# Study of vibration signals using CWTFT and STFT for belt changes diagnostic

Donald Rotimbo, Mourad Lamraoui, Mohamed El Badaoui, François Guillet, Marius Béka

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Donald Rotimbo, Mourad Lamraoui, Mohamed El Badaoui, François Guillet, Marius Béka. Study of vibration signals using CWTFT and STFT for belt changes diagnostic. 18th Symposium Vibrations, SHocks & NOise (VISHNO), Jun 2014, Aix en Provence, France. hal-04183702

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Study of vibration signals using CWTFT and STFT

for belt changes diagnostic

Donald Rotimbo<sup>a,b,a*</sup>, Mourad Lamraoui<sup>a</sup>, Mohamed El Badaoui<sup>a</sup>, François Guillet<sup>a</sup> and Marius Béka<sup>b</sup>

<sup>a</sup> Université de Lyon, F-42023, Saint-Etienne, France

Université Jean Monnet de Saint-Etienne, F-42000, Saint-Etienne, France

LASPI, F-42334, IUT de Roanne, France

<sup>b</sup>Ecole Polytechnique de Masuku, USTM,B.P. 941, Franceville, Gabon

{rotimboromarick,lamraoui.mourad,mt_beka}@yahoo.fr

{guillet,mohamed.elbadaoui}@univ-st-etienne.fr

## Highlights

- We study vibration signals with Continuous Wavelet Transform based on the Fourier Transform (CWTFT) and Short-Time Fourier Transform (STFT). This study shows the benefits of time-frequency representation for the analysis of non-stationary signals such as location, extraction and monitoring of interesting frequency bands. The new approach is belt changes diagnostic. The energy to kurtosis ratio calculated along scales (frequencies) from CWTFT and STFT coefficients allows a good discrimination.

## 1. Introduction

Vibration analysis is often used to monitor the operation of rotating machines. It provides crucial information for the detection, identification and location of a fault. Traditional techniques, ranging from the power spectral density of the cepstrum, to the amplitude demodulation, actually possible to produce a robust diagnostic. Their main limitation is their definition around the assumption of stationarity

The time-frequency analysis is very effective for the characterization of non-stationary signals, with tools such as the wavelet transform [1] or the Short-Term Fourier Transform [2]. The continuous wavelet transform (CWT) is known for its many benefits. It has different levels in time and location in frequency that can be adapted to the phenomenon to be characterized. Its applications are varied. In meteorology, it indicated a change in the surface temperature of the sea [3]. In denoising, it has improved the quality of microscopic images [4]. In the mechanical field, it has widely been used as a discrimination tool [5] for the trouble diagnosis of bearings [6]. The STFT does not have the key strengths of the CWT. However, its ease implementation has made a very interesting tool for monitoring. Studies have shown its strengths in the filtering [7], the extraction of random components [8] or the detection of bearing defects [9].

<footnote>* Corresponding author. Tel.: +33-6-09-79-80-22.</footnote>

<footnote>E-mail address: donald.romarick.rotimbo.mbourou@univ-st-etienne.fr</footnote>
In this paper, we want to compare these two time-frequency tools for the diagnosis of belt tension changes in during power transmission. Belt drives systems are found in many sectors: automotive industry, mining career, appliances ... Several studies have attempted to model the forces when belt transmits power to other rotating elements. Some have studied this type of transmission system consists of two pulleys and others have added a tensioner [10, 11]. These investigations have shown that the belt was subjected to lateral, longitudinal, transverse and torsional vibrations. However, the transverse vibrations are the most harmful and then retain more attention. They can be treated as impulsive phenomena [12]. The transverse vibrations associated with the rotating pulleys eccentricity leads to a belt-pulley coupling phenomenon, at the contact points [13]. In the presence of tensioner, we must add the belt-tensioner coupling [11]. These two couplings operate with the belt tension and result in the spectral domain by a frequency modulation around rotating frequencies of driven pulleys [13,14,15] and tensioner [11]. It is noted that these frequencies are very similar of the first natural frequencies of belt. It is possible to use this information to monitor the voltage.

The time-frequency representation has already been applied to the estimation of the belt state. It has provided critical information on the tension loss [16] or even the quality of the belt-pulley coupling [17]. Before showing the contribution of these time-frequency tools in our work and the innovative aspect, we will return the theoretical aspects.

## 2. Continuous Wavelet Transform using Fast Fourier Transform (CWTFT)

The signal study by the Fourier Transform (FT) gives an overall idea of the frequencies present in it. It is not possible to know over time changes or fluctuations in the presence. The Wavelet analysis provides superior results because it is capable to give a time evolution of a factors number related to the operating conditions. It usually overrides the other analysis techniques because it reveals more discontinuities, trends or breakpoints [18,19].

