OPTIMIZATION OF MACHINING PARAMETERS IN MILLING By Kulbir Singh B.E. (Mechanical Engineering) South Gujarat University M.Sc. (Applied Mathematics) Simon Fraser University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1992 © Kulbir Singh, 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract The cost of machining for milling is dependent on machining parameters such as spindle speed and feed per tooth. The competitiveness of manufacturing industries can be increased by optimization of machining parameters. A scientific method for the optimization of machining parameters for workpieces of continously varying radial widths is proposed in this thesis. The necessary mathematics for the proposed procedure is derived for both single pass and multi-pass milling operations. The computational results obtained 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 also suggested. The best cutting directions for a number of workpieces of known geometry are ascertained on the basis of computational results. ii Table of Contents ii Abstract^ List of Figures^ viii List of Tables^ ix Nomenclature Acknowledgement^ 1 xii Introduction 1 1.0.1^Thesis Outline ^ 2 Literature Review 3 The Economics of Milling Process 4 6 18 3.1 Cost Equation ^ 21 3.2 Tool Life Equation ^ 22 3.3 Process Constraints ^ 25 3.3.1^Tooth Breakage Constraint ^ 26 3.3.2^Shank Breakage Constraint ^ 26 3.3.3^Power and Torque Constraints ^ 29 3.3.4^Chatter Constraint ^ 29 3.3.5^Surface finish Constraint ^ 30 iii 4 Optimization of Single Pass Milling Operations with Irregular Workpieces^ 31 4.1 Mathematical Formulation ^ 4.1.1 Evaluation of Machining Time ^ 5 Optimization of Multi-Pass Milling Operations ^ 5.1 Mathematical Formulation ^ 32 42 53 54 5.1.1 Case 1: dr min < dr < s 2 ^58 5.1.2 Case 2: s 2 < dr < si ^58 5.1.3 Case 3: s i < dr < dr ^59 5.1.4 Evaluation of Cost ^60 5.2 Evaluation of Centre to Centre Tool Traverse Distance ^ 72 5.2.1 Case 1: drn.„ < dr < s 2 5.2.2 Case 2: s 2 < dr ^72 < si ^74 6 Optimal Cutting Direction for N-sided Polygonal Surfaces^75 6.1 Mathematical Formulation ^ 75 6.1.1 Analysis of Results ^94 6.1.2 Conclusions ^94 Bibliography^ 95 A Coordinate Transformation ^ 97 A.0.3 From Old to New ^ 97 A.0.4 From New to Old ^ 98 B Pseudo-code for single pass milling operations^ iv 103 C Pseudo-code for two-pass milling operations D Pseudo-code for optimal cutting direction E Flow Chart for Single Pass Optimization v ^ ^ ^ 110 120 139 List of Figures 1.1 Schematic Diagram of a CNC System ^ 2 1.2 Adaptive Machine Tool Control System Block Diagram ^ 2 2.1 Boundary representation of a polyhedron ^ 10 2.2 Block Diagram of Feedback Control Loop (Tlusty) ^ 15 2.3 Block Diagram of Adaptive Control Loop (Altintas) ^ 17 3.1 Basic Geometry of the Milling Process 19 3.2 Variation of total variable cost with cutting speed ^ 20 3.3 Variation of machining time with cutting speed 20 3.4 Mohr-circle Diagram for Shank Stresses ^ 26 4.1 Generalized Workpiece Geometry (Top view) ^ 31 4.2 Influence of Tool Radius on Cost for a Generalized Workpiece 35 4.3 Influence of Tool Radius on Cost for a Triangle 4.4 Optimum Feeds and Tool Traverse Distance For a Generalized Workpiece 37 4.5 Optimum Feeds and Tool Traverse Distance For a Triangle ^ 38 4.6 Optimum Speeds and Tool Traverse Distance For a Generalized Workpiece 39 4.7 Optimum Speeds and Tool Traverse Distance For a Triangle ^ 40 4.8 Geometry of Portion ABEF (Top View) ^ 44 4.9 Geometry of Portion BCDE (Top View) ^ 48 5.1 S i < dr 5.2 s 2 < ci,. < s i (Top view) < dr . ^ (Top view) ^ ^ vi ^ ^ ^ 36 54 55 5.3 dr n , ,„ < d, < s 2 (Top view) ^55 5.4 Cost Characteristics of a Rectangle ^60 5.5 Cost Characteristics of a Triangle ^61 5.6 Cost Characteristics of an Equilateral Triangle ^62 5.7 Cost Characteristics of a Symmetric Generalized Workpiece ^ 63 5.8 Cost Characteristics of an Unsymmetric Generalized Workpiece ^ 64 5.9 Cost Characteristics of a Rectangle (varying tool diameter) ^65 5.10 Cost Characteristics of a Triangle (varying tool diameter) ^66 5.11 Cost Characteristics of an Equilateral Triangle (varying tool diameter) ^67 5.12 Cost Characteristics of a Symmetric Generalized Workpiece (varying tool diameter) ^68 5.13 Cost Characteristics of an Unsymmetric Generalized Workpiece (varying tool diameter) ^69 5.14 Geometry for Centre to Centre Tool Traverse Distance ^ 72 6.1 Geometry of a Generalized 6-sided Polygon (Top view) ^ 75 ^ 6.2 Cutting Direction 0 (Top view) 76 6.3 Swept angle for a 5-sided Symmetric Polygon ^ 6.4 Cost characteristics for a 5-sided Unsymmetric Polygon 6.5 Cost characteristics for a 5-sided Unsymmetric Polygon 6.6 Cost characteristics for a 5-sided Unsymmetric Polygon 6.7 Cost characteristics for a 5-sided Symmetric Polygon 6.8 Cost characteristics for a 5-sided Symmetric Polygon 6.9 Cost characteristics for a 5-sided Symmetric Polygon 6.10 Cost characteristics for a Non-equilateral Triangle 6.11 Cost characteristics for a Non-equilateral Triangle vii 78 ^ 81 ^ 82 ^ 83 ^ 84 ^ 85 ^ 86 ^ 87 ^ 88 6.12 Cost characteristics for a Non-equilateral Triangle ^89 6.13 Cost characteristics for an Equilateral Triangle ^90 6.14 Cost characteristics for an Equilateral Triangle ^91 6.15 Cost characteristics for an Equilateral Triangle ^92 A.1 Coordinate Transformation:0 < 1,0 < a ^ 99 A.2 Coordinate Transformation:0 < 1,0 > a ^ 99 0> a ^ 100 A.4 Coordinate Transformation:0 > 1,0 > a ^ 100 A.5 Coordinate Transformation:0 = 7r, 0> a ^ 101 A.3 Coordinate Transformation:0 = 1, viii List of Tables 4.1 Table of constants ^34 5.1 Machining costs for single pass milling operations ^71 ix Nomenclature K3 :^specific cutting pressure a:^axial depth of cut st:^feed per tooth smax:^ maximum allowable feed per tooth Z:^number of tooth N:^spindle r.p.m. d:^radial width of cut R:^tool radius v:^tool traverse rate Seq :^equivalent feed per tooth 0:^instantaneous angle of immersion swept angle of cut Ch:^machine cost rate Ct :^tool cost tool change time X:^thermal fatigue parameter Er :^range of thermal strain parameter t e :^cooling time M:^moment on the cutter T:^torque on the cutter Tmax:^maximum allowable torque on the cutter Ormax:^ maximum allowable shank stress Pmax:^maximum allowable power At:^sampling interval Ft :^tangential force FT :^radial force Fx :^cutting force in the x-direction Fy :^cutting force in the y-direction xi Acknowledgement I am indebted to Dr. Ian Yellowley for accepting me as his graduate student. I would like to thank him for his valuable help and guidance during my studies. I am also thankful to Yetvart Hosepyan and Philip Richard Pottier for making the work environment pleasant and enjoyable with their barbed humour. Their help in the use of various computer software is also acknowledged. Finally, I take this opportunity to express my feelings of gratitude for all my friends especially the "Michael Saliba Lunch Gang". Their friendship and company provided a much needed respite from a "dreary" world of academics. xii J 4-5 , e) av )WTS r.5 H ow 3 bres Tv urtft 11 - . . 76)-r cPJ-Itv5 otcli *57 6 ' e) ci f T-r3 Tia I I Chapter 1 Introduction Low 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 these demands. Untended or minimally manned machining centres are the most versatile form of computerised manufacturing. These systems offer a significant technological advancement in terms of quality, design, time and costs, and may increase the user's competitive advantage. The most advanced automatic manufacturing systems utilize computers as an integral part of their control. The era of advanced automation started with the introduction of numerical controlled machine tools. The term numerical control is commonly used for programmable automation; a demand for high accuracy in manufacturing and a desire to reduce the production time as well as the necessity to produce complex geometries were the primary motivations for the development of these machines. Numerical controlled machines are hardware based machines which use electronic hardware and digital circuit technology. In order to increase the flexibility of these systems controllers based upon general purpose computers rather than specialised hardware were introduced. The machines which use these controllers are called CNC (Computer Numerical control) machines. The schematic diagram of a CNC system is shown in Fig. (1.1). These machine tools use a computer to control the machine tool and eliminate some of the hardware circuits in the control cabinet of an NC machine tool. CNC systems are quite flexible 1 Chapter 1. Introduction^ 2 Computer I Interpolator 1 Servo Amplifier Controller Encoder —,—,,— Machine Tool •■■gk..,■,.. Tachometer Axis Motor ^ Figure 1.1: Schematic Diagram of a CNC System Microcomputer based adaptive controller Feed Depth of cut ^--*Speed Machine tool control system Machine tool ^a. Feedback data from sensors Figure 1.2: Adaptive Machine Tool Control System Block Diagram Chapter 1. Introduction^ 3 and because of the declining costs of minicomputers and microcomputers the number of CNC systems has increased tremendously in last few decades. The full potential of CNC systems can be realized only if there are realistic strategies of prescribing the operating parameters like speeds and feedrates of the machine tool. Quite often, the prescription of these parameters is based on the experience and knowledge of the 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 into consideration. Therefore, the estimate of operating parameters tends to be conservative, which results in an under-utilization of machines and production losses. This common drawback of CNC systems can be overcome to some extent by the adaptive control strategy. In adaptive control, the operating parameters automatically adapt themselves to the existing conditions of machining in real time. Adaptive control systems for machine tools 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 within process and system constraints. The performance index can be an economic function such as the process cost or profit. In ACC systems the maximum possible machining parameters within a prescribed region bounded by process and system constraints are selected. In reality most ACC systems use a single easily measured parameter as the only variable. 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 of adaptive machine tool control. The capability of the computing equipment to do all the calculations in real-time is also crucial for the performance of adaptive control systems. Chapter 1. Introduction^ 4 Most of the objective functions used for the optimization of machining parameters are non-linear functions. Direct search techniques or hill climbing methods are required for the numerical solution of these problems. These techniques are time consuming and quite 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 machining parameters 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 optimal machining parameters for milling with various workpiece geometries. Both single pass and multi-pass milling operations are considered. Keeping in view the strong influence of cutting direction on the optimization procedure, an algorithm for the computation of an optimal cutting direction is also suggested. Some researchers have proposed the adaptive control of the feedrate based on the maximum allowable force. This method is compared with the strategy proposed in this thesis. 