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Optimization of machining parameters in milling Singh, Kulbir 1992

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OPTIMIZATION OF MACHINING PARAMETERS IN MILLINGByKulbir SinghB.E. (Mechanical Engineering)South Gujarat UniversityM.Sc. (Applied Mathematics)Simon Fraser UniversityA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1992© Kulbir Singh, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department ofThe University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractThe cost of machining for milling is dependent on machining parameters such as spin-dle speed and feed per tooth. The competitiveness of manufacturing industries can beincreased by optimization of machining parameters. A scientific method for the opti-mization of machining parameters for workpieces of continously varying radial widths isproposed in this thesis. The necessary mathematics for the proposed procedure is de-rived for both single pass and multi-pass milling operations. The computational resultsobtained on the basis of derived mathematical formulation are analysed and discussed.The cutting direction has a considerable influence on the cost of machining. Therefore,an algorithm to determine the influence of cutting direction on machining cost is alsosuggested. The best cutting directions for a number of workpieces of known geometryare ascertained on the basis of computational results.iiTable of ContentsAbstract^ iiList of Figures^ viiiList of Tables^ ixNomenclatureAcknowledgement^ xii1 Introduction 11.0.1^Thesis Outline^ 42 Literature Review 63 The Economics of Milling Process 183.1 Cost Equation ^ 213.2 Tool Life Equation 223.3 Process Constraints ^ 253.3.1^Tooth Breakage Constraint ^ 263.3.2^Shank Breakage Constraint 263.3.3^Power and Torque Constraints ^ 293.3.4^Chatter Constraint ^ 293.3.5^Surface finish Constraint 30iii4 Optimization of Single Pass Milling Operations with Irregular Work-pieces^ 314.1 Mathematical Formulation ^  324.1.1 Evaluation of Machining Time ^  425 Optimization of Multi-Pass Milling Operations^ 535.1 Mathematical Formulation ^  545.1.1 Case 1: dr min < dr < s2  ^585.1.2 Case 2: s2 < dr < si  ^585.1.3 Case 3: s i < dr < dr  ^595.1.4 Evaluation of Cost  ^605.2 Evaluation of Centre to Centre Tool Traverse Distance ^ 725.2.1 Case 1: drn.„ < dr < s2  ^725.2.2 Case 2: s2 < dr < si  ^746 Optimal Cutting Direction for N-sided Polygonal Surfaces^756.1 Mathematical Formulation ^  756.1.1 Analysis of Results  ^946.1.2 Conclusions  ^94Bibliography^ 95A Coordinate Transformation^ 97A.0.3 From Old to New  97A.0.4 From New to Old ^  98B Pseudo-code for single pass milling operations^ 103ivC Pseudo-code for two-pass milling operations^ 110D Pseudo-code for optimal cutting direction^ 120E Flow Chart for Single Pass Optimization^ 139vList of Figures1.1 Schematic Diagram of a CNC System ^ 21.2 Adaptive Machine Tool Control System Block Diagram ^ 22.1 Boundary representation of a polyhedron ^ 102.2 Block Diagram of Feedback Control Loop (Tlusty) ^ 152.3 Block Diagram of Adaptive Control Loop (Altintas) 173.1 Basic Geometry of the Milling Process ^ 193.2 Variation of total variable cost with cutting speed ^ 203.3 Variation of machining time with cutting speed 203.4 Mohr-circle Diagram for Shank Stresses ^ 264.1 Generalized Workpiece Geometry (Top view) 314.2 Influence of Tool Radius on Cost for a Generalized Workpiece ^ 354.3 Influence of Tool Radius on Cost for a Triangle ^ 364.4 Optimum Feeds and Tool Traverse Distance For a Generalized Workpiece 374.5 Optimum Feeds and Tool Traverse Distance For a Triangle ^ 384.6 Optimum Speeds and Tool Traverse Distance For a Generalized Workpiece 394.7 Optimum Speeds and Tool Traverse Distance For a Triangle ^ 404.8 Geometry of Portion ABEF (Top View) ^ 444.9 Geometry of Portion BCDE (Top View) 485.1 S i < dr < dr. (Top view) ^ 545.2 s 2 < ci,. < s i (Top view) 55vi5.3 dr,n,„ < d, < s 2 (Top view)  ^555.4 Cost Characteristics of a Rectangle  ^605.5 Cost Characteristics of a Triangle  ^615.6 Cost Characteristics of an Equilateral Triangle  ^625.7 Cost Characteristics of a Symmetric Generalized Workpiece ^ 635.8 Cost Characteristics of an Unsymmetric Generalized Workpiece ^ 645.9 Cost Characteristics of a Rectangle (varying tool diameter)  ^655.10 Cost Characteristics of a Triangle (varying tool diameter)  ^665.11 Cost Characteristics of an Equilateral Triangle (varying tool diameter)^675.12 Cost Characteristics of a Symmetric Generalized Workpiece (varying tooldiameter)  ^685.13 Cost Characteristics of an Unsymmetric Generalized Workpiece (varyingtool diameter)  ^695.14 Geometry for Centre to Centre Tool Traverse Distance ^ 726.1 Geometry of a Generalized 6-sided Polygon (Top view) ^ 756.2 Cutting Direction 0 (Top view)  ^766.3 Swept angle for a 5-sided Symmetric Polygon ^  786.4 Cost characteristics for a 5-sided Unsymmetric Polygon  ^816.5 Cost characteristics for a 5-sided Unsymmetric Polygon  ^826.6 Cost characteristics for a 5-sided Unsymmetric Polygon  ^836.7 Cost characteristics for a 5-sided Symmetric Polygon  ^846.8 Cost characteristics for a 5-sided Symmetric Polygon  ^856.9 Cost characteristics for a 5-sided Symmetric Polygon  ^866.10 Cost characteristics for a Non-equilateral Triangle  ^876.11 Cost characteristics for a Non-equilateral Triangle  ^88vii6.12 Cost characteristics for a Non-equilateral Triangle  ^896.13 Cost characteristics for an Equilateral Triangle  ^906.14 Cost characteristics for an Equilateral Triangle  ^916.15 Cost characteristics for an Equilateral Triangle  ^92A.1 Coordinate Transformation:0 < 1,0 < a ^  99A.2 Coordinate Transformation:0 < 1,0 > a  99A.3 Coordinate Transformation:0 = 1, 0 > a ^  100A.4 Coordinate Transformation:0 > 1,0 > a  100A.5 Coordinate Transformation:0 = 7r, 0 > a ^  101viiiList of Tables4.1 Table of constants  ^345.1 Machining costs for single pass milling operations  ^71ixNomenclatureK3 :^specific cutting pressurea:^axial depth of cutst:^feed per toothsmax:^maximum allowable feed per toothZ:^number of toothN:^spindle r.p.m.d:^radial width of cutR:^tool radiusv:^tool traverse rateSeq :^equivalent feed per tooth0:^instantaneous angle of immersionswept angle of cutCh:^machine cost rateCt :^tool costtool change timeX:^thermal fatigue parameterEr :^range of thermal strain parameterte :^cooling timeM:^moment on the cutterT:^torque on the cutterTmax:^maximum allowable torque on the cutterOrmax:^maximum allowable shank stressPmax:^maximum allowable powerAt:^sampling intervalFt :^tangential forceFT :^radial forceFx :^cutting force in the x-directionFy :^cutting force in the y-directionxiAcknowledgementI am indebted to Dr. Ian Yellowley for accepting me as his graduate student. I would liketo thank him for his valuable help and guidance during my studies. I am also thankful toYetvart Hosepyan and Philip Richard Pottier for making the work environment pleasantand enjoyable with their barbed humour. Their help in the use of various computersoftware is also acknowledged. Finally, I take this opportunity to express my feelings ofgratitude for all my friends especially the "Michael Saliba Lunch Gang". Their friendshipand company provided a much needed respite from a "dreary" world of academics.xiiJ 4-5 , e) av )WTS r.5 H ow3-bres. Tv. urtft 1176)-r cPJ-Itv5 otcli *576 ' e) ci f T-r3 Tia I IChapter 1IntroductionLow costs and high productivity are important requirements of competitive economies.Sophisticated manufacturing systems, such as Flexible Manufacturing Systems ( FMS )and Computer Integrated Manufacturing ( CIM ), are quite effective in meeting thesedemands. Untended or minimally manned machining centres are the most versatileform of computerised manufacturing. These systems offer a significant technologicaladvancement in terms of quality, design, time and costs, and may increase the user'scompetitive advantage.The most advanced automatic manufacturing systems utilize computers as an integralpart of their control. The era of advanced automation started with the introduction ofnumerical controlled machine tools. The term numerical control is commonly used forprogrammable automation; a demand for high accuracy in manufacturing and a desireto reduce the production time as well as the necessity to produce complex geometrieswere the primary motivations for the development of these machines. Numerical con-trolled machines are hardware based machines which use electronic hardware and digitalcircuit technology. In order to increase the flexibility of these systems controllers basedupon general purpose computers rather than specialised hardware were introduced. Themachines which use these controllers are called CNC (Computer Numerical control) ma-chines. The schematic diagram of a CNC system is shown in Fig. (1.1). These machinetools use a computer to control the machine tool and eliminate some of the hardwarecircuits in the control cabinet of an NC machine tool. CNC systems are quite flexible1I Interpolator Controller ServoAmplifier1 Encoder —,—,,— MachineToolTachometerAxis Motor ^•■■gk..,■,..Chapter 1. Introduction^ 2ComputerFigure 1.1: Schematic Diagram of a CNC SystemFeedMicrocomputerbasedadaptivecontroller^--*-SpeedFeedback datafrom sensorsDepthof cutMachinetoolcontrolsystem ^a.MachinetoolFigure 1.2: Adaptive Machine Tool Control System Block DiagramChapter 1. Introduction^ 3and because of the declining costs of minicomputers and microcomputers the number ofCNC systems has increased tremendously in last few decades.The full potential of CNC systems can be realized only if there are realistic strategies ofprescribing the operating parameters like speeds and feedrates of the machine tool. Quiteoften, the prescription of these parameters is based on the experience and knowledge ofthe part programmer. For the prevention of tool breakage and the safety of machine tool,the most adverse machining conditions, (which might not occur in reality), are taken intoconsideration. Therefore, the estimate of operating parameters tends to be conservative,which results in an under-utilization of machines and production losses. This commondrawback of CNC systems can be overcome to some extent by the adaptive controlstrategy. In adaptive control, the operating parameters automatically adapt themselvesto the existing conditions of machining in real time. Adaptive control systems for machinetools can be divided in two categories:• Adaptive control with optimization (ACO)• Adaptive control with constraints (ACC)In ACO systems, the extremum of a specified performance index is obtained withinprocess and system constraints. The performance index can be an economic functionsuch as the process cost or profit. In ACC systems the maximum possible machiningparameters within a prescribed region bounded by process and system constraints areselected. In reality most ACC systems use a single easily measured parameter as the onlyvariable. The block diagram of an adaptive machine tool control is shown in Fig. (1.2).Accurate measurement of tool wear or tool life is important for the implementation ofadaptive machine tool control. The capability of the computing equipment to do all thecalculations in real-time is also crucial for the performance of adaptive control systems.Chapter 1. Introduction^ 4Most of the objective functions used for the optimization of machining parameters arenon-linear functions. Direct search techniques or hill climbing methods are required forthe numerical solution of these problems. These techniques are time consuming andquite often it is not possible to find the optimal solution in real-time adaptive control.The implementation of adaptive control becomes easier if the optimization of machiningparameters is carried out beforehand and the optimum values are stored in a database.The primary focus of this thesis is to suggest procedures to determine the optimalmachining parameters for milling with various workpiece geometries. Both single passand multi-pass milling operations are considered. Keeping in view the strong influence ofcutting direction on the optimization procedure, an algorithm for the computation of anoptimal cutting direction is also suggested. Some researchers have proposed the adaptivecontrol of the feedrate based on the maximum allowable force. This method is comparedwith the strategy proposed in this thesis.1.0.1 Thesis OutlineA brief literature review on optimization of machining parameters, adaptive control andtool path planning for milling process is presented in chapter 2 of this thesis. Chapter 3discusses the economics of a milling process. The tool life equation, the objective functionand mathematical relations for the constraints are discussed in that chapter. Chapter4 suggests a scientific basis for the selection of machining parameters for a workpieceof a given geometry. The optimization of machining parameters for single pass millingoperations and a mathematical formulation for the evaluation of machining time is alsoincluded in that chapter.Since, in reality, a given cutter can only cut a certain maximum radial width ofcut, (because of physical or dynamic constraints), some practical guidelines have beensuggested for the subdivision of the