# Dynamics Modeling and Load-Sharing Performance Optimization of Concentric Face Gear Split-Torque Transmission Systems

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Neural Network Surrogate Models for Time-Varying Mesh Stiffnesses

#### 2.2. Two Types of Meshing States

_{1}(t) is the meshing stiffness for regular meshing, k

_{2}(t) is the meshing stiffness for tooth-back meshing, and f

_{δ}is the relative displacement of the gear pair.

_{in}

_{1-L}= k(t). When the number of pinion teeth is even, no phase difference exists among the meshing pairs driven by the pinion. When the number of pinion teeth is odd, the phase difference between the rest of the meshing pairs and the Input Gear 1–lower face gear meshing pair is determined with the following method:

_{M-L}and k

_{in}

_{1-L}is 0. In contrast, the phase difference between the meshing stiffness k

_{M-U}and k

_{in}

_{1-L}is T/2, where T is the meshing period.

_{M-L}and k

_{in}

_{1-L}is T/2. At this time, the phase difference between k

_{M-U}and k

_{in}

_{1-L}is 0.

#### 2.3. Nonlinear Dynamic Model

_{3}. The z-axis of the upper face gear is in the same direction as the z-axis of the lower face gear.

_{1}is the pinion distribution angle; r

_{1}and r

_{f}are the distances from the meshing point to the pinion and face gear axes, respectively; and e

_{16}is the normal static transmission error of the gear pair, which is obtained from Equation (3).

_{0}is the constant value of the comprehensive transmission error; e

_{r}is the amplitude; ω

_{h}is the meshing angular frequency of the gear pair; and φ

_{r}is the initial phase.

_{6}and y

_{6}, expressed as:

_{17}of the meshing point between Input Gear 1 and the upper face gear in the meshing line direction is expressed as:

_{26}and δ

_{27}of Input Gear 2 with the lower and upper face gears in the direction of the meshing line, respectively, are:

_{36}and δ

_{37}of Idler 1 with the lower and upper face gears in the direction of the meshing line, respectively, are:

_{46}and δ

_{47}of Idler 2 with the lower and upper face gears in the direction of the meshing line, respectively, are:

_{56}and δ

_{57}of the tail gear with the lower and upper face gears in the direction of the meshing line, respectively, are:

_{m}is the meshing damping.

_{m}is the meshing damping, which is expressed as:

_{av}is the mean mesh stiffness; and m

_{eq}

_{,i}is the equivalent mass of the gear.

#### 2.4. Optimization Model Based on MPA

_{i}is the elite matrix; Prey

_{i}is the prey matrix; ⊗ is the term-by-term multiplication operator; P is equal to 0.5; and R is the rand() function.

_{L}is a random number with a Lévy distribution; $CF={(1-\mathrm{Iter}/\mathrm{MaxIter})}^{\left(2\cdot \mathrm{Iter}/\mathrm{MaxIter}\right)}$; and n is the number of prey.

_{1}and r

_{2}are the index subscripts of the prey.

## 3. Verification

## 4. Numerical Results and Discussion

_{m}/300 and 200T

_{m}, respectively. The dynamic parameters and support stiffnesses used in this study are shown in Table 2 and Table 3, respectively.

#### 4.1. Influence of Support Stiffness on Load-Sharing Coefficient under Different Loads

_{in}for a single input gear varied between 500 and 1500 Nm, and the support stiffness factor ζ

_{i}(i = in1, in2, id1, id2, t, l, u) for each gear ranged from 0.1 to 10. While analyzing the effect of the support stiffness of a single gear on the load-sharing factor for a given load, the support stiffness of the remaining gears was fixed.

_{in}

_{1}in the upper left region is the largest, which indicates that increasing the support stiffness and decreasing the load increase κ

_{in}

_{1}. Similar to prior results [38], the support stiffness of the input gear has a stronger effect on the load-sharing behavior of the input gear. However, the distribution of the load-sharing coefficient in Figure 14b is different from that in Figure 14a. As the support stiffness changes from 0.1 to 10, the load-sharing coefficient κ

_{in}

_{2}shows an overall decreasing trend. In addition, Figure 14c shows that the smaller support stiffness of Input Gear 1 is more favorable for load equalization between the idlers.

