# Crack Size Identification for Bearings Using an Adaptive Digital Twin

^{*}

## Abstract

**:**

## 1. Introduction

- The first contribution is about bearing vibration signal modeling. The combination of mathematical vibration bearing signal modeling, Gaussian Process Regression (GPR), input-output Laguerre filter, and fuzzy approach, MGPRLF, is used for bearing vibration signal modeling.
- The second contribution is proposed to adaptive digital twin. A combination of MGPRLF and proposed observer (hence is a combination of PI observer, Lyapunov robust technique, and adaptive fuzzy algorithm) is recommended to design proposed adaptive digital twin. This proposed technique is suggested to prepare the vibration signals for easier and higher-accuracy classification.
- A combination of the resulting adaptive digital twin and a machine learning (SVM) algorithm is recommended for signal classification and crack size identification.

## 2. Dataset

## 3. Proposed Scheme

#### 3.1. Adaptive Digital Twin

#### 3.2. Residual Signal Computation

#### 3.3. Signal Classification

## 4. Experimental Result

#### 4.1. Signal Modeling and Estimation Using the ADT Results

#### 4.2. Fault Pattern Recognition (Crack Identification)

#### 4.3. Crack Size Identification

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

ADT | Adaptive Digital Twin |

SVM | Support Vector Machine |

GPR | Gaussian Process Regression |

PI | Proportional Integral |

NC | Normal Condition |

IF | Inner Fault |

hp | horsepower |

GPRLF | The combination of GPRL and fuzzy approach |

MGPRLF-PI | PI observer with MGPRLF modeling |

MGPRLF-ARPI (ADT) | The combination of MGPRLF-RPI observer and adaptive approach that is called adaptive digital twin |

${X}_{i}\left(k\right)$ | The measurable vibration signal |

${Y}_{GPR}\left(k\right)$ | The signal modeled by the GPR technique |

$({\delta}_{i},\left(\left({\delta}_{o}{)}^{T}\left({x}_{n}\right)\right)\right)$ | The coefficient of signal modeling using the GPR algorithm |

$\alpha $ | Signal variance |

$k$ | Kernel width |

${Y}_{GPRL}\left(k\right)$ | The modeled signal by the GPRL method |

CoG | Center of Gravity |

${e}_{GPRLF}\left(k\right)$ | The error of signal modeling using the GPRLF algorithm |

${Y}_{f}\left(k\right)$ | The modeled signal using the fuzzy algorithm to improve the accuracy and flexibility |

${\delta}_{f}$ | The coefficient of the modeled signal using the fuzzy algorithm |

${\mathbb{Z}}_{D\left(q\right)}$ | The mass of bearing matrices |

${\mathbb{N}}_{D}\left(q,\dot{q}\right)$ | A nonlinear term for modeling the bearing |

${\theta}_{RF}$ | The effect of the roller fault |

${\theta}_{OF}$ | The effect of the outer fault |

The number of rollers in the bearing | |

${\delta}_{XD}\left({X}_{D}\left(k\right),{X}_{Di}\left(k\right)\right)$ | The nonlinear term of the bearing using mathematically based vibration modeling |

${X}_{M}$ | The state of the vibration signal modeling using the mathematical approach |

${\delta}_{YD}$ | The coefficient |

${Y}_{GPRLF}\left(k\right)$ | The modeled signal by the GPRLF method |

${X}_{MGPRLF-PI}\left(k\right)$ | The state of the bearing signal estimation using the MGPRLF-PI technique |

${Y}_{raw}\left(k\right)$ | The original raw signals that are collected by the vibration sensor |

${\upsilon}_{\gamma}\left(e,X\left(k\right),\varphi \left(k\right)\right)$ | The Lyapunov function |

${\eta}_{\gamma}\left(e\right)\varphi \left(k\right)$ | Differentiable function of the uncertainty (unknown) condition |

${X}_{MGPRLF-RPI}\left(k\right)$ | The state of the bearing signal estimation using the MGPRLF-RPI technique |

