=K*, time by level �3, Ψ*q*ð Þ *tk*, *m* —FGH *q*—FGH order, *m* is the scale parameter (similar to the scale parameter in the wavelet transform). In fact, (3) is the decomposition of a fragment *Sptrn*ð Þ *tk* in space Ψ*q*ð Þ *tk*, *m* . Gauss-Hermite functions have the following mathematical notation:

$$\Psi\_q(t\_k, m) = H\_q\left(\frac{t\_k}{2^m}\right) \exp\left(-0.5\left(\frac{t\_k}{2^m}\right)^2\right),\tag{4}$$

where *Hq tk* 2*m* � �—Hermite *q*—order polynomial.

Then, (2) is the inverse Hermite transform, and the reproduction accuracy *Sptrn*ð Þ *tk* is determined by the expression:

$$Err(m,q) = \frac{\sum\_{k=-K}^{K} \left(\frac{1}{\sqrt{2^m}} \sum\_{q=0}^{Q} \mathcal{W}\left(q, \ \mathbf{q}\_{\text{cf}}\right) A\_q \Psi\_q(\mathbf{t}\_k, \ m) - \mathcal{S}\_{\text{ptr}}(\mathbf{t}\_k)\right)^2}{\sum\_{k=-K}^{K} \mathcal{S}\_{\text{ptr}}\left(\mathbf{t}\_k\right)^2} \mathbf{100\%, \tag{5}$$

where *W q*, *<sup>q</sup>*cf � � is the smoothing filter in FGH space. When the series is truncated in FGH space, the reference signal has oscillations, and this effect is analogous to the

*Principles of Diagnosing: The Technical Condition of the Bearings of the Gas Turbine… DOI: http://dx.doi.org/10.5772/intechopen.108400*

"Gibbs phenomenon" in harmonic analysis. To weaken it, you can use low-pass filtering in the FGH space, based on well-known approximations of the frequency characteristics: Gauss, Butterworth, and Bessel.

The error *Er m*ð Þ , *q* can be set by the researcher and is determined by the maximum FGH order—*Q* .

The key feature of the signal according to (2) is that it is built from a real discrete record of the signal, so we can take into account any local features of the process under study.

In accordance with the scheme of **Figure 1**, the next operation is the integration or calculation of the correlation integral. The correlation integral is defined in the theory of optimal filtering and searches for similar fragments ~ *Sptrn*ð Þ *tk*, *m* is the process under study *Sin*ð Þ *tk* . In this paper, the correlation integral is built on the basis of the wavelet transform:

$$R\_{out}(t\_n, m) = \sum\_{k=0}^{N} \mathcal{S}\_{in}(t\_k) \tilde{\mathcal{S}}\_{\text{prtr}}(m, t\_k - t\_n), \tag{6}$$

where *N* is the maximum number of samples in the signal *Sin*ð Þ *tk* . As you can see, with a large number of readings, calculation (6) is computationally labor-intensive. Therefore, it makes sense to move from spatial integration in the time domain to multiplication in the frequency domain or matched filtering. Then the scheme of **Figure 1** can be transformed into the following form (**Figure 2**):

The signal goes directly to the matched filtering (*MF*) block. With matched filtering, an increase in the detection rate is associated with the following circumstances:

1. the spectral image of the Gauss-Hermite functions is known;


**Figure 1.**

*Block diagram of the processing method in the temporary area.*

**Figure 2.**

*Flowchart of the processing method in the time-frequency domain.*

It was obtained [12] the transmission coefficient of the filter that selects a fragment of the signal, consisting of a set of features:

$$\dot{K}(f\_n, m) = \sum\_{q=0}^{Q} j^q \sqrt{\frac{\sqrt{\pi}}{q! 2^{q-1}}} \exp\left(-0, \, 5 (2^m)^2 f\_n^{-2}\right) H\_q\left(2^m f\_n\right),\tag{7}$$

where *f <sup>n</sup>* ¼ *n=N*Δ*t*, *n* ¼ 0,1, … *N* � 1 .

The signal after matched filtering the cross-correlation function enters the threshold device (*TD*), where, based on a priori statistical information, peaks exceeding a certain threshold value are selected. The median value can be used as a threshold. After threshold processing, the resulting one enters the block of transformation (*BT*), at the output of which we have the rhythmogram of the process (*Tnum*).