_{in}

_{1}in the lower left corner is the largest and then gradually decreases with increasing support stiffness and load. The heatmap distribution of the load-sharing coefficient in Figure 15b is similar to that in Figure 14a, and the influence of the support stiffness of Input Gear 2 on κ

_{in}

_{2}is more apparent. The effect of the support stiffness of Input Gear 2 on κ

_{id}is similar to that shown in Figure 14c, indicating that κ

_{id}decreases with a higher load and increases with a larger support stiffness.

_{in}

_{1}, κ

_{in}

_{2}, and κ

_{id}. Notably, the effect of the idler support stiffness on κ

_{id}is more prominent. Therefore, a smaller support stiffness of the idlers results in a more uneven load distribution. The higher the support stiffness of the idlers, the more balanced the load of the input gear. However, excessive support stiffnesses ζ

_{id}

_{1}and ζ

_{id}

_{2}increase the load-sharing coefficient κ

_{id}. Therefore, from the perspective of load equalization, choosing the support stiffness at the idlers needs further analysis. Additionally, the influence trend of the support stiffness of the idler gear on the load-sharing coefficient of the input gear is similar to that previously reported [37,38].

_{in}

_{1}, κ

_{in}

_{2}, and κ

_{id}. By increasing the support stiffness of the tail gear at a specific input load, the load-sharing coefficients κ

_{in}

_{1}, κ

_{in}

_{2}, and κ

_{id}decrease to varying degrees. Consequently, a higher support stiffness of the tail gear is conducive to lowering the load-sharing factor and balancing the load carrying of each meshing pair.

_{u}increases the load-sharing coefficient of Input Gear 1, thus aggravating the uneven load distribution at Input Gear 1. Nevertheless, the increase in ζ

_{u}benefits the load balance of the idlers. In summary, the change in the support stiffness of the upper face gear, to a certain extent, can adjust the balance load condition of the system, but it must be matched with the support stiffness of other gears.

