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Article

Predictive Analysis of Crack Growth in Bearings via Neural Networks

by
Manpreet Singh
1,†,
Dharma Teja Gopaluni
1,†,
Sumit Shoor
1,†,
Govind Vashishtha
2,3,* and
Sumika Chauhan
2
1
School of Mechanical Engineering, Lovely Professional University, Phagwara 144411, India
2
Faculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, Poland
3
Department of Mechanical Engineering, Graphic Era Deemed to be University, Dehradun 248002, India
*
Author to whom correspondence should be addressed.
†
This paper is an extended version of our paper published in International Conference on Production and Industrial Engineering, Jalandhar, India, 10–12 March 2023.
Machines 2024, 12(9), 607; https://doi.org/10.3390/machines12090607 (registering DOI)
Submission received: 29 July 2024 / Revised: 26 August 2024 / Accepted: 30 August 2024 / Published: 1 September 2024
(This article belongs to the Special Issue Data-Driven Approaches regarding Dynamic Modelling, Diagnostics and Prognostics of Complex Mechanical Structures)
Browse Figures
Versions Notes

Abstract

:
Machine learning (ML) and artificial intelligence (AI) have emerged as the most advanced technologies today for solving issues as well as assessing and forecasting occurrences. The use of AI and ML in various organizations seeks to capitalize on the benefits of vast amounts of data based on scientific approaches, notably machine learning, which may identify patterns of decision-making and minimize the need for human intervention. The purpose of this research work is to develop a suitable neural network model, which is a component of AI and ML, to assess and forecast crack propagation in a bearing with a seeded crack. The bearing was continually run for many hours, and data were retrieved at time intervals that might be utilized to forecast crack growth. The variables root mean square (RMS), crest factor, signal-to-noise ratio (SNR), skewness, kurtosis, and Shannon entropy were collected from the continuously running bearing and utilized as input parameters, with the total crack area and crack width regarded as output parameters. Finally, utilizing several methodologies of the Neural Network tool in MATLAB, a realistic ANN model was trained to predict the crack area and crack width. It was observed that the ANN model performed admirably in predicting data with a better degree of accuracy. Through analysis, it was observed that the SNR was the most relevant parameter in anticipating data in bearing crack propagation, with an accuracy rate of 99.2% when evaluated as a single parameter, whereas in multiple parameter analysis, a combination of kurtosis and Shannon entropy gave a 99.39% accuracy rate.