Continuous Wavelet Transform [20] is defined as:

$$CWT(\tau, s) = \int_{-\infty}^{\infty} x(t) \frac{1}{\sqrt{s}} \psi^* \left( \frac{t-\tau}{s} \right) dt \quad (1)$$

Where $\tau$ is the translation factor, $s$ the scale factor, $\psi$ the mother wavelet and $\frac{1}{\sqrt{s}}$ the complex conjugate. The $\frac{1}{\sqrt{s}}$ ratio normalizes wavelets energy at all scales.

CWT can also be interpreted as a filter based on the signal frequency. Thus, it is possible to rewrite the CWT as an Inverse Fourier Transform such that:

$$CWTFT(\tau, s) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{x}(\omega) \sqrt{s}(\hat{\psi}(s\omega))^* e^{j\omega\tau} d\omega, \quad \omega = 2\pi f \quad (2)$$

With $\hat{x}(\omega)$ and $\hat{\psi}(\omega)$, the Fourier transforms of the $x$ signal and $\psi$ wavelet. Equation (2) indicates that as the CWT is the Fourier Transform of a Fourier transforms product. The $t$ time variable of the inverse Fourier Transform becomes the $\tau$ translation factor of the CWT. Therefore, it is possible to obtain the CWT from the inverse of the Fourier Transform (whose calculation is faster in practice): it is the CWTFT [3].

The calculation of the wavelet coefficients (WC) with the CWT depends essentially on the definition of the mother wavelet. It is expanded and contracted with changing the $s$ scale parameter. By varying $s$, this contributes to change the $f_c$ center frequency of the wavelet and the window size. In this case, instead of frequencies, which are the scales represent the results in the wavelet analysis. $\tau$, the translation factor, shows where the wavelet on the time axis: by varying the wavelet is shifted to the signal. A $s$ constant scaling factor and $\tau$ variable translation factor, the lines of the map time-scales are determined. A variable scale factor $s$ and $\tau$ constant translation factor, the columns of the map time-scales can be calculated.
It should be noted that a wavelet function has its own $f_c$ center frequency at each scale. There is an inverse proportionality relation between the scale $s$ and the associated frequency. Thus, the large scales correspond to low frequencies (overview) and small scales at high frequencies (detailed view). The quality of the time-frequency map is such that the decrease of the $s$ scale, thus a shorter window, increases the time resolution and the frequency resolution reduces (good frequency resolution for high frequencies). In contrast, the increase in the $s$ scale, therefore a longer window, time resolution decreases and increases the frequency resolution (high temporal resolution for low frequencies). In our case, this is the Morlet wavelet is used, defined as follows:

$$ \psi(t) = g(t)e^{-j2\pi f_c t}, \quad g(t) = \sqrt{\pi f_b} e^{\frac{t^2}{4f_b}} \qquad (3) $$

Where the $f_c$ is center frequency and $f_b$, the bandwidth factor. These are the two parameters to be changed. The link between the frequencies and scales for the Morlet wavelet, is as follows:

$$ f = \frac{f_c}{s} \qquad (4) $$

Despite its strengths, the CWT often has limitations in computing time or trouble to always follow the same frequency band comprising the major part of information when faults arise. It may be advantageous to have a time-frequency map linearly distributed along the frequency.

### 3. Short-Time Fourier Transform (STFT)

The Short-Time Fourier Transform is a method of time-frequency analysis, in which we compute the Fourier Transform of the original signal on a windowed part (with shift along the time axis of the window). Specifically, the $x$ original signal is windowed by a $w$ window of defined size that is centered at $\tau$ time. The Fourier Transform is determined from the windowed signal. We obtain the frequency content of the signal within a range of windowed time : the time-frequency map.

STFT [2.22] for a mathematical expression:

$$ \text{STFT}(\tau, f) = \int_{-\infty}^{\infty} x(t)w^*(t-\tau)e^{-j\omega t}dt, \quad \omega = 2\pi f \qquad (5) $$

Where * represents the complex conjugate.

The quality of the chosen time-frequency map mainly depends on the window. A small window gives good time resolution : in this case, some frequencies can not be clearly identified. In contrast, a larger window provides a temporal resolution of less quality. However, it is possible to better distinguish non-identifiable frequencies in the previous case. It should be noted the difficulty of having both a good time resolution and frequency resolution [21].

