Design-oriented optimization of transmission ratio allocation in two-stage bevel-spur gearboxes considering contact fatigue strength and oil-bath lubrication constraints

Abstract

Abstract

Abstract Transmission ratio allocation significantly influences gear dimensions, contact stress levels, lubrication feasibility, and the overall compactness of multi-stage gearboxes. However, lubrication constraints are commonly treated as post-design checks rather than being incorporated directly into the ratio allocation process, which can lead to designs that satisfy strength requirements but fail to ensure adequate lubrication conditions. This study proposes an analytical–optimization framework for transmission ratio allocation in two-stage bevel–spur gearboxes operating under oil-bath lubrication. The framework simultaneously considers balanced contact fatigue strength utilization between gear stages, oil-immersion feasibility, and minimization of gearbox axial length. An analytical feasible domain for transmission ratio distribution is first established by combining contact stress relations with geometric oil-immersion constraints. Sequential Quadratic Programming (SQP) is then employed to determine the optimal ratio allocation under these coupled constraints. To improve engineering applicability, a surrogate regression model based on a face-centered central composite design (FCCCD) is developed to enable rapid prediction of near-optimal solutions without repeated numerical optimization. A 4 kW gearbox case study is used to demonstrate the proposed methodology, and finite element analysis (FEA) is conducted to validate the analytical results through contact stress simulation. The results show that lubrication feasibility significantly restricts the admissible transmission ratio range and influences the optimal gearbox configuration. The resulting design charts provide practical guidance for rapid preliminary gearbox design.