Call For Paper - Upcoming Conferences
Research Article | Open Access | Download PDF
Volume 13 | Issue 5 | Year 2026 | Article Id. IJME-V13I5P106 | DOI : https://doi.org/10.14445/23488360/IJME-V13I5P106

Geometry-Driven Vibro-Structural Optimization and Predictive Validation of a Three-Stage Helical Gearbox Casing under Harmonic Excitation

Ronak D. Gandhi, Hiral H. Parikh

Received Revised Accepted Published
12 Feb 2026 22 Mar 2026 22 Apr 2026 29 May 2026

Citation :

Ronak D. Gandhi, Hiral H. Parikh, "Geometry-Driven Vibro-Structural Optimization and Predictive Validation of a Three-Stage Helical Gearbox Casing under Harmonic Excitation," International Journal of Mechanical Engineering, vol. 13, no. 5, pp. 83-104, 2026. Crossref, https://doi.org/10.14445/23488360/IJME-V13I5P106

Abstract

The structural sustainability of multi-stage helical gearboxes is largely influenced by casing geometry, material stiffness, and dynamic excitation effects. Premature structural deterioration may occur due to resonant stress amplification and excessive deformation induced by harmonic gear–mesh excitations under severe operating conditions. A methodological framework for predictive validation and geometry-driven vibro-structural optimization of three-stage helical transmission casings under harmonic excitation is presented. Three geometric configurations - Rectangular Top And Bottom (RTRB), Curved Top And Rectangular Bottom (CTRB), and curved top and curved bottom (CTCB) with casing thicknesses of 5, 7, and 9 mm and materials of Structural Steel A36 (S.S.A36), AISI 1050 High Carbon Steel (AISI 1050 H.C.S.), and Al–Ti–SiC Metal Matrix Composites (Al–Ti–SiC M.M.C.) are studied. Through finite element-based modal and harmonic response analyses, natural frequency shifts, total deformation, and equivalent von Mises stress are examined. Geometry alteration notably changes modal stiffness and resonance separation margin. Overall, the CTRB with 9 mm S.S.A36 shows the best dynamic behavior, reducing total deformation and equivalent stress by about 97.2% and 80.3%, respectively, compared to the 5 mm RTRB under the same harmonic load. Upon validation, Taguchi L27 DOE quantifies the effect of geometry, thickness, and material properties, while regression and ANN predict responses within the design space and at intermediate thicknesses (5.5–8.5 mm). Regression exhibited lower deviation (11–16% and 12–16%) than ANN (17–24% and 15–32%). The integrated computational–statistical framework provides a systematic approach for geometry-driven optimization under harmonic excitation, improving vibration resistance and structural integrity of high-torque transmission systems.

Keywords

Geometry-driven optimization, Harmonic response, Predictive modeling, Taguchi method, Vibro-structural analysis.

