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Robust Multicriteria Optimization of Surface Location Error and Material Removal Rate in High-Speed Milling under Uncertainty

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ROBUST MULTICRITERIA OPTIMIZATI ON OF SURFACE LOCATION ERROR AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER UNCERTAINTY By MOHAMMAD H. KURDI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Mohammad H. Kurdi

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To my Mom and Dad.

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iv ACKNOWLEDGMENTS I would like to thank my advisor Dr. T ony Schmitz for his advice and generous financial support of my resear ch. I would like to thank Dr. Haftka for his expert advice and inspiring questions. I woul d like to thank Dr. Mann for in troducing me to the field of time finite elements. I would like to thank the committee members Dr. Schmitz, Dr. Haftka, Dr. Mann, Dr. Schuller and Dr. Ak cali for their advice, time and effort. In completing my research I was lucky to be a member of the Machine Tool Research Center where I had th e opportunity to work with intelligent and hard working graduate students. I would like to thank all fellow members for their helpful suggestions and interactions. Also, I would like to thank Ms. Christine Schmitz for taking the time to edit the dissertation draft. I would like to thank my wife Caro lina for her continued support and encouragement. I would like to thank my da ughters Alanis and Alia for bringing laughter and joy to my life. Finally, I would like to thank my mom and dad for their endless encouragement and support.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix NOMENCLATURE........................................................................................................xiv ABSTRACT....................................................................................................................xvi i CHAPTER 1 INTRODUCTION........................................................................................................1 Justification of Work....................................................................................................1 Literature Review.........................................................................................................2 Optimization in Machining....................................................................................2 High-speed Milling Optimization..........................................................................3 Multi-objective Optimization................................................................................3 Stability and Surface Location Error.....................................................................4 Scope of Work..............................................................................................................6 2 MULTI-OBJECTIVE OPTIMIZATION.....................................................................8 Fundamental Concepts in Mu lti-Objective Optimization.............................................8 Single and Multi-objective Optimization..............................................................8 Definition of Multi-Objective Optimization Problem.........................................10 Definition of Terms.............................................................................................10 Pareto Optimality.................................................................................................11 Multi-objective Optimization Methods......................................................................12 Methods with a Priori Articulation of Preference s using a Utility Function......13 Weighted global criteria method..................................................................14 Weighted sum method..................................................................................14 Exponential weighted criterion....................................................................15 Weighted product method............................................................................15 Conjoint analysis..........................................................................................15 Methods with a Priori Articulation of Preference s without using a Utility Function...........................................................................................................16

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vi Lexicographic method..................................................................................16 Goal programming methods.........................................................................16 Methods for an a Posteriori Articulation of Preferences....................................16 Bounded objective function method............................................................17 Normal boundary intersection (NBI) method..............................................17 Normal constraint (NC) method...................................................................18 Homotopy method........................................................................................18 Choice of Optimization Method..........................................................................18 3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM..............................20 Introduction.................................................................................................................20 Milling Problem..........................................................................................................20 Milling ModelEquation Chapter 3 Section 1.......................................................20 Solution Method..................................................................................................21 Problem Specifics................................................................................................22 Stability Boundary...............................................................................................22 Surface location error and stability boundary: C1 discontinuity..................23 TFEA convergence.......................................................................................24 Optimization Method..................................................................................................26 Particle Swarm Optimization Technique.............................................................26 Sequential Quadratic Programming ( SQP ).........................................................27 Problem Formulation..................................................................................................27 Problem Statement...............................................................................................27 Tradeoff Method..................................................................................................28 Robust Optimization............................................................................................29 Problem solution..........................................................................................29 Reformulation of problem............................................................................32 Bi-objective space........................................................................................37 Selection of spindle speed perturbation bandwidth......................................38 Case Studies.........................................................................................................41 Radial immersion ( a )....................................................................................41 Chip load ( c ).................................................................................................42 Discussion...................................................................................................................46 4 UNCERTAINTY ANALYSIS...................................................................................47 Milling Model.............................................................................................................48 Stability and Surface Location Error Analysis....................................................50 Bi-section Method Convergence Criterion..........................................................50 Number of Elements............................................................................................50 Numerical Sensitivity Analysis..................................................................................51 Truncation Error..................................................................................................51 Step Size..............................................................................................................52 Case Studies................................................................................................................52 Stability Sensitivity Analysis......................................................................................58 Surface Location Error Sensitivity Analysis..............................................................61

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vii Uncertainty of Stability Bounda ry and Surface Location Error.................................63 Input Parameters Correlation Effect....................................................................63 Monte Carlo Simulation......................................................................................64 Sensitivity Method...............................................................................................65 Latin Hyper-Cube Sampling Method..................................................................68 Robust Optimization under Uncertainty.....................................................................69 Discussion...................................................................................................................70 5 EXPERIMENTAL RESULTS...................................................................................72 Cutting Force Coefficients..........................................................................................72 Milling Forces.....................................................................................................72 Experimental Procedure......................................................................................74 Covariance Matrix (Linear Multi-Response Model)..................................................80 Compliant Tool M odal Parameters.............................................................................84 Stability Lobe Validation....................................................................................87 Stability Lobe Uncertainty..................................................................................87 Experimental Procedure......................................................................................90 Results.................................................................................................................91 Pareto Front Validation...............................................................................................93 Pareto Front Simulation Results..........................................................................93 Experimental Procedure and Results...................................................................95 Conclusions.......................................................................................................101 6 SUMMARY..............................................................................................................103 Robust Optimization Algorithm...............................................................................103 Limitations and Future Research..............................................................................104 APPENDIX A TIME FINITE ELEMENT ANALYSIS..................................................................106 B MATLAB CODE......................................................................................................115 LIST OF REFERENCES.................................................................................................177 BIOGRAPHICAL SKETCH...........................................................................................185

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viii LIST OF TABLES Table page 1 Classification of solutions........................................................................................13 2 Cutting conditions and modal parameters for the tool used in optimization simulations...............................................................................................................30 3 Cutting conditions, modal parameters and cu tting force coefficients used in biobjective space simulations......................................................................................37 4 Cutting conditions, modal parameters and cutting force coefficients used in radial immersion case study.....................................................................................41 5 Milling cutting conditions, modal paramete rs and cutting force coefficients used in chip load study case.............................................................................................43 6 Cutting force coefficients, modal parameters and cutting conditions of milling process......................................................................................................................49 7 Cutting coefficients for 1 insert endmill for slotting cutting tests............................77 8 Up milling cutting coefficients for 12% radial immersion......................................78 9 Estimated cutting force coefficients and their correlation matrix for 7475 aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30 degree helix angle.....................................................................................................83 10 Tool modal parameters in x and y -directions...........................................................85 11 Correlation coefficient matrix for modal parameters...............................................85 12 Surface location error cutti ng conditions for two Pareto optimal designs with no uncertainty considered............................................................................................100

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ix LIST OF FIGURES Figure page 1 (a) Typical Pareto front in the criteria space (b) Design variables x1 and x2, and constraint in the design space.....................................................................................9 2 Pareto optimality and domination relation...............................................................13 3 Schematic of 2-DOF milling tool.............................................................................20 4 Surface location error and its absolute.....................................................................24 5 A typical stability boundary.....................................................................................24 6 Convergence of stability constraint for 5% radial immersion and different spindle speeds for an 18 mm axial depth.................................................................25 7 Schematic of milling cutting conditions and various types of milling operations...28 8 Stability, SLE f and M RRfcontours with optimum points overlaid............................31 9 A typical optimum point found; optimum point sensitivity with respect to spindle speed is apparent..........................................................................................31 10 Perturbed average of SLE f validation as optimizati on criterion that avoids spindle speed sensitive SLE f ...................................................................................33 11 Stability, SLE f and M RRf contours with optimum Pareto front points found using PSO and SQP (average perturbed spindle speed formulation).................................34 12 Pareto front showing optimum poi nts found using three optimization algorithms/formulations; the same trends are apparent............................................35 13 Variations in the eigenvalues, surfac e location error, and removal rate for PSO and SQP optima, where M RRf is the objective for both.............................................36 14 Average surface location error contour s for 300 rpm bandwidth perturbation, stability boundary and material removal rate (see Table 3).....................................38 15 Feasible domain........................................................................................................39

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x 16 Contour lines corresponding to constant spindle speed in feasible region of biobjective space.........................................................................................................39 17 Average surface location error contours for 100 and 300 rpm band width, stability boundary and materi al removal rate contours............................................40 18 Pareto front for spindle speed and axia l depth as design variables with radial immersion 0.508 mm, compared to the case where radial immersion is added as a third design variable..............................................................................................43 19 Pareto front using chip load as a third design variable compared to spindle speed and axial depth as design variables..........................................................................44 20 Stability, perturbed average SLE f and M RRf contours with optimum Pareto front points found using 100 rpm and 400 rpm bandwidth...............................................45 21 Schematic of 2-D milling model..............................................................................49 22 The effect of error limit in the bi section method on nume rical noise in the sensitivity calculation...............................................................................................53 23 Sensitivity of SLE with respect to Kx.......................................................................54 24 Comparison between 2nd and 4th order central difference formulas.........................55 25 The logarithmic derivative of axial depth with respect to inpu t parameters versus step size percentage..................................................................................................56 26 The variation of axial depth blim with respect to a 10% change in nominal input parameters................................................................................................................57 27 The variation of blim with respect to a 10% change in Kt and Kn. The sensitivity of blim with respect to each parameter is superimposed. Linearity of blim(Xi) can be observed (see Table 6).........................................................................................57 28 Sensitivity of axial depth blim to changes in modal mass M and modal stiffness K in the x and y -directions (see Table 6)..................................................................58 29 Sensitivity of axial depth blim to changes in modal damping C in the x and y directions..................................................................................................................59 30 Sensitivity of axial depth blim to changes in spindle speed. The spindle speed sensitivity is compared here to the modal mass and stiffness in y -direction............60 31 Sensitivity of axial depth blim to changes in force cutting coefficients in the tangential Kt and normal directions Kn.....................................................................60

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xi 32 Sensitivity of surface location error SLE to changes in modal parameters in y direction....................................................................................................................61 33 Sensitivity of SLE to cutting force coefficients........................................................62 34 Sensitivity of SLE to spindle speed and radial depth of cu......................................62 35 Confidence in stability b oundary due to input parameters uncertainties using Monte Carlo simulation............................................................................................65 36 Uncertainty boundary in axial depth limit using two standard deviation confidence interval...................................................................................................66 37 Uncertainty in axial depth using sensitivity and Monte Carlo methods..................67 38 Surface location error uncertainty with two standard deviation confidence interval on the nominal SLE .....................................................................................68 39 Example simulation of cutting forces to facilitate proper selection of dynamometer............................................................................................................75 40 Work-piece, dynamometer and tool setup................................................................76 41 Cutting coefficient in tangential direction ( Kt).........................................................77 42 Cutting coefficient in normal direction ( Kn).............................................................78 43 Simulated and measured forces fo r 0.12 mm/tooth chip load and 1000 rpm...........79 44 Simulated and measured cutting forces for 0.2mm/tooth chip load, b =0.4 mm and 5000 rpm............................................................................................................79 45 Simulated and measured forces at 20 krpm and b =0.4 mm for slotting...................80 46 Modal analysis test equipment typically used in machine tool structures...............86 47 Frequency response function measurement of tool in x -direction...........................86 48 Frequency response function measurement of tool in y -direction...........................87 49 Boxplot of stability lobes boundary uncertainty......................................................89 50 Histograms of axial depth limit distri butions for various spindle speeds................89 51 Probability plot of axial depth lim it distribution at 10000 rpm spindle speed.........90 52 Schematic of stability tests for partia l radial immersion cutting conditions............91

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xii 53 Stability lobe generated using mean values of input parameters with experimental results overlaid, also shown the boxplot co rresponding to each spindle speed used in the measurements..................................................................92 54 Fast Fourier Transform (FFT) of sound signals for selected stability tests.............93 55 Stability boundary using m ean values in the input pa rameters Pareto optimal designs are overlaid for two cases: mean va lues and uncertain input parameters...94 56 Pareto Front of perturbed average SLE and MRR The Pareto Front with uncertainty in axial depth is compar ed to the one with no uncertainty....................95 57 Surface location error experiment schematic...........................................................97 58 Measured surface location error of b= 2.12 mm and the reference dimension (A) error.......................................................................................................................... 98 59 Measured surface location error of b= 4.45 mm and the reference dimension (A) error.......................................................................................................................... 98 60 Boxplot of SLE uncertainty at spindle speed s for 4.45 mm axial depth case...........99 61 Measured surface location error of b=4.45 mm case...............................................99 62 Surface location error of preferred design conditi ons with no uncertainty considered in the optimization. Optimum spindle speeds are indicated in the figure......................................................................................................................100 63 Slotting cut with time in the cut divided into two elements...................................112