1. Introduction

Bearings, the unsung heroes of countless mechanical systems, are subjected to immense stresses and fatigue during operation [1,2]. These stresses can lead to the insidious growth of microscopic cracks, ultimately culminating in catastrophic failure. Understanding the intricate interplay between crack dimensions and propagation dynamics is crucial for ensuring the reliability and longevity of bearing systems [3,4,5].
This analysis delves into the complex world of crack propagation in bearings, aiming to shed light on the factors influencing crack growth and the crucial role of crack dimensions [6,7,8]. By meticulously examining the relationship between crack length, depth, and orientation with respect to the applied stresses and material properties, this study seeks to unveil the mechanisms governing crack evolution [9,10,11]. The findings of this investigation will hold significant implications. For instance, by accurately forecasting the rate of crack propagation, engineers can establish reliable estimations of bearing lifespan under varying operating conditions. Understanding the influence of crack dimensions on failure modes can guide the development of bearings with enhanced resistance to fatigue and crack propagation [12,13]. The early detection and monitoring of crack dimensions can enable timely interventions, preventing catastrophic failures and ensuring safe operation [14,15]. Ren et al. [16] investigated the fracture behaviour of buried pipelines with corrosion defects subjected to seismic loads. Utilizing the extended finite element method, their study identified a critical crack tip angle near 5° in the corrosion area where maximum stress values occur. Xie et al. [17] delved into the complex influence of crack coupling on fatigue crack propagation in pipelines, moving beyond qualitative observations to provide a quantitative approach for predicting remaining useful life (RUL). Parsania et al. [18] investigated the complex interplay of multiple cracks in an infinite plate, focusing on the interaction between a main crack and an adjacent crack under various loading conditions. Utilizing the J-integral and finite element analysis, their study revealed that the presence of an adjacent crack can significantly influence the stress intensity factor (SIF) of the main crack, leading to intensifying, protective, or neutral effects depending on the relative position and orientation of the cracks.
This exploration into the realm of crack dimensions and their role in bearing failure will pave the way for a deeper comprehension of bearing behaviour, ultimately contributing to the design of more robust and reliable mechanical systems [19,20]. To address these issues, researchers have focussed on machine learning (ML) techniques, as they are more efficient, less costly, and more adaptable due to their involvement-free nature. In general, machine learning approaches are classified into supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning (RL) [21,22,23]. The artificial neural network (ANN) imitates the neurons of the human brain. The ANN architecture consists of input layers, hidden layers, and output layers.
In this work, a bearing with a seeded crack is taken and run continuously for several hours, with readings taken at regular intervals of time to determine the crack state. Furthermore, the measurements are trained and analyzed in MATLAB’s neural network workspace to estimate the crack dimensions after properly training the system. The readings are obtained from the bearing while it is operating constantly, and these readings are taken with various combinations of the sets of inputs while retaining the outputs as total area and widths. These reading inputs and outputs are trained in the ANN model to test the accuracy of the prediction for a certain time period or reading.
The ANN model is developed in MATLAB, and the network type is a feed-forward back propagation neural network, since backpropagation is a learning approach that operates on a multilayer feed-forward neural network [24,25]. It continually learns a set of weights for predicting the class label. The training model is Levenberg–Marquardt optimization (TRAINLM). Furthermore, TRAINLM is a network training function that uses Levenberg–Marquardt optimization to update weight and bias variables [26,27]. Although it requires more memory than other algorithms, TRAINLM is frequently the fastest backpropagation method in the toolbox and is strongly recommended as a first-choice supervised technique [28,29].
Additionally, the LOGSIG transfer function is applied in the process, as A = logsig(N) takes a matrix of net input vectors, N, and returns the S-by-Q matrix, A, of the elements of N compressed into [0, 1] [30,31]. The LOGSIG function is a transfer function. Transfer functions compute the output of a layer based on its net input. This LOGSIG is used for the hidden layers and the PURELIN function is used for the output layer so that the regression graph will be linear and can measure the graph easily and accurately [32].
Various factors are used in the field of machine health monitoring and maintenance to analyze data obtained from machines in order to identify any irregularities or potential issues. The data obtained from a bearing with a seeded crack is analyzed in this work based on the parameters RMS, crest factor, SNR, skewness, kurtosis, and Shannon entropy. The parameters listed above are utilized to extract characteristics from data obtained from the bearing with the seeded crack that may be used to identify possible faults in the machine and anticipate crack growth.
The main contributions of this study are as follows:
  • This research work successfully developed a practical artificial neural network (ANN) model capable of predicting crack area and width in a bearing with a seeded crack.
  • The model utilizes key parameters like RMS, crest factor, SNR, skewness, kurtosis, and Shannon entropy, reflecting the practical application of these metrics in bearing condition monitoring.
  • This study highlights the significant influence of SNR on predicting crack propagation when using it as a single input parameter.
  • The combination of kurtosis and Shannon entropy achieved an even higher accuracy, demonstrating the potential for enhancing predictive performance through the integration of multiple parameters.
  • This study validates the use of ANN models for predicting crack propagation in bearings, showcasing their ability to achieve high accuracy in real-world applications.

2. Research Methodology

2.1. Data Extraction

The bearing with the seeded crack was run continuously for several hours to extract some of the characteristics that may cause the crack to grow. A pictorial view of the seeded cracks at different hours is shown in Figure 1. RMS, crest factor, SNR, skewness, kurtosis, and Shannon entropy were the parameters which were analyzed to assess the propagation of cracks in the bearing. The data of these parameters are tabulated in Table 1.

2.2. Application of Neural Network

The data tabulated in Table 1 were imported into MATLAB 2021a and the proposed neural network model was trained. Keeping the time duration constant, the rest of the parameters were trained independently to see which factor was more affected by crack propagation by comparing the projected data output to the actual data [9,11,12,13,14,15,16,17].
Furthermore, the overall area and width of the crack were considered as the goal data. The ANN was trained using the TRAINLM algorithm for each parameter independently, and the expected output was simulated by feeding the algorithm sample data; the predicted total area and crack width were then compared to the actual data. The number of hidden layers was set to 10 when carrying out this study. The architecture used in this study is shown in Figure 2. This study was repeated for each parameter separately, and the ANN model with the lowest error in projected values was regarded to be the most influential factor in crack propagation.
Moreover, the method provides a regression plot for the input and output data, which establishes a regression line between two parameters and aids in the visualization of their linear correlations. The regression plot is commonly characterized as R, the plot that is formed comes with an R value, which stands for the regression value, and this value is regarded to be best and most accurate if R square equals 1 [10].

