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2 0 07 年第7 期 工艺与装备 文章编号: l 0()l 一 2 2 6 5 ( 2 (X)7 ) 0 7 一 的9 3 一 0 3卧式加工中心配置回转夹具回转中心位置误差的检验李生斌， 董欣胜( 大连机床集团有限责任公司 自 动化所， 辽宁 大连1 1 662 0)摘要: 在卧式加工中心上配置回转夹具是国内外机床制造技术的发展方向。卧式加工中心配置回转夹具主要是由夹具体、 伺服电机、 转台和液压系统等装置组成的一种复杂的夹紧结构， 适用于汽车行业缸体、 缸盖等零件的加工， 在卧式加工中心装配调试工作中， 经常遇到卧式加工中 心配置回转夹具( 转台+夹具) 专机， 而确定回转夹 具回转中心位里成为该专机调试工作的技术关键， 该文主要研究回转夹具回转中心的确定， 夹具主定位面( 含定位销) 误差的判断、 调整及其机床三个坐标轴运动之间的关系 。关键词: 卧式加工中心; 回转夹具; 夹具主定位面中图分类号: T G65 9文献标识码: A The T es t o f H o ri z o n 加 I Ma c hi ni n gC e n t e r D i s pose s S 初ng F i xt u rea nd G y r a t i o nC e n te r LIS h e n g 一 b i n ， D O N GX i n 一 she n g( A u t o m at i o ni n s t i t u t e t e c h n i q u e d e P artm e n t D al i a nM a c h i n e T ool G r l u pC o rp， D al i a n U a o n i n g l l 6620， C h i n a )A bst r a c t : D is P o s e t h e s w in g fi xt ure o nh o rizo ni alm ac h in in g cen t eri s t h e d e v eloP in g d ir ee t io no f b o t hh er eand abro ad r n a n u fa c t u n n g t e c h n o fo 留. 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T h i s artic lei s r n 别 叮 fys t u dy t h e d e fmit i o n o f s w in g fi xt ure and留ratio n c en t er ， t h e erro r j udg m e n t an d adju s t r O e n t o f flXture mamfo c at in g s u r fa c e ( in c hi d e lo c at in g P in ) ， an dt h erelatfono f 3c o o rdinatea 石 smo t ion .K e y words : h o n Z o n t allT 比 c h in in g c enter ; s w in g 6 xt ure ; 行 狱 ure IT 坦 mfo catin g s u ri 勿 c e1 问题的提出 卧式加工中心配置回转夹具主要有两种典型结构， 一种是卧式回 转夹具， 如图1 所示。图1 中回转夹具的组成主要有转台、 夹具体、 四个主定位面及两个定位销等; 另一种是水平回转夹具， 如图2 所示。除水平转台外， 其余与图1 所示基本相同。 上述典型结构的加工中心专机在出厂合格证中除三坐标单元的几何精度规定外， 还有如下要求: ( 1)夹具主定位面与X轴方向的平行， 检验示意图如图3 ， 其允差: 0 . 02/ 全长;定位销夹具体 主定位面图1 臣 卜 式回转夹具图2 水平回转夹具 ( 2 ) 夹具主定位面与2轴方向的平行， 检验示意图如图4 ， 其允差: 0 . 02/ 全长; ( 3 ) 两定位销中心线与X轴方向的平行， 检验示意图如图5 ， 其允差: 0 . 0 2 / 全长;收稿日 期: 2 007 一 05一 1 仇修回日 期: 2 (X)7 一 06一 18作者简介: 李生斌， 男， 任职于大连机床集团有限责任公司， ( E 一 m ail) dxs9 27dl . cn。9 3 . ，.， 臼 卜 卜 ， 劲士 月 2 曰 2 二 刁二 友 拍可组合机床与自动化加工技术在 0 “ 位蓦 律 尹4 宕 曰- - 一一 - 一 一厂 、一一二 一 - 一 一肠凡 彦 ，在 1 8 0 位图3 夹具主定位面与X轴移动平行在 0 。 位寥了 厂 嗯 多- - 一 - 一 - 一 一户 一- 一 - 一 - 一 一钾 尸 几 聋 曰在 1 8 0 。 位图4 夹具主定位面与2轴移动平行数) : 2 / 测量长度; 表示: X轴移动与回转中心线平行误差值; 公式n: ( 0 。 位测量仪表读数 + 1 8 0 “ 位测量仪表读数) + 2 / 测量长度; 表示夹具主定位面与回转中心线的平行误差值。 现将本实例实测值代人公式 1 计算得: 仁 一 0 . 03-( 一 0 . 0 3 ) * 2 / 全长= 0 / 全长; 表示X轴移动与回转中心线平行误差为0 / 全长。 将实测值代人公式 n计算得: 一 0 . 03 +( -0 . 03) + 2 / 全长=一 0 . 0 3/全长; 表示主定位面与回 转在 0 。 位 X 6 口-卜- - 一 - 一 - 一 一介丘一龙。/在 1 8 0 0位中心线平行误差为0 . 0 3 / 全长。 误差简图为图 6( 注: 止 表示表头朝下，寸 表示表头朝上) 。 ( 3 ) 对策: 将误差计算值与出厂合格证比较( 略) 。 一 0 . 0 3- - - 一 气1 8 0 0位夹具回转中心线 图5 两定位销中心线与X轴移动平行 在总装交检几何精度时， 以坐标单元x 、 Y 、 2三个坐标移动来检验与夹具主定位面( 或两定位销中心线)的平行误差: 可能出现如下问题: 转台0 “ 位时在合格证允差范围之内， 而转台转到1 80。 位时， 其平行超差;转台0 “ 位时其平行超差， 转到 1 8 0 “ 位时平行仍然超差。出现这样情况， 不但几何精度不合格， 同时影响加工精度， 因此必须认真分析， 查找原因， 采取相应措施。2 个案分析 在实际精度检验中， 会出现很多不同的检验结果。为了把错综复杂的情况分析清楚， 现归纳出一些实例进行分析。 本文主要以卧式转台回转夹具( 一面两销定位) 为例， 见图1 ， 将夹具主定位面( 1 、 2 、 3 、 4 点) 和定位销( 5 、6 点) 检测出现的情况分别进行分析。2 . 1 主定位面1 、 2 、 3 、 4 点检测及分析 实例 1 : 当X轴移动与回转中心线平行但主定位面与回转中心线不平行时， 见图3 所示。 ( 1)检测: 转台在0 “ 位时， 移动X坐标轴， 检测朝上的主定位面1 、 2 、 3 、 4点， 测量仪表在 1 、 2面上的读数为0 ， 测量仪表在3 、 4 面上的读数为一 0 . 03; 将转台转到1 80。 