Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash
Abstract
:1. Introduction
2. Model and Methodology
2.1. Time-Varying Gear Backlash Based on Fractal Theory
2.2. Dynamic Model of the Gear System
3. Model Validation and Adaptability
3.1. Model Experimental Verification
3.2. Universal Adaptability of Model
4. Result and Analysis
4.1. Effect of Frictional Excitation on the Gear System
4.2. The Coupling Effect of Frictional Excitation and Fractal Excitation
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Alphabets | |
Initial normal pressure angle | |
Time-varying normal pressure angle | |
Initial normal backlash of gears | |
Time-varying normal position angle | |
Gear time-varying normal backlash | |
Time-varying mesh damping | |
Support damping in the , and directions | |
The time-varying center distance | |
Initial Normal center distance of gears | |
The fractal dimension | |
Dynamic mesh force of the tooth pair | |
Dynamic friction force of the tooth pair | |
The component of meshing force on the , , and axes | |
The component of fiction force on the , , and axes | |
The characteristic-scale coefficient | |
Moment of inertia with respect to the mass center | |
Time-varying mesh stiffness | |
Support stiffness in the , and directions | |
Bearing stiffness in the , and directions | |
Shaft stiffness in the , and directions | |
l | Projection distance of theoretical meshing line |
Initial tooth width | |
Projection length of base circle tooth pitch | |
Time-varying contact surface tooth width | |
Mass of gear | |
Gear contact ratio | |
Number of pairs of instantaneously engaged teeth | |
Base pitch | |
Generalized force vector | |
Pitch radius of pinion and gears | |
Average surface roughness | |
Radius of the base circle | |
Total dissipation function | |
Moment arm of the friction force for the pair | |
Tooth tip circle radius of gears | |
Input and output working torque | |
Kinetic energy of gear system | |
System engagement cycle | |
Gear friction torque | |
Energy generated of gear system | |
The viscosity coefficient of the gear | |
The sliding speed | |
The speed of entraining motion | |
Normal force of tooth surface contact | |
Contact wire length | |
Helical angle of helical gear | |
Base helix angle | |
Transmission error along the LOA direction | |
Translational displacement along the LOA direction | |
Transmission error along the OLOA direction | |
Translational displacement along the OLOA direction | |
Torsional vibration displacement of pinion and gear | |
Coefficient of friction | |
Mesh damping ratio | |
Support damping ratio | |
Time-varying normal meshing angle | |
Off-line-of-action | |
Dynamic transmission error | |
Line-of-action | |
Electrohydrodynamic lubrication |
Appendix A
References
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Parameters | Pinion/Gear |
---|---|
30/53 | |
30 | |
20 | |
20 | |
2.5 | |
20 | |
2.5 | |
79.83/141.04 | |
86.60/153.00 | |
12.01 | |
2.65 | |
160 | |
50 | |
25/30 | |
Parameters | Pinion/Gear |
---|---|
23/47 | |
501.6/8748 | |
0.78/3.28 | |
2.5 | |
8.0 | |
20 | |
30 | |
120 | |
0.1 | |
1.0 | |
Damping coefficient of the shaft | 0.05 |
0.025 |
Driving Speed (RPM) | Support Stiffness (Nm) | |||||
---|---|---|---|---|---|---|
(1 × 106) | (1 × 106) | (6 × 106) | (6 × 106) | (1.2 × 107) | (1.2 × 107) | |
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Mo, G.; Liu, C.; Liu, G.; Liu, F. Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash. Machines 2025, 13, 262. https://doi.org/10.3390/machines13040262
Mo G, Liu C, Liu G, Liu F. Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash. Machines. 2025; 13(4):262. https://doi.org/10.3390/machines13040262
Chicago/Turabian StyleMo, Guoyin, Chengyu Liu, Guimian Liu, and Fuhao Liu. 2025. "Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash" Machines 13, no. 4: 262. https://doi.org/10.3390/machines13040262
APA StyleMo, G., Liu, C., Liu, G., & Liu, F. (2025). Improved Nonlinear Dynamic Model of Helical Gears Considering Frictional Excitation and Fractal Effects in Backlash. Machines, 13(4), 262. https://doi.org/10.3390/machines13040262