1.0.1 Thesis Outline A brief literature review on optimization of machining parameters, adaptive control and tool path planning for milling process is presented in chapter 2 of this thesis. Chapter 3 discusses the economics of a milling process. The tool life equation, the objective function and mathematical relations for the constraints are discussed in that chapter. Chapter 4 suggests a scientific basis for the selection of machining parameters for a workpiece of a given geometry. The optimization of machining parameters for single pass milling operations and a mathematical formulation for the evaluation of machining time is also included in that chapter. Since, in reality, a given cutter can only cut a certain maximum radial width of cut, (because of physical or dynamic constraints), some practical guidelines have been suggested for the subdivision of the total machining surface of the workpiece. Chapter 5 Chapter 1. Introduction^ 5 contains the multiple pass optimization problem for milling operations. Finally, in chapter 6 the problem of determination of the influence of cutting direction on cost has been addressed, an attempt has been made to determine the cutting direction that will minimize the cost for a generalized workpiece of given dimensions. Chapter 2 Literature Review The 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 study of milling process. Adaptive control and tool path planning also have a considerable influence on the process cost of milling. However, relatively little attention has been paid to this topic by manufacturing engineering researchers. This chapter presents a brief overview 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 for milling and have developed a suitable algorithm for its solution. They consider variable cost as the single objective function. An algorithm which minimizes this objective function within four inequality constraints is used to suggest the optimum values of cutter diameter, feed per tooth and peripheral velocity for a given geometry of shoulder. An important conclusion for the selection of cutter diameter and number of teeth is expressed as follows: "In general, it would seem that it is preferable to either use the smallest diameter of cutter available which is capable of machining the required width, or the first larger radius having a greater number of teeth." Chang and Wysk [2] have proposed an optimization criterion based on the discrete transformation method. The objective function for this study is not process cost but 6 Chapter 2. Literature Review ^ 7 production time which is expressed as follows: TT = T, + Tt L^uL TT Vf + VfP+ 1 An, d, R, Z) where Tc^actual cutting time Tt^tool change time per unit of work TT^production time per piece (idle time excluded) L^length of work piece u^unit tool change time (could be a function of R and Z) Vf^cutter traverse speed n^spindle rotation rate d^radial depth of cut R^cutter radius Z^number of teeth p^constant coefficient The 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 function contains five decision variables Vf , n, d, R and Z. Some of these variables like spindle speed and number of teeth belong to discrete sets such as ST, and S. This fact is used for the discrete transformation of the objective function. Consequently only three variables are left and the solution procedure is drastically simplified. ^ Chapter 2. Literature Review ^ 8 The optimization procedure suggested by Chang and Wysk is a good analytical exercise but, somehow, falls short of practical requirements. Minimization of production time is a desirable goal for those economies which have a scarcity of goods. The minimization of production time is a primary manufacturing objective in very rare situations. In a modern, free-market, with fiercly competitive economies the real challenge is of reducing costs and not of reducing production time. Also, the model proposed by Chang and Wysk does not take into account important constraints such as the tooth breakage and shank breakage constraint. An optimization procedure which does not safegaurd the cutting tool from breakage is of limited practical use. Chatter and the physical dimensions of the tool impose a limit on the maximum radial width for a single pass. In cases where the amount of stock to be removed in a rough milling 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. The following mathematical expression is used for the tool life equation: Tr, = 360^C6 0, Xn1SVa where ^C6^is a constant Cbs^is the swept angle of cut in radian ^X^is the thermal fatigue parameter ^Seq^is the equivalent chip thichness in mm ^V^is the peripheral velocity in mm per sec. a^is the axial depth of cut in mm a, b, m^are positive exponents (2.2) ^ Chapter 2. Literature Review^ 9 A similar tool life equation is discussed in detail in the next chapter. The cutting time per unit length for each pass is given by tco(i )^ A C7f2 ( d,) ( 1-A ) f1 (di ) ^(2.3) f1 (d) and f2 (d) are obtained from the following relations: S f2(d) < C3 VT'S fi (d)A = C2 where ^C7^is a constant V^is the cutting velocity ^T ^is the tool life ^S ^is the feed/rev. ^C2^is a constant ^C3^is a constant ^d ^is the radial width and 0<a<#<1^ (2.4) The objective function is reduced to the following mathematical form: n E(f2(d1)(1 0)fi(di)') - 0<d,< dmax E = dTOT i=1 0<a<#<1 ^ (2.5) ^ ^ Chapter 2. Literature Review ^ 10 B A A B^ B^ D C B C C Figure 2.1: Boundary representation of a polyhedron where ^d,^is the radial depth for the ith pass Lax ^( ^dTOT ^ ^ is the chatter limited depth is the total amount of stock to be removed The objective function is minimized and an optimal solution is obtained. The optimal solution has the property that all passes except one should be taken at the maximum allowable width of cut, with the one other pass used to remove the required remaining amount of stock. Some researchers have reasoned that the optimal metal removal rate should not be modelled 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 tool with the job, thus resulting in process optimization. Wang et al. [5] presented a mathematical model for computing an optimal tool cutter path for face milling. They utilized this model to identify the minimum length of cut Chapter 2. Literature Review ^ 11 for face milling flat surfaces. At the highest level a polyhedron is represented as a set of surfaces. Then each surface is broken down into line segments which compose the surface. The lowest level consists of the starting and end points of vectors which are the vertices of surfaces. An illustration of boundary representation of a polyhedron is shown in Fig. (2.1). The equation of set of surfaces of a polyhedron would be of the following form: AIX + B l Y + C i Z < (or >)D i A2X + B2 Y + C2Z < (Or >)D2 ANX + BNY + CNZ < (or >)DN Each of the above surfaces can be represented in terms of their boundary vectors as follows: AT1XT + BT1YT < (or >)DTA. AT2XT + BT2YT < (or >)DT2 ATNXT + BTNYT < (or >)DTN An N-sided polygon is divided into (N-2) triangles and the length of cut L m for each triangle is computed from an analytical expression. Finally, the lengths L m for all the (N-2) triangles are added together to obtain the total tool travel length. This procedure Chapter 2. Literature Review ^ 12 is used for a range of sweep angles from 0 to 180 degrees and the angle corresponding to the minimum tool travel length is thus determined. The same authors in a second paper [5] have examined the two commonly used approaches of stair case and window frame milling. Again a boundary representation scheme is used for the transference of input data about the workpiece geometries. A polyhedron is represented as a set of surfaces and each surface can be identified by the equation of its edges. Edges are recognized by their starting or end points Once the part geometry has been defined, the impact of the selection of a starting point and cutting orientation on tool path is studied for both stair case and window frame milling. The conclusions of this study are summarized as follows: 1. In window frame milling, the selection of a starting point does not significantly affect the length of cut, although a small amount of variation exists. 2. The cutting orientation in stair case milling produces a significant impact on the length of cut. The average variation is on the order of 5 — 10 percent. The worst case can be as much as 100 percent. 3. There appears to be no correlation between the optimal cutting orientation and other control parameters, such as tool diameter and number of edges. 4. Based on the experimental results, the optimal length of cut generated by stair case milling is better than that generated by window frame milling. However, the average results from stair case milling are sometimes worse than those of window frame milling. 5. From the experimental results, the authors observed that, for stair case milling of regular polygons, the optimal cutting orientation is normally parallel to the longest edge of a given polygon. Chapter 2. Literature Review^ 13 Prabhu and Wang [6] have also developed a mathematical model representing the total tool path on an N-sided convex polygon surface. They have considered the staircase type of tool path. The mathematical formulation is complex and can not be solved by standard analytical or numerical methods. They have proposed an algorithm to find an optimal solution between 0 and 180 degrees. The conclusions of this study are stated as follows: 1. For a triangle the optimum sweep angle seems to be the one which makes the sweep path parallel to the largest side of the triangle, which is consistent with the findings of Wang et al. [5]. 2. For a square or rectangle or parallelogram the optimum sweep angle is the one that makes 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 the largest 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 the only optimization criterion. The authors of these studies have indicated that shortest length of cut would minimise the tool wear. It is, however, not clear why these researchers have chosen this objective. In most practical situations minimization of process cost is the foremost consideration and in some rare cases minimization of production time is the primary goal of manufacturing planning. Minimization of tool wear may not fulfil any of these objectives. It has been proven by Yellowley [1] and Chang et al. [2] that both process cost and production time depend on the machining parameters. Therefore, a study on optimal tool path planning without any regard to machining parameters and constraints may not be too useful for manufacturing engineering. ^ 14 Chapter 2. Literature Review^ It has been mentioned in chapter 1 that the adaptive control of machine tools is an effective way of selecting machining parameters and reducing costs. Several researchers have 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 the workpiece traverse rate: Fact(t) = Cv act (t — T) Fact (s) = Cv act (s)e " - where T^is T the tooth period 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. The proportionality constant C is expressed as follows: C = Kba^ (2.6) where ^K^depends on the workpiece material ^b ^is the axial depth of cut a^is the radial depth of cut The force error is evaluated as follows: F Man — Fact of = ^ Fmam (2.7) Chapter 2. Literature Review^ a = 0(e/) = 15 - Fag ., F = f(v) d DDA. z N/C Servo z vs Table If vc ,„„, !kern • ,...., w■ra...i. N/C Servo Y DDA y Mini Computer Software Interpolator Figure 2.2: Block Diagram of Adaptive Control Loop (Tlusty) where Fnom^is the desired cutting force which should be kept constant by adapting feedrate The 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 comparison is the relative force error e f . Based on this force error a desired change of velocity is expressed as acceleration a = 0(ef). Integration of this expression gives us the commanded velocity v. The block diagram of the adaptive control scheme proposed by Tlusty is shown in Fig. (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 parameters. Daneshmend [10] used a similar strategy for the turning process. Due to Chapter 2. Literature Review^ 16 cutting 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 valid for dc servo controlled machine tools. Moreover, none of these above mentioned studies took into account the inevitable nonlinearities associated with systems of these kind. Altintas et al. [10] developed an adaptive control strategy based on linear dynamics of the plant with simple nonlinearities. The milling process to be controlled is considered to have two cascaded dynamic processes. The time-invariant feed drive servo control and the time variant cutting process dynamics. The discrete transfer function of the feed drive servo is expressed as: c(k)[mm/tooth] Gs(z)^ u(k)[count/8] kp z -1 (1 + ziz -1 ) C 3 (z) =^ 1 + p i z -1 where kp,^are constants The time variant cutting process dynamics is expressed by the following discrete transfer function: f3z -1 Fp (k) = ^ic(k) + 1 1 + az ^+ az - - 1 (2.8) The process parameters a, and -y are time varying and functions of the workpiece geometry. The maximum cutting forces are regulated by estimating the time varying parameters a, and -y at each sampling period. The method of Normalized Recursive Least Square is used to estimate these parameters. The block diagram of the adaptive control scheme proposed by Altintas is shown in Fig. (2.3). None of these works on adaptive force control have indicated the method of calculating the maximum reference force. It is also not certain how the feedrates obtained on the Chapter 2. Literature Review^ 17 AD . F, ■.......40.