时， 移动X坐标轴， 检测朝下的主定位面1 、2 、 3 、 4 点， 测量仪表在1 、 2 面上读数为0 ， 测量仪表在3 、 4面上的读数为一 0 . 03。 ( 2 ) 计算与判断: 对于这种检测方法， 推理归纳出几个误差计算公式: 公式 1 : ( 0 “ 位测量仪表读数一 1 80“ 位测量仪表读 9 4 一 滋 00位一X图6 主定位面与回转中心线平 行误差0 . 03 时误差简图 如果误差计算值超差， 可以采取如下方法处理: 一种方法是修磨定位面1 、 2 点， 去掉0 . 03m m ; 另一种方法是加高定位面3 、 4 点0 . 03m m 。 实例2 : 当X 轴移动与回转中心线不平行， 但主定位面与回转中心线平行时， 见图3 。 ( 1)检测: 转台在0 “ 位时， 移动X坐标轴， 检测朝上的主定位面1 、 2 、 3 、 4点， 测量仪表在 1 、 2面上的读数为0 ， 测量仪表在3 、 4 面上的读数为一 0 . 03; 将转台转到1 80“ 时， 移动x轴坐标轴， 检测朝下的主定位面1 、 2 、 3 、 4点， 测量仪表在1 、 2 面上的读数为0 ， 测量仪表在3 、 4 面上的读数为+ 0 . 03。 ( 2 ) 计算与判断 现将本实例实测值代人公式1 计算得: 一 0 . 03-( + 0 . 03) 令 2 / 全长=一 0 . 0 3 / 全长; 表示X轴移动与回转中心线平行误差值为0 . 03/ 全长。 将实测值代人公式n 计算得: 一 0 . 0 3 + ( + 0 . 0 3 ) + 2 / 全长 = 0 / 全长; 表示主定位面与回转中心线平行误差为0/全长。 误差简图为图7 。 ( 3 ) 对策: 将误差计算值与 出厂合 格 证 比较( 略) 。子 -1 8 0 0位夹具回转中心线 一 石 龙 吞 00位一一-卜图7 主定位面与回转中心 线平行误差为 0时误 差简图2 0 0 7年第7 期 . ， .叫 日， 卜 竺 习士 通 汾 J2 二. 刁 习 斌拍 即 如果误差计算值超差， 只需调整夹具底座下面安装平面， 使回转中心线与X坐标轴移动平行即可。 实例3 : 当X轴移动与回转中心线不平行， 且夹具主定位面与回转中心线也不平行时， 见图3 。 ( 1)检测: 转台在0 “ 位时， 移动X坐标轴， 检测朝上的主定位面1 、 2 、 3 、 4 点， 测量仪表在1 、 2 面上的读数为0 ， 测量仪表在3 、 4面上的读数为一 0 . 08; 将转台转到1 80“ 时， 移动x坐标轴， 检测朝下的主定位面1 、2 、 3 、 4 点， 测量仪表在1 、 2 面上的读数为0 ， 测量仪表在3 、 4 面上的读数为一 0 . 0 1 。 ( 2 ) 计算与判断 现将本实例实测值代人公式1 计算得: 一 0 . 08 -( 一 0 . 01) + 2 / 全长 =一 0 . 0 3 5 / 全长， 表示X轴移动很复杂: 其中有夹具体刚度不足， 也有可能是夹具回转轴两端安装不同轴， 紧固端面是否与回 转中心垂直等， 采取逐个排除法一一解决。2 . 2 分析定位销5 、 6 ， 见图5所示 针对定位销5 、 6 的分析可能出现情况: ( 1)当X轴轴向与在 1 8 0 0 位一 众 。 屯3 点1 、2 点夹具回转中心线+ 0 . 。 3 厂 4 点_一X3 点 一 0 . 0 3在1 5 0 位4 点+ 0 . 0 3图9 主定位面与中心线平行误 差0 ， 06 时误差简图与回 转中心线平行误差为 将实测值代人公式n计算 得:一0 . 08 +( 一 0 。 0 1 ) + 2 / 全长 二-0 . 045 / 全长， 表示主定位面与回转中心线平行误差为0 . 0 4 5 / 全长。 误差简图为图8 。 ( 3 ) 对策: 从本示例可以清楚地判断出两种不0 . 0 3 5 / 全长。图8 主定位面与回转中心线 平行误差为0 . 0 45 时误 差简图同的误差交错在一起: 一种是夹具回转中心线与X轴移动间安装误差; 另一种是夹具主定位面与回转中心线间误差， 可采用上述实例 1 、 实例2 的对策分别解决。 实例4 : 当夹具体主定位平面发生扭曲变形时， 见图3 、 图4 所示。 ( 1)检测: 转台在0 “ 位时， 移动X坐标轴， 检测朝上的主 定位面1 、 2 、 3 、 4 点， 测量仪表在1 、 2 面上的读数为0 ， 测量仪表在3 、 4 面上的读数也为0;转台在1 8 00时， 移动X坐标轴， 检测朝下的主定位面1 、 2 、 3 、 4点， 测量仪表在1 、 2 面上的读数为0 ， 测量仪表在3 面上的读数为一 0 . 03， 测量仪表在4 面上的读数为十 0 . 03。 ( 2 ) 计算与判断 按X轴方向为基准， 1 点与3 点连线朝上倾斜 -0 . 0 3 / 全长， 而2 、 4 点连线朝下倾斜十 0 . 03/ 全长。 按2 轴方向为基准， 1 、 2 点连线为0 ， 而3 、 4 点连线为0 . 0 6 / 全长， 反映出3 、 4 点与1 、 2 点发生了“ 扭曲变形” 。 误差简图为图9 。 ( 3 ) 对策: 出现“ 扭曲变形” 这种现象， 分析其原因也回转中心线平行， 但两定位销连线与回转中心线不平行时; ( 2 ) 当X轴轴向与回转中心线不平行， 但两定位销对回转中心线平行时; ( 3 ) 当X轴轴向与回转中心线不平行时， 且两定位销连线与回转中心线也不平行时。 针对定位销的检测方法， 推理归纳出如下误差计算公式: 公式nl: ( 0 。 位测量仪表读数+l80。 位测量仪表读数)于 2/测量长度， 表示X 轴移动与回 转中心 线平行 误差; 公式W: ( 0 “ 位测量仪表读数一 1 80“ 位测量仪表读数) : 2 / 测量长度， 表示两定位销连线与回转中心线平行误差， 具体的检测、 计算与判断及对策请参照主定位面的检测与分析。3 结论 卧式加工中心配置回转夹具， 虽然种类繁多， 但都与上述的典型示例大同小异， 原理基本相同， 因此， 搞清楚上述实例后， 遇到再难的问题也会迎刃而解。 以上归纳出的一些公式实用方便( 应用这些公式时， 应注意检测方法是否一致) ， 便于统一认识， 有利于提高装配制造技术， 使产品质量上升到一个新的台阶，提高效率， 取得良 好的经济效益。 参考文献【 1 加工中心检验条件 第一部分 卧式和带附加主轴头机床几 何精度检验( 水平2 轴) J B/T8 7 7 1 . 1 1 9 9 8 . 2 形位和位置公差的代号及其注法 G Bll 82一 8 0.【 3 王广彦， 等. 加工中心在线测头在自由曲线检测中心应用 J . 组合机床与自 动化加1技术， 2 (X)3 ( 7 ) : 3 0 一 3 2 .【 41 田 延岭， 张大卫， 同兵. 