基于遗传算法的滚动轴承的多目标优化设计翻译

1、Multi-objective design optimisation of rolling bearings using genetic algorithms 基于遗传算法的滚动轴承的多目标优化设计Shantanu Guptaa Rajiv Tiwarib, and Shivashankar B. Naira, aDepartment of Computer Science and Engineering计算机科学与工程系Indian Institute of Technology Guwahati, 印度理工学院 GuwahatiGuwahati 781039, Assam, India

2、印度阿萨姆邦，781039，GuwahatibDepartment of Mechanical Engineering 机械工程系 Indian Institute of Technology Guwahati, 印度理工学院 GuwahatiGuwahati 781039, Assam, India 印度阿萨姆邦，781039，GuwahatiReceived 8 March 2006;  revised 6 September 2006;  accepted 2 October 2006.  Available online 28 December 2006.

3、 Abstract 摘要The design of rolling bearings has to satisfy various constraints, e.g. the geometrical, kinematics and the strength, while delivering excellent performance, long life and high reliability. This invokes the need of an optimal design methodology to achieve these objectives collectively, i

4、.e. the multi-objective optimisation. In this paper, three primary objectives for a rolling bearing, namely, the dynamic capacity (Cd), the static capacity (Cs) and the elastohydrodynamic minimum film thickness (Hmin) have been optimized separately, pair-wise and simultaneously using an advanced mul

5、ti-objective optimisation algorithm: NSGA II (non-dominated sorting based genetic algorithm). These multiple objectives are performance measures of a rolling bearing, compete among themselves giving us a trade-off region where they become “simultaneously optimal”, i.e. Pareto optimal. A sensitivity

6、analysis of various design parameters has been performed, to see changes in bearing performance parameters, and results show that, except the inner groove curvature radius, no other design parameters have adverse affect on performance parameters. 滚动轴承的设计，要满足各种制约因素，如几何、 运动学和强度，同时还要提供优异的性能、 长寿命和高可靠性。这

7、将调用优化设计方法来实现这些目标集体，即多目标优化。在本文中，对于一个滚动轴承的三个主要目标，即动态能力（CD），静态电容（Cs），流体最小膜厚度（HMIN）已被单独、成对和同时优化，使用先进的多目标优化算法：NSGA II（非支配排序遗传算法）。这些多目标用于滚动轴承的性能衡量，而彼此之间的竞争性给予我们一个权衡范围，使他们成为“同时最优”，即帕累托最优。对各种设计参数的灵敏度进行了分析，看看轴承性能参数的变化，结果表明，除内沟曲率半径，没有其他设计参数对性能参数有不利影响。Keywords: Rolling bearings; Multi-objective evolutionary op

8、timisation; NSGA II; Mechanical design; Sensitivity analysis 关键词：滚动轴承；多目标演进优化；NSGA II;机械设计；灵敏度分析Article Outline 文章目录1. Introduction 绪论2. Macro-geometries of rolling bearings 滚动轴承的宏观结构3. Problem formulation of rolling bearing design 滚动轴承的设计问题3.1. Design parameters 设计参数3.2. Objective functions 目标功能3.2

9、.1. Dynamic capacity (Cd) 动态能力3.2.2. Elastohydrodynamic minimum film thickness (Hmin) 弹的最小膜厚度3.2.3. Static capacity (Cs) 静态能力3.3. Constraints 约束条件4. Multi-objective optimization 多目标优化5. Application and results 应用程序及结果5.1. NSGA II implementation and application NSGA实现与应用5.2. Parametric sensitivity an

10、alysis 参数灵敏度分析5.3. Contributions 6. Conclusions 总结Appendix A. Appendix 附录Appendix B. Sensitivity of Hmin with Q References 参考文献1. Introduction 绪论Rolling bearings are widely used as an important component in the most of the mechanical and aerospace engineering applications. Be it development of the h

11、ouse-hold appliance, automotive, space, aeronautical, micro- or nano-machine applications, all of them have given a boost to the advancement in the design technology of rolling bearings. This motivated design engineers to come up with a design technology that gives long lasting, more efficient and h

12、ighly reliable bearing designs. These objectives are hard to satisfy, thus making it a numerically challenging problem. Furthermore, there is a need to optimize them collectively. The numerical toughness and a need to optimize them collectively warrant an application of the evolutionary multi-object

