To record vibration signals, sensors are connected to the channels of the signal analyzer-recorder. In our case, it is the CoCo-80X signal analyzer from National Instruments (see Figure 1 ).

The test bench allowed us to experiment with twelve tests with different configurations of defects carried out on the gear and on the bearing mounted on the bench. Table 1 below shows the tests carried out.

1) Scalar indicators

Temporal domain analysis of vibration signals is a method of analyzing stationary signals in the time domain. This type of analysis makes it possible to highlight the presence of faults in machines. Indeed, the presence of faults and, if applicable, their severity in rotating machines is detected. Temporal domain methods are based on the statistical analysis of the collected signal and use well-known scalar indicators to monitor the evolution of a quantity derived from the power or peak amplitude of the signal.

These indicators include, among others, the RMS (effective value) is calculated by:

$\text{RMS}=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^{N}X{\left(i\right)}^2}}{N}}$ (1)

The Kurtosis is obtained by:

$\text{KURT}=\frac{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(X\left(t\right)-\bar{x}\right)}^4}{N}}{{\left(\frac{{{\displaystyle \sum }}_{i=1}^N{\left(s\left(i\right)-\bar{x}\right)}^2}{N}\right)}^2}$ (2)

The Crest factor is calculated by:

$\text{FC}=\frac{\text{Sup}\left|X\left(i\right)\right|}{\sqrt{\frac{{\displaystyle {\sum }_{i=1}^{N}X{\left(i\right)}^2}}{N}}}$ (3)

The peak-to-peak value of the vibration signal is calculated by:

$\text{ACC}=\text{ACC}\max -\text{ACC}\min$ (4)

The peak value of the vibration signal is obtained by:

$\text{AC}=\text{ACC}\max$ (5)

And the K factor is obtained by:

$K=\text{Acccrete}\ast \text{Acceff}=\text{Sup}\left|X\left(i\right)\right|\times \sqrt{\frac{{\displaystyle {\sum }_{i=1}^{N}X{\left(i\right)}^2}}{N}}$ (6)

To detect a defect, we look at the value of the indicators, if there is a change in the signal, then there will be a variation in the indicator. For example, a bearing in good condition generates a vibration signal with a Kurtosis close to 3. For a degraded bearing, with flaking, indentations or significant clearances, the shape of the distribution of the signal amplitude is modified and the Kurtosis is greater than 3.

2) Combined scalar indicators

In this subject, we proposed in [14] a new indicator obtained by linear combination of two traditional scalar indicators such as $I=aX+bY+c$, with a, b and c constants, and X and Y are traditional scalar indicators. The results showed that the best combined scalar indicator is obtained by the linear combination of RMS and crest factor:

$\text{Ic}=-0.0099356\ast \text{RMS}+0.44901\ast \text{FC}-2.0601$ (7)

Avec a = –0.0099356, b = 0.44901 and c = –2.0601.

Tests were performed with defects on the bearing and gear created on a test bench. The results of defect detection by the indicators are presented in the results and discussion section.

1) Spectral analysis or (Fourier analysis of the signal)

The vibration signal taken from a rotating machine is very complex, and originates from different parts of the machine. The Fourier transform is a mathematical calculation tool [15] that allows these complex signals to be processed appropriately, to decompose them into several basic sinusoidal components and to represent them in the form of an “Amplitude-frequency” spectrum [16] . The Fast Fourier Transform (FFT) developed by James Cooley and John Tukey requires a low calculation time to perform the discrete Fourier transform. The fast Fourier transform is widely implemented in diagnostic systems for rotating machines. Formulas (8) and (9) represent the Fourier transform and its discrete variant respectively.

$X\left(f\right)={\displaystyle {\int }_{-\infty }^{+\infty }x\left(t\right){\text{e}}^{-i2\pi ft}\text{d}t}$ (8)

$X\left(f\right)={\displaystyle {\sum }_{k=0}^{N-1}x\left(t\right)\cdot {\text{e}}^{-i2\pi f\frac{k}{N}}}$ (9)

2) Cepstral analysis

In its most common definition, the cepstrum of a signal is the result of the inverse Fourier transform of the decimal logarithm of the Fourier transform. Algebraically, the operation of cepstral analysis allows performing a Fourier transform on the spectrum of the signal in decibels (dB) (Equation (10)).