纳米的xy微定位平台的设计与研 究【 J 组合机床与自 动化加工技术， 2 加5 ( 1 2 ) : 83一 85. ( 编辑 李秀敏)Int J Adv Manuf Technol (2001) 17:471476 2001 Springer-Verlag London LimitedComparative Analysis of Conventional and Non-ConventionalOptimisation Techniques for CNC Turning ProcessR. Saravanan1, P. Asokan2and M. Sachithanandam21Department of Mechanical Engineering, Shanmugha College of Engineering, Thanjavur, India; and2Department of ProductionEngineering and Management Studies, Regional Engineering College, Tirchirapalli, IndiaThis paper describes various optimisation procedures forsolving the CNC turning problem to find the optimum operatingparameters such as cutting speed and feedrate. Total pro-duction time is considered as the objective function, subject toconstraints such as cutting force, power, toolchip interfacetemperature and surface roughness of the product. Conven-tional optimisation techniques such as the Nelder Meadsimplex method and the boundary search procedure, andnon-conventional techniques such as genetic algorithms andsimulated annealing are employed in this work. An example isgiven to illustrate the working procedures for determining theoptimum operating parameters. Results are compared and theirperformances are analysed.Keywords: Boundary search procedure; Genetic algorithm;NelderMeadsimplexmethod;Optimisation;Simulatedannealing1.IntroductionOptimisation of operating parameters is an important step inmachiningoptimisation,particularlyforoperatingCNCmachine tools. With the general use of sophisticated and high-cost CNC machines coupled with higher labour costs, optimumoperating parameters are desirable for producing the parteconomically. Although there are handbooks that provide rec-ommended cutting conditions, they do not consider the econ-omic aspect of machining and also are not suitable for CNCmachining. The operating parameters in this context are cuttingspeed, feedrate, depth of cut, etc., that do not violate any ofthe constraints that may apply to the process and satisfyobjective criteria such as minimum machining time or minimummachining cost or the combined objective function of machin-ing time and cost.Correspondence and offprint requests to: R. Saravanan, Departmentof Mechanical Engineering, Shanmugha College of Engineering, Than-javur 613402, Tamilnadu, India. E-mail: saradharaniKMachining optimisation problems have been investigated bymany researchers. Gilbert 18 presented a theoretical analysisof optimisation of the process by using two criteria: maximumproduction rate and minimum machining cost. Since the resultsobtained by using these two different criteria are always differ-ent, a maximum profit rate, which yields a compromise resulthas been used in subsequent investigations. These early studieswere, however, limited to single-pass operations without con-sideration of any constraints. Subsequent studies included vari-ous operating constraints in the optimisation models and severaltechniques have been tried for optimisation 17. In order toaccount for the stochastic nature of tool use, various probabilis-tic approaches have also been proposed. Since multipass oper-ation is often preferred to single-pass operation for economicreasons, efforts have been made to determine optimal machin-ing conditions for multipass operations 2,8,9.So far, machining optimisation problems