13、ive optimisation. Objective functions for optimisation are the dynamic capacity (Cd), the static capacity (Cs) and the elastohydrodynamic minimum film thickness (Hmin). Due to the aforementioned toughness of the problem, there have been very few attempts at optimizing these objectives, simultaneousl

14、y. 滚动轴承作为一个重要元件更多的被广泛应用在机械和航空航天工程领域。随着它在家电，汽车，航空，航天，微纳米机器应用方面的发展，从而推动了滚动轴承设计技术的进步。这促使设计工程师提出一种设计技术，可以提供持久的，更有效的和高度可靠的轴承设计。这些目标很难满足，从而使它成为在数字上具有挑战性的问题。此外，还要能集体的进行优化。数值的强健和需要集体优化使得渐进的多目标优化得以应用。优化的目标函数是动态能力（CD），静态电容（Cs），流体最小膜厚度（HMIN）。由于上述问题的困难，已经很少尝试同时优化这些目标。Several research works have been reported on

15、 optimisation of various machine elements, however, very few literatures are available on the optimisation of rolling bearings. Asimow 1 used the NewtonRaphson method for the optimum design of the length and the diameter of a journal bearing, which was supporting a given load at a given speed. The o

16、bjective function was to minimize a weighted sum of the frictional loss and the shaft twist. Seireg and Ezzat 2 utilized a gradient-based search to optimise the bearing length, the radial clearance and the average viscosity of the lubricant. 有许多的研究工作报道各种机械零件优化，然而，能用于滚动轴承优化的文献却很少。阿西穆 1 利用牛顿迭代法长度的优化设计

17、和滑动轴承的直径，这是支持一个给定的负载在一个给定的速度。目标函数为最小化的摩擦损失和轴扭转的加权总和。Seireg和伊扎特 2 利用基于梯度的搜索优化支承长度，径向间隙和润滑油的粘度平均值。The objective function was chosen to minimize a weighted sum of the quantity of lubricant fed to the bearing and its temperature rise. Maday 3 and Wylie and Maday 4 used bounded variable methods of the ca

18、lculus of variable, to determine the optimum configuration for hydrodynamic bearings. The design criterion was chosen that to maximize the load carrying capacity of the bearing. Seireg 5 reviewed some illustrative examples of the use of optimisation techniques, in the design of mechanical elements a

19、nd systems. These include gears, journal bearings, rotating discs, pressure vessels, shafts under bending and torsion, beams subjected to the longitudinal impact and problems of the elastic contact and load distributions.Hirani et al. 6 proposed a design methodology for an engine journal bearing. Th

20、e procedure of selection of the diametral clearance and the bearing length was described by limiting the minimum film thickness, the maximum pressure and the maximum temperature. All the aforementioned literatures were concerned mainly with the journal bearing design. However, internal geometries of

21、 journal bearings are far simple as compared to rolling bearings. 目标函数是选择减少润滑油供给量的加权总和的轴承的温升。Maday 3 和 4 怀利和Maday使用变量的微积分有界变量的方法，以确定最佳配置的流体动力轴承。设计标准的选择，以最大限度地提高轴承的承载能力。seireg 5 回顾了一些说明性的例子使用的优化技术，在机械元件和系统的设计。这些包括齿轮，轴承，旋转盘，压力容器，弯曲和扭转载荷作用下的轴，在弹性接触和负荷分布的纵向冲击和梁Hirani等问题。 6 提出了一种内燃机滑动轴承的设计方法。的径向间隙和轴承长度选择

22、的程序是通过限制最小油膜厚度，最大压力和最高温度。上述所有的文献主要关注的是滑动轴承设计。然而，相比，滚动轴承的轴承内部几何形状是远远简单。Changsen 7 described a design method by using a gradient-based numerical optimisation technique for rolling bearings. He proposed, five objective functions for design of rolling bearings: the maximum fatigue life, the maximum wear

23、 life, the maximum static load rating, the minimum frictional moment and the minimum spin to roll ratio. The concept of optimisation of the multi-objective of rolling bearings was also proposed. Only the basic concepts and solution techniques of an optimisation problem were introduced without any il

