基于遺傳算法的滾動軸承的多目標優(yōu)化設計翻譯

1、Multi-objective design optimisation of rolling bearings using genetic algorithms 基于遺傳算法的滾動軸承的多目標優(yōu)化設計Shantanu Guptaa Rajiv Tiwarib, and Shivashankar B. Naira, aDepartment of Computer Science and Engineering計算機科學與工程系Indian Institute of Technology Guwahati, 印度理工學院 GuwahatiGuwahati 781039, Assam, India

2、印度阿薩姆邦，781039，GuwahatibDepartment of Mechanical Engineering 機械工程系 Indian Institute of Technology Guwahati, 印度理工學院 GuwahatiGuwahati 781039, Assam, India 印度阿薩姆邦，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. 滾動軸承的設計，要滿足各種制約因素，如幾何、 運動學和強度，同時還要提供優(yōu)異的性能、 長壽命和高可靠性。這

7、將調用優(yōu)化設計方法來實現(xiàn)這些目標集體，即多目標優(yōu)化。在本文中，對于一個滾動軸承的三個主要目標，即動態(tài)能力（CD），靜態(tài)電容（Cs），流體最小膜厚度（HMIN）已被單獨、成對和同時優(yōu)化，使用先進的多目標優(yōu)化算法：NSGA II（非支配排序遺傳算法）。這些多目標用于滾動軸承的性能衡量，而彼此之間的競爭性給予我們一個權衡范圍，使他們成為“同時最優(yōu)”，即帕累托最優(yōu)。對各種設計參數(shù)的靈敏度進行了分析，看看軸承性能參數(shù)的變化，結果表明，除內溝曲率半徑，沒有其他設計參數(shù)對性能參數(shù)有不利影響。Keywords: Rolling bearings; Multi-objective evolutionary op

8、timisation; NSGA II; Mechanical design; Sensitivity analysis 關鍵詞：滾動軸承；多目標演進優(yōu)化；NSGA II;機械設計；靈敏度分析Article Outline 文章目錄1. Introduction 緒論2. Macro-geometries of rolling bearings 滾動軸承的宏觀結構3. Problem formulation of rolling bearing design 滾動軸承的設計問題3.1. Design parameters 設計參數(shù)3.2. Objective functions 目標功能3.2

9、.1. Dynamic capacity (Cd) 動態(tài)能力3.2.2. Elastohydrodynamic minimum film thickness (Hmin) 彈的最小膜厚度3.2.3. Static capacity (Cs) 靜態(tài)能力3.3. Constraints 約束條件4. Multi-objective optimization 多目標優(yōu)化5. Application and results 應用程序及結果5.1. NSGA II implementation and application NSGA實現(xiàn)與應用5.2. Parametric sensitivity an

10、alysis 參數(shù)靈敏度分析5.3. Contributions 6. Conclusions 總結Appendix A. Appendix 附錄Appendix B. Sensitivity of Hmin with Q References 參考文獻1. Introduction 緒論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. 滾動軸承作為一個重要元件更多的被廣泛應用在機械和航空航天工程領域。隨著它在家電，汽車，航空，航天，微納米機器應用方面的發(fā)展，從而推動了滾動軸承設計技術的進步。這促使設計工程師提出一種設計技術，可以提供持久的，更有效的和高度可靠的軸承設計。這些目標很難滿足，從而使它成為在數(shù)字上具有挑戰(zhàn)性的問題。此外，還要能集體的進行優(yōu)化。數(shù)值的強健和需要集體優(yōu)化使得漸進的多目標優(yōu)化得以應用。優(yōu)化的目標函數(shù)是動態(tài)能力（CD），靜態(tài)電容（Cs），流體最小膜厚度（HMIN）。由于上述問題的困難，已經很少嘗試同時優(yōu)化這些目標。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. 有許多的研究工作報道各種機械零件優(yōu)化，然而，能用于滾動軸承優(yōu)化的文獻卻很少。阿西穆 1 利用牛頓迭代法長度的優(yōu)化設計

