# Improvement on Meshing Stiffness Algorithms of Gear with Peeling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Algorithm Model of Meshing Stiffness

#### 2.1. Calculate Hertz Contact Stiffness and Wheel Stiffness

_{f}. The formula can be expressed as [25]:

#### 2.2. Calculate Bending Stiffness, Shear Stiffness, and Compression Stiffness

_{b}, K

_{s}, K

_{a}respectively represent the bending stiffness, shear stiffness, and compression stiffness.

## 3. Improvement on Meshing Stiffness Algorithms of Gear with Peeling

#### 3.1. Establishing a Calculation Model of Peeling Failure Meshing Stiffness

_{1}, K

_{2}, K

_{3}, K

_{h}, and K

_{fi}, and the formulas are:

#### 3.2. Solve Peeling Failure Meshing Stiffness

_{b}, shear stiffness K

_{s}, and compression stiffness K

_{a}can be calculated with:

_{b}, shear stiffness K

_{s}, and compression stiffness K

_{a}can be integrated with:

## 4. The Meshing Stiffness of Gear with Variable Peeling Parameter

#### 4.1. Variable Depth Variable Meshing Stiffness

#### 4.2. Variable Width Variable Meshing Stiffness

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Cui, L.L.; Li, B.B.; Ma, J.F.; Jin, Z. Quantitative trend fault diagnosis of a rolling bearing based on Sparsogram and Lempel-Ziv. Measurement
**2018**, 128, 410–418. [Google Scholar] [CrossRef] - Song, L.Y.; Wang, H.Q.; Chen, P. Vibration-Based Intelligent Fault Diagnosis for Roller Bearings in Low-Speed Rotating Machinery. IEEE Trans. Instrum. Meas.
**2018**, 67, 1887–1899. [Google Scholar] [CrossRef] - Cui, L.L.; Wang, X.; Xu, Y.G.; Jiang, H.; Zhou, J.P. A novel Switching Unscented Kalman Filter method for remaining useful life prediction of rolling bearing. Measurement
**2019**, 135, 678–684. [Google Scholar] [CrossRef] - Han, T.; Jiang, D.; Zhao, Q.; Wang, L.; Yin, K. Comparison of random forest, artificial neural networks and support vector machine for intelligent diagnosis of rotating machinery. Trans. Inst. Meas. Control
**2018**, 40, 2681–2693. [Google Scholar] [CrossRef] - Cui, L.; Huang, J.; Zhang, F. Quantitative and localization diagnosis of a defective ball bearing based on vertical-horizontal synchronization signal analysis. IEEE Trans. Ind. Electron.
**2017**, 64, 8695–8705. [Google Scholar] [CrossRef] - 6. Hao, Y.S.; Song, L.Y.; Wang, M.Y.; Cui, L.I.; Wang, H. Underdetermined Source Separation of Bearing Faults Based on Optimized Intrinsic Characteristic-Scale Decomposition and Local Non-Negative Matrix Factorization. IEEE Access
**2019**, 7, 11427–11435. [Google Scholar] [CrossRef] - Cui, L.; Wang, J.; Lee, S. Matching Pursuit of an Adaptive Impulse Dictionary for Bearing Fault Diagnosis. J. Sound Vib.
**2014**, 333, 2840–2862. [Google Scholar] [CrossRef] - Wang, H.; Li, S.; Song, L.Y. A novel convolutional neural network based fault recognition method via image fusion of multi-vibration-signals. Comput. Ind.
**2019**, 105, 182–190. [Google Scholar] [CrossRef] - Cui, L.; Huang, J.F.; Zhang, F.B. HVSRMS localization formula and localization law: Localization diagnosis of a ball bearing outer ring fault. Mech. Syst. Signal Process.
**2019**, 120, 608–629. [Google Scholar] [CrossRef] - Wang, H.Q.; Ren, B.Y.; Song, L.Y. A Novel Weighted Sparse Representation Classification Strategy based on Dictionary Learning for Rotating Machinery. IEEE Trans. Instrum. Meas.
**2019**. [Google Scholar] [CrossRef] - Cui, L.L.; Yao, T.C.; Zhang, Y.; Gong, X.Y.; Kang, C.H. Application of pattern recognition in gear faults based on the matching pursuit of a characteristic waveform. Measurement
**2017**, 104, 212–222. [Google Scholar] [CrossRef] - Hao, Y.; Song, L.; Cui, L.; Wang, H. A three-dimensional geometric features-based SCA algorithm for compound faults diagnosis. Measurement
**2019**, 134, 480–491. [Google Scholar] [CrossRef] - Cui, L.; Jin, Z.; Huang, J.; Wang, H. Fault severity classification and size estimation for ball bearings based on vibration mechanism. IEEE Access
**2019**. [Google Scholar] [CrossRef] - Wang, H.; Wang, P.; Song, L. A Novel Feature Enhancement Method based on Improved Constraint Model of Online Dictionary Learning. IEEE Access
**2019**, 7, 17599–17607. [Google Scholar] [CrossRef] - Al-Meshari, A.; Al-Zahrani, E.; Diab, M. Failure analysis of cooling fan gearbox. Eng. Fail. Anal