24、lustrations. Objective functions proposed for optimisation of rolling bearings are nonlinear in nature, moreover, associated with the geometric and kinematics constraints. 畅森 7 描述了一种设计方法，采用基于梯度的数值优化技术的滚动轴承。他提出了五个目标函数，对滚动轴承的设计：最大的疲劳寿命，最大磨损寿命，最大额定静载荷，最小摩擦力矩和滚比最低的自旋。对滚动轴承的多目标优化的概念也被提出。只有基本概念和优化问题的求解技术，

25、介绍了没有任何插图。目标函数的优化提出了滚动轴承在本质上是非线性的，此外，与几何和运动学约束的关联。Choi and Yoon 8 used GAs in optimizing the automotive wheel-bearing unit, by considering the maximization of life of the unit as the objective function. Periaux 9 discussed in detail the application of GAs to the aeronautics and turbo machineries. Ch

26、akraborthy et al. 10 described a design optimisation problem of rolling element bearings with five design parameters, by using GAs based on requirements of the longest fatigue life. They presented bearing internal geometrical parameters resulted from the optimised design of different bearing boundar

27、y dimensions. Main limitations of the method were that the use of the single objective function and some of constraints were unrealistic. Assembling angles were assumed and values of other constraint constants were chosen (arbitrary) fixed to solve the optimisation problem. Recently, Rao and Tiwari

28、11 developed a rolling bearing design methodology with the improved and realistic constraints for the single objective optimisation with the help of GAs. Choi Yoon 8 和优化汽车轮毂轴承单元用气，考虑机组的寿命最大化为目标函数。periaux 9 详细讨论了气体对航空涡轮机械中的应用。chakraborthy等人。【10】描述一个滚动轴承的优化设计问题的五个设计参数，通过使用天然气的基础上持续时间最长的疲劳寿命的要求。他们提出了轴承

29、内部几何参数导致不同轴承外形尺寸的优化设计。该方法的主要限制是在单一的目标函数的使用和一些约束是不现实的。安装角度被认为和其他约束常数的值被选择（任意）固定的解决优化问题。最近，饶和Tiwari 11 开发滚动轴承的设计方法与改进的和现实的制约与气体的帮助的单目标优化。A work on the multi-objective optimisation for the design of rolling bearings 12 does a weighted combination of these individual objective functions namely the dynam

30、ic capacity, the static capacity and the minimum film thickness. The multi-objective problem is converted into a scalar optimisation problem. This work made use of the deterministic as well as stochastic algorithms, for solving the constraint scalar optimisation problem. As the deterministic approac

31、h the interior penalty function method was used, while the simulated annealing and genetic algorithms were used as stochastic approaches. This way of combining the multiple competing objectives and optimisation of the obtained scalar objective has some prominent disadvantages (1) a single run of the

32、 algorithm will give only one trade-off point, (2) solution points on a non-convex trade-off front cannot be obtained, and (3) no criteria for choosing weights for each of the objective function exists. The work proposed in this paper handles each of these problems. 一对滚动轴承的 12 做这些个体目标函数加权组合，即动态能力设计的

33、多目标优化的工作，静容量和最小油膜厚度。多目标优化问题转化为标量优化问题。采用确定性以及随机算法，用于求解约束标量优化问题。为确定性方法，采用内点罚函数法，而模拟退火和遗传算法被用来作为随机方法。这种方式相结合的多的竞争目标和所得到的单目标优化具有一些突出的缺点（1）单次运行的算法将只有一个平衡点，（2）在非凸交易前不能得到的溶液分，和（3）没有标准选择权值目标函数的存在。本文提出的工作处理这些问题。Any weight based combination approach has drawbacks like unknown selection of weights for differen

34、t objectives, one point obtained in one run, and incapability to explore non-convex regions of the trade-off front, i.e. Pareto front. Thus there is need to use a better multi-objective (evolutionary) algorithm (MOEA) for solving this problem. Coello gave a survey of the MOEA in 13. Another survey w

35、as given in the doctoral thesis of Zitzler 14. Because of NSGA-IIs (Elitist non-dominated sorting based genetic algorithm II) 15 low computational requirements, elitist approach, and parameter-less sharing approach; it was chosen as the algorithm for determination of trade-off between competing perf

