Optimal Design of a Two-Stage Helical Gearbox Using NSGA-II and MAIRCA: A Trade-Off between Performance and Axial Length

Authors

  • Van Thanh Dinh East Asia University of Technology, Trinh Van Bo Street, Hanoi City 12000, Vietnam
  • Duc Binh Vu Viet Tri University of Industry, 09 Tien Son Street, Viet Tri City 35100, Vietnam
  • Ngoc Pi Vu Thai Nguyen University of Technology, 3/2 Street, Tich Luong ward, Thai Nguyen City 251750, Vietnam
  • Manh Cuong Nguyen Thai Nguyen University of Technology, 3/2 Street, Tich Luong ward, Thai Nguyen City 251750, Vietnam
Volume: 15 | Issue: 4 | Pages: 26100-26106 | August 2025 | https://doi.org/10.48084/etasr.11758

Received: 27 April 2025 | Revised: 21 June 2025 | Accepted: 28 June 2025 | Online: 2 August 2025


Corresponding author: Manh Cuong Nguyen

Abstract

This study presents a Multi-Objective Optimization (MOO) approach for the design of a two-stage helical gearbox, aiming to simultaneously maximize the transmission efficiency and minimize the overall axial length. The optimization framework integrates the Non-dominated Sorting Genetic Algorithm II (NSGA-II) to generate a diverse Pareto front of optimal solutions, and the Multi-Attribute Ideal-Real Comparative Analysis (MAIRCA) method to support decision-making by ranking the Pareto-optimal alternatives. A physics-based mathematical model of gearbox geometry, load capacity, and efficiency was established to evaluate the performance of each candidate design. The design variables include the gear ratio distribution between the stages and the face width coefficients, subject to practical constraints, such as the gear strength and contact ratios. The results show a clear trade-off between performance and compactness, and demonstrate the effectiveness of combining NSGA-II with MAIRCA for identifying optimal gearbox designs under conflicting objectives. This approach offers valuable insights for gearbox designers aiming to balance performance and spatial constraints in modern mechanical systems. This is the first study to apply a combined NSGA-II and MAIRCA technique for the bi-objective optimization of a two-stage helical gearbox with the goals of maximizing efficiency and minimizing the length. The proposed method successfully determines the optimal design parameters across a range of transmission ratios, thereby facilitating the early-stage selection of suitable gear ratios in preliminary gearbox design.

Keywords:

two-stage helical gearbox, multi-objective optimization, NSGA-II, MAIRCA, gear ratio, face width coefficient, efficiency, gearbox length, MCDM

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References

K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, vol. 6, no. 2, pp. 182–197, Apr. 2002. DOI: https://doi.org/10.1109/4235.996017

H. Wang and H. P. Wang, "Optimal engineering design of spur gear sets," Mechanism and Machine Theory, vol. 29, no. 7, pp. 1071–1080, Oct. 1994. DOI: https://doi.org/10.1016/0094-114X(94)90074-4

C. Gologlu and M. Zeyveli, "A genetic approach to automate preliminary design of gear drives," Computers & Industrial Engineering, vol. 57, no. 3, pp. 1043–1051, Oct. 2009. DOI: https://doi.org/10.1016/j.cie.2009.04.006

V. Savsani, R. V. Rao, and D. P. Vakharia, "Optimal weight design of a gear train using particle swarm optimization and simulated annealing algorithms," Mechanism and Machine Theory, vol. 45, no. 3, pp. 531–541, Mar. 2010. DOI: https://doi.org/10.1016/j.mechmachtheory.2009.10.010

M. Patil, P. Ramkumar, and K. Shankar, "Multi-Objective Optimization of Spur Gearbox with Inclusion of Tribological Aspects," Journal of Friction and Wear, vol. 38, no. 6, pp. 430–436, Nov. 2017. DOI: https://doi.org/10.3103/S1068366617060101