Due to the fact that the principles of the wavelet transformation are incorporated in the processing method, the rhythmogram can be refined by varying the scale parameter or the filter band (7):

$$F(t\_k, m) = \sqrt{E\_t} - \left| \frac{1}{2\pi\sqrt{a}} \sum\_{n=0}^{N-1} \dot{\mathcal{S}}\_{in}(f\_n) \dot{\mathcal{K}}\_q(f\_n, \ m) \exp\left(\frac{j2\pi f\_n t\_k}{N}\right) \right|,\tag{8}$$

As a result, a functional or a surface is formed, on which there are many extrema (minimums) corresponding to the number of reference signals in the process under study. The processing process is reduced to solving a multi-extremal problem. This problem is solved using the steepest descent method, where a priori information about the rhythm of the system under study can serve as starting points. However, in real problems, such information is not always available. Therefore, as a starting point for the steepest descent algorithm, it is advisable to use the coordinates of the maxima of the cross-correlation function determined at an early stage without changing the scale parameter. As a result of such processing, we obtain refined information about the location of the reference signal and, consequently, a more accurate process dynamics or rhythmogram.

#### **2.2 Signal processing without a priori information**

The records of three signals (*S*1, *S*2, *S*3) are shown in **Figure 3**. The sample size is 30,000 readings. In accordance with the considered theory, it is necessary to isolate the standard from the signal under study. The standard is a fragment of the recording, the dynamics (rhythm) of which we want to trace throughout the entire time of the study. The standard reflects the distinctive features of the waveform, which can mean both the correct operation of the device, and, conversely, a faulty one. The choice of standard should be carried out taking into account the opinion of a specialist or an expert in the field of diagnostics. Since we do not have a priori information on the characteristic features of the signal, we will conduct a preliminary spectral analysis of the signals to select the standard.

Take the value of the sampling frequency equal to 1. Then, for a sample of 30,000 samples, the frequency step in the spectral region will be 1/30,000 rel. units. The recording spectra are shown in **Figure 4**. The figure shows that the maximum frequency components in the signal spectra are concentrated in the range of approximately 0.015–0.02 rel. units, which corresponds to approximately 50–70 samples in the signal record.

*Principles of Diagnosing: The Technical Condition of the Bearings of the Gas Turbine… DOI: http://dx.doi.org/10.5772/intechopen.108400*

**Figure 3.** *Recordings of vibration signal of bearings: N—count number.*

The ideal mechanism functions cyclically, which is determined by the characteristic spectral component. In the discrete spectrum of the signal of such a mechanism, the reciprocal of the period has a maximum value. In real records, the spectrum is blurred (**Figure 4**), but its maximum can be identified with the average pulse repetition period.

After analyzing the spectra, it can be established that the signal *S*<sup>2</sup> is a faulty operation of the bearing, since many spectral peaks are observed in the spectrum. Bearing signals *S*<sup>1</sup> and *S*<sup>2</sup> are of the greatest interest due to the fact that it is impossible to say for sure from the spectral pattern whether they are in good order or not. Let us choose a signal fragment from the record as a reference, since its main spectral components are present both in the signal spectrum and in the signal spectrum.

After analyzing the record, a fragment (from 6020 to 6080 samples) was selected, shown in **Figure 5**, since similar waveforms occur throughout the entire signal record.

In **Figure 5**, characteristic fragments with a duration of approximately 50–70 samples are clearly visible. The next stage of processing is the construction of a support function (SF) based on FGH. In essence, SF is a mirror image impulse response of a complex quasi-matched filter (7). As a result of multiplying the complex conjugate transfer coefficient of the quasi-matched filter and the spectrum of the signal under study, after the inverse Fourier transform, we obtain the crosscorrelation function, fixing its maxima, we can build a rhythmogram.

**Figure 4.** *Spectra of the studied signals.*

**Figure 5.** *Fragment of the record S*1*.*

The position of the extrema of cross-correlation function can be refined by varying the scale parameter when constructing the filter (8). Thus, the problem of diagnosing a system is reduced to solving a multi-extremal problem, since the cross-correlation function is a complex surface with many local maxima. From a methodological point of view, it is more convenient to look for local minima. This transition can be made using the Cauchy-Bunyakovsky inequality (**Figure 6**).