have been investi-gated by many researchers using conventional techniques suchas geometric programming 3,4, convex programming 10,and the Nelder Mead simplex search method 1,2. Ruy Mes-quita et al. 7 have also used a conventional technique (directsearch method combination of Hook and Jeeves and randomsearch) for optimisation of multipass turning. A mathematicalmodel proposed by S. J. Chen was used in this work. It isobserved that the conventional methods are not robust because:1. The convergence to an optimal solution depends on thechosen initial solution.2. Most algorithms tend to become stuck on a suboptimal sol-ution.3. An algorithm efficient in solving one machining optimisationproblem may not be efficient in solving a different machin-ing optimisation problem.4. Computational difficulties in solving multivariable problems(more than four variables).5. Algorithms are not efficient in handling multiobjective func-tions.6. Algorithms cannot be used on a parallel machine.To overcome the above problems, non-conventional tech-niques are used in this work for solving the optimisation of472R. Saravanan et al.the turning process. In earlier work, a standard optimisationprocedure using a genetic algorithm 11 was developed tosolve different machining optimisation problems such as turn-ing, face milling and surface grinding 12. In this work,in addition to a GA, the simulated annealing technique hasbeen tried.In this work, the mathematical model proposed by Agapiou1,2 is adopted for finding the optimum operating parametersfor CNC turning operations. Total production time is consideredas the objective function, subject to constraints such as cuttingforce, power, toolchip interface temperature, and surfaceroughness of the product. Conventional optimisation techniquessuch as the Nelder Mead simplex method and the boundarysearch procedure, and non-conventional techniques such asgenetic algorithms and simulated annealing are used in thiswork. An example is given to illustrate how these proceduresare used to determine the optimum operating parameters.Results are compared and their performances are analysed.2.The Optimisation Problem 12.1Objective FunctionThe total time required to machine a part is the sum of thetimes necessary for machining, tool changing, tool quick return,and workpiece handling:Tu= tm+ tcs(tm/T) + tR+ thThe cutting time per pass (tm) = PDL/1000 V f. Taylors toollife equation is:V fa1doca2Ta3= KThis Eq. is valid over a region of speed and feed for whichthe tool life (T) is obtained.2.2Operating Parameters2.2.1FeedrateThe maximum allowable feed has a pronounced effect on boththe optimum spindle speed and production rate. Feed changeshave a more significant impact on tool life than depth of cutchanges. The system energy requirement reduces with feed,since the optimum speed becomes lower. Therefore, the largestpossible feed consistent with the allowable machine power andsurface finish is desirable, in order for a machine to be fullyused. It is often possible to obtain much higher metal removalrates without reducing tool life by increasing the feed