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xiv NOMENCLATURE slotting tranformation matrix modal damping in -direction modal damping in -direction number of elements () vector of objective x yA Cx Cy E Fx 0functions utopia point average cutting force average cutting force in x-direction average normal cutting force in x-direction x xc xeF F F F F average normal edge cutting force in x-direction average cutting force in y-direction average tangential cutting force in y-direction average tangential edy yc ycF F F ge cutting force in y-direction Identity matrix of size Q cutting force coefficients matrix defined in Appendix A tangential cutting force coefficient t nK KQIcK normal cutting force coefficient edge tangential cutting force coefficient edge normal cutting foce coefficient modal stiffness in -direction te ne x yK K Kx K modal stiffness in -direction sample size modal mass in -direction modal mass in -direction material removal rate numbex yy L Mx My MRR N r of teeth on the cutting tool particle swarm optimization PSO

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xv 2Q number of experimental runs cutter radius adjusted coefficient of determination surface location error sequential quadratic proadjR R SLE SQPthgramming used for utility function expanded uncertainty feasible design space two-element position vector milling model i input pe iU U X Xt X th iarameter vector of observations in i response feasible criterion space Z matrix of rank radial depth of cut axial depi iiY Z Qpp a b th of cut set of goals for objective functions maximum stable axial depth chip load deviation from the goals constraint limitj lim jb b c d eMRR f 0 cutting force coefficients vector defined in Appendix A surface location error objective function material removal rate objective set of inequality conSLE MRR jf f g straints absolute value of maximum characteristic multiplier set of equality constraints step size used to estimate numerical derivative parametelg h h pr in exponential weighted criterion rank of matrix number of response variables correlation coefficient between and combined standard ii xy cpZ r rxy u uncertainty vector of weights (preferences) -direction w x x

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xvi vector of design variables -direction parameter used in homotopy method vector of unknown constant parameters unbiased estimate x yy i of spindle speed perturbation(half of bandwidth) absolute error limit random error vector associated with response variance-covariancethi st ex matrix system characteristic multipliers radial depth angle radial depth angle at start of cut radial depth angle at end of cut spindle speed covariance matrix element of covariance matrix estimate of tooth passing period dampinij ijiji,j g factor

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xvii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ROBUST MULTICRITERIA OPTIMIZATI ON OF SURFACE LOCATION ERROR AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER UNCERTAINTY By Mohammad H. Kurdi August 2005 Chair: Tony L. Schmitz Cochair: Raphael T. Haftka Major Department: Mechanic al and Aerospace Engineering High-speed milling ( HSM ) provides an efficient method for accurate discrete part fabrication. However, successful implementa tion requires the selection of appropriate operating parameters. Balancing the multiple process requirements, including high material removal rate, maximum part accurac y, chatter avoidance, and adequate surface finish, to arrive at an optimum solution is difficult without the ai d of an optimization framework. Despite the attractive gain in productivity that HSM offers, full realization of the benefits is dependent on the proper selection of cutting parameters. Parameters selected must achieve the required productivity while maintaining an acceptable accuracy. Milling models are used to aid in the proper selecti on of these cutting parameters. They provide information on whether a cutting condition is stable and/or predict the surface accuracy. However, this selection is rather tedious, costly and time consuming and might not even

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xviii provide an optimum solution. Parameters are se lected based on experi ence until a point is found that provide the productivity and su rface accuracy required. Difficulties encountered in this selection process incl ude sensitivity of surface accuracy to cutting parameters, uncertainties in several parameters in the milling model and the computational effort needed to account for stability and surface accuracy. Therefore, balancing the multiple requirements, including high material removal rate, minimum surface location error and chatter avoidance, to arrive at an optimum solution is difficult without the aid of op timization techniques. In this dissertation a robust optimizati on algorithm that accounts for the inherent process uncertainty and surface location erro r sensitivity is developed. Two optimization criteria are considered, namely, surface locati on error and material removal rate under the stability constraint. The trade off curve of surface location error versus material removal rate is calculated for the m ean values of input parameters, as well as for a confidence level in the stability boundary. An experime ntal validation of the robust optimization algorithm is also conducted, including an experi mental validation of the variation of the cutting forces as a function of spindle speed. The confidence level in the axial depth limit and surface location error prediction is found us ing two methods: 1) sensitivity analysis; and 2) sampling methods. The sensitivity st udy highlights the most significant factors affecting process stability a nd surface location error. The effect of input parameters correlation is included in the confidence le vel predictions using Monte Carlo and Latin Hyper-Cube sampling methods.

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1 CHAPTER 1 INTRODUCTION Justification of Work Intense competition in manufacturing places a continuous demand on developing cost-effective manufacturing processes w ith acceptable dimensional accuracy. Highspeed milling, HSM offers these benefits provided appropriate operating parameters are selected. Some typical applica tions include, but are not limited to, orthopedic surgery [1], end milling (pocketing) of airframe panels [2 ] and ball end milling of stamping dies [3, 4] in automotive manufacturing. Equation Chapter 1 Section 1 Despite the attractive gain in productivity that HSM offers, full realization of the benefits is dependent on the proper selection of cutting parameters. Parameters selected must achieve the required productivity while maintaining an acceptable accuracy. Milling models are used to aid in the proper selecti on of these cutting parameters. They give us information on whether a cutting condition is stable and/or they predict the surface accuracy. However, this selection is rather tedious, costly and time consuming and might not even provide an optimum solution. Parameters are selected based on experience until a point is found that provides the productivity and surface accuracy required. Difficulties encountered in this selection process incl ude sensitivity of surface accuracy to cutting parameters, uncertainties in several parameters in the milling model and the computational effort needed to account for stability and surface accuracy. Therefore, balancing the multiple requirements, including high material removal rate, M RR f

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2 minimum surface location error SLE f and chatter avoidance, to arrive at an optimum solution, is difficult without the aid of optimization techniques. Literature Review The literature review proceeds with a su mmary of previous implementations of optimization methods in machining, with par ticular attention to high-speed milling and multi-objective optimization. Also, a review of milling models for stability and surface location error is provided. Optimization in Machining Previous research in machining pr ocess optimization [5] has focused on mathematical modeling approaches to determin e optimal cutting parameters with regard to various objective functions. Three main obj ectives have been recognized: 1) maximum production rate or minimum cycle time [6-9]; 2) minimum cost [10-21]; and 3) maximum profit [12, 22], or a combined criterion ba sed on a weighted sum of these [23, 24]. The machining optimization problem can be formulated using deterministic and probabilistic approaches [11, 25] Several optimization techniques were used to handle both formulations. For the deterministic a pproach they include linear and nonlinear programming techniques [9, 15, 26, 27], while for the probabilistic approach chanceconstrained programming can be used [17, 28]. Other optimization techniques used in machining include graphical optimization [12, 22], polynomial geometric programming [6, 18-20, 29, 30], geometric programming [1 0] based on quadratic posylognomials (QPL) [31], goal programming with linear [32, 33] and nonlinear [34] goals, fuzzy optimization [35], and global search methods such as particle sw arm optimization [21] and simulated annealing [16].

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3 The machining optimization literature can al so be classified according to different constraints and design variab les handled. Several authors [7, 14] considered cost optimization for single-pass milling and turning [10, 17, 19, 20, 29]. The range of constraints considered are machine tool constr aints, such as cutting speed and feed rate, tool dynamics constraints such as cutti ng force, power and stability, and product constraints such as surface roughness. In reference, [17] some of the constraints considered are of probabilistic nature. Also, multi-pass peripheral and end milling to maximize production rate are considered [8] unde r a range of constraints with relevance to rough milling such as the machine tool limiting power, torque, f eed force and feedspeed boundaries while in anothe r work. In addition to the previous constraints, arbor rigidity and deflection are used [6]. High-speed Milling Optimization Few references are found on optimization of high-speed milling. The concept of adaptive learning (polynomial ne twork) [16] is used to construct a machining model. Simulated annealing was then used to mi nimize production cost for rough high-speed machining operations for three cutting condi tion parameters namely cutting speed, chip load and axial depth of cut. A similar study was done for low speed milling [21] where an artificial neural netw ork was used to build the machining model. However, particle swarm optimization was used to optimize production cost under machine, tool and product constraints. Multi-objective Optimization Multi-objective optimization (MOO) addre sses the issue of competing objectives using concepts first introduced by Edgewo rth [36] then expanded and developed by Pareto [37], the French-Italian economist who established an optimality concept in the

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4 field of economics based on multiple objectives A Pareto front [38] is generated that allows designers to trade off one objective against another. In the area of machining, Jha [24] st udied two objective f unction optimization based on cost and rate of production where example constraints were machine power, cutting speed limitations, depth of cut, and ta ble feed. The two objectives were combined using weights. Koulams [28] studied single-pass machining c onsidering the influence of tool chatter failure where a tool failure proba bility function effect was added as a penalty cost function to the objective function. Stability and Surface Location Error As explained earlier, th e full exploitation of HSM demands mathematical models to predict stability and surface location error. An unstable milling process is caused by a phenomenon called chatter. Among the first to describe chatter is Taylor, [39] who described chatter as the most obscure and de licate of all problems facing the machinist. Chatter [40] is a self-excited vibration that occurs if the chip width is too large with respect to the dynamic stiffness of the syst em. It causes undulations in the machined surface (poor surface finish) and could result in tool breakage. Extensive work has been done to generate stability bounda ries or lobes. The lobes define a region below which chatter is nonexistent. Two approaches are used to generate these lobes: 1) analytical [41] with a continuous cutting mode l or with an interrupted cu tting model [42]; and 2) time domain simulation [43, 44]. Surface location error is defined as the e rror in the placement of the milling cutter teeth when the surface is generated. This error depends on the interaction of workpiece/tool dynamic stiffness and the cutting fo rces. The correct prediction of this error depends on correct prediction of the cutting forces and resu lting deflections. Mechanistic

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5 models can be used to estimate these for ces. The cutting force is found by summing the forces acting on incremental sections of a he lical cutting edge [45, 46], then the surface location error is computed based on the static stiffness of the tool [47]. However, the effect of the deflection of the cutter on the cutting forces is not included. In an improvement of the previous model, the static deflection is fed back to correct the cutting forces [48, 49]. A more realistic regenerative force model [50] considered the effect of undulations in the surface gene rated by previous tooth passag e on the next tooth passage. In this model the dynamic deflection of th e tool imprints waviness on the generated surface. Using time domain simulation, surface lo cation error, cutting forces and stability lobes are predicted. An improvement on this model considered [42, 51-53] interrupted cutting as a factor influencing the stabili ty lobes and surface location error. A newly developed method uses time finite element an alysis (TFEA) to model the governing time delayed differential equation [ 54-58]. Regenerative cutting forces and dynamic deflection of the tool are all implicitly included in the governing differential equation. The advantage of this method is that it conc urrently provides surface location error and stability information on the milling process in a semi-analytical manner. In this method the governing differential equa tion is modeled by dividing th e time in the cut into a number of elements, where displacement and velocity continuity are enforced between each element. A discrete linear map is fo rmed by mapping the time in the cut to free vibration. The eigenvalues of the discrete map determine the stability boundaries, whereas fixed points of the dynamic ma p determine surface location error (SLE f ).

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6 Scope of Work The purpose of this dissertation is to us e optimization as a tool to efficiently determine preferred and r obust operating conditions in HSM considering multiple objectives. Although known optimization methods and machining models will be applied, there are a number of innovative aspects of this research. Firs t, proper formulation of the objective functions to account for practical application of the preferred conditions is necessary. The formulation should account for uncertainty in the milling model and sensitivity of objective(s) to process variab les. Uncertainty has not previously been considered. Second, two objectives are simulta neously optimized: su rface locat ion error SLE f and material removal rate, M RR f Stability and side bounds of design variables are considered as constraints. Prior research ha s focused only on the empi rical tool life, not the unavoidable milling dynamics and the inherent limitations they impose. The tradeoff curve (Pareto front) [38, 59] of M RR f and SLE f is generated based on nominal experimental model parameters. Experimental case studies are conducted to verify the validity of the Pareto front. The uncertainty in the milling model is addressed using Monte Carlo simulation and/or se nsitivity analysis, where a confidence interval is applied to the stability limit. The uncertainty of diffe rent input parameters such as cutting force coefficients, tool/work-piece dynamic parameters and milling process parameters are considered in the uncertainty prediction. This uncertainty is used in the selection of a robust design that would allow a venue for the practical application of the stability lobe theory at the shop floor.

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7 The dissertation organization proceeds as follows: Chapter 2 gives a general description of multi-objective optimization; Chapter 3 describes Pareto front generation formulation of the optimization problem, opt imization methods and case studies; Chapter 4 provides the uncertainty analysis of stab ility and surface location error; Chapter 5 describes the robust optimization algorithm and presents some practical case studies to verify stability lobes and selected design points on the Pareto front. Chapter 6 summarized the results and outlines future work in this area.