$c\left(t\right)=F^{-1}\left[\log \left({\left|X\left(f\right)\right|}^2\right)\right]$ (10)

$X\left(f\right)=F\left[X\left(t\right)\right]$ (11)

where c(t) is the cepstrum, F is the Fourier transform and F^-1 is the inverse transform.

The purpose of the cepstrum is to recognize and analyze all the periodic structures present in the spectrum. It allows defining indicators adapted for the early detection of defects inducing at different stages of evolution, vibrational energies that indicators from traditional methods would have difficulty in highlighting. The cepstrum and its derivatives illustrate the amplitudes of the components whose frequencies are associated with the repetition periods of the shocks caused by the faults of the monitored machine. It is used to identify gear faults and, to a lesser extent, bearing faults [17] .

3) Envelope analysis

Envelope analysis allows us to look at the evolution of a harmonic on a given frequency to identify abnormal and repetitive shocks. It consists of filtering the signal and then processing it by the Hilbert transform, and then by the inverse Fourier transform. The signal obtained (envelope) allows us to calculate a spectrum called the envelope spectrum. The Hilbert transform of a time signal x is defined by the following equation [15] :

$\tilde{x}\left(t\right)=H\left(x\left(t\right)\right)=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{x\left(\tau \right)}{t-\tau }\text{d}\tau }$ (12)

$\tilde{x}\left(t\right)=x\left(t\right)\times \frac{1}{\pi \tau }$ (13)

where H is the Hilbert transform.

4) Multiresolution wavelet analysis

Multiresolution wavelet analysis is a signal analysis technique that has been widely used recently for fault diagnosis in rotating machines. It was used in vibration analysis by Djebala et al. [5] - [8] , among other things, to overcome the problem of random noise that can be associated with low fault severity.

Multiresolution wavelet analysis decomposes the original signal into wavelets and then performs an approximation to reconstruct the signal for better localization in time and frequency. It allows for amplitude and phase modulation. In our study, we will perform a level 3 wavelet decomposition (see Figure 2 ) available in MATLAB®, then analyze the spectrum of the reconstructed signal to search for possible faults.

Temporal analysis has been studied in particular by exploiting scalar indicators. We have chosen to compare on several signals, some relevant indicators, in order to evaluate their fault detection capacity and their severity according to the type of fault.

As we mentioned previously, temporal analysis makes it possible to highlight the presence of faults in machines. Machine operating parameters, such as speed or load, can influence the result of fault detection. Figure 3 and Figure 4 present the results of the RMS [18] and the combined scalar indicator that we proposed in [14] .

In this figure, we can see that when there is meshing, the signal is modified. When the gear is defective, we can clearly see a modification of the average value of the acceleration, which increases considerably because of the shock generated by the broken tooth on the pinion.

The results in Figure 4 show that the combined scalar indicator given by Equation (7) above detects defects in all twelve configurations of defect. This result of the combined scalar indicators is a notable advance in the field of signal analysis in the time domain, because we manage to detect defects in several configurations. However, simple scalar indicators are limited in detecting defects in different configurations. Some are better in detecting shock defects (like Kurtosis or RMS), others are better in detecting bearing defects (like K factor or Crest factor) and others are better in detecting gear defects (Kurtosis for example).

There are fifteen combined scalar indicators obtained that we proposed by Boukar and Abakar Moussa [14] . But the best combined scalar indicator is given by Equation (7). Figure 4 below shows the result of this indicator on twelve tests carried out.

In this figure, we show that the value of the indicator obtained is greater than the value of initial indicator fixed. In this condition, the fault is detected when Ic > Iinitial.

Gear loading reduces the influence of the defect on the Kurtosis. Its value actually decreases. This shows that the vibrations are attenuated and stabilized when the gear is loaded. Figure 5 shows the influence of the load on the Kurtosis (a) and the RMS (b). Unlike the Kurtosis, whose value decreases when the load increases, the RMS on the other hand decreases ( Figure 5 ). It is indeed shown that when the load increases, the Kurtosis decreases and thus the bearing defect is better detected. Figure 5 shows the evolution of the indicators according to the load in the case of the test with a “Defective cage bearing + Healthy gear”.

To monitor a rotating machine, it is essential to know the geometry and kinematics of this machine. Thus, in the frequency domain, to detect defects on a bearing or a gear, it is necessary to know the characteristic frequencies of each component of the machine.

In this section, given the large number of figures per method and per test, we deemed it necessary to present the results of only three tests (configuration 10, configuration 11, and configuration 12).

