# Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors

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## Abstract

**:**

## 1. Introduction

## 2. FE Model Description

#### 2.1. Tooth Contact Analysis Considering the Assembly Error

**r**

_{1}and

**r**

_{2}, and the tooth surface normal directions

**n**

_{1}and

**n**

_{2}of the pinion and the gear are obtained, respectively. The detail of the tooth surface derivation can refer to Appendix A. The following coordinate transformations are used to assemble the two gears into the same mounting coordinate system:

**M**

_{hi}is the transformation matrix from the cutter head coordinate system to the blank coordinate system;

**L**

_{hi}is the first three-order sub-matrix of

**M**

_{hi}; ψ

_{ci}, s

_{i}and θ

_{i}are the rotation angle of the cradle, the distance from the point on the theoretical tool tip to the tool profile, and the rotation angle of the cutter head, respectively.

**M**

_{hi}can be expressed as:

_{p}is the pinion axial error; ΔA

_{g}is the gear axial error. The assembly errors are expressed in Figure 2. φ

_{1}and φ

_{2}are the initial installation angles of the pinion and the gear, respectively; Σ is the shaft angle.

_{h}-y

_{h}-z

_{h}, the two meshing gear teeth should have collinear normal vectors and the same coordinate values at the contact point:

_{1}, other variables can be determined. Then, the contact trajectory and the non-load transmission error (NLTE) of the SBGP in the rotation range of ψ

_{1}= (−π/z

_{1}, π/z

_{1}) are obtained. The transmission error between the two gears can be expressed as:

_{10}and ϕ

_{20}are the initial rotation angles of the pinion and the gear, respectively; ϕ

_{1}and ϕ

_{2}are the real-time rotation angles of the pinion and the gear, respectively; z

_{1}and z

_{2}are the numbers of teeth of the pinion and the gear, respectively.

_{g}become, the larger the value of NLTE becomes, and the smaller the value of ΔA

_{p}becomes, the larger the value of NLTE becomes. Therefore, in the process of assembling the SBGP, it is necessary to control these errors to obtain a smaller value of ΔE, ΔΣ and ΔA

_{g}, and a larger value of ΔA

_{p}to ensure that the NLTE values are as small as possible.

#### 2.2. FE Modelling of the SBGP

_{p1}of the master node along the rotation direction is obtained. The mesh stiffness value of the SBGP can be obtained by the following equation:

_{p}is the pitch circle radius of the SBGP. It is worth noting that the TVMS obtained in this paper is entirely caused by the contact deformation of the SBGP, so the changing of NLTE should be considered additionally in the subsequent dynamic analysis.

#### 2.3. Tooth Root Crack Fault Simulation

## 3. Results and Discussion

#### 3.1. Influence of the Assembly Errors on the Tooth Root Stress Distribution

#### 3.1.1. Offset Error

#### 3.1.2. Shaft Angle Error

#### 3.1.3. Pinion Axial Error

_{p}on the maximum root stress value and its position is illustrated in Figure 12. It can be seen from the figure that the smaller the value of ΔA

_{p}, the greater the maximum root stress. However, no matter how much ΔA

_{p}changes, it does not affect the position where the maximum root stress occurs.

#### 3.1.4. Gear Axial Error

_{g}on the maximum root stress value of the pinion and its position along the tooth width. It can be seen from the figure that with the increase in ΔA

_{g}, the position of the maximum tooth root stress of the pinion moves towards the toe direction. However, within the range of ΔAg = −0.1 mm to 0.1 mm, the maximum tooth root stress value only increased by 6.5 MPa. This indicates that the maximum root stress of the pinion is insensitive to the change in ΔA

_{g}.

_{g}has the least. Within the scope of ensuring the reasonable root stress of the mating gear, the changes of ΔE, ΔΣ and ΔA

_{p}should be especially controlled. The smaller the value of ΔE and ΔΣ are, the better it is to obtain a small root stress value. The larger the value of ΔA

_{p}becomes, the better it is to obtain a small root stress value. No matter how the assembly error changes in this study, the maximum root stress always appears around the middle of the tooth width and deviates from the toe side. This shows that the root crack fault of the pinion is most likely to occur around this zone.