A compromise is necessary. When we want to study the low-frequency components having a longer time period, it is more interesting to try a high resolution spectrum (ie a size large enough window). Conversely, the frequency components, which can materialize during a very short time, requires much greater frequency resolution (and thus a relatively small window size).

When the estimation procedure of the time-frequency map is optimized, it is necessary to define an appropriate setting that will allow to select the frequency bands comprising the majority of useful information.

### 4. Energy to Kurtosis ratio criterion (EKRC)

The Relative Wavelet Energy (RWE) is a good estimate of density along a time-scale map. It can help in particular to detect a particular phenomenon. The RWE provides energy information along specific frequency bands.
This is a test that measures the degree of similarity between the segments of a signal [23]. At each level of resolution, depending on the selected wavelet parameters for calculating wavelet coefficients, the energy presents the signal energy content. It can be calculated as:

$$E(s) = \sum_{i=1}^{k} |WC_{s,i}|^2 \qquad (6)$$

Where k is the wavelet coefficients number at each scale s and WCs,i represents the wavelet coefficient at position i of s scale on the time-scale map.

According to the phenomenon under study, RWE can sometimes lack robustness. We now define the criterion energy to spectral kurtosis ratio as more suitable for selecting significant scales in the best time-frequency map.

The spectral kurtosis has some ability to identify regions with a specific interest and locate [24]. It can be used to characterize and localize the transient components in the frequency domain. Thus, it is an interesting indicator that can provide a suitable bandwidth (eg for demodulation) for the study of stationary and non-stationary signals.

The spectral kurtosis for a definition:

$$SK(s) = \frac{E[|X(s)^4|]}{E[|X(s)^2|]^2} - 2 \qquad (7)$$

Where E[|X(s)ⁿ|] is the n-order spectral moment of the X(n) non-stationary signal.

The energy and spectral kurtosis have been used for the selection of specific regions in the time-frequency map, in order to achieve a vibration diagnosis [9,26].

We use the following criteria:

$$ESK(s) = \frac{E(s)}{SK(s)} \qquad (8)$$

The ESK(s) energy to spectral kurtosis ratio, at each scale s, summed scales (frequency) the richest information in the time-frequency map. We apply for the first time and with the CWTFT and STFT.

# 5. Results and discussion

## 5.1. Test bench

<figure><img src="image_2.png" /></figure>
Fig. 1. Test bench of V-belt drive.

Experimental data come from a test bench Fig.1. We have a V-belt in two distinct states: stretched and relaxed. This belt serves to drive a driven pulley, by a motor (drive pulley) rotating at constant speed. During operation, particular attention is paid to the belt spans beats. The stretched side is of the belt portion located on the tensioner side. In contrast, we have the slack side. Belt beats (impulse phenomena [12]) can significantly degrade the transmission power and accelerate wear. Therefore, a tensioner is located between the two pulleys. It also ensures proper tension during operation.

The vibration signals of our study are acquired from a piezoelectric strain gauge. This sensor is arranged on the driven pulley shaft, closer to the rotational vibration. The strain gauge advantage relative to the accelerometer is its best sensitive to the structure deformation [14]. This is interesting for our study because we work at slow speed. All information acquisition are summarized in Table 1.

In the following, we make a comparative study of vibration signals coming from strain gauge with a V-belt at stretched (figures in blue) and relaxed (figures in red) state. As a first step, we calculate statistical parameters from time signals. We present some interesting indicators reflecting an overall change in belt tension. In second stage, the spectral analysis proposes the use of a suitable discriminatory parameter. There is a frequency modulation around the driven pulley and tensioner rotating frequencies [13,14,15]. These frequencies are close to the belt natural modes. Then, the time-frequency map confirms the previous observation on the on modulation and gives a more accurate view of the major events in our impulsed signals. The CWTFT and results are presented in the last part.

## 5.2. Time analysis

A signals time view of order is given in FIGS Fig.1 (a) - (c). Two horizontal arrows show the dominant cycles duration : belt cycle at 0.81 s and pulley cycle at 0.32 s. The vibration amplitudes of power transmission with belt stretched Fig.1 (a) is about 0.8 (oscillations in the [-0.3,0.5]). These amplitudes are divided by two and about 0.4 (oscillations in the [-0.15,0.25]), in the belt relaxed case (Fig.1 (c)). With the decrease in belt tension, appearance changes of the time signals. Cycles belt disappear and we see more clearly the driven pulley rotation cycles.