3. Results and Discussion

3.1. Single Parameters

Root Mean Square (RMS)

Using the RMS and time duration values as input parameters and the total area and breadth as targets, an ANN model was constructed, and values were forecasted using that model by providing some sample data as the input. Table 2 shows the projected values as well as the actual data.
The predicted values were created using the ANN model, and the accuracy of the trained model was calculated using the actual values. The trained model’s accuracy was calculated by taking the average of all the values. The findings indicate that RMS as a single parameter had an accuracy of 94.2%. The accuracy was computed using the formula below [10].
Accuracy = 1 − (Predicted value − Actual value)/Predicted value
The ANN model was built with the crest factor and time duration values as input parameters and the total area and breadth as targets, and values were projected using this model by supplying some sample data as the input. Table 3 displays both the predicted and actual numbers.
By repeating the accuracy calculation method from the previous stage, the crest factor exhibited 97.4% accuracy. The same procedure was followed for the remaining parameters, namely SNR, skewness, kurtosis, and Shannon entropy, and the predicted data, as well as the actual data’s accuracy, are shown in Table 4. The regression graphs for all the parameters were also obtained through ANN, as shown in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.
It can be observed from Table 4 that the SNR was 99.2%, skewness was 98.6%, kurtosis was 97.1%, and Shannon entropy was 97.8% accurate. Furthermore, the analysis is carried out by conducting various permutations and combinations of the input parameters. In this study, the number of input parameters was reduced to three, with the time duration staying constant while the other parameters were shuffled in various combinations.

3.2. Multiple Parameters

To validate the efficacy of the proposed method, the analysis of multiple parameters was also carried out.

3.2.1. RMS and Crest Factor

An ANN model was built with the RMS, crest factor, and time duration values as input parameters and the total area and breadth as targets, and values were projected using that model by supplying some sample data as the input. Table 5 displays both the predicted and actual numbers.
It can be observed from Table 5 that the RMS and crest factor showed 99.18% accuracy on average. The regression plot obtained through ANN and the performance curve are also shown in Figure 9 and Figure 10, respectively.

3.2.2. RMS and SNR

In another case, the RMS, SNR, and time duration values were taken as input parameters and the total area and breadth were considered as research goals. An ANN model was developed, and values were projected using that model by feeding some sample data as the input. Table 6 shows both the expected and actual results.
It can be observed from Table 6 that the RMS and SNR showed 95.14% accuracy on average. The regression plot and the performance plot of this analysis are shown in Figure 11 and Figure 12.

3.2.3. RMS and Skewness

An ANN model was constructed with the RMS, skewness, and time duration values as input parameters and the total area and breadth as targets, and values were projected using this model by feeding it some sample data. Table 7 displays both the predicted and actual outcomes.
It can be observed from Table 7 that the RMS and skewness showed 98.60% accuracy on average. The regression plot and the performance plot for this analysis are shown in Figure 13 and Figure 14, respectively.

3.2.4. RMS and Kurtosis

The RMS, kurtosis, and time duration data were used as input parameters in a neural network model, and total area and width were used as goals. This model was then used to create predictions by inputting sample data. Table 8 displays both the expected and actual results.
The RMS and kurtosis showed 98.94% accuracy on average, as shown in Table 8. The regression plot and the performance plot for this analysis are shown in Figure 15 and Figure 16, respectively.

3.2.5. RMS and Shannon Entropy

The RMS, Shannon entropy, and time duration values were used as input parameters in the development of the ANN model, with total area and width as goals. This model was then used to predict outcomes after being fed sample data. Table 9 shows the projected and actual results.
The RMS and Shannon entropy showed 98.86% accuracy on average, as shown in Table 9. The regression plot and the performance plot for this analysis are shown in Figure 17 and Figure 18, respectively.

3.2.6. Remaining Combinations

The same procedure was followed for the remaining parameters, namely the crest factor, SNR, skewness, kurtosis, and Shannon entropy, with all possible permutations and combinations, and the predicted data, as well as the actual data’s accuracy, are shown in Table 10.
The accuracy of all possible combinations of the input parameters is shown in Table 10. From Table 10, it is clear that the parameters kurtosis and Shannon entropy show 99.39% accuracy when taken as multi-parameters simultaneously. Furthermore, it is obvious that the multiple-parameter input has slightly higher accuracy than the single-parameter input.

4. Conclusions

After analyzing the findings of our investigation, we can infer that the ANN model performed admirably in predicting data with a better degree of accuracy. While there was some fluctuation for each parameter, our analysis indicates that SNR was discovered to be the most relevant parameter in anticipating data in bearing crack propagation, with an accuracy rate of 99.2% when evaluated as a single parameter. Our investigation was carried out by incorporating the multiple characteristics, and it was discovered that kurtosis and Shannon entropy, when combined, had a 99.39% accuracy rate. This finding suggests that increasing the number of input parameters can enhance the accuracy of predicting the outcome. In this investigation, the trainlm function was used to predict bearing crack propagation. As a result of these findings, it can be concluded that the ANN model, when combined with numerous input parameters, can provide a more accurate forecast of bearing crack propagation, which has practical applications in the field of machine health monitoring and maintenance. In future, an additional study can be conducted by raising the number of input factors to three or four, which may result in a higher accuracy rate in predicting outputs.