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8 CHAPTER 2 MULTI-OBJECTIVE OPTIMIZATION Fundamental Concepts in Mu lti-Objective Optimization Optimization is an engineering discipline wh ere extreme values of design criteria are sought. However, quite often there are multip le conflicting criteria that need to be handled. Satisfying one of thes e criteria comes at the expens e of another. Multi-objective optimization deals with such conflicting objectives. It provides a mathematical framework to arrive at an optimal design state which accommodates the various criteria demanded by the application. Equation Chapter 2 Section 1 This chapter begins with a comparison of singleand multiple-objective optimization. Next, the definition of the multi-objective optimization problem and terms are explained. Then, a summary of multi-obj ective optimization methods is presented. Finally, reasons are given for the choice of the multi-objective optimization method. Single and Multi-objective Optimization In single objective optimization one is faced with the problem of finding the optimum of the objective function. For ex ample considering the decision making involved in an investment (Fi gure 1). There are several possi ble designs in the feasible domain (A, B and 1-6). These designs are ma pped from the design space Figure 1 (b) into the criteria space Figure 1 (a). In the design space ther e are two design variables x1 (spindle speed) and x2 (axial depth) where the fe asible domain is limited by the

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9 constraint. If we are only con cerned about profit with no rega rd to risk (profit is our single objective), then poin t B would correspond to the maximum profit optimum design. A risk averse investor would choose risk as an objective func tion. The optimum design for the risk objective would correspond to point A. Depending on the objective function, constraints, and design variab les, different techniques are used to solve for the singleobjective optimum. However, in multi-objective optimizati on, a vector of objectives needs to be optimized. For the investment exam ple, two objectives are considered. In this two objective case there is no unique optimum, rather a set of optimum solutions is found. In Figure 1, for instance, points A, B and 5-6 are all candidate solutions. Depending on the decision makers risk aversi on, a single solution can be chosen from that set. ProfitRisk A B 1 2 3 4 $1000 $5000 10% 90% Feasible domain 5 6 Feasible domain x 1 x 2 1 2 3 4 B 5 A 6 criteria space design space constraint (a) (b) Figure 1. (a) Typical Pareto front in the criteria space (b) Design variables x1 and x2, and constraint in the design space. The similarity between singleand multi -objective optimization makes it possible to use the same optimization algorithms as for the single-objective case. The only

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10 required modification is to transform the multiobjective problem into a single one. This may be accomplished in a number of ways, such as introducing a vector of preferences, w, to get a single objective as a weighted su m, or by solving one of the objectives for a different set of limits on the other objectives [6062]. In any case, a set of optimal solutions are found rather than a single one. It is worth noting, however, that when the objective functions are non-conf licting, the optimal set reduces to a single solution rather than a set. This can be related to the comm odity example. For instance, if we want to maximize both cost and quality, th en solution B is the only one. Definition of Multi-Objective Optimization Problem The mathematical representation of the multi-objective optimization problem is formulated as follows: 12 ,,..., 0, 1,2,..., 0, 1,2,..., T k j lMinimizeFxFxFxFx subjecttogxjm hxle (2.1) where subscript k denotes the number of objective functions F, m is the number of inequality constraints and e is the number of equa lity constraints; and n E x is the vector of design variables, where n is the number of independent design variables. Definition of Terms The feasible design space (inference space), X i s defined as the set of design variables that satisfy th e constraint set, or 0, 1,2,...,; and 0, 1,2,..., .jlxgxjmhxle (2.2)

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11 The feasible criterion space, Z (often called the cost space or attainable set) i s defined as the set of cost functions Fx such that x maps to a point in the feasible design space X or FxxX The preferences refer to the decision makers opi nion in terms of points in the criterion space. The preferences can be set a priori (before solution set is obtained) or a posteriori (after solution set is obtained). The preference function is an abstract function of points in the criterion space which perfectly satisfies the decision makers preferences. The utility function is an amalgamation of indivi dual utility functions of each objective that approxim ates the preference function, whic h typically cannot be expressed in mathematical form. The formation of a utility function requires insight into the physical aspects of each objective. This may require finding the Pareto front (explained next) in order to properly formulate the utility function. A utopia point is a point0kFZ that satisfies 0 ii F minimumFxxX for each 1,2...,. ik Pareto Optimality The multi-objective optimization problem has more than one global optimum. The predominant concept in defining an optimal point is that of Pareto optimality [37] which is defined as follows: a point, x X is Pareto optimal if there does not exist another point, x X such that FxFx and iiFxFx for at least one function. That is the set of Pareto optimal points dom inates any other optimal set. This can be defined by the domination relation [60], where a vector 1 x dominates a vector 2 x if: 1 x is

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12 at least as good as 2 x for all the objectives, and 1 x is strictly better than 2 x for at least one objective. To better understand the domin ation relation, or Pareto optimality, an example is provided [63] (Figure 2). A two-objective problem of maximizing f1 and minimizing f2 is addressed. Table 1 presents the set of solutions, classified with respect to each other. A solution P is designated as +,or = depending on whether it is better, worse or equal to a solution Q for the corres ponding objective. For example, comparing solutions A and B, we find that solution A is worse for f1 (maximizing f1) compared to B, therefore it is designate d as (-) for objective f1. Also, comparing objective f2 we find that solution A is worse than B (-). Now for a solution to belong to the non-dominated set it must be as good as the other solutions for bot h objectives and it must be strictly better for at least one objective. Considering solution A in Figure 2 we see it is worse than all other solutions (dominated); solution B is also wo rse than C for both objectives (dominated). Solution C is not dominated by point E (couple (+,-) at the intersection of the row E and the column C) and it does not dominate point E (couple (-,+) at the in tersection of the row C and the column E), therefore points C a nd E are non-dominated. Solution D is worse than C for both objectives theref ore solution D is dominated. Multi-objective Optimization Methods As explained earlier the solution to a multi-objective optimization problem is a Pareto optimal set that gives a tradeoff between the different objective functions considered. Depending on the decision makers pr eferences, a solution is selected from that set. Therefore multi-objective optimizati on methods can be categorized according to how the designer articulates his preferences (by order or by importance of objectives). This includes three cases: a priori a posteriori, and progressive articulation of

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13 preferences. A brief overview of the methods us ed is outlined. For a detailed description of the methods the reader is referred to reference 64. Table 1. Classification of solutions Solutions A B C D E A (-,-) (-,-) (-,-) (-,-) B (+,+) (-,-) (-,=) (-,=) C (+,+) (+,+) (+,+) (-,+) D (+,+) (+,=) (-,-) (-,=) E (+,+) (+,=) (+,-) (+,=) 2 4 6 8 10 12 14 16 1 2 3 4 5 6 f 1f 2 C E D B A Figure 2. Pareto optimality and domination relation. Methods with a Priori Articulation of Preferences using a Utility Function In these methods, the decision makers pref erences are incorporated as parameters in terms of a utility function a priori Typically these parameters can be coefficients, exponents, constraint limits, etc. These para meters determine the tradeoff of objectives before implementation of the optimization method. The optimum solution found would

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14 reflect the tradeoff made a priori Depending on whether the solu tion found turned out to satisfy the preferences or not, the decision ma ker can re-adjust the parameters to get a better solution. However the beau ty of these methods is that they do not require doing a multi-objective optimization problem since the a priori preferences and utility function reduce the optimization to a single one. Weighted global criteria method In this method, all objective functions are combined to form a single utility function. The weighted global criteri on is a type of utility function U in which parameters are used to model preferences. The simplest form of a general utility function can be defined as 1, 0, ork P iii iUwFxFxi (2.3) 1, 0,k P iii iUwFxFxi (2.4) where w is the vector of weights set by the decision maker such that 0 w and 11k i iw. The difference between the two above fo rmulations is rela ted to conditions required for Pareto optimality. Complete discussion can be found in reference [64]. Weighted sum method This is a special case of the weighted gl obal criteria method in which the exponent P is equal to one; that is, 1.k ii iUwFx (2.5) The method is easy to implement and guara ntees finding the Pareto optimal set, provided the objective function sp ace is convex. However, a uni formly distributed set of

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15 weights does not necessarily find a uniformly di stributed Pareto optimal set, which makes it difficult to obtain a Pareto solution in a desired region of the objective space. Exponential weighted criterion It is defined as follows: 11,i ik pFx pw iUee (2.6) where the argument of the summation repr esents an individual utility function for iFx. Weighted product method To avoid transforming objective functions with similar significance and different order of magnitude, one may consider the following formulation [65]: 1,iw k i iUFx (2.7) where iw are weights indicating the relative si gnificance of the obj ective functions. Conjoint analysis This method [66, 67] uses a concept borro wed from marketing, where a product is characterized by a set of attr ibutes, with each attribute having a set of levels. An aggregated utility function is developed by direct interaction with the customer/designer; the designer is asked to rate, rank orde r, or choose a set of product bundles. In engineering design studies, we can assume th at people will choose their most preferred product alternative. Conjoint anal ysis takes these sets of attrib utes and converts them into a utility function that specifies the preferen ces that the customer has for all of the products attributes and attri bute levels. The advantage of this method is that it automatically takes into account marginal dimi nishing utility (i.e., no cost is expended in a design that does not really have practical utility).

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16 Methods with a Priori Articulation of Preferences wi thout using a Utility Function Lexicographic method Here the objective functions are arranged in a descendi ng order of importance [68]. The highest preference objective is optimized with no regard to the other objectives, and then a single objective problem is solved consecutively (in order of preference of objectives) for a set of limits on the optimums of previously solved for objectives. This can be defined as 1,2,...,1,1, 1,2,...,i jjjMinimizeFx subject toFxFxjii ik (2.8) where i represents the functions position in the preferred sequence and j j F x represents the optimum of the jth objective function found in the jth iteration. Goal programming methods Here, goals j b are specified for each objective function jFx [69]. Then the total deviation from the goals, 1 k j jd, is optimized, where j d is the deviation from the goal j b for the jth objective. Methods for an a Posteriori Articulation of Preferences The inability of the decision maker to set preferences a priori in terms of a utility function makes it necessary to generate a Pareto optimal set after which an a posteriori articulation of preferences is made; such methods are sometimes referred to as cafeteria or generate-first-choose-later. These methods however requ ire the generation of the Pareto optimal set which may be prohibitively time consuming. It is worth noting that repeatedly solving the weighted sum approaches presented earl ier can be used to find the

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17 entire Pareto optimal solution for convex cr iteria space; however, these methods fail to provide an even distribution of points that can accurat ely represent the Pareto optimal set. Bounded objective function method In this method [70], the single mo st important objective function, sFx is minimized, while all other objective functions are added as constraints with lower and upper bounds such that 1,2,...,,iiilFxikis A variation of this method is the constraint [71] or trade-off method in which the lower bound il is excluded and the Pareto optimal set is obtained using a systematic variation of i This method is particularly useful in finding the Pareto optimal solution for convex or non-convex objective spaces alike. However, c hoice of the constraint vector must lie within the minimum and maximum of the objective functi on considered; otherwise, no feasible solution will be found. Also the distribution of the Pareto optimal solution will usually be non-uniform for the objective function(s) minimized. Normal boundary intersection (NBI) method This method provides a means for obtaining an even distribution of Pareto optimal points for a consistent variation in parameter vector of weights [72, 73], even with a nonconvex Pareto optimal set (a deficiency found in weighted sum method). For each parameter weight the NBI problem is solved to find an optimum point that intersects the criteria feasible space boundary, however for non-convex problems, some of the solutions found can be non Pareto optimal. Details of the method can be found in the references.

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18 Normal constraint (NC) method This method uses normalized objective functi ons with a Pareto filter to eliminate non-Pareto optimal solutions [74]. The indi vidual minima of th e normalized objective functions are used to construct the vertices of the utopia hyper-plane A sample of evenly distributed points on the ut opia hyper-plane is found from a linear combination of the vertices with consistently varied weights in criterion space. Each Pareto optimal point is found by solving a separate normalized si ngle-objective function with additional inequality constraints for the rema ining normalized objective functions. Homotopy method In this method the convex combin ation of bi-objective functions 121 f f is optimized for an initial value of the parameter Then homotopy curve tracking methods are used to generate the Pare to optimal solution curve for 0,1 whenever the curve is smooth [75, 76] or even non-sm ooth [62, 77] at points co rresponding to changes in the set of active constraints. Choice of Optimization Method The ease of implementation of the -constraint method [71] for a bi-objective problem makes it a good candidate method. In this method, one of the objectives is optimized for systematic variation of limits (12,...,i ) on the second objective. A uniform distribution of the Pareto optimal se t can be found for the constrained objective. There is no limitation on the convexity or nonconvexity of the objective space in finding the Pareto optimal set. However, choice of the constraint set of limits (12,...,i ) must lie within the minimum and maximum of the objective function consid ered; otherwise, no feasible solution would be found. In our case the material removal rate ( M RR f ) and

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19 surface location error (SLE f ) are the bi-objective criteria. The material removal rate objective would be a better choice for the c onstrained objective, since the set of limits (12,...,i ) of M RR f constraint can be more easily c onstructed according to designers preference, whereas that would be difficult for the SLE f objective.