#### 3.2. Crack Fault Analysis of the SBGP

#### 3.2.1. TVMS Due to the Crack Fault

#### 3.2.2. Dynamic Simulation

_{bi}, I

_{bix}, I

_{biy}and I

_{biz}are the mass and moments of inertia of the gear i (i = 1,2 represents the bevel pinion and bevel gear, respectively). k

_{bix}, k

_{biy}, k

_{biz}, k

_{bi}

_{θx}, k

_{bi}

_{θy}and k

_{bi}

_{θz}are the supporting stiffness of gear i in six directions, respectively. k

_{b1b2}is the TVMS of the SBGP and NLTE is the NLTE of the SBGP accordingly. Considering the NLTE in the system means that only when the dynamic projected displacement of the SBGP exceeds the value of NLTE, the contact of the SBGP can occur. The derivation of the projection displacement vector of the SBGP

**δ**

_{b1b2}can be seen in Ref. [34]. The superscript k of

**δ**

^{k}

_{b1b2}denotes the k-th element of vector

**δ**

_{b1b2}.

**q**

_{b1b2}is the displacement vector consisting of 12 degrees of freedom of the two gears. Due to the introduction of the shaft components, the driving torque T

_{b1}originally added to the rotation direction of the pinion is transferred to the drive node of the input shaft. At the same time, the slave torque T

_{b2}originally added in the rotation direction of the gear is transferred to the slave node of the intermediate shaft. The schematic of the grouping of the system stiffness matrix and the damping matrix is illustrated in Figure 17.

_{i}and β

_{i}are the damping coefficients of the structure i;

**M**

_{i}and

**K**

_{i}are the mass matrix and stiffness matrix of the structure i, which are solved by the Timoshenko beam theory. The damping of the supporting bearings is considered as contact values. The mesh damping of the SBGP is described as:

_{b1}and m

_{b2}are the mass of the bevel pinion and the bevel gear, respectively.

_{b1b2}and NLTE are interpolated according to the time series relationship to obtain the mesh stiffness and NLTE values under each Newmark integration step. After solving for a whole sampling time, the dynamic response of the system is obtained.

#### 3.2.3. Response Analysis Due to the Crack Fault of the SBGP

_{m}and its harmonics. Under the crack fault condition, every time the pinion revolts one period T

_{bp}, the faulty tooth participates in meshing for two mesh periods. There are fluctuations at intervals of T

_{bp}in the time-domain waveform of Case 4. The fault causes the mutation waveform of three meshing cycles. In the faulty amplitude spectrum, under the influence of the crack fault, sidebands with the rotation frequency of the pinion f

_{bp}as the interval appear on both sides of the meshing frequency and its harmonics.

#### 3.2.4. Sensitivity Analysis of Statistical Indicators under Healthy Condition

#### 3.2.5. Crack Fault Detection under the Influence of the Offset Errors

## 4. Conclusions

- (1)
- Through static analysis, the value of the maximum tooth root stress and its position is discussed considering the assembly errors. It is found that the position of the maximum tooth root stress appears in the middle of the tooth width. The value is influenced by the assembly errors. To avoid excessive tooth root stress of the pinion, the changes in errors ΔE, ΔΣ and ΔA
_{p}should be strictly controlled. A smaller ΔE, ΔΣ, and a larger ΔA_{p}are preferred to ease the pinion tooth root stress. - (2)
- The dynamic response of the SBGP with the pinion tooth root crack fault is obtained by introducing the faulty TVMS curve as the excitation. When a crack fault occurs on the pinion, every time the pinion revolves one cycle, the faulty tooth participates in meshing for two meshing periods. There are fluctuations at intervals of the pinion rotation period in the time-domain waveform. Each time the faulty tooth participates in meshing, there are three meshing cycle mutations in the time-domain waveform. In the faulty amplitude spectrum, under the influence of the crack fault, sidebands with the rotation frequency of the pinion as the interval appear on both sides of the meshing frequency and its harmonics.
- (3)
- Through the analysis of statistical indicators. The sensitive indicators for identifying the root crack of the pinion are obtained. They are the A, P, SMR, C, I, and L in the time-domain, and F12, F14, F16, F17, F18, F19, F20, F21, F22 and F23 in the frequency-domain. These indicators can be used to monitor and diagnose crack faults in the SBGP system under the assembly error free condition. Moreover, under the interference of offset error, the time-domain indicators A, P, C, I and L, and the frequency-domain indicators F12, F18 and F19 still maintain a good judgment threshold for fault information, so these indicators can be used as the indicators for diagnosing crack faults in the presence of offset errors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{g}-x

_{g}-y

_{g}-z

_{g}can be expressed in the blank coordinate system through the following coordinate transformation:

_{g}is the rotation angle of the cutter; S

_{r2}is radial distance; q

_{2}is the cradle angle; X

_{B2}is the sliding base; E

_{2}is the blank offset; γ

_{m2}is the machine root angle; X

_{2}is the machine center to back; ψ

_{2}is the blank rotation angle; ψ

_{c2}is the cradle rotation angle; and ψ

_{c2}= ψ

_{2}/i

_{2,}i

_{2}is the velocity ratio. The detail of the coordinate transformation matrix

**ROTY**(θ),

**ROTZ**(θ),

**TRNX**(δ),

**TRNY**(δ), and

**TRNZ**(δ) are expressed as:

**Figure A1.**The layout of the machine tool coordinate for machining bevel gear by the generating method: (

**a**) Front view; (

**b**) General view.

_{p}-x

_{p}-y

_{p}-z

_{p}can be expressed in the blank coordinate system through the following coordinate transformation:

_{p}is the rotation angle of the cutter; i is the tilt angle; j is the swivel angle; S

_{r1}is radial distance; q

_{1}is the cradle angle; X

_{B1}is the sliding base; E

_{1}is the blank offset; γ

_{m1}is the machine root angle; X

_{1}is the machine center to back; ψ

_{1}is the blank rotation angle; ψ

_{c1}is the cradle rotation angle; and ψ

_{c1}= ψ

_{1}/i

_{1,}i

_{1}is the velocity ratio.

**Figure A2.**The layout of the machine tool coordinate for machining bevel pinion by the tilt method: (

**a**) Front view; (

**b**) General view.

_{i}-x

_{i}-y

_{i}-z

_{i}, (i = p, g denotes cutter machining the bevel pinion and the bevel gear, respectively), the flank and the transition zone can be expressed by:

_{i}is the distance from the tooth flank to the tooth tip; the “±” corresponds to the concave side and the convex side, respectively; γ

_{j}is the central angle between any point on the transition arc and T

_{0j}, where T

_{0j}is the point of tangent between the tooth surface and the transition zone. When machining the concave surface of the gear, the value range of γ

_{j}is [0, π/2 + α

_{0j}], and when machining the convex surface of the gear, the value range of γ

_{j}is [−π/2 + α

_{0j}, 0].

_{i}-x

_{i}-y

_{i}-z

_{i}, (i = 1, 2 denotes the blank of the bevel pinion and the bevel gear, respectively), the tooth surface equation can be expressed as:

**r**

_{2}satisfies the following relation:

_{M}, y

_{M}) of any point (see Figure A4) satisfies the following equations:

_{m1}-x

_{m1}-y

_{m1}-z

_{m1}, the position of the cutter can be expressed as:

**r**

_{m1}satisfies the following relation:

_{M}, y

_{M}) of any point (see Figure A4) satisfies the following equations:

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**Figure 3.**Contact trajectories and NLTEs variation due to the assembly errors: (

**a**) ΔE; (

**b**) ΔΣ; (

**c**) ΔA

_{g}; (

**d**) ΔA

_{p}.

**Figure 6.**Bevel pinion crack propagation path [6].

**Figure 7.**Node replacement method to generate crack section: (

**a**) Node replacement example; (

**b**) Crack characterization parameters.

**Figure 9.**Contour plot of the von Mises stress distribution of the pinion: (

**a**) 3D view; (

**b**) Plane view.

**Figure 10.**Variation of tooth root stress in three mesh cycles considering the variation of offset error.

**Figure 11.**Variation of tooth root stress in three mesh cycles considering the variation of shaft angle error.

**Figure 12.**Variation of tooth root stress in three mesh cycles considering the variation of pinion axial error.

**Figure 13.**Variation of tooth root stress in three mesh cycles considering the variation of gear axial error.

**Figure 15.**TVMS and the initial gaps of the SBGP: (

**a**) TVMS versus different crack levels; (

**b**) The initial gaps during two meshing cycles.

**Figure 20.**Fluctuations of time-domain indicators relative to the healthy condition for different crack degrees.

**Figure 21.**Fluctuations of frequency-domain indicators relative to the healthy condition for different crack degrees.

**Figure 22.**TVMS curves of SBGP considering assembly errors and crack faults: (

**a**) TVMS in health conditions with offset errors; (

**b**) TVMS curves in the condition of ΔE = −0.1 mm with different crack degrees; (

**c**) TVMS curves in the condition of ΔE = −0.05 mm with different crack degrees; (

**d**) TVMS curves in the condition of ΔE = 0 mm with different crack degrees; (

**e**) TVMS curves in the condition of ΔE = 0.05 mm with different crack degrees; (

**f**) TVMS curves in the condition of ΔE = 0.1 mm with different crack degrees.

**Figure 23.**Under the influence of offset error, the fluctuation of time-domain indicators relative to their respective healthy condition values under different degrees of cracks.