Author Contributions

M.S.: data curation and writing—original draft; D.T.G.: data curation and writing—original draft; S.S.: writing—review and editing; G.V.: data curation, software, writing—original draft, methodology and supervision; S.C.: data curation, software, writing—original draft, and methodology. All authors have read and agreed to the published version of the manuscript.

Funding

No funding has been received for this work.

Data Availability Statement

Data can be made available upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Pictorial view of cracks (a) at 90 h (b), at 192 h, and (c) at 285 h.
Figure 1. Pictorial view of cracks (a) at 90 h (b), at 192 h, and (c) at 285 h.
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Figure 2. The architecture of NN used in the analysis.
Figure 2. The architecture of NN used in the analysis.
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Figure 3. Regression plot for the RMS.
Figure 3. Regression plot for the RMS.
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Figure 4. Regression plot for the crest factor.
Figure 4. Regression plot for the crest factor.
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Figure 5. Regression plot for the SNR.
Figure 5. Regression plot for the SNR.
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Figure 6. Regression plot for skewness.
Figure 6. Regression plot for skewness.
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Figure 7. Regression plot for kurtosis.
Figure 7. Regression plot for kurtosis.
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Figure 8. Regression plot for Shannon entropy.
Figure 8. Regression plot for Shannon entropy.
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Figure 9. Regression plot for the RMS and crest factor.
Figure 9. Regression plot for the RMS and crest factor.
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Figure 10. Performance plot for the RMS and crest factor.
Figure 10. Performance plot for the RMS and crest factor.
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Figure 11. Regression plot for the RMS and SNR.
Figure 11. Regression plot for the RMS and SNR.
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Figure 12. Performance plot for the RMS and SNR.
Figure 12. Performance plot for the RMS and SNR.
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Figure 13. Regression plot for the RMS and skewness.
Figure 13. Regression plot for the RMS and skewness.
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Figure 14. Performance plot for the RMS and skewness.
Figure 14. Performance plot for the RMS and skewness.
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Figure 15. Regression plot for the RMS and kurtosis.
Figure 15. Regression plot for the RMS and kurtosis.
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Figure 16. Performance plot for the RMS and kurtosis.
Figure 16. Performance plot for the RMS and kurtosis.
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Figure 17. Regression plot for the RMS and Shannon entropy.
Figure 17. Regression plot for the RMS and Shannon entropy.
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Figure 18. Performance plot for the RMS and Shannon entropy.
Figure 18. Performance plot for the RMS and Shannon entropy.
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Table 1. Data retrieved from the bearing.
Table 1. Data retrieved from the bearing.
Time Duration in HoursRMS
(m/s2)		Crest FactorSNRSkewnessKurtosisShannon Entropy
(107)		Input Variables
Total Area (mm2)Width (mm)
08.3477.125622.980.1895.5542.4224.861.82
1010.226.821520.770.1686.4923.5924.861.82
208.0876.245722.330.2085.5272.0224.861.82
308.4728.038421.70.1845.7492.2724.861.82
408.4857.079422.460.225.6262.2724.861.82
508.4488.282820.710.2285.9132.2624.86251.82