anddecreasing the speed. In general, the maximum feed in aroughing operation is limited by the force that the cutting tool,machine tool, workpiece and fixture are able to withstand. Themaximum feed in a finish operation is limited by the surfacefinish requirement and can often be predicted to a certaindegree, based on the surface finish and tool nose radius.2.2.2Cutting SpeedCutting speed has a greater effect on tool life than either depthof cut (or) feed. When compared with depth of cut and feed,the cutting speed has only a secondary effect on chip breaking,when it varies in the conventional speed range. There arecertain combinations of speed, feed, and depth of cut whichare preferred for easy chip removal which are mainly dependenton the type of tool and workpiece material. Charts providingthe feasible region for chip breaking as a function of feedversus depth of cut are sometimes available from the toolmanufacturers for a specific insert (or) tool, and can be incor-porated in the optimisation systems.2.3Constraints1. Maximum and minimum permissible feed rates, cuttingspeed, and depth of cut:fmin# f # fmaxVmin# V # Vmaxdocmin# doc # docmax2. Power limitation:0.0373 V0.91f0.78doc0.75# Pmax3. Surface roughness limitations especially for finish pass:14 785 V1.52f1.004doc0.25# Ramax4. Temperature constraint:74.96 V0.4f0.2doc0.105# umax5. Cutting force constraint:844 V0.1013f0.725doc0.75# Fmax3.Conventional Optimisation Techniques3.1Boundary Search Procedure (BSP)To start with, the depth of cut (doc) is fixed and the feedrate(f) is assumed to be at the maximum allowable limit. Thevalues of “doc” and “f” are substituted in power and tempera-ture constraint equations and the corresponding cutting speeds(Vpand Vt) are calculated. Further the obtained values “f” and“V” are accepted or modified according to constraint violationand objective function minimisation using the bounding phaseand interval halving method the flowchart of which is givenin Fig. 1.3.2Nelder Mead Simplex Method (NMS) 13In this method, three points are used in the initial simplex. Ateach iteration, the worst point (Xh) in the simplex is found. Atfirst, the centroid (Xo) of all but the worst point is determined.Thereafter, the worst point in the simplex is reflected aboutthe centroid and the new point (Xr) is found. If the constraintsare not violated at this point the reflection is considered tohave taken the simplex to a good region in the search space.Thus, an expansion (Xe) along the direction from the centroidto the reflected point is performed. On the other hand, if theconstraints are violated at the reflected point, a contraction(Xcont) in the direction from the centroid is made. The procedureConventional and Non-Conventional Optimisation Techniques473Fig. 1. Flowchart of “boundary search procedure”.Table 1. By boundary search procedure.NumberdocVfTU12.0119.770.7622.8422.5112.950.7622.9333.0114.020.6803.1143.5118.280.5823.3454.0122.130.5093.5964.5124.450.4523.8475.0125.100.4064.10is repeated until the termination criteria are met. The followingequations are used in the method:Xo= (Xl+ Xg)/2Xr= 2Xo XhXe= (1 + g) Xo g XhXcont= (1 + b) Xo b XhTable 2. By Nelder Mead simplex method.NumberdocVfTU12.0118.320.7502.8722.5113.130.7382.9733.0108.670.6603.1543.5117.500.5463.4454.0126.250.4733.6964.5125.420.4633.8475.0126.350.4204.10Where, g = expansion coefficient (2.0); b = contraction coef-ficient (0.5).The flowchart is given in Fig. 2.4.Non-Conventional OptimisationTechniques4.1Genetic Algorithms (GA) 1417GAs form a class of adaptive heuristics based on principlesderived from the dynamics of natural population genetics. Thesearching process simulates the natural evaluation of biologicalcreatures and turns out to be an intelligent exploitation of arandom search. A candidate solution (chromosomes) is rep-resented by an appropriate sequence of numbers. In manyapplications the chromosome is simply a binary string of 0and 1. The quality is the fitness function which evaluates achromosome with respect to the objective function of theoptimisationproblem.Aselectedpopulationofsolution(chromosome) initially evolves by employing mechanismsmodelled after those currently believed to apply in genetics.Generally, the GA mechanism consists of three fundamentaloperations: reproduction, crossover and mutation. Reproductionis the random selection of copies of solutions from the popu-lation according to their fitness value, to create one or moreoffspring. Crossover defines how the selected chromosomes(parents) are recombined to create new structures (offspring)for possible inclusion in the population. Mutation is a randommodification of a randomly selected chromosome. Its functionis to guarantee the possibility of exploring the space of sol-utions for any initial population and to permit the escape froma zone of local minimum. Generally, the decision of thepossible inclusion of crossover/mutation offspring is governedby an appropriate filtering system. Both crossover and mutationoccur at every cycle, according to an assigned probability. Theaim of the three operations is to produce a sequence ofpopulations that, on average, tends to improve.4.1.1The Optimisation Procedure Using GAStep 1. Choose a coding to represent problem parameters, aselection operator, a crossover operator and a mutation operator.Choose a population size n, crossover probability pc, andmutation probability pm. Initialise a random population ofstrings of size l. Choose a maximum allowable generationnumber tmax. Set t = 0.474R. Saravanan et al.Fig. 2. Flowchart of “Nelder Mead simplex method”.Step 2. Evaluate each string in the population.Step 3. If t . tmax(or) other termination criteria is satisfied,terminate.Step 4. Perform reproduction on the population.Step 5. Perform crossover on random pairs of strings.Step 6. Perform bitwise mutation.Step 7. Evaluate strings in the new population. Set t = t+1and go to step 3.End4.1.2GA ParametersSample size= 20Crossover probability= 0.8Mutation probability= 0.05Number of generations= 1004.2Simulated Annealing (SA) 16The simulated annealing procedure simulates the annealingprocess to achieve the minimum function value in a minimis-ation problem. The