**Figure 24.**Under the influence of offset error, the fluctuation of frequency-domain indicators relative to their respective healthy condition values under different degrees of cracks.

Parameter | Bevel Pinion | Bevel Gear |
---|---|---|

Tooth number z_{1}/z_{2} | 17 | 81 |

Modulus m (mm)/Shaft angle Σ (°)/Mean spiral angle β (°) | 5.6/90/35 | |

Direction of rotation | Left-handed | Right-handed |

Face width b (mm) | 60 | |

Mean cone distance R (mm) | 201.741 | |

Pitch angle δ (°) | 11.8530 | 78.1470 |

Root angle δ_{f} (°) | 10.9321 | 76.1944 |

Face angle δ_{a} (°) | 13.8056 | 79.0679 |

Addendum height h_{a} (mm) | 6.8480 | 2.6720 |

Dedendum height h_{f} (mm) | 3.7250 | 7.9010 |

Parameter | Concave | Convex |
---|---|---|

Cutter point radius r_{01} (mm) | 226.47 | 230.86 |

Pressure angle α_{01} (°) | −18.75 | 21.25 |

Root fillet radius ρ_{01} (mm) | 1 | 1 |

Machine center to back X_{1} (mm) | −5.237 | 8.545 |

Sliding base X_{B1} (mm) | 15.217 | 13.183 |

Blank offset E_{1} (mm) | 3 | 3.5 |

Radial distance S_{r}_{1} (mm) | 197.762 | 209.634 |

Machine root angle γ_{m1} (°) | 7.7936 | 10.05 |

Cradle angle q_{1} (°) | 71.6258 | 67.3267 |

Tilt Angle i (°) | 2.7343 | 3 |

Swivel angle j (°) | 286.2802 | 227.2839 |

Velocity ratio i_{1} | 4.6799 | 5.2574 |

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

Cutter point radius r_{02} (mm) | 229.975(concave)/227.225(convex) |

Pressure angle α_{02} (°) | −19(concave)/21(convex) |

Root fillet radius ρ_{02} (mm) | 1.6 |

Machine center to back X_{2} (mm) | 0 |

Sliding base X_{B2} (mm) | 0 |

Blank offset E_{2} (mm) | 0 |

Radial distance S_{r}_{2} (mm) | 200.091 |

Machine root angle γ_{m2} (°) | 76.1944 |

Cradle angle q_{2} (°) | 69.3682 |

Velocity ratio i_{2} | 1.0212 |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

λ (mm) | 29.2 | 29.2 | 29.2 | 29.2 |

δ (mm) | 4.43 | 8.46 | 12.48 | 16.51 |

χ (mm) | 1.78 | 4.45 | 6.23 | 8.02 |

Name | Equation | Name | Equation |
---|---|---|---|

Average (A) | $\overline{x}=\frac{1}{N}{\displaystyle \sum _{n=1}^{N}}x(n)$ | Crest (C) | $C=\frac{{x}_{\mathrm{p}}}{{x}_{\mathrm{rms}}}$ |

Standard deviation (STD) | ${\sigma}_{x}=\sqrt{\frac{1}{N-1}{\displaystyle \sum _{n=1}^{N}}{[x(n)-\overline{x}]}^{2}}$ | Impulse (I) | $I=\frac{{x}_{\mathrm{p}}}{\overline{x}}$ |

Square mean root (SMR) | ${x}_{\mathrm{r}}={\left(\frac{1}{N}{\displaystyle \sum _{n=1}^{N}\sqrt{\left|x\left(n\right)\right|}}\right)}^{2}$ | Clearance (L) | $L=\frac{{x}_{\mathrm{p}}}{{x}_{\mathrm{r}}}$ |

Root mean square (RMS) | ${x}_{\mathrm{rms}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{n=1}^{N}}{x}^{2}(n)}$ | Peak-to-peak (PP) | $\mathrm{PP}=\mathrm{max}\left(x(n)\right)-\mathrm{m}\mathrm{in}\left(x(n)\right)$ |

Peak (P) | ${x}_{\mathrm{p}}=\mathrm{max}|x(n)|$ | Skewness (S) | $S=\frac{{\displaystyle \sum _{\mathrm{n}=1}^{N}}{[x(n)-\overline{x}]}^{3}}{(N-1){\sigma}_{x}^{3}}$ |