608.4567.388823.40.2295.5492.2624.8651.82
708.4496.554920.990.2235.6182.2524.86751.82
808.3657.249223.450.2235.5422.1924.961.82
908.3086.41822.340.2215.6032.1724.961.8286
1008.2517.028521.540.2165.7522.1324.961.8373
1208.36.423722.20.2185.6792.1625.361.895
1408.6016.720421.80.2256.1472.3725.561.92
1609.2646.888120.350.1685.5212.826.04452.1
1708.796.058820.290.185.7282.4826.45862.4766
1807.6813.35723.60.1198.5372.0326.9783.1198
18416.911.78824.10.077.4313.927.6093.2798
1888.114.384522.10.179.532.0727.9873.4399
1927.9711.43922.90.047.671.9828.2753.6975
1968.2414.1009240.1111.62.1928.3454.0525
20025.0210.238219.670.1297.31230.728.7784.867
204229.504519.40.097.532828.8355.16
20825.49.464422.40.117.152028.8925.4545
21225.712.75419.50.168.632628.9495.7482
21624.98.896618.180.1788.01730.529.0076.042
22013.89.112224.30.127.097.2728.8857.0116
2241610.081521.20.167.6315.228.9457.0983
22915.2811.041223.310.0788.110.229.157.231
23219.211.251722.50.2610.215.929.1727.341
23916.287.77421.910.1416.4811.729.2027.531
24819.219.161217.670.1516.13512.229.3247.761
25914.867.341118.420.17076.8249.5829.5568.631
267.58.7239.170521.70.144.9672.6629.5648.987
274.59.158.330119.510.0416.6533.0631.0199.362
278.59.1789.632921.20.1156.68213.2633.3729.761
284.59.60310.24719.470.0147.2883.0936.51810.401
2859.11410.439619.350.0097.9542.7440.15710.967
Table 2. Analyzed data with the RMS as the input parameter.
Table 2. Analyzed data with the RMS as the input parameter.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time DurationRMS (m/s2)Total Area (mm2)Width (mm)Total Area
(mm2)		Width
(mm)
08.34724.861.8224.86001.82100.0%
1010.2224.861.8224.86001.82100.0%
208.08724.861.8224.86001.82100.0%
308.47224.861.8224.86001.82100.0%
408.48524.861.8224.86001.82100.0%
508.44824.86251.8224.86011.82100.0%
608.45624.8651.8224.86041.82100.0%
708.44924.86751.8224.86191.8200100.0%
808.36524.961.8224.87081.820099.8%
908.30824.961.828624.90761.820299.7%
1008.25124.961.837325.00261.824999.6%
1208.325.361.895325.34881.878799.5%
1408.60125.561.9225.57711.940199.4%
1609.26426.04452.126.04102.087899.7%
1708.7926.45862.476626.28262.439198.9%
1807.6826.9783.119827.61823.079898.2%
18416.927.6093.279824.86003.277794.4%
1888.127.9873.439927.99533.459699.7%
1927.9728.2753.697528.27493.680099.8%
1968.2428.3454.052528.10863.843296.9%
20025.0228.7784.86724.864.862792.1%
2042228.835255.1624.865.154692.0%
20825.428.89255.454524.865.434691.7%
21225.728.949755.748224.865.740791.7%
21624.929.0076.04224.866.043191.6%
22013.828.8857.011624.866.893891.1%
2241628.9457.098324.867.090691.7%
22915.2829.157.23124.867.238491.3%
23219.229.17257.34124.867.519490.1%
23916.2829.2027.53124.867.439690.7%
24819.2129.3247.76124.867.760791.0%
25914.8629.5568.63124.868.629690.5%
267.58.72329.5648.98724.868.986290.5%
274.59.1531.0199.36224.869.222686.9%
278.59.17833.3729.76124.869.764482.9%
284.59.60336.51810.40124.8610.398676.5%
2859.11440.15710.96724.8610.960069.2%
Table 3. Analyzed data with the crest factor as input parameter.
Table 3. Analyzed data with the crest factor as input parameter.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time
Duration		Crest FactorTotal
Area (mm2)		Width
(mm)		Total
Area (mm2)		Width
(mm)
07.125624.861.8224.86001.82100.0%
106.821524.861.8224.86001.82100.0%
206.245724.861.8224.86001.82100.0%
308.038424.861.8224.86001.82100.0%
407.079424.861.8224.86001.82100.0%
508.282824.86251.8224.86001.82100.0%
607.388824.8651.8224.86001.82100.0%
706.554924.86751.8224.86001.8200100.0%
807.249224.961.8224.86001.820099.8%
906.41824.961.828625.64151.820098.4%
1007.028524.961.837324.95931.820099.5%
1206.423725.361.895325.77921.820097.1%
1406.720425.561.9225.89871.820096.6%
1606.888126.04452.126.04451.820092.3%
1706.058826.45862.476626.45771.820082.0%
18013.35726.9783.119824.86003.121895.7%