In the current configuration of our power transmission, two dominant phenomena occur during operation: the transverse vibration of belt spans (impulse phenomena) and the rotating pulley vibration. The transverse vibration amplitudes of spans depend on the belt tension (speed impact is negligible at low speeds). When the tension increases, the deflection amplitude of the slack portion increases also. The differences in amplitude between the dominant modes and other parts of the system decreases. The distinction of the different components in the system becomes less obvious. The tension induces a mode coupling of the belt spans associated with driven pulley. This coupling is less important with a relaxed belt (low tension). [11]
<table><thead><tr><th>Sensor type</th><th>Piezoelectric strain gauge</th></tr></thead><tbody><tr><td>Acquisition time (s)</td><td>30</td></tr><tr><td>Sampling frequency (Hz)</td><td>400</td></tr></tbody></table>

<table><caption>xTab. 1. Acquisition informations.</caption><thead><tr><th>Rotating element</th><th>Rotating Period (s)</th><th>Rotating Frequency (Hz)</th></tr></thead><tbody><tr><td>V-belt</td><td>0.81</td><td>1.23</td></tr><tr><td>Driven pulley</td><td>0.32</td><td>3.08</td></tr><tr><td>Tensioner</td><td>0.15</td><td>6.28</td></tr><tr><td>Driving motor</td><td>0.11</td><td>8.61</td></tr></tbody></table>

<table><caption>Tab. 2. Comparison between stretched and relaxed V-belt with statistics parameters.</caption><thead><tr><th rowspan="2">Statistics parameters</th><th rowspan="2">Acquisition time (s)</th><th colspan="2">Belt type</th><th rowspan="2">Decreasing ratio (%)</th></tr><tr><th>Stretched V-belt</th><th>Relaxed V-belt</th></tr></thead><tbody><tr><td rowspan="3">Mean</td><td>5</td><td>0.0043</td><td>0.0053</td><td><strong>+23.25</strong></td></tr><tr><td>15</td><td>0.0050</td><td>0.0048</td><td><strong>-4</strong></td></tr><tr><td>30</td><td>0.0051</td><td>0.0050</td><td><strong>-1.9</strong></td></tr><tr><td rowspan="3">Variance</td><td>5</td><td>0.0002</td><td>0.0001</td><td><strong>-50</strong></td></tr><tr><td>15</td><td>0.0002</td><td>0.0001</td><td><strong>-50</strong></td></tr><tr><td>30</td><td>0.0002</td><td>0.0001</td><td><strong>-50</strong></td></tr><tr><td rowspan="3">Skewness (absolute value)</td><td>5</td><td>0.2004</td><td>0.0470</td><td><strong>-76.54</strong></td></tr><tr><td>15</td><td>0.1271</td><td>0.0162</td><td><strong>-87.25</strong></td></tr><tr><td>30</td><td>0.1489</td><td>0.0002</td><td><strong>-99.86</strong></td></tr><tr><td rowspan="3">Kurtosis</td><td>5</td><td>3.1971</td><td>2.6028</td><td><strong>-18.58</strong></td></tr><tr><td>15</td><td>3.0054</td><td>2.6123</td><td><strong>-13.07</strong></td></tr><tr><td>30</td><td>2.9749</td><td>2.5474</td><td><strong>-14.37</strong></td></tr></tbody></table>

Table. 2 shows the statistical parameters (mean, variance, skewness and kurtosis) calculated on vibration signals. The signals acquisition duration processed from 5, from 15 to 30 s to test the robustness of the selected criteria. We want to verify that the criteria do not change with the size of the time window analysis. However, we strive to obtain a sufficient number of machinery cycles in each case.

Averages are very similar for the two belts states. We note a low variance value indicating a good distribution of signal samples around their mean value; cycles are not very far from each other. The variance decreases to 50% in relaxed belt. Which is close to the time analysis Fig. 1 (a) - (b). The tension drop results in lower deflection amplitude of belt spans. There is a phenomena reduction in belt heartbeats (↔ impulse phenomena decrease) and therefore of the transverse vibration decreases. The system becomes more stable. The main changes (around 80%) are observed with the skewness. System stability, with the impulse phenomena limitation, leads to a new form of the vibration signal. There is then an increase in the similarity cycles (driven pulley). Vertical symmetry increases leading to (much closer to 0) low skewness values. The kurtosis values decrease is consistent (about 15%) in the relaxed belt case. The kurtosis is sensitive to impulse phenomena. These are seen better in high tension (stretched belt).
<figure><img src="image_3.png" /><figcaption>Fig. 2. Time representation (1ère colonne) and PSD en decibel (2ème colonne): stretched V-belt (courbe en blue), relaxed V-belt (courbe en rouge); fréquence fondamentale et harmoniques de la poulie menée □, fréquence de rotation du tendeur ○, fréquences de modulation Δ.</figcaption></figure>

A combination of variance and skewness criteria could indicate an overall belt relaxation (with piezoelectric strain gauge signal). While the variance is insensitive to acquisition time, we note an increase in the skewness. We can conclude that the acquisition time has an influence on the outcome of the combined endpoint. A cycles number of the driven pulley 40 is adapted to order (purchase 32 s).