Waveform (W) | $W=\frac{{x}_{\mathrm{rms}}}{\overline{x}}$ | Kurtosis (K) | $K=\frac{{\displaystyle \sum _{\mathrm{n}=1}^{N}{[x(n)-\overline{x}]}^{4}}}{(N-1){\sigma}_{x}^{4}}$ |

Name | Equation | Name | Equation |
---|---|---|---|

F_{12} | ${F}_{12}=\frac{1}{K}{\displaystyle \sum _{k=1}^{K}}s(k)$ | F_{18} | ${F}_{18}=\sqrt{{\displaystyle \sum _{k=1}^{K}{f}_{k}^{2}s(k)}/{\displaystyle \sum _{k=1}^{K}s(k)}}$ |

F_{13} | ${F}_{13}=\sqrt{\frac{1}{K-1}{\displaystyle \sum _{k=1}^{K}}{\left[s(k)-{F}_{12}\right]}^{2}}$ | F_{19} | ${F}_{19}=\sqrt{{\displaystyle \sum _{k=1}^{K}{f}_{k}^{4}s(k)}/{\displaystyle \sum _{k=1}^{K}{f}_{k}^{2}s(k)}}$ |

F_{14} | ${F}_{14}={\displaystyle \sum _{k=1}^{K}{\left[s(k)-{F}_{12}\right]}^{3}}/(K-1){F}_{13}^{3}$ | F_{20} | ${F}_{20}={\displaystyle \sum _{k=1}^{K}{f}_{k}^{2}}s(k)/\sqrt{{\displaystyle \sum _{k=1}^{K}s}(k){\displaystyle \sum _{k=1}^{K}{f}_{k}^{4}}s(k)}$ |

F_{15} | ${F}_{15}={\displaystyle \sum _{k=1}^{K}{\left[s(k)-{F}_{12}\right]}^{4}}/(K-1){F}_{13}^{4}$ | F_{21} | ${F}_{21}=\frac{{F}_{17}}{{F}_{16}}$ |

F_{16} | ${F}_{16}={\displaystyle \sum _{k=1}^{K}{f}_{k}}s(k)/{\displaystyle \sum _{k=1}^{K}s}(k)$ | F_{22} | ${F}_{22}={\displaystyle \sum _{k=1}^{K}{\left({f}_{k}-{F}_{16}\right)}^{3}s(k)}/(K-1){F}_{17}^{3}$ |

F_{17} | ${F}_{17}=\sqrt{\frac{1}{K-1}{\displaystyle \sum _{k=1}^{K}}{\left({f}_{k}-{F}_{16}\right)}^{2}s(k)}$ | F_{23} | ${F}_{23}={\displaystyle \sum _{k=1}^{K}{\left({f}_{k}-{F}_{16}\right)}^{4}s(k)}/(K-1){F}_{17}^{4}$ |

ΔE = −0.1 mm | ΔE = −0.05 mm | ΔE = 0 mm | ΔE = 0.05 mm | ΔE = 0.1 mm | |
---|---|---|---|---|---|

λ (mm) | 27.2 | 28.0 | 29.2 | 30.4 | 30.8 |

ΔE = −0.1 mm | ΔE = −0.05 mm | ΔE = 0 mm | ΔE = 0.05 mm | ΔE = 0.1 mm | |
---|---|---|---|---|---|

Relative Error Max/Min (%) | |||||

Case 1 | 0.02/1.31 | 0.03/1.20 | 0.03/1.02 | 0.07/0.95 | 0.09/1.34 |

Case 2 | 0.14/2.13 | 0.18/2.06 | 0.35/2.06 | 0.40/2.22 | 1.23/1.63 |

Case 3 | 0.30/3.46 | 0.31/3.50 | 0.75/3.75 | 0.71/4.14 | 1.79/3.60 |

Case 4 | 0.60/5.23 | 0.65/5.34 | 1.25/5.89 | 1.29/6.52 | 2.53/5.97 |

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## Share and Cite

**MDPI and ACS Style**

Han, H.; Ma, H.; Wang, H.; Zhu, J.; Li, Z.; Liu, Z.
Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors. *Machines* **2022**, *10*, 929.
https://doi.org/10.3390/machines10100929

**AMA Style**

Han H, Ma H, Wang H, Zhu J, Li Z, Liu Z.
Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors. *Machines*. 2022; 10(10):929.
https://doi.org/10.3390/machines10100929

**Chicago/Turabian Style**

Han, Hongzheng, Hui Ma, Haixu Wang, Jiazan Zhu, Zhanwei Li, and Zimeng Liu.
2022. "Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors" *Machines* 10, no. 10: 929.
https://doi.org/10.3390/machines10100929