${\upsilon}_{\gamma}\left({e}_{MGPRLF},{X}_{MGPRLF-RPI}\left(k\right),{\varphi}_{MGPRLF-RPI}\left(k\right)\right)$ | The Lyapunov function to increase the robustness of the proposed algorithm |

${X}_{ADT}\left(k\right)$ | The state of the bearing signal estimation using the proposed ADT technique |

${\varphi}_{ADT}$ | The uncertainty estimation using the proposed ADT algorithm |

${\delta}_{ADT-New}$ | The adaptive (update) coefficient for tuning the proposed ADT estimator |

${R}_{ADT}\left(k\right)$ | Residual signal using proposed ADT method |

ADT + SVM | The combination of ADT and SVM |

MGPRLF-PI + SVM | The combination of MGPRLF-PI and SVM |

RMS | Root Means Square |

CWRUBD | Case Western Reserve University Bearing Dataset |

FIE | Fuzzy Inference Engine |

RPM | Rotation Per Minute |

RF | Roller Fault |

OF | Outer Fault |

GPRL | The combination of GPR and Laguerre technique |

MGPRLF | The combination of mathematical modeling and GPRLF |

MGPRLF-RPI | The combination of MGPRLF-PI observer and Lyapunov approach |

${X}_{GPR}\left(k\right)$ | The state of the bearing signal modeling using the GPR technique |

${e}_{GPR}\left(k\right)$ | The error of signal modeling using the GPR algorithm |

${\u2102}_{GPR}$ | The covariance matrix using the GPR technique |

$\epsilon $ | Noise variance |

${e}_{GPRL}\left(k\right)$ | The error of signal modeling using the GPRL algorithm |

${X}_{GPRL}\left(k\right)$ | State of the bearing signal modeling using the GPRL technique |

${\u2102}_{GPRL}$ | The covariance matrix using the GPRL algorithm |

${X}_{GPRLF}\left(k\right)$ | The state of the bearing signal modeling using the GPRLF technique |

${Y}_{GPRLF}\left(k\right)$ | The modeled signal by the GPRLF method |

${\u2102}_{GPRLF}$ | The covariance matrix using the GPRLF algorithm |

${F}_{D\left(q\right)}$ | The external source forces |

$\ddot{q}$ | The acceleration vibration signal that is measured by a vibration sensor |

${\theta}_{D}$ | And the unknown condition (hence is called uncertainty) |

${\theta}_{IF}$ | The effect of the inner fault |

${\phi}_{\alpha}$ | The angular velocity of rotor |

${\theta}_{f}$ | The difference between two reference angular positions |

${\chi}_{XD}\left({X}_{D}\left(k\right),{X}_{Di}\left(k\right)\right)$ | The uncertainty term of the bearing using mathematically based vibration modeling |

${Y}_{M}\left(k\right)$ | The modeled vibration signal using the mathematical technique |

${X}_{GPRLF}\left(k\right)$ | The state of the bearing signal modeling using the MGPRLF technique |

${Y}_{MGPRLF-PI}\left(k\right)$ | The estimated signal by the MGPRLF-PI method |

${\varphi}_{MGPRLF-PI}$ | The uncertainty estimation using the MGPRLF-PI algorithm |

${\delta}_{PI}$ | The coefficient of PI observer |

${\mathbb{R}}_{\gamma}\left(e,X\left(k\right)\right)$ | The Hamilton–Jacobi discrimination |

${Y}_{MGPRLF-RPI}\left(k\right)$ | The estimated signal by the MGPRLF-RPI method |

${\varphi}_{MGPRLF-RPI}$ | The uncertainty estimation using the MGPRLF-RPI algorithm |

${\delta}_{RPI}$ | The coefficient of the RPI technique |

${Y}_{ADT}\left(k\right)$ | The estimated signal by the proposed ADT method |

${\upsilon}_{\gamma}\left({e}_{MGPRLF},{X}_{ADT}\left(k\right),{\varphi}_{ADT}\left(k\right)\right)$ | The effect of the Lyapunov function to improve the robustness in the proposed ADT algorithm |

${R}_{ADT}{(k)}_{rms}$ | The RMS resampled residual signal using proposed ADT |

T | The number of windows |

MGPRLF-RPI + SVM | The combination of MGPRLF-RPI and SVM |

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**Figure 1.**The testbed of CWRUBD to collect the data [27].