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20 CHAPTER 3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM Introduction In this chapter, a description of the milling problem and solution method used to solve the mathematical model is presented. Two optimization met hods of interest are briefly described. These methods are then applied to the multi-objective optimization problem and a discussion of results is provided. Equation Chapter 3 Section 1 Milling Problem Milling Model Equation Chapter 3 Section 1 The schematic for a two degree-of-freedom (2-DOF) milling process is shown in Figure 3 (repeated here). With the assumption of either a compliant tool or a structure, a summation of forces gives the following equation of motion: k y c y k x c x x y Figure 3. Schematic of 2-DOF milling tool

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21 ,00 0() ()()() + 00 0() ()()() mc kFt xtxtxt xx xx mc kFt ytytyt yy yy (3.1) where the terms mx, cx, kx and my, cy, ky are the modal mass, viscous damping, and stiffness terms, and Fx, and Fy are the cutting forces in the x and y directions, respectively. A compact form of the milling process can be found by considering the chip thickness variation and forces on each tooth (a detailed derivation is provided in references [54-58] and Appendix A): ()()()()()() X tXtXttbXtXt+ftb o MCKK c (3.2) where T X txtyt is the two-element position vector and M, C, and K are the 2x2 modal mass, damping, and stiffness matrices, Kc and 0 f (function of the cutting force coefficients) are defined in Appendix A, b is the axial depth of cut, = 60/( N ) is the tooth passing period in seconds, is the spindle speed given in rev/min (rpm), and N is the number of teeth on the cutting tool. As shown in Eq. (3.2), the milling model is dependent on modal parameters of the tool/w ork-piece combination and the cutting force coefficients. Solution Method As described in Chapter 1, a solution of Eq. (3.2) can be completed using numerical time-domain simulation [43, 44, 50] or the semi-analytical TFEA [54-58]. Compared to the first approach, TFEA can obtain rapid process performance calculations of surface location error,SLEf, and stability. The computational ef ficiency of TFEA compared to conventional time-domain simulation methods ma kes it the most attractive candidate for use in the optimization formulation. In this method a discrete linear map is generated that

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22 relates the vibration while the tool is in the cut to free vibratio n out of the cut. Stability of the milling process can be determined using the eigenvalues of the dynamic map, while surface location error (see Appendix A) is found from the fixed points of the dynamic map. Details can be found in references [54-58] An added advantage of TFEA is that it provides a clear and distinct definition of stability boundaries (i.e., eigenvalues of the milling equation with an absolute value greate r than one identify unstable conditions). Problem Specifics In this section, the calculation of the st ability boundary is analyzed, the continuity of surface location error and stability boundary is addressed, TFEA convergence is described, and sensitivity of the milling model to cutting force coefficients is defined. Stability Boundary In order to find the axial depth limit, blim, of neutral stability at corresponding input parameters, the bi-section me thod is used in the TFEA algorithm to solve for blim at which the maximum characteristic multiplie r is equal to one (stability limit) max1g (3.3) where is the eigenvalues of the dynamic map. An absolute error is used as a criterion for convergence 1 ii ibb b (3.4) where corresponds to the error tolerance and ibis the root corresponding to max1 at iteration i. The value of is set based on the numerical accuracy required in the calculation of blim. A value of 13 e can be adequate for the calculation of blim.

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23 Surface location error and stability boundary: C1 discontinuity Correct use of an optimization method de pends on its limitations. Gradient-based methods, for example, depend on C1 continuity (the first derivative of the function is continuous) of the objective functions ( M RRf and SLE f ) and stability constraint (Eq. (3.3)). The objective M RRf is defined analytically in Eq. (3 .6), where it is clear that it is C1 continuous. However, the SLE f and stability ( g ) functions are only found numerically using TFEA. A graphical descri ption of both functions provides some insight into the continuity of these functions. Figure 4 depicts the variation of SLEf and SLE f as a function of spindle speed for a typica l set of cutting parameters. Although SLEf is C1 continuous in the region where it is defined (stable region), SLE f is C1 discontinuous. This can be easily verified analy tically by considering the functions ()fxx and () 0 and 0 fxxxforxxforx The absolute function is clearly C1 discontinuous at0 x The same argument can be made for the near-zero SLE f range shown in Figure 4. In Figure 5 the variati on of stability function g versus spindle speed shows lobe peaks where C1 (slope) discontinuity of g is also observed. C1 discontinuity makes convergence of gradient-based optimization al gorithms near the di scontinuity rather difficult. This requires the use of multiple initial guesses in order to converge to even a local optimum.

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24 10 12 14 16 18 20 1 0 1 2 3 4 5 6 7 fSLE ( m) ( x 103 rpm ) fSLE |fSLE| Figure 4. Surface location error and its absolu te. Discontinuity of the absolute surface location error is apparent in the lower insert. 1 1.1 1.2 1.3 1.4 1.5 0.5 1 1.5 2 2.5 3 3.5 ( x 10 3 rpm )b (mm) Figure 5. A typical stabil ity boundary. The cusps where 1C discontinuity in the stability boundary are depicted. TFEA convergence The convergence of TFEA depends on the cu tting parameters. A higher number of elements must be used when convergence is not achieved. Either SLE f or the stability boundary g can be used to check for convergence. A typical procedure to test for the convergence of finite element meshes is to compare the change in the estimated value ( g or SLE f ) as the number of elements is increased (mesh refinement). In Figure 6, the

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25 dependence of convergence on the spindle speed is shown for a randomly selected cutting condition of 5% radial immersi on (percentage of radial depth of cut to tool diameter) and 18 mm axial depth. As seen in Figure 6, the flawed convergence for a small number of elements (=1) would give the impression of a sufficient number of elements. However, further increasing the number of elements (=12) shows poor convergence for the low speed. This can be due to the fact that as th e spindle speed decreases, the time in the cut increases, which requires a higher number of elements to achieve convergence. The fact that the optimization algorithm will pick milling parameters within the design space makes it necessary to choose a rather high number of elements to ensure convergence anywhere in the design space. However, a pena lty in computational time is incurred. 0 10 20 30 40 -20 0 20 40 No. of elementsg 1 2 4 6 0 10 20 30 40 50 No. of elementsg 1 20 25 30 35 40 0 2 4 6 8 10 No. of elementsg 1 10 15 20 0 1 2 3 x 104 No. of elementsg 1 500 (rpm) 1000 Figure 6. Convergence of stability constraint fo r 5% radial immersion and different spindle speeds for an 18 mm axial depth. We can see that convergence at lower speeds require substantially more number of elements.

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26 Optimization Method Optimization methods can be categorized according to the searching method used to find the optimum [78]. They are either direct where only the values of the objective function and constraints are used to guide the search strategy, or gradient-based, where first and/or second order de rivatives guide the search process. Particle swarm optimization (PSO) and sequential quadratic programming (SQP) will be used to test the feasibility of both methods, respec tively, for the problem at hand. Particle Swarm Optimization Technique Particle swarm optimization is an evolutionary comput ation technique developed by Kennedy and Eberhart [79, 80]. It can be used for solving si ngle or multi-objective optimization problems. To find the optimum so lution, a swarm of pa rticles explores the feasible design space. Each particle keeps tr ack of its own personal best (pbest) fitness and the global best (gbest) fitness achieved during design space exploration. The velocity of each particle is updated toward its pbest and the gbest positions. Acceleration is weighted by a random term, with separa te random numbers being generated for acceleration toward pbest and gbest. In order to accommodate constraints, Xiaohui et al. [79] presented a modified particle swarm optimi zation algorithm, where PSO is started with a group of feasible solutions and a feasibility function is used to check if the newly explored solutions satisfy all the constraints. All the particles keep onl y those feasible solutions in their memory while discarding infeasible ones.

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27 Sequential Quadratic Programming ( SQP ) The basic idea of this me thod is that it transforms the nonlinear optimization problem into a quadratic sub-problem around the initial guess. The nonlinear objective function and constraints are transformed into their quadratic and linear approximations. The quadratic problem is then solved iterati vely and the step size is found by minimizing a descent function along the search direction. Standard optimizati on algorithms may be used to solve the quadratic sub-problem. Usually SQP leads to identification of only lo cal optima. In order to better converge to the global optimum, a number of initial guesses is used to scan the design space and the optimum of th ese local optima is clos e to the global optimum. Problem Formulation In this section, the multi-objective optimization problem is defined and then a description of the tradeoff met hod is given. The problem soluti on is then presented in the order it has been addressed in the robust optimization section. Finally, discussion of the simulation results is provided. Problem Statement The problem of minimizing surface location error SLE f and maximizing material removal rate M RRfis stated as follows: ,,,,,,,,,, : ,,,,1SLEMRRminfabcNfabcN subjecttogbmaxabN (3.5) where g is the stability constraint obtain ed from the dynamic map eigenvalues,SLEf is found from the fixed points, and the mean M RRfis given as: MRRfabcN (3.6)

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28 where a, b, c, N and are radial depth of cut, axial de pth of cut, feed per tooth (chip load), number of teeth, and spindle speed, re spectively (Figure 7). From Eq. (3.5) it can be seen that only the stability constraint is not a function of the feed per tooth. In Eq.(3.5) SLEf and M RRf are explicitly stated as a function of cutting conditions (a, b, c, N and ). This reflects the relative ease by which th ese conditions can be adjusted to achieve optimality of the objectives. Chip load, c N=2 Axial depth, b Radial depth of cut, a x y Slotting x y Up milling x y Down milling ex ex = st =0 st =0 ex = st a R Figure 7. Schematic of milling cutting conditi ons and various types of milling operations. Tradeoff Method To address the multi-objective problem the constraint method is used, where the two-objective problem is tran sformed into a single objectiv e problem of minimizing one objective with a set of different limits on the second objective. E ach time the single objective problem is solved, the second objective is constrained to a specific value until a sufficient set of optimum points are found. Thes e are used to generate the Pareto front [38] of the two objectives. In the case that SLEf is chosen as the objective function to be minimized then Eq. (3.5) is transformed to:

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29 ,,,,, : ,,,,, 1 ... ,,,,,,1,SLE MRRiminfabcN s ubjecttofabcNeforik gabNmaxabN for a series of k selected limits (e) on MRRf. (3.7) Where the cutting conditions: a, b, c, N and are the design variables. On the other hand if M RRf is chosen as the objective function to be maximized, then Eq. (3.5) is transformed to: ,,,,, : ,,,,, 1 ... ,,,,,,1,MRR SLEiminfabcN s ubjecttofabcNeforik gabNmaxabN for a series of k selected limits (e) on SLEf. (3.8) It should be noted that a pplying Eq. (3.7) using the SQP method is more straightforward than Eq. (3.8). The reason is that in order to use a number of initial guesses along the SLE f contour in Eq. (3.8), the ax ial depth corresponding to that SLE f needs to be found, whereas in Eq. (3.7) the ax ial depth can be explic itly expressed found as shown in Eq. (3.6). Robust Optimization Problem solution In the first iteration of th e problem, only axial depth (b) and spindle speed ( ) are considered as design variables. Other cutting conditions are held fixed (Table 2) for a down milling cut. Modal parameters for a single degree-of-freedom tool with one dynamic mode in x and y directions are used (Table 2). The nominal values of the tangential (Kt) and normal (Kn) cutting force coefficients are 550 N/mm2 and 200 N/mm2, respectively. The SQP method is used to find the Pareto front using the formulation in Eq. (3.7). Here SLE f is minimized for a set of limits on M RRf. As mentioned earlier, the

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30 SQP method is a local search method that is highly dependent on C1 continuity of the objective function and constraints. To obtain a global optimum, a number of initial guesses are used along each M RRf constraint limit. A set of optimum points are obtained for these initial guesses. The minimum of thes e optimum points is nominated as a global optimum. The number of initial guesses is incr eased and another run of the optimization simulation is made to check the validity of that global optimum. Table 2. Cutting conditions and modal parameters for Tool used in optimization simulations M (kg) C (Ns/m) K (N/m) 0.0560 00.061 3.940 03.86 6 61.52100 01.6710 Tool diameter (mm) c (mm) a (mm) N 19.05 0.178 0.76 2 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 550 x 106 200 x 106 0 0 In this formulation, the minimum SLE f points were found to favor spindle speeds where the tooth passing frequency is equal to an integer fraction of the systems natural frequency (Figure 8), which corresponds to the most flexible mode (these are the traditionally-selected best speeds which are lo cated near the lobe peaks in stability lobe diagrams). BecauseSLEfcan undergo large change s in value for small perturbations in at these optimum points, the formulation provid ed in Eqs. (3.7) and (3.8) leads to optima which are highly sensitive to spindle speed vari ation (Figure 8) To show the sensitivity of these optimum points, a typical optimum point is superimposed on a graph of SLE f vs. in Figure 9. It is seen that th e optimum point is located in a highSLEfslope region.