total machining surface of the workpiece. Chapter 5Chapter 1. Introduction^ 5contains the multiple pass optimization problem for milling operations.Finally, in chapter 6 the problem of determination of the influence of cutting directionon cost has been addressed, an attempt has been made to determine the cutting directionthat will minimize the cost for a generalized workpiece of given dimensions.Chapter 2Literature ReviewThe examination of the economics of the milling process is an important research topic.Optimization of machining parameters is a highly desirable goal of any economic studyof milling process. Adaptive control and tool path planning also have a considerableinfluence on the process cost of milling. However, relatively little attention has been paidto this topic by manufacturing engineering researchers. This chapter presents a briefoverview of some of the work done in the areas of optimization of machining parameters,tool path planning and adaptive control for milling process.Yellowley and Desmit [1] have modelled the single pass optimization problem formilling and have developed a suitable algorithm for its solution. They consider variablecost as the single objective function. An algorithm which minimizes this objective func-tion within four inequality constraints is used to suggest the optimum values of cutterdiameter, feed per tooth and peripheral velocity for a given geometry of shoulder. Animportant conclusion for the selection of cutter diameter and number of teeth is expressedas follows:"In general, it would seem that it is preferable to either use the smallest diameterof cutter available which is capable of machining the required width, or the first largerradius having a greater number of teeth."Chang and Wysk [2] have proposed an optimization criterion based on the discretetransformation method. The objective function for this study is not process cost but6Chapter 2. Literature Review^ 7production time which is expressed as follows:TT = T, + TtL^uLVf + VfP+ 1 An, d, R, Z)whereTc^actual cutting timeTt^tool change time per unit of workTT^production time per piece (idle time excluded)L^length of work pieceu^unit tool change time (could be a function of R and Z)Vf^cutter traverse speedn^spindle rotation rated^radial depth of cutR^cutter radiusZ^number of teethp^constant coefficientThe tool life equation used in the study is of the following form:TL, = VI f (n, d, R, Z)^ (2.1)It is clear from the expression for production time per piece that the objective functioncontains five decision variables Vf , n, d, R and Z. Some of these variables like spindlespeed and number of teeth belong to discrete sets such as ST, and S. This fact isused for the discrete transformation of the objective function. Consequently only threevariables are left and the solution procedure is drastically simplified.TTChapter 2. Literature Review^ 8The optimization procedure suggested by Chang and Wysk is a good analytical exer-cise but, somehow, falls short of practical requirements. Minimization of production timeis a desirable goal for those economies which have a scarcity of goods. The minimizationof production time is a primary manufacturing objective in very rare situations. In amodern, free-market, with fiercly competitive economies the real challenge is of reducingcosts and not of reducing production time. Also, the model proposed by Chang and Wyskdoes not take into account important constraints such as the tooth breakage and shankbreakage constraint. An optimization procedure which does not safegaurd the cuttingtool from breakage is of limited practical use.Chatter and the physical dimensions of the tool impose a limit on the maximum radialwidth for a single pass. In cases where the amount of stock to be removed in a roughmilling operation exceeds the allowable width, there is a need for a multipass operation.Yellowley and Gunn [3] have examined the problem of multipass milling operations. Thefollowing mathematical expression is used for the tool life equation:Tr, =360^C60, Xn1SVa(2.2)where^C6^is a constant^Cbs^is the swept angle of cut in radian^X^is the thermal fatigue parameterSeq^is the equivalent chip thichness in mm^V^is the peripheral velocity in mm per sec.a^is the axial depth of cut in mma, b, m^are positive exponentsChapter 2. Literature Review^ 9A similar tool life equation is discussed in detail in the next chapter.The cutting time per unit length for each pass is given bytco(i )^C7f2 (d,) ( 1-A )f1 (di )A^(2.3)f1 (d) and f2 (d) are obtained from the following relations:S f2(d) < C3VT'S fi (d)A = C2where^C7^is a constant^V^is the cutting velocity^T ^is the tool lifeS ^is the feed/rev.^C2^is a constant^ 3^is a constantd ^is the radial widthand0<a<#<1^ (2.4)The objective function is reduced to the following mathematical form:nE(f2(d1)(1 -0)fi(di)')^(2.5)0<d,< dmaxE = dTOTi=10 < a < # < 1A^A B^B DChapter 2. Literature Review^ 10BCBCCFigure 2.1: Boundary representation of a polyhedronwhere^d,^is the radial depth for the ith pass^( Lax^is the chatter limited depth^dTOT^is the total amount of stock to be removedThe objective function is minimized and an optimal solution is obtained. The optimalsolution has the property that all passes except one should be taken at the maximumallowable width of cut, with the one other pass used to remove the required remainingamount of stock.Some researchers have reasoned that the optimal metal removal rate should not bemodelled independently of the cutter path selected for the operation. The choice of a,proper cutter path can reduce the tool wear due to the lesser engagement of the toolwith the job, thus resulting in process optimization.Wang et al. [5] presented a mathematical model for computing an optimal tool cutterpath for face milling. They utilized this model to identify the minimum length of cutChapter 2. Literature Review^ 11for face milling flat surfaces. At the highest level a polyhedron is represented as a setof surfaces. Then each surface is broken down into line segments which compose thesurface. The lowest level consists of the starting and end points of vectors which are thevertices of surfaces. An illustration of boundary representation of a polyhedron is shownin Fig. (2.1). The equation of set of surfaces of a polyhedron would be of the followingform:AIX + Bl Y + CiZ < (or >)D iA2X + B2 Y + C2Z < (Or >)D2ANX + BNY + CNZ < (or >)DNEach of the above surfaces can be represented in terms of their boundary vectors asfollows:AT1XT + BT1YT < (or >)DTA.AT2XT + BT2YT < (or >)DT2ATNXT + BTNYT < (or >)DTNAn N-sided polygon is divided into (N-2) triangles and the length of cut Lm for eachtriangle is computed from an analytical expression. Finally, the lengths Lm for all the(N-2) triangles are added together to obtain the total tool travel length. This procedureChapter 2. Literature Review^ 12is used for a range of sweep angles from 0 to 180 degrees and the angle corresponding tothe minimum tool travel length is thus determined.The same authors in a second paper [5] have examined the two commonly used ap-proaches of stair case and window frame milling. Again a boundary representation schemeis used for the transference of input data about the workpiece geometries. A polyhedronis represented as a set of surfaces and each surface can be identified by the equation ofits edges. Edges are recognized by their starting or end pointsOnce the part geometry has been defined, the impact of the selection of a startingpoint and cutting orientation on tool path is studied for both stair case and windowframe milling. The conclusions of this study are summarized as follows:1. In window frame milling, the selection of a starting point does not significantlyaffect the length of cut, although a small amount of variation exists.2. The cutting orientation in stair case milling produces a significant impact on thelength of cut. The average variation is on the order of 5 — 10 percent. The worstcase can be as much as 100 percent.3. There appears to be no correlation between the optimal cutting orientation andother control parameters, such as tool diameter and number of edges.4. Based on the experimental results, the optimal length of cut generated by staircase milling is better than that generated by window frame milling. However, theaverage results from stair case milling are sometimes worse than those of windowframe milling.5. From the experimental results, the authors observed that, for stair case milling ofregular polygons, the optimal cutting orientation is normally parallel to the longestedge of a given polygon.Chapter 2. Literature Review^ 13Prabhu and Wang [6] have also developed a mathematical model representing thetotal tool path on an N-sided convex polygon surface. They have considered the stair-case type of tool path. The mathematical formulation is complex and can not be solvedby standard analytical or numerical methods. They have proposed an algorithm to findan optimal solution between 0 and 180 degrees. The conclusions of this study are statedas follows:1. For a triangle the optimum sweep angle seems to be the one which makes the sweeppath parallel to the largest side of the triangle, which is consistent with the findingsof Wang et al. [5].2. For a square or rectangle or parallelogram the optimum sweep angle is the one thatmakes the sweep parallel to any one of its sides.3. If the square is divided into two triangles the sweep of the tool path parallel to thelargest side of the triangles does not give an optimal solution.4. A series of local optima of the objective function make global optimization difficult.The above mentioned tool path planning studies for milling use length of cut as theonly optimization criterion. The authors of these studies have indicated that shortestlength of cut would minimise the tool wear. It is, however, not clear why these researchershave chosen this objective. In most practical situations minimization of process cost isthe foremost consideration and in some rare cases minimization of production time isthe primary goal of manufacturing planning. Minimization of tool wear may not fulfilany of these objectives. It has been proven by Yellowley [1] and Chang et al. [2] thatboth process cost and production time depend on the machining parameters. Therefore,a study on optimal tool path planning without any regard to machining parameters andconstraints may not be too useful for manufacturing engineering.Chapter 2. Literature Review^ 14It has been mentioned in chapter 1 that the adaptive control of machine tools is aneffective way of selecting machining parameters and reducing costs. Several researchershave proposed the adaptive control of machine tools by varying the feed rate adaptivaly,and keeping the cutting forces below the limiting value.Tlusty et al. [7] proposed the following relationhip between the cutting force and theworkpiece traverse rate:Fact(t) = Cvact (t — T)Fact (s) = Cvact (s)e-"whereT^is the tooth periodT represents the time delay between velocity change and force change. The above re-lations are for actual values of these parameters and not for commanded values. Theproportionality constant C is expressed as follows:C = Kba^ (2.6)where^K^depends on the workpiece materialb ^is the axial depth of cut^a^is the radial depth of cutThe force error is evaluated as follows:F Man — Facto f = ^Fmam(2.7)- Fag.,=a = 0(e/) F = f(v)Chapter 2. Literature Review^ 15dDDA.zvc,„„,w■ra...i.,....,•DDA yN/C ServozN/C ServoYMini ComputerSoftware Interpolator!kernvs Table IfFigure 2.2: Block Diagram of Adaptive Control Loop (Tlusty)whereFnom^is the desired cutting force which shouldbe kept constant by adapting feedrateThe actual cutting force is measured by the dynamometer attached to the spindle.This force signal is compared with the nominal force F„,,, n . The result of the comparisonis the relative force error e f . Based on this force error a desired change of velocity is ex-pressed as acceleration a = 0(ef). Integration of this expression gives us the commandedvelocity v.The block diagram of the adaptive control scheme proposed by Tlusty is shown inFig. (2.2).Tomizuka et al. [9] based their study on the model reference adaptive control method.The milling process was treated as a first order dynamic process with time varying pa-rameters. Daneshmend [10] used a similar strategy for the turning process. Due toChapter 2. Literature Review^ 16cutting process to a simple time varying gain with a time invariant pole.Tomizuka et al. [8] treated the dynamics of the feed drive as a gain which is not validfor dc servo controlled machine tools. Moreover, none of these above mentioned studiestook into account the inevitable nonlinearities associated with systems of these kind.Altintas et al. [10] developed an adaptive control strategy based on linear dynamicsof the plant with simple nonlinearities. The milling process to be controlled is consideredto have two cascaded dynamic processes. The time-invariant feed drive servo control andthe time variant cutting process dynamics. The discrete transfer function of the feeddrive servo is expressed as:c(k)[mm/tooth]Gs(z)^u(k)[count/8]kpz -1 (1 + ziz -1 ) C3 (z) =^1 + pi z -1wherekp,^are constantsThe time variant cutting process dynamics is expressed by the following discrete transferfunction:f3z -1Fp(k) = i^c(k) + 11 + az -^+ az - 1(2.8)The process parameters a, and -y are time varying and functions of the workpiecegeometry. The maximum cutting forces are regulated by estimating the time varyingparameters a, and -y at each sampling period. The method of Normalized RecursiveLeast Square is used to estimate these parameters. The block diagram of the adaptivecontrol scheme proposed by Altintas is shown in Fig. (2.3).None of these works on adaptive force control have indicated the method of calculatingthe maximum reference force. It is also not certain how the feedrates obtained on theChapter 2. Literature Review^ 17ADF, Controller u ServoRu = T F,. — S F, C•(z)■.......40.