# A Novel Physics-Informed Hybrid Modeling Method for Dynamic Vibration Response Simulation of Rotor–Bearing System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physics-Based Dynamic Vibration Model of Rotor–Bearing System

#### 2.1. Preliminary

#### 2.2. Numerical Simulation Implementation Using Physics-Based Model

## 3. Physics-Informed Hybrid Modeling Method

#### 3.1. Description of Simulated and Measured Vibration Datasets

#### 3.2. Construction of Vibration Generation Network

#### 3.3. Construction of Data Mapping Network

## 4. Experimental Verification

#### 4.1. Experimental Setup

#### 4.2. Numerical Analysis of Physics-Based Dynamic Vibration Model

#### 4.3. Generation of Simulated Vibration Samples and Their Validation

#### 4.4. Performance Analysis of Physics-Informed Hybrid Modeling Method

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${m}_{1}$, ${m}_{4}$ | equivalent masses of rotor at O${}_{1}$ and O${}_{4}$ |

${m}_{2}$, ${m}_{3}$ | masses of discs |

${c}_{s1}$, ${c}_{s2}$ | damping of rotor at O${}_{1}$ and O${}_{4}$ |

${c}_{d1}$, ${c}_{d2}$ | damping of rotor at O${}_{2}$ and O${}_{3}$ |

${k}_{1}$, ${k}_{2}$, ${k}_{3}$, ${k}_{4}$ | stiffness of rotor at O${}_{1}$, O${}_{2}$, O${}_{3}$ and O${}_{4}$ |

g | acceleration due to gravity |

${F}_{bx}$, ${F}_{by}$ | supporting forces of bearing in radial x and y directions |

${k}_{b}$ | Hertz elastic coefficient |

j | index of rolling elements |

${N}_{b}$ | number of rolling elements |

${\theta}_{j}$ | rotating angular at time t of the jth rolling element |

${\delta}_{j}$ | contact deformation of the jth rolling element |

$\mu $ | initial radial clearance of rolling bearing |

${x}_{in}$, ${y}_{in}$ | displacements of the inner race in radial x and y directions |

${x}_{out}$, ${y}_{out}$ | displacements of the outer race in radial x and y directions |

$\beta $ | switching variable indicating the health status of the bearing |

${\delta}_{d}$ | displacement excitation caused by outer-race bearing fault |

${\varphi}_{0}$ | initial angular of defect area |

${\varphi}_{d}$ | span angular of defect area |

${\delta}_{dmax}$ | maximum value of the displacement excitation |

r | radius of rolling element |

L | length of defect area |

${k}_{eq}$ | stiffness coefficient of rolling bearing |

${c}_{eq}$ | damping coefficient of rolling bearing |

${F}_{preload}$ | preload of spring |

${u}_{0}$ | displacement caused by preload |

${F}_{j}$ | normal contact force |

d | penetration depth |

$\varsigma $ | force exponent |

K | equivalent contact stiffness |

C | equivalent contact damping |

${x}_{j}$ | distance from geometry center of the jth rolling element to raceway |

${v}_{s}$ | slip velocity |

${\mu}_{0}$ | coefficient of friction |

${\mu}_{s}$ | static coefficient of friction |

${\mu}_{d}$ | dynamic coefficient of friction |

${V}_{s}$ | static transition velocity |

${V}_{d}$ | dynamic transition velocity |

$\omega $ | rotor angular speed |

W | width of defect area |

${\lambda}_{1},{\lambda}_{2}$ | optional weight coefficients |

T | discrete Fourier transform |

${\omega}_{cage}$ | rotating speed of cage |

${\omega}_{roller}$ | rotating speed of rolling element |

${r}_{c}$ | center-circle radius of rolling element |

$\gamma $ | ratio of rolling element radius r to its center-circle radius ${r}_{c}$ |