18411.78827.6093.279824.86003.277394.4%
18814.384527.9873.439924.86003.377792.8%
19211.43928.2753.697524.86003.694993.1%
19614.100928.3454.052524.86004.054193.0%
20010.238228.7784.86728.79830744.905099.6%
2049.504528.835255.1628.854349465.144899.8%
2089.464428.89255.454528.895215375.492799.6%
21212.75428.949755.748224.865.740491.7%
2168.896629.0076.04228.962909786.079099.6%
2209.112228.8857.011629.001152166.420995.2%
22410.081528.9457.098329.049989626.831097.9%
22911.041229.157.23129.104919027.246999.8%
23211.251729.17257.34129.127293767.432399.3%
2397.77429.2027.53129.182873497.526399.9%
2489.161229.3247.76129.205995168.009598.2%
2597.341129.5568.63129.56001338.6280100.0%
267.59.170529.5648.98729.696942648.780998.6%
274.58.330131.0199.36231.013053869.351199.9%
278.59.632933.3729.76132.192296149.718798.0%
284.510.24736.51810.40136.5025892110.495999.5%
28510.439640.15710.96736.8225726110.539893.4%
Table 4. Accuracy percentage of remaining parameters.
Table 4. Accuracy percentage of remaining parameters.
S. No.Input ParametersAccuracy (in %)
1SNR99.2
2Skewness98.6
3Kurtosis97.1
4Shannon entropy (107)97.8
Table 5. Analyzed data with the RMS and crest factor as input parameters.
Table 5. Analyzed data with the RMS and crest factor as input parameters.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time
Duration		RMS (m/s2)Crest FactorTotal
Area (mm2)		Width
(mm)		Total
Area (mm2)		Width
(mm)
08.3477.12567.125624.861.8224.86299.99%
1010.226.82156.821524.861.8224.860100.00%
208.0876.24576.245724.861.8224.86599.98%
308.4728.03848.038424.861.8224.86999.97%
408.4857.07947.079424.861.8224.87599.94%
508.4488.28288.282824.86251.8224.88699.91%
608.4567.38887.388824.8651.8224.90499.84%
708.4496.55496.554924.86751.8224.93499.73%
808.3657.24927.249224.961.8224.98199.92%
908.3086.4186.41824.961.828625.05199.64%
1008.2517.02857.028524.961.837325.15199.24%
1208.36.42376.423725.361.895325.46899.58%
1408.6016.72046.720425.561.9225.94098.54%
1609.2646.88816.888126.04452.125.75398.87%
1708.796.05886.058826.45862.476626.86998.47%
1807.6813.35713.35726.9783.119827.47598.19%
18416.911.78811.78827.6093.279827.43799.37%
1888.114.384514.384527.9873.439927.87299.59%
1927.9711.43911.43928.2753.697527.98698.97%
1968.2414.100914.100928.3454.052528.43699.68%
20025.0210.238210.238228.7784.86728.75299.91%
204229.50459.504528.835255.1628.78599.83%
20825.49.46449.464428.89255.454528.89799.98%
21225.712.75412.75428.949755.748228.92999.93%
21624.98.89668.896629.0076.04228.09996.77%
22013.89.11229.112228.8857.011628.90099.95%
2241610.081510.081528.9457.098329.03699.69%
22915.2811.041211.041229.157.23129.22099.76%
23219.211.251711.251729.17257.34129.29499.59%
23916.287.7747.77429.2027.53129.13199.76%
24819.219.16129.161229.3247.76129.34399.93%
25914.867.34117.341129.5568.63129.51699.87%
267.58.7239.17059.170529.5648.98730.49296.96%
274.59.158.33018.330131.0199.36229.42594.58%
278.59.1789.63299.632933.3729.76133.38099.98%
284.59.60310.24710.24736.51810.40135.73497.81%
2859.11410.439610.439640.15710.96738.66396.14%
Table 6. Analyzed data with the RMS and SNR as input parameters.
Table 6. Analyzed data with the RMS and SNR as input parameters.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time
Duration		RMS (m/s2)SNRTotal
Area (mm2)		Width
(mm)		Total
Area (mm2)		Width
(mm)
08.34722.9822.9824.861.8224.860100.00%
1010.2220.7720.7724.861.8224.860100.00%
208.08722.3322.3324.861.8224.860100.00%
308.47221.721.724.861.8224.860100.00%
408.48522.4622.4624.861.8224.860100.00%
508.44820.7120.7124.86251.8224.86099.99%
608.45623.423.424.8651.8224.86099.98%
708.44920.9920.9924.86751.8224.86099.97%
808.36523.4523.4524.961.8224.96399.99%
908.30822.3422.3424.961.828624.86199.60%
1008.25121.5421.5424.961.837324.86099.60%
1208.322.222.225.361.895325.359100.00%
1408.60121.821.825.561.9225.95098.50%