J. Alam and S. Panda, "A comprehensive study on multi-objective design optimization of spur gear," Mechanics Based Design of Structures and Machines, vol. 51, no. 9, pp. 5272–5298, Sep. 2023. DOI: https://doi.org/10.1080/15397734.2021.1996246

A. Kumar, P. Ramkumar, and K. Shankar, "Multi-objective 3-stage wind turbine gearbox (WTG) with tribological constraint," Mechanics Based Design of Structures and Machines, vol. 52, no. 8, pp. 5263–5289, Aug. 2024. DOI: https://doi.org/10.1080/15397734.2023.2249987

C. A. Coello Coello, D. A. Van Veldhuizen, and G. B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems. Boston, MA: Springer US, 2007.

R. C. Sanghvi, A. S. Vashi, H. P. Patolia, and R. G. Jivani, "Multi-Objective Optimization of Two-Stage Helical Gear Train Using NSGA-II," Journal of Optimization, vol. 2014, pp. 1–8, 2014. DOI: https://doi.org/10.1155/2014/670297

M. Hu, J. Zhu, L. Gong, Z. Lu, and H. Liu, "Multi-Objective Optimization of an Aeroengine Accessory Gearbox Transmission Based on a Heuristic Algorithm," Journal of Aerospace Engineering, vol. 38, no. 2, Mar. 202. DOI: https://doi.org/10.1061/JAEEEZ.ASENG-5155

K. Daoudi, E. M. Boudi, and M. Abdellah, "Genetic Approach for Multiobjective Optimization of Epicyclical Gear Train," Mathematical Problems in Engineering, vol. 2019, no. 1, Jan. 2019, Art. no. 9324903, https://doi.org/10.1155/2019/9324903. DOI: https://doi.org/10.1155/2019/9324903

Q. Zhang, S. Xu, Y. Yuan, X. Yan, D. Huang, and J. Yu, "Multi-objective Optimization Design and Dynamic Characteristic Analysis based on Planetary Gear Transmission," Journal of Network Intelligence, vol. 8, no. 3, pp. 639–657, 2023.

M. Patil, P. Ramkumar, and S. Krishnapillai, "Multi-Objective Optimization of Two Stage Spur Gearbox Using NSGA-II," in SAE Technical Paper Series, 400 Commonwealth Drive, Warrendale, PA, United States, Jul. 2017, vol. 1, Art. no. 1939. DOI: https://doi.org/10.4271/2017-28-1939

L. Qi, J. Zhou, and H. Xu, "Multi-objective optimization of gearbox based on panel acoustic participation and response surface methodology," Journal of Low Frequency Noise, Vibration and Active Control, vol. 41, no. 3, pp. 1108–1130, Sep. 2022. DOI: https://doi.org/10.1177/14613484221091075

E. S. Maputi and R. Arora, "Multi-objective optimization of a 2-stage spur gearbox using NSGA-II and decision-making methods," Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 42, no. 9, pp. 1–22, Sep. 2020. DOI: https://doi.org/10.1007/s40430-020-02557-2

Y. Lei, L. Hou, Y. Fu, J. Hu, and W. Chen, "Research on vibration and noise reduction of electric bus gearbox based on multi-objective optimization," Applied Acoustics, vol. 158, Jan. 2020, Art. no. 107037. DOI: https://doi.org/10.1016/j.apacoust.2019.107037

K. Deb and S. Jain, "Multi-Speed Gearbox Design Using Multi-Objective Evolutionary Algorithms," Journal of Mechanical Design, vol. 125, no. 3, pp. 609–619, Sep. 2003. DOI: https://doi.org/10.1115/1.1596242

R. Ananthapadmanabhan, S. A. Babu, K. Hareendranath, C. Krishnamohan, S. Krishnapillai, and K. A, "Investigation on Multiple Algorithms for Multi-Objective Optimization of Gear Box," IOP Conference Series: Materials Science and Engineering, vol. 149, Sep. 2016, Art. no. 012049. DOI: https://doi.org/10.1088/1757-899X/149/1/012049

