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

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.