${\omega}_{0}$ | rotating speed of bearing outer race |

${f}_{outer}$ | fault characteristic frequency of outer-race fault |

## References

- Zhu, W.; Lin, H.; Sun, W.; Wei, J. Vibration performance of traction gearbox of a high-speed train: Theoretical analysis and experiments. Actuators
**2023**, 12, 103. [Google Scholar] [CrossRef] - Jafarian, M.; Nazarzadeh, J. Spectral analysis for diagnosis of bearing defects in induction machine drives. IET Electr. Power Appl.
**2019**, 13, 340–348. [Google Scholar] [CrossRef] - Zhao, D.; Li, J.; Cheng, W.; Wen, W. Bearing multi-fault diagnosis with iterative generalized demodulation guided by enhanced rotational frequency matching under time-varying speed conditions. ISA Trans.
**2023**, 133, 518–528. [Google Scholar] [CrossRef] [PubMed] - Zhu, D.; Gao, Q.; Sun, D.; Lu, Y. A detection method for bearing faults using complex-valued null space pursuit and 1.5-dimensional teager energy spectrum. IEEE Sens. J.
**2020**, 20, 8445–8454. [Google Scholar] [CrossRef] - Rezamand, M.; Kordestani, M.; Orchard, M.E.; Carriveau, R.; Ting, D.S.K.; Saif, M. Improved remaining useful life estimation of wind turbine drivetrain bearings under varying operating conditions. IEEE Trans. Ind. Inform.
**2020**, 17, 1742–1752. [Google Scholar] [CrossRef] - Wrzochal, M.; Adamczak, S. The problems of mathematical modelling of rolling bearing vibrations. Bull. Pol. Acad. Sci.-Tech. Sci.
**2020**, 68, 1363–1372. [Google Scholar] - Singh, S.; Howard, C.; Hansen, C. An extensive review of vibration modelling of rolling element bearings with localised and extended defects. J. Sound Vibr.
**2015**, 357, 300–330. [Google Scholar] [CrossRef] - McFadden, P.; Smith, J. Model for the vibration produced by a single point defect in a rolling element bearing. Bull. Pol. Acad. Sci.-Tech. Sci.
**1984**, 96, 69–82. [Google Scholar] [CrossRef] - Su, Y.; Lin, S. On initial fault detection of a tapered roller bearing: Frequency domain analysis. J. Sound Vibr.
**1992**, 155, 75–84. [Google Scholar] [CrossRef] - Liu, J. A dynamic modelling method of a rotor-roller bearing-housing system with a localized fault including the additional excitation zone. J. Sound Vibr.
**2020**, 469, 115144. [Google Scholar] [CrossRef] - Wang, N.; Liu, M.; Yao, J.; Ge, P.; Wu, H. Numerical study on unbalance response of dual-rotor system based on nonlinear bearing characteristics of active magnetic bearings. Actuators
**2023**, 12, 86. [Google Scholar] [CrossRef] - Li, Y.; Cao, H.; Niu, L.; Jin, X. A general method for the dynamic modeling of ball bearing-rotor systems. J. Manuf. Sci. Eng.-Trans. ASME
**2015**, 137, 021016. [Google Scholar] [CrossRef] - Brouwer, M.; Sadeghi, F.; Ashtekar, A.; Archer, J.; Lancaster, C. Combined explicit finite and discrete element methods for rotor bearing dynamic modeling. Tribol. Trans.
**2015**, 58, 300–315. [Google Scholar] [CrossRef] - Mishra, C.; Samantaray, A.K.; Chakraborty, G. Ball bearing defect models: A study of simulated and experimental fault signatures. J. Sound Vibr.
**2017**, 400, 86–112. [Google Scholar] [CrossRef] - Liu, J.; Ni, H.; Li, X.; Xing, Q.; Pan, G. A simulation analysis of ball bearing lubrication characteristics considering the cage clearance. J. Tribol.
**2023**, 145, 044301. [Google Scholar] [CrossRef] - Yu, Y.; Gao, H.; Zhou, S.; Pan, Y.; Zhang, K.; Liu, P.; Yang, H.; Zhao, Z.; Madyira, D.M. Rotor faults diagnosis in PMSMs based on branch current analysis and machine learning. Actuators
**2023**, 12, 145. [Google Scholar] [CrossRef] - Popescu, T.; Aiordachioaie, D. Fault detection of rolling element bearings using optimal segmentation of vibrating signals. Mech. Syst. Signal Proc.
