Optimization of transmission ratios in a three-stage spur-helical gearbox using an analytical sizing algorithm and differential evolution

• Combines modern Differential Evolution algorithm with classical Analytical Sizing algorithm. • Gearbox volume optimization with contact stress and oil-bath lubrication constraint. • Lookup table of optimal gear ratios for a 3-stage gearbox. • Validate the lookup table’s feasibility by applying it to practical gearbox design. • Providing the regression equation of the optimal transmission ratio facilitates computer programming, design automation, prediction, and extrapolation. Designing three-stage gearboxes that combine a high overall transmission ratio with a compact envelope remains a notable challenge for both manufacturers and design engineers. Consequently, the study frames transmission ratio allocation as an objective optimization that (i) uniform contact-fatigue strength across stages, (ii) secures oil-bath lubrication, and (iii) minimises gearbox volume. The solution strategy consists of two sequential steps. First, an Analytical Sizing Algorithm (ASA) is used to delimit the minimum-maximum bounds for the first and second stage transmission ratios. Next, the Differential Evolution (DE) meta heuristic searches those bounds to identify the optimum ratio set that yields the smallest volume. Finally, regression analysis is employed to build surrogate equations for stage ratios, allowing a ready to use lookup table for designers. The study establishes feasible ratio ranges for the high speed and intermediate stages and formulates an objective model whose decision variables are the individual stage ratios. The approach is validated through an industrial case study. Results show that the optimised ratio distribution balances contact stresses between stages and maintains satisfactory lubrication while meeting the minimum volume constraint.