■FONLP--"4 Estimator.c CuttingprocessFigure 2.3: Block Diagram of Adaptive Control Loop (Altintas)basis of this strategy are kept below the tooth breakage constraint. Therefore, thereis a need to conduct a closer investigation of adaptive force control strategy so that acomparison between this strategy and the optimization strategy proposed in this thesiscan be made. However, it has been decided not to include the research related to thistopic in this thesis.Chapter 3The Economics of Milling ProcessThere are two kinds of industries in a free market economy - competitive and closed. Thecompetitiveness of an industry can only be established by thoroughly studying all thedetails of the manufacturing cost and finding ways and examining means of reducing thiscost. The cost of machining can only be reduced by the proper selection of machiningparameters. The best selection is made when the value chosen for these parameters issuch that cost is minimized or profit maximized.Milling is an important metal cutting process, but relatively little attention has beenpaid to the economics of this process. The basic geometry of the milling process is shownin the Fig. (3.1). The machining parameters of interest in a milling process are:• Tool radius R• Spindle rotational speed N• Feed per tooth s t• Radial width of cut d• Axial depth of cut a• Number of teeth ZOptimization of these parameters can be based on several different objectives. Someof these objectives are:18Chapter 3. The Economics of Milling Process^ 19Figure 3.1: Basic Geometry of the Milling Process• Minimization of process time• Minimization of process cost• Maximization of profitDepending upon specific circumstances any one of these criteria may be important.Each criterion will typically lead to the selection of different conditions. Barrow haspresented an analysis for each of the above mentioned objectives. Minimization of processcost is the most common objective and, therefore, the present work is based on thisobjective.The cost of producing a part can be divided into fixed and variable costs. Fixedcosts are independent of the machining process; these costs consist of machine centre setup costs and raw material costs. Therefore, the goal of an economic model which is ofinterest to manufacturing engineers is to minimize the variable costs. Figs. (3.2) and(3.3) show the variation of total variable cost and machining time with cutting speedNon productivecostMachiningcostTool costToial costTime^Tool change timeTotal timeMachiningtimeNon productivetimeChapter 3. The Economics of Milling Process^ 20V1Cutting speedFigure 3.2: Variation of total variable cost with cutting speed1/2Cutting speedFigure 3.3: Variation of machining time with cutting speedChapter 3. The Economics of Milling Process^ 21respectively. These graphs indicate that the velocity giving the minimum cost is lessthan the velocity giving minimum time corresponding to maximum production rate.3.1 Cost EquationYellowley [1] has proposed an equation for the cost of milling per unit length at constantwidth and depth of cut. This equation can be expressed as follows:Ci = ^ + ^ -t-Ch^Ct , ChTctV VTL VTL(3.1)whereC1^is the process cost per unit length in dollarsCh^is the machine cost rate in dollars per secondCt^is the tool cost in dollarsTa^is the tool change time in secondsv^is the tool traverse rate in mms per secondTL,^is the tool life in secondsThe values of economic parameters used in the above equation are site specific andare therefore dependent on the shop/machine/tool combination.The machine cost rate includes labour, plant operating costs and machine operatingcosts. Tool change cost is calculated by multiplying tool change time by machining costrate. Tool cost is determined by multiplying the tool life fraction used in the machiningoperation by the total cost of the tool.Chapter 3. The Economics of Milling Process^ 223.2 Tool Life EquationMilling is an extremely complex process. It is not only a discontinuous process, but isalso affected by both mechanical and thermal shock. Milling is also characterized by avariable chip thickness during cut. Research workers in Germany are of the belief thatthe mechanical effects are more important whereas Japanese and Soviet workers considerthermal effects to be the more prevalent. Yellowley [11] has suggested that a realistictool life equation can be obtained by considering only the thermal effects provided thatonly one mode of milling is considered and chip formation at exit is not problematic.There are many ways of defining the useful tool life but the most common criterionis related to flank wear. A tool is considered to have reached the end of its life when itreaches the maximum limit of wearland width V. The rate of change of the width ofwearland with respect to time can be approximated by the following relation:di7B 1713'^ ,_dt^Tr,whereTr,^is the tool life corrresponding to VI;The above equation may be used to evaluate an expression for the equivalent feed inmilling in the following manner:The relation between tool life and equivalent chip thickness is assumed to be of thefollowing form:hTt, = C^ (3.3)whereh^is the chip thicknessk^is a constant with value quite close to unity^(3.4)(3.2)Chapter 3. The Economics of Milling Process^ 23Therefore,CTL = —hConsequently, the rate of change of wear land width takes the following form:dVB VI; hdt^CIntegrating the above expression to evaluate the total wear in a swept angle, we obtain:q = Cvi; / 3 s t sin OdOoThe expression for average wear rate is :vA  ^vi; pp.^O0,^CO, ./o st sin OdYellowley has proposed the concept of this so called equivalent feed rate to combine theinfluence of cutter diameter, width of cut and feed per tooth on the milling process. Theequivalent feed rate is defined as that constant feed rate which will yield the same averagewear rate as the variable chip thickness in milling. Using this concept of equivalent feedin Eq. (3.6), the average wear rate can also be expressed as:dVB^vb. seq( dt )ave = CFrom Eqs. (3.8) and (3.9), we obtain:Vi; Se gvi; 0 .C — co , 10 St sin NO(3.9)(3.10)(3.5)(3.6)(3.7)(3.8)Orst JO.S„^.= —^sin OdO0(3.11)The influence of the intermittent nature of the milling process on tool life can be wellrepresented by the thermal fatigue parameter, introduced by Yellowley [11] as :X = ED (NO^ (3.12)27r^Cl71 , = Os X m .5;19 VP aq (3.14)Chapter 3. The Economics of Milling Process^ 24where^Er^range of thermal strain parameter^N^rotational speedx^ratio of total cycle time to time in cutIt is clear from the above relation that the thermal fatigue parameter is dependenton both the range of thermal strain and the number of thermal strain cycles per unitcutting time. The range of thermal strain is a function of heating and cooling time. Thevalues for this parameter do not vary too much for high speed steels and carbide toolmaterials for the same heating and cooling times. The mathematical expression for therange of thermal strain parameter is as follows:Er = 39log t c — 23log th + 37.5^ (3.13)wheretc^cooling timeth^heating timeBoth the concepts of equivalent feed and thermal fatigue parameter have been usedin the tool life equation which is based on the allowable amount of flank wear in the tool.Some of the process constraints are aimed at preventing catastrophic failure of the tool.The active tool life of a milling cutter is defined as :whereC1^is a constantChapter 3. The Economics of Milling Process^ 25^0,^is the swept angle of cut in radian^X^is the thermal fatigue parameterSeq^is the equivalent chip thichness in mmV^is the peripheral velocity in mm per sec.a^is the axial depth of cut in mmm, n, p, q^are positive exponentsThis equation is only valid when there is no chip sticking and when the lag angle betweenleading and trailing edges is very small. We have not considered the effect of mechanicalshock caused by entry and exit conditions in the above equation. Fortunately, in processeswhere chip sticking does not occur, the entry and exit conditions do not affect tool lifein milling. The effect of the tool/workpiece materials is reflected in the values of theconstants in the equation.3.3 Process ConstraintsThe physical properties of the work/tool pair, the capacity of the driving motor, thedynamics of the machining process (e.g.chatter) and part design specifications such assurface finish impose constraints on the values of machining parameters. Therefore, anyrealistic economic model must take into account these constraints. The constraints whichcan influence the economics of the milling process are listed below:• Tooth Breakage Constraint• Shank Breakage Constraint• Power and Torque Constraint• Chatter ConstraintChapter 3. The Economics of Milling Process^ 26• Surface Finish ConstraintThese constraints will be discussed one by one.3.3.1 Tooth Breakage ConstraintTooth breakage is defined as the loss of a significant portion of the edge of an individualtooth. A catastrophic tooth breakage will eventually result in damage to the workpieceand the machine. Therefore, in order to avoid this catastrophic tooth breakage it is essen-tial to control the maximum cutting stress experienced by the cutting teeth. Yellowleyhas defined the tooth breakage constraint limit as:St sin 95.9 < sma.^ (d < R)^(3.15)St < Smas^ (d > R)^(3.16)where08^is the swept angle in radians.Smax^is the maximum allowable feed per tooth in mm.s t^is the feed per tooth in mm.3.3.2 Shank Breakage ConstraintThe shank of a tool may fail under the combination of bending and torsional workingloads. To avoid failure, the maximum tensile stress allowed on the shank must be keptunder a critical value. This critical value depends on the geometry and mechanicalproperties of the shank. Let a represent the normal stresses on the tool and T be theshearing stress. This state of stress can be represented by Mohr-circle diagram as shownin Fig. (3.4). The maximum tensile stress is given by the distance OA on this diagram.OC + CAOC + AP — PCcrChapter 3. The Economics of Milling Process^ 27T7Figure 3.4: Mohr-circle Diagram for Shank StressesIt is clear from the geometry of this figure thatOA =OA =OC =PC =AP =PX =Therefore,OA =OA =a + NA T ) 2 + ( ci )2  — cli; + NA T )2 + ( i )2From elementary mechanics of material we know that for a circular cross section,MR0" = IChapter 3. The Economics of Milling Process^ 28TRT = Jwhere^M^is the moment on the cutterT^is the torque on the cutter^J ^is the polar moment of inertiaI ^is the moment of inertia^R^is the shank radiusHence,MR 1.1 TR^MROA =^+ (^ )2 + (^ )221 ^' 21After some simplification we obtain:2croA = wR3 [M + (M2 +7,2)2]Therefore, the mathematical expression for the torque constraint is:27rR3 [111 + (M2 + T 2 ) ] < Grmax(3.17)(3.18)(3.19)whereAmax^is the max. allowable tensile stress on the tool shankFor a specific tool with defined geometry and material, the tensile stress on the shankcan be controlled by varying the depth of cut (a), peripheral cutting speed (V) and cuttertraverse rate (v).Chapter 3. The Economics of Milling Process^ 293.3.3 Power and Torque ConstraintsPower and torque are machine constraints which are imposed by the maximum capacityof the motor. The violation of these constraints may cause a serious damage to the powerdrive, spindle shaft or workpiece. These constraints can be represented by the followinginequality relations:Kavd < Pm.^ (3.20)K aR( v )d < Tmax^ (3.21)where^K^is the specific cutting pressure^Pmax 3 Tmax^are the maximum allowable power and torque3.3.4 Chatter ConstraintA chatter threshold limits the width of cut (d) and the depth of cut (a). This thresholdmust not be exceeded if instability of the milling process is to be avoided. It is extremelydifficult to study the effect of width of cut on the occurrence of instability in milling.This is mainly due to the following reasons:• The width of cut influences both the magnitude and direction of the average resul-tant force.