In the next section, the frequency analysis results will be studied in the hope of determining a more precise criterion for discrimination.

## 5.3. Spectral analysis

The power spectral density (PSD) of two belt configurations are shown in Figures 1(b) - (d). In the stretched belt case, certain frequencies are not well discernible. However, we find the frequency peak of driven pulley rotation at 3.08 Hz and many harmonics (□). There is also the frequency peak of the tensioner rotation at 6.28 Hz tension (○), which coincides with first harmonic frequency of driven pulley rotation. Finally, the modulation frequency peaks are identifiable (Δ) on each side of those rotations of the driven pulley and tensioner (respectively ± 0.6 and ± 1.3 Hz).

In operation, the transverse vibration of belt spans, coupled with the driven pulley rotation, excite the belt natural frequencies. The major part of the natural frequency increases as the belt tension increases (stretched belt case). This is understandable because the natural frequencies depend on the system steady state. The coupling phenomenon existence leads to the system instabilities (excitation due to large deflection amplitudes in belt spans) and thus to changes in natural frequencies. Note that the natural frequency is also influenced by the belt rotation speed. The speed decrease causes a non-linear decrease of the natural frequencies of the dominant modes.
<table><caption>Tab. 3. Acquisition parameters.</caption><thead><tr><th></th><th>V-belt type</th><th>Stretched V-belt</th><th>Relaxed V-belt</th><th>Decreasing ratio (%)</th></tr></thead><tbody><tr><td rowspan="6">Driven pulley</td><td>Frequency peak (Hz)</td><td>3</td><td>3.1</td><td></td></tr><tr><td>Magnitude of frequency peak</td><td><strong>-44.09</strong></td><td><strong>-48.18</strong></td><td><strong>-9.27</strong></td></tr><tr><td>1<sup>st</sup> modulating frequency peak (Hz)</td><td>2.4</td><td>2.5</td><td></td></tr><tr><td>Magnitude of 1<sup>st</sup> modulating frequency peak</td><td><strong>-53.31</strong></td><td><strong>-82.85</strong></td><td><strong>-55.41</strong></td></tr><tr><td>Difference between frequency peak and its 1<sup>st</sup> modulating frequency peak (Hz)</td><td>0.6</td><td>0.6</td><td></td></tr><tr><td>Ratio between amplitudes of frequency peak and its 1<sup>st</sup> modulating frequency peak</td><td><strong>0.82</strong></td><td><strong>0.58</strong></td><td><strong>-29.26</strong></td></tr><tr><td rowspan="6">Tensioner</td><td>Frequency peak (Hz)</td><td>6.1</td><td>6.2</td><td></td></tr><tr><td>Magnitude of frequency peak</td><td><strong>-54.07</strong></td><td><strong>-54.39</strong></td><td><strong>-0.5</strong></td></tr><tr><td>1<sup>st</sup> modulating frequency peak (Hz)</td><td>4.8</td><td>4.9</td><td></td></tr><tr><td>Magnitude of 1<sup>st</sup> modulating frequency peak</td><td><strong>-60.12</strong></td><td><strong>-86.6</strong></td><td><strong>-44.04</strong></td></tr><tr><td>Difference between frequency peak and its 1<sup>st</sup> modulating frequency peak (Hz)</td><td>1.3</td><td>1.3</td><td></td></tr><tr><td>Ratio between amplitudes of frequency peak and its 1<sup>st</sup> modulating frequency peak</td><td><strong>0.89</strong></td><td><strong>0.62</strong></td><td><strong>-30.33</strong></td></tr><tr><td colspan="2">Global ratio (driven pulley ratio to tensioner ratio)</td><td><strong>0.82 / 0.89 = 0.9196</strong></td><td><strong>0.58 / 0.62 = 0.9259</strong></td><td><strong>+0.6</strong></td></tr></tbody></table>

The modulating peaks of the driven pulley rotation frequency are related to the coupling transverse vibrations of belts spans and with driven pulley rotation [14]. PSD give us new information. There is also another coupling belt spans transversal vibration with the belt tensioner rotation. These elements are used to define a second discrimination criteria in able 3. We calculate the amplitudes ratio of first modulating frequency peak of driven pulley and tensioner rotation.