**2019**, 116, 370–391. [Google Scholar] [CrossRef] - Yan, T.; Wang, D.; Xia, T.; Xi, L. A generic framework for degradation modeling based on fusion of spectrum amplitudes. IEEE Trans. Autom. Sci. Eng.
**2020**, 19, 308–319. [Google Scholar] [CrossRef] - Wang, Y.; Yang, M.; Li, Y.; Xu, Z.; Wang, J.; Fang, X. A multi-input and multi-task convolutional neural network for fault diagnosis based on bearing vibration signal. IEEE Sens. J.
**2021**, 21, 10946–10956. [Google Scholar] [CrossRef] - Wang, H.; Li, C.; Du, W. Coupled hidden Markov fusion of multichannel fast spectral coherence features for intelligent fault diagnosis of rolling element bearings. IEEE Trans. Instrum. Meas.
**2021**, 70, 1–10. [Google Scholar] [CrossRef] - Shao, S.; Wang, P.; Yan, R. Generative adversarial networks for data augmentation in machine fault diagnosis. Comput. Ind.
**2019**, 106, 85–93. [Google Scholar] [CrossRef] - Zhou, Q.; Hu, Y.; Liu, J. A novel assessable data augmentation method for mechanical fault diagnosis under noisy labels. Measurement
**2022**, 198, 111114. [Google Scholar] - Liu, S.; Jiang, H.; Wu, Z.; Li, X. Data synthesis using deep feature enhanced generative adversarial networks for rolling bearing imbalanced fault diagnosis. Mech. Syst. Signal Proc.
**2022**, 163, 108139. [Google Scholar] [CrossRef] - Jeong, H.; Bai, J.; Batuwatta-Gamage, C.P.; Rathnayaka, C.; Zhou, Y.; Gu, Y. A physics-informed neural network-based topology optimization (PINNTO) framework for structural Optimization. Eng. Struct.
**2023**, 278, 115484. [Google Scholar] [CrossRef] - Chaiprabha, K.; Chancharoen, R. A deep trajectory controller for a mechanical linear stage using digital twin concept. Actuators
**2023**, 12, 91. [Google Scholar] [CrossRef] - Li, L.; Li, Y.; Du, Q.; Liu, T.; Xie, Y. ReF-nets: Physics-informed neural network for Reynolds equation of gas bearing. Comput. Meth. Appl. Mech. Eng.
**2022**, 391, 115484. [Google Scholar] [CrossRef] - Thelen, A.; Zhang, X.; Fink, O.; Lu, Y.; Ghosh, S.; Youn, B.D.; Todd, M.D.; Mahadevan, S.; Hu, C.; Hu, Z. A comprehensive review of digital twin—Part 1: Modeling and twinning enabling technologies. Struct. Multidiscip. Optim.
**2022**, 65, 354. [Google Scholar] [CrossRef] - Lai, X.; Wang, S.; Guo, Z.; Zhang, C.; Sun, W.; Song, X. Designing a shape-performance integrated digital twin based on multiple models and dynamic data: A boom crane example. J. Mech. Des.
**2021**, 143, 071703. [Google Scholar] [CrossRef] - Piltan, F.; Kim, J. Crack size identification for bearings using an adaptive digital twinn. Sensors
**2021**, 21, 5009. [Google Scholar] [CrossRef] - Qin, Y.; Wu, X.; Luo, J. Data-model combined driven digital twin of life-cycle rolling bearing. IEEE Trans. Ind. Inform.
**2021**, 18, 1530–1540. [Google Scholar] [CrossRef] - Wang, J.; Ye, L.; Gao, R.X.; Li, C.; Zhang, L. Digital Twin for rotating machinery fault diagnosis in smart manufacturing. Int. J. Prod. Res.
**2019**, 57, 3920–3934. [Google Scholar] [CrossRef] - MSC ADAMS Reference Manual; MSC Software Corp.: Newport Beach, CA, USA, 2012; Available online: https://www.mscsoftware.com/product/adams (accessed on 21 February 2017).
- Giesbers, J. Contact Mechanics in MSC Adams—A Technical Evaluation of the Contact Models in Multibody Dynamics Software MSC Adams. Bachelor’s Thesis, University of Twente, Enschede, The Netherlands, 2012. [Google Scholar]
- Tu, W.; Liang, J.; Yu, W.; Shi, Z.; Liu, C. Motion stability analysis of cage of rolling bearing under the variable-speed condition. Nonlinear Dyn.
**2023**, 111, 11045–11063. [Google Scholar] [CrossRef]