• The width of cut influences the frequency content of the milling force signal andthe basic frequencies are dependent on the cutter diameter and number of teeth.It is therefore extremely difficult to formulate a realistic chatter constraint without signif-icant specific machine tool and work/tool data. Chatter is neglected in the optimizationstudies.Chapter 3. The Economics of Milling Process^ 303.3.5 Surface finish ConstraintSurface finish imposes a constraint on the depth of cut and cutter traverse rate, especiallyfor fine finishing. However, its influence is usually not considered in the study of roughmilling operations. A theoretical estimate could of course be made based upon thekinematics of the process. This would not in most cases be indicative of actual finishbecause of radial run out, adhered material and dynamic effects. Therefore, surface finishconstraints are also not included in the optimization studies.Chapter 4Optimization of Single Pass Milling Operations with Irregular WorkpiecesThe milling operation is an intermittent cutting process. The rotating cutter with oneor more cutting teeth comes in contact with a translating workpiece producing a chipof variable thickness. The machining parameters for the milling process which can beoptimized are mentioned in chapter 3, and are once again listed below :• Tool radius R• Spindle rotational speed N• Feed per tooth s t• Radial width of cut d• Axial depth of cut a• Number of teeth ZYellowley and Desmit [1] have developed an algorithm for the selection of tool di-ameter, feed per tooth and peripheral velocity for a shoulder of given geometry. Manyworkpiece surfaces however have polygonal, circular or elliptic geometries. The radialwidths for these workpieces vary continously throughout the tool traverse length becauseof which the optimum machining conditions also keep changing. The problem of de-termining the optimum machining parameters for these workpieces is, therefore, quitecomplex but of immense practical value.3112 ----414u1' Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces32Figure 4.1: Generalized Workpiece Geometry (Top view)In this chapter we present a model which can be used for the selection of machiningparameters for a large variety of workpiece geometries. The general method and strategyon which the model is based is applied on a few specific workpiece geometries. However,the same approach is also valid for other geometries not discussed in this thesis.4.1 Mathematical FormulationLet us consider a workpiece geometry as shown in the Fig. (4.1). It is assumed that themaximum radial width encountered in the workpiece does not exceed the chatter limitand a tool which can machine the workpiece in a single pass is available in the machineshop. Also for simplicity the axial depth of cut is kept constant throughout this workunless otherwise specified.For process economy and real-time process control, the feed rate needs to be variedwith changing radial width as the tool traverse the workpiece length. This is done at aseries of sampling intervals. It is assumed that the tool traverses the whole workpiecewhereE xkk=1k2ZOt I -160^ .(E stkNk)11d2 — d312k=1d2 — d1Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces33length^+ /2 ) in n sampling intervals.letdi^be the radial width for the ith intervalx2^be the distance traversed by the tool in the ith intervalAt^is the sampling periodNi^is the R.P.M. for the ith intervals ty^is the feed per tooth for the ith intervalHere i varies from 1 to n. The whole workpiece length can be written as a summation ofthe distances traversed in all the sampling intervals as follows:11 + 12 E xt^ (4.1)Mathematically di can be obtained by one of the following two expressions:=^ki(E xk )^ (4.2)k=1di = d2 — k2[E(xk — 10]^(4.3)k=1It is clear from Eqs. (4.2) and (4.3) that the radial width at each sampling interval isdependent on the workpiece geometry and the history of spindle speed and machine feed.Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces34Using Eq. (3.1) the process cost of machining the ith interval of length x, can beexpressed as:Ct ChTct Cpi = (Ch^) tTL.^TL.(4.4)whereTL ,^is the tool life for the conditions at the ith intervalIn terms of the basic machining parameters, the Eq. (4.4) can be expressed as :Cp,^f(di,^ sty)^ (4.5)It has been proved by Yellowley that the highest allowable feed always results in theminimum cost. Therefore, for minimization of cost, we select the maximum allowablefeed without exceeding the limit imposed by any of the constraints. Now consider thetooth breakage constraint, Eqs. (3.14) and (3.15). For any milling process, the tool ispre-selected and hence the tool radius is constant. Therefore, the feed per tooth is eithera constant or a function of the radial width of cut as shown below:St, = Sin=^(d < R)Or^St, = g(di) (d > R)s maxg(d,) = ^s in(cos -1 (1 — 14))It has been shown earlier that the radial width of each time interval is a functionof the workpiece geometry and the history of spindle speed and machine feed. It canbe evaluated by using Eq. (4.2) or (4.3). Therefore, the feed which satisfies the torqueconstraint, Eq. (3.20), can be obtained from the above expressions. Equation (3.19) canthen be used to find the spindle speed within the power constraint which results in aminimum process cost.Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces35Symbol ValueCh 0.005 dollars/secCt .0396.R.ZTct 120 secC1 1179.36m 2n 1p 2q 0.5Pmax 7.5 K.W.K, 4140 N/sq. mm.Ti 0.3smar 0.2 mmarnax 1242 N/sq. mm.Table 4.1: Table of constantsThe process cost of machining the whole workpiece is :Cp E^(4.6)For minimum process cost for the whole workpiece we will have :CPmin E( CPi. )7n n^ (4.7)i =1The influence of cutter radii and number of teeth on cost for a generalized workpieceand a triangular workpiece when feeds and speeds are kept optimal for each intervalis shown in Figs.(4.2) and (4.3) respectively. The optimal feeds and speeds for both ageneralized workpiece and a triangular workpiece are graphically shown in Figs. (4.4),(4.5), (4.6) and (4.7). The values of economic parameters, tool life constant, tool lifeexponents, machine constraints, cutting constants and tool material properties used inthis thesis are listed in table 4.1. It is obvious from Figs. (4.2) and (4.3) that it ispreferable to use the smallest diameter cutter available which is capable of machining25 301^1^I^1^110^15^205 35Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces36dl =10,d2=20,d3=15,1i =30,12=20Radius in mmFigure 4.2: Influence of Tool Radius on Cost for a Generalized Workpiece10.504.543.532.521.5Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces37d'a =0,d2=20,d3=0,11=30,12=205^10^15^20^25^30^35Radius in mmleigure 4.3: Influence of Tool Radius on Cost for a Triangle0.20.270.260.250.240.230.220.21Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces38dl =10,d2=20,d3=15,11=30,12=200^10^20^30^40^50^60Tool traverse distance in mmFigure 4.4: Optimum Feeds and Tool Traverse Distance For a Generalized WorkpieceChapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces39d1=0,d2=20,d3=0,11=30,12=200.650.60.550.50.450.40.350.30.250.20R=28R=2010^20^30^40^50^60Tool traverse distance in mmFigure 4.5: Optimum Feeds and Tool Traverse Distance For a Triangle620600580560540520Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces40dl =10,d2=20,d3=15,11=30,12=20660640500R=20R=22 ^......,__.__._.......„,_......,-R=244800R=26I^I^I^1^I10^20^30^.40^50^60Tool traverse distance in mmFigure 4.6: Optimum Speeds and Tool Traverse Distance For a Generalized WorkpieceChapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces4ldl =0,d2=20,d3=0,11=30,12=20700 650 600550500 450 R=284003503000^10^20^30^40^50^60Tool traverse distance in mmR=20R=24Figure 4.7: Optimum Speeds and Tool Traverse Distance For a TriangleChapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces42the workpiece or the next larger diameter with more number of teeth. These graphs alsoindicate that the cost of machining a larger axial depth is higher as would intuitively beexpected. Graphs (4.4) and (4.5) indicate that the allowable feed per tooth decreaseswith increasing radial depths and also the value of allowable feed per tooth for each radialwidth increases with the tool diameter. No clear trend is available from the graphs ofoptimum spindle speeds.It is thus clear that with the help of the optimization strategy we have employed, itis possible to determine the optimal values of feed and speed for any kind of workpiecefor any cutting direction if we can obtain the radial widths either as a function of theworkpiece geometry or by some other means. The optimal values of feed and speed willresult in an optimal process cost.4.1.1 Evaluation of Machining TimeThe estimation of total processing time is important for efficient process planning andscheduling of manufacturing activities. The total processing time consists of the following:• Set up time• Loading unloading time• Machining process timeMachining process time consists of manual time and machining time. An analyticalexpression for the machining time in milling when the feed is maintained at an optimalvalue can be derived as follows:Machining time is a function of the workpiece geometry, tool diameter, number ofteeth and the tool traverse rate. The tool diameter and the number of teeth can beChapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces43selected on the basis of previous analysis and availability. The tool traverse rate is de-pendent on the machining constraints. Therefore, it is possible to mathematically evalu-ate the machining time for many workpiece geometries by making use of the constraintrelations.Let us again consider the geometry of Fig. (4.1). Tooth breakage can be avoided byrestricting the maximum chip thickness encountered to some constant value according tothe inequality relations of Eqs. (3.14) and (3.15). Rewriting those relations, we haveSt sin 0, < Smax^ (d < R)^(4.8)St < Smax^ (d > R)^(4.9)where0,^is the swept angle in radians.smax^is the maximum allowable feed per tooth in mm.s t^is the feed per tooth in mm.Case 1:d < RThe tooth breakage constraint for this case is governed by equation (4.8).Rewriting equa-tion (4.8)s t sin 0, < smaxThis equation gives:sst < maxsin 0,(d < R)^(4.10)(4.11)The tool traverse rate in mms per second can be obtained from the relation:dx s tNZdt^60(4.12)Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces44whereN^is the spindle r.p.m.Z^is the number of teeth.For optimum conditions, feed per tooth should be as large as possible. Therefore (4.11)assumes the form: 3 maxSt = •sin 0,Combining (4.12) and (4.13) we obtain:dx smax N Z(4.13)(4.14)dt^60 sin 0.A constant spindle R.P.M. which does not exceed the constraints can be determinedfor the given workpiece geometry. The selection of the tool is made before the startof machining so that the tool radius R and the number of teeth Z are constant duringmachining. The maximum permissible value of smax is also constant for a tool/workpiececombination. Therefore, the tool traverse rate can be represented by the relation:dx _ C^dt^sin 0.wheresmaxNZ^C =^60From (4.15), we obtain the following relation:dt = C—1 sin 0,,dx(4.15)(4.16)(4.17)We can integrate this equation to evaluate the time taken to traverse a specified length.lett i^is the time taken to traverse length d it2^is the time taken to traverse length / 2Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces45 d3ft---- 11 '-'•••0•1Figure 4.8: Geometry of Portion ABEF (Top View)Then for portion ABEF Fig. (4.8),we obtain:jot i^1dt C—^sin 0,dxo (4.18)The swept angle O a can be represented in mathematical form as follows:= cos -1 (1 — )1)^ (4.19)The radial depth of tool at an arbitrary position during its traverse can be found fromthe following expression:d =^kixThe slope k1 can be represented by the following relation:kl d2 — d1=(4.20)(4.21)11hereis the distance traversed by the toolChapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces46Using Eqs. (4.19), (4.20) and (4.21) in Eq. (4.18) we obtain the following expression:letel^ 1 fii^+ kix ]dxdt = C o sin cos -1 [1 (4.22)m 1 =m2d11— Rk1RIntegrating the left hand side of Eq. (4.22) and using the expressions for m 1 and m2 inthat equation, we obtain:1 ill^-1t i = —^sin cos (m i — m2x)dxC o(4.23)letcos "(m i — m2x) = y l^(4.24)thenm2 x^cosy].1 ,x = ---(m i --cosy l )m2sin y i dy idxm2Substituting the variable y i in place of x by using the above relations, equation (4.23)takes the following form:= ^1t i sin2 yi dyi2CmThe new limits of integration can be evaluated as follows:when(4.25)x = 0yl = cos 1 m1Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces47and whenX = 11y i = cos -1 (m i — m2 1 1 )Equation (4.25) can also be written in the form:ti^f (1 — cos 2yi)dyi= ^2Cm2Integrating the right hand side of this expression, we obtain:(4.26)ti = 1,^sin 2yi 2Cm2lYi )2(4.27)Applying the limits of integration to the above expression we obtain:1t i =2Cm2 [cos'^2(m i — m2 /1 ) — —1sin(2(cos' (m i — m2 1 1 )))1— cos -1 m l + 2— sin 2(cos -1 m l )]Rearranging the terms,12Cm21^ [sin (2 cos -1 m1 ) — sin 2(cos -1 (m i — m211))]+ 4Cm 2The time of traverse for any distance x between zero and 11 can be obtained from thefollowing expression:1tx = ^2Cm2 [cos -1 (m 1 — m 2x) — 2—sin 2(cos -1 (m i — m 2x)))1— cos -1 m l + —2 sin 2(cos -1 m l )]wheretx^is the time of traverse for a distance x between length 1 1x^is the distance traversed(4.28)ti [cos l (Mi — m 2 11 ) — cos 1 m l ]Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces48let1=2Cm2cos' (m i. — m2x)=M12 = COS -1 M1M13 = Sin[2(M11)]M14 = sin[2(m 12 )](4.29)With these substitutions, equation (4.28) becomes:\ M3 ,tx = M3(M11 — m12) + ^ (m14 — m13)2 (4.30)The above expression is valid for any distance x between zero and 11Similarly we can integrate equation (4.17) for portion BCDE Fig (4.9) as follows:fo^Ct2 dt = —1 f12 sin O a dxowhere(A8^is the swept angle as beforeFrom Fig. (4.9), the radial depth at an arbitrary point along length / 2 can be determinedfrom the following relation:d = d2 — k2 x^ (4.32)whered2 — d3k2 —  ^ (4.33)/2Substituting the above expressions for d and k 2 in equation (4.31), we have:fot2 dt C o= I 112 sin[cos'^d2 Rk2 x(1 ^ )]dx^(4.34)M3mi i(4.31)d2letilirr.