References

  1. Beom-Soo Kim et al., “An Improved Gear Wear Model Considering Coupled Effects of Surface Roughness and Friction,” Social Science Research Network, pp. 1-54, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  2. Martin Hofstetter et al., “Multi-Objective System Design Synthesis for Electric Powertrain Development,” IEEE Transportation and Electrification Conference and Expo, pp. 286-292, 2018.
    [    CrossRef] [Google Scholar] [Publisher Link]
  3. Edmund S. Maputi, and Rajesh Arora, “Design Optimization of a Three Stage Transmission Using Advanced Optimization Techniques,” International Journal for Simulation and Multidisciplinary Design Optimization, vol. 10, no. 8, pp. 1-11, 2019.
    [    CrossRef] [Google Scholar] [Publisher Link]
  4. David F. Thompson, Shubhagm Gupta, and Amit Shukla, “Trade-Off Analysis in Minimum Volume Design of Multistage Spur Gear Reduction Units,” Mechanism and Machine Theory, Elsevier Science, vol. 35, no. 5, pp. 609-627, 2000.
    [    CrossRef] [Google Scholar] [Publisher Link]
  5. A. Kahraman, and R. Singh, “Interactions between Time-Varying Mesh Stiffness and Clearance Non-Linearities in a Geared System,” Journal of Sound and Vibration, vol. 146, no. 1, pp. 135-156, 1991.
    [    CrossRef] [Google Scholar] [Publisher Link]
  6. Chung Woo-Jin et al., “Improved Analytical Model for Calculating Mesh Stiffness and Transmission Error of Helical Gears Considering Trochoidal Root Profile,” Mechanism and Machine Theory, vol. 163, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  7. Yu Wang et al., “Dynamic Response Analysis of Helical Gear Systems Considering Periodic Surface Waviness Deviation,” Tribology International, vol. 215, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  8. Xiaohe Deng, “Analysis and Prediction of Gear Fatigue Life,” IOP Conference Series: Earth and Environmental Science, vol. 252, no. 2, pp. 1-5, 2019.
    [    CrossRef] [Publisher Link]
  9. Vasim Bashir Maner, M. M. Mirza, and Shrikant Pawar, “Design, Analysis and Optimization for Foot Casing of Gearbox,” Proceedings of 3rd IRF International Conference, pp. 35–38, 2014.
    [    Google Scholar] [Publisher Link]
  10. Tianfei Ma et al., “Fatigue Life Estimation of a Light Truck Gearbox Housing using Multi-Body Dynamics and Finite Element Method,” Proceedings of International Conference on Electronic & Mechanical Engineering and Information Technology, pp. 3707-3710, 2011.
    [    CrossRef] [Google Scholar] [Publisher Link]
  11. Zhengyan Zhang et al., “The Optimization Design of Triple Gearbox Assembled with Spiral-Bevel and Helical-Spur Gears,” IEEE 10th International Conference on Computer Aided Industrial Design and Conceptual Design, pp. 2078-2081, 2009.
    [    CrossRef] [Google Scholar] [Publisher Link]
  12. Shivaji V. Gawali et al., “Effect of Coefficient of Asymmetry on Strength and Contact Ratio of Asymmetric Helical Gear,” International Journal of Scientific Research in Science, Engineering and Technology, vol. 3, no. 1, pp. 144-150, 2017.
    [    Google Scholar] [Publisher Link]
  13. Yi Guo et al., “Vibro-Acoustic Propagation of Gear Dynamics in a Gear Bearing Housing System,” Journal of Sound and Vibration, vol. 333, no. 22, pp. 5762-5785, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  14. R.G. Parker, S.M. Vijayakar, and T. Imajo, “Non-Linear Dynamic Response of a Spur Gear Pair: Modelling and Experimental Comparison,” Journal of Sound and Vibration, vol. 237, no. 3, pp. 435-455, 2000.
    [    CrossRef] [Google Scholar] [Publisher Link]
  15. Takanori Ide et al., “Structural Optimization Methods and Techniques to Design Light and Efficient Automatic Transmission of Vehicles with Low Radiated Noise,” Structural and Multidisciplinary Optimization, vol. 50, pp. 1137-1150, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  16. Katsumi Inoue et al., “Optimum Stiffener Layout for the Reduction of Vibration and Noise of Gearbox Housing,” Journal of Mechanical Design, vol. 124, no. 3, pp. 518-523, 2012.
    [    CrossRef] [Google Scholar] [Publisher Link]
  17. A. Naess, F.E. Kolnes, and E. Mo, “Stochastic Spur Gear Dynamics by Numerical Path Integration,” Journal of Sound and Vibration, vol. 302, no. 4-5, pp. 936-950, 2007.
    [    CrossRef] [Google Scholar] [Publisher Link]
  18. S. Jyothirmai et al., “A Finite Element Approach to Bending, Contact & Fatigue Stress Distribution in Helical Gear Systems,” Procedia Materials Science, vol. 6, pp. 907-918, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  19. He Dai et al., “An Improved Analytical Model for Gear Mesh Stiffness Calculation,” Mechanism and Machine Theory, vol. 159, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  20. Beom-Soo Kim et al., “Optimization of Gearbox Housing Shape for Agricultural UTV Using Structural–Acoustic Coupled Analysis,” Scientific Reports, vol. 14, pp. 1-16, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  21. Beom-Soo Kim et al., “Vibration Analysis of Gearboxes for Agricultural UTV Using A Reduced-Order Model,” Journal of the Korean Society of Manufacturing Process Engineers, vol. 18, no. 8, pp. 8-17, 2019. [CrossRef] [Google Scholar] [Publisher Link]