**2012**, 20, 166–172. [Google Scholar] [CrossRef] - Amarnath, M.; Chandramohan, S. Experimental investigations of surface wear assessment of spur gear teeth. J. Vib. Control
**2012**, 7, 1009–1024. [Google Scholar] [CrossRef] - Subramanian, R.B.; Srinivasan, K. Vibration analysis of an influence of groove in the bottom land of a spur gear. J. Vib. Control
**2014**, 6, 847–858. [Google Scholar] [CrossRef] - Cui, L.L.; Wang, X.; Wang, H.Q.; Wu, N. Improved Fault Size Estimation Method for Rolling Element Bearings Based on Concatenation Dictionary. IEEE Access
**2019**, 7, 22710–22718. [Google Scholar] [CrossRef] - Song, L.Y.; Wang, H.Q.; Chen, P. Step-by-step Fuzzy Diagnosis Method for Equipment Based on Symptom Extraction and Trivalent Logic Fuzzy Diagnosis Theory. IEEE Trans. Fuzzy Syst.
**2018**, 26, 3467–3478. [Google Scholar] [CrossRef] - Lin, J.; Robert, G.P. Mesh stiffness variation instabilities in two-stage gear systems. J. Vib. Acoust.
**2002**, 124, 68–76. [Google Scholar] [CrossRef] - Li, R.Z. Ring-wheel strength design material. Japan Society of Mechanical Engineers. Mach. Tool Ind. Ed.
**1984**, 56–58. [Google Scholar] - Weber, C. The deformation of loaded gears and the effect on their load-carrying capacity. In Scientific and Industrial Research in British; EReport No.3; Department of Scientific and Industrial Research: New Delhi, India, 1949. [Google Scholar]
- Yang, D.C.H.; Lin, J.Y. Hertzian damping, tooth friction and bending elasticity in gear impact dynamics. J. Mesh. Trans. Auto. Des.
**1987**, 109, 189–196. [Google Scholar] [CrossRef] - Wang, Q.B.; Zhang, Y.M. A model for analyzing stiffness and stress in a helical gear pair with tooth profile errors. J. Vib. Control
**2017**, 23, 272–289. [Google Scholar] [CrossRef] - Wang, X. Study on the Calculation of Gear Crack Dynamics and Gear Tooth Crack Stiffness. Master’s Thesis, Chongqing University, Chongqing, China, 2012. [Google Scholar]
- Mohammed, O.D.; Rantatalo, M.; AIDANPÄÄ, J.O. Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis. Eng. Fail. Anal.
**2013**, 34, 235–251. [Google Scholar] [CrossRef] - Tian, X.H. Dynamic Simulation for System Response of Gearbox Including Localized Gear Faults. Master’s Thesis, University of Alberta, Edmonton, AB, Canada, 2004. [Google Scholar]
- Zhao, S.B.; Hong, R.J. Function law calculation Calculation peeling breakdown time at the time of ringing. Mach. Des. Manuf.
**2014**, 10, 171–176. [Google Scholar] - Del Rincon, A.F.; Viadero, F.; Iglesias, M.; De-Juan, A.; Garcia, P.; Sancibrian, R. Effect of cracks and pitting defects on gear meshing. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2012**, 226, 2805–2815. [Google Scholar] [CrossRef] - Shao, Y.M.; Wang, X.L. Contact surface time hardness surface surface peeling dynamics model robustness application. J. Vib. Shock
**2014**, 33, 8–14. [Google Scholar] - Fernandez-del-Rincon, A.; Garcia, P.; Diez-Ibarbia, A.; de-Juan, A.; Iglesias, M.; Viadero, F. Enhanced model of gear transmission dynamics for condition monitoring applications: Effects of torque, friction and bearing clearance. Mech. Syst. Signal Process.
**2017**, 85, 445–467. [Google Scholar] [CrossRef] - Del Rincon, A.F.; de-Juan, A.P.; Garcia, P.A.; Diez-Ibarbia, A.; Viadero, F. Planetary transmission load sharing: Manufacturing errors and system configuration study Iglesias. Mech. Mach. Theory
**2017**, 111, 21–38. [Google Scholar] - Saxena, A.; Parey, A. Time varying mesh stiffness calculation of spur gear pair considering sliding friction and peeling defects. Eng. Fail. Anal.
**2016**, 70, 200–211. [Google Scholar] [CrossRef] - Yang, D.C.H.; Sun, Z.S. A Rotary Model for Spur Gear Dynamics. J. Mech. Transm. Autom. Des.
**1985**, 107, 529–535. [Google Scholar] [CrossRef] - Steward, J.H. Elastic analysis of load distribution in wide-faced spur gears. Master’s Thesis, University of Newcastle, Galan, Australian, 1989. [Google Scholar]
- Chen, Z.; Shao, Y. Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Eng. Fail. Anal
**2011**, 18, 2149–2164. [Google Scholar] [CrossRef] - Mohammed, O.D.; Rantatalo, M.; Aidanpää, J.O. Vibration signal analysis for gear fault diagnosis with various crack progression scenarios. Mech. Syst. Signal Process.
**2013**, 41, 176–195. [Google Scholar] [CrossRef]