36、ormances, i.e. to generate the Pareto front 16 and 17. 任何基于权重的组合方法存在不同的目标权重未知的选择，在一个运行中获得一点，不能探索权衡前非凸区域，即帕累托面前。因此，有必要使用更好的多目标（进化）算法（MOEA）为解决这一问题。Coello给在 13 经济部的一项调查。另一项调查是在Zitzler 14 的博士论文了。由于NSGA-II的（精英非支配排序遗传算法II基础） 15 低的计算要求，精英的方法，和参数不共享的方法；它作为测定之间的竞争性能折衷算法，即产生帕累托前沿 16 和 17 。The present paper is

37、 organized as follows; Section 2 briefs the basic geometry of rolling bearings. The mathematical model of the problem as a set of objective functions, design parameters and constraints are described in Section 3. Section 4 introduces the concept of the multi-objective optimisation and it discusses w

38、hether a deterministic or a stochastic approach is more appropriate for this problem. Section 5 details the application methodology, optimised results and the sensitivity analysis. Section 6 concludes the present work, and is followed by important references. Results obtained are satisfactory, and g

39、ive a good insight into the trade-offs among the performance measures of rolling bearings. Apart from the numerical significance of the obtained optimal solution, these results can help us in better understanding the parameters behind the effective designing of rolling bearings. 本文的组织如下；2节介绍了滚动轴承的基本

40、几何。问题的数学模型作为目标函数的一组，在3节中所描述的设计参数和约束条件。4节介绍了多目标优化的概念，并讨论了是否确定的或随机的方法，这个问题更为合适。5节的具体应用方法，优化结果和敏感性分析。6部分总结了目前的工作，其次是重要的参考资料。得到的结果是令人满意的，并提供一个良好的洞察权衡滚动轴承的性能指标。除了得到的最优解的数值的意义，这些结果可以帮助我们更好的理解在滚动轴承的有效参数设计。2. Macro-geometries of rolling bearings 滚动轴承的宏观设计Rolling bearings appear to have a simple outer geomet

41、ry, but their internal geometry can have varying effects on the amount of stresses, deflections and load distributions it can handle. Therefore, the internal geometry has a direct effect on the performance and the life of a bearing. Fig. 1 shows the common nomenclature of a typical rolling bearing.滚

42、动轴承出现有一个简单的外部几何，但其内部的几何形状可以有不同的应力的影响，挠度和荷载分布可以处理。因此，内部的几何形状对性能和轴承的寿命直接影响。图1显示了典型的滚动轴承常见的命名Fig. 1. Macro-geometries of a radial ball bearing. 一个向心球轴承的宏观结构。In the crudest form, the geometry of a bearing can be defined by three boundary dimensions, namely, the bore diameter (d), the outer diameter

43、 (D) and the bearing width (Bw). These boundary dimensions have been standardised. Parameters that help to define the complete internal geometry of a given rolling bearing (i.e. for a given boundary dimensions) are the ball diameter (Db), the pitch diameter of the bearing (Dm), the inner and outer r

44、aceway curvature coefficients (fi and fo), and number of rolling elements (Z). 在最原始的形式，轴承的几何可以由三边尺寸，即定义，内径（D），外直径（D）和轴承宽度（BW）。这些边界尺寸已标准化。参数，有助于确定一个给定的滚动轴承的完整的内部几何（即，对于一个给定的边界尺寸）是球的直径（DB），轴承的内径（DM），内、外滚道曲率系数（FI FO），和滚动元件数（Z）。3. Problem formulation of rolling bearing design 滚动轴承的设计问题We seek to find ou

45、t the complete internal geometry (i.e. the ball and pitch diameters, the inner and outer raceway curvature coefficients, and number of rolling elements) of a bearing (as specified by standard bearing boundary dimensions), while optimizing its performance characteristics and overall life. Presence of

46、 more than one objective makes the problem come into the domain of multi-objective optimisation. 我们试图找出完整的内部几何（即球直径和间距，内、外滚道曲率系数，和滚动元件数）的轴承（标准轴承外形尺寸指定），而优化其性能特点和整体生活。一个以上的目标的存在使问题为多目标优化的域。Any constrained multi-optimisation optimisation problem is essentially composed of three components, namely, des

47、ign parameters, objective functions, and constraints (for defining feasible design parameter space). We discuss in brief, these components of the present problem in following sections. 任何约束多优化优化问题基本上是由三部分组成，即，设计参数，目标函数，约束（确定可行的设计参数空间）。我们简要地讨论，这些组件将在以后的章节中存在的问题。3.1. Design parameters 设计参数The design p

