# Influence of Coexistence of Pitting and Cracking Faults on a Two-Stage Spur Gear System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Calculating Gear Meshing Stiffness Model

#### 2.1. Spur Gear at the Perfect State

_{x}of the tooth can be calculated as follows:

_{b}produces bending and shear effects, while the axial force ${\mathrm{F}}_{\mathrm{a}}$ produces axial compression and bending effects. Couple M was used to illustrate the bending effect ${\mathrm{F}}_{\mathrm{a}}$. It is expressed as follows:

_{2}is half the angle of the tooth base.

_{1}is expressed as follows:

#### 2.2. Cracking and Pitting into the Tooth Surface

**h**is the height of the section at point A of the crack, which can be written as follows:

_{a}#### 2.2.1. Derivation of Mesh Stiffness for Cracked Gears

#### 2.2.2. Derivation of Mesh Stiffness for Pitted Gears

_{s}, and Hertzian contact stiffness are deduced as follows:

_{h}as follows:

#### 2.2.3. Derivation of Total Effective Gear Mesh Stiffness under Coexistence of Pitting and Cracking

_{h(pit)}, ${\mathrm{k}}_{{\mathrm{b}}_{1}(\mathrm{p}\mathrm{i}\mathrm{t}+\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{k})}$

**,**${\mathrm{k}}_{{\mathrm{s}}_{1}(\mathrm{p}\mathrm{i}\mathrm{t}+\mathrm{c}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{k})}$, and ${\mathrm{k}}_{{\mathrm{a}}_{1}(\mathrm{p}\mathrm{i}\mathrm{t})}$ in sequence. As a result, the expression for total effective mesh stiffness for a two-stage spur gear is:

#### 2.3. Gear Mesh Stiffness Evaluation with Coupled Pitting and Cracking on the Tooth Surface

^{7}to 7.772 × 10

^{7}N/m at a period of 21.45 s. However, for a moderate cracking of 29% on the tooth root, the meshing stiffness decreases from 7.772 × 10

^{7}to 6.9 × 10

^{7}N/m at a period of 22.31 s, which implies significant damage to the major components of the gear vibration signal.

^{7}to 1.732 × 10

^{7}N/m. Pitting and cracking together enhance gear bending fatigue and increase the weakening of the fractured tooth by dramatically lowering gear mesh stiffness, and it was noted that the crack fault is more dominant on the gear mesh stiffness than the pitting.

## 3. Spur Gear Dynamic Response

_{m}) revolution per minute, which carried a pinion of mass 1, a base circle radius 1, and moment of inertia 1 (m

_{1}, R

_{1}, J

_{1}) driven by an electric motor via an input coupling joint, which has the following characteristics: a torsional stiffness (k

_{p}) and a damping coefficient (c

_{p}); shaft 2 (intermediary), which carried a wheel of mass 2, base circle radii 2, within a moment of inertia 2 (m

_{2}, R

_{2}, J

_{2}), and carried a pinion of mass 3, base circle radius 3, and moment inertia 3 (m

_{3}, R

_{3}, J

_{3}); and shaft 2 (output), rotating at (T

_{L}) revolution per minute, which carried a wheel of mass 4, a base circle radius 4, and moment of inertia 4 (m

_{4}, R

_{4}, J

_{4}), and was connected to a torsional stiffness (k

_{g}) and a damping coefficient (c

_{g}) characterizing the load through an output coupling joint, respectively. All bearings are represented by k

_{1}, vertical stiffness, and c

_{1}, vertical damping on the input bearing; k

_{2}= k

_{3}, vertical stiffness, and c

_{2}= c

_{3}, vertical damping on the intermediary bearing; and k4, vertical stiffness, and c4, vertical damping on the output bearing. Figure 6b,c shows two pairs of spur gears, each with a pinion and a gear. The first stage (b) and second stage (c) are properly coupled, and the two-surface gear contact is subjected to torsional stiffness and damping forces generated by the gear meshing stiffness k

_{t1}and damping coefficient c

_{t1}in Figure 6b and the gear meshing stiffness k

_{t2}and damping coefficient c

_{t2}in Figure 6c.

#### 3.1. Motion of Dynamic Model

_{1}, y

_{2}, y

_{3}, and y

_{4}from the lateral vibrations developed on bearings and six angular rotations as follows: driving motor ${\mathsf{\theta}}_{\mathrm{m}}$, pinion ${\mathsf{\theta}}_{1}$, wheel ${\mathsf{\theta}}_{2}$, pinion ${\mathsf{\theta}}_{3}$, wheel ${\mathsf{\theta}}_{4}$, and load ${\mathsf{\theta}}_{\mathrm{L}}$ from the torsional vibrations.

#### 3.2. Numerical Analysis of a Two-Stage Gearbox System with Coupled Pitting and Cracking on the Tooth Surface

^{2}. Additionally, three sub-harmonics, a 25% pitting, and sidebands in the frequency range of 80.08 to 800 Hz are present.

^{2}between 4 and 8 s. The fluctuation in peak amplitudes suggests the onset of crack-induced tooth surface damage at 25%.

^{2}.

^{2}.

## 4. Experimental Model

^{−2}and an acceleration sensor range of 200 g to 200 g.