slow cooling phenomenon of the annealingprocess is simulated by controlling a temperature-like parameterintroduced using the concept of the Boltzmann probabilitydistribution. According to the Boltzmann probability distri-bution, a system at thermal equilibrium at a temperature T hasits energy distributed probabilistically according to P(E) =exponent of (DE/kT), where k is the Boltzmann constant. Thisexpression suggests that a system at a high temperature hasan almost uniform probability of being at any energy state,but at a low temperature it has a small probability of beingat a high energy state. Therefore, by controlling the temperatureT and assuming that the search process follows the Boltzmannprobability distribution, the convergence of an algorithm canbe controlled.Conventional and Non-Conventional Optimisation Techniques475Table 3. By genetic algorithm.NumberdocVfTU12.0118.910.7642.8522.5114.150.6443.1233.0114.490.6653.1343.5120.610.5313.4654.0106.160.5653.5164.5104.800.4543.9675.0110.580.4354.14Table 4. By simulated annealing.NumberdocVfTU12.0120.390.7532.8522.5112.570.7612.9333.0116.680.6483.1543.5118.210.5823.3454.0122.050.5073.5964.5125.410.4483.8575.0126.280.4004.12The above procedure can be used in the function minimis-ation of the total production time in the CNC turning optimis-ation problem. The algorithm begins with an initial point x1(V1, f1) and a high temperature T. A second point x2(V2, f2)is created at random in the vicinity of the initial point and thedifference in the function values (DE) at these two points iscalculated. If the second point has a smaller function value,the point is accepted; otherwise the point is accepted with aprobability exp (DE/T). This completes one iteration of thesimulated annealing procedure. In the next generation, anotherpoint is created at random in the neighbourhood of the currentpoint and the Metropolis algorithm is used to accept or rejectthe point. In order to simulate the thermal equilibrium at everytemperature, a number of points are usually tested at a parti-cular temperature, before reducing the temperature. The algor-ithm is terminated when a sufficiently small temperature isobtained or a small enough change in function values is found.4.2.1The Optimisation Procedure Using SAStep 1. Choose an initial point x1(V1, f1), a terminationcriteria e.Table 5. The total production time obtained from different techniques and percentage deviation of time with reference to BSP methods.NMSGASABSPNumberdocTUTU% devTU% devTU% dev12.02.842.87+1.02.85+0.42.85+0.422.52.932.97+1.03.12+1.02.93033.03.113.15+1.03.13+1.03.15+1.043.53.343.44+3.03.46+3.03.34054.03.593.69+3.03.51+3.03.59064.53.843.88+1.03.96+1.03.85+0.375.04.104.23+3.04.14+3.04.12+0.5Set T a sufficiently high value, number of iterations to beperformed at a particular temperature n, and set t = 0.Step 2. Calculate a neighbouring point x2. Usually, a randompoint in the neighbourhood is created.Step 3.If DE = E(x2) E(x1) , 0, set t = t + 1;Else create a random number r in the range (0,1). Ifr # exp (DE/T) set t = t+1;Else go to step 2.Step 4.If ux2 x1u , e and T is small, terminate.Else if (t mod n) = 0 then lower T according to acooling schedule.Goto step 2;Else goto step 2.5.Data of the Problem 1L = 203 mmD = 152 mmVmin= 30 m min1Vmax= 200 m min1fmin= 0.254 mm rev1fmax= 0.762 mm rev1Ramax(rough) = 50 mmRamax(finish) = 20 mmPmax= 5 kWFmax= 900 Numax= 500Cdocmin(r)= 2.0 mmdocmax(r)= 5.0 mmdocmin(f)= 0.6 mmdocmax(f)= 1.5 mma1 = 0.29a2 = 0.35a3 = 0.25K = 193.3tcs= 0.5 min/edgetR= 0.13 min/passth= 1.5 min/pieceCo= Rs3.50/minCt= Rs17.50/edge6.Optimisation ResultsTables 1 to 4 show the operating parameters such as cuttingspeed, feed rate and total production time obtained usingconventional and non-conventional techniques.Table 5 shows the total production time obtained fromdifferent techniques and percentage deviation of time withreference to BSP method.7.ConclusionIn this work, optimisation of the CNC turning process issolved using conventional and non-conventional optimisationtechniques. An example is given to illustrate the workingprocedures of these techniques in solving the above problem.It is observed that a boundary search procedure is able to findthe exact answer, but it is not flexible enough to include more476R. Saravanan et al.variables and equations. The results obtained by simulatedannealing are comparable with the boundary search procedureand the flexibility of the method needs further investigation.The Nelder Mead simplex method deviates from the boundarysearch method by 1%3%, but with simple modification it canbe extended to other machining problems by including onemore variable. The genetic Algorithm method also deviatesfrom boundary search method by 1%3%, but by adopting asuitable coding system, this can be used to solve any type ofmachining optimisation problem such as milling, cylindricalgrinding, surface grinding, etc., by including any number ofvariables.References1. J. S. Agapiou, “The optimization of machining operations basedon a combined criterion, Part 1: The use of combined objectivesin single pass operations”, Transactions ASME, Journal of Engin-eering for Industry, 114, pp. 500507, 1992.2. J. S. Agapiou, “The optimization of machining operations basedon a combined criterion, Part 2: Multipass operations”, Trans-actions ASME, Journal of Engineering for Industry, 114, pp. 508513, 1992.3. D. S. Ermer, “Optimization of the constrained machining econom-ics problem by geometric programming”, Transactions ASME,Journal of Engineering for Industry, pp. 10671072, 1971.4. B. Gopalakrishnan and Faiz Al-Khayyal, “Machine parameterselection for turning with constraints: an analytical approach basedon geometric programming”, International Journal of ProductionResearch, pp. 18971908, 1991.5. D. Y. Jang and A. Seireg, “Machining parameter optimizationfor specified surace conditions”, Transactions ASME, Journal ofEngineering for Industry, 114, pp. 254257, 1992.6. Y. C. Shin and Y. S. Joo, “Optimization of machining conditionswith practical constraints”, International Journal of ProductionResearch, 30, pp. 29072919, 1992.7. Ruy Mesquita, E.
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