1609.26420.3520.3526.04452.126.044100.00%
1708.7920.2920.2926.45862.476626.459100.00%
1807.6823.623.626.9783.119827.96096.49%
18416.924.124.127.6093.279827.608100.00%
1888.122.122.127.9873.439927.987100.00%
1927.9722.922.928.2753.697528.12099.45%
1968.24242428.3454.052528.344100.00%
20025.0219.6719.6728.7784.86724.86084.24%
2042219.419.428.835255.1624.86084.01%
20825.422.422.428.89255.454528.892100.00%
21225.719.519.528.949755.748224.86083.55%
21624.918.1818.1829.0076.04224.86083.32%
22013.824.324.328.8857.011625.14885.14%
2241621.221.228.9457.098331.52991.80%
22915.2823.3123.3129.157.23129.149100.00%
23219.222.522.529.17257.34129.172100.00%
23916.2821.9121.9129.2027.53129.56998.76%
24819.2117.6717.6729.3247.76124.86082.04%
25914.8618.4218.4229.5568.63133.78387.49%
267.58.72321.721.729.5648.98727.38492.04%
274.59.1519.5119.5131.0199.36231.018100.00%
278.59.17821.221.233.3729.76128.19781.65%
284.59.60319.4719.4736.51810.40136.518100.00%
2859.11419.3519.3540.15710.96731.54472.69%
Table 7. Analyzed data with the RMS and skewness as input parameters.
Table 7. Analyzed data with the RMS and skewness as input parameters.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time
Duration		RMS (m/s2)SkewnessTotal Area (mm2)Width
(mm)		Total
Area (mm2)		Width
(mm)
08.3470.1890.18924.861.8224.89099.88%
1010.220.1680.16824.861.8224.91199.80%
208.0870.2080.20824.861.8224.89099.88%
308.4720.1840.18424.861.8224.89499.86%
408.4850.220.2224.861.8224.90099.84%
508.4480.2280.22824.86251.8224.90799.82%
608.4560.2290.22924.8651.8224.91699.80%
708.4490.2230.22324.86751.8224.92799.76%
808.3650.2230.22324.961.8224.94299.93%
908.3080.2210.22124.961.828624.96499.99%
1008.2510.2160.21624.961.837324.99299.87%
1208.30.2180.21825.361.895325.12499.06%
1408.6010.2250.22525.561.9225.47299.65%
1609.2640.1680.16826.04452.126.19299.44%
1708.790.180.1826.45862.476626.67499.19%
1807.680.1190.11926.9783.119826.94799.88%
18416.90.070.0727.6093.279827.84599.15%
1888.10.170.1727.9873.439927.72499.05%
1927.970.040.0428.2753.697528.57498.95%
1968.240.110.1128.3454.052528.21699.54%
20025.020.1290.12928.7784.86729.08698.94%
204220.090.0928.835255.1628.81599.93%
20825.40.110.1128.89255.454530.96393.31%
21225.70.160.1628.949755.748229.15799.29%
21624.90.1780.17829.0076.04228.64998.75%
22013.80.120.1228.8857.011628.97899.68%
224160.160.1628.9457.098328.82699.59%
22915.280.0780.07829.157.23129.60298.47%
23219.20.260.2629.17257.34130.70195.02%
23916.280.1410.14129.2027.53129.25799.81%
24819.210.1510.15129.3247.76129.19899.57%
25914.860.17070.170729.5568.63129.58099.92%
267.58.7230.140.1429.5648.98729.65899.68%
274.59.150.0410.04131.0199.36232.71894.81%
278.59.1780.1150.11533.3729.76129.70087.64%
284.59.6030.0140.01436.51810.40139.24993.04%
2859.1140.0090.00940.15710.96739.55798.48%
Table 8. Analyzed data with the RMS and kurtosis as input parameters.
Table 8. Analyzed data with the RMS and kurtosis as input parameters.
Input ParametersActual/Target ValuesPredicted ValuesAccuracy (in %)
Time
Duration		RMS (m/s2)KurtosisTotal Area (mm2)Width
(mm)		Total Area (mm2)Width
(mm)
08.3475.5545.55424.861.8224.860100.00%
1010.226.4926.49224.861.8224.860100.00%
208.0875.5275.52724.861.8224.860100.00%
308.4725.7495.74924.861.8224.860100.00%
408.4855.6265.62624.861.8224.860100.00%
508.4485.9135.91324.86251.8224.86099.99%
608.4565.5495.54924.8651.8224.86099.98%
708.4495.6185.61824.86751.8224.86099.97%
808.3655.5425.54224.961.8224.86099.60%
908.3085.6035.60324.961.828624.86099.60%
1008.2515.7525.75224.961.837324.86399.61%
1208.35.6795.67925.361.895325.17099.25%
1408.6016.1476.14725.561.9225.85898.85%
1609.2645.5215.52126.04452.125.98199.76%
1708.795.7285.72826.45862.476626.43899.92%
1807.688.5378.53726.9783.119826.91399.76%
18416.97.437.4327.6093.279828.37597.30%
1888.19.539.5327.9873.439928.00299.95%