■ Controller Ru = T F,. S F, — u Servo c C•(z) Cutting process FONL -- "4 Estimator P Figure 2.3: Block Diagram of Adaptive Control Loop (Altintas) basis of this strategy are kept below the tooth breakage constraint. Therefore, there is a need to conduct a closer investigation of adaptive force control strategy so that a comparison between this strategy and the optimization strategy proposed in this thesis can be made. However, it has been decided not to include the research related to this topic in this thesis. Chapter 3 The Economics of Milling Process There are two kinds of industries in a free market economy - competitive and closed. The competitiveness of an industry can only be established by thoroughly studying all the details of the manufacturing cost and finding ways and examining means of reducing this cost. The cost of machining can only be reduced by the proper selection of machining parameters. The best selection is made when the value chosen for these parameters is such that cost is minimized or profit maximized. Milling is an important metal cutting process, but relatively little attention has been paid to the economics of this process. The basic geometry of the milling process is shown in 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 Z Optimization of these parameters can be based on several different objectives. Some of these objectives are: 18 Chapter 3. The Economics of Milling Process^ 19 Figure 3.1: Basic Geometry of the Milling Process • Minimization of process time • Minimization of process cost • Maximization of profit Depending upon specific circumstances any one of these criteria may be important. Each criterion will typically lead to the selection of different conditions. Barrow has presented an analysis for each of the above mentioned objectives. Minimization of process cost is the most common objective and, therefore, the present work is based on this objective. The cost of producing a part can be divided into fixed and variable costs. Fixed costs are independent of the machining process; these costs consist of machine centre set up costs and raw material costs. Therefore, the goal of an economic model which is of interest 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 speed Chapter 3. The Economics of Milling Process ^ Tool cost Toial cost Machining cost Non productive cost V1 Cutting speed Figure 3.2: Variation of total variable cost with cutting speed Time^Tool change time Total time Machining time Non productive time 1/2 Cutting speed Figure 3.3: Variation of machining time with cutting speed 20 Chapter 3. The Economics of Milling Process^ 21 respectively. These graphs indicate that the velocity giving the minimum cost is less than the velocity giving minimum time corresponding to maximum production rate. 3.1 Cost Equation Yellowley [1] has proposed an equation for the cost of milling per unit length at constant width and depth of cut. This equation can be expressed as follows: Ch^Ct , ChTct Ci = ^ + ^ -tV VTL VTL (3.1) where C1^is the process cost per unit length in dollars Ch^is the machine cost rate in dollars per second Ct^is the tool cost in dollars Ta^is the tool change time in seconds v^is the tool traverse rate in mms per second TL,^is the tool life in seconds The values of economic parameters used in the above equation are site specific and are therefore dependent on the shop/machine/tool combination. The machine cost rate includes labour, plant operating costs and machine operating costs. Tool change cost is calculated by multiplying tool change time by machining cost rate. Tool cost is determined by multiplying the tool life fraction used in the machining operation by the total cost of the tool. Chapter 3. The Economics of Milling Process ^ 22 3.2 Tool Life Equation Milling is an extremely complex process. It is not only a discontinuous process, but is also affected by both mechanical and thermal shock. Milling is also characterized by a variable chip thickness during cut. Research workers in Germany are of the belief that the mechanical effects are more important whereas Japanese and Soviet workers consider thermal effects to be the more prevalent. Yellowley [11] has suggested that a realistic tool life equation can be obtained by considering only the thermal effects provided that only 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 criterion is related to flank wear. A tool is considered to have reached the end of its life when it reaches the maximum limit of wearland width V. The rate of change of the width of wearland with respect to time can be approximated by the following relation: di7 ^B ,_ 1713' dt^Tr, (3.2) where Tr,^is the tool life corrresponding to VI; The above equation may be used to evaluate an expression for the equivalent feed in milling in the following manner: The relation between tool life and equivalent chip thickness is assumed to be of the following form: hTt, = C^ (3.3) where h^is the chip thickness k^is a constant with value quite close to unity ^(3.4) Chapter 3. The Economics of Milling Process ^ 23 Therefore, C TL = — h (3.5) Consequently, the rate of change of wear land width takes the following form: dVB VI; h dt^C (3.6) Integrating the above expression to evaluate the total wear in a swept angle, we obtain: q = vi; / s t sin OdO C o 3 (3.7) The expression for average wear rate is : ^vi; pp.^st sin Od O vA 0,^CO, ./o (3.8) Yellowley has proposed the concept of this so called equivalent feed rate to combine the influence of cutter diameter, width of cut and feed per tooth on the milling process. The equivalent feed rate is defined as that constant feed rate which will yield the same average wear rate as the variable chip thickness in milling. Using this concept of equivalent feed in Eq. (3.6), the average wear rate can also be expressed as: dVB^vb. seq ( dt )ave = C (3.9) From Eqs. (3.8) and (3.9), we obtain: (3.10) st JO. S„^ = —^sin OdO . 0 (3.11) co Or 1 0. Vi; Se gvi; 0 St sin NO C — , The influence of the intermittent nature of the milling process on tool life can be well represented by the thermal fatigue parameter, introduced by Yellowley [11] as : X = ED (NO^ (3.12) ^ Chapter 3. The Economics of Milling Process^ 24 where Er^range of thermal strain parameter ^N^rotational speed ^x^ratio of total cycle time to time in cut It is clear from the above relation that the thermal fatigue parameter is dependent on both the range of thermal strain and the number of thermal strain cycles per unit cutting time. The range of thermal strain is a function of heating and cooling time. The values for this parameter do not vary too much for high speed steels and carbide tool materials for the same heating and cooling times. The mathematical expression for the range of thermal strain parameter is as follows: Er = 39log t c — 23log t h + 37.5^ (3.13) where t c^cooling time t h^heating time Both the concepts of equivalent feed and thermal fatigue parameter have been used in 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 : 711, = 27r^Cl O s X m .5; 9 VP aq 1 where C1^is a constant (3.14) ^ Chapter 3. The Economics of Milling Process^ 25 0,^is the swept angle of cut in radian ^X^is the thermal fatigue parameter ^Seq^is the equivalent chip thichness in mm V^is the peripheral velocity in mm per sec. a^is the axial depth of cut in mm m, n, p, q^are positive exponents This equation is only valid when there is no chip sticking and when the lag angle between leading and trailing edges is very small. We have not considered the effect of mechanical shock caused by entry and exit conditions in the above equation. Fortunately, in processes where chip sticking does not occur, the entry and exit conditions do not affect tool life in milling. The effect of the tool/workpiece materials is reflected in the values of the constants in the equation. 3.3 Process Constraints The physical properties of the work/tool pair, the capacity of the driving motor, the dynamics of the machining process (e.g.chatter) and part design specifications such as surface finish impose constraints on the values of machining parameters. Therefore, any realistic economic model must take into account these constraints. The constraints which can influence the economics of the milling process are listed below: • Tooth Breakage Constraint • Shank Breakage Constraint • Power and Torque Constraint • Chatter Constraint Chapter 3. The Economics of Milling Process^ 26 • Surface Finish Constraint These constraints will be discussed one by one. 3.3.1 Tooth Breakage Constraint Tooth breakage is defined as the loss of a significant portion of the edge of an individual tooth. A catastrophic tooth breakage will eventually result in damage to the workpiece and the machine. Therefore, in order to avoid this catastrophic tooth breakage it is essential to control the maximum cutting stress experienced by the cutting teeth. Yellowley has defined the tooth breakage constraint limit as: St sin 95.9 < sma. St < Smas ^ ^ (d < R)^(3.15) (d > R)^(3.16) where 0 8^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 Constraint The shank of a tool may fail under the combination of bending and torsional working loads. To avoid failure, the maximum tensile stress allowed on the shank must be kept under a critical value. This critical value depends on the geometry and mechanical properties of the shank. Let a represent the normal stresses on the tool and T be the shearing stress. This state of stress can be represented by Mohr-circle diagram as shown in Fig. (3.4). The maximum tensile stress is given by the distance OA on this diagram. Chapter 3. The Economics of Milling Process^ 27 T 7 Figure 3.4: Mohr-circle Diagram for Shank Stresses It is clear from the geometry of this figure that OA = OC + CA OA = OC + AP — PC OC = cr PC = AP = PX = Therefore, OA = a + NA T ) 2 + ( ci ) 2 — cli OA = ; + NA T )2 + i ( )2 From elementary mechanics of material we know that for a circular cross section, 0" = MR I Chapter 3. The Economics of Milling Process ^ T = 28 TR J where ^M^is the moment on the cutter ^T^is the torque on the cutter ^J ^is the polar moment of inertia ^I ^is the moment of inertia ^R^is the shank radius Hence, MR 1.1 TR^MR OA =^+ ( ^ )2 + ( ^ )2 21 ^' 21 (3.17) After some simplification we obtain: croA = 2 wR3 [M + (M2 +7,2)2] (3.18) Therefore, the mathematical expression for the torque constraint is: 2 7rR3 [ 111 + (M 2 + T 2 ) ] < Gr max (3.19) where Amax ^is the max. allowable tensile stress on the tool shank For a specific tool with defined geometry and material, the tensile stress on the shank can be controlled by varying the depth of cut (a), peripheral cutting speed (V) and cutter traverse rate (v). Chapter 3. The Economics of Milling Process ^ 29 3.3.3 Power and Torque Constraints Power and torque are machine constraints which are imposed by the maximum capacity of the motor. The violation of these constraints may cause a serious damage to the power drive, spindle shaft or workpiece. These constraints can be represented by the following inequality relations: Kavd < Pm.^ v K aR( v )d < Tmax^ (3.20) (3.21) where ^K^is the specific cutting pressure ^Pmax 3 Tmax^are the maximum allowable power and torque 3.3.4 Chatter Constraint A chatter threshold limits the width of cut (d) and the depth of cut (a). This threshold must not be exceeded if instability of the milling process is to be avoided. It is extremely difficult 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 resultant force. • The width of cut influences the frequency content of the milling force signal and the basic frequencies are dependent on the cutter diameter and number of teeth. It is therefore extremely difficult to formulate a realistic chatter constraint without significant specific machine tool and work/tool data. Chatter is neglected in the optimization studies. Chapter 3. The Economics of Milling Process^ 30 3.3.5 Surface finish Constraint Surface finish imposes a constraint on the depth of cut and cutter traverse rate, especially for fine finishing. However, its influence is usually not considered in the study of rough milling operations. A theoretical estimate could of course be made based upon the kinematics of the process. This would not in most cases be indicative of actual finish because of radial run out, adhered material and dynamic effects. Therefore, surface finish constraints are also not included in the optimization studies. Chapter 4 Optimization of Single Pass Milling Operations with Irregular Workpieces The milling operation is an intermittent cutting process. The rotating cutter with one or more cutting teeth comes in contact with a translating workpiece producing a chip of variable thickness. The machining parameters for the milling process which can be optimized 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 Z Yellowley and Desmit [1] have developed an algorithm for the selection of tool diameter, feed per tooth and peripheral velocity for a shoulder of given geometry. Many workpiece surfaces however have polygonal, circular or elliptic geometries. The radial widths for these workpieces vary continously throughout the tool traverse length because of which the optimum machining conditions also keep changing. The problem of determining the optimum machining parameters for these workpieces is, therefore, quite complex but of immense practical value. 