In the two considered cases (relaxed and stretched belt), Table 3 shows that the frequency peaks amplitudes of driven pulley rotation (-44.09 and -48.18 respectively) and idler rotation (-54.07 and -54.39 respectively) are very close. The decay rate was respectively 9.27% and 0.7% for each rotating element. However, there are significant changes in the peaks amplitudes of first modulating frequency of driven pulley rotation (-53.31 and -82.85 respectively) and idler rotation (-86.6.85 and -60.12 respectively). The decay rate is respectively 55.41% and 44.04% for each rotating element. The ratio of the frequency peak amplitude of rotation and the first modulating peak amplitude decay almost equivalent to the components (29.26% and 30.33% respectively). This indicator shows that the tension loss in the belt has a proportional effect on the belt-pulley and belt-tensioner coupling. Note that no modulation to the driven pulley rotation frequency is 0.6 Hz, 1.3 Hz to against the tensioner rotational frequency. This corresponds to the energy flow through the belt. [13]

Frequency analysis showed the frequency modulation due to the two belt-pulley and belt-tensioner coupling phenomena. We hope that the time-frequency analysis will provide a robust and local information and phenomena.
5.4. Time-frequency analysis with CWTFT and STFT

Figure 5 shows CWTFT and STFT results. The CWTFT gives a better representation of events contained in the vibration signal in the low-frequency for both belt states. With CWTFT, we see clearly two periodic phenomena (Fig. 3 (a) - (c)). They have an impulsive character. The most visible (red) has a period of 1.66 s (0.6 Hz). It is linked to the rotating frequency to 3.02 Hz (driven pulley). This is the belt-pulley coupling. The second (green) returns all 0.76 s (1.3 Hz). It is linked to the rotating frequency to 6.05 Hz (tensioner). This is the belt-tensioner coupling. Periodic amplification seems to occur every 12.5 s (0.07 Hz). It suggests the existence of another unidentified phenomenon. Belt tension changes lose their impulsive features. The STFT is a time-frequency map of which the frequency distribution is linear (Figure 3 (b) - (d)). The resolution is worse and we can not easily distinguish existing phenomena in the signal, for both belt states. However, major changes occur in the low frequencies.

The ESK criteria results is displayed in Figure 4. Same frequency peaks are selected in the STFT for both belt states (Table 4). These peaks move slightly in the CWTFT. The CWTFT has a better temporal resolution at low frequencies. It better represents the signal time-frequency evolution. A high tension, the transverse vibrations of the belt spans are impulse phenomena that excite the belt natural frequencies, more phenomena couplings already present (Fig. 4 (a)). With the tension loss (relaxed belt case), some modes are not excited (Fig. 4 (c)). Both phenomena couplings are then better visible. This observation is not visible on the STFT because of the linearity of frequency scales.

Considering the CWTFT, we note a decrease of 69.93% from the frequency peak related to belt pulley coupling and 81.36% of the frequency peak related to belt-tensioner coupling. The STFT, the frequency peak related to belt-pulley coupling also decreased by 59.25%. It is the same goes for the frequency peak of belt-tensioner coupling (66.03%). The amplitudes ratio of two frequency peaks coupling is higher for CWTFT (3.94 and 6.38 respectively for the stretched and relaxed belt states) and the STFT (1.89 and 2.27 respectively for the stretched and relaxed belt states).
<figure><img src="image_4.png" /><figcaption>Fig. 3. Modulus of CWTFT (1ère colonne) and spectrogram with window 512 (2ème colonne) : stretched V-belt (a)-(b), relaxed V-belt (c)-(d).</figcaption></figure>