**Figure 2.**The combination of proposed adaptive digital twin and machine learning for the bearing crack detection and size identification.

**Figure 5.**Error of signal estimation using proposed ADT algorithm: (I) NC, (II) RF, (III) IF, and (IV) OF.

**Figure 6.**The RMS resampled residual signals using the proposed ADT algorithm: (I) for the NC, RF, IF, and OF and (II) zoom view.

**Figure 12.**The average boxplots of crack identification fluctuation (NR, RF, IF, and OF) of the ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM techniques to test the reliability and robustness with 20 tests.

**Figure 13.**The RMS resampled roller fault residual signal using the proposed ADT algorithm for crack size identification.

**Figure 14.**The RMS resampled inner fault residual signal using the proposed ADT algorithm for crack size identification.

**Figure 15.**The RMS resampled outer fault residual signal using the proposed ADT algorithm for crack size identification.

**Figure 16.**The average accuracies of crack size identification for the RF, IF, and OF using the proposed ADT+SVM scheme.

**Figure 17.**The average accuracies of crack size identification for the RF, IF, and OF using the proposed MGPRLF-RPI+SVM scheme.

**Figure 18.**The average accuracies of crack size identification for the RF, IF, and OF using the proposed MGPRLF-PI+SVM scheme.

**Figure 19.**The average boxplots of crack size identification fluctuation (RF, IF, and OF) of the ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM to test the reliability and robustness of tests conducted 20 times.

Information | Detail |
---|---|

Power of the induction motor | 2 [hp] |

Bearing rotating speeds | 1730 [RPM]; 1750 [RPM]; 1772 [RPM]; 1797 [RPM] |

Crack sizes | 0.007 [inches]; 0.014 [inches]; 0.021 [inch] |

Sampling rate frequency | 48 [KHz] |

Type of bearing | 6205-2RS JEM SKF |

Number of rollers | 9 |

Roller’s stiffness | $5.96\times {10}^{7}\left(\frac{\mathrm{N}}{\mathrm{m}}\right)$ |

Outer’s stiffness | $1.31\times {10}^{5}\left(\frac{\mathrm{N}}{\mathrm{m}}\right)$ |

Shaft’s stiffness | $23.3\times {10}^{6}\left(\frac{\mathrm{N}}{\mathrm{m}}\right)$ |

Outer’s Mass | $2.7\left(\mathrm{Kg}\right)$ |

Shaft’s Mass | $1.36\left(\mathrm{Kg}\right)$ |

Defect depth | $2\left(\mathrm{mm}\right)$ |

Pitch diameter | $39.04\left(\mathrm{mm}\right)$ |

Roller diameter | $7.940\left(\mathrm{mm}\right)$ |

Classes | Motor Torque Load [hp] | Crack Sizes [inch] |
---|---|---|

NC | 0,1,2,3 | - |

RF | 0,1,2,3 | 0.007; 0.014; 0.021 |

IF | 0,1,2,3 | 0.007; 0.014; 0.021 |

OF | 0,1,2,3 | 0.007; 0.014; 0.021 |

Conditions | Number of Training Samples | Number of Testing Samples |
---|---|---|

Crack Identification | ||

NC | 900 | 300 |

RF | 900 | 300 |

IF | 900 | 300 |

OF | 900 | 300 |

Size Identification for RF | ||

0.007-inch | 300 | 100 |

0.014-inch | 300 | 100 |

0.021-inch | 300 | 100 |

Size Identification for IF | ||

0.007-inch | 300 | 100 |

0.014-inch | 300 | 100 |

0.021-inch | 300 | 100 |

Size Identification for OF | ||

0.007-inch | 300 | 100 |

0.014-inch | 300 | 100 |

0.021-inch | 300 | 100 |

**Table 4.**The proposed algorithm uses a combination of the adaptive digital twin (ADT) and SVM for fault diagnosis of the bearing.