31 Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces32 u1' 12 ----414 Figure 4.1: Generalized Workpiece Geometry (Top view) In this chapter we present a model which can be used for the selection of machining parameters for a large variety of workpiece geometries. The general method and strategy on 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 Formulation Let us consider a workpiece geometry as shown in the Fig. (4.1). It is assumed that the maximum radial width encountered in the workpiece does not exceed the chatter limit and a tool which can machine the workpiece in a single pass is available in the machine shop. Also for simplicity the axial depth of cut is kept constant throughout this work unless otherwise specified. For process economy and real-time process control, the feed rate needs to be varied with changing radial width as the tool traverse the workpiece length. This is done at a series of sampling intervals. It is assumed that the tool traverses the whole workpiece Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces33 length^+ / 2 ) in n sampling intervals. let di^be the radial width for the ith interval x 2^be the distance traversed by the tool in the ith interval At^is the sampling period Ni^is the R.P.M. for the ith interval s ty^is the feed per tooth for the ith interval Here i varies from 1 to n. The whole workpiece length can be written as a summation of the 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 x k )^ (4.2) k=1 di = d2 — k2[E(xk — 1 0] ^ (4.3) k=1 where Ex k=1 k ZOt I -1 ^ .(E stkNk) 60 k=1 d2 — d1 11 k2 d 2 — d3 12 It is clear from Eqs. (4.2) and (4.3) that the radial width at each sampling interval is dependent on the workpiece geometry and the history of spindle speed and machine feed. Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces34 Using Eq. (3.1) the process cost of machining the ith interval of length x, can be expressed as: Ct ChTct Cpi = (Ch^) t TL.^TL. (4.4) where TL ,^is the tool life for the conditions at the ith interval In 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 the minimum cost. Therefore, for minimization of cost, we select the maximum allowable feed without exceeding the limit imposed by any of the constraints. Now consider the tooth breakage constraint, Eqs. (3.14) and (3.15). For any milling process, the tool is pre-selected and hence the tool radius is constant. Therefore, the feed per tooth is either a constant or a function of the radial width of cut as shown below: St, = S in=^(d Or^St, = < R) g(di)^ (d > R) s max g(d,) = ^ si n(cos -1 (1 — 14)) It has been shown earlier that the radial width of each time interval is a function of the workpiece geometry and the history of spindle speed and machine feed. It can be evaluated by using Eq. (4.2) or (4.3). Therefore, the feed which satisfies the torque constraint, Eq. (3.20), can be obtained from the above expressions. Equation (3.19) can then be used to find the spindle speed within the power constraint which results in a minimum process cost. Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces35 Symbol Ch Ct Tct C1 m n p q Pmax K, Ti smar arnax Value 0.005 dollars/sec .0396.R.Z 120 sec 1179.36 2 1 2 0.5 7.5 K.W. 4140 N/sq. mm. 0.3 0.2 mm 1242 N/sq. mm. Table 4.1: Table of constants The 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( C Pi. 7n n ) ^ (4.7) i =1 The influence of cutter radii and number of teeth on cost for a generalized workpiece and a triangular workpiece when feeds and speeds are kept optimal for each interval is shown in Figs.(4.2) and (4.3) respectively. The optimal feeds and speeds for both a generalized 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 life exponents, machine constraints, cutting constants and tool material properties used in this thesis are listed in table 4.1. It is obvious from Figs. (4.2) and (4.3) that it is preferable to use the smallest diameter cutter available which is capable of machining Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces36 dl =10,d2=20,d3=15,1i =30,12=20 5 1^1^I^1^1 10^15^20 25 30 Radius in mm Figure 4.2: Influence of Tool Radius on Cost for a Generalized Workpiece 35 Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces37 d'a =0,d2=20,d3=0,11=30,12=20 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 5 ^ ^ ^ ^ ^ ^ 15 20 25 30 35 10 Radius in mm leigure 4.3: Influence of Tool Radius on Cost for a Triangle Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces38 dl =10,d2=20,d3=15,11=30,12=20 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0 ^ 10 ^ 20 ^ 30 ^ 40 ^ 50 ^ 60 Tool traverse distance in mm Figure 4.4: Optimum Feeds and Tool Traverse Distance For a Generalized Workpiece Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces39 d1=0,d2=20,d3=0,11=30,12=20 0.65 0.6 0.55 R=28 0.5 0.45 0.4 0.35 R=20 0.3 0.25 0.2 0 10 ^ 20 ^ 30 ^ 40 ^ 50 ^ Tool traverse distance in mm Figure 4.5: Optimum Feeds and Tool Traverse Distance For a Triangle 60 Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces40 dl =10,d2=20,d3=15,11=30,12=20 660 640 R=20 620 600 580 R=22 ^......,__.__._.......„,_......,- 560 540 R=24 520 500 480 0 R=26 I^I^I^1^I 10^20^30^.40^50^60 Tool traverse distance in mm Figure 4.6: Optimum Speeds and Tool Traverse Distance For a Generalized Workpiece Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces4l dl =0,d2=20,d3=0,11=30,12=20 700 650 R=20 600 550 R=24 500 R=28 450 400 350 300 0^10^20^30^40^50^60 Tool traverse distance in mm Figure 4.7: Optimum Speeds and Tool Traverse Distance For a Triangle Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces42 the workpiece or the next larger diameter with more number of teeth. These graphs also indicate that the cost of machining a larger axial depth is higher as would intuitively be expected. Graphs (4.4) and (4.5) indicate that the allowable feed per tooth decreases with increasing radial depths and also the value of allowable feed per tooth for each radial width increases with the tool diameter. No clear trend is available from the graphs of optimum spindle speeds. It is thus clear that with the help of the optimization strategy we have employed, it is possible to determine the optimal values of feed and speed for any kind of workpiece for any cutting direction if we can obtain the radial widths either as a function of the workpiece geometry or by some other means. The optimal values of feed and speed will result in an optimal process cost. 4.1.1 Evaluation of Machining Time The estimation of total processing time is important for efficient process planning and scheduling of manufacturing activities. The total processing time consists of the following: • Set up time • Loading unloading time • Machining process time Machining process time consists of manual time and machining time. An analytical expression for the machining time in milling when the feed is maintained at an optimal value can be derived as follows: Machining time is a function of the workpiece geometry, tool diameter, number of teeth and the tool traverse rate. The tool diameter and the number of teeth can be Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces43 selected on the basis of previous analysis and availability. The tool traverse rate is dependent on the machining constraints. Therefore, it is possible to mathematically evaluate the machining time for many workpiece geometries by making use of the constraint relations. Let us again consider the geometry of Fig. (4.1). Tooth breakage can be avoided by restricting the maximum chip thickness encountered to some constant value according to the inequality relations of Eqs. (3.14) and (3.15). Rewriting those relations, we have St sin 0, < St < Smax Smax^ ^ (d < R)^(4.8) (d > R)^(4.9) where 0,^is the swept angle in radians. s max^is the maximum allowable feed per tooth in mm. s t^is the feed per tooth in mm. Case 1:d < R The tooth breakage constraint for this case is governed by equation (4.8).Rewriting equation (4.8) s t sin 0, < s max (d < R)^(4.10) This equation gives: st < s max sin 0, (4.11) The tool traverse rate in mms per second can be obtained from the relation: dx s t NZ dt^60 (4.12) ^ Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces44 where N^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 max St = • sin 0, (4.13) Combining (4.12) and (4.13) we obtain: dx s max N Z dt^60 sin 0. (4.14) A constant spindle R.P.M. which does not exceed the constraints can be determined for the given workpiece geometry. The selection of the tool is made before the start of machining so that the tool radius R and the number of teeth Z are constant during machining. The maximum permissible value of s max is also constant for a tool/workpiece combination. Therefore, the tool traverse rate can be represented by the relation: dx _ C dt^sin 0. (4.15) where s max NZ ^C = ^ 60 (4.16) From (4.15), we obtain the following relation: 1 dt = C — sin 0,,dx (4.17) We can integrate this equation to evaluate the time taken to traverse a specified length. let t i^is the time taken to traverse length d i t 2^is the time taken to traverse length / 2 Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces45 d3 ft---- 11 '-'•••0•1 Figure 4.8: Geometry of Portion ABEF (Top View) Then for portion ABEF Fig. (4.8),we obtain: jo t i^1 dt C o —^ sin 0,dx (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 from the following expression: d =^kix (4.20) The slope k 1 can be represented by the following relation: kl = d2 — d 1 11 here is the distance traversed by the tool (4.21) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces46 Using Eqs. (4.19), (4.20) and (4.21) in Eq. (4.18) we obtain the following expression: el^ dt = 1 fii^ C o sin cos -1 [1 + kix ]dx (4.22) let m1 = 1— d1 R k1 m2 R Integrating the left hand side of Eq. (4.22) and using the expressions for m 1 and m 2 in that equation, we obtain: 1 ill^ t i = —^sin cos-1(m i — m2 x)dx C o (4.23) let cos "(m i — m2x) = y l^(4.24) then m 2 x^cosy]. 1, x = ---(m i --cosy l ) m2 sin y i dy i dx m2 Substituting the variable y i in place of x by using the above relations, equation (4.23) takes the following form: 1 yi dyi t i^sin2 =^ Cm 2 The new limits of integration can be evaluated as follows: when x yl =0 = cos 1 m1 (4.25) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces47 and when = 11 X y i = cos -1 (m i — m 2 1 1 ) Equation (4.25) can also be written in the form: ti^ = ^ 2Cm 2 f (1 — cos 2yi)dyi (4.26) Integrating the right hand side of this expression, we obtain: ti = 1 ,^sin 2yi lYi^) 2 2Cm2 (4.27) Applying the limits of integration to the above expression we obtain: ti = 1 2Cm2 — cos -1 1 — m 2 / 1 ) — — sin(2(cos' (m i — m 2 1 1 ))) ^2 1 m l + — sin 2(cos -1 m l )] [cos' (m i 2 Rearranging the terms, 1 [cos l (Mi — m 2 1 1 ) — cos 1 m l ] 2Cm2 1 + ^ [sin (2 cos -1 m 1 ) — sin 2(cos -1 (m i — m211))] 4Cm 2 ti The time of traverse for any distance x between zero and 1 1 can be obtained from the following expression: tx 1 = ^ [cos -1 (m 1 — m 2 x) — —sin 2(cos -1 (m i — m 2 x))) 2Cm2 2 1 — cos -1 m l + — 2 sin 2(cos -1 m l )] where t x^is the time of traverse for a distance x between length 1 1 x^is the distance traversed (4.28) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces48 let M3 = mi i = 1 2Cm 2 cos' (m i . — m 2 x) M12 = COS -1 M1 M13 = Sin[2(M11)] M14 = sin[2(m 12 )] (4.29) With these substitutions, equation (4.28) becomes: t x = M3(M11 — \ M3 , m12) + ^ (m14 — m13) 2 (4.30) The above expression is valid for any distance x between zero and 11 Similarly we can integrate equation (4.17) for portion BCDE Fig (4.9) as follows: t2 1 f12 dt = — fo^ C o sin O a dx (4.31) where (A8 ^ is the swept angle as before From Fig. (4.9), the radial depth at an arbitrary point along length / 2 can be determined from the following relation: d = d 2 — k 2 x^ (4.32) where d2 — d 3 ^ /2 Substituting the above expressions for d and k 2 in equation (4.31), we have: k2 — f o t2 112 = I dt C o (4.33) ^d2 k 2 x ^ )]dx^(4.34) R sin[cos' (1 Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces49 d2 ilirr.