48、arameter vector can be written as: 设计参数向量可以写为：X=Dm,Db,Z,fi,fo,KDmin,KDmax,e,(1)where,(2)Parameters that define bearing internal geometries are Dm, Db, Z, fi, and fo (see Appendix A for the nomenclature). Whereas, KDmin, KDmax, , e, and are part of constraints 11 (refer Section Sections 3.2 and 3.3)

49、and do not directly represent any measurement of the bearing internal geometries. The latter are usually kept constant while designing bearings 7, but for the present case these secondary parameters are also considered as variables. This has been made possible due to the flexibility and the robustne

50、ss offered by the adopted GA based approach. All angles are measured in radians, dimensions in millimetres with the exception of the minimum film thickness (Hmin) that is measured in micrometers, and forces in Newton (N). Assembly angle (o) of a bearing (see Fig. 2) also forms an important constrain

51、t on the number of rolling elements. Based on the geometrical derivation presented in 11, one could arrive at the following formula for the assembly angle,参数定义的轴承内部几何形状的DM，DB，Z，FI，FO（见附录A的命名）。然而，kdmin，kdmax，E，和是制约 11 （参见第3.2节和3.3）和不直接代表任何测量轴承内部几何形状。后者通常是在设计轴承 7 保持不变，但在目前的情况下，这些次要的参数也被视为变量。这可能是由于的灵活性

52、和所采用的基于遗传算法的方法提供的鲁棒性。所有的测量角度弧度，毫米与最小油膜厚度的例外尺寸（HMIN），是测量微米，和军队在牛顿（N）。装配角度（O）的轴承（见图2）也形成在滚动元件的数量的一个重要约束。基于 11 提出的几何推导，可以得出以下公式进行装配角度，(3)where,T=D-d-2Db.(4)Fig. 2. A rolling bearing showing the assembly angle. 滚动轴承的装配角度3.2. Objective functions 目标参数As mentioned earlier, there are three important pe

53、rformance measures of a rolling bearing. These are namely, the dynamic capacity (Cd), the minimum film thickness (Hmin), and the static capacity (Cs). All of them have to be simultaneously maximized, for getting the best performance of the bearing. These performance parameters are discussed in more

54、detail in the following subsections. 如前所述，有三个重要的滚动轴承性能的措施。这些都是即，动态能力（CD），最小油膜厚度（HMIN），和静态电容（Cs）。所有的人都是同时最大化，使轴承的最佳性能。这些性能参数在下面的小节中更详细地讨论。3.2.1. Dynamic capacity (Cd) 动态能力Among different objectives for rolling bearings, the dynamic capacity (Cd) is the most important one, as this directly forms the b

55、asis for longest fatigue life of a bearing. The dynamic capacity, also known as the dynamic load rating, is defined as “the constant radial load, which a group of apparently identical bearings can endure for a rating life of one million revolutions of the inner ring (for a stationary load and the st

56、ationary outer ring)”. It is expressed as 7:不同的目标之间的滚动轴承，动态能力（CD）是最重要的，因为这直接形成最长的疲劳寿命的轴承的基础上。动态能力，也被称为动态负载率，定义为“恒定的径向负荷，这一组显然相同的轴承可以承受一百万次革命的内圈额定寿命（对于一个固定的负载和固定的外环）”。它表现为 7 ：(5)where,(6)=Dbcos/Dm.(7)Here the factor, , is not an independent parameter. For the current discussion, the deep groove ball

57、bearing has been considered, for which the contact angle, , is zero. Therefore,  = Db/Dm. On careful inspection of Eq. (5), it could be observed that the dynamic capacity depends on (2/3)rd to the power of number of roller and 1.8th to the power the diameter of the ball. Hence, during opti

58、misation, we expect that the maximum possible ball diameter would give us better dynamic capacity. Furthermore, with bigger ball diameter, less number of balls would be accommodated in a given space. The dynamic capacity is derived on the basis of the maximum octahedral shear stress that occurs at the subsurface of the contact zone, between trolling elements and races. Hence, it should be noted that the constraint related to the strength against the shear stress will not appear, explicitly, in constraint section. 这里的因素，不是一个独立的参数。对于目前的讨论，深沟球轴承