#### 4.2. Optimization Analysis

## 5. Conclusions

_{in}

_{1}, κ

_{in}

_{2}, and κ

_{id}and need to be comprehensively considered when adjusting the load-sharing performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- White, G. Helicopter transmission arrangements with split-torque gear trains. In NASA. Lewis Research Center Advanced Power Transmission; Transmission Research, Inc.: Cleveland, OH, USA, 1983. [Google Scholar]
- Kish, J.G. Sikorsky Aircraft Advanced Rotorcraft Transmission (ART) Program-Final Report; NASA CR-191079; NASA Lewis Research Center: Washington, DC, USA, 1993.
- Krantz, T.L. Dynamics of a Split Torque Helicopter Transmission. Master’s Thesis, Cleveland State University, Cleveland, OH, USA, 1994. [Google Scholar]
- Krantz, T.L.; Rashidi, M.; Kish, J.G. Split torque transmission load sharing. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**1992**, 208, 137–148. [Google Scholar] [CrossRef][Green Version] - Heath, G.F.; Bossler, R.B., Jr. Advanced Rotorcraft Transmission (ART) Program-Final Report; National Aeronautics and Space Administration: Washington, DC, USA, 1993; pp. 1–206.
- Lewicki, D.G.; Handschuh, R.F.; Heath, G.F. Evaluation of Carburized and ground Face Gears. J. Am. Helicopter Soc.
**2000**, 45, 118–124. [Google Scholar] [CrossRef][Green Version] - Lewicki, D.G.; Heath, G.F.; Filler, R.R.; Slaughter, S.C.; Fetty, J. RDS-21 face-gear surface durability tests. In Proceedings of the American Helicopter Society 63rd Annual Forum, Virginia Beach, VA, USA, 1–3 May 2007. [Google Scholar]
- Lewicki, D.G.; Dempsey, P.J.; Heath, G.F. Gear Fault Detection Effectiveness as Applied to Tooth Surface Pitting Fatigue Damage. In Proceedings of the American Helicopter Society 65th Annual Forum, Grapevine, TX, USA, 27–29 May 2009. [Google Scholar]
- Heath, G.F.; Slaughter, S.C.; Fisher, D.J. Helical face gear development under the enhanced rotorcraft drive system program. In Proceedings of the 67th Annual Forum and Technology Display (Forum 67), Virginia Beach, VA, USA, 3–5 May 2011. (AHS 2011-000270). [Google Scholar]
- Kish, J. Comanche drive system. In Proceedings of the American Helicopter Society Rotary Wing Propulsion Specialists Meeting, Williamsburg, VA, USA, 25–28 October 1993. [Google Scholar]
- Heath, G.F.; Slaughter, S.C.; Morris, M.T. Face gear development under the rotorcraft drive system for the 21st century program. In Proceedings of the 65th Annual Forum Proceedings-AHS International, Grapevine, TX, USA, 27–29 May 2009; Volume 2, pp. 1011–1030. [Google Scholar]
- Mo, S.; Yue, Z.X.; Feng, Z.Y. Analytical investigation on load sharing characteristics for face gear split flow system. J. Huazhong Univ. Sci. Technol. Nat. Sci. Ed.
**2020**, 48, 23–28. (In Chinese) [Google Scholar] [CrossRef] - Zhao, N.; Wang, R.F.; Tao, L. Load Sharing of Parallel Shaft Split Torque Transmission System. Adv. Mater. Res.
**2012**, 490–495, 2231–2235. [Google Scholar] [CrossRef] - Dong, H.; Duan, L.L.; Zhang, J.A. Load-Sharing Characteristics of Power-Split Transmission System Based on Deformation Compatibility and Loaded Tooth Contact Analysis. Int. J. Aerosp. Eng.
**2015**, 2015, 305808. [Google Scholar] [CrossRef][Green Version] - Fu, C.X.; Zhao, N.; Zhao, Y.Z. Load Sharing Multiobjective Optimization Design of a Split Torque Helicopter Transmission. Math. Probl. Eng. Theory Methods Appl.
**2015**, 2015 Pt 20, 381010. [Google Scholar] [CrossRef] - Dong, H.; Liu, Z.Y.; Zhao, X.L.; Hu, Y.H. Research on static load sharing characteristics of power split two-stage five-branching star gearing drive system. J. Vibroeng.
**2019**, 21, 11–27. [Google Scholar] [CrossRef][Green Version] - Hu, Z.; Tang, J.; Wang, Q. Investigation of nonlinear dynamics and load sharing characteristics of a two-path split torque transmission system. Mech. Mach. Theory
**2020**, 152, 103955. [Google Scholar] [CrossRef] - Liu, X.; Fang, Z.D.; Jia, H.; Zhao, N.; Sheng, Y. Investigation of Load Sharing and Dynamic Load Characteristics of a Split Torque Transmission System with Double-Helical Gear Modification. Shock. Vib.
**2021**, 2021, 9912148. [Google Scholar] [CrossRef] - Jin, G.H.; Yang, H.Y.; Zhu, R.P. Tooth surface friction and its influence on dynamic transmission error of double power input transmission system. J. Vibroeng.
**2018**, 20, 1937–1954. [Google Scholar] [CrossRef][Green Version] - Jin, G.H.; Xiong, Y.P.; Gui, Y.F.; Zhu, R.P. Sensitive Parameter and Its Influence Law on Load Sharing Performance of Double Input Split Torque Transmission System. J. Vib. Eng. Technol.
**2017**, 5, 583–595. [Google Scholar] - Lin, T.J.; Ran, X.T. Nonlinear vibration characteristic analysis of a face-gear drive. J. Vib. Shock.
**2012**, 31, 25–31. [Google Scholar] - Litvin, F.L.; Zhang, Y.; Wang, J.; Bossler, R.B.; Chen, Y.D. Design and Geometry of Face-Gear Drives. J. Mech. Des.
**1992**, 114, 642–647. [Google Scholar] [CrossRef] - Litvin, F.L.; Egelja, A.; Tan, J.; Heath, G. Computerized design, generation and simulation of meshing of orthogonal offset face-gear drive with a spur involute pinion with localized bearing contact. Mech. Mach. Theory
**1998**, 33, 87–102. [Google Scholar] [CrossRef] - Litivin, F.L.; Alfonso, F.; Laudio, Z.Z. Design, generation and TCA of new type of asymmetric face-gear drive with modified geometry. Comput. Methods Appl. Mech. Eng.
**2001**, 190, 5837–5865. [Google Scholar] [CrossRef] - Litvin, F.L.; Nava, A.; Fan, Q.; Fuentes, A. New geometry of worm face gear drives with conical and cylindrical worms: Generation, simulation of meshing, and stress analysis. Comput. Methods Appl. Mech. Eng.
**2002**, 191, 3035. [Google Scholar] [CrossRef] - Handschuh, R.F.; Lewicki, D.G.; Heath, G.F. Experimental Evaluation of Face Gears for Aerospace Drive System Applications; National Aeronautics and Space Administration Cleveland Oh Lewis Research Center: Cleveland, OH, USA, 1996.
- Handschuh, R.F.; Lewicki, D.G.; Bossler, R.B. Experimental testing of prototype face gears for helicopter transmissions. Proc. Inst. Mech. Eng.
**1994**, 208, 129. [Google Scholar] [CrossRef][Green Version] - Pias, R.; Turro, S. Face-Gear Transmission Assembly with Floating Balance Pinions. U.S. Patent 5,974,911, 2 November 1999. [Google Scholar]
- Jin, G.H.; Ren, W.; Zhu, R.P. Influence of backlash on load sharing and dynamic load characteristics of twice split torque transmission system. J. Vib. Eng. Technol.
**2019**, 7, 565–577. [Google Scholar] [CrossRef] - Mo, S.; Yue, Z.; Feng, Z.; Shi, L.; Zou, Z.; Dang, H. Analytical investigation on load-sharing characteristics for multi-power face gear split flow system. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2020**, 234, 676–692. [Google Scholar] [CrossRef] - Mo, S.; Song, Y.; Feng, Z. Research on dynamic load sharing characteristics of double input face gear split-parallel transmission system. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2022**, 236, 2185–2202. [Google Scholar] [CrossRef] - Dong, J.; Hu, Z.; Tang, J. Investigation of the Vibration Features and Dynamic Load Sharing Characteristics of Concentric Face Gear Torque Split Transmission. J. Comput. Nonlinear Dyn.
**2021**, 16, 071003. [Google Scholar] [CrossRef] - Zhao, N.; Li, W.; Hu, T. Quasistatic load sharing behaviours of concentric torque-split face gear transmission with flexible face gear. Math. Probl. Eng.
**2018**, 2018, 6568519. [Google Scholar] [CrossRef][Green Version] - Dong, J.; Wang, Q.; Tang, J. Dynamic characteristics and load-sharing performance of concentric face gear split-torque transmission systems with time-varying mesh stiffness, flexible supports and deformable shafts. Meccanica
**2021**, 56, 2893–2918. [Google Scholar] [CrossRef] - Gong, F.; Zhu, R.P.; Li, P.J. Analysis of Nonlinear Vibration Characteristics of the Concentric Face-Gear Split-Torque Transmission System. Math. Probl. Eng.
**2022**, 2022, 1977367. [Google Scholar] [CrossRef] - Faramarzi, A.; Heidarinejad, M.; Mirjalili, S. Marine Predators Algorithm: A nature-inspired metaheuristic. Expert Syst. Appl.
**2020**, 152, 113377. [Google Scholar] [CrossRef] - Li, W.; Zhao, N.; Lin, Y.H.; Qiu, P.Y. Study on Dynamic Load Sharing Characteristics of the Concentric Torque Split Face Gear Transmission. J. Xi’an Jiaotong Univ.
**2020**, 54, 1–9. (In Chinese) [Google Scholar] - Gong, F.; Zhu, R.; Li, P. Analysis of Load-Sharing and Contact Characteristics of the Concentric Face Gear Split-Torque Transmission System with Elastic Supports. Appl. Sci.
**2022**, 12, 4894. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of concentric face gear split-torque transmission system: (