^12 ........41 Figure 4.9: Geometry of Portion BODE (Top View) let 7125 = 1 — d2 R- ms = k2 ii Integrating left hand side of Eq. (4.34) and using m 5 and m 6 in that equation, we obtain: 1 /2 . t 2 = — sm[cos' (m 5 + m6 x)]dx C (4.35) let COS -1 (1ns + M6X ) = Y2 ^ then m 5 + msx = cos y 2 1, z = —kcos y 2 — m 5 ) ms sin y 2 dy 2 dr = ^ M6 (4.36) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces50 Substituting the variable y 2 in place of x in Eq. (4.35) we have: t2^ =^ 1 f Cm6 J (sin 2 y2)42 (4.37) The new limits of integration can be evaluated as follows: when x = 0 y 2 = cos -1 m 5 and when X = 12 Y2 = COS -1 (M5 + m612) Writing Eq. (4.37) in a slightly different form: t2 1 ^f ( 1 — cos 2y2 )42 2Cm 6 i L--- (4.38) Integrating the right hand side of this expression, we obtain: 1 t2 = ^ ( sin 2y2 2Cm6 2 Y2) Applying the limits of integration : r t2^ = ^ [ sin 2CM6 2 2 ( cos -1 (m 5 + m612)) 1 — cos -1 (m 5 + m6 /2 ) — 2 sin 2(cos' (m 5 )) + cos -1 m 5 ] Rearranging the terms, 1^r t2 =_ ^ [COS-1 m5 — COS -1 (M6 + 7/2612 )1 2CM6 1 + 4Cm6 [sin 2(cos -1 (m 5 + m 6 / 2 ) — sin 2(cos -1 m 5 )] (4.39 ) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces5l The time of traverse for any distance x between 1 1 and 1 2 can be obtained from the following expression: 1 ^ [cos -1 m 5 — cos -1 (m 5 m6x)] tx^ 2Cm 6 1 [sin 2(cos -1 (m 5 m 6 x)) — sin 2(cos -1 m 5 )1 ^ 4C m 6 where t a,^is the time of traverse for a distance x between length 1 1 and length 1 2 . x^is the distance traversed. let M7 1 2c1 m66 M17^cos-1(m5 + M6X) M18 = cos M19 Sin[2(M17)] 1 m5 M20 = sin[2(mis)] Substituting the new nomenclature in equation (4.40) we have: M7(M18 — m17) Ti/7 1 +^m20) 2 (4.40) The above equation is valid for any distance between 1 1 and 12 Case2:d > R The tooth breakage constraint for this case is governed by the following equation: St < S mas ^ (d > R)^(4.41) Chapter 4. Optimization of Single Pass Milling Operations with Irregular Workpieces52 let the operating feed s t0 be slightly less than the maximum allowable feed s m .., then the tool traverse rate in mms per second can be obtained from the relation: dx _ s to NZ dt ^60 (4.42) here s t0^is a constant operating speed Integration of the above expression yields: t. x C (4.43) where t x^is the time of traverse for a distance x between length (l i + 1 2 ). x^is the distance traversed. and s to NZ C -= 60 (4.44) It is clear from Eqs. (4.30), (4.40) and (4.43) that the machining time for milling is a function of the workpiece geometry, tool diameter, number of teeth and the tool traverse rate. For many industries such as aircraft industry increasing the production rate is a highly desirable goal. The analytical expressions for machining time can be used to determine the machining parameters which minimize the production time for specific workpiece geometries. Since minimization of process cost is the focus of this thesis it has been decided not to explore the criterion of minimization of production time any further. Chapter 5 Optimization of Multi-Pass Milling Operations The maximum allowable radial width of cut for a single pass is dependent on the chatter constraint and the size of the milling cutters available in the machine shop. When the radial width of cut for machining is larger than the maximum allowable value, there is a need 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 each pass. 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 established an optimal selection procedure for the radial width of cut for each pass. This chapter deals 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. Rewriting Eq. (4.5) CPC = 34) (5.1) It was shown in chapter 3 that the objective for a single pass milling operation is to minimize the value of C pz for each sampling interval. The radial width of the workpiece d, 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, which does not exceed the limit imposed by the power constraint and results in the minimum cost is selected for each sampling interval. The process cost of machining the whole 53 ^ Chapter 5. Optimization of Multi-Pass Milling Operations^ 54 workpiece is obtained from the following expression: Cp E Cpi^ (5.2) 2=1 For a multipass operation the radial width di for each interval can be varied by changing the subdivision of the machining area. Our objective is then to minimize the following expression: k ni = min. E( E f (dt, , Art,' i =1 i 3 =1 S ts j )^ (5.3) Subscript i denotes the number of intervals and j denotes the number of passes. For example represents the radial width at the ith interval and jth pass. Mathematical formulation and computational results for two pass milling operations are presented below. These results will then be used to make conclusions about a general multi-pass milling operation. 5.1 Mathematical Formulation The generalized workpiece of Fig. (4.1) can be subdivided into three different geometric shapes depending on the maximum radial width dr selected for the first pass. A suitable range 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 pass be the upper limit in the range of max. radial width of the first pass A i ,^be the workpiece length for the first pass ^A2^be the workpiece length for the second pass Chapter 5. Optimization of Multi-Pass Milling Operations ^ 55 T First pass Second pass Figure 5.1: S i < d . < dr . (Top view) y here dq = da — dy, = rs, nl 4.4 i n2 A2 X is = i2 =1 Let us introduce the following geometric constants: s i = d2 — 32 = d2 — d3 Figs. (5.1), (5.2) and (5.3) show the three different ways in which the workpiece can be subdivided when d, moves in its range from dims ,, to d,„,... It is obvious that with the 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 dr within the selected range and evaluate the machining cost. We can thus ascertain the best subdivision of the workpiece for minimum machining cost. Chapter 5. Optimization of Multi-Pass Milling Operations^ .1•••••■ S ■•••■ First pass MINN& ■1,. ■■•10^WENN. ••■■• .1■ND ■■•■■/ d, Second pass Figure 5.2: s 2 < d,. < s i (Top view) T First pass Second pass Figure 5.3: ds.„, < dr < 32 (Top view) 82 56 Chapter 5. Optimization of Multi-Pass Milling Operations ^ 57 Using Eq. (4.4), the process cost of machining the ith interval of jth pass of length x ii can be expressed as: Cpi3 = (Ch Ct^ChTct )At m^m IL, 3^1 L ‘3 (5.4) where is the tool life for the conditions at the ith interval of jth pass At^is the sampling period^ (5.5) In terms of the basic machining parameters, Eq. (5.4) can be expressed as: = f(di , Ars a , St, )^ (5.6) The process cost of machining the whole workpiece is : k n3 Cp E(E C j=1 . 3 =1 pii ^(5.7) For minimum process cost, we should have: k^113 Cp mn -7-- DE( )min ^ (5. 8) j=1 i.,=1 In order to determine the influence of d,. on cost, we should be able to express the machining 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 spindle speed and machine feed. In addition, in a two-pass milling operation, the radial width would also depend on the way the workpiece has been subdivided . In other words the radial width can be represented as a function of dr , workpiece geometry and the history of spindle speed and feed. Workpiece geometry and the history of spindle speed and feed are known beforehand. Therefore, it is possible to determine the influence of dr on cost by using Eq.(5.6). Let us mathematically derive the relationships for the radial width for the three cases represented by figures (5.1), (5.2) and (5.3). ^ Chapter 5. Optimization of Multi-Pass Milling Operations^ 58 5.1.1 Case 1: dr ,„,„ < dr < 32 For this case the radial width of the first pass is given by one of the following two relations: di, -=^ E xk) (0 < k=1 dr E xk < —) k=1 d,.^d , dr^a1-1^ v-, r k < — —) k l^k=1^ lk2 di l = dr k2(^xk) —< k=1 x k where Z Ot a -1 3th Nk) 60 (^ k=1 k=1^ E xk kl d2 = 11 d2 — d3 k2 = 12 The radial width of the second pass can be obtained from one of the following three relations: =2-1^ i2-1 di e =^+ ki( E xk)^(o < E xk ,^dr , Li — —) k=1 k=1^ die = d2 — dr^ d dr , (/1 —^<^xk < /1 + k=1 i2-1^ (d 2 — dr) — k2(E 4^22-1 xk)^(/1^< k=1^ = E xk 2 ) < /1 + 12) k2^k=1 4 4 + k-2 k1 AZ = + 5.1.2 Case 2: s 2 < dr < s1 For this case the radial width of the first pass is given by one of the following two relations: 2, -1-1 d„ = k i (^( o < E Xk < --k; ) k=1^ k=1 Chapter 5. Optimization of Multi-Pass Milling Operations ^ ^ dr^dr ti _ i d1 = (d3 — (d 2 — dr)) k2^(xk –^( k=1^ Xk < k J. 12;1 + 1 2) where E k=1 Z At " -1 ^s t ,N,) 60 •( k=1 Xk = k2 d2 — d 1 11 d2 d3 12 The expressions for the radial width of the second pass are : i2 -1^ 22-1 dr d22 =^+ ki(^x,)^<^< /, – ^— -dr ) k=1^ k=1 d^12-1 ^<^x, < + / 2 ) ^— d22 = d 2 — dr k=1 The length of the workpiece for the two passes : dr kJ. A 2 = 11 -1- 1 2 5.1.3 Case 3: s 1 < dr < dr„,.. The radial width of first pass can be obtained by one of the following two relations: -1^ ^21 21-1 di ,^(d i – (d 2 – dr)) + ki(E^<^Xk < li) ^k=1^ ^it -1^ – /1)^ k=1 -1 ti (ll <^Xk < 1 1 + d21 = (d3 — (d2 — dr)) — k2^(xk ^k=1^ k=1 where 1 2) 59 Chapter 5. Optimization of Multi-Pass Milling Operations^ —1 Z At 60 21-1 ^ •(E s th Nk ) 60 k=1 Xk k=1 d2— k1 d1 11 d2 d3 k2 12 The radial width of the second pass is: die = d2 —^(0 <^< /1 + 1 2) ^ (5.9) k=1 The length of the workpiece for the two passes : = 1 1 + 12 2 = 1 1 + 12 5.1.4 Evaluation of Cost Feed per tooth for a constant tool radius can be obtained from the tooth breakage constraint from the following relations: Sti j = Smax Or ^ ^ st, i = g(c1, 2 ) ^ (d (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 constraint. Eqs. (5.4), (5.6) and (5.7) can then be used to find that spindle speed within the power constraint which gives the minimum machining cost. The cost characteristics for a rectangle, triangle, an equilateral triangle, a symmetric and an unsymmetric generalized 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 Chapter 5. Optimization of Multi-Pass Milling Operations ^ dl =d2=d3=20,I1=30,12=20 6 Pass 1 5 Pass 2 4 T. cost 3 2 1 0 0 ^ ^ ^ 16 24 8 Max. radial width of first pass in mm Figure 5.4: Cost Characteristics of a Rectangle 61 Chapter 5. Optimization of Multi-Pass Milling Operations ^ 62 dl =d3=0,d2=20,11 =30,12=20 3.5 Pass 1 3 Pass 2 2.5 --X-T. cost 2 1.5 1 0.5 0 0 ^ ^ ^ 16 24 8 Max. radial width of first pass in mm Figure 5.5: Cost Characteristics of a Triangle Chapter 5. Optimization of Multi-Pass Milling Operations ^ 63 dl=d3=0,d2=20,I1=11.547,12=11.547 Pass 1 Pass 2 X 1 T. cos t 0.8 0.6 0.4 0.2 0 8 16 24 Max. radial width of first pass in mm Figure 5.6: Cost Characteristics of an Equilateral Triangle Chapter 5. Optimization of Multi-Pass Milling Operations ^ 64 dl =1 0,d2=20,d3=1 0,11 =30,12=30 6 Pass 1 5 X Pass 2 4 X T. cost 3 2 1 0 0 ^ ^ ^ 24 16 8 Max. radial width of first pass in mm Figure 5.7: Cost Characteristics of a Symmetric Generalized Workpiece Chapter 5. Optimization of Multi-Pass Milling Operations ^ 65 dl =10,d2=20,d3=1 5,11 =30,12=20 5 4.5 Pass 1 X 4 Pass 2 3.5 X 3 T. cost 2.5 2 1.5 1 0.5 0 Max. radial width of first pass in mm Figure 5.8: Cost Characteristics of an Unsymmetric Generalized Workpiece Chapter 5. Optimization of Multi-Pass Milling Operations ^ d1=20,d2=20,d3=20,11=30,12=20 3.5 3 Total cost 2.5 1.5 1 Pass 1 /." Pass 2 0.5 0 I ^I 1 0^4^8^12 16 20 Max. radial width of first pass in mm Figure 5.9: Cost Characteristics of a Rectangle (varying tool diameter) 24 66 Chapter 5. Optimization of Multi-Pass Milling Operations ^ d1=0,d2=20,d3=0,11=30,12=20 2.5 Total cost 1.5 Pass 2 0.5 Pass 1 4^8^12^16^20^24 Max. radial width of first pass in mm Figure 5.10: Cost Characteristics of a Triangle (varying tool diameter) 67 Chapter 5. Optimization of Multi-Pass Milling Operations ^ dl =0,d2=20,d3=0,11=11.54,12=11.54 4 ^ ^ ^ ^ ^ 12 16 20 24 8 Max. radial width of first pass in mm Figure 5.11: Cost Characteristics of an Equilateral Triangle (varying tool diameter) 68 Chapter 5. Optimization of Multi-Pass Milling Operations ^ 69 dl =10,d2=20,d3=10,11=30,12=30 0 ^ 4^8^12 ^ 16 ^ 20 ^ 24 Max. radial width of first pass in mm Figure 5.12: Cost Characteristics of a Symmetric Generalized Workpiece (varying tool diameter) Chapter 5. Optimization of Multi-Pass Milling Operations ^ 70 dl =10,d2=20,d3=15,11=30,12=20 4 3.5 3 2.5 2 1.5 1 0.5 0 0 ^ 4^8^12 ^ 16 ^ 20 ^ 24 Max. radial width of first pass in mm Figure 5.13: Cost Characteristics of an Unsymmetric Generalized Workpiece (varying tool diameter) Chapter 5. Optimization of Multi-Pass Milling Operations^ 71 feeds are selected in an optimal manner. Three different cutter radii are used and the axial 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 . In actual practice, a tool with a diameter equal to the larger of the two values of d,. and dq is 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 are listed in table (5.1). The following conclusions can be drawn from the above mentioned data : • Single pass operations should be preferable to multi pass milling operations provided the chatter limit is not encountered during the single pass operation. • The cost increases at a faster rate with increasing radial widths for smaller radial widths. • The cost decreases at a much slower rate with decreasing radial widths for larger radial widths. • The machining cost for an area with larger radial widths will be less than that of an equal area with smaller radial widths. • It is preferable to first machine the area with largest radial widths of cut within the 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 ^ Workpiece geometry Unsymmetric generalized Symmetric generalized Unsymmetric triangle Equilateral triangle Rectangle d1 10 10 0 0 20 d2 20 20 20 20 20 d3 15 10 0 0 20 11 30 30 30 11.54 30 12 20 30 20 11.54 20 72 Cost in cents 2.208859 2.708165 2.036704 0.660608 2.122573 Table 5.1: Machining costs for single pass milling operations 5.2 Evaluation of Centre to Centre Tool Traverse Distance The machining time is dependent on the centre to centre tool traverse distance. This distance is a function of the workpiece geometry and the tool radius. The accurate evaluation 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 distance is included in this section for the sake of completeness. 