  22. Ashwani Kumar et al., “Free Vibration Analysis of Truck Transmission Housing Based on FEA,” Procedia Materials Science, vol. 6, pp. 1588-1592, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  23. S. Mahendran, K.M. Eazhil, and L. Senthil Kumar, “Design and Analysis of Composite Helical Gear,” International Journal of Research of Science, vol. 1, no. 6, pp. 42-53, 2014.
    [    Google Scholar] [Publisher Link]
  24. Kunal Menavlikar et al., “Design and Topology Optimization of Two Stage Gearbox for All Terrain Vehicles,” International Journal of Innovative Research in Science, Engineering and Technology, vol. 8, no. 2, pp. 927-935, 2019.
    [    CrossRef] [Google Scholar] [Publisher Link]
  25. Mingxuan Liang et al., “Topology Optimization of Transmission Gearbox under Multiple Working Loads,” Advances in Mechanical Engineering, vol. 10, no. 11, pp. 1-7, 2018.
    [    CrossRef] [Google Scholar] [Publisher Link]
  26. Miryam B. Sánchez, José I. Pedrero, and Miguel Pleguezuelos, “Critical Stress and Load Conditions for Bending Calculations of Involute Spur and Helical Gears,” International Journal of Fatigue, vol. 48, pp. 28-38, 2013.
    [    CrossRef] [Google Scholar] [Publisher Link]
  27. K. Mujiburrahman et al., “Design and Analysis of E-Glass Gearbox Housing in Tractor and Optimization of Its Design Parameters,” Materials Today: Proceedings, vol. 49, no. 8, pp. 3696-3704, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  28. S. Padmanabhan, V. Srinivasa Raman, and M. Chandrasekaran, “Optimization of Gear Reducer Using Evolutionary Algorithm,” Materials Research Innovations, vol. 18, no. 6, pp. 378-383, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  29. S. Pandey, N.N. Singh, and P.K. Sinha, “Modeling, Design & Analysis of Differential Gear Box through FEM, Solidwork & Ansys Benchwork 14.0,” International Journal of Engineering Sciences & Research Technology, vol. 6, no. 7, pp. 887-894, 2017.
    [    CrossRef] [Google Scholar] [Publisher Link]
  30. Mitesh Patel, and A.V. Patil, “Study About Stress and Deformation of 3 Stage Helical Gearbox Casing,” International Journal of Advance Research in Engineering, Science & Technology, vol. 2, no. 7, pp. 65-71, 2015.
    [    Google Scholar] [Publisher Link]
  31. S. Patel Mitesh, “Stress Analysis and Design Modification of 3 Stage Helical Gear Box Casing,” International Journal for Scientific Research & Development, vol. 1, no. 9, pp. 2027-2028, 2013.
    [    Google Scholar] [Publisher Link]
  32. Francesco Pizzolante et al., “Combining the Asymptotic Numerical Method with the Harmonic Balance Method to Capture the Nonlinear Dynamics of Spur Gears,” Mechanical Systems and Signal Processing, vol. 214, pp. 1-14, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  33. P.D. Patel, and D.S. Shah, “Steady State Thermal Stress Analysis of Gearbox Using FEM,” International Journal of Mechanical and Industrial Engineering, vol. 2, no. 4, pp. 26-30, 2012.
    [    Google Scholar]
  34. Asmita Patil et al., “Design and Analysis of a Gearbox for an All-Terrain Vehicle,” International Journal of Innovative Research in Technology, vol. 4, no. 11, pp. 690-701, 2018.
    [    Google Scholar] [Publisher Link]
  35. Maruti Patil, P. Ramkumar, and K. Shankar, “Multi-Objective Optimization of the Two Stage Helical Gearbox with Tribological Constraints,” Mechanism and Machine Theory, vol. 138, pp. 38-57, 2019.
    [    CrossRef] [Google Scholar] [Publisher Link]
  36. Swapnil J. Patil, Vipin B. Singh, and Amit M. Pawa, “Design and Vibration Analysis for Shaft with Gear Mountings Using Finite Element Analysis,” International Advanced Research Journal in Science, Engineering and Technology, vol. 4, no. 1, pp. 30-33, 2017.
    [    CrossRef] [Google Scholar] [Publisher Link]
  37. Tiancheng Ouyang et al., “Cavitation Mechanism of High-Speed Helical Gears Induced by Vibration,” Tribology International, vol. 193, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  38. Real Technocast Limited, Rajkot, 2025. [Online]. Available: https://rajkotcastingandforginghub.com/industry/real-techno-cast/
  39. Santosh S. Patil et al., “Contact Stress Analysis of Helical Gear Pairs Including Frictional Coefficients,” International Journal of Mechanical Sciences, vol. 85, pp. 205-211, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  40. Smita Pawar, and Avinash Lavnis, “Improvement in Design of Gearbox Housing (Code No: MFO225DR) through Static Analysis,” International Journal of Advance Research, Ideas and Innovations in Technology, vol. 5, no. 2, pp. 2023-2026, 2017.
