Robust optimum design of tapered roller bearings based on maximization of fatigue life using evolutionary algorithm

Abstract

For satisfactory performance of rolling element bearings, they must be designed for long fatigue life. As the fatigue life of the bearings is directly affected by its dynamic capacity, therefore the latter has been chosen as an objective function, which is to be optimized using a robust design optimization. Robust design improves the quality of a product by optimising its mean performance and minimising variation in its performance that incurred due to uncertainties present with the product. Also at the same time, it maintains the design feasibility under realistic and probabilistic constraints. The uncertainties considered in this bearing design problem is the tolerance in bearing dimensions since bearings cannot be manufactured with the exact dimension. In this work, tapered roller bearings has been designed in such a way that it will have maximum dynamic capacity but along with it, it will have minimum variation in its dynamic capacity due to variation in basic dimensions. Therefore, objective of the problem is not to only maximize the dynamic capacity but also to minimise its variation to an extent level. A nonlinear constrained optimization problem has been formulated with single objective function, thirty realistic constraints and eleven design variables. Constraints are basically based on geometrical parameters and strength criterion. Out of total eleven design variables, five are geometrical parameters that directly affect the dynamic capacity and six are constraints parameters, which present in the constraints and affects the possible feasible design space. Real coded genetic algorithm has been used as an optimization tool and the result obtained are compared with the standard bearings in catalogue and that are indicating a drastic improvement in the bearing life. Their life is found to be on an average 1.5 times of the life of standard bearings in catalogues. A constraints violation study has been carried out to prioritise the constraints for faster convergence. A convergence study and sensitivity analysis have also been carried out. Convergence study ensured the global optima in the feasible design space. To perform sensitivity analysis, the variation of dynamic capacity with respect to all basic design variables, i.e. bearing pitch diameter, roller mean diameter, roller effective length and the nominal contact angle have been calculated. These are found to be a positive value indicating that dynamic capacity has a positive variation with respect to them. The same has been calculated for bearing 30,204 considering positive tolerance, the least variation is found 0.00018 times dynamic capacity, which is with respect to the bearing pitch diameter. Whereas, the most variation is 0.212 times dynamic capacity, which is with respect to the nominal contact angle. This shows that the bearing dynamic capacity is highly sensitive to the variation in nominal contact angle and least sensitive to the variation in bearing pitch diameter. Therefore, while manufacturing, the manufacturer should have more attention on contact angle. Finally, for visualisation, radial dimensions of optimized bearing 30,204 with positive tolerance have been drawn, showing no interference among the dimensions, which indicates a geometrically feasible design. Hence, the present paper allows a designer to design rolling bearings with maximum fatigue life and minimum variation in its performance due to tolerance accounted during manufacturing. The present problem is not limited to the bearings only, but can also be applied in optimization of any machine components (e.g. gears, cams, springs, etc.) having significant uncertainties in their materials or geometrical parameters such as modulus, thickness, residual strain, density, etc.