#### 4.1. Experimental Results

#### 4.2. Feature Extraction of Experimental Results

_{r}and f

_{e}) are significantly affected. It has been observed that the sidebands and harmonic amplitudes of pitted and cracked gears can be directly identified from the RPM–frequency map, as shown in Figure 19c.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${m}_{1}$ | Pinion mass at the first stage |

${m}_{2}$ | Wheel (gear) mass at the first stage |

${m}_{3}$ | Pinion mass at the second stage |

${m}_{4}$ | Gear mass at the second stage |

${x}_{1}$ | The linear displacement of the pinion at the first stage |

${x}_{2}$ | The linear displacement of the wheel at the second stage |

${x}_{1}$ | The linear displacement of the pinion at the first stage |

${x}_{2}$ | The linear displacement of the wheel at the second stage |

${y}_{1}$ | The linear displacement of the pinion at the first stage |

${y}_{2}$ | The linear displacement of the wheel at the first stage |

${y}_{3}$ | The linear displacement of the pinion at the second stage |

${y}_{4}$ | The linear displacement of the wheel at the second stage |

${J}_{1}$ | Mass moment of inertia of the pinion at the first stage |

${J}_{2}$ | Mass moment of inertia of the gear at the first stage |

${J}_{3}$ | Mass moment of inertia of the pinion at the second stage |

${J}_{4}$ | Mass moment of inertia of the gear at the second stage |

${J}_{m}$ | Mass moment of inertia of the motor |

${J}_{L}$ | Mass moment of inertia of the load |

${k}_{{y}_{1}}$ | Stiffness of the input bearing in y-direction at the first stage |

${k}_{{y}_{2}}$ | Stiffness of the input bearing in y-direction at the second stage |

${k}_{{y}_{3}}$ | Stiffness of the output bearing in y-direction at the first stage |

${k}_{{y}_{4}}$ | Stiffness of the output bearing in y-direction at the second stage |

${k}_{p}$ | Torsional stiffness of the input shaft coupling |

${k}_{g}$ | Torsional stiffness of the output shaft coupling |

${k}_{t}$ | Gear meshing stiffness |

${c}_{g}$ | Torsional damping of the output shaft coupling |

${c}_{p}$ | Torsional damping of the input shaft coupling |

${c}_{t}$ | Gear meshing damping |

${\theta}_{1}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{\theta}_{2}$ | The angular displacement of the pinion and gear at the first stage |

${\theta}_{3}\hspace{0.17em}and\hspace{0.17em}{\theta}_{4}$ | The angular displacement of the pinion and gear at the second stage |

${R}_{1}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{R}_{2}$ | Base circle radius of pinion and wheel at the first stage |

${R}_{3}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{R}_{4}$ | Base circle radius of pinion and wheel at the second stage |

${C}_{{y}_{1}}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{C}_{{y}_{2}}$ | Damping of the input bearing and output bearing at the first stage |

${C}_{{y}_{3}}\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{C}_{{y}_{4}}$ | Damping of the input bearing and output bearing at the second stage |

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**Figure 2.**Isometric profile of the tooth showing: (

**a**) crack; (

**b**) circular pit; and (

**c**) coexistence of pitting and cracking.

**Figure 3.**Top view of the tooth showing: (

**a**) crack; (

**b**) circular pit; and (

**c**) coexistence of pitting and cracking.

**Figure 6.**Schematic diagram of the two-stage gearbox with 10 DOF. (

**a**) Gear transmission system, (

**b**) First stage contact, (

**c**) Second stage (Pinion-wheel).

**Figure 11.**3D-Waterfall: (

**a**) healthy gear, (

**b**) cracked gear, (

**c**) pitted gear, (

**d**) pitted/cracked gear.

**Figure 18.**Spectrogram of gear: (

**a**) perfect, (

**b**) cracked tooth, (

**d**) pitted tooth, and (

**c**) pitted–cracked tooth.

Parameters | Value | |
---|---|---|

Driving Gear (Pinion) ^{1,3} | Driven Gear (Wheel) ^{2,4} | |

Young Module (E) [Pa] | 2.068 × 10^{11} | 2.068 × 10^{11} |

Pressure angle (°) | 20 | 20 |

Poisson’s ratio | 0.3 | 0.3 |

Number of teeth Z_{1} = Z_{3} (pinion) and Z_{2} = Z_{4} (gear) | 30 | 90 |

Base circle radius of a pinion R_{1} = R_{3} [mm] and gear R_{2} = R_{4} [mm] | 30.1 | 76.1 |

Mass m_{1} (pinion) and m_{2} (gear) [kg] | 0.96 | 2.88 |

Meshing stiffness of bearings k_{1}, k_{3} (pinion) = k_{2}, k_{4} (gear) [N.s/m] | 6.56 × 10^{7} | 6.56 × 10^{7} |

Damping coefficient of bearings c_{1}, c_{3} (pinion) = c_{2}, c_{4} (gear) [N.s/m] | 1.8 × 10^{5} | 1.8 × 10^{5} |

Torsional stiffness of coupling k_{p} (pinion) = k_{g} (gear) [N.s/m] | 4.4 × 10^{4} | 4.4 × 10^{4} |

Damping coefficient of coupling c_{p} (pinion) = c_{g} (gear) [Nm. s/rad] | 5 × 10^{5} | 5 × 10^{5} |

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## Share and Cite

**MDPI and ACS Style**

Happi, K.H.Y.; Kouejou, B.X.T.; Alugongo, A.A.
Influence of Coexistence of Pitting and Cracking Faults on a Two-Stage Spur Gear System. *Vibration* **2023**, *6*, 195-217.
https://doi.org/10.3390/vibration6010013

**AMA Style**

Happi KHY, Kouejou BXT, Alugongo AA.
Influence of Coexistence of Pitting and Cracking Faults on a Two-Stage Spur Gear System. *Vibration*. 2023; 6(1):195-217.
https://doi.org/10.3390/vibration6010013

**Chicago/Turabian Style**

Happi, Kemajou Herbert Yakeu, Bernard Xavier Tchomeni Kouejou, and Alfayo Anyika Alugongo.
2023. "Influence of Coexistence of Pitting and Cracking Faults on a Two-Stage Spur Gear System" *Vibration* 6, no. 1: 195-217.
https://doi.org/10.3390/vibration6010013