5.2.1 Case 1 :d* min <^s2 From the geometry of Fig. (5.14), the following relations can be obtained: = Z3 = Z4 = Z5 = /9 1^(5.10) L6 = 90 — pq = sin L6 pq^Rsin(90 — 0 1 ) pq^R cos 0 1 (5.11) as^d2 — (d1 + dr) qa = R — pq^ qa nq sin L5 (5.12) Chapter 5. Optimization of Multi-Pass Milling Operations ^ 73 Figure 5.14: Geometry for Centre to Centre Tool Traverse Distance nq qa sin 0 1 R 2 + nq 2 = on 2 xn 2 ^ (5.13) (5.14) on e —R 2 (5.15) Using the above relations, we obtain: xn 2 R2 + nq 2 — R2 ( qa )2 sin 0 1 ( R R cos 0 1 )2 xn 2 sin 0 1 R R cos 9 1 xn = sin 0 1 1 — cos 0 1 xn = R( ^ ) sin 0 1 xn 2 — — where 9 1 = tan-1( d2 — dl ) 11 Chapter 5. Optimization of Multi-Pass Milling Operations^ 74 Similarly ym =-- R( 1 — cos 92 sin 92 ^) (5.16) where 12 d3 ) 82 = tan -1 ( d2 -- The total centre to centre distance moved by the tool for the first pass is: L1 = A i + xn + ym^ (5.17) where 4 4 Ai = ^ + tan 81 tan 9 2 5.2.2 Case 2: s2 < di. < si From Fig. (5.14), it is clear that the centre to centre tool traverse distance for the first pass is: Li =-- A i + xn + R^ (5.18) where Ai . ^ 4 + 12 tan 8 1 The centre to centre distance moved by the tool for the second pass for all the three cases is: L2 = A2 + 2R A2 = 1 1 --IF 12 Chapter 6 Optimal Cutting Direction for N-sided Polygonal Surfaces The selection of cutting direction is an extremely important issue which has a considerable effect on a machining process. Tool wear is dependent on the tooliworkpiece engagement time which is directly influenced by the selection of direction of tool motion. The cutting direction also determines the geometry of the machining surface and hence the other parameters 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 this chapter the selection of the best cutting direction for minimum machining cost for an N-sided polygon is discussed. 6.1 Mathematical Formulation Let y:) ^ 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, = ni i (x — x,) i = 1,2,3,...n where mi = Yi Yi+ i xi — i = 1,2,3, ...(n — 1) 75 Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^76 Figure 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 sampling intervals and the distance traversed in the jth sampling interval is x 3 . then la = Ex^ ; (6.2) also = 4, + 2R9^(6.3) where ^Lax ^ is the maximum workpiece length in a given direction ^Ra^is the cutting tool radius The cutting tool can be represented by the equation of a circle as follows: _^(y gi)2 = j = 1,2,3, ...p^1 Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^77 Figure 6.2: Cutting Direction 9 (Top view) where (h,, gi) are p 1 tool centres for p sampling intervals Our aim, as before, is to determine the cutting parameters such as spindle speed and feed for each sampling period and to evaluate the machining cost. The machining cost can then be obtained for a range of cutting directions and the cutting direction which gives the minimum machining cost is finally selected. The cutting direction is defined as the angle which the direction of tool traverse makes with the positive x-axis as shown in the Fig.(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 the edges of the workpiece. There can be a maximum of 2n points of intersection between a circle and n straight lines. Out of all the points of intersection only two points of intersection which fall on the boundary of the workpiece represent the entry and exit Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^78 points. The angle these two points subtend at the centre of the circle is the swept angle of cut for a particular position and the moving direction of the tool. let be the cutting direction (x:, y:)^be the vertices of the polygon in the new coordinate system (1 3 , g31 )^be the tool centres in the new coordinate system i = 1,2,3,...n j = 1,2,3,...p+ 1 The mathematical formulas for coordinate transformation are derived in appendix A. The maximum width of the workpiece dmax perpendicular to the cutting direction 8 is 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=1 The maximum workpiece length / max in the 8 direction is the largest of the following distances: 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^79 Td 3 A Figure 6.3: Swept angle for a 5-sided Symmetric Polygon abs(x: — x: +3 ), i = 1,2,3,...(n — 3) a b.5 ( — X n ^i = 1 let ), (a, 21 , Y2 ) ^(4, di), (c'2, d'2) ^ ^ be the coordinates corresponding to the maximum width be the coordinates corresponding to the maximum workpiece length Re^be the tool radius for the 9 direction ^ Re =^ (6.4) (6.5) The coordinates of the tool centre at the start of machining (WI , gl) can be determined ^ Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^80 from the expressions: h'1 = c11 — R hi= C l2 - R g1 = b'b' 1^2 2 The subsequent tool centres can be obtained from the following relations: E 3 -1 hj =^ Xk k=-1 3 h; = hi — -1 E Xk k=1 g 1 j = 2,3,4,...p+1 Two relations for^and 14 are for two sides of the workpiece. The points of intersection between the tool and the workpiece edges can be obtained from the following relations: /2i-1 - —Pi +^— 4cea 2ai i = 1,2,3,...n —^4(347 11/2i^2ai i^1,2,3,...n where = 2m:b: — 2m:g'j — 2h 3. = (1):) 2 (gil ) 2 + (14) 2 — 2b:g'j — ai = 1 + (m:)2 RB Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^81 b: = y i — mi x i Y: Y:+1 xi — —,:+1 i = 1,2,3,...n Out of these 2n points of intersection, a pair of points which lie on the boundary of the polygon and fall in the cutting region are selected. These two points correspond to the entry and exit positions. let (e'3 f) 3 . be the coordinates of the entry point be the coordinates of the exit point The distance between these two point is: De^\^f^j)2 L6 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 of triangles as follows: cos ( 2R e2 — DI ) Once we know the swept angle of cut for a particular sampling interval, we can find the optimal feed, spindle speed and cost by making use of the optimization strategy described earlier. Costs for all the sampling intervals can be summed to obtain the total cost of machining at a cutting direction of O. This procedure has been used to compute costs for a range of angles from 0 degree to 180 degrees. The graphs between cutting direction and the cost for a few shapes are shown in Figs. (6.4)-(6.11). Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces^82 dl =10,d2=20,d3=15,11=30,12=20 5.5 5 4.5 4 3.5 3 2.5 2 0 i ^ 1 40 1^1 80 1^I^I^I ^1 120^160 Cutting direction in degree Figure 6.4: Cost characteristics for a 5-sided Unsymmetric Polygon 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^83 dl =1 0,d2=20,d3=15,11 =30,12=20 6.5 6 5.5 5 R=dmax 4.5 4 ^^ 1^I^'^I 1 ^ 1 ^ ^ 40 80 120^160 Cutting direction in degree Figure 6.5: Cost characteristics for a 5-sided Unsymmetric Polygon 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^84 dl =1 0,d2=20,d3=1 5,11 =30,12=20 5.2 4.8 4.6 4.4 4.2 Ly 40^80^120^160^20C Cutting direction in degree Figure 6.6: Cost characteristics for a 5-sided Unsymmetric Polygon Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^85 d1=10,d2=20,d3=10,I1=30,12=30 5.5 4.5 R=0.55dmax 3.5 ^ 1 ^ I^'^I^'^I^ ' ^ 40 80^120^160 Cutting direction in degree Figure 6.7: Cost characteristics for a 5-sided Symmetric Polygon 20( Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^86 dl =10,d2=20,d3=10,I1=30,12=30 7.5 6.5 R=dmax i^I^I^I^1 120^160 Cutting direction in degree Figure 6.8: Cost characteristics for a 5-sided Symmetric Polygon 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^87 dl =10,d2=20,d3=10,11 =30,12=30 6.2 6 5.8 5.6 5.4 5.2 5 4.8 4.6 R=1.5dmax 1 0^40^80^120^160^20C Cutting direction in degree Figure 6.9: Cost characteristics for a 5-sided Symmetric Polygon Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^88 d1=0,d2=20,d3=0,11=30,12=20 3.4 3.2 2.8 2.6 2.4 2.2 40 80 11[1^i 120^160 Cutting direction in degree Figure 6.10: Cost characteristics for a Non-equilateral Triangle 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^89 d1=0,d2=20,d3=0,11=30,12=20 4.4 4.2 3.8 3.6 3.4 3.2 3 0 I^'^I^r^I^I 40^80^120^160 Cutting direction in degree Figure 6.11: Cost characteristics for a Non-equilateral Triangle 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^90 dl =0,d2=20,d3=0,11=30,12=20 5.2 4.8 4.6 4.4 4.2 I 80 I I I I 120^160 Cutting direction in degree Figure 6.12: Cost characteristics for a Non-equilateral Triangle 1 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^91 dl =0,d2=20,d3=0,11=11.54,12=11.54 1.8 1.6 1.4 1.2 1 0.8 0 ^ ' I ^ I ^ '^ I ^ ' ^ I ^ ^ 40 80^120^160 20C Cutting direction in degree Figure 6.13: Cost characteristics for an Equilateral Triangle Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^92 dl =0,d2=20,d3=0,11 =11.54,12=11.54 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 0 40 I 80 I^III! 120^160 Cutting direction in degree Figure 6.14: Cost characteristics for an Equilateral Triangle 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^93 dl =0,d2=20,d3=0,I1=11.54,12=11.54 2.4 2.3 vv 2.2 2.1 2 1.9 1.8 R=1.5dmax 1.7 1.6 0 1 40 1^'^I^'^1^' 80^120^160 Cutting direction in degree Figure 6.15: Cost characteristics for an Equilateral Triangle 20C Chapter 6. Optimal Cutting Direction for N-sided Polygonal Surfaces ^94 6.1.1 Analysis of Results The 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 radial width encountered during machining; the cutting directions of 0 degree and 180 degree 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 the maximum radial width encountered during machining; the cutting direction of 90 degree 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 width encountered during cut are shifted upwards. In other words, the cost increases for the selection of larger cutters. 6.1.2 Conclusions This chapter describes a general method of evaluating the best cutting direction for a workpiece of known geometry. Even though the results have been obtained for only a few workpieces under certain site specific conditions, the method is valid for almost any workpiece geometry. A change in the assumptions and site specific conditions would alter the results. Software based on the proposed mathematical formulation can be developed which would give the best cutting direction for any set of conditions and any workpiece geometry. 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 a Discrete 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 Machining Operations, International Journal of Production Research, Vol. 27, 1989, pp. 15731588. [4] Chang, H., Wysk, R.A., Wang, H., An Analytical Approach to Optimize NC Tool Path 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 for Face 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 for Optimizing NC Tool Path for Face Milling of Flat Convex Polygonal Surfaces, International Journal of Production Research, Vol. 28, 1990, pp. 101-130. [7] Tlusty, J., Elbestawi, M.A.A., Analysis of Transients in an Adaptive Control Servomechanism for Milling with Constant Force, Transactions of ASME, 1977, pp. 766-772. 95 Bibliography^ 96 [8] Tomizuka, M., Oh, J.H., Dornfield, D.A., Model Reference Adaptive Control of the Milling 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 in Turning, 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 Journal of 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 Peripheral Milling, International Journal of Machine Tool Design and Research, Vol. 16, 1976, pp. 1-12. Appendix A Coordinate Transformation Let the coordinate system X -Y be rotated by an angle of 9 degrees. The new coordinate system is X' - Y' represented by X' and Y' axis. The coordinate (x, y) in the old system is (x', y') in the new sytem. The line joining the origin with the coordinate (x, y) make an angle of a. The coordinates (x, y) and (x', y') can be mathematically related by the following equations (see Figs. 6.5- 6.9): A.0.3 From Old to New 9<1 0<a Y a = tan' - x y = p sin a y P = sin a / Y = p sin(a - 0) x' = p cos(a - 0) 9<I 8>a Y a = tan -1 x 97 Appendix A. Coordinate Transformation ^ y = p sin a P , y sin a Y i = — p cos(2 — (a — 0)) X 8 i = psin(2 — (a — 0)) = 12 0<a xI = y yI = X 0 = 7r 0<a I Y = X i = — y - x A.0.4 From New to Old 0<1 0<a tan(a — 8) = Y1 x' a = tan -1 y ' 0 x' Y1 = sin(a — 0) p Y1 P= sin(a — 9) y = p sin a x = p cos a 98 Appendix A. Coordinate Transformation^ < 9 > a tan(7; — (0 — a)) — — (0 2 — a)) = abs(tan -1 ( a^ p — 2 abs(tan -1 ( -7)) ^sin(; — (0 — a)) p = ^ sinq — (0 — a) y^p sin a x = p cos a B = 22 <a y y= <a —Y X^- X 99 Appendix A. Coordinate Transformation^ Figure A.1: Coordinate Transformation:8 < 1,8 < a Y x • Figure A.2: Coordinate Transformation:8 < I, 8 > 100 Appendix A. Coordinate Transformation^ fe,—......