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31 5 10 15 20 25 30 0 2 4 6 8 10 12 (x 103 rpm)b (mm)51 01 52 02 53 04 05 0 m7 0 6 0 0 m m3/ s5 5 05 0 04 5 04 0 03 5 03 0 02 5 02 0 01 5 01 0 05 0 Stability Lobe Optimum points Figure 8. Stability, SLE f and M RRfcontours with optimum points overlaid. The figure shows that optimum points occur in regi ons sensitive to spindle speed variation (Table 2). 4 6 8 10 12 14 -5 0 5 10 fSLE (m) (x103 rpm) Surface location error Spindle speed sensitive optimum Figure 9. A typical optimum point found; op timum point sensitivity with respect to spindle speed is apparent (Table 2).

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32 Reformulation of problem The optimization problem was redefined in order to avoid convergence to spindle speed-sensitive optima. Two approaches were applied: 1) an addi tional constraint was added to the SLE f slope; and 2) the SLEf objective was redefined as the average of three perturbed spindle speeds. The la tter proved to be more robust than the former. This is due to the difficulty in setting the value of the SLE f slope constraint a priori. The spindle speed perturbed form of the problem transforms Eqs. (3.7) and (3.8) to ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbgbgb for a series of selected limi ,MRRts (e) on f (3.9) and ,,, 3 ,, : 1 ... ,,,1,MRR ifbfbfb SLESLESLEminfb s ubjecttoforik gbgbgb for a series of selected limits () on a ,SLEverage perturbedf (3.10) where is the spindle speed perturbation selected by the designer (a typical value for our analyses was 50 rpm). A study of spindle speed perturbation selecti on is provided in the next section. The validity of the perturbed SLE f average as a convergence criteria can be seen in Figure 10. In this figure the perturbed average SLE f is plotted with SLE f where points A and B correspond to highly and moderately spindle speed-sensitive SLE f respectively.

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33 The average perturbed SLE f at point A (high slope point) is shown to be higher than at point B. Therefore, using the perturbed average SLE f as an objective function criteria can avoid convergence to spindle speed sensitive SLE f (such as SLE f region near point A). 10.5 11 11.5 12 12.5 13 13.5 14 0.2 0.4 0.6 0.8 1 1.2 fSLE ( m) x x A B (x 103 rpm) Perturbed average fSLE fSLE around point A relatively more sensitive |f SLE | region Figure 10. Perturbed average of SLE f validation as optimizati on criterion that avoids spindle speed sensitive SLE f Shown in the figure are points A (close to steep slope region of SLE f ) and B (close to moderate slope region of SLE f ), the perturbed average of SLE f near A is higher than at B. Therefore, using the perturbed average as an optimum criterion is valid. The SQP method is used to solve Eqs. (3.9) and (3.10). In case Eq. (3.9) is implemented then initial guesses of and b (design variables) are made along the M RRf contour. In the other case (Eq. (3.10)) the initial guesses of and b are made along SLE f

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34 contour. The number of initial guesses along th e constraint is made such that convergence is towards a global optimum. The initial guesse s for the spindle speed are increased in 625 rpm increments for the corresponding sp indle speed range considered. Also, the PSO method is used to solve Eq. (3.8). When using PSO, the optimum points do not tend to converge to spindle speed sens itive optimums. Therefore, th ere is no need to solve the reformulated form of the problem in PSO. This leads to a fewer number of evaluations of SLE f and is a computati onally more efficient optimization method. A comparison of the three optimization sc hemes is shown in Figure 11 and Figure 12. Figure 11 shows the optima for each approach superimposed on the corresponding stability lobe diagram. In Fi gure 12, the Pareto fronts for the three methods are shown. The optimum points found using the two SQP formulations closely agree with the PSO method (Figure 12). Figure 11. Stability, SLE f and M RRf contours with optimum Pa reto front points found using PSO and SQP (average perturbed spindle speed formulation). The figure shows that optimum points are not in regions sensitive to spindle speed (Table 2).

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35 50 100 150 200 250 300 350 400 450 0 2 4 6 8 10 12 14 16 18 20 fMRR (mm3/s)perturbed average |fSLE| (m) SQP SLE objective SQP MRR objective PSO MRR objective Figure 12. Pareto front showing optimum points found using three optimization algorithms/formulations; the same trends are apparent. However, the SQP methods required additional computational time (Table 2). Although the PSO points show the same trend, some improvement in the fitness is still possible relative to the SQP results. Because the PSO search inherently avoided optimum points that are spindle speed inse nsitive, there is no need to use average perturbed fSLE as with SQP, which leads to a decreased number of fSLE evaluations in PSO. However, narrow optimum poin ts may go undetected when using PSO. As noted, when comparing the Pareto fr onts in Figure 12, it is seen that the PSO approach did not converge to the same fitness as SQP method. A check of the optimum points which correspon d to a value of SLE f = 4 m, for example, shows that PSO converged to 100 mm3/s, while SQP converged to 150 mm3/s. To better understand this

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36 result, the design space was divide d between the two design vectors, b and for SQP and PSO using a factor, a, that was normalized between 0 and 1. The PSO and SQP optimums were normalized to a = 0 and 1, respectively. Ne xt, the stability constraint ( g ), M RRf, and SLE f were plotted against that rati o. In Figure 13 it is seen that discontinuities exist in the SLE f constraint and the first de rivative of the eigenvalue constraint within this region. Although PSO is not significantly affected by a discontinuity in the derivative constraint, it can be affected by a discontinuity of the SLE f constraint, where the discontinuity tends to narrow the search region of the swarm. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 1 1.1 ag PSO optimum at 4 m SQP optimum 4 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 afSLE m PSO optimum at 4 m SQP optimum 4 m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 120 140 160 afMRR mm3/s PSO optimum at 4 m SQP optimum 4 m Figure 13. Variations in the eigenvalues, su rface location error, and removal rate for PSO and SQP optima, where M RRf is the objective for both. Th e discontinuities in the surface location error cause PSO to not converge on the SQP optimum.

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37 Bi-objective space In this section, the bi-objective domain (the feasible space of the objective functions) of average perturbed SLE f and M RRf for the set of input parameters listed in Table 3 for an up milling case is provided. Figure 14 shows the objective contours in the design space of spindle speed ( ) and axial depth (b). The respective bi-objective space is shown in Figure 15 and Figure 16. In Figure 15 the contours of constant axial depth are shown, while the contours of constant sp indle speeds are shown in Figure 16. These figures give an idea of the feasible design a nd bi-objective space. It can be seen that the bi-objective feasible sp ace can be non-convex (not all poi nts on a straight line connecting two points in the feasible domain belong to th at domain). This makes the choice of using the tradeoff method as a multi-objective optimi zation approach a suitable one, since this method can handle both convex and non-conve x problems. A good observation can be made from Figure 15, where it can be seen that for the high M RRf region with high b values, the relative sensitivity of SLE f increases compared to the lower M RRf region. Table 3. Cutting conditions, modal parameters and cutting force coefficients used in biobjective space simulations M (kg) C (Ns/m) K (N/m) 0.440 00.35 830 090 6 64.45100 03.5510 Tool diameter (mm) c (mm) a (mm) N 25.4 0.1 21.8 1 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 700 x 106 20 x 106 46 x 103 33 x 103

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38 Selection of spindle speed perturbation bandwidth In Figure 10, it was shown that the average perturbation of SLE f provided an adequate optimization criteria. However, the choice of the spindle speed perturbation step size or bandwidth,2 depends on the designer pref erence. Any spindle speed perturbation in SLE f would avoid convergence to sensitive SLE f optima. Depending on the machining center spindle drive accuracy, the perturba tion bandwidth can be set accordingly. The average perturbed SLE f contours of 100 and 300 rpm bandwidth are shown in Figure 17 (use Table 3 paramete rs). The high slope region of average SLE f in the 100 rpm bandwidth case is repl aced by higher values of average SLE f making the optimization formulation favor insensitive spindle speed SLE f (x 103 rpm)b (mm)2 53 5 4 55 56 51 4 01 2 01 0 0701 051 0153 05 0 2 70 02 5 0 023 002 1 0 01 9 0 01 8 0 01 6 0 01 4 0 01 2 0 01 0 0 09 0 07 0 05 0 03 0 0 10 12 14 16 18 20 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Stability boundary fMRR mm3/s perturbed average |fSLE| m Zig zag line indicate the limit of the domain where SLE is calculated Figure 14. Average surface location error co ntours for 300 rpm bandwidth perturbation, stability boundary and material removal rate (see Table 3).

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39 perturbed average |f SLE | ( mm)4 7 m m 4 3 3 9 3 73 5 3 5 3 53 12 72 5 2 32 11 91 71 31 11 10 7 500 1000 1500 2000 2500 20 40 60 80 100 120 140 160 Feasible domain Pareto front f MRR (mm 3 /s) Figure 15. Feasible domain. C ontour lines corresponding to c onstant axial depth in the stable region in the bi-objective space (see Table 3). fMRR (mm3/s)perturbed average |fSLE| ( m)1515.114.6 x103 rpm15.415.610.116 13.9 14.6 12.2 11.219.6 18.314.917 500 1000 1500 2000 2500 20 40 60 80 100 120 140 160 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 Figure 16. Contour lines correspond ing to constant spindle speed in feasible region of biobjective space (see Table 3).

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40 (x 103 rpm)b (mm) 2 3 5 710142440598572534328 17 13 9 5 1700150013001100900700500300100 rpm bandwidth 10 12 14 16 18 20 1 2 3 4 5 6 (x 103 rpm)b (mm) 2 3579142030426181695746362719 368 1700150013001100900700500 300 300 rpm bandwidth 10 12 14 16 18 20 1 2 3 4 5 Stability boundary fSLE m fMRR mm3/s Figure 17. Average surface location error contours for 100 and 300 rpm band width, stability boundary and ma terial removal rate contours (see Table 3).

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41 Case Studies As opposed to the previous anal ysis of two design variables ( and b), two cases of an added third design variable were anal yzed. The first one was for radial immersion (a) and the second one was for chip load (c). These cases are compared to the two design variable case. Radial immersion ( a ) Previous simulations consid ered spindle speed and axia l depth of cut as design variables. Another simulation was completed using radial immersion as a third design variable for an up milling cut. It was compared to a two design variable case where radial immersion was held constant at 0.508 mm in a 25.4 mm to ol (Table 4). Figure 18 shows the Pareto front for these two cases. It is seen that adding radial immersion as a third design variable improved the value of perturbed average SLE f with respect to the constant radial immersion case. The optimum ra dial immersion found was 0.58 mm for all optimum points up to 500 mm3/s. In both simulations the same spindle speed perturbation (170 rpm) was used. As seen in Figure 18, a better calculation of the Pareto front (smoother than Figure 12) is found by using sm all increments in the spindle speed (each 100 rpm) initial guesses. However, the SLE f found in Figure 18 appear to be unrealistically small which ma y warrant further analysis. Table 4. Cutting conditions, modal parameters and cutting force coefficients used in radial immersion case study M (kg) C (Ns/m) K (N/m) 0.250 00.23 34.40 027.0 6 1.30100 6 01.2010 Tool diameter (mm) c (mm) a (mm) N 25.4 0.1 0.508 2 Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m) 700 x 106 210 x 106 0 0

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42 Chip load ( c ) To study the effect of chip load on surf ace location error, it is added as a third design variable in addition to spindle speed a nd axial depth. The parameters used in this study are listed in Table 5 fo r a down milling case. For the two design variable case (0.1 mm/tooth chip load), the Pare to optimal points are found fo r two different bandwidths, 100 rpm and 400 rpm, respectively (F igure 20). It is noted that of the optimum points is almost constant up to 700 mm3/s ( 31,325 rpm) where it changes to another almost constant (29,500 rpm) for the higher M RRf range. Also the effect of bandwidth size does not show significant effect on the optimum points found. Figure 19 shows the Pareto front for a constant chip load of 0.1 mm/tooth compared to th e three design variable case, where the chip load (3rd design variable) side constraint s are from 0.01 mm/tooth to 0.2 mm/tooth. An improvement in the average perturbed SLE f can be seen. It should be noted here that for the latter case, is also found to be consta nt (31,325 rpm same as two design variable case) while the chip load incr eased from 0.16 to 0.2 mm/tooth. The effect of adding the chip load is therefore seen as an improvement in th e fitness of average perturbed SLE f objective, where further improvement is possible while eliminating the need to switch to a lower speed (29,500 rpm) where the SLE f error is much higher. This explains the agreement between the two desi gn variables case and three design variable case in the M RRf range below 600 mm3/s. When higher M RRf is needed the two design variable case fails to account for the M RRf constraint at the same spindle speed. However the three design variable case (with chip load) can accommodate this by increasing the chip load while keeping the spindl e speed unchanged. This makes the SLE f in the three design variable case substantially lower.