^12 ........41Figure 4.9: Geometry of Portion BODE (Top View)Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces49d27125 = 1 — R-k2ms = iiIntegrating left hand side of Eq. (4.34) and using m5 and m6 in that equation, we obtain:1 /2 .t2 = C— sm[cos' (m 5 + m6x)]dx (4.35)letCOS -1 (1ns + M6X ) = Y2^ (4.36)thenm5 + msx = cos y21 ,z = —kcos y2 — m5 )smsin y2dy2dr = ^M6Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces50Substituting the variable y 2 in place of x in Eq. (4.35) we have:t2^Cm J=^1 f(sin2 y2)426The new limits of integration can be evaluated as follows:when(4.37)x = 0y2 = cos -1 m5and whenX = 12Y2 = COS -1 (M5 + m612)Writing Eq. (4.37) in a slightly different form:t2 L--- 1 ^f ( 1 — cos 2y2 )422Cm6 iIntegrating the right hand side of this expression, we obtain:1t2 = ^2Cm6( sin22y2 Y2)Applying the limits of integration :r2t2^2CM6= ^ [ sin 2 (cos -1 (m5 + m612))1— cos -1 (m5 + m6 /2 ) — 2 sin 2(cos' (m5 )) + cos -1 m5 ]Rearranging the terms,1^rt2 =_ ^ [COS-1 m5 — COS -1 (M6 + 7/2612 )12CM6+ 14Cm6 [sin 2(cos -1 (m 5 + m6 /2 ) — sin 2(cos -1 m 5 )](4.38)(4.39 )Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces5lThe time of traverse for any distance x between 1 1 and 1 2 can be obtained from thefollowing expression:2Cm614C^[sin 2(cos -1 (m5 m6x)) — sin 2(cos -1 m5 )1m6whereta,^is the time of traverse for a distance x between length 1 1 and length 12 .x^is the distance traversed.12c1 m66M17^cos-1(m5 + M6X)cos 1 m5M18 =Sin[2(M17)]M19M20 = sin[2(mis)]Substituting the new nomenclature in equation (4.40) we have:Ti/7 1M7(M18 — m17) +^m20)2The above equation is valid for any distance between 1 1 and 12(4.40)Case2:d > RThe tooth breakage constraint for this case is governed by the following equation:tx^1^ [cos -1 m 5 — cos -1 (m 5 m6x)]letM7St < Smas^ (d > R)^(4.41)Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces52let the operating feed s t0 be slightly less than the maximum allowable feed sm.., thenthe tool traverse rate in mms per second can be obtained from the relation:dx _ s toNZdt ^60heres t0^is a constant operating speedIntegration of the above expression yields:xt. C(4.42)(4.43)wheretx^is the time of traverse for a distance x between length (l i + 12 ).x^is the distance traversed.andC s toNZ-=  60(4.44)It is clear from Eqs. (4.30), (4.40) and (4.43) that the machining time for milling is afunction of the workpiece geometry, tool diameter, number of teeth and the tool traverserate. For many industries such as aircraft industry increasing the production rate isa highly desirable goal. The analytical expressions for machining time can be used todetermine the machining parameters which minimize the production time for specificworkpiece geometries. Since minimization of process cost is the focus of this thesis it hasbeen decided not to explore the criterion of minimization of production time any further.Chapter 5Optimization of Multi-Pass Milling OperationsThe maximum allowable radial width of cut for a single pass is dependent on the chatterconstraint and the size of the milling cutters available in the machine shop. When theradial width of cut for machining is larger than the maximum allowable value, there is aneed for multi-pass milling operations. This need often arises in rough milling operations.For these cases, it is therefore necessary to select an appropriate radial width for eachpass. This must be carried out in a manner which minimizes cost. Yellowley and Gunn [2]have given a mathematical formulation for multipass milling operations, and establishedan optimal selection procedure for the radial width of cut for each pass. This chapterdeals with the optimal selection of radial widths for various workpiece geometries.Let us once again consider the generalized workpiece, Fig (4.1), of chapter 3. Rewrit-ing Eq. (4.5)CPC = 34) (5.1)It was shown in chapter 3 that the objective for a single pass milling operation is tominimize the value of Cpz for each sampling interval. The radial width of the workpieced, for each sampling interval is fixed. For a constant cutter radius the feed per tooth s t ,is obtained from the tooth breakage and torque constraints. The spindle speed N, whichdoes not exceed the limit imposed by the power constraint and results in the minimumcost is selected for each sampling interval. The process cost of machining the whole53Chapter 5. Optimization of Multi-Pass Milling Operations^ 54workpiece is obtained from the following expression:Cp E Cpi^ (5.2)2=1For a multipass operation the radial width di for each interval can be varied by changingthe subdivision of the machining area. Our objective is then to minimize the followingexpression:k ni= min. E( E f (dt, , Art,' S tsj )^ (5.3)i=1 i3 =1Subscript i denotes the number of intervals and j denotes the number of passes. Forexample represents the radial width at the ith interval and jth pass. Mathemati-cal formulation and computational results for two pass milling operations are presentedbelow. These results will then be used to make conclusions about a general multi-passmilling operation.5.1 Mathematical FormulationThe generalized workpiece of Fig. (4.1) can be subdivided into three different geometricshapes depending on the maximum radial width dr selected for the first pass. A suitablerange for the selection of max. radial width of the first pass is chosen.let^d,.^be the maximum radial width of the first pass^dq^be the maximum radial width of the second pass^R^be the radius of the cutter for both the passes^dr,nin^be the lower limit in the range of max. radial width of the first passbe the upper limit in the range of max. radial width of the first passA i ,^be the workpiece length for the first pass^A2^be the workpiece length for the second passChapter 5. Optimization of Multi-Pass Milling Operations^ 55TFirst passSecond passFigure 5.1: S i < dy. < dr. (Top view)heredq = da — dy,nrl= 4.4 s i,n2A 2 = X isi2 =1Let us introduce the following geometric constants:s i = d2 —32 = d2 — d3Figs. (5.1), (5.2) and (5.3) show the three different ways in which the workpiece canbe subdivided when d, moves in its range from dims,, to d,„,... It is obvious that withthe change in the value of dr , the dimensions of each subdivision also undergo a change.Therefore, our aim is to determine the dimensions of each part at every value of drwithin the selected range and evaluate the machining cost. We can thus ascertain thebest subdivision of the workpiece for minimum machining cost..1•••••■d,Chapter 5. Optimization of Multi-Pass Milling Operations^ 56S First pass■•••■ MINN& ■1,. ■■•10^WENN. ••■■• .1■ND ■■•■■/Second passFigure 5.2: s2 < d,. < s i (Top view)First passT8 2Second passFigure 5.3: ds.„, < dr < 32 (Top view)Chapter 5. Optimization of Multi-Pass Milling Operations^ 57Using Eq. (4.4), the process cost of machining the ith interval of jth pass of lengthxii can be expressed as:Ct^ChTctCpi3 = (Ch m^m )AtIL,3^1 L ‘3(5.4)whereis the tool life for the conditions at the ith interval of jth passAt^is the sampling period^ (5.5)In terms of the basic machining parameters, Eq. (5.4) can be expressed as:= f(di , Arsa , St, )^ (5.6)The process cost of machining the whole workpiece is :k n3Cp E(E Cpii^(5.7)j=1 .3 =1For minimum process cost, we should have:k^113Cpmn -7-- DE(j=1 i.,=1In order to determine the influence of d,. on cost, we should be able to express themachining cost as a function of dr . It has been shown earlier (Eqs. (4.2) and (4.3))that the radial width is a function of the workpiece geometry and the history of spindlespeed and machine feed. In addition, in a two-pass milling operation, the radial widthwould also depend on the way the workpiece has been subdivided . In other words theradial width can be represented as a function of dr , workpiece geometry and the historyof spindle speed and feed. Workpiece geometry and the history of spindle speed and feedare known beforehand. Therefore, it is possible to determine the influence of dr on costby using Eq.(5.6).Let us mathematically derive the relationships for the radial width for the three casesrepresented by figures (5.1), (5.2) and (5.3).)min^ (5. 8)Chapter 5. Optimization of Multi-Pass Milling Operations^ 585.1.1 Case 1: dr ,„,„ < dr < 32For this case the radial width of the first pass is given by one of the following two relations:di, -=^E xk)k=1di l = dr k2(^xk)k=1dr(0 < E xk < —)k=1dr^a1-1^d,.^dr ,— < v-, x k < — —)kl^k=1^klk2whereZOt a -1E xk 60 (^3th Nk)k=1^ k=1d21 1d2 — d312The radial width of the second pass can be obtained from one of the following threerelations:=2-1^ i2-1^,^dr ,die =^+ ki( E xk) (o < E xk Li — —)k=1^ k=1d dr ,die = d2 — dr (/1 —^<^xk < /1 + 2)k=1i2-1^ 4^22-1(d2 — dr) — k2(E xk)^(/1^< E xk < /1 + 12)k=1 k2^k=14 4= k-1+ k-2AZ = +5.1.2 Case 2: s 2 < dr < s1For this case the radial width of the first pass is given by one of the following two relations:2, -1-1d„ = k i (^(o < E Xk < k--; )k=1 k=1kl =k2 =E Xkk=1Z At " -160 •(^s t,N,)k=1Chapter 5. Optimization of Multi-Pass Milling Operations^ 59t i _ i dr^drd1 = (d3 — (d2 — dr)) k2^(xk –^( Xk < J. + 1 2)kk=1 12;1where =d2 — d111k2d2 d312The expressions for the radial width of the second pass are :i2 -1^22-1d22 =^+ ki(^x,) <^< /, – —dr )k=1 k=1^-drd^12-1^— ^<^x, < + /2 )k=1The length of the workpiece for the two passes :drkJ.A2 = 11 -1- 125.1.3 Case 3: s 1 < dr < dr„,..The radial width of first pass can be obtained by one of the following two relations:^21 -1^ 21-1di ,^(di – (d2 – dr)) + ki(E <^Xk < li)^k=1 k=1it -1^ ti -1d21 = (d3 — (d2 — dr)) — k2^(xk – /1)^(ll <^Xk < 11 + 12)^k 1 k=1whered22 = d2 — drChapter 5. Optimization of Multi-Pass Milling Operations^ 6060^•(E s th Nk )Z At 21-1k=1d2— d111d2 d312The radial width of the second pass is:die = d2 —^(0 <^< /1 + 1 2)^(5.9)k=1The length of the workpiece for the two passes := 11 + 122 = 11 + 125.1.4 Evaluation of CostFeed per tooth for a constant tool radius can be obtained from the tooth breakageconstraint from the following relations:Stij = Smax^ (dOr^st,i = g(c1,2 ) (d > R)Smax g(dii)^cl,sin(cos -1 (1 — Ti9)The feed calculated from the above formula is reduced until it satisfies the torque con-straint. Eqs. (5.4), (5.6) and (5.7) can then be used to find that spindle speed withinthe power constraint which gives the minimum machining cost. The cost character-istics for a rectangle, triangle, an equilateral triangle, a symmetric and an unsymmetricgeneralized workpiece are shown in Figs. (5.4), (5.5), (5.6), (5.7) and (5.8) respectively.These cost characteristics are for a two pass milling operation when the speeds and the—1k=1 X kk 1k26510342Pass 1Pass 2T. costChapter 5. Optimization of Multi-Pass Milling Operations^ 61dl =d2=d3=20,I1=30,12=200^8^16^24Max. radial width of first pass in mmFigure 5.4: Cost Characteristics of a Rectangle3.532.510.5021.5Pass 1Pass 2--X--T. costChapter 5. Optimization of Multi-Pass Milling Operations^ 62dl =d3=0,d2=20,11 =30,12=200^8^16^24Max. radial width of first pass in mmFigure 5.5: Cost Characteristics of a Triangle8 16 2410.80.60.40.20Chapter 5. Optimization of Multi-Pass Milling Operations^ 63dl=d3=0,d2=20,I1=11.547,12=11.547Pass 1Pass 2XT. cos tMax. radial width of first pass in mmFigure 5.6: Cost Characteristics of an Equilateral Triangle6510342Pass 1XPass 2XT. costChapter 5. Optimization of Multi-Pass Milling Operations^ 64dl =1 0,d2=20,d3=1 0,11 =30,12=300^8^16^24Max. radial width of first pass in mmFigure 5.7: Cost Characteristics of a Symmetric Generalized WorkpieceChapter 5. Optimization of Multi-Pass Milling Operations^ 65dl =10,d2=20,d3=1 5,11 =30,12=20 54.543.532.521.510.50Pass 1XPass 2XT. costMax. radial width of first pass in mmFigure 5.8: Cost Characteristics of an Unsymmetric Generalized Workpiece3.532.5 Total costPass 22010.501.5I^I 10^4 8^12/."Pass 116 24Chapter 5. Optimization of Multi-Pass Milling Operations^ 66d1=20,d2=20,d3=20,11=30,12=20Max. radial width of first pass in mmFigure 5.9: Cost Characteristics of a Rectangle (varying tool diameter)Total cost Pass 2Pass 12.51.50.5Chapter 5. Optimization of Multi-Pass Milling Operations^ 67d1=0,d2=20,d3=0,11=30,12=204^8^12^16^20^24Max. radial width of first pass in mmFigure 5.10: Cost Characteristics of a Triangle (varying tool diameter)Chapter 5. Optimization of Multi-Pass Milling Operations^ 68dl =0,d2=20,d3=0,11=11.54,12=11.544^8^12^16^20^24Max. radial width of first pass in mmFigure 5.11: Cost Characteristics of an Equilateral Triangle (varying tool diameter)Chapter 5. Optimization of Multi-Pass Milling Operations^ 69dl =10,d2=20,d3=10,11=30,12=300^4^8^12^16^20^24Max. radial width of first pass in mmFigure 5.12: Cost Characteristics of a Symmetric Generalized Workpiece (varying tooldiameter)43.5310.502.521.5Chapter 5. Optimization of Multi-Pass Milling Operations^ 70dl =10,d2=20,d3=15,11=30,12=200^4^8^12^16^20^24Max. radial width of first pass in mmFigure 5.13: Cost Characteristics of an Unsymmetric Generalized Workpiece (varyingtool diameter)Chapter 5. Optimization of Multi-Pass Milling Operations^ 71feeds are selected in an optimal manner. Three different cutter radii are used and theaxial depth of cut is kept constant for simplicity.In the above discussion a tool of constant radius has been used for all values of dr . Inactual practice, a tool with a diameter equal to the larger of the two values of d,. and dqis capable of machining both the parts at a lower cost. This feature is included in graphs(5.9), (5.10), (5.11), (5.12) and (5.13).The machining costs for single pass operations for a few workpiece geometries arelisted in table (5.1). The following conclusions can be drawn from the above mentioneddata :• Single pass operations should be preferable to multi pass milling operations pro-vided the chatter limit is not encountered during the single pass operation.