Before that, we will present the resonance test results used to determine the resonance frequencies, which can be distinguished from the defect frequencies on the bearing and gear.

To analyze defects in the frequency domain, the frequencies obtained from the frequency spectrum of each signal are examined. The presence of a defect on a bearing or gear is highlighted by the appearance of its characteristic frequency and several of its harmonics. It is also necessary to separate the natural frequencies of the structure, i.e. the resonance frequencies from the characteristic frequencies of the defects. To know the resonance frequencies, a resonance test is carried out on the test bench.

Figure 6 below shows the resonance frequencies of the structure, at motor shaft rotation frequencies of 0 Hz (machine at rest), 5 Hz, 10 Hz and 15 Hz.

In general, we avoid these resonance frequencies coinciding with the rotation frequency of the machine to avoid a possible catastrophe of the rotating machine. These are the frequencies of 8 HZ, 14 Hz, 18 Hz, 114 Hz, 178 Hz, etc.

In the following figures, the results obtained by Fourier spectral analysis are presented. Figure 7 shows the spectrum signal of “defective cage bearing at 10 Hz”.

In Figure 7 above, it is the signal of the defective bearing on the cage, the gear not being coupled. The cage defect of frequency 3.81 Hz is not detected, because the frequency of the cage does not appear. Depending on the size of the defect, the Fourier analysis of the signal does not detect the defect. Also, as there was the problem of shaft misalignment mentioned previously and which constitutes a mask for this defect. Figure 8 shows the spectrum signal of “defective cage bearing and healthy gear at 8.5 Hz”.

In Figure 8 above, there is the defect on the cage and a healthy gear without defect. The cage defect, which has a frequency of 3.24 Hz, is not detected, but a probable shaft misalignment is detected, highlighted by the presence of the rotation frequency and its harmonics.

Figure 9 above shows the result of the signal analysis with a defect on the cage and on the gear. The cage defect, which has a frequency of 3.24 Hz, is not always detected, but the gear defect with a frequency of 8.5 Hz and its harmonics.

In the following figures, the results obtained by envelope analysis of configurations 10, 11, and 12 are presented in Figures 10-12 .

In Figure 10 above, the cage defect is not detected by the envelope. The frequency of the cage defect is a modulating frequency masked by the shaft alignment problem, which is the carrier frequency.

In Figure 11 , when the gear is healthy and with a defective bearing, the envelope analysis has made it possible to highlight the cage defect, which corresponds to the regular spacing of 3 Hz between the frequency peaks.

Figure 12 above shows that when there are two defects, i.e. two distinct frequencies in the signal, it is the modulating frequency that stands out (cage defect). The carrier frequency (gear defect) is not visible.

In conclusion, signal envelope analysis can highlight the modulating frequencies in a multiple defect configuration. When there is a single defect, this is not a problem; it is detected for both the bearing and the gear.

In the following figures below, the results obtained by multiresolution wavelet analysis of configurations 10, 11, and 12 are presented in Figures 13-15 .

In Figure 13 , the wavelet analysis revealed a frequency (62 Hz) that is a multiple of the ball frequency of 7.62 Hz (approximately 8 times 7.62 Hz). This may be a ball-spalling problem caused by the cage defect. The 118 Hz frequency is the resonant frequency of the structure.

Figure 14 shows the rotation frequency and its harmonics. The cage defect is not detected when there is meshing. The frequencies of 8 Hz, 18 Hz, 118 Hz, etc. are the natural frequencies of the system.

As in the previous figure, the analysis failed to detect the cage defect.

In the following figures below, the results obtained by cepstral analysis of configurations 10, 11 and 12 are presented in Figures 16-18 .

In the figure above, we note the presence of harmonics of the cage defect frequency of 3.81 Hz (7 Hz, 14 Hz, etc.). Cepstral analysis allows the detection of simple cage defects.

In the figure above, there are two defects: a cage defect and a gear defect. Cepstral analysis does not clearly detect either of these defects. However, in principle, the function of cepstral analysis is to highlight harmonics in a signal.

In the figure above, when there are two or more defects (multiple defects, in bearing and gear), cepstral analysis is unable to detect them.

In conclusion, cepstral signal analysis, in the case of our work, was able to detect simple bearing defects or shaft misalignment. However, it has limitations when it comes to detecting multiple defects in a system. Low-amplitude defects are not detected when there are two or more defects in a signal.