**a**) 3D schematic diagram; (

**b**) 2D schematic diagram.

**Figure 4.**Mesh stiffness model of the lower face gear: (

**a**) neural network model; (

**b**) verification of generalization capability.

**Figure 5.**Mesh stiffness model of upper face gear: (

**a**) neural network model; (

**b**) verification of generalization capability.

**Figure 6.**Difference in meshing stiffness between fixed and varying load: (

**a**) lower face gear; (

**b**) upper face gear.

**Figure 7.**Difference in meshing force between fixed and varying load: (

**a**) lower face gear; (

**b**) upper face gear.

**Figure 8.**Difference in meshing stiffness between face gear driving and pinion driving: (

**a**) direction of meshing; (

**b**) difference in meshing stiffness.

**Figure 13.**Comparison of meshing force between proposed model and finite element model: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idlers; (

**d**) tail gear.

**Figure 14.**Influence of support stiffness of Input Gear 1 on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 15.**Influence of support stiffness of Input Gear 2 on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 16.**Influence of support stiffness of Idler 1 on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 17.**Influence of support stiffness of Idler 2 on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 18.**Influence of support stiffness of tail gear on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 19.**Influence of support stiffness of lower face gear on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 20.**Influence of support stiffness of upper face gear on load-sharing coefficient: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear.