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43 0 100 200 300 400 500 600 0 0.005 0.01 0.015 0.02 0.025 0.03 fMRR (mm3/s)Perturbed average |fSLE| ( m) radial immersion as a third design variable radial immersion = 0.508 mm Figure 18. Pareto front for spindle speed and axial depth as design variables with radial immersion 0.508 mm, compared to the case where radial immersion is added as a third design variable. The optimum radial immersion for the latter case is 0.58 mm up to 500 mm3/s (see Table 4). Table 5. Milling cutting conditions, modal para meters and cutting force coefficients used in chip load study case M (kg) C (Ns/m) K (N/m) 0.0270 00.03 70 02 6 61.0100 01.610 Tool diameter (mm) c (mm) a (mm) N 12.7 0.1 0.635 2 Kt (N/m2) Kn (N/m2) Kte (N/m)Kne (N/m) 600 x 106 180 x 106 0 0

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44 0 200 400 600 800 1000 0 5 10 15 20 25 30 35 fMRR (mm3/s) Perturbed average |fSLE| (m) chip load constant chip load 3rd design c=0.16 to 0.2 mm/tooth Figure 19. Pareto front using chip load as a third design variable compared to spindle speed and axial depth as design variables. For the three design variable case, an improvement in the average surface loca tion error can be seen (see Table 5).

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45 (rpm)b (mm)264610163052122446468121620242832812182432425468844814 208122444 12001150 1100 1050950850 8007507006005505004504003502503006501000 90020015010050 10 15 20 25 30 35 40 2 4 6 8 10 12 14 16 18 Average fSLE ( m) for 400 rpm bandwidth fMRR (mm3/s) Stability boundary Optimum points 400 rpm bandwidth Optimum points 100 rpm bandwidth Figure 20. Stability, perturbed average SLEf, and MRRf contours with optimum Pareto front points found using 100 rpm and 400 rpm bandwidth. This case study shows the di fficulty in selecting op timum points based on experience (Table 5).

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46Discussion The formulations provided in Eqs. (3.9) and (3.10) proved adequate in finding the Pareto optimal set insensitive to spindle speed variation, provided an appropriate number of initial guesses is made. Also, the Eq. (3.9) formulation is easier to apply using the SQP method, where the initial guesses are made along the M RRf contour. The generation of the Pareto front for the multi-design variable case can be rather time-consuming. However, if the designer is given that freedom of choice, it might be a necessity. For example, the effect of adding chip load or radial immersion as a third design variable gave a substantial impr ovement in the surf ace location error in comparison to the two design variable case. This is counterintuit ive to using a lower value of c or a as means of reducing the surface location error. The effect of spindle speed perturbation bandwidth on the sensitivity of optimum points is rather complex. Qualitatively, in Figure 17 it is shown that increasing the bandwidth from 100 rpm to 300 rpm had the sa me effect of increasing the value of SLE f near the sensitive region. Further investiga tion is needed to establish a quantitative relation between bandwidth and sensitivity of optimum points.

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47 CHAPTER 4 UNCERTAINTY ANALYSIS In Chapter 3, optimization was used to find preferable designs for two objectives: material removal rate (MRR) and surface location error [48, 81, 82] (SLE), with a Pareto front, or tradeoff curve, found for the two competing objectives. Although the milling model used in the optimization algorithm is de terministic (time finite element analysis), uncertainties in the input parameters to the model limit the confidence in the optimum predictions. These input parameters include cutting force coefficients (materialand process-dependent), tool modal parameters and cutting conditions. By accounting for these uncertainties it is possible to arrive at a robust optimum operating condition. In previous studies [83-85], uncertainty in the milling process was handled from a control perspective. The uncertainty in the cutting force was accommodated using a control system. The force c ontroller was designed to comp ensate for known process effects and accounted for the force-feed nonlinea rity inherent in metal cutting operations. In this study, the uncertainties in the milling model are estimated using sensitivity analysis and Monte Carlo simulation. This en ables selection of a preferred design that takes into account the inhere nt uncertainty in the model a priori. This chapter begins with a description of the milling model and continues with a discussion of stability lobes and surface loca tion error analysis w ith regard to their numerical accuracy. Sensitivity analysis is discussed in the next section. Then, case studies for the numerical accuracy of the sensi tivities of the maximum stable axial depth, blim, and SLE are presented for a typical two degree-of -freedom tool. This enables us to

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48 carry out the stability lobe and surface locati on error sensitivity anal ysis in the next two sections. Sensitivity is used to determ ine the effect of input parameters on blim and SLE. This enables the determination of which pa rameter(s) is the highe st contributor to stability enhancement and SLE reduction. The uncertainties in blim and SLE predictions are then calculated using two methods 1) the Monte Carlo simulation; and 2) the use of numerical derivatives of the system characte ristic multipliers to determine sensitivities. The uncertainty in axial depth effects a reduction in the MRR, and the SLE uncertainty provides bounds on SLE mean expected value. This allo ws robust optimization that takes into consideration both performance and uncertainty. Equation Chapter 4 Section 1 Milling Model A schematic of a two degree-of-freedom milling tool is shown in Figure 21. The tool/work-piece dynamics and cutting forces are used to formulate the governing delay differential equation for the system. Solution of the delay differen tial equation is found using time finite element analysis (TFEA) [54-56]. This method provides the means for predicting the milling process stability and quality (SLE). However, the uncertainty in the input parameters to the solution method pl aces an uncertainty on the stability and SLE prediction. These parameters are divided into two groups; 1) uncerta inty from lack of knowledge of the tool modal matrices, K, C and M, and the cutting force coefficients (mechanistic force model); and 2) uncertaint y in other machining parameters, such as spindle speed, chip load and radial depth. To estimate the parameters in the former, modal testing is used to measure the dyna mic parameters while cutting tests are completed to estimate the cutting force coefficients. In the modal parameter estimation the peak amplitude method is used to fit th e measured frequency response function. In this method [86, 87], the peak of the ma gnitude of the frequency response function

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49 corresponds to the natura l frequency. From this the half power frequencies are used to estimate the damping ratio. Table 6 lists th e mean modal values for 25.4 mm diameter endmill having a 12 helix angle with 114 mm ove rhang length and the corresponding cutting force coefficients for 6061 aluminum (assuming a mechanistic force model, see Chapter 5). The cutting conditions are also list ed in the table. These parameters will be used in the simulations in this chapter for a down milling cut. K x x K y C x C y y Feed SLE Figure 21. Schematic of 2-D milling model. Surface location error (SLE) due to phasing between cutting force and tool displacement is also shown. Table 6. Cutting force coefficients, modal parameters and cutting conditions of milling process. M (kg)K (N/m x106)C (N.s/m)x 0.44 4.45 830.030 y 0.44 3.55 90.90.036Kt(N/m2 x106)Kn(N/m2 x106)Kne(N/m x103)6001806 Tool diameter (mm)radial dept h,a (mm)chip load, c (mm/tooth) 25.4 0.5080.1Kte(N/m x103)12 N 1

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50 Stability and Surface Location Error Analysis The stability lobes are used to repres ent the stable space of axial depth (b) and spindle speed of a milling process. In TFEA [54-57], a discrete map is used to match the tool-free vibration while out of the cut, w ith the tool vibration in the cut. The system characteristic multipliers ( ) of the map provide the st able cutting zone where max is less than one. TFEA provides a field of max in the design space of b and The limit of stability, blim can be found using root-finding numerical techniques. Here we use the bisection root-finding method. The convergence cr iterion of the bi-section method should account for the amplification of numerical noi se induced by sensitivity estimation. It should be noted that the number of elemen ts affects the accuracy of the estimation. For calculation of SLE in TFEA, the numerical noise is only due to the number of elements. In this section we will discuss th e effect of both the convergence criterion and the number of elements on th e sensitivity estimation of blim and SLE. Bi-section Method Convergence Criterion As described in Chapter 3 the axial depth limit, blim, was calculated using the bisection method (Eq. (3.4)). Although a relativel y large value of can be adequate for the calculation of the stability lobe s, a tighter limit is needed to calculate the sensitivities. This is attributed to amplification of nume rical noise in the deri vative calculation. This comparison is made in the Case Studies section. Number of Elements The accuracy of TFEA pred iction of stability and SLE is highly dependent on the number of elements used. The effect of the number of elements is even more apparent

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51 when calculating the sensitivity of the prediction, where a higher number of elements is needed to eliminate numerical noise from the sensitivity calculation. Numerical Sensitivity Analysis The sensitivity of axial de pth to input parameters / bX i is cumbersome to compute analytically using the TFEA method; therefore, a numerical derivative is used by implementing a small perturbation. Factors which affect accurate calculation of sensitivity to inputs include: 1) central difference truncation error; and 2) step size selection. Theref ore, a balance needs to be achieved in determining the sensitivity that provides a stable estimate of the sensitivity while maintaining computational efficiency. In the following, we describe these factors and their consideration in the calculati on of stability and SLE sensitivities. Truncation Error The central difference method is used in the sensitivity calculation. The formula for this method is 2 11, 2ibb b Oh Xh (4.1) where h denotes the step size in input parameter Xi, 1ibbXh 1ibbXh and O(h2) is the 2nd order truncation error. A higher order formula with 4th order truncation error O(h4) can also be used. However, as show n in Eq. (4.2), it is two times more computationally expensive than Eq. (4.1), 4 211288 12ibbbb b Oh Xh (4.2)

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52 In order to help decide whether the highe r truncation error formula need be applied (Eq. (4.2)), the sensitivity of blim with respect to modal stiffness Kx is calculated as a function of step size h This comparison is made in the Case Studies section. Step Size The step size, h in Eqs. ((4.1) and (4.2)) should be carefully chosen. This is especially important when there is numerical noise in the calculated blim due to the convergence criterion (Eq. (1)). The step si ze should be large enough to be out of the numerical noise range, however, not so larg e that the non-linear va riation in the output ( blim or SLE ) takes effect. The following section illustrates this idea. Case Studies In this section, numerical estimations of the sensitivity are made based on different variations of convergence criterion, number of elements, sensitivity analysis formula (Eq. (4.1) and Eq. (4.2)), and step size. The co mparisons are made for a 10 krpm spindle speed, 10 elements and 4310 x unless otherwise noted. The logarithmic derivative can be used in making these comparisons by evaluati ng the percentage of change in an output (axial depth, b ) due to a percentage change in the input, Xi. It is expressed as ln lni iib X b X bX (4.3) To illustrate the effect of convergence cr iterion, the logarithmic derivative of blim with respect to Mx (the X direction modal mass) is calculated for two error limits as a function of step size percentage %/100iihXX see Figure 22. It can be seen that a tighter error limit nearly eliminates the num erical noise in the derivative calculation.

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53 The effect of the number of elements on SLE sensitivity is illustrated in Figure 23, where the SLE sensitivity with respect to Kx is calculated. The /xSLEK is used to illustrate the effect of the number of elements because it is known that the SLE does not depend on the Kx stiffness (tool feeding direction being the xaxis). Therefore /0xSLEK which would amplify and illustrate more clearly the effect of the number of elements on the sensitivity estimation. The higher number of elements provides a larger stable region of sensitivity It should be noted that the 2nd order finite difference method is used in this sensitivity comparison and the bi-section convergence criterion is not applicable here since SLE is found from fixed points of the dynamic map (see Eq. (A.18) in Appendix A) when the cutting conditions provide a stable cut. 0 0.1 0.2 0.3 0.4 0.5 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 %hln(blim)/ln(Mx) = 3e-4 = 3e-10 2nd order central difference E=10 Figure 22. The effect of error limit in the bisection method on numerical noise in the sensitivity calculation (see Table 6).

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54 0 0.1 0.2 0.3 0.4 0.5 -6 -4 -2 0 2 4 6 8 10 12 14 x 10-23 %hSLE/Kx (m2/N) E = 10 E = 30 E = 50 Figure 23. Sensitivity of SLE with respect to Kx. The higher number of elements, E, provides more stable sensitivity estimation. The second order finite difference formula is used here (see Table 6). Figure 24 shows the effect of the central difference truncation error. A finite step size percentage is needed to reach a stable value of the derivative for both formulas. It can be seen that Eq. (4.2) gives a wider ra nge of step sizes at which the sensitivity calculation is stable. However, the improved stability range, or reduction in numerical noise, is not significant to sacrifice co mputational efficiency for its usage.

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55 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 x 10-9 %hblim/Kx (m2/N) 2nd Order central difference 4th order central difference Convergence limit = 3e-4 E=10 Figure 24. Comparison between 2nd and 4th order central difference formulas. The 4th order formula shows a wider stable regi on for step size, but higher computation time (see Table 6). The importance of step size selection can be illustrated by Figure 25, which shows the logarithmic derivative of ax ial depth with respect to input parameters versus step size percentage. It can be seen that the step size should be chosen high e nough to be out of the numerical noise range but not so high so that the non-linear variation is included (in this range of % h only is non-linear). The figure also i ndicates the relative sensitivity of axial depth to each input parameter, spindl e speed having the largest effect followed by modal mass and stiffness.