—..^yg .--.....■.0.1 Figure A.3: Coordinate Transformation:0 = 1,0 > a (::------- 2 1 ,■44 ..„... Figure A.4: Coordinate Transformation:0 > 1 ,6 > a ; 101 Appendix A. Coordinate Transformation^ Figure A.5: Coordinate Transformation:0 = 7r, 8 > a 102 Appendix B Pseudo-code for single pass milling operations ;Inputs: Geometrical parameters of workpiece^d2, d3 , 1 1 , / 2 Economic parameters^:Ch, Ct , Tct Tool life constant and exponents ^m, n, p, q Machine constraints^:Pmax,14 Cutting constants^ :Ks, r i Range of cutter radii^:Rtnin) 'Lax Small increment in cutter radius^:AR Range of spindle speeds^:Vnun, Vmax Cutting conditions^ :smax, a Tool material properties ^:o-max, etc. Milling type^ :Upmilling or downmilling Sampling interval^ :At ;Outputs: Cutter radius and Machining cost ^:R, Cpmin ;Variables: Bigk — R ma7A-RR "lin ^ ;Number of steps to cover the range of cutter radii 103 Appendix B. Pseudo-code for single pass milling operations ^ Rt = Rm " ,T Rmin ;Increase in radius after which the number of teeth change 1 k2 = d2 — Ch 11 ;Radial width slope upto length d2 — d3 12 ;Radial width slope after length Pmao(• 7 5 4, 10 6 ) C .a i = 0 to Bigk^ ;Power constraint constant ;Loop for range of radii R(i 1) =^iAR^;Next cutter radius if Rmin < R(i 1) <^Rt)^;Selection of cutter teeth then z=2 if ( 1?-mt. + R t ) < R(i + 1) < (Rmin + 2Rt) - then z =3 if (Rm.+ 2Rt ) R(i +1) < (Rm. + 3Rt) then z=4 if R(i + 1) > (R„,,„ 3R d ) then z=6 L(i 1) = 6R(i 1)^ ;Flute length 104 Appendix B. Pseudo-code for single pass milling operations ^ 105 Tmax = .75 * 106 * Prn- " R( s + ' )^;Maximum allowable Torque Cp = 0.0^ ;Initialize the process cost x = 0.0^ ;Initialize total tool traverse distan Delta= 500^ ;Arbitrary number for sampling ;intervals j = 1 to Delta^ ;Loop for sampling intervals if x < 1 1^;Evaluation of radial depth of cut then d(j) = else d(j) = d2 — k 2 (x — 1 1 ) 0 (j) = cos -1 (1^Rd(a( 3+)1) )^ ;Swept angle of cut if d(j) = 0 ;Initial radial depth for triangle 3 then d(j) = 2 31 if R(i + 1) > d(j) ;Feed per tooth then ;from tooth breakage constraint st(i) = sin(;:(j)) s else st(j) = sma. ( ^7 1) = Ka az (2 ) F=(j) = 7(j)[(1 — cos(205(j))) + ri( 2 46 .(.7) Cutting force constant sin(20 5 (j)))] ;Average cutting forces Appendix B. Pseudo-code for single pass milling operations ^ 106 Fib) = -6)(7'1(1 — cos( 2 0.(j))) ( 2 080) — sin( 2 0.(i)))] ;for upmilling Fx(j) = 7(i)[ri( 2 03(i) — sin( 2 080))) — (1 — cos( 2 0.(i)))] ;Average cutting forces _EVA = -Y(7)tri( 1 — cos( 2 09(i))) ( 2 08(j) — sin( 2 03(j)))] ;for downmilling VF2 + FR(j) = ;Resultant cutting force M(j) = FR (j)(gi + 1) — ;Resulting moment on the cutter T(j) = Yd(i) ;Resulting torque - 0- (j) = p R(+ 1 ) . 1u(i) Vm(i) 2 + T(i) 2 i ;Shank stresses ;Shank breakage constraint if crU) < Cfmax then goto 41 ;Check torque constraint else s max = s max — 0.01 ;Reduce maximum feed goto 31 ;Calculate feed again 41 if Tmax > T(j) ;Torque constraint check then goto 51 ;Calculate equivalent feed else Srnas Smax — 0.01 goto 31 0 I ;Reduce maximum feed ;Calculate feed again st(j)d(j) • \^ Ni-1-1)08(i) Nmax(j) = 2 71-VA(7 1-2 1 - ;Equivalent feed per tooth ;Range of ) Appendix B. Pseudo-code for single pass milling operations 107 ;r.p.m. = 27,-v R(:+ 1) rmax • — _62L IV^(3 P ^ ;Maximum r.p.m. allowed by StO,Z ;Power constraint AN = 100 ;Initial increment for r.p.m. —1‘1""' 61 Omega=integer value of -Arma°AN ;Upper limit of the index for r.p.m. optimization loop k = 0 to Omega ;Loop for optimization of r.p.m. N(k 1) =^kAN ;Next spindle r.p.m. while Nmaxp > N(k 1) ;Power constraint check N(k + 1) = N(k + 1) — 1 ;Reduce spindle r.p.m. (k + 1) = Th (k + 1) 2rR(i+1)N(k+1) 60 ;Spindle peripheral velocity *R(i+1)41.(j ) = 60000V(k+1) s0000.n(i+i)(22 -0.(3)) Te (k + 1) -=^ V (k+1) - smallx(k 1 ) ^ ;Heating time in ms ;Cooling time in ms ;Ratio of total cycle time to actual cutting time per cycle Er (k + 1) = 39 log(Tc (k + 1)) — 23 log(Th (k + 1)) + 37.5 ;Range of thermal strain X(k 1) = Er(k 1 )(N(k+i).x6somali(k-1-1) )4^ ;Thermal fatigue parameter Appendix B. Pseudo-code for single pass milling operations^ Tak 1) = 0^ 2:(.j) X(k+1)ntSeq (j)nV(k+1)Paq ACp (k + 1) = (Ch 71,+1 ) c ^TChTt /(-1- 1) 108 ;Tool life )At ;Process cost for one interval if k > 1 then^ ;Search of optimum r.p.m. if ACp (k + 1) > ACp (k) then^;by comparing the cost if 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.m AN =^ and new r.p.m. increment goto 61^ ;New upper limit of loop index else Nmin = N(1) AN = 1 if k = 0 then ACman ACp (k + 1)^ ;Initialization of minimum cost ^ ;Initialization of optimum r.p.m. Nmin o N(k 1) if AC(k + 1) < AC,„„ then^Comparison with previous minimum cost AC,,, in = AC(k + 1)^ ;New minimum cost N„,,„. = 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 ^ Cp = Cp AX(i) = ACmin ^ st(j)1`61(T in o z pt x = x As(j) ^ ^ ;Summation of costs ;Distance moved ;Total tool traverse distance if x > l l + 1 2 then^ ;Check whether the goto 91^ ;whole workpiece is machined else 81 next j^ ;Next sampling interval 91 Cp,..(i + 1) = Cp^;Cost with the selected radius next i^ ;Plot Cpm , for different cutter radii R(i 1) ;stop end ;Next tool radius 109 Appendix C Pseudo-code for two-pass milling operations ;Inputs: Geometrical parameters of workpiece^d2, d3, 4,12 Economic parameters ^:Ch, Ct,Tct Tool life constant and exponents Machine constraints^:Pmax Cutting constants Range of maximum radial width^:d7 min7 di.maa for the first pass Small increment in maximum radial width of first pass Tool geometry^ :R, z Range of spindle speeds^:Vmin, Vmax Cutting conditions^ :smax, a Tool material properties^:amax, etc. Milling type^ :Upmilling or downmilling Sampling interval^ :At ;Outputs: Maximum radial width of^CPmsn first pass and Machining cost 110 Appendix C. Pseudo-code for two-pass milling operations ^ ;Variables: Bigk = drma. ;Number of steps to cover the range of max. radial width of first pass 31^d2 dl ;Geometric 32 = d2 — d3 ;constants L d2 -d1 ;Radial width slope upto length d i d2 -d3 ;Radial width slope after length 1 1 ^ — k2— Pm ..(.75*10 6 ) Ksa ;Power constraint constant i = 0 to Bigk^;Loop for range of maximum radial width of the first pass dr (i + 1) = dr „,„ + iAdr^;Next maximum radial width of the first pass L = 6R^ Truax = . 75 * 10 6 * P. (=+1) ;Flute length ;Maximum allowable Torque Cp = 0.0^ ;Initialize the process cost x = 0.0^ ;Initialize total tool traverse distance flag = 1^ ;Initialization of pass number Delta= 1000^;Arbitrary number for sampling intervals j = 1 to Delta^;Loop for sampling intervals 111 Appendix C. Pseudo-code for two-pass milling operations ^ ;Case 1 if 4,, < dr (i + 1) < 82 then d (i+1) Al =^ka ;Length of first pass A2 = 11 + 1 2 ;Length of second pass if x < 4(i+1) and flag= 1 then ;Radial depth of first d(j) = kix if dr(ki+1) < x < a l and flag= 1 then d(j) = k2 (x^dr (ki 1+1) ) ;pass of machining if x <^dr(kii+1)) and flag= 2 then d(j)^+ k i x if (1 1^dr(kil-1-1) < x < (A 2^dr(ki2+1) ;Radial depth of second ) and flag= 2 ; d(j) d2 — dr (i + 1)^ if x > (A2^dr(ki2+1) ) and flag= 2 then ;pass of machining 112 Appendix C. Pseudo-code for two-pass milling operations ^ d(j)^(d2 — dr (i + 1)) — k 2 Ex — (li + 4(12: 1) )} ;Case 2 if 8 2 < dr (i + 1) < then dr (i-F1) , /2 Al —^1k1 ;Length of first pass A2 = 11 + 1 2 ;Length of second pass if x < 4(1+1) and flag= 1 then d ( i) = k i x if d,(i-1-1) kl ;Radial depth of first < x < A i and flag= 1 then d(j) = dr(i + 1) — k2(x — ;pass of machining if x < (1 1^dr (kli+1) ) and flag= 2 then ;Radial depth of second d(j) = d1 k i x if x^dr(i+i) ) and flag= 2 d(j) d 2 — dr (i + 1)^ ;pass of machining 113 Appendix C. Pseudo-code for two-pass milling operations^ ;Case 3 if .9 1 < dy.(i + 1) < d,.. then Ai = 11 + 12 ^ ;Length of first pass A2 - 1 1 + 1 2 ^ ;Length of second pass if x < 1 1 and flag= 1 then d(j) =d 1 — (d 2 — dr (i + 1)) + kix^;Radial depth of first if / 1 < x < A i and flag= 1 then d(j) = d3 — (d2 — dr (i + 1)) — k 2 (x — l i ) ;pass of machining if x < A2 and flag= 2 ;Radial depth of second then d(j) = d2 — dr (i + 1) ;pass of machining if d 1 = d2 = d 3 ;Rectangular workpiece if x < A i and flag= 1 ;Radial depth of first then d(j) = dr (i + 1) ;pass of machining 114 Appendix C. Pseudo-code for two-pass milling operations ^ if x < A2 and flag= 2^ 115 ;Radial depth of second then d(j) = d2 — cl,(i + 1)^ 0 8 (j)= cos -1 (1 ;pass of machining — 14)^;Swept angle of cut if d(j) = 0^ ;Initial radial depth for triangle then d(j) = 2 31 if R > d(j) ;Feed per tooth then ;from tooth breakage constraint st(j) = sui(46.(7)) else st(i) = 3 rnax 7(i) = K„azw 2 at(j) Cutting force constant Fx (j) = -y(j)[(1 — cos(20,(j))) r i (20.(j) — sin(20,(j)))] ;Average cutting forces Fib) = 7(j)[ri (1 — cos( 2 0.(1))) ( 2 0.(j) — sin( 2 0.0)))] ;for upmilling Fx (j) 7(j)[ri( 2 0.(j) — sin( 2 0.(i))) — ( 1 — cos( 2 0.(j)))} ;Average cutting forces Fy(i) = "Y(2)[ri( 1 — cos( 2 0.(j))) ( 2 0.(j) — sin( 2 0.(3)))] ;for downmilling FR(j) -= VF2 + ^ ^ M(j) = FR(i)(L ^ T(i) =7d(j) ;Resultant cutting force ;Resulting moment on the cutter ;Resulting torque Appendix C. Pseudo-code for two-pass milling operations ^ u( 3) = pi 2R ,M( j)VM(j) 2 TW 2 if Q(j) < umax ;Shank stresses ;Shank breakage constraint then goto 41 ;Check torque constraint else Smax = Smax — 0.01 ;Reduce maximum feed goto 31 ;Calculate feed again 41 if Tmax > T (i)^ ;Torque constraint check then goto 51^ ;Calculate equivalent feed else Sm ax = Sm ax 0.01^ ;Reduce maximum feed goto 31^ S e q()) = at(i)d(7) 313) ;Calculate feed again ;Equivalent feed per tooth Nmax(i) ^ ;Range of Nmin (•) — v2; ;r.p.m. Nmax,(i) = 06:3(7)1.^ ;Maximum r.p.m. allowed by ;Power constraint AN = 100^ ;Initial increment for r.p.m. 61 Omega=integer value of Nma'a-NNmin ;Upper limit of the index for r.p.m. optimization loop 116 Appendix C. Pseudo-code for two-pass milling operations ^ 117 k = 0 to Omega^ ^ N(k +1)^+ LAN ;Loop for optimization of r.p.m. while Nm a .,, > N(k + 1) ;Power constraint check N(k +1) = N(k +1) —1 ;Reduce spindle r.p.m. V(k + 1) .= 211RN61(3k+i) ;Spindle peripheral velocity 600004444j) Th (k + 1)^v(h+i) ;Heating time in ms R(271 0.4.i)) Tc (k + 1) = 600004 ^V(k+1) ;Cooling time in ms smallx(k + 1) = ^ ;Ratio of total cycle time to , -- ;Next spindle r.p.m. actual cutting time per cycle E,.(k + 1) = 39log(Tc (k + 1)) — 23 log(Th (k 1)) + 37.5 ;Range of thermal strain X(k + 1) = E,.