17、和滑動軸承的直徑，這是支持一個給定的負載在一個給定的速度。目標函數(shù)為最小化的摩擦損失和軸扭轉的加權總和。Seireg和伊扎特 2 利用基于梯度的搜索優(yōu)化支承長度，徑向間隙和潤滑油的粘度平均值。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. 目標函數(shù)是選擇減少潤滑油供給量的加權總和的軸承的溫升。Maday 3 和 4 懷利和Maday使用變量的微積分有界變量的方法，以確定最佳配置的流體動力軸承。設計標準的選擇，以最大限度地提高軸承的承載能力。seireg 5 回顧了一些說明性的例子使用的優(yōu)化技術，在機械元件和系統(tǒng)的設計。這些包括齒輪，軸承，旋轉盤，壓力容器，彎曲和扭轉載荷作用下的軸，在彈性接觸和負荷分布的縱向沖擊和梁Hirani等問題。 6 提出了一種內燃機滑動軸承的設計方法。的徑向間隙和軸承長度選擇

22、的程序是通過限制最小油膜厚度，最大壓力和最高溫度。上述所有的文獻主要關注的是滑動軸承設計。然而，相比，滾動軸承的軸承內部幾何形狀是遠遠簡單。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. 暢森 7 描述了一種設計方法，采用基于梯度的數(shù)值優(yōu)化技術的滾動軸承。他提出了五個目標函數(shù)，對滾動軸承的設計：最大的疲勞壽命，最大磨損壽命，最大額定靜載荷，最小摩擦力矩和滾比最低的自旋。對滾動軸承的多目標優(yōu)化的概念也被提出。只有基本概念和優(yōu)化問題的求解技術，

25、介紹了沒有任何插圖。目標函數(shù)的優(yōu)化提出了滾動軸承在本質上是非線性的，此外，與幾何和運動學約束的關聯(lián)。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 和優(yōu)化汽車輪轂軸承單元用氣，考慮機組的壽命最大化為目標函數(shù)。periaux 9 詳細討論了氣體對航空渦輪機械中的應用。chakraborthy等人?！?0】描述一個滾動軸承的優(yōu)化設計問題的五個設計參數(shù)，通過使用天然氣的基礎上持續(xù)時間最長的疲勞壽命的要求。他們提出了軸承

29、內部幾何參數(shù)導致不同軸承外形尺寸的優(yōu)化設計。該方法的主要限制是在單一的目標函數(shù)的使用和一些約束是不現(xiàn)實的。安裝角度被認為和其他約束常數(shù)的值被選擇（任意）固定的解決優(yōu)化問題。最近，饒和Tiwari 11 開發(fā)滾動軸承的設計方法與改進的和現(xiàn)實的制約與氣體的幫助的單目標優(yōu)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. 一對滾動軸承的 12 做這些個體目標函數(shù)加權組合，即動態(tài)能力設計的

33、多目標優(yōu)化的工作，靜容量和最小油膜厚度。多目標優(yōu)化問題轉化為標量優(yōu)化問題。采用確定性以及隨機算法，用于求解約束標量優(yōu)化問題。為確定性方法，采用內點罰函數(shù)法，而模擬退火和遺傳算法被用來作為隨機方法。這種方式相結合的多的競爭目標和所得到的單目標優(yōu)化具有一些突出的缺點（1）單次運行的算法將只有一個平衡點，（2）在非凸交易前不能得到的溶液分，和（3）沒有標準選擇權值目標函數(shù)的存在。本文提出的工作處理這些問題。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. 任何基于權重的組合方法存在不同的目標權重未知的選擇，在一個運行中獲得一點，不能探索權衡前非凸區(qū)域，即帕累托面前。因此，有必要使用更好的多目標（進化）算法（MOEA）為解決這一問題。Coello給在 13 經濟部的一項調查。另一項調查是在Zitzler 14 的博士論文了。由于NSGA-II的（精英非支配排序遺傳算法II基礎） 15 低的計算要求，精英的方法，和參數(shù)不共享的方法；它作為測定之間的競爭性能折衷算法，即產生帕累托前沿 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. 本文的組織如下；2節(jié)介紹了滾動軸承的基本