*Principles of Diagnosing: The Technical Condition of the Bearings of the Gas Turbine… DOI: http://dx.doi.org/10.5772/intechopen.108400*

**Figure 6.**

*Topographic picture of the cross-correlation function of a signal S*<sup>1</sup> *fragment with a variation of the scale parameter.*

Such a procedure (**Figure 6**) resembles the process of scale variation in the wavelet transform. The key difference between the two processing approaches is that at the output of the method under consideration, we have the cross-correlation function that allows us to estimate the degree of similarity of the SF and the reference, and with wavelet processing, the spectral distribution. The position of the minima of the obtained surface can be found, for example, by the steepest descent method.

As a result of such processing, we have rhythmograms and scatterograms.

To construct the rhythmogram, the cross-correlation function was subjected to threshold processing. The processing results are shown in **Figure 7**.

When constructing the diagram, the minimum positive value from the set of maxima of the cross-correlation function signal *S*<sup>1</sup> was taken as the threshold.

Rhythmograms look like a random process with an average value. It is approximately equal to 50 samples and gives an estimate of the average period of the processed signals.

Rhythmogram surges indicate deceleration (upsurges) or acceleration (downsurges) of the mechanism. Upsurges commensurate with the average value of the rhythmogram indicate that the algorithm skipped a cycle due to the fact that the maximum cross-correlation function does not exceed the set threshold. There are few such outliers in **Figure 7a** and **c**, while there are quite a lot of them in **Figure 7b**. In the presence of noise, downward spikes may appear, commensurate with the average value of the rhythmogram, due to the appearance of false maxima. There are no such outliers in **Figure 7**.

Preliminary visual analysis suggests that the bearings, the signals of which are shown in **Figure 1a** and **c**, are in good condition, and the bearing with the signal *S*<sup>2</sup> has a defect. Let us confirm these preliminary considerations with quantitative estimates.

The rhythmogram can be considered as a discrete signal that can be processed by one of the traditional methods. You can, for example, get the spectrum of the rhythmogram. **Figure 8** shows the smoothed spectra of rhythmograms after lowfrequency filtering. As can be seen, the signal *S*<sup>2</sup> spectrum stands out significantly compared to the signal spectra *S*<sup>1</sup> and *S*3. For a quantitative assessment, we will perform a statistical analysis of rhythmograms. We calculate the mean, standard deviation, mode and median, as well as the minimum and maximum values. The calculated parameters are presented in **Table 1.**

**Figure 7.**

*Rhythmograms of signals, m is the number of the cross-correlation function maximum, T*1,*T*2,*T*3*—the duration of the intervals between the maxima of the cross-correlation function.*

#### **Figure 8.**

*Spectra of rhythmograms: Sp\_T1—spectrum of T*<sup>1</sup> *rhythmogram, Sp\_T2— spectrum of T*<sup>2</sup> *rhythmogram, Sp\_T3 spectrum of T*<sup>3</sup> *rhythmogram.*

*Principles of Diagnosing: The Technical Condition of the Bearings of the Gas Turbine… DOI: http://dx.doi.org/10.5772/intechopen.108400*


#### **Table 1.**

*Statistical parameters of rhythmograms.*

In medical practice, the diagnosis of pathology is based on the value of standard deviation. The standard deviation values given in **Table 1** differ by 10%. The signal *S*<sup>2</sup> has the highest standard deviation value. The median and mode of recording signals *S*<sup>1</sup> and *S*<sup>3</sup> have close numerical values of the statistical parameters, which are close to the average. This indicates that the distribution of outliers relative to the mean value is close to symmetrical. For the signal *S*2, downward surges (jerks and bumps) predominate. Next, consider the scatterograms of GTE bearing signals.

Scatterogram is a geometric method. In practice, the scatterogram has the shape of an ellipse stretched along the bisector **Figure 9** [12].

#### **Figure 9.**

*Scatterograms of signals,Tn Tn + 1—duration of the previous and subsequent intervals between the maximum cross-correlation.*

According to the scatterogram, one can judge the quasi-periodicity of the signal under study. The more clustered the points are, the less the quasi-periodicity.

The shift of the points to the right along the coordinate axis reflects a decrease in the rhythm, while the shift to the left reflects an increase. If the points are far from the whole population, then this may indicate a defect.

Based on the totality of statistical estimates of rhythmograms and the scatter of points in scatterograms, we can assume that the bearings *S*<sup>1</sup> and *S*<sup>3</sup> are both in good condition, and *S*2—with a high degree of probability—are faulty.