(k 1)(N(k+i)*x6somall(k-14)).12, ^;Thermal fatigue parameter TL (k + 1) = 462,,(73 ) X(k-1-1)mSegniqk+1)Pa9 ^ ACp (k + 1) = (Ch ;Tool life ^ )At^;Process cost for one interval Appendix C. Pseudo-code for two-pass milling operations ^ if k > 1 then^;Search of optimum r.p.m. if ACp (k + 1) > ACp (k) then ;by comparing the cost if 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.m AN —and new r.p.m. increment io goto 61^ ;New upper limit of loop index else Nmin = N(1) AN = 1 if k 0 then ^ ;Initialization of minimum cost ACTn in ACp (k + 1) ^ Nmin . = N(k +1) ;Initialization of optimum r.p.m. if AC(k + 1) < AC,,,„ then Comparison with previous minimum cost ACT„,„ = AC(k 1)^;New minimum cost Nminc, = N(k + 1)^New optimum r.p.m. next k^ ;Try the next r.p.m. Cp Cp + 0 Cmin^;Summation of costs Ax ( i ) = 60min a z 8t(i)N At ^;Distance moved ^ ;Total tool traverse distance x = x Ax(j) if x > .A i and flag= 1 then ^;End of first pass ^ ;Start of second pass flag= 2 118 Appendix C. Pseudo-code for two-pass milling operations^ 119 x = 0^ ;Initialization of tool traverse distance if x > A2 and flag= 2 then^ ;End of second pass goto 91 next j^ ;Next sampling interval 91 Cp„„,ji + 1) = C,^ ;Cost with the selected radius next i^ ;Next maximum radial width for ;first pass ;Plot Cp„„„, for maximum radial widths dr (i + 1) ;of first pass stop end Appendix D Pseudo-code for optimal cutting direction ;Inputs: Geometrical parameters of workpiece ^d2, d3 , / 1 , /2 Economic parameters ^:Ch, Ct,Td Tool life constant and exponents^rn, n, p, q Machine constraints^ Cutting constants^ Vi :K,,r1 Range of cutting orientations ^:em..) Borax Small increment in cutting orientation :At9 Tool geometry^ :z Number of sides of polygon ^:BigL Co-ordinates of one vertex ^:x(3), y(3) Shape of workpiece^ :Shape Range of spindle speeds^:Vmtn) Vmax Cutting conditions^ :smas, a Tool material properties^:Amax, etc. Milling type^ :Upmilling or downmilling Sampling interval^ :At ;Outputs: Cutting direction^ :0, Cp ,„„ and Machining cost 120 Appendix D. Pseudo-code for optimal cutting direction^ ;Variables: Bigk = Integer of ;Number of steps to cover the range Borax Bmin oe of cutting orientations Pmax(.75*10 6 ) ;Power constraint constant El^Kitt ;Prescribing the coordinates of polygons ; if shape = 1 then^ x(1) ;Five sided polygon + 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) = d 1 + y(3) Y( 3 ) = Y( 3 ) y(4) = y( 3 ) y(5) = d3 + y(3) if shape = 2 then^ x(1) = 1 1 + x(3) ;Three sided polygon 121 Appendix D. Pseudo-code for optimal cutting direction^ x(2) = x(3) x(4) = 1 1 + / 2 + x(3) x(5) = l l + / 2 + x(3) y(1) d2 + y(3) y(2) = y(3) Y( 3 ) = Y( 3) i = 0 to Bigk ;Loop for range of cutting orientations Od(i + 1) °min iA0 ;Cutting orientation in degrees 0(i + 1) . d „,„ = 0.0 i lrOd(i+1) 0+ ;Cutting orientation in radians ;Maximum workpiece width ;perpendicular to cutting direction lmax = 0.0 ;Maximum workpiece length ;along cutting direction L = 0 to BigL^;Loop for transformation of co-ordinates a(L) tan -1 0)^;Angles of lines through origin ;and vertices in radians 122 Appendix D. Pseudo-code for optimal cutting direction^ P(L) — si4a(L(1), ;Lengths of lines joining origin and vertices ;Transformation of vertex co-ordinates if 0(i + 1) < 1 and O(i -I- 1) < a(L) then y'(L) = P(L) * sin(a(L) — 9(i + 1)) if 0(i + 1) < li and 0(i +1) > a(L) then y'(L) = —P(L) * sin(9(i + 1) — a(L)) if 0(i + 1) < 1- then x'(L) = P(L) * cos(abs(a(L) — 0(i + 1)) if 0(i + 1) = I then x'(L) = y(L) y'(L) = x(L) if 9(i + 1) > 2 and 0(i + 1) < r then if 12(-- > (0(i + 1) — a(L)) then y'(L) = —P(L) * cosg- — (0(i + 1) — a(L))) x'(L) = P(L) * sing — (9(i + 1) — a(L))) 123 Appendix D. Pseudo-code for optimal cutting direction^ if 0(i + 1) > z and 0(i + 1) < it then if ; < (8(i + 1) — a(L)) then y'(L) = —P(L) * cos(0(i 1) — a(L) — x'(L) = —P(L) * sin(0(i + 1) — a(L) — if 8(i + 1) = 7r then y'(L) = —y(L) x'(L) = —x(L) ;Maximum workpiece width and length if L > 1 then y20(L) = abs(y'(L) — y'(L — 1)) x20(L) = abs(x'(L) — x'(L — 1)) if y20(L) > d„nax then dmax = y20(L) bi = y'(L) ;Co-ordinates corresponding b2 = y'(L — 1) .to the maximum width x'(L) a2 = xi(L — 1) if x20(L) > / max then /„„„ = x20(L) 4 = xPrime(L)^ 4 = xl(L — 1)^ ;Co-ordinates corresponding ;to the maximum length 124 Appendix D. Pseudo-code for optimal cutting direction ^ if L > 2 then y21(L) = abs(y'(L) — y'(L — 2)) x21(L) = abs(x'(L) — x'(L — 2)) if y21(L) > dma . then dmax = y21(L) bl = y'(L) 14 = y'(L — 2) di = x'(L) a2 = x'(L — 2) if x21(L) > L as then = x21(L) e1 = x'(L) e, 2 x 1 (L — 2) if shape = 1 then if L > (BigL — 2) then y22(L) = abs(y'(L) — y'(L — 3)) x22(L) = abs(x'(L) — x'(L — 3)) if y22(L) > dm. then dmaz = y22(L) 1/1 = y'(L) b12 = yi(L — 3) al = x'(L) 125 Appendix D. Pseudo-code for optimal cutting direction ^ a2 = x'(L — 3) if x22(L) > / mai then /max = x22(L) cii = x'(L) 4 = x'(L — 3) if shape = 1 then if L > (BigL — 1) then y23(L) = abs(y'(L) — y'(L — 4)) ; x23(L) = abs(x'(L) — x'(L — 4)) ; if y23(L) > dmax then dmax = 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'^ 2 = x'(L — 4) next L Ra(i + 1) = dlaz^;Tool radius for full immersion 126 Appendix D. Pseudo-code for optimal cutting direction^ 127 R(i + 1) = 1.1* Ro (i + 1)^;10 percent tolerance 9i(2 + 1) _ 61 6z if ei < c'2 then 114(i + 1) = c'i — R(i + 1) else 14(i + 1) = c'2 — R(i + 1) 19 (i + 1) = 14(i + 1) + 1„,„ x + 2R(i + 1) ;Slope of edges of polygon L = 1 to BigL if L < Big L then sden(L) = x'(L + 1) — x'(L) ;Denominator in slope expression snum(L) = y'(L + 1) — y'(L) ;Numerator in slope expression else sden(L) = x'(1) — x'(L) snum(L) = y 1 (1) — y'(L) if abs(sden(L)) < abs(1) then m'(L) = 200^ ;Edge is perpendicular to cutting direction if abs(snum(L)) < abs(1) then^;Edge is parrallel to cutting direction Appendix D. Pseudo-code for optimal cutting direction^ m/(L) = 0 if abs(sden(L)) > abs(1) then if abs(snum(L)) > abs(1) then mi( L ) = sZen um(C,L) ) b'(L) = y'(L) — mf(L) * x'(L)^;y intercept for edges next L L(i + 1) = 6R(i 1)^;Flute length Tmaz = .75 * 10 6 * 1: " R(1+1) vl ;Maximum allowable Torque Cp = 0.0 ;Initialize the process cost Delta= 1000 ;Arbitrary number for sampling intervals s ti = 0.02 ;Initial feed = 200 ;Initial speed x= at Niz 60 ;Initial distance moved j = 1 to Delta^;Loop for sampling intervals h'i (j) = 14(i + 1) + x ;Subsequent co-ordinates of tool centre = 91(i + 1 ) 3 = —1000 ;Initialization of y co-ordinate of exit point f; = 1000 ;Initialization of y co-ordinate of entry point countp = 0.0 ;Counter of entry and exit points 128 Appendix D. Pseudo-code for optimal cutting direction ^ 129 ;Points of intersection L = 1 to BigL if m'(L) = 200 then bh(j) = —2 * gii (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) = 1 disc(j) = bh(j) 2 — 4* ah(j) * chp(j) ko = 1 to 2^ if m'(L) 200 then if disc(j) > 0 then if ko = 1 then 9p(ko) = -bh(j)+"disc(2) hp(k o ) = m'(L) * gp(1) b'(L) else 9p(ko) — -bh(i)--Vcfisc(j) 2*ah(j) hp(k o ) = m'(L) * gp(k o )d- b'(L) if m'(L) = 200 then if disc(j) > 0 then if k o = 1 then gp(k o ) = x'(L) ;Loop for entry and exit points Appendix D. Pseudo-code for optimal cutting direction hp(ko)^ ^ --bh(J)+^diac(j) 2 else 9134( 1c.)^x'(L) -1'k-0- N/disco) hp(k o ) if disc(j) < 0 then gp(ko ) = 1000000 hp(k o ) = 1000000 gp(ko ) = integer of gp(k o ) hp(ko ) = integer of hp(k o ) if L < Big L then if x'(L) > x'(L + 1) then fxlim = x'(L) ;Upper and lower sxlim = xi(L -I- 1) ;limits of abscissas for edges else fxlim = x'(L 1) ;Upper and lower limits sylim = x'(L) ;of ordinates for edges if y'(L) > y'(L + 1) then fylim = y'(L) sylim = y'(L + 1) else fylim = yi(L + 1) sylim = y'(L) if L =BigL then 130 Appendix D. Pseudo-code for optimal cutting direction^ if x'(L) > x'(1) then fxlim = x'(L) sxlim = x'(1) else fxlim = x'(1) sxlim = x'(L) if y'(L) < y'(1) then fylim = Y l ( 1 ) sylim = y'(L) else fylim = y'(L) sylim = yl(1) if gp(k,,)> (j) then^ if m'(L) 200 and m'(L) L Check for entry 0 then^;and exit points if gpint(k o ) < fxlim and gpint(k„)> sxlim then if hpint(k o ) < fylim and hpint(kc,)> sylim then if abs(hp(k o ) — (m'(L) * gp(k 0 ) b' (L))) < 2 then countp = countp + 1 ; Increase the count by one if hp(k,,) >^then vij = hp(k o )^ = gp(ko )^ ; Update the ;exit point 131 Appendix D. Pseudo-code for optimal cutting direction^ if hp(k o ) < f; then f; = hp(ko )^ ;Update the e'i = gp(ko )^ ;entry point goto 28 if gp(ko ) > 11,;(j) then if m'(L) = 200 then if gpint(k o ) = f xlim and gpint(k o ) sxlim then if hpint(k o ) < fylim and hpint(k o ) > sylim then countp countp 1 if hp(k o ) > v; then hp(k o ) u?3 if hp(k o ) < u", then = hp(ko) e, gp(ko) goto 28 if gp(ko ) > 114(j) then if m'(L) = 0 then if gpint(k o ) < fxlim and gpint(ko) > sxlim then if hpint(k o ) fylim and hpint(ko ) sylim then countp countp + 1 132 Appendix D. Pseudo-code for optimal cutting direction^ if hp(k o ) >^then • = hp(ko ) = gp(k o ) if hp(ko ) < ,f; then = hp(k o ) eij gp(ko ) next ko next L if countp = 0 then ;No tool workpiece glp(j) = 0.0 ;engagement hlp(j)^0.0 g2p(j) = 0.0 h2p(j) = 0.0 if countp = 1 then ;Tool just touches glp(j) = 0.0 ;the workpiece but no hlp(j) = 0.0 ;cutting g2p(j) = 0.0 h2p(j) = 0.0 if countp > 1 then ;Entry and exit points glp(j) = hlp(j) = 133 Appendix D. Pseudo-code for optimal cutting direction^ 134 g2p(j) = h2p(j) = f; dp(j) = V(glp(j) — g2p(j)) 2 (hlp(j) — h2p(j)) 2 ;Distance between entry and exit points f- --111;(i) 2 exper(j)^2 *-11i.-Z 0,(j) = cos -1 (exper(j))^ ;Swept angle of cut if O s (j) = 0 then next j d(j) = R(i + 1)(1 — cos(0 8 (j))) ;Radial width if d(j) = 0 ;Initial radial depth for triangle then d(j) = 2 31 if R > d(j) ;Feed per tooth then ;from tooth breakage constraint st(i) else st(j) = sma. 1, (1) = K a;: (3) . Cutting force constant Fx (j)^7(j)[(1 — cos(20.(j))) ri(20.(j) — sin( 2 0.(j)))1 ;Average cutting forces Fy (j)^'Y(j)[ri(1 — cos( 2 0.(j))) — ( 2 0.(j) — sin( 2 0.(j)))] ;for upmilling Appendix D. Pseudo-code for optimal cutting direction ^ 135 Fz (j) = -y(j)[ri( 2 0.(i) — sin( 2 0.(i))) — ( 1 — cos ( 2 0.(j)))] ;Average cutting forces Fy (j) = 7(i)[ri(1 — cos( 2 0.(j))) ( 2 0.(i) — sin ( 2 0.(j)))] ;for downmilling FR(j) = VF2^ M(j) = FR(j)(L —^ T(j) = 'yd(j) ^ 5 (j) = pi2R,^ M (j)VM(j)2+ T(j) 2^ if Q(j) < Grmax^ ;Resultant cutting force ;Resulting moment on the cutter ;Resulting torque ;Shank stresses ;Shank breakage constraint then goto 41^ ;Check torque constraint else Smacc = Smaz — 0.01^ goto 31^ 41 if Tmaz > T(j) ;Reduce maximum feed ;Calculate feed again ;Torque constraint check then goto 51 ;Calculate equivalent feed else Smax = sm. — 0.01 goto 31 ;Reduce maximum feed ;Calculate feed again Appendix D. Pseudo-code for optimal cutting direction ^ at (7 )d(i) Se q0^ i. ;Equivalent feed per tooth 3 Nmax(i) vm 2.-r ;Range of Nmin (j) = ^ ;r.p.m. ;Maximum r.p.m. allowed by Nniaxp(i ) .6t()z^ ;Power constraint ;Initial increment for r.p.m. AN = 100^ 61 Omega=integer value of Nnial-NNnwn ;Upper limit of the index for r.p.m. optimization loop k = 0 to Omega ^ N(k + 1) = Nmin + LAN ;Loop for optimization of r.p.m. ^ ;Next spindle r.p.m. while N,„„, > N(k + 1)^;Power constraint check N(k + 1) = N(k 1) — 1^;Reduce spindle r.p.m. V(k + 1) = 2R-RN(k+1) 60 Th (k + 1 ) = 60000*R42,(j) V( k+ 1) Tc (k + 1) = s0000.R(2,95.(;)) v(k+i) ;Spindle peripheral velocity ;Heating time in ms; ;Cooling time in ms 136 Appendix D. Pseudo-code for optimal cutting direction ^ smallx(k + 1) =^ 137 ;Ratio of total cycle time to actual cutting time per cycle .E,.(k + 1) = 39log(Tc (k + 1)) — 231og(Th(k + 1)) + 37.5 ;Range of thermal strain X(k + 1) = Er (k + 1 )(N(k+1)*znall(k-l-ly ;Thermal fatigue parameter 71L(k + 1) = 02.(7) x (k-I-1)", Seq (3)°V(k-1-1)Pa 9 ;Tool life c t^chTct )At ACp(k + 1 ) = (Ch _L Tdk+i) + Ti( + 1 )1 ;Process cost for one interval if k > 1 then ;Search of optimum r.p.m. if ACp (k + 1) > ACp (k) then ;by comparing the cost if AN > 1 then ;with the previous value, if ACp (k + 1) > ACp (k) then ;by comparing the cost if AN > 1 then ;with the previous value, N(k 1) ;narrowing the speed range, Nmin = N(k — 1) ;setting new limits for r.p.m AN _ AN and new r.p.m. increment goto 61 ;New upper limit of loop index Nmax = 10 else Nmin = N(1) AN = 1 if k = 0 then Appendix D. Pseudo-code for optimal cutting direction ^ OCmin = OCp(k+1) Arm in . N(k 1) ^ ^ 138 ;Initialization of minimum cost ;Initialization of optimum r.p.m. if AC(k + 1) < AC„„„ then^Comparison with previous minimum cost AC„, in = LIC(k + 1)^ ;New minimum cost Nmino = N(k + 1)^ New optimum r.p.m. next k^ ;Try the next r.p.m. Cp = Cp ACmin^ A x u) 6Lninoz st(j)N X == X + AX(i) A t^ ^ ;Summation of costs ;Distance moved ;Total tool traverse distance if x > (1 8 (i + 1) — R(i + 1)) then^;Machining finished goto 91 next j^ ;Next sampling interval 91 Cp „,,,(i + 1) = Cp^;Cost with the selected radius next i^ ;Plot Cp,nin for cutting directions O d (i + 1) stop end ;Next cutting direction Appendix E Flow Chart for Single Pass Optimization INPUT RANGE OF RADIUS WORKPIECE GEOMETRY CONSTRAINTS COST INFORMATION METAL CUTTING DATA INPUT CALCULATE FEED FROM TOOTH BREAKAGE CON. IS TORQUE OR SHANK STRESS EXCEEDED? INCREMENT RADIUS YES REDUCE FEED NO CALCULATE OPTIMAL VELOCITY YES CALCULATE COST NO IS CUTTER RADIUS LAST? YES END 139 REDUCE VELOCITY
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Optimization of machining parameters in milling Singh, Kulbir 1992
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Title | Optimization of machining parameters in milling |
Creator |
Singh, Kulbir |
Date Issued | 1992 |
Description | The cost of machining for milling is dependent on machining parameters such as spindle speed and feed per tooth. The competitiveness of manufacturing industries can be increased by optimization of machining parameters. A scientific method for the optimization of machining parameters for workpieces of continously varying radial widths is proposed in this thesis. The necessary mathematics for the proposed procedure is derived for both single pass and multi-pass milling operations. The computational results obtained 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 also suggested. The best cutting directions for a number of workpieces of known geometry are ascertained on the basis of computational results. |
Extent | 4006179 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080870 |
URI | http://hdl.handle.net/2429/2257 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1992-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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