**Figure 1.**Wheel stiffness parameters [36].

**Figure 2.**The force diagram of the tooth [26].

**Figure 6.**Normal Meshing Stiffness and Peeling Meshing Stiffness. (

**a**) is the meshing stiffness curve of spalling gear and healthy gear in the meshing process, The blue curve is the meshing stiffness of the healthy gear, and the red curve is the meshing stiffness of the stripped gear; (

**b**) is a local enlarged view of gear fitting stiffnes.

Parameters | Drive Wheel | Driven Wheel |
---|---|---|

Modulus | 5 | 5 |

Number of teeth | 19 | 48 |

Pressure angle | 20° | 20° |

Addendum coefficient | 1 | 1 |

Tip clearance coefficient | 0.25 | 0.25 |

Elastic Modulus | 2.06 × 1011 | 2.06 × 1011 |

Poisson’s ratio | 0.3 | 0.3 |

Tooth width | 20 | 20 |

Ai | Bi | Ci | Di | Ei | Fi | |
---|---|---|---|---|---|---|

${\mathit{L}}^{\mathbf{*}}\mathbf{\left(}\mathit{h}\mathbf{,}{\mathit{\theta}}_{\mathit{f}}\mathbf{\right)}$ | −5.574e−5 | −1.9986e−3 | −2.3015e−4 | 4.7702e−3 | 0.0271 | 6.8045 |

${\mathit{M}}^{\mathbf{*}}\mathbf{\left(}\mathit{h}\mathbf{,}{\mathit{\theta}}_{\mathit{f}}\mathbf{\right)}$ | 60.111e−5 | 28.100e−3 | −83.431e−4 | −9.9256e−3 | 0.1624 | 0.9086 |

${\mathit{P}}^{\mathbf{*}}\mathbf{\left(}\mathit{h}\mathbf{,}{\mathit{\theta}}_{\mathit{f}}\mathbf{\right)}$ | −50.952e−5 | 185.50e−3 | 0.0538e−4 | 53.300e−3 | 0.2895 | 0.9236 |

${\mathit{Q}}^{\mathbf{*}}\mathbf{\left(}\mathit{h}\mathbf{,}{\mathit{\theta}}_{\mathit{f}}\mathbf{\right)}$ | −6.2042e−5 | 9.0889e−3 | −4.0964e−4 | 7.8297e−3 | −0.1472 | 0.6904 |

${\mathit{W}}_{1}$ | ${\mathit{W}}_{2}$ | ${\mathit{W}}_{3}$ | ${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ${\mathit{Y}}_{3}$ | $\mathit{a}$ |
---|---|---|---|---|---|---|

7 mm | 2 mm | 7 mm | 2.5 mm | 1.0 mm | 4.33 mm | 45° |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cui, L.; Liu, T.; Huang, J.; Wang, H.
Improvement on Meshing Stiffness Algorithms of Gear with Peeling. *Symmetry* **2019**, *11*, 609.
https://doi.org/10.3390/sym11050609

**AMA Style**

Cui L, Liu T, Huang J, Wang H.
Improvement on Meshing Stiffness Algorithms of Gear with Peeling. *Symmetry*. 2019; 11(5):609.
https://doi.org/10.3390/sym11050609

**Chicago/Turabian Style**

Cui, Lingli, Tongtong Liu, Jinfeng Huang, and Huaqing Wang.
2019. "Improvement on Meshing Stiffness Algorithms of Gear with Peeling" *Symmetry* 11, no. 5: 609.
https://doi.org/10.3390/sym11050609