A Multidimensional Health Indicator Based on Autoregressive Power Spectral Density for Machine Condition Monitoring
<p>Bands definition and computation of the nominal AR spectrum through healthy data.</p> "> Figure 2
<p>Online monitoring procedure.</p> "> Figure 3
<p>Time–frequency analysis of acceleration signal in the absence of failure and at a maximum speed of 1797 rpm (Healthy0): (<b>a</b>) Spectrogram (<b>b</b>) magnitude of Fourier Synchrosqueezed Transform, (<b>c</b>) Fourier power spectrum, (<b>d</b>) integral over time of local instantaneous squeezed frequencies in the TF plane. Vertical red lines identify the boundaries of the bands obtained by the procedure.</p> "> Figure 4
<p>Time–frequency analysis of acceleration signal in the absence of failure and at a speed of 1750 rpm (Healthy2): (<b>a</b>) Spectrogram (<b>b</b>) magnitude of Fourier Synchrosqueezed Transform, (<b>c</b>) Fourier power spectrum, (<b>d</b>) integral over time of local instantaneous squeezed frequencies in the TF plane. Vertical red lines identify the boundaries of the bands obtained by the procedure.</p> "> Figure 5
<p>CWRU bearing test rig: (1) electric motor, (2) torque transducer/encoder, (3) dynamometer. Accelerometers are located at the housing of both drive end (4) and fan end (5) bearings.</p> "> Figure 6
<p>CWRU DE vibration signals sampled at 48 kHz (motor load 0 hp): (<b>a</b>) healthy, (<b>b</b>) ball fault (0.007 inches), (<b>c</b>) ball fault (0.014 inches), (<b>d</b>) ball fault (0.021 inches).</p> "> Figure 7
<p>Estimation of the AR order <span class="html-italic">p</span>: (<b>a</b>) FPE criterion (<b>b</b>) MDL criterion. The red star shows the values of FPE and MDL associated with the order <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>54</mn> </mrow> </semantics></math>, which is double the number of peaks <math display="inline"><semantics> <msub> <mi>N</mi> <mi>p</mi> </msub> </semantics></math> estimated through the FSST procedure.</p> "> Figure 8
<p>Evolution of the SISSD indicator in the four different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math> as a function of the signal frames: (1) healthy, (2) BF (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (3) BF (<math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> in), (4) BF (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 9
<p>Evolution of the MSISSD indicator in the four different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics></math> for all the defined frequency bands as a function of the signal frames. Subfigures (<b>a</b>–<b>f</b>) refer to the subbands 1–6 defined in <a href="#sensors-24-04782-t002" class="html-table">Table 2</a>. The picture associated with Band <span class="html-italic">i</span> reports the evolution of the <span class="html-italic">i</span>-th entry of the MSISSD indicator. For every picture, the four conditions are (1) healthy, (2) BF (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (3) BF (<math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> in), (4) BF (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 10
<p>Evolution of the SISSD indicator in the four different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>}</mo> </mrow> </semantics></math> as a function of the signal frames: (1) healthy, (2) IRF (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (3) IRF (<math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> in), (4) IRF (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 11
<p>Evolution of the MSISSD indicator in the four different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>}</mo> </mrow> </semantics></math> for all the defined frequency bands as a function of the signal frames. Subfigures (<b>a</b>–<b>f</b>) refer to the subbands 1–6 defined in <a href="#sensors-24-04782-t002" class="html-table">Table 2</a>. The picture associated with Band <span class="html-italic">i</span> reports the evolution of the <span class="html-italic">i</span>-th entry of the MSISSD indicator. For every picture, the four conditions are (1) healthy, (2) IRF (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (3) IRF (<math display="inline"><semantics> <mrow> <mn>0.014</mn> </mrow> </semantics></math> in), (4) IRF (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 12
<p>Evolution of the SISSD indicator in the three different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>12</mn> <mo>}</mo> </mrow> </semantics></math> as a function of the signal frames: (1) healthy, (2) ORF orthogonal (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (4) ORF orthogonal (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 13
<p>Evolution of the MSISSD indicator in the four different conditions associated with the set of signals <math display="inline"><semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>12</mn> <mo>}</mo> </mrow> </semantics></math> for all the defined frequency bands as a function of the signal frames. Subfigures (<b>a</b>–<b>f</b>) refer to the subbands 1–6 defined in <a href="#sensors-24-04782-t002" class="html-table">Table 2</a>. The picture associated with Band <span class="html-italic">i</span> reports the evolution of the <span class="html-italic">i</span>-th entry of the MSISSD indicator. For every picture, the four conditions are (1) healthy, (2) ORF orthogonal (<math display="inline"><semantics> <mrow> <mn>0.007</mn> </mrow> </semantics></math> in), (4) ORF orthogonal (<math display="inline"><semantics> <mrow> <mn>0.021</mn> </mrow> </semantics></math> in).</p> "> Figure 14
<p>Confusion matrices associated with two classification experiments: (<b>a</b>) 0 hp load, <math display="inline"><semantics> <mrow> <mn>50</mn> <mo>%</mo> </mrow> </semantics></math> training ratio, worst case (accuracy <math display="inline"><semantics> <mrow> <mn>98.41</mn> <mo>%</mo> </mrow> </semantics></math>), (<b>b</b>) 1 hp load, <math display="inline"><semantics> <mrow> <mn>50</mn> <mo>%</mo> </mrow> </semantics></math> training ratio, accuracy <math display="inline"><semantics> <mrow> <mn>100</mn> <mo>%</mo> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. A Multidimensional Health Indicator Based on AR Spectrum
3. Condition Monitoring Procedure
3.1. Time–Frequency Representation of the Signal
3.2. Partitioning of the Fourier Axis in Subbands
3.3. Online Monitoring Procedure
- Collected data of the signal of interest (vibration, current, etc.) are segmented into (overlapping or not) frames of N samples.