**Figure 22.**Dynamic meshing force curve before and after optimization: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idler gear–upper face gear; (

**d**) idler gear–lower face gear.

**Figure 23.**Meshing force of each gear pair by finite element method: (

**a**) Input Gear 1; (

**b**) Input Gear 2; (

**c**) idlers.

Parameter | Pinion | Face Gear |
---|---|---|

Number of teeth | 21 | 142 |

Normal modulus (mm) | 3.9 | 3.9 |

Pressure angle (deg) | 25 | 25 |

Tooth width (mm) | 51 | 49 |

k_{x} (N/mm) | k_{y} (N/mm) | k_{z} (N/mm) | k_{θx} (Nmm/rad) | k_{θy} (Nmm/rad) | |
---|---|---|---|---|---|

Input gear | 4.0 × 10^{5} | 4.0 × 10^{5} | / | / | / |

Idler gear | 9.4 × 10^{5} | 9.4 × 10^{5} | / | / | / |

Tail gear | 7.7 × 10^{5} | 7.7 × 10^{5} | / | / | / |

Upper face gear | 3.2 × 10^{6} | 3.2 × 10^{6} | 1.8 × 10^{6} | 4.8 × 10^{10} | 4.8 × 10^{10} |

Lower face gear | 1.5 × 10^{7} | 1.5 × 10^{7} | 1.26 × 10^{7} | 3.2 × 10^{11} | 3.2 × 10^{11} |

System Parameter | Symbol/Unit | Value |
---|---|---|

Mass of pinion | m_{p}/kg | 1.95 |

Mass of face gear | m_{6}/kg | 19.95 |

m_{7}/kg | 48.80 | |

Inertia moment of pinion | J_{p}/kg$\xb7$m^{2} | 2.31 × 10^{−3} |

Inertia moment of face gear | J_{6x}, J_{6y}/kg$\xb7$m^{2} | 0.84 |

J_{6z}/kg$\xb7$m^{2} | 1.49 | |

J_{7x}, J_{7y}/kg$\xb7$m^{2} | 1.52 | |

J_{7z}/kg$\xb7$m^{2} | 2.48 | |

Supporting damping of pinion | c_{px}, c_{p}_{y}/N$\xb7$s$\xb7$m^{−1} | 0.8 × 10^{4} |

Supporting damping of face gear | c_{6x}, c_{6y}, c_{6z}/N$\xb7$s$\xb7$m^{−1} | 1.2 × 10^{4} |

c_{6θx}, c_{6θy}/N$\xb7$m$\xb7$s$\xb7$rad^{−1} | 0.9 × 10^{4} | |

c_{7x}, c_{7y}, c_{7z}/N$\xb7$s$\xb7$m^{−1} | 1.2 × 10^{4} | |

c_{7θx}, c_{7θy}/N$\xb7$m$\xb7$s$\xb7$rad^{−1} | 0.9 × 10^{4} | |

Damping ratio | $\mathsf{\xi}$ | 0.05 |

Normal gear backlash | b_{h}/mm | 0.02 |

Static transmission error | e_{0}/mm | 0 |

e_{r}/mm | 0.01 |

Parameter | ζ_{in}_{1} | ζ_{in}_{2} | ζ_{id}_{1} | ζ_{id}_{2} | ζ_{t} | ζ_{l} | ζ_{u} |
---|---|---|---|---|---|---|---|

Value | 0.10 | 0.10 | 6.32 | 9.99 | 3.82 | 0.58 | 5.93 |

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## Share and Cite

**MDPI and ACS Style**

Gong, F.; Zhu, R.; Wang, Q. Dynamics Modeling and Load-Sharing Performance Optimization of Concentric Face Gear Split-Torque Transmission Systems. *Appl. Sci.* **2023**, *13*, 4352.
https://doi.org/10.3390/app13074352

**AMA Style**

Gong F, Zhu R, Wang Q. Dynamics Modeling and Load-Sharing Performance Optimization of Concentric Face Gear Split-Torque Transmission Systems. *Applied Sciences*. 2023; 13(7):4352.
https://doi.org/10.3390/app13074352

**Chicago/Turabian Style**

Gong, Fei, Rupeng Zhu, and Qibo Wang. 2023. "Dynamics Modeling and Load-Sharing Performance Optimization of Concentric Face Gear Split-Torque Transmission Systems" *Applied Sciences* 13, no. 7: 4352.
https://doi.org/10.3390/app13074352