Optimal weight design problem of spur gears

Minh H. Nguyen, Nguyen Anh My, L. Q. Vĩnh, Vo Thanh Binh
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Abstract

Gear is one of the most common and important components in machinery. Evaluation on durability of gears plays crucial role in the assessment of the whole system reliability and service life. For other parts like shafts, the gears also act as loads. Therefore, dimensions and weight of the gears should be reduced as much as possible, contributing the size and weight reduction of the whole systems, which is essential to be cost-effectiveness. The current research focuses on optimal weight design problem of spur gears, such that the weight is minimized under the constraints taken from working conditions. The weight is a function of six variables, i.e. face width, shaft diameter of pinion, shaft diameter of gear, number of teeth on pinion, module and hardness. Constraints are derived based on AGMA standard and engineering handbooks, including the bending strength, the surface fatigue strength, the interference condition, the condition for uniform load distribution, the torsional strength of shaft on pinion and gear, and the center distance. The set of optimum design variables is determined by the heuristic algorithm Grey Wolf Optimizer (GWO). The accuracy and efficiency of the GWO in the optimal weight design problem of spur gears are assessed based on comparison with other popular methods, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Simulated Annealing (SA). It is noted that in previous works, some of the constraints are still violated. Therefore, a penalty term is taken into the objective function, such that any set of design variables that violates constraints will be considered as ``unfit'' by the algorithm. It is demonstrated that using the proposed approach by current work, the optimal weight and the corresponding set of design variable are very close to reference data. Yet the advantage of the proposed approach is exhibited in the fact that all of the constraints are satisfied.
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正齿轮重量优化设计问题
齿轮是机械中最常见和最重要的部件之一。齿轮的耐久性评估在整个系统可靠性和使用寿命的评估中起着至关重要的作用。对于其他部件,如轴,齿轮也作为负载。因此,齿轮的尺寸和重量应尽可能减少,有助于整个系统的尺寸和重量的减少,这是必不可少的成本效益。目前的研究重点是正齿轮的重量优化设计问题,如何在给定的工况约束下使正齿轮的重量最小化。权重是六个变量的函数,即面宽度,小齿轮轴直径,齿轮轴直径,小齿轮上的齿数,模块和硬度。根据AGMA标准和工程手册推导了约束条件,包括弯曲强度、表面疲劳强度、干涉条件、载荷均匀分布条件、轴对小齿轮的扭转强度和中心距离。最优设计变量集由启发式算法灰狼优化器(GWO)确定。通过与遗传算法(GA)、粒子群算法(PSO)和模拟退火算法(SA)的比较,评价了GWO算法在正齿轮重量优化设计中的精度和效率。值得注意的是,在以前的作品中,仍然违反了一些约束。因此,在目标函数中加入一个惩罚项,使得任何一组违反约束的设计变量都将被算法视为“不适合”。目前的研究表明,采用该方法得到的最优权值和相应的设计变量集与参考数据非常接近。然而,该方法的优点在于它满足了所有的约束条件。
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