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56 0 0.5 1 1.5 2 -4 -3 -2 -1 0 1 2 h%ln(blim)/ln(Xi) Kx Cx Mx Kt Kn Convergence limit =3e-4 E=10 Figure 25. The logarithmic derivative of axia l depth with respect to input parameters versus step size percentage (see Table 6). From Figure 24 and Figure 25 it can be seen that h= 0.2% provides a stable sensitivity estimation. To verify that a typical step size of 0.2%, convergence limit 4310 E=10, and the 2nd order finite difference a pproximation give correct calculation of sensitivity, the variations of b to modal parameters and cutting coefficients are plotted in Figure 26 and Fi gure 27, respectively. Also, the slope predicted using Eq. (4.1) with h=0.2% is superimposed on the same plot. The suitable selection of h is indicated by the tangency of the predicted slope to the functional variation. On the other hand, it can be seen that wh en the variation is linear, th e linear approximation can be accurate for a large variation of the input parameter.

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57 90 95 100 105 110 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Xi/Xiblim (mm) Ky Cy My Sensitivity Prediction Figure 26. The variation of axial depth blim with respect to a 10% change in nominal input parameters. The sensitivity of blim with respect to each pa rameter is superimposed. Linearity and non-linearity of blim(Xi) can be observed (see Table 6). 90 95 100 105 110 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Xi/Xi (x 100)blim (mm) Kt Kn Sensitivity Prediction Figure 27. The variation of blim with respect to a 10% change in Kt and Kn. The sensitivity of blim with respect to each parameter is superimposed. Linearity of blim(Xi) can be observed (see Table 6).

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58 Stability Sensitivity Analysis In this section, calculations of the sensitivity of blim to the input parameters are provided. The parameters used in the sensitiv ity calculations are provided in Table 7. In Figure 28 a comparison between the sensitivities of stiffness, K, and modal mass, M, are compared in the x (feed) and y-directions of the tool. As can be seen in the figure, the sensitivities in the x and y-directions are comparable in magnitude; however, the sensitivity in the y-direction is inaccurate near discontinuities in the system characteristic multipliers. This will be explained in the Un certainty section with a graphic depicting these discontinuities. Table 7. Parameters used in sensitivity analysis. h (%) E Central difference 0.2 10 2nd order 4310 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 (rpm x103)ln(blim)/ln(Xi) Kx Ky Mx My C1 discontinuities in blim Figure 28. Sensitivity of axial depth blim to changes in modal mass M and modal stiffness K in the x and y-directions (see Table 6).

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59 In Figure 29, the effect of damping on the stability is shown to be minimal compared to the modal stiffness and mass. This is a somewhat counter-intuitive result, but can be explained by regene ration (undulations in the cut surface experienced by the tooth in the current cut that are caused by the tooth vibration in the previous cut), which is a primary physical phenomenon that causes instability. The modal mass and stiffness have a great effect on the systems natural fr equency, which has a significant effect on regeneration. This also explai ns the result shown in Figure 30, where the sensitivity of axial depth blim to a change in spindle speed is significant and comparable to modal mass and stiffness. The effect of cutting force co efficients is shown in Figure 31, where the tangential cutting force coefficient, Kt, has more effect on the axial depth limit than the normal direction coefficient, Kn. 5 10 15 20 -30 -20 -10 0 10 20 30 (x 103)ln(blim)/ln(Xi) Kx Cx Cy C1 discontinuities in blim Figure 29. Sensitivity of axial depth blim to changes in modal damping C in the x and ydirections. The damping sensitivity is compared to modal stiffness sensitivity in the x-direction (see Table 6).

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60 5 10 15 20 -250 -200 -150 -100 -50 0 50 100 150 (rpm x103)ln(blim)/ln(Xi) Ky My rpm C1 discontinuities in blim Figure 30. Sensitivity of axial depth blim to changes in spindle speed. The spindle speed sensitivity is compared here to the modal mass and stiffness in y-direction (see Table 6). 5 10 15 20 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0 4 (rpm x103)ln(blim)/ln(Xi) Kt Kn Figure 31. Sensitivity of axial depth blim to changes in force cutting coefficients in the tangential Kt and normal directions Kn. Higher sensitivity can be seen for Kt (see Table 6).

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61 Surface Location Error Sensitivity Analysis The sensitivity of surface location error, SLE, to changes in input parameters is examined here. The parameters listed in Table 6 are used with b=1 mm and down milling case. In Figure 32, the sensitivity of SLE to changes in modal parameters in the ydirection is shown. Again, it can be seen that changes in Ky and My contribute more than Cy to a change in SLE. In Figure 33, the effect of cutt ing force coefficients is shown, where it is observed that the highest contributors to SLE sensitivity are Kt and Kte. Also, in Figure 34, SLE sensitivity to spindle speed and radial depth, rstep, is shown. Substantial sensitivity to spindle speed can be s een. This is due to the dependence of SLE on the relationship between the tool point frequency response and the selected spindle speed. As the spindle speed changes, it tracks different parts of the response. 5 10 15 20 0 50 100 150 200 250 (rpm x 103)Sensitivity Xi(SLE)/(Xi) (m) Ky My Cy Figure 32. Sensitivity of surface location error SLE to changes in modal parameters in ydirection (see Table 6).

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62 5 10 15 20 0 5 10 15 20 (rpm x103)Xi SLE/ Xi ( m) Kt Kn Kte Kne Figure 33. Sensitivity of SLE to cutting force coefficients (see Table 6). 5 10 15 20 0 100 200 300 400 500 (x 103)Xi (SLE)/ (Xi) ( m) rstep Figure 34. Sensitivity of SLE to spindle speed and radial depth of cut (see Table 6).

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63 Uncertainty of Stability Bounda ry and Surface Location Error Input Parameters Correlation Effect The correlation between the input parameters can have significant effect on the prediction of uncertainty. Neglecting the correl ation can result in e rroneous estimation of the uncertainty, especially when the input para meters are highly corr elated. Inclusion of the covariance matrix between pa rameters is necessary in this case. The input parameters can be classified into thr ee groups: dynamic modal paramete rs of the tool (work-piece assumed rigid), cutting force coeffici ents and machining parameters (e.g., radial step and spindle speed). In Chapter 5, estimation of th e correlation between parameters of the first two groups is explained and used in the uncertain ty prediction. The combined standard uncertainty uc can be found using sensitivities of output (blim or SLE) to input parameters. For the case of axial depth limit, uc is given as [88]: 2 1 2 limlimlim lim 1112,mmm ciij iii iijbbb ubuXuXX XXX (4.4) where u(Xi) refers to the standard unce rtainty in the input parameter Xi, u(Xi,Xj) is the estimated covariance between parameters Xi and Xj,. and m is the number of input parameters. The degree of the correlation between Xi and Xj is characterized by the correlation coefficient ,ij ij ijuXX rXX uXuX (4.5) In the Monte Carlo and Latin Hype-Cube sampling methods (described next), the multivariate normal distribution can be used to estimate the confidence level, in which case the covariance matrix between parameters controls the random sampling procedure.

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64 Monte Carlo Simulation The combined standard uncertainty, uc, of the stability boundary ( blim) and surface location error ( SLE ) can be predicted using Monte Carl o simulation. In this method, a random sample of size n is selected from the populati on of each input parameter. A normal distribution of the input parame ters is assumed. In the sample n the nominal value and standard deviation of each input pa rameter are used to generate the sample. The axial depth limit and surface location error are then calculated using TFEA for each point in the sample. The standa rd deviation of the predicted blim and SLE is then calculated from sample output for the range of spindle speeds of interest. It should be noted here that in doing so, no correlation between the input parameters is assumed, which is a common, and sometimes erroneous approach. To illustrate the effect of uncertainty in the input parameters on stability boundary uncertainty, standard uncertainties of 5% 0.5%, 0.001% and 0.5% are assigned to nominal values of the cutting force coeffici ents, modal parameters, radial step, and spindle speed, respectively. Th e values of the standard uncer tainties assigned correspond to practical variation in th e parameters involved. The parameters are assumed to be uncorrelated here. A sample size of 1000 is used. The stability boundary uncertainty is found, as shown in Figure 35, for one sta ndard deviation interval around the neutral stability boundary.

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65 5 10 15 20 2 4 6 8 10 12 14 (rpm x103)blim (mm) mean one std mean mean + one std Figure 35. Confidence in stability boundary due to input parameters uncertainties using Monte Carlo simulation (see Table 6). Sensitivity Method The combined standard uncertainty uc in axial depth limit while neglecting correlation between input parameters can be obtained from Eq. (4.4) as 2 lim lim 1 m ci i ib ubuX X (4.6) where u(Xi) refers to the standard unce rtainty in the input parameter Xi (same used for Monte Carlo method), and m is the number of input pa rameters. Although this relation assumes no correlation between input paramete rs it should be noted that cutting force

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66 coefficients ( Kt, Kn, Kte, Kne) and modal parameters ( K C M ) may be correlated in practice. The same standard uncertainty is assumed in the input parameters as in previous sections and the confidence level in axial depth limit is calculated for an interval of 2 uc( blim). Figure 36 shows the close agreement found using the two methods. However, it should be noted that the sensitivity method can be inaccurate near points where the function ( blim) is C1 discontinuous. Figure 37 shows the direct correspondence between the inaccurate sensitivity and C1 discontinuity in The C1 discontinuity in blim leads to inaccurate estimation of uc(blim) (see Eq. (4.6)). 5 10 15 20 0 2 4 6 8 10 12 14 16 18 20 (x 103)blim (mm) Sensitivity Monte Carlo Nominal Figure 36. Uncertainty boundary in axial depth limit using two standard deviation confidence interval. Uncertainty is calcul ated using sensitivity method and Monte Carlo method (see Table 6).

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67 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 uc (b) (mm) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -1 0 1 Real max( ) 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0.5 1 (x 103)Imag max( ) Derivative method Monte Carlo Simulation Figure 37. Uncertainty in axial depth us ing sensitivity and Monte Carlo methods. Inaccuracies in the sensitivity method can be seen near C1 discontinuity in the real and imaginary part of system characteristic multipliers (see Table 6). It should be noted here that predicting the uncertainty by Eq. (4.6) uses a linear approximation. The standard uncertainties assumed earlier are small where the linear approximation is still valid. However, if the uncertainties in the input parameters are large, then that linear approximation is no longer valid. In this case, simple random sampling methods (such as Monte Carl o simulation) are more appropriate. The surface location error uncertainty is found similarly using both methods. However, as shown earlier (see Figure 32 and Figure 34), the SLE sensitivities are accurate and do not depend on the characteristic multipliers continuity. Since the SLE is only defined for stable cutting conditions (see Eq. (A.18) in Appendix A) and explains

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68 the close prediction of uncertainty in SLE using sensitivity and Monte Carlo methods (Figure 38). 5 10 15 20 -15 -10 -5 0 5 10 15 20 25 30 (x 103)SLE 2uc(m) Monte Carlo method Sensitivity method Figure 38. Surface location error uncertainty with two standard deviation confidence interval on the nominal SLE Close agreement is observed (see Table 6). Latin Hyper-Cube Sampling Method This method was originally proposed as a variance reduction technique [89] in which the estimated variance is asymptotical ly lower than with simple random sampling (Monte Carlo method) [90, 91]. That is, for a sample size L this method gives a lower estimate of the output variance than is possi ble with the Monte Carlo method. The basic idea of this method is that each value (or range of values) of a variab le is represented in the sample, no matter which value turns out to be the most important. In this way, the

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69 sampling distribution is divided into a number of strata w ith a random selection inside each stratum. The Latin Hyper-Cube method will be used in Chapter 5 for predicting the standard combined uncertainty of the stabili ty and surface location e rror cutting tests in that chapter. Robust Optimization under Uncertainty In order to account for uncertainty in the ax ial depth stability limit, the safety factor design analogy is used here. The deterministic optimization algorithm implemented in Chapter 3 (Eq. (3.9)), repeated here, can be modified to account for the axial depth uncertainty. ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbgbgb for a series of selected li ,MRRmits (e) on f (4.7) Therefore, the axial depth b used in the stability constraint is set equal to an uncertainty inflated value. That is, b is replaced by b+ Ue, where ecUkub is the expanded uncertainty, k is a factor that estimates the uncertainty confidence interval and uc(b) is the combined standard uncertainty in th e axial depth. Therefore Eq. (4.7) becomes ,,, 3 : ,, 1 ... ,,,1,MRRi eeefbfbfb SLESLESLEmin subjecttofbeforik gbUgbUgbU for a series of ,MRRselected limits (e) on f (4.8)

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70 Discussion In this chapter, the sensitivities of axial depth limit and surface location error to model input uncertainties were studied. Numerical estimation of the sensitivities can be challenging, where several factors contribute to the accuracy of the estimation. Step size is one of the significant factors that affect the accuracy of the estimation. The sensitivity analysis aids in identify ing the relative contribution of the milling model input parameters to the sensitivity of either axial depth limit or surface location error. For the case of axial depth, the sp indle speed, followed by modal stiffness and mass, is the most significant contributor. In the case of cutting force coefficients, the tangential cutting force coefficient is found to contribute more to the sensitivity than the normal cutting force coefficient. As for the surface location error sensitivity, the same trend can be observed. However, for the cut ting force coefficients, the edge tangential cutting force coefficient signi ficantly contributes to the SLE The uncertainty in axial depth and surf ace location error was predicted using two methods: the sensitivity method and the Mont e Carlo simulation approach. Comparable agreement is shown. However, the sensitivit y method is more efficient computationally. For example, in the case of SLE uncertainty prediction, Monte Carlo simulation required 9.34 hours, while the sensitivity method needed only 0.26 hours (36 times more efficient). It is noted that for the uc(SLE) case, when the milling parameters are well into the stable region, the accuracy of the sensitivity method is not sacrificed at the cost of efficiency as is the case for uc(b) at discontinuities in the characteristic multipliers. Finally, the optimization algorithm introdu ced in Chapter 3 was modified to account for confidence levels in the axial dept h limit. This allows robust optimization to

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71 account for inherent uncertainty in the mean va lues of the input parameters. In Chapter 5 an implementation of this algorithm is demonstrated.