**Figure 4.**Geometric models of bearing. (

**a**) Normal bearing without fault, (

**b**) defected bearing with outer-race fault.

**Figure 5.**The proposed physics-informed hybrid modeling method. (

**a**) Overall architecture, (

**b**) diagram of Cartesian coordinate system involved in ADAMS.

**Figure 6.**Flow chart of hybrid modeling method based on vibration generation and data mapping networks.

**Figure 7.**The structure diagram of two networks. (

**a**) The vibration generation network, (

**b**) the data mapping network.

**Figure 9.**Simulated speed signals of different bearing components when rotor rotates at 2000 rpm. (

**a**) Cage, (

**b**) rolling element.

**Figure 10.**Comparison of vibration signals of a rotor–bearing system under a speed-up condition from 0 rpm to 4000 rpm. (

**a**) The measured signal, (

**b**) the simulated signal, (

**c**) their spectrum results.

**Figure 11.**Comparison of vibration responses solved using Equations (2)–(6) and simulated by Equations (7)–(9).

**Figure 12.**Comparison of simulated and measured vibration signals of rotor–bearing system with bearing outer-race fault at 2000 rpm. (

**a**) Acceleration response, (

**b**) frequency spectrum.

**Figure 13.**Comparison of signals generated using vibration generation network at 1200 rpm. (

**a**) Network with traditional Loss 1, (

**b**) network with improved Loss 2.

**Figure 14.**Comparison of signals generated using vibration generation network with Loss 2 and obtained via simulation at 1250 rpm. (

**a**) Time-domain signals, (

**b**) frequency-domain signals.

**Figure 15.**Comparison of the simulated vibration signal obtained using proposed hybrid modeling method and the signal measured under the constant-speed condition. (

**a**) Vibration acceleration response, (

**b**) frequency spectrum.

**Figure 16.**Comparison of the simulated vibration signal obtained using a WGAN network and the measured signal under the constant-speed condition. (

**a**) Vibration acceleration response, (

**b**) frequency spectrum.

**Figure 17.**Comparison of the simulated vibration signal obtained using the proposed hybrid modeling and the measured signal under a variable-speed condition. (

**a**) Vibration acceleration response, (

**b**) frequency spectrum.

**Figure 18.**Comparison of the simulated vibration signal obtained by WGAN network and the measured signal under the variable-speed condition. (

**a**) Vibration acceleration response, (

**b**) frequency spectrum.

1st Body | 2nd Body | Constraint Type |
---|---|---|

Base | Ground | Fixed joint |

Bearing pedestal | Base | Spring-damping |

Outer race | Bearing pedestal | Fixed joint |

Inner race | Rotor | Fixed joint |

Rolling element | Outer race, inner race | Contact force |

Rolling element | Cage | Spherical joint |

Parameter Type | Parameter | Value |
---|---|---|

Opecrating conditions | Rotor angular speed ($\omega $/rpm) | 1000–2000 |

Bearing health status ($\beta $) | 0, 1 | |

Process-dependent parameters | Width of defect area (W/mm) | 1 |

Length of defect area (L/mm) | 0.5 | |

Observed Cartesian generalized coordinates (mm) | (45, 4.5, 0) | |

Simulation parameters | Length per sample (s) | 1 |

Step size per sample | 1/2000 |

Network Layer | Output Size | Operator |
---|---|---|

Linear1 | 11 × 1 × 128 | Leaky_ReLU |

Linear2 | 11 × 1 × 256 | Leaky_ReLU |

Linear3 | 11 × 1 × 512 | Leaky_ReLU |

Linear4 | 11 × 1 × 1000 | Tanh |

Module | Network Layer | Output Size | Operator |
---|---|---|---|

Encoder | Conv1 | 11 × 3 × 501 | ReLU, MaxPool |

Conv2 | 11 × 6 × 249 | ReLU, MaxPool | |

Linear1 | 11 × 512 | ReLU | |

Linear2 | 11 × 256 | ReLU | |

Linear3 | 11 × 10 | – | |

Decoder | Linear4 | 11 × 128 | Leaky_ReLU |

Linear5 | 11 × 256 | Leaky_ReLU | |

Linear6 | 11 × 512 | Leaky_ReLU | |

Linear7 | 11 × 1000 | Tanh |

Parameter | Numerical Value |
---|---|

Length of shaft (mm) | 460 |

Radius of shaft (mm) | 5 |

Radius of disc (mm) | 38 |

Thickness of disc (mm) | 18 |

Diameter of rolling element (mm) | 4.76 |

Diameter of inner race (mm) | 10 |

Diameter of outer race (mm) | 30 |

Number of rolling elements | 8 |

Contact angle (${}^{\circ}$) | 49.3 |

Operating Condition | Dataset | Rotating Speed (rpm) | Number of Samples |
---|---|---|---|

Constant-speed | Simulated dataset | 1900 | 120 |

Measured dataset | 1900 | 120 | |

Variable-speed | Simulated dataset | 1800 1900 2000 | 120 120 120 |

Measured dataset | 1800 1900 2000 | 120 10 120 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhu, M.; Peng, C.; Yang, B.; Wang, Y.
A Novel Physics-Informed Hybrid Modeling Method for Dynamic Vibration Response Simulation of Rotor–Bearing System. *Actuators* **2023**, *12*, 460.
https://doi.org/10.3390/act12120460

**AMA Style**

Zhu M, Peng C, Yang B, Wang Y.
A Novel Physics-Informed Hybrid Modeling Method for Dynamic Vibration Response Simulation of Rotor–Bearing System. *Actuators*. 2023; 12(12):460.
https://doi.org/10.3390/act12120460

**Chicago/Turabian Style**

Zhu, Mengting, Cong Peng, Bingyun Yang, and Yu Wang.
2023. "A Novel Physics-Informed Hybrid Modeling Method for Dynamic Vibration Response Simulation of Rotor–Bearing System" *Actuators* 12, no. 12: 460.
https://doi.org/10.3390/act12120460