40、幾何。問題的數(shù)學模型作為目標函數(shù)的一組，在3節(jié)中所描述的設計參數(shù)和約束條件。4節(jié)介紹了多目標優(yōu)化的概念，并討論了是否確定的或隨機的方法，這個問題更為合適。5節(jié)的具體應用方法，優(yōu)化結果和敏感性分析。6部分總結了目前的工作，其次是重要的參考資料。得到的結果是令人滿意的，并提供一個良好的洞察權衡滾動軸承的性能指標。除了得到的最優(yōu)解的數(shù)值的意義，這些結果可以幫助我們更好的理解在滾動軸承的有效參數(shù)設計。2. Macro-geometries of rolling bearings 滾動軸承的宏觀設計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.滾

42、動軸承出現(xiàn)有一個簡單的外部幾何，但其內部的幾何形狀可以有不同的應力的影響，撓度和荷載分布可以處理。因此，內部的幾何形狀對性能和軸承的壽命直接影響。圖1顯示了典型的滾動軸承常見的命名Fig. 1. Macro-geometries of a radial ball bearing. 一個向心球軸承的宏觀結構。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，軸承的幾何可以由三邊尺寸，即定義，內徑（D），外直徑（D）和軸承寬度（BW）。這些邊界尺寸已標準化。參數(shù)，有助于確定一個給定的滾動軸承的完整的內部幾何（即，對于一個給定的邊界尺寸）是球的直徑（DB），軸承的內徑（DM），內、外滾道曲率系數(shù)（FI FO），和滾動元件數(shù)（Z）。3. Problem formulation of rolling bearing design 滾動軸承的設計問題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. 我們試圖找出完整的內部幾何（即球直徑和間距，內、外滾道曲率系數(shù)，和滾動元件數(shù)）的軸承（標準軸承外形尺寸指定），而優(yōu)化其性能特點和整體生活。一個以上的目標的存在使問題為多目標優(yōu)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. 任何約束多優(yōu)化優(yōu)化問題基本上是由三部分組成，即，設計參數(shù)，目標函數(shù)，約束（確定可行的設計參數(shù)空間）。我們簡要地討論，這些組件將在以后的章節(jié)中存在的問題。3.1. Design parameters 設計參數(shù)The design p

48、arameter vector can be written as: 設計參數(shù)向量可以寫為：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,參數(shù)定義的軸承內部幾何形狀的DM，DB，Z，F(xiàn)I，F(xiàn)O（見附錄A的命名）。然而，kdmin，kdmax，E，和是制約 11 （參見第3.2節(jié)和3.3）和不直接代表任何測量軸承內部幾何形狀。后者通常是在設計軸承 7 保持不變，但在目前的情況下，這些次要的參數(shù)也被視為變量。這可能是由于的靈活性

52、和所采用的基于遺傳算法的方法提供的魯棒性。所有的測量角度弧度，毫米與最小油膜厚度的例外尺寸（HMIN），是測量微米，和軍隊在牛頓（N）。裝配角度（O）的軸承（見圖2）也形成在滾動元件的數(shù)量的一個重要約束。基于 11 提出的幾何推導，可以得出以下公式進行裝配角度，(3)where,T=D-d-2Db.(4)Fig. 2. A rolling bearing showing the assembly angle. 滾動軸承的裝配角度3.2. Objective functions 目標參數(shù)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，有三個重要的滾動軸承性能的措施。這些都是即，動態(tài)能力（CD），最小油膜厚度（HMIN），和靜態(tài)電容（Cs）。所有的人都是同時最大化，使軸承的最佳性能。這些性能參數(shù)在下面的小節(jié)中更詳細地討論。3.2.1. Dynamic capacity (Cd) 動態(tài)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:不同的目標之間的滾動軸承，動態(tài)能力（CD）是最重要的，因為這直接形成最長的疲勞壽命的軸承的基礎上。動態(tài)能力，也被稱為動態(tài)負載率，定義為“恒定的徑向負荷，這一組顯然相同的軸承可以承受一百萬次革命的內圈額定壽命（對于一個固定的負載和固定的外環(huán)）”。它表現(xiàn)為 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. 這里的因素，不是一個獨立的參數(shù)。對于目前的討論，深溝球軸承