1927.977.677.6728.2753.697529.44696.02%
1968.2411.611.628.3454.052527.28396.11%
20025.027.3127.31228.7784.86728.93599.46%
204227.537.5328.835255.1628.84699.96%
20825.47.157.1528.89255.454528.96699.75%
21225.78.638.6328.949755.748228.86499.70%
21624.98.0178.01729.0076.04228.95799.83%
22013.87.097.0928.8857.011629.16899.03%
224167.637.6328.9457.098328.88999.81%
22915.288.18.129.157.23129.10499.84%
23219.210.210.229.17257.34130.25096.44%
23916.286.486.4829.2027.53129.09299.62%
24819.216.1356.13529.3247.76129.29799.91%
25914.866.8246.82429.5568.63129.29999.12%
267.58.7234.9674.96729.5648.98726.82289.78%
274.59.156.6536.65331.0199.36231.45298.62%
278.59.1786.6826.68233.3729.76131.97595.63%
284.59.6037.2887.28836.51810.40136.29299.38%
2859.1147.9547.95440.15710.96739.88399.31%
Table 9. Analyzed data with the RMS and Shannon entropy as input parameters.
Table 9. Analyzed data with the RMS and Shannon entropy as input parameters.
Input Parameters Actual/Target ValuesPredicted ValuesAccuracy (in %)
Time DurationRMS (m/s2)Shannon Entropy (107)Total Area
(mm2)		Width
(mm)		Total Area
(mm2)		Width
(mm)
08.3472.422.4224.861.8224.861100.0%
1010.223.593.5924.861.8224.86299.99%
208.0872.022.0224.861.8224.861100.0%
308.4722.272.2724.861.8224.86299.99%
408.4852.272.2724.861.8224.86399.99%
508.4482.262.2624.86251.8224.863100.0%
608.4562.262.2624.8651.8224.865100.0%
708.4492.252.2524.86751.8224.867100.0%
808.3652.192.1924.961.8224.87199.64%
908.3082.172.1724.961.828624.87799.67%
1008.2512.132.1324.961.837324.89199.72%
1208.32.162.1625.361.895324.98798.51%
1408.6012.372.3725.561.9225.43899.52%
1609.2642.82.826.04452.126.61497.86%
1708.792.482.4826.45862.476627.02597.90%
1807.682.032.0326.9783.119827.32498.73%
18416.913.913.927.6093.279828.52096.81%
1888.12.072.0727.9873.439927.70798.99%
1927.971.981.9828.2753.697527.82198.37%
1968.242.192.1928.3454.052527.87598.31%
20025.0230.730.728.7784.86728.51699.08%
20422282828.835255.1628.97599.52%
20825.4202028.89255.454528.97399.72%
21225.7262628.949755.748229.00699.81%
21624.930.530.529.0076.04228.79899.27%
22013.87.277.2728.8857.011627.91096.51%
2241615.215.228.9457.098328.51698.49%
22915.2810.210.229.157.23128.89899.13%
23219.215.915.929.17257.34128.48697.59%
23916.2811.711.729.2027.53129.34999.50%
24819.2112.212.229.3247.76129.49599.42%
25914.869.589.5829.5568.63130.12598.11%
267.58.7232.662.6629.5648.98729.74999.38%
274.59.153.063.0631.0199.36232.18096.39%
278.59.17813.2613.2633.3729.76133.54499.49%
284.59.6033.093.0936.51810.40137.42797.57%
2859.1142.742.7440.15710.96738.14294.72%
Table 10. Accuracy percentages of all the remaining parameters in combinations.
Table 10. Accuracy percentages of all the remaining parameters in combinations.
S. No.Input ParametersAccuracy (in %)
1Crest Factor and SNR96.51%
2Crest Factor and Skewness97.64%
3Crest Factor and Kurtosis98.87%
4Crest Factor and Shannon Entropy97.36%
5SNR and Skewness98.17%
6SNR and Kurtosis97.81%
7SNR and Shannon Entropy96.08%
8Skewness and Kurtosis98.41%
9Skewness and Shannon Entropy97.45%
10Kurtosis and Shannon Entropy99.39%
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Singh, M.; Gopaluni, D.T.; Shoor, S.; Vashishtha, G.; Chauhan, S. Predictive Analysis of Crack Growth in Bearings via Neural Networks. Machines 2024, 12, 607. https://doi.org/10.3390/machines12090607

AMA Style

Singh M, Gopaluni DT, Shoor S, Vashishtha G, Chauhan S. Predictive Analysis of Crack Growth in Bearings via Neural Networks. Machines. 2024; 12(9):607. https://doi.org/10.3390/machines12090607

Chicago/Turabian Style

Singh, Manpreet, Dharma Teja Gopaluni, Sumit Shoor, Govind Vashishtha, and Sumika Chauhan. 2024. "Predictive Analysis of Crack Growth in Bearings via Neural Networks" Machines 12, no. 9: 607. https://doi.org/10.3390/machines12090607

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