- For each signal frame, an AR model of order p is estimated by using the LS approach, and the associated PSD is computed. The current PSD and the reference one are then used to compute the multivariate health indicator MSISSD (17).
4. Results
- A portion of the healthy signal is used to define the frequency bands, , to select a proper order p of the AR model, to estimate an AR model of the selected order, and to compute the associated reference PSD through (4). Note that this offline step is the same for every set as it involves only the (same) healthy signal.
- The remaining part of the healthy data and the faulty data sequences are segmented into frames of N = 20,000 samples. For each signal frame, an AR model of order p is estimated by using the LS approach, and the associated PSD is computed. The current PSD and the reference one are then used to compute the health indicator. Both the scalar spectral distance SISSD (13) (that does not use the frequency bands but only the whole spectrum) and the multidimensional distance MSISSD (17) are considered.
Evaluation of the MSISSD Indicator in Fault Classification
- The remaining part of the healthy signal and the faulty signals are segmented into frames of N = 20,000 samples. For each signal frame, an AR model of order p is estimated by using the LS approach, and the associated PSD is computed. The current PSD and the reference one are then used to compute the multivariate spectral distance MSISSD (17).
- At the end of step 2, we have a set of MSISSD points for each of the 14 classes representing the different conditions. For every class, the related MSISSD points are labeled with the corresponding class number and divided into a training set and a validation set. As regards the number of points of the training and validation sets, both the and fractions are considered.
- The classification task is performed by means of the support vector machine (SVM) classifier with a linear kernel and one-versus-one approach. To this end, the MATLAB [47] function “fitcecoc” was employed.
5. Conclusions
- The initial parameter setting step is carried out on the healthy signal only once.
- It can be applied to different types of acquired signals (vibration, current, torque, etc.) and to different types of machine components (bearings, gearboxes, shafts, etc.) from real industrial contexts without the need to interrupt the operation of the monitored system.
- It is not designed for a specific case, even though only vibration signals and roller bearings have been considered in this paper. Moreover, the successful use of the AR PSD for other signal/components, like current/bearings [14], vibration/gears [17], PLC torque/shaft [20], can be seen as further case studies where the MSISSD indicator can be successfully applied.
- The multivariable nature of the indicator may improve the detection of subtle faults with respect to spectrum-based scalar indicators.
- It allows one to perform the fault classification task without requiring a huge amount of data, unlike modern data-based approaches.
- There is no need to precisely identify the characteristic frequencies of faults, since it is only important to highlight the emergence of changes in the spectra.
- All steps involve operations that are not computationally critical, so the method can also be adopted as part of monitoring edge-computing conditions, allowing for early detection of failures.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Signal Class Number | Type of Signal |
---|---|
1 | Normal (healthy) |
2 | BF ( in) |
3 | BF ( in) |
4 | BF ( in) |
5 | IRF ( in) |
6 | IRF ( in) |
7 | IRF ( in) |
8 | ORF centred ( in) |
9 | ORF centred ( in) |
10 | ORF centred ( in) |
11 | ORF orhtogonal ( in) |
12 | ORF orthogonal ( in) |
13 | ORF opposite ( in) |
14 | ORF opposite ( in) |
Band Number | Frequency Range (kHz) |
---|---|
1 | |
2 | |
3 | |
4 | |
5 | |
6 |
Motor Load (hp) | Training Set (%) | Accuracy (Mean) | Accuracy (Worst) | No of Runs with Accuracy |
---|---|---|---|---|
0 | 132/200 | |||
81/200 | ||||
1 | 200/200 | |||
200/200 | ||||
2 | 197/200 | |||
152/200 | ||||
3 | 76/200 | |||
28/200 |
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Diversi, R.; Speciale, N. A Multidimensional Health Indicator Based on Autoregressive Power Spectral Density for Machine Condition Monitoring. Sensors 2024, 24, 4782. https://doi.org/10.3390/s24154782
Diversi R, Speciale N. A Multidimensional Health Indicator Based on Autoregressive Power Spectral Density for Machine Condition Monitoring. Sensors. 2024; 24(15):4782. https://doi.org/10.3390/s24154782
Chicago/Turabian StyleDiversi, Roberto, and Nicolò Speciale. 2024. "A Multidimensional Health Indicator Based on Autoregressive Power Spectral Density for Machine Condition Monitoring" Sensors 24, no. 15: 4782. https://doi.org/10.3390/s24154782
APA StyleDiversi, R., & Speciale, N. (2024). A Multidimensional Health Indicator Based on Autoregressive Power Spectral Density for Machine Condition Monitoring. Sensors, 24(15), 4782. https://doi.org/10.3390/s24154782