• The cost increases at a faster rate with increasing radial widths for smaller radialwidths.• The cost decreases at a much slower rate with decreasing radial widths for largerradial widths.• The machining cost for an area with larger radial widths will be less than that ofan equal area with smaller radial widths.• It is preferable to first machine the area with largest radial widths of cut withinthe chatter limit and take the remaining area in next passes.• The cost of machining is highest when the whole area is divided in two equal halves.Chapter 5. Optimization of Multi-Pass Milling Operations^ 72Workpiece geometry d1 d2 d3 11 12 Cost in centsUnsymmetric generalized 10 20 15 30 20 2.208859Symmetric generalized 10 20 10 30 30 2.708165Unsymmetric triangle 0 20 0 30 20 2.036704Equilateral triangle 0 20 0 11.54 11.54 0.660608Rectangle 20 20 20 30 20 2.122573Table 5.1: Machining costs for single pass milling operations5.2 Evaluation of Centre to Centre Tool Traverse DistanceThe machining time is dependent on the centre to centre tool traverse distance. Thisdistance is a function of the workpiece geometry and the tool radius. The accurateevaluation of this distance is important for a realistic path planning and machine control.The detailed mathematics required to evaluate the centre to centre tool traverse distanceis included in this section for the sake of completeness.5.2.1 Case 1 :d*min <^s2From the geometry of Fig. (5.14), the following relations can be obtained:= Z3 = Z4 = Z5 = /9 1^(5.10)L6 = 90 —pq = sin L6pq^Rsin(90 — 0 1 )pq^R cos 0 1 (5.11)as^d2 — (d1 + dr)qa = R — pq^ (5.12)qasin L5nqChapter 5. Optimization of Multi-Pass Milling Operations^ 73Figure 5.14: Geometry for Centre to Centre Tool Traverse Distanceqasin 0 1R2 + nq2xn2^one —R2nqon2= (5.13)(5.14)(5.15)Using the above relations, we obtain:xn2xn2xn2R2 + nq 2 — R2(  qa  )2sin 0 1 ( R — R cos 01 )2sin 0 1R — R cos 9 1xn = sin 0 1xn = R(1 — cos 0 1^ )sin 0 1where91 = tan-1(d2 — dl  )11Chapter 5. Optimization of Multi-Pass Milling Operations^ 74Similarlyym =-- R(1 — cos 92^)sin 92(5.16)where82 = tan -1 ( d2 -- d3 )12The total centre to centre distance moved by the tool for the first pass is:L1 = A i + xn + ym^ (5.17)where4 4Ai = t^an 81 + tan 925.2.2 Case 2: s2 < di. < siFrom Fig. (5.14), it is clear that the centre to centre tool traverse distance for the firstpass is:Li =-- A i + xn + R^ (5.18)whereAi . 4^ + 12tan 8 1The centre to centre distance moved by the tool for the second pass for all the three casesis:L2 = A2 + 2RA2 = 11 --IF 12Chapter 6Optimal Cutting Direction for N-sided Polygonal SurfacesThe selection of cutting direction is an extremely important issue which has a considerableeffect on a machining process. Tool wear is dependent on the tooliworkpiece engagementtime which is directly influenced by the selection of direction of tool motion. The cuttingdirection also determines the geometry of the machining surface and hence the otherparameters like machine feed and spindle speed. The influence of workpiece geometry,spindle speed and machine feed on cost was discussed in the preceding chapters. In thischapter the selection of the best cutting direction for minimum machining cost for anN-sided polygon is discussed.6.1 Mathematical FormulationLety:)^i =1,2,3,...n^ (6.1)be the n vertices of an n-sided convex polygon.The equation of n edges of the polygon can be obtained from the following expressions:y — y, = nii (x — x,)i = 1,2,3,...nwhereYi Yi+ imi =xi —i = 1,2,3, ...(n — 1)75Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^76Figure 6.1: Geometry of a Generalized 6-sided Polygon (Top view)It is assumed that the tool centre to centre workpiece length l e is traversed in p samplingintervals and the distance traversed in the jth sampling interval is x3 .thenla = E x ;^ (6.2)also= 4, + 2R9^(6.3)where^Lax^is the maximum workpiece length in a given direction^Ra^is the cutting tool radiusThe cutting tool can be represented by the equation of a circle as follows:_^(y gi)2 =j = 1,2,3, ...p^1Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^77Figure 6.2: Cutting Direction 9 (Top view)where(h,, gi) are p 1 tool centres for p sampling intervalsOur aim, as before, is to determine the cutting parameters such as spindle speed andfeed for each sampling period and to evaluate the machining cost. The machining cost canthen be obtained for a range of cutting directions and the cutting direction which givesthe minimum machining cost is finally selected. The cutting direction is defined as theangle which the direction of tool traverse makes with the positive x-axis as shown in theFig.(6.12). As the tool traverses the length la , the swept angle of cut changes continously.It is, therefore, necessary to determine the swept angle of cut for each sampling interval.This can be accomplished geometrically and is discussed below.The points of engagement and disengagement of the tool with the workpiece lie on theedges of the workpiece. There can be a maximum of 2n points of intersection betweena circle and n straight lines. Out of all the points of intersection only two points ofintersection which fall on the boundary of the workpiece represent the entry and exitChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^78points. The angle these two points subtend at the centre of the circle is the swept angleof cut for a particular position and the moving direction of the tool.letbe the cutting direction(x:, y:)^be the vertices of the polygon in the new coordinate system(1 3 , g 13 )^be the tool centres in the new coordinate systemi = 1,2,3,...nj = 1,2,3,...p+ 1The mathematical formulas for coordinate transformation are derived in appendix A.The maximum width of the workpiece dmax perpendicular to the cutting direction 8is the largest of the following distances:abs(y:— y:+1 ) i = 1, 2,3,... (n — 1)abs(y:— y:+2 ) i = 1, 2, 3, . . . (n — 2)abs(y: — y:+3 ) i= 1, 2, 3, . . . (n — 3)abs(y: — y:+(n_ i) )^i = 1The maximum workpiece length / max in the 8 direction is the largest of the followingdistances:abs(x: — x:+i ), i = 1, 2, 3, . . . (n — 1)abs(x: — x:+2 ), i = 1, 2, 3, . . . (n — 2)Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^79Td3AFigure 6.3: Swept angle for a 5-sided Symmetric Polygonabs(x: — x: +3), i = 1,2,3,...(n — 3)a b.5 ( — X n^i = 1let^, ), (a, 12 , Y2 )^be the coordinates corresponding to the maximum width^(4, di), (c'2, d'2)^be the coordinates corresponding to the maximum workpiece length^Re^be the tool radius for the 9 direction^ (6.4)Re =^ (6.5)The coordinates of the tool centre at the start of machining (WI , gl) can be determinedChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^80from the expressions:h'1 = c1 — Rhi= Cl2 - Rb'b'1^2 g1 = 2The subsequent tool centres can be obtained from the following relations:3 -1hj =^E Xkk=-13 -1h; = hi — E Xkk=1g1j = 2,3,4,...p+1Two relations for^and 14 are for two sides of the workpiece.The points of intersection between the tool and the workpiece edges can be obtainedfrom the following relations: —Pi +^— 4cea/2i-1 - 2aii = 1,2,3,...n—^4(347^11/2i^2aii^1,2,3,...nwhereai= 2m:b: — 2m:g'j — 2h3.= (1):) 2 (g li ) 2 + (14) 2 — 2b:g'j — RB= 1 + (m:)2Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^81b: = y i — mix iY: Y:+1 xi — —,:+1i = 1,2,3,...nOut of these 2n points of intersection, a pair of points which lie on the boundary of thepolygon and fall in the cutting region are selected. These two points correspond to theentry and exit positions.let(e' . f)3 3 be the coordinates of the entry pointbe the coordinates of the exit pointThe distance between these two point is:De^\ f^j)2L6 3 (6.6)The angle these two points subtend at the centre of the tool is the swept angle of cut.This is shown in Fig. (6.3). The swept angle can be obtained from the cosine law oftriangles as follows:cos ( 2R2e — DI)Once we know the swept angle of cut for a particular sampling interval, we canfind the optimal feed, spindle speed and cost by making use of the optimization strategydescribed earlier. Costs for all the sampling intervals can be summed to obtain the totalcost of machining at a cutting direction of O.This procedure has been used to compute costs for a range of angles from 0 degreeto 180 degrees. The graphs between cutting direction and the cost for a few shapes areshown in Figs. (6.4)-(6.11).i^1401^1801^I^I^I^1120^1605.554.543.532.520 20CChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^82dl =10,d2=20,d3=15,11=30,12=20Cutting direction in degreeFigure 6.4: Cost characteristics for a 5-sided Unsymmetric PolygonChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^83dl =1 0,d2=20,d3=15,11 =30,12=20 6.565.554.54R=dmax1^1^1^I^'^I40 80 120 160^20CCutting direction in degreeFigure 6.5: Cost characteristics for a 5-sided Unsymmetric Polygon5.24.24.6Ly4.84.4Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^84dl =1 0,d2=20,d3=1 5,11 =30,12=2040^80^120^160^20CCutting direction in degreeFigure 6.6: Cost characteristics for a 5-sided Unsymmetric PolygonR=0.55dmax1^I^'^I^'^I^'40 80 120 160 20(5.54.53.5Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^85d1=10,d2=20,d3=10,I1=30,12=30Cutting direction in degreeFigure 6.7: Cost characteristics for a 5-sided Symmetric PolygonChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^86dl =10,d2=20,d3=10,I1=30,12=307.56.5R=dmaxi^I^I^I^1120^160 20CCutting direction in degreeFigure 6.8: Cost characteristics for a 5-sided Symmetric PolygonR=1.5dmax1Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^87dl =10,d2=20,d3=10,11 =30,12=300^40^80^120^160^20CCutting direction in degreeFigure 6.9: Cost characteristics for a 5-sided Symmetric Polygon6.265.85.65.45.254.84.62.82.62.42.23.43.240 8011[1^i120^160 20CChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^88d1=0,d2=20,d3=0,11=30,12=20Cutting direction in degreeFigure 6.10: Cost characteristics for a Non-equilateral TriangleChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^89d1=0,d2=20,d3=0,11=30,12=204.43.83.63.43.2304.2I^'^I^r^I^I40 80 120 160 20CCutting direction in degreeFigure 6.11: Cost characteristics for a Non-equilateral Triangle5.24.24.6I I I I 1120^160I80 20C4.84.4Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^90dl =0,d2=20,d3=0,11=30,12=20Cutting direction in degreeFigure 6.12: Cost characteristics for a Non-equilateral Triangle0I^I^'^I^'^I^'40 80 120 160 20C11.80.81.41.61.2Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^91dl =0,d2=20,d3=0,11=11.54,12=11.54Cutting direction in degreeFigure 6.13: Cost characteristics for an Equilateral Triangle2.22.121.91.81.71.61.51.40 40I80I^III!120^160 20CChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^92dl =0,d2=20,d3=0,11 =11.54,12=11.54Cutting direction in degreeFigure 6.14: Cost characteristics for an Equilateral TriangleChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^93dl =0,d2=20,d3=0,I1=11.54,12=11.542.42.32.22.121.91.81.71.60R=1.5dmax140v v1^'^I^'^1^'80 120 160 20CCutting direction in degreeFigure 6.15: Cost characteristics for an Equilateral TriangleChapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^946.1.1 Analysis of ResultsThe following conclusions can be drawn from the computational results:1. Symmetric workpieces give symmetric graphs with small computational errors.2. When the tool radius is slightly larger (R = 0.554.) than the maximum radialwidth encountered during machining; the cutting directions of 0 degree and 180degree are the optimal directions for all the workpieces.3. When the tool radius is twice (R = dn.) or thrice as large (R = 1.54,„x ) as themaximum radial width encountered during machining; the cutting direction of 90degree is the optimal direction for all the workpieces.4. The results are dependent on the tool diameter and workpiece geometry.5. The graphs for larger tool diameter with respect to the maximum radial widthencountered during cut are shifted upwards. In other words, the cost increases forthe selection of larger cutters.6.1.2 ConclusionsThis chapter describes a general method of evaluating the best cutting direction for aworkpiece of known geometry. Even though the results have been obtained for only afew workpieces under certain site specific conditions, the method is valid for almost anyworkpiece geometry. A change in the assumptions and site specific conditions would alterthe results. Software based on the proposed mathematical formulation can be developedwhich would give the best cutting direction for any set of conditions and any workpiecegeometry.Bibliography[1] Yellowley, I., Wong, A., Desmit, B., The economics of Peripheral Milling, Proc. 6th.North American Metalworking Research Conference, 1978, pp. 388-394.