Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors
Abstract
:1. Introduction
2. FE Model Description
2.1. Tooth Contact Analysis Considering the Assembly Error
2.2. FE Modelling of the SBGP
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
3.1.4. Gear Axial Error
3.2. Crack Fault Analysis of the SBGP
3.2.1. TVMS Due to the Crack Fault
3.2.2. Dynamic Simulation
3.2.3. Response Analysis Due to the Crack Fault of the SBGP
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 ΔAp should be strictly controlled. A smaller ΔE, ΔΣ, and a larger ΔAp 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
References
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Parameter | Bevel Pinion | Bevel Gear |
---|---|---|
Tooth number z1/z2 | 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 δf (°) | 10.9321 | 76.1944 |
Face angle δa (°) | 13.8056 | 79.0679 |
Addendum height ha (mm) | 6.8480 | 2.6720 |
Dedendum height hf (mm) | 3.7250 | 7.9010 |
Parameter | Concave | Convex |
---|---|---|
Cutter point radius r01 (mm) | 226.47 | 230.86 |
Pressure angle α01 (°) | −18.75 | 21.25 |
Root fillet radius ρ01 (mm) | 1 | 1 |
Machine center to back X1 (mm) | −5.237 | 8.545 |
Sliding base XB1 (mm) | 15.217 | 13.183 |
Blank offset E1 (mm) | 3 | 3.5 |
Radial distance Sr1 (mm) | 197.762 | 209.634 |
Machine root angle γm1 (°) | 7.7936 | 10.05 |
Cradle angle q1 (°) | 71.6258 | 67.3267 |
Tilt Angle i (°) | 2.7343 | 3 |
Swivel angle j (°) | 286.2802 | 227.2839 |
Velocity ratio i1 | 4.6799 | 5.2574 |
Parameter | Value |
---|---|
Cutter point radius r02 (mm) | 229.975(concave)/227.225(convex) |
Pressure angle α02 (°) | −19(concave)/21(convex) |
Root fillet radius ρ02 (mm) | 1.6 |
Machine center to back X2 (mm) | 0 |
Sliding base XB2 (mm) | 0 |
Blank offset E2 (mm) | 0 |
Radial distance Sr2 (mm) | 200.091 |
Machine root angle γm2 (°) | 76.1944 |
Cradle angle q2 (°) | 69.3682 |
Velocity ratio i2 | 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) | Crest (C) | ||
Standard deviation (STD) | Impulse (I) | ||
Square mean root (SMR) | Clearance (L) | ||
Root mean square (RMS) | Peak-to-peak (PP) | ||
Peak (P) | Skewness (S) | ||
Waveform (W) | Kurtosis (K) |
Name | Equation | Name | Equation |
---|---|---|---|
F12 | F18 | ||
F13 | F19 | ||
F14 | F20 | ||
F15 | F21 | ||
F16 | F22 | ||
F17 | F23 |
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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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
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 StyleHan, 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