1: | Adaptive Digital Twin DesignImplement the state-space GPR algorithm, Equation (1). |

2: | Improve the robustness of autoregressive technique by combining autoregressive algorithm with the Laguerre filter. Equations (3) and (4) |

3: | Improve the accuracy and flexibility of GPRL using the GPRLF algorithm, Equation (9). |

4: | Mathematical modeling of the bearing, Equation (19). |

5: | Improve the performance of modeling in the digital twin using the MGPRLF algorithm, Equation (20). |

6: | Implement the combination of PI observer and MGPRLF for signal estimation, Equations (21) and (22). |

7: | Improve the robustness of MGPRLF-PI for signal estimation using MGPRLF-RPI, Equations (24) and (25). |

8: | signal estimation using the proposed ADT, Equations (26) and (27). |

9: | Residual Signal ComputationCompute the residual signals, Equation (29). |

Residual Signals Fault Classification | |

10: | Compute the resampled RMS residual signals, Equation (30). |

11: | Perform classification of the resampled RMS residual signals using the SVM [20,23]. |

**Table 5.**The average accuracies of crack identification using ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM techniques.

Classes | ADT +SVM (%) | MGPRLF-RPI +SVM (%) | MGPRLF-RPI +SVM (%) |
---|---|---|---|

NC | 100 | 100 | 100 |

RF | 98 | 90 | 82 |

IF | 92 | 83 | 63 |

OF | 93 | 88 | 75 |

Average | 95.75 | 90.25 | 80 |

**Table 6.**The average accuracies of the roller crack size identification using the ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM techniques.

Size (inch) | ADT +SVM (%) | MGPRLF-RPI +SVM (%) | MGPRLF-RPI +SVM (%) |
---|---|---|---|

0.007 | 98 | 90 | 80 |

0.014 | 96 | 88 | 82 |

0.021 | 98 | 89 | 80 |

Average | 97.33 | 89 | 80.67 |

**Table 7.**The average accuracies of the inner crack size identification using the ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM techniques.

Size (inch) | ADT +SVM (%) | MGPRLF-RPI +SVM (%) | MGPRLF-RPI +SVM (%) |
---|---|---|---|

0.007 | 98 | 90 | 82 |

0.014 | 98 | 86 | 81 |

0.021 | 99 | 90 | 84 |

Average | 98.33 | 88.67 | 82.33 |

**Table 8.**The average accuracies of outer crack size identification using the ADT+SVM, MGPRLF-RPI+SVM, and MGPRLF-PI+SVM techniques.

Size (inch) | ADT +SVM (%) | MGPRLF-RPI +SVM (%) | MGPRLF-RPI +SVM (%) |
---|---|---|---|

0.007 | 97 | 91 | 80 |

0.014 | 99 | 88 | 80 |

0.021 | 99 | 88 | 86 |

Average | 98.33 | 89 | 82 |

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## Share and Cite

**MDPI and ACS Style**

Piltan, F.; Kim, J.-M.
Crack Size Identification for Bearings Using an Adaptive Digital Twin. *Sensors* **2021**, *21*, 5009.
https://doi.org/10.3390/s21155009

**AMA Style**

Piltan F, Kim J-M.
Crack Size Identification for Bearings Using an Adaptive Digital Twin. *Sensors*. 2021; 21(15):5009.
https://doi.org/10.3390/s21155009

**Chicago/Turabian Style**

Piltan, Farzin, and Jong-Myon Kim.
2021. "Crack Size Identification for Bearings Using an Adaptive Digital Twin" *Sensors* 21, no. 15: 5009.
https://doi.org/10.3390/s21155009