[2] Chang, H., Wysk, R.A., Davis, R.P., Milling Parameter Optimization through aDiscrete Variable Transformation, International Journal of Production Research,Vol. 20, 1982, pp. 503-516.[3] Yellowley, I., Gunn, E.A., The optimal Subdivision of Cut in Multi-pass MachiningOperations, International Journal of Production Research, Vol. 27, 1989, pp. 1573-1588.[4] Chang, H., Wysk, R.A., Wang, H., An Analytical Approach to Optimize NC ToolPath Planning for Face Milling Flat Convex Polygonal Surfaces, IIE Transactions,Vol. 20, 1988, pp. 325-332.[5] Chang, H., Wysk, R.A., Wang, H., On the Efficiency of NC Tool Path Planning forFace Milling Operations, Journal of Engineering for Industry, Vol. 109, 1987, pp.370-376.[6] Prabhu, P.V., Gramopadhye, A.K., Wang, H., A general Mathematical Model forOptimizing NC Tool Path for Face Milling of Flat Convex Polygonal Surfaces, In-ternational Journal of Production Research, Vol. 28, 1990, pp. 101-130.[7] Tlusty, J., Elbestawi, M.A.A., Analysis of Transients in an Adaptive Control Ser-vomechanism for Milling with Constant Force, Transactions of ASME, 1977, pp.766-772.95Bibliography^ 96[8] Tomizuka, M., Oh, J.H., Dornfield, D.A., Model Reference Adaptive Control of theMilling Process, Control of Manufacturing Processes and Robotics Systems ASME,1983, pp. 55-63.[9] Daneshmend, L.K., Pak, H.A., Model Reference Adaptive Control of Feed Force inTurning, ASME Journal of Dynamic Systems, Measurement and Control, Vol. 108,1986, pp. 215-222.[10] Altintas, Y., Spence, A., CAD Assisted Adaptive Control for Milling, ASME Journalof Dynamic Systems, Measurement,and Control, Vol. 113, 1991, pp. 444-451.[11] Yellowley, I., Barrow, G., The influence of Thermal Cycling on Tool Life in PeripheralMilling, International Journal of Machine Tool Design and Research, Vol. 16, 1976,pp. 1-12.Appendix ACoordinate TransformationLet the coordinate system X -Y be rotated by an angle of 9 degrees. The new coordinatesystem is X' - Y' represented by X' and Y' axis. The coordinate (x, y) in the old systemis (x', y') in the new sytem. The line joining the origin with the coordinate (x, y) makean angle of a. The coordinates (x, y) and (x', y') can be mathematically related by thefollowing equations (see Figs. 6.5- 6.9):A.0.3 From Old to New9 < 10 < aa = tan' -Yxy = p sin ay P = sin a/Y = p sin(a - 0)x' = p cos(a - 0)9 < I8 > aa = tan -1 -Yx97yPY iXiAppendix A. Coordinate Transformation^ 98= p sin ay ,sin a= — p cos(2 — (a — 0))= psin(2 — (a — 0))8 = 120 < aIx = yyI = X0 = 7r0 < aIY = — yXi = - xA.0.4 From New to Old0 < 10 < atan(a — 8) =a =Y 1 =pP =y =Y 1x'tan -1 y '  0x'sin(a — 0)Y1 sin(a — 9)p sin ax = p cos aAppendix A. Coordinate Transformation^ 99<9 > atan(7; — (0 — a))—2 — (0 — a)) = abs(tan -1 (a^— 2 abs(tan -1 ( -7))p^sin(; — (0 — a))p = ^sinq — (0 — a)y^p sin ax = p cos aB = 2< ayy =—YX^-X< aFigure A.1: Coordinate Transformation:8 < 1,8 < aYx•Appendix A. Coordinate Transformation^ 100Figure A.2: Coordinate Transformation:8 < I, 8 >fe,—......—..^yg .--.....■.0.1Figure A.3: Coordinate Transformation:0 = 1,0 > a(::------- 21 ..„...,■44Appendix A. Coordinate Transformation^ 101Figure A.4: Coordinate Transformation:0 > ;1 ,6 > aAppendix A. Coordinate Transformation^ 102Figure A.5: Coordinate Transformation:0 = 7r, 8 > aAppendix BPseudo-code for single pass milling operations;Inputs:Geometrical parameters of workpiece^d2, d3 , 11 , /2Economic parameters^:Ch, Ct , TctTool life constant and exponents^m, n, p, qMachine constraints^:Pmax,14Cutting constants :Ks, riRange of cutter radii^:Rtnin) 'LaxSmall increment in cutter radius^:ARRange of spindle speeds^:Vnun, VmaxCutting conditions :smax, aTool material properties^:o-max, etc.Milling type^ :Upmilling or downmillingSampling interval :At;Outputs: Cutter radius and Machining cost^:R, Cpmin;Variables: Bigk — Rma7A-RR"lin^ ;Number of steps to cover the rangeof cutter radii103d2 — Ch 1 11k2 = d2 — d3 12Pmao(• 75 4, 106 )C .aRt = Rm" ,T Rmin ;Increase in radius after whichthe number of teeth change;Radial width slope upto length;Radial width slope after length;Power constraint constantAppendix B. Pseudo-code for single pass milling operations^ 104i = 0 to Bigk^ ;Loop for range of radiiR(i 1) =^iAR^;Next cutter radiusif Rmin < R(i 1) <^Rt)^;Selection of cutter teeththenz = 2if ( -1?-mt. + R t ) < R(i + 1) < (Rmin + 2Rt)thenz = 3if (Rm.+ 2Rt ) R(i +1) < (Rm. + 3Rt)thenz = 4if R(i + 1) > (R„,,„ 3Rd )thenz = 6L(i 1) = 6R(i 1)^ ;Flute lengthAppendix B. Pseudo-code for single pass milling operations^ 105Tmax = .75 * 106 * Prn-"R( s+ ' )^;Maximum allowable TorqueCp = 0.0^ ;Initialize the process costx = 0.0 ;Initialize total tool traverse distanDelta= 500^ ;Arbitrary number for sampling;intervalsj = 1 to Delta^ ;Loop for sampling intervalsif x < 11^;Evaluation of radial depth of cutthend(j) =elsed(j) = d2 — k2 (x — 11 )03 (j) = cos -1 (1^Rd(a( +3)1) )^ ;Swept angle of cutif d(j) = 0thend(j) = 231 if R(i + 1) > d(j)thenst(i) = ssin(;:(j))elsest(j) = sma.^7 (1) = Kaaz (2 )F=(j) = 7(j)[(1 — cos(205(j))) + ri(2 46.(.7);Initial radial depth for triangle;Feed per tooth;from tooth breakage constraintCutting force constantsin(20 5 (j)))] ;Average cutting forcesAppendix B. Pseudo-code for single pass milling operations^ 106Fib) = -6)(7'1(1 — cos( 20.(j))) ( 2 080) — sin( 20.(i)))]Fx(j) = 7(i)[ri( 2 03(i) — sin( 2080))) — (1 — cos( 20.(i)))]_EVA = -Y(7)tri( 1 — cos(209(i))) (208(j) — sin( 203(j)))]FR(j) = VF2 +M(j) = FR(j)(gi + 1) —T(j) = -Yd(i)0-(j) = piR(+ 1 ) . 1u(i)Vm(i)2 + T(i) 2if crU) < Cfmaxthengoto 41elsesmax = smax — 0.01goto 3141 if Tmax > T(j)thengoto 51elseSrnas Smax — 0.01goto 31;for upmilling;Average cutting forces;for downmilling;Resultant cutting force;Resulting moment on the cutter;Resulting torque;Shank stresses;Shank breakage constraint;Check torque constraint;Reduce maximum feed;Calculate feed again;Torque constraint check;Calculate equivalent feed;Reduce maximum feed;Calculate feed again0 I • \^st(j)d(j) Ni-1-1)08(i)Nmax(j) = 2 71-VA(7-1-2 1 );Equivalent feed per tooth;Range of= 27,-vR(:+1) ;r.p.m.AN = 10061 Omega=integer value of -Arma° —1‘1""'ANrmax • — _62LPIV^(3 StO,Z ;Maximum r.p.m. allowed by;Power constraint;Initial increment for r.p.m.;Upper limit of the indexfor r.p.m. optimization loopAppendix B. Pseudo-code for single pass milling operations^ 107k = 0 to OmegaN(k 1) =^kANwhile Nmaxp > N(k 1)N(k + 1) = N(k + 1) — 12rR(i+1)N(k+1) (k + 1) = 60Th (k + 1) = 60000*R(i+1)41.(j)V(k+1)s0000.n(i+i)(22- -0.(3)) Te (k + 1) -=^V (k+1);Loop for optimization of r.p.m.;Next spindle r.p.m.;Power constraint check;Reduce spindle r.p.m.;Spindle peripheral velocity;Heating time in ms;Cooling time in mssmallx(k 1 )^ ;Ratio of total cycle time toactual cutting time per cycleEr (k + 1) = 39 log(Tc (k + 1)) — 23 log(Th (k + 1)) + 37.5 ;Range of thermal strainX(k 1) = Er(k 1 )(N(k+i).x6somali(k-1-1) )4^;Thermal fatigue parameterAppendix B. Pseudo-code for single pass milling operations^ 108Tak 1) = 0^2:(.j) X(k+1)ntSeq (j)nV(k+1)Paq ;Tool lifeChTtACp(k + 1) = (Ch 71,+1)^T/(-1-c1) )At ;Process cost for one intervalif k > 1 then^ ;Search of optimum r.p.m.if ACp(k + 1) > ACp(k) then^;by comparing the costif AN > 1 then^ ;with the previous value,Nmax N(k + 1) ;narrowing the speed range,Nmin N(k — 1)^ ;setting new limits for r.p.mAN =^ and new r.p.m. incrementgoto 61 ;New upper limit of loop indexelseNmin = N(1)AN = 1if k = 0 thenACman ACp(k + 1)^ ;Initialization of minimum costNmino N(k 1) ;Initialization of optimum r.p.m.if AC(k + 1) < AC,„„ then^Comparison with previous minimum costAC,,,in = AC(k + 1)^ ;New minimum costN„,,„. = N(k + 1) New optimum r.p.m.next k^ ;Try the next r.p.m.Appendix B. Pseudo-code for single pass milling operations^ 109Cp = Cp ACmin^ ;Summation of costsAX(i) = st(j)16`1(Tino z pt^ ;Distance movedx = x As(j) ;Total tool traverse distanceif x > l l + 12 then^ ;Check whether thegoto 91^ ;whole workpiece is machinedelse81 next j^ ;Next sampling interval91 Cp,..(i + 1) = Cp^;Cost with the selected radiusnext i^ ;Next tool radius;Plot Cpm, for different cutter radii R(i 1);stopendAppendix CPseudo-code for two-pass milling operations;Inputs:Geometrical parameters of workpiece^d2, d3, 4,12Economic parameters^:Ch, Ct,TctTool life constant and exponentsMachine constraints^:PmaxCutting constantsRange of maximum radial width^:d7 min7 di.maafor the first passSmall increment in maximumradial width of first passTool geometry^ :R, zRange of spindle speeds^:Vmin, VmaxCutting conditions :smax, aTool material properties^:amax, etc.Milling type^ :Upmilling or downmillingSampling interval :At;Outputs: Maximum radial width of^CPmsnfirst pass and Machining cost110Appendix C. Pseudo-code for two-pass milling operations^ 111;Variables: Bigk = drma.31^d2 dl32 = d2 — d3L^d2 -d1 —d2 -d3 k2—Pm..(.75*106 ) K sa;Number of steps to cover the rangeof max. radial width of first pass;Geometric;constants;Radial width slope upto length d i;Radial width slope after length 1 1;Power constraint constanti = 0 to Bigk^;Loop for range of maximumradial width of the first passdr (i + 1) = dr„,„ + iAdr^;Next maximum radial widthof the first passL = 6R^ ;Flute lengthTruax = . 75 * 106 * P. (=+1)  ;Maximum allowable TorqueCp = 0.0^ ;Initialize the process costx = 0.0 ;Initialize total tool traverse distanceflag = 1^ ;Initialization of pass numberDelta= 1000^;Arbitrary number for sampling intervalsj = 1 to Delta^;Loop for sampling intervalsAppendix C. Pseudo-code for two-pass milling operations^ 112;Case 1if 4,, < dr (i + 1) < 82thend (i+1) Al =^kaA2 = 11 + 1 2;Length of first pass;Length of second passif x < 4(i+1) and flag= 1thend(j) = kixif dr(ki+1) < x < a l and flag= 1thend(j) = k2 (x^dr (ki1+1) )if x <^dr(kii+1)) and flag= 2thend(j)^+ ki x;Radial depth of first;pass of machining;Radial depth of secondif (11^dr(kil-1-1) < x < (A 2^dr(ki2+1) ) and flag= 2 ;d(j) d2 — dr (i + 1) ;pass of machiningif x > (A2^dr(ki2+1) ) and flag= 2thenkl < x < A i and flag= 1if d,(i-1-1)Appendix C. Pseudo-code for two-pass milling operations^ 113d(j)^(d2 — dr (i + 1)) — k2 Ex — (li + 4(12: 1) )};Case 2if 82 < dr (i + 1) <thendr (i-F1) ,/2Al —^1-k1A2 = 11 + 12if x < 4(1+1) and flag= 1thend(i) = ki xthend(j) = dr(i + 1) — k2(x —if x < (11^dr (kli+1) ) and flag= 2thend(j) = d1 ki x;Length of first pass;Length of second pass;Radial depth of first;pass of machining;Radial depth of secondif x^dr(i+i)) and flag= 2d(j) d2 — dr (i + 1)^ ;pass of machiningAppendix C. Pseudo-code for two-pass milling operations^ 114;Case 3if .9 1 < dy.(i + 1) < d,..thenAi = 11 + 12^ ;Length of first passA2 - 11 + 12 ;Length of second passif x < 11 and flag= 1thend(j) =d1 — (d2 — dr (i + 1)) + kix^;Radial depth of firstif /1 < x < A i and flag= 1thend(j) = d3 — (d2 — dr (i + 1)) — k2 (x — li) ;pass of machiningif x < A2 and flag= 2thend(j) = d2 — dr (i + 1)if d 1 = d2 = d3if x < A i and flag= 1thend(j) = dr (i + 1);Radial depth of second;pass of machining;Rectangular workpiece;Radial depth of first;pass of machining3rnax31 if R > d(j)thenst(j) = sui(46.(7))elsest(i) =7(i) = K„azwat(j)2 ;Feed per tooth;from tooth breakage constraintCutting force constantAppendix C. Pseudo-code for two-pass milling operations^ 115if x < A2 and flag= 2^ ;Radial depth of secondthend(j) = d2 — cl,(i + 1)^ ;pass of machining08 (j)= cos -1 (1 — 14) ;Swept angle of cutif d(j) = 0^ ;Initial radial depth for trianglethend(j) = 2Fx(j) = -y(j)[(1 — cos(20,(j))) r i (20.(j) — sin(20,(j)))] ;Average cutting forcesFib) = 7(j)[r i (1 — cos( 20.(1))) ( 20.(j) — sin(20.0)))] ;for upmillingFx(j) 7(j)[ri( 20.(j) — sin( 20.(i))) — ( 1 — cos(20.(j)))} ;Average cutting forcesFy(i) = "Y(2)[ri( 1 — cos( 20.(j))) ( 20.(j) — sin( 20.(3)))] ;for downmillingFR(j) -= VF2 +^ ;Resultant cutting forceM(j) = FR(i)(L ;Resulting moment on the cutterT(i) =7d(j)^ ;Resulting torqueAppendix C. Pseudo-code for two-pass milling operations^ 116u(3) = pi2R,M(j)VM(j) 2 TW 2if Q(j) < umaxthengoto 41elseSmax = Smax — 0.01goto 31;Shank stresses;Shank breakage constraint;Check torque constraint;Reduce maximum feed;Calculate feed again41 if Tmax > T (i)^ ;Torque constraint checkthengoto 51^ ;Calculate equivalent feedelseSmax = Smax 0.01^ ;Reduce maximum feedgoto 31^ ;Calculate feed againS eq()) = at(i)d(7)313)Nmax(i) ^Nmin (•) — v2;;Equivalent feed per tooth;Range of;r.p.m.Nmax,(i) = 06:3(7)1.^ ;Maximum r.p.m. allowed by;Power constraintAN = 100^ ;Initial increment for r.p.m.61 Omega=integer value of Nma'a-NNmin ;Upper limit of the indexfor r.p.m. optimization loopAppendix C. Pseudo-code for two-pass milling operations^ 117k = 0 to Omega^ ;Loop for optimization of r.p.m.N(k +1)^+ LAN^ ;Next spindle r.p.m.while Nma.,, > N(k + 1)N(k +1) = N(k +1) —1V(k + 1) .