Q. Yao, "Multi-objective optimization design of spur gear based on NSGA-II and decision making," Advances in Mechanical Engineering, vol. 11, no. 3, Mar. 2019. DOI: https://doi.org/10.1177/1687814018824936

D. Miler, D. Žeželj, A. Lončar, and K. Vučković, "Multi-objective spur gear pair optimization focused on volume and efficiency," Mechanism and Machine Theory, vol. 125, pp. 185–195, Jul. 2018. DOI: https://doi.org/10.1016/j.mechmachtheory.2018.03.012

W. Xin, Y. Zhang, Y. Fu, W. Yang, and H. Zheng, "A multi-objective optimization design approach of large mining planetary gear reducer," Scientific Reports, vol. 13, no. 1, Oct. 2023, Art. no. 18640. DOI: https://doi.org/10.1038/s41598-023-45745-5

P. X. Yi, L. J. Dong, and Y. X. Chen, "The Multi-Objective Optimization of the Planet Carrier in Wind Turbine Gearbox," Applied Mechanics and Materials, vol. 184–185, pp. 565–569, Jun. 2012. DOI: https://doi.org/10.4028/www.scientific.net/AMM.184-185.565

J. H. Kim, B. S. Kim, D. Im, J. H. Park, D. J. Moon, and Y. J. Park, "Optimization of high-speed angular contact ball bearing for aircraft gearbox utilizing an evolutionary multi-objective algorithm, NSGA-III," Advances in Mechanical Engineering, vol. 16, no. 9, Sep. 2024. DOI: https://doi.org/10.1177/16878132241272197

M. Gobbi, G. Mastinu, and M. Caudano, "Stochastic Multi-Objective Optimisation of a Gearbox Synchroniser and Selector Mechanism," in Design Engineering, Volumes 1 and 2, Washington, DC, USA, Jan. 2003, pp. 113–124. DOI: https://doi.org/10.1115/IMECE2003-43005

L. D. Bao, V. D. Binh, D. V. Thanh, K. M. Nguyen, and L. X. Hung, "Multi-Objective Optimization of a Two-stage Helical Gearbox with Double Gears in the First Stage using MARCOS," Engineering, Technology & Applied Science Research, vol. 14, no. 6, pp. 18245–18251, Dec. 2024. DOI: https://doi.org/10.48084/etasr.8865

T. Chat and L. Van Uyen, Design and Calculation of Mechanical Transmissions Systems, vol. 1. Hanoi, Vietnam: Educational Republishing House, 2007.

D. Jelaska, Gears and Gear Drives, 1st ed., New Tork, NY: Wiley, 2012. DOI: https://doi.org/10.1002/9781118392393

X. H. Le and N. P. Vu, "Multi-Objective Optimization of a Two-Stage Helical Gearbox Using Taguchi Method and Grey Relational Analysis," Applied Sciences, vol. 13, no. 13, Jun. 2023, Art. no. 7601. DOI: https://doi.org/10.3390/app13137601

K. Sri, S. Hartati, A. Harjoko, and R. Wardoyo, Fuzzy Multi-Attribute Decision Making (Fuzzy MADM), Yogyakarta, Indonesia: Graha Ilmu, 2006.

D. Van Thanh, B. T. Danh, V. Le The Anh, T. H. Danh, H. X. Tu, and N. M. Cuong, "Multi-objective Optimization of a Two-Stage Helical Gearbox to Increase Gearbox Efficiency and Decrease Gearbox Length," in Lecture Notes in Networks and Systems, Cham: Springer Nature Switzerland, 2024, pp. 195–211. DOI: https://doi.org/10.1007/978-3-031-62235-9_20

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[1]
V. T. Dinh, D. B. Vu, N. P. Vu, and M. C. Nguyen, “Optimal Design of a Two-Stage Helical Gearbox Using NSGA-II and MAIRCA: A Trade-Off between Performance and Axial Length”, Eng. Technol. Appl. Sci. Res., vol. 15, no. 4, pp. 26100–26106, Aug. 2025.

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