<figure><img src="image_5.png" /></figure>
<table><caption>Fig. 4. ESK criterion along frequencies for modulus of CWTFT (1ère colonne) and spectrogram with window 512 (2ème colonne) : stretched V-belt (a)-(b), relaxed V-belt (c)-(d).</caption><thead><tr><th colspan="2">V-belt type</th><th>Stretched<br>V-belt</th><th>Relaxed<br>V-belt</th><th>Decreasing<br>ratio<br>(%)</th><th>Stretched<br>V-belt</th><th>Relaxed<br>V-belt</th><th>Decreasing<br>ratio<br>(%)</th></tr><tr><th colspan="2">Time-frequency analysis type</th><th>CWTFT</th><th>CWTFT</th><th>CWTFT</th><th>STFT</th><th>STFT</th><th>STFT</th></tr></thead><tbody><tr><td rowspan="2">Driven<br>pulley</td><td>Frequency peak (Hz)</td><td>2.822</td><td>3.025</td><td></td><td>3.125</td><td>3.125</td><td></td></tr><tr><td>Magnitude of frequency peak</td><td><strong>109.3</strong></td><td><strong>32.97</strong></td><td><strong>-69.83</strong></td><td><strong>114.1</strong></td><td><strong>46.49</strong></td><td><strong>-59.25</strong></td></tr><tr><td rowspan="2">Tensioner</td><td>Frequency peak (Hz)</td><td>6.05</td><td>6.713</td><td></td><td>6.25</td><td>6.25</td><td></td></tr><tr><td>Magnitude of frequency peak</td><td><strong>27.7</strong></td><td><strong>5.161</strong></td><td><strong>-81.36</strong></td><td><strong>60.1</strong></td><td><strong>20.41</strong></td><td><strong>-66.03</strong></td></tr><tr><td colspan="2">Ratio between amplitudes of driven pulley frequency peak and tensioner frequency peak</td><td><strong>3.94</strong></td><td><strong>6.38</strong></td><td><strong>+61.92</strong></td><td><strong>1.89</strong></td><td><strong>2.27</strong></td><td><strong>+20.1</strong></td></tr></tbody></table>

<caption>Tab. 4. Peak-to-peak ratios for modulation study.</caption>

## 6. Conclusion

In our paper, we presented several indicators of belt tension changes. Some research has shown that the belt tension variation had a significant impact on the belt spans transverse vibration. This results in the coupling quality at the contact points belt-pulley and belt tensioner (in the presence of tensioner). In the time domain, the tension diminution reduces belt spans beats and therefore a decrease in the signal vibration amplitude. On the frequency domain, we observe two modulation around the driven pulley and tensioner rotation frequencies. These frequencies conterminant also with the belt natural frequencies. Investigations suggested estimating the belt-pulley coupling degree (tension degree) from the amplitudes ratio of frequency peak of the driven pulley rotation on its modulating peaks.

We conducted an experimental study of belt drives on a test bench in the presence of a tensioner. In our study at slow speed, we considered two transmission configurations : stretched and relaxed belt. From a piézolelectrique strain gauge, we acquired vibration data.

The real signals analysis confirmed the previous theoretical and experimental works results. The time analysis showed a decrease in the vibration level with the tension loss in the belt. The time cycles corresponding to pulley rotation become increasingly distinct and ressemblance increases. The statistical results have allowed us to propose two new global indicators : the variance and skewness. The acquisition time influences their effectiveness.

In the frequency analysis, we actually found two modulations around the pulley and tensioner rotation frequencies. The estimate of the belt-pulley coupling degree from the amplitudes ratio of the frequency peak the driven pulley rotation on its modulating peaks is adapted. We have proposed two new indicators. The first also considers the belt-tensioner coupling degree. The second is the ratio of two frequency estimators mentioned above. Our analysis showed that the belt-pulley and belt-tensioner coupling degrees are very close. The change in the belt tension seems to have a repercussion on the quite similar in driven pulley and tensioner. This coupling decreases as the tension decreases.

The comparative study of time-frequency maps obtained with CWTFT and STFT were really the novelty of this work. The results suggest that CWTFT has a better time resolution at low frequencies. We were able to observe the belt-pulley and belt-tensioner couplings. However, it is difficult to follow the interesting rotation frequency when the tension decreases in the belt because the maximum peaks move. Should follow a wide frequency band. The cause is certainly the belt natural frequencies evolution. The STFT does not forces the previous tool. Phenomena representation is quite poor. However, the peaks corresponding to the maximum rotation frequency of interest do not change position when the tension decreases. It characterized by changes in the natural frequencies correctly. For monitoring, it is more
concise and summarize relevant information provided to choose the right parameters. On the other hand, it is easier to calculate than the CWTFT.

We expect the outlook to see the speed impact on the results presented in this paper, with two relevant tools : cepstrum and cepstrogram.

## References

1. O. Rioul and M. Vetterli, *Wavelets and signal processing*, IEEE SP Magazine, October 1991, p. 14-28.

2. J. B. Allen, *Short Time Spectral analysis, synthesis, and modification by discrete Fourier Transform*, IEEE Transactions on Acoustics, Speech, and Signal Processing, June 1977, 25 (2) : p. 235-238.

3. C. Torrence and G.P. Compo, *A pratical guide to wavelet analysis*, Bull. Am. Meteorol. Soc., 1998, 79 : p. 61-78. [Available : http://paos.colorado.edu/research/wavelets/].