    [    Google Scholar] [Publisher Link]
  41. G. Raghavendra Setty et al., “Modeling and Dynamic Analysis of Gearbox Casing using Finite Element Analysis,” International Journal of Innovative Research in Science, Engineering & Technology, vol. 5, no. 6, pp. 11835-11847, 2016.
    [    Google Scholar] [Publisher Link]
  42. Mehmet Sarıtaş, Özgür Gölbol, and Paşa Yayla, “Finite Element Stress Analysis of Three-Stage Gearbox,” Nigde Omer Halisdemir University Journal of Engineering Sciences, vol. 10, no. 2, pp. 784-790, 2021.
    [    CrossRef] [Google Scholar] [Publisher Link]
  43. H. Singh and D. Kumar, “Effect of Face Width of Spur Gear on Bending Stress using AGMA and ANSYS,” International Journal for Simulation and Multidisciplinary Design Optimization, vol. 11, no. 23, pp. 1-8, 2020.
    [    CrossRef] [Google Scholar] [Publisher Link]
  44. Gwan-Hee Son et al., “Optimization of the Housing Shape Design for Radiated Noise Reduction of an Agricultural Electric Vehicle Gearbox,” Applied Sciences, vol. 10, no. 23, pp. 1-, 2020.
    [    CrossRef] [Google Scholar] [Publisher Link]
  45. Takanori Ide et al., “Structural Optimization Methods of Nonlinear Static Analysis with Contact and Its Application to Design Lightweight Gear Boxes of Automatic Transmission of Vehicles,” Structural and Multidisciplinary Optimization, vol. 53, pp. 1383-1394, 2016.
    [    CrossRef] [Google Scholar] [Publisher Link]
  46. Eiichirou Tanaka et al., “Vibration and Sound Radiation Analysis for Designing a Low Noise Gearbox with a Multi-Stage Helical Gear System,” JSME International Journal Series C Mechanical Systems, Machine Elements and Manufacturing, vol. 46, no. 3, pp. 1178-1185, 2023.
    [    CrossRef] [Google Scholar] [Publisher Link]
  47. B. Venkatesh, S.V. Prabhakar Vattikuti, and S. Deva Prasad, “Investigate the Combined Effect of Gear Ratio, Helix Angle, Face Width and Module on Bending and Compressive Stress of Steel Alloy Helical Gear,” Procedia Materials Science, vol. 6, pp. 1865-1870, 2014.
    [    CrossRef] [Google Scholar] [Publisher Link]
  48. J. Venkatesh and P.B.G.S.N. Murthy, “Design and Structural Analysis of High Speed Helical Gear Using ANSYS,” International Journal of Engineering Research and Applications, vol. 4, no. 3, pp. 01-05, 2014.
    [    Google Scholar] [Publisher Link]
  49. B.S. Vikhe, “Design and Analysis of Industrial Gearbox Casing,” International Research Journal of Engineering and Technology, vol. 3, no. 11, pp. 1379-1383, 2014.
    [    Google Scholar] [Publisher Link]
  50. Zoltan-Iosif Korka, and Nicoleta Gillich, “Modal Analysis of Helical Gear Pairs with Various Ratios and Helix Angles,” Romanian Journal of Acoustics and Vibration, vol. 14, no. 2, pp. 91-96, 2017.
    [    Google Scholar] [Publisher Link]
  51. F.L. Liao et al., “Dynamic Characteristics of Helical-Gear Systems Considering Axial Mesh Force Components,” International Journal of Mechanical Sciences, vol. 304, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  52. Fulin Liao et al., “Analysis of Nonlinear Dynamics of a Gear Transmission System Considering Effects of the Extended Tooth Contact,” Machines, vol. 13, no. 2, pp. 1-26, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  53. Yeping Yuan, Wen Zhang, and Junguo Wang, “Vibration Power Flow of Nonlinear Periodic Dynamic Response in Helical Gears Based on Harmonic Balance Method,” Nonlinear Dynamics, vol. 113, pp. 22555-22584, 2025.
    [    CrossRef] [Google Scholar] [Publisher Link]
  54. Yancong Li et al., “Dynamic Analysis of the Helical Gear Transmission System in Electric Vehicles with a Large Helix Angle,” Machines, vol. 11, no. 7, pp. 1-2023.
    [    CrossRef] [Google Scholar] [Publisher Link]
  55. Xiao Wu et al., “A Revised Method to Calculate Time-Varying Mesh Stiffness of Helical Gear,” MATEC Web of Conferences, vol. 355, pp. 1-9, 2022.
    [    CrossRef] [Google Scholar] [Publisher Link]
  56. Roee Cohen et al., “A Novel Method for Helical Gear Modeling with an Experimental Validation,” Nonlinear Dynamics, vol. 112, pp. 1-19, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  57. Iulian Lupea, Mihaiela Lupea, and Adrian Coroian, “Helical Gearbox Defect Detection with Machine Learning Using Regular Mesh Components and Sidebands,” Sensors, vol. 24, no. 11, pp. 1-25, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]
  58. Fei Liu et al., “Study on the Vibration Characteristics of Worm Helical Gear Drive,” Mechanism and Machine Theory, vol. 191, 2024.
    [    CrossRef] [Google Scholar] [Publisher Link]