= 211RN61(3k+i)600004444j) Th (k + 1)^v(h+i)600004, R(271--0.4.i))Tc(k + 1) = ^V(k+1)smallx(k + 1) = ^;Power constraint check;Reduce spindle r.p.m.;Spindle peripheral velocity;Heating time in ms;Cooling time in ms;Ratio of total cycle time toactual cutting time per cycleE,.(k + 1) = 39log(Tc (k + 1)) — 23 log(Th(k 1)) + 37.5 ;Range of thermal strainX(k + 1) = E,.(k 1)(N(k+i)*x6somall(k-14)).12,^;Thermal fatigue parameterTL (k + 1) = 462,,(37 ) X(k-1-1)mSegniqk+1)Pa9^ ;Tool lifeACp(k + 1) = (Ch   ^ )At^;Process cost for one intervalAppendix C. Pseudo-code for two-pass milling operations^ 118if k > 1 then^;Search of optimum r.p.m.if ACp(k + 1) > ACp (k) then ;by comparing the costif AN > 1 then^;with the previous value,Nmaz = N(k + 1) ;narrowing the speed range,N„,,,, = N(k — 1)^;setting new limits for r.p.mAN —and new r.p.m. incrementiogoto 61^ ;New upper limit of loop indexelseNmin = N(1)AN = 1if k 0 thenACTnin ACp (k + 1)^;Initialization of minimum costNmin. = N(k +1) ;Initialization of optimum r.p.m.if AC(k + 1) < AC,,,„ then Comparison with previous minimum costACT„,„ = AC(k 1)^;New minimum costNminc, = N(k + 1)^New optimum r.p.m.next k^ ;Try the next r.p.m.Cp Cp + 0 Cmin^;Summation of costsAx (i ) = 8t(i)N60mina z At^;Distance movedx = x Ax(j)^;Total tool traverse distanceif x > .A i and flag= 1 then^;End of first passflag= 2^ ;Start of second passAppendix C. Pseudo-code for two-pass milling operations^ 119x = 0^ ;Initialization of tool traverse distanceif x > A2 and flag= 2 then^ ;End of second passgoto 91next j^ ;Next sampling interval91 Cp„„,ji + 1) = C,^ ;Cost with the selected radiusnext i^ ;Next maximum radial width for;first pass;Plot Cp„„„, for maximum radial widths dr (i + 1);of first passstopendAppendix DPseudo-code for optimal cutting direction;Inputs: Geometrical parameters of workpiece^d2, d3 , /1 , /2Economic parameters^:Ch, Ct,TdTool life constant and exponents^rn, n, p, qMachine constraints^ ViCutting constants :K,,r1Range of cutting orientations^:em..) BoraxSmall increment in cutting orientation :At9Tool geometry^ :zNumber of sides of polygon^:BigLCo-ordinates of one vertex :x(3), y(3)Shape of workpiece^ :ShapeRange of spindle speeds :Vmtn) VmaxCutting conditions^ :smas, aTool material properties^:Amax, etc.Milling type^ :Upmilling or downmillingSampling interval :At;Outputs:Cutting direction^ :0, Cp,„„and Machining cost120Appendix D. Pseudo-code for optimal cutting direction^ 121;Variables: Bigk = Integer ofPmax(.75*106 )El^Kitt;Prescribing the coordinates of polygons ;if shape = 1 then^ ;Five sided polygonx(1) + x(3)x(2) = x(3)x(3) = x(3)x(4) = l l + /2 + x(3)x(5) = 11 + / 2 + x(3)Y(1) = d2 + y(3)y(2) = d1 + y(3)Y( 3 ) = Y( 3 )y(4) = y(3 )y(5) = d3 + y(3)if shape = 2 then^ ;Three sided polygonx(1) = 11 + x(3)Borax Bmin oe ;Number of steps to cover the rangeof cutting orientations;Power constraint constantAppendix D. Pseudo-code for optimal cutting direction^ 122x(2) = x(3)x(4) = 11 + /2 + x(3)x(5) = ll + /2 + x(3)y(1) d2 + y(3)y(2) = y(3)Y( 3 ) = Y( 3 )i = 0 to BigkOd(i + 1) °min iA00(i + 1) . lrOd(i+1)0  di„,„ = 0.0lmax = 0.0;Loop for range of cuttingorientations;Cutting orientation in degrees;Cutting orientation in radians;Maximum workpiece width;perpendicular to cutting direction;Maximum workpiece length;along cutting directionL = 0 to BigL^;Loop for transformation of co-ordinatesa(L) tan -10)^;Angles of lines through origin;and vertices in radiansAppendix D. Pseudo-code for optimal cutting direction^ 123P(L) — si4a(L(1),;Transformation of vertex co-ordinatesif 0(i + 1) < 1 and O(i -I- 1) < a(L) theny'(L) = P(L) * sin(a(L) — 9(i + 1))if 0(i + 1) < li and 0(i +1) > a(L) theny'(L) = —P(L) * sin(9(i + 1) — a(L))if 0(i + 1) < 1- thenx'(L) = P(L) * cos(abs(a(L) — 0(i + 1))if 0(i + 1) = I thenx'(L) = y(L)y'(L) = x(L)if 9(i + 1) > 2 and 0(i + 1) < r thenif 12(-  > (0(i + 1) — a(L)) theny'(L) = —P(L) * cosg- — (0(i + 1) — a(L)))x'(L) = P(L) * sing — (9(i + 1) — a(L)));Lengths of lines joiningorigin and verticesAppendix D. Pseudo-code for optimal cutting direction^ 124if 0(i + 1) > z and 0(i + 1) < it thenif ; < (8(i + 1) — a(L)) theny'(L) = —P(L) * cos(0(i 1) — a(L) —x'(L) = —P(L) * sin(0(i + 1) — a(L) —if 8(i + 1) = 7r theny'(L) = —y(L)x'(L) = —x(L);Maximum workpiece width and lengthif L > 1 theny20(L) = abs(y'(L) — y'(L — 1))x20(L) = abs(x'(L) — x'(L — 1))if y20(L) > d„nax thendmax = y20(L)bi = y'(L)b2 = y'(L — 1)x'(L)a2 = xi(L — 1);Co-ordinates corresponding.to the maximum widthif x20(L) > /max then/„„„ = x20(L)4 = xPrime(L)^ ;Co-ordinates corresponding4 = xl(L — 1) ;to the maximum lengthAppendix D. Pseudo-code for optimal cutting direction^ 125if L > 2 theny21(L) = abs(y'(L) — y'(L — 2))x21(L) = abs(x'(L) — x'(L — 2))if y21(L) > dma. thendmax = y21(L)bl = y'(L)14 = y'(L — 2)di = x'(L)a2 = x'(L — 2)if x21(L) > Las then= x21(L)e1 = x'(L)e2 , x 1 (L — 2)if shape = 1 thenif L > (BigL — 2) theny22(L) = abs(y'(L) — y'(L — 3))x22(L) = abs(x'(L) — x'(L — 3))if y22(L) > dm. thendmaz = y22(L)1/1 = y'(L)b12 = yi(L — 3)al = x'(L)Appendix D. Pseudo-code for optimal cutting direction^ 126a2 = x'(L — 3)if x22(L) > /mai then/max = x22(L)ci = x'(L)4 = x'(L — 3)if shape = 1 thenif L > (BigL — 1) theny23(L) = abs(y'(L) — y'(L — 4)) ;x23(L) = abs(x'(L) — x'(L — 4)) ;if y23(L) > dmax thendmax = y23(L)bii. = y'(L)1/2 = y'(L — 4)al = x'(L)a2 = x'(L — 4)if x23(L) > /mar then/,,„,„ = x23(L)cI = x'(L)c ='^x'(L — 4)2next LRa(i + 1) = dlaz^;Tool radius for full immersionAppendix D. Pseudo-code for optimal cutting direction^ 127R(i + 1) = 1.1* Ro (i + 1)^;10 percent tolerance9i(2 + 1) _ 61 6zif ei < c'2 then114(i + 1) = c'i — R(i + 1)else14(i + 1) = c'2 — R(i + 1)19 (i + 1) = 14(i + 1) + 1„,„x + 2R(i + 1);Slope of edges of polygonL = 1 to BigLif L < Big L thensden(L) = x'(L + 1) — x'(L)snum(L) = y'(L + 1) — y'(L)elsesden(L) = x'(1) — x'(L)snum(L) = y 1(1) — y'(L);Denominator in slope expression;Numerator in slope expressionif abs(sden(L)) < abs(1) thenm'(L) = 200^ ;Edge is perpendicular to cutting directionif abs(snum(L)) < abs(1) then^;Edge is parrallel to cutting directionAppendix D. Pseudo-code for optimal cutting direction^ 128m/(L) = 0if abs(sden(L)) > abs(1) thenif abs(snum(L)) > abs(1) thenmi(L ) = sZenum(C,L))b'(L) = y'(L) — mf(L) * x'(L)^;y intercept for edgesnext LL(i + 1) = 6R(i 1)^;Flute length= .75 * 10 6 * 1:"R(1+1) vlCp = 0.0Delta= 1000s ti = 0.02= 200x = at Niz60;Maximum allowable Torque;Initialize the process cost;Arbitrary number for sampling intervals;Initial feed;Initial speed;Initial distance movedTmazj = 1 to Delta^;Loop for sampling intervalsh'i (j) = 14(i + 1) + x= 91(i + 1 )3 = —1000f; = 1000countp = 0.0;Subsequent co-ordinates of tool centre;Initialization of y co-ordinate of exit point;Initialization of y co-ordinate of entry point;Counter of entry and exit pointsAppendix D. Pseudo-code for optimal cutting direction^ 129;Points of intersectionL = 1 to BigLif m'(L) = 200 thenbh(j) = —2 * gi (j)chp(j) = (x'(L)) 2 (h'j (j)) 2 (g (j))2 — R(i 1) 2 —2 * xi(L) * h/i (j)ah(j) = 1disc(j) = bh(j) 2 — 4* ah(j) * chp(j)ko = 1 to 2^ ;Loop for entry and exit pointsif m'(L) 200 thenif disc(j) > 0 thenif ko = 1 then9p(ko) =-bh(j)+"disc(2) hp(ko ) = m'(L) * gp(1) b'(L)else9p(ko) —-bh(i)--Vcfisc(j)2*ah(j)hp(ko ) = m'(L) * gp(ko )d- b'(L)if m'(L) = 200 thenif disc(j) > 0 thenif ko = 1 thengp(ko ) = x'(L)Appendix D. Pseudo-code for optimal cutting direction^ 130hp(ko)^--bh(J)+^diac(j)2else9134( 1c.)^x'(L)hp(ko ) -1'k-0- N/disco)if disc(j) < 0 thengp(ko ) = 1000000hp(ko ) = 1000000gp(ko ) = integer of gp(ko )hp(ko ) = integer of hp(ko )if L < Big L thenif x'(L) > x'(L + 1) thenfxlim = x'(L)sxlim = xi(L -I- 1)elsefxlim = x'(L 1)sylim = x'(L)if y'(L) > y'(L + 1) thenfylim = y'(L)sylim = y'(L + 1)elsefylim = yi(L + 1)sylim = y'(L);Upper and lower;limits of abscissas for edges;Upper and lower limits;of ordinates for edgesif L =BigL thenAppendix D. Pseudo-code for optimal cutting direction^ 131if x'(L) > x'(1) thenfxlim = x'(L)sxlim = x'(1)elsefxlim = x'(1)sxlim = x'(L)if y'(L) < y'(1) thenfylim = Y l ( 1 )sylim = y'(L)elsefylim = y'(L)sylim = yl(1)if gp(k,,)> (j) then^ Check for entryif m'(L) 200 and m'(L) L 0 then^;and exit pointsif gpint(ko ) < fxlim and gpint(k„)> sxlim thenif hpint(ko ) < fylim and hpint(kc,)> sylim thenif abs(hp(ko ) — (m'(L) * gp(k0 ) b' (L))) < 2 thencountp = countp + 1 ;Increase the count by oneif hp(k,,) >^thenvij = hp(ko ) ; Update the= gp(ko )^ ;exit pointAppendix D. Pseudo-code for optimal cutting direction^ 132if hp(ko ) < f; thenf; = hp(ko )^ ;Update thee'i = gp(ko ) ;entry pointgoto 28if gp(ko ) > 11,;(j) thenif m'(L) = 200 thenif gpint(ko ) = f xlim and gpint(ko ) sxlim thenif hpint(ko ) < fylim and hpint(ko ) > sylim thencountp countp 1if hp(ko ) > v; thenhp(ko )u?3if hp(ko ) < u", then= hp(ko)e, gp(ko)goto 28if gp(ko ) > 114(j) thenif m'(L) = 0 thenif gpint(ko ) < fxlim and gpint(ko) > sxlim thenif hpint(ko) fylim and hpint(ko ) sylim thencountp countp + 1Appendix D. Pseudo-code for optimal cutting direction^ 133if hp(ko ) >^then •= hp(ko )= gp(ko )if hp(ko ) < ,f; then= hp(ko )eij gp(ko )next konext Lif countp = 0 thenglp(j) = 0.0hlp(j)^0.0g2p(j) = 0.0h2p(j) = 0.0;No tool workpiece;engagementif countp = 1 thenglp(j) = 0.0hlp(j) = 0.0g2p(j) = 0.0h2p(j) = 0.0;Tool just touches;the workpiece but no;cuttingif countp > 1 then ;Entry and exit pointsglp(j) =hlp(j) =Appendix D. Pseudo-code for optimal cutting direction^ 134g2p(j) =h2p(j) = f;dp(j) = V(glp(j) — g2p(j)) 2 (hlp(j) — h2p(j)) 2exper(j)^2 *-1 i.-f-Z--111;(i) 2 ;Distance between entryand exit points0,(j) = cos -1 (exper(j))^ ;Swept angle of cutif Os (j) = 0 thennext jd(j) = R(i + 1)(1 — cos(08 (j)))if d(j) = 0thend(j) = 231 if R > d(j)thenst(i)elsest(j) = sma.1, (1) = K.a;: (3);Radial width;Initial radial depth for triangle;Feed per tooth;from tooth breakage constraintCutting force constantFx(j)^7(j)[(1 — cos(20.(j))) ri(20.(j) — sin( 20.(j)))1 ;Average cutting forcesFy(j)^'Y(j)[ri(1 — cos( 20.(j))) — ( 20.(j) — sin( 20.(j)))] ;for upmillingAppendix D. Pseudo-code for optimal cutting direction^ 135Fz(j) = -y(j)[ri(20.(i) — sin( 20.(i))) — ( 1 — cos (20.(j)))] ;Average cutting forcesFy(j) = 7(i)[ri(1 — cos( 20.(j))) ( 20.(i) — sin (20.(j)))] ;for downmillingFR(j) = VF2^ ;Resultant cutting forceM(j) = FR(j)(L — ;Resulting moment on the cutterT(j) = 'yd(j)^ ;Resulting torque5(j) = pi2R,^ M(j)VM(j)2+ T(j) 2^ ;Shank stressesif Q(j) < Grmax^ ;Shank breakage constraintthengoto 41^ ;Check torque constraintelseSmacc = Smaz — 0.01^ ;Reduce maximum feedgoto 31^ ;Calculate feed again41 if Tmaz > T(j)thengoto 51elseSmax = sm. — 0.01goto 31;Torque constraint check;Calculate equivalent feed;Reduce maximum feed;Calculate feed againSeq0^at (7 )d(i) i. 3 ;Equivalent feed per toothNmax(i) vm2.-rNmin (j) = ^ ;r.p.m.;Range ofAppendix D. Pseudo-code for optimal cutting direction^ 136) .6t()z^ ;Maximum r.p.m. allowed byNniaxp(i ;Power constraintAN = 100^ ;Initial increment for r.p.m.61 Omega=integer value of Nnial-NNnwn ;Upper limit of the indexfor r.p.m. optimization loopk = 0 to Omega^ ;Loop for optimization of r.p.m.N(k + 1) = Nmin + LAN^;Next spindle r.p.m.while N,„„, > N(k + 1)^;Power constraint checkN(k + 1) = N(k 1) — 1 ;Reduce spindle r.p.m.V(k + 1) = 2R-RN(k+1)60 ;Spindle peripheral velocityTh(k + 1 ) = 60000*R42,(j) V(k+ 1)Tc (k + 1) = s0000.R(2,95.(;)) v(k+i);Heating time in ms;;Cooling time in msAppendix D. Pseudo-code for optimal cutting direction^ 137smallx(k + 1) =^ ;Ratio of total cycle time toactual cutting time per cycle.E,.(k + 1) = 39log(Tc (k + 1)) — 231og(Th(k + 1)) + 37.5 ;Range of thermal strainX(k + 1) = Er (k + 1 )(N(k+1)*znall(k-l-ly71L(k + 1) = 02.(7) x(k-I-1)", Seq (3)°V(k-1-1)Pa 9ct^chTACp(k + 1 ) = (Ch _L Tdk+i) + Ti( + 1ct)1)At;Thermal fatigue parameter;Tool life;Process cost for one intervalif k > 1 thenif ACp(k + 1) > ACp(k) thenif AN > 1 thenif ACp (k + 1) > ACp (k) thenif AN > 1 thenNmax = N(k 1)Nmin = N(k — 1)AN _ AN10goto 61elseNmin = N(1)AN = 1if k = 0 then;Search of optimum r.p.m.;by comparing the cost;with the previous value,;by comparing the cost;with the previous value,;narrowing the speed range,;setting new limits for r.p.mand new r.p.m. increment;New upper limit of loop indexAppendix D. Pseudo-code for optimal cutting direction^ 138OCmin = OCp(k+1)^;Initialization of minimum costArmin. N(k 1) ;Initialization of optimum r.p.m.if AC(k + 1) < AC„„„ then^Comparison with previous minimum costAC„,in = LIC(k + 1)^ ;New minimum costNmino = N(k + 1) New optimum r.p.m.next k^ ;Try the next r.p.m.Cp = Cp ACmin^ ;Summation of costsAxu) st(j)N6Lninoz  A t ;Distance movedX == X + AX(i)^ ;Total tool traverse distanceif x > (18 (i + 1) — R(i + 1)) then^;Machining finishedgoto 91next j^ ;Next sampling interval91 Cp„,,,(i + 1) = Cp^;Cost with the selected radiusnext i^ ;Next cutting direction;Plot Cp,nin for cutting directions Od(i + 1)stopendNOCALCULATE OPTIMALVELOCITYAppendix EFlow Chart for Single Pass OptimizationINPUTINPUTCALCULATE FEEDFROMTOOTH BREAKAGE CON.RANGE OF RADIUSWORKPIECE GEOMETRYCONSTRAINTSCOST INFORMATIONMETAL CUTTING DATAINCREMENTRADIUSISTORQUE OR SHANKSTRESS EXCEEDED? REDUCEFEEDYES YESREDUCEVELOCITYCALCULATECOSTNO ISCUTTER RADIUSLAST?YESEND139

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