4. U. Bal, *Dual tree complex wavelet transform based denoising of optical microscopy images*. Follow Journal Biomedical Optics Express, 2012, vol. 3 (12).

5. R. Yan and R. X. Gao, *Approximate Entropy as a diagnostic tool for machine health monitoring*, Mechanical Systems and Signal Processing, February 2007, vol. 21 issue 2 : p. 824-839.

6. A. Kumar and R. Kumar, *Adaptive wavelet based signal processing scheme for detecting localized defects in rolling element of taper roller bearing*, Proceedings of Surveillance 7 International Conference, October 29-30, 2013.

7. B. Chen, Z. Yan and W. Chen, *Defect detection for wheel-bearings with time-spectral kurtosis and entropy*. Entropy, 2014, 16(3) : p. 607-626.

8. J. Antoni and R.B. Randall, *Unsupervised noise cancellation for vibration signals - Part 2 - a Novel STFT algorithm*, Mechanical Systems and Signal Processing, 2004, 18 (3) : p. 103-117.

9. M. Cocconcelli, R. Zimroz, R. Rubini and W. Bartelmus, *STFT based spectral kurtosis and energy distribution approach for ball bearing fault detection in varying speed motor*, Proceedings of Surveillance 6 International Conference, France, 2011.

10. R.S Beikmann, N.C. Perkins and A.G. Ulsoy, *Design and analysis of automotive serpentine belt drive systems for steady state performance*, ASME Journal of Mechanical Design, 1997, 119 : p. 162-168.

11. L. Kong and R.G. Parker, *Coupled belt-pulley vibration in serpentine drives with belt bending stiffness*, ASME Journal of Applied Mechanics, 2004, 71 : p. 109-119.

12. K. W. Wang and C. D. J. Mote, *Band/wheel system vibration under impulsive boundary excitation*. Journal of Sound and Vibration, 1987. 115(1) : p. 203-216

13. A. G. Ulsoy, *Coupling between spans in the vibration of axially moving materials*, Journal of Vibration, Acoustics, Stress and Reliability in Design, 1986, 108 : p. 207-212.

14. W.-T. Rim and H.-J. Kim, *Identification of tension in a belt-driven system by analysing flexural vibrations*. Mechanichal Systems and Signal Processing, 1994, 8(1) : p. 199-213.

15. K. W. Wang and C. D. J. Mote, *Vibration coupling analysis of and/wheel mechanical systems*, Journal of Sound Vibration, 1986, 109 : p.237-258.

16. S. Fang, B. HU and S. LIU, *The analysis on fire smoke exhausting robot vibration based on Wavelet Transform*, Advanced Materials Research, 2011, Vols. 328-330 : p. 1887-1891.

17. L. Notini, A. Al-habaibeh, F. Zorraiassatine, R.M. Parkin, M.R. Jackson and J. Coy, *Condition monitoring of belt based power transmission systems using low resolution thermal imaging*. Proceedings of ICOM 2003 Mechatronics Conference, UK, June 2003.

18. I. Daubechies, *Ten lectures on Wavelets*, PA : Society for industrial and applied Mathematics (SIAM), Philadelphia, 1992.

19. M. Farge, *Wavelet Transforms and their application to turbulence*, Annual Review of Fluid Mechanics, 1992, vol. 24 : p. 395-458.

20. S. Mallat, *A Wavelet tour of Signal Processing*, CA : Academic Press, 1998.

21. R. J. E. Merry, *Wavelet theory and applications : a litterature study*. Technische Universiteit Eindhoven, 2005,
[Available : http://alexandria.tue.nl/repository/books/612762.pdf]

22. M. G. E. Schneiders. *Wavelets in control engineering*, Master’s thesis, Eindhoven University of Technology, August 2001.

23. O.A. Rosso, S. Blanco, J. Yordanova, V. Kolev, A. Figliola, M. Schurmann and E. Basar, *Wavelet entropy : a new tool for analysis of short duration brain electrical signals*, Journal of Neuroscience Methods, 2001, 105 : p. 65-75

24. J. Antoni, *The spectral kurtosis : a useful tool for characterising non-stationary signals*, Mechanical Systems and Signal Processing, 2006, 20(1) : p. 282-307.

25. R.B Randall and J. Antoni, *Rolling element bearing diagnostics-A tutorial*. Mech. Syst. Signal Process, 2011, 25 : p. 485-520.

26. R. X. Gao and R. Yan, *Wavelets Theory and Applications for Manufacturing*, Springer, 2010.