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72 CHAPTER 5 EXPERIMENTAL RESULTS The milling model accuracy depends on reli able determination of cutting force coefficients and tool or work-piece modal parameters. These values are found experimentally and their uncertainties c ontribute to the uncertainty of the model prediction. In this chapter, the experime ntal procedure used to determine these parameters is described and then the op timization algorithm is executed using the experimentally determined input parameters to find the Pareto optimal points. Another set of experiments is completed to validate/i nvalidate these optimal points. Using the optimization algorithm, the strength and weakness of the mathematical model or solution method can be obtained. Equation Chapter 5 Section 1 Cutting Force Coefficients Milling Forces The average milling forces during one tooth period in the x and y -directions are [92, 93], cos22sin2sincos 82 2sin2cos2cossin 82ex s t ex stxtntene ytnteneNbcNb FKKKK NbcNb FKKKK (5.1) where teK and neK are the tangential and normal edge cutting force coefficients, respectively. In slotting tests (see Figure 7), the entry and exit angles of the cutter are

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73 0st and ex respectively. The average forces per tooth period for this case are found to be: 4 4xnne ytteNbNb FKcK NbNb FKcK (5.2) Equation (5.2) can be written as a function of chip load ( c ) as: ,,.qqcqeFFcFqxyz (5.3) The experimental procedure consists of completing multiple cutting tests at varying chip loads and recording the cutting forces. For each chip load increment, the average cutting forces in the x and y -directions are measured, and th en a linear regression of the data points is made to extract the cutting coefficients using Eqs. (5.2) and (5.3): 4 4 .ycye tte xcxe nneFF KK NbNb FF KK NbNb (5.4) For radial immersions less than the cutter diameter, the entry and exit angles differ from the slotting case. For up-milling (see Figure 7) the entry and exit angles of the cutter are 0st and 1cos1exa R where a is the radial depth of cut. Substituting in Eq. (5.1) gives: cos212sin2sincos1 82xtexnexexteexneexNbcNb FKKKK (5.5) Factoring Eq. (5.5) in terms of chip load c gives:

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74 xxcxeFFcF (5.6) cos212sin2 8 sincos1. 2xctexnexex xeteexneexNb FKK Nb FKK (5.7) Similarly, the following equations are obtained for the y-direction. 2sin2cos21cos1sin 82ytexexnexteexneexNbcNb FKKKK (5.8) yycyeFFcF (5.9) 2sin2cos21 8 cos1sin 2yctexexnex yeteexneexNb FKK Nb FKK (5.10) Writing Eqs. (5.7) and (5.10) in matrix form to solve for the cutting coefficients the final equation can be expressed as shown in Eq. (5.11). cos212sin2 00 2sin2cos21 00 8 sincos1 00 1cossin 00 2exexex xc t exexex yc n xe te exex ye ne exexNb F K F K F K Nb F K (5.11) The same procedure can be used to solve for the cutting coefficients in the downmilling case (Figure 7). Experimental Procedure Proper selection of a suitable dynamometer to measure the dynamic cutting forces is important. Some of the factors that need to be addressed are the calibration range of the

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75 dynamometer and its dynamic response. Simulation of the cutting forces helps in addressing the issue of cutting force magnit ude range. Using time-domain simulation of the cutting forces and approximate cutting coe fficient values, an estimate of the typical range of cutting forces can be found. Euler integration is used to solve for the tool displacement during the cut in the 2nd order differential equation (Eq. (3.2)) and find the corresponding cutting forces in the x and y-directions. An example is shown in Figure 39. It is seen that a dynamometer with the 0 kN to 5 kN range is acceptable, although the force levels are relatively small compared to the full scale value. A Kistler 9257A dynamometer with 5 kN range was available for these tests. One requirement for this dynamometer is that the cutting force is ap plied to the dynamometer not more than 25 mm above the top surface of the dynamometer. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 -100 -50 0 50 Time (s)Fx (N) 0 0.1 0.2 0.3 0.4 0.5 -50 0 50 100 Time (s)Fy (N) Figure 39. Example simulation of cutting forc es to facilitate proper selection of dynamometer.

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76 A 25 mm thick 6061-T6 aluminum work-piece was sized to 100 mm x 85 mm, then faced and drilled to fit the dynamometer hole pattern as shown in Figure 40. Slotting cutting tests were made for a 25.4 mm diameter end mill with a 145 mm overhang and a single 12 helix insert for chip load range of 0.1-0.24 mm/tooth in 0.02 mm/tooth steps. The cutting forces in x and y-directions were measured for each chip load using an axial depth of 0.4 mm. Two sets of measurements were made for a 1000 rpm spindle speed. To address the influence of spindle speed on cutt ing coefficients, the above two sets were repeated for {5000, 10000, 15000 and 20000} rpm. The average value of the measured cutting forces was inserted into Eq. (5.4) to solve for the cutting coefficients. Average cutting coefficients of the two sets of meas urements at each spindle speed are listed in Table 8. Work-piece D y namomete r Figure 40. Work-piece, dynamometer and tool setup

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77 A regression analysis of the cutting force coefficients as a function of spindle speed was carried out. For Kt and Kn, a linear regression with logarithmic transformation of spindle speed indicates a stat istically significant relation with a P-Value of less than 0.007. Figure 41 and Figure 42 show the tr end line for this regression for both Kt and Kn, respectively. For the edge cutting force coefficients Kne and Kte the regression doesnt indicate a significant statistical relation between Kne or Kte and spindle speed. The PValue for the slope of the regressi on was 0.39 and 0.55, respectively. Table 8. Cutting coefficients for 1 insert endmill for slotting cutting tests (krpm) Kt (N/mm2 ) Kn (N/mm2) Kte (N/mm) Kne (N/mm) 1 1321 379 28 32 5 832 183 47 34 10 841 62 37 38 15 655 34 52 33 20 670 65 37 26 y = -504.17x + 1284.9 R2 adj = 0.91 0 200 400 600 800 1000 1200 1400 00.511.5 log10( (rpm) x10 3 )Kt (N/mm2) Figure 41. Cutting coefficient in tangential direction (Kt)

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78 y = -268.84x + 369.14 R2adj = 0.93 0 50 100 150 200 250 300 350 400 00.20.40.60.811.21.4 log10( rpm) x103)Kn (N/mm2) Figure 42. Cutting coefficient in normal direction (Kn) A similar set of measurements were ma de using partial radial immersion (up milling) for a 15000 rpm spindle speed. Equatio n (5.11) was used to find the cutting coefficients in this case. The results are provided in Table 9. Table 9. Up milling cutting coefficients for 12% radial immersion Kt( N/mm2) Kn( N/mm2) Kte( N/mm) Kne( N/mm) 833 431 6 8 To verify the fit, the cutting coeffici ents obtained were used in a time-domain simulation of the cutting forces. The measured forces were then ove rlaid on the simulated forces. Figure 43 shows a case for 0.12 mm/t ooth chip load and 1000 rpm. Also Figure 44 and Figure 45 show the fit for high er spindle speeds of 5000 and 20000 rpm, respectively.

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79 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -40 -20 0 20 40 60 80 Time (secs)Fx (N) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -50 0 50 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 43. Simulated and measured forces for 0.12 mm/tooth chip load and 1000 rpm. 0 0.02 0.04 0.06 0.08 0.1 0.12 0 50 100 Time (secs)Fx (N) 0 0.02 0.04 0.06 0.08 0.1 0.12 -80 -60 -40 -20 0 20 40 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 44. Simulated and measured cutti ng forces for 0.2mm/tooth chip load, b=0.4 mm and 5000 rpm.

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80 1.01 1.015 1.02 1.025 1.03 1.035 -20 0 20 40 60 80 100 Time (secs)Fx (N) 1.01 1.015 1.02 1.025 1.03 1.035 -50 0 50 Time (secs)Fy (N) Simulated force Measured force Simulated force Measured force Figure 45. Simulated and measur ed forces at 20 krpm and b=0.4 mm for slotting. Covariance Matrix (Linear Multi-Response Model) The regression analysis performed in th e previous section is a single response analysis. However, the measured re sponses are the forces in both the x and y-directions during a single measurement (dynamometer). Obviously this is a multi-response measurement. Therefore analysis of the da ta should take into consideration the multivariate nature of the data. The interrelationship existing between the variables could render univariate investigations meaningl ess. The development for a multi-response model follows the description in [94]. If we let Q be the number of experimental runs and r be the number of response variables measured for each setting (two in our case, i.e., Fx and Fy) of a group of variables (chi p load only in our case). The ith response model can be written in vector form as

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81 1,2,...,iiiiYZir (5.12) where Yi is an 1Qvector of observations in the ith response, Zi is an iQp matrix of rank pi (for the simple linear model pi = 2), i is a 1ip vector of unknown constant parameters, and i is an 1Q random error vector associated with ith response. The assumption of simple linear regr ession apply here, that is 0iE and iiiQVar I However, the covariance matrix be tween the responses is not zero, 1,2,..., ,1,2,..,;iiiQ ijijQVarir Covijrij I I (5.13) We denote the rr covariance matrix whose (i,j) th element is ,1,2,...,ijijr by For the case of two responses, Eq. (5.12) can be written in matrix form as: 01 111 121 11 02 222 121 12ZQQQ QQQYZ Y 0 0 (5.14) where 12 1 11Q Q Z Zc (5.15) where c represents the chip load vector (see Eq. (5.3)) and the left hand side vector of Eq. (5.14) represents the observed average cutting forces in the x and y-directions. From Eq. (5.13) it can be seen that has the following variance-covariance matrix, Var Q I (5.16) where is a symbol for the direct (or Kron ecker) product of matrices. The direct product of two matrices and Q I both of size rr gives an 22rr matrix which is

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82 partitioned as ij Q I where ij is the (i,j)the element of matrix The best linear unbiased estimate of is given by [95] 1 11 ZZZY (5.17) where Y is the left hand side of Eq. (5.14) The variance-covariance matrix of the estimated vector is 1 1 Var ZZ (5.18) Since is usually unknown, it is estimated using the following equation [95] 11 ,1,2,...,iNiiiiNjjjjj ijYY Q ijr IZZZZIZZZZ (5.19) It should be noted that ij is computed from the residual vectors which result from ordinary least-squares fit of the ith and jth single response models to their respective data sets. Using this estimate for in Eq.(5.19), an estimate of the variance of can be obtained. The cutting force coefficients ar e determined using a linear transformation KA (5.20) where the matrix A for slotting (see Eq. (5.4)) is 000 4 000 000 4 000 Nb Nb A Nb Nb (5.21)

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83 Therefore the variance-covari ance matrix of cutting force coefficients can be found as '. VarKAVarA (5.22) Using the procedure outlined above, the cutting force coefficients and their corresponding correlation matrices are calculat ed and listed in Table 10 for cutting tests carried out according to the same procedure de scribed earlier, noting that the correlation matrix is obtained directly from the covari ance matrix (see Eq. (4.5)) As indicated in Table 10, a high correlation between Kt and Kte and Kn and Kne is found. This high correlation is justified since both of th e corresponding cutting coefficients ( Kt and Kte or Kn and Kne) are obtained from the same regression fit and cutting force direction. However, a small correlation between the x and y -directions of the forces is found (between Kt and Kn or Kne) which may be due to experimental error. Table 10. Estimated cutting force coeffici ents and their correlation matrix for 7475 aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30 degree helix angle. Kt (N/m2)Kn (N/m2)Kte (N/m)Kne (N/m) Mean 8.41E+082.53E+081.27E+041.01E+04 Standard deviation 2.19E+072.66E+071.70E+032.07E+03 Coefficient of variation 0.030.110.130.20 P Value 2.E-088.E-053.E-043.E-03 Correlation Coeff. MatrixKneKnKteKt Kne 1.00 Kn -0.931.00 Kte -0.130.121.00 Kt 0.12-0.13-0.931.00

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84 Compliant Tool Modal Parameters The cutting tests were conducted on a Makino V55 vertical milling machining center located at Techsolve, an Ohio-bas ed not-for-profit ma nufacturing research organization. The cutting tool was a 12 .7 mm diameter solid carbide end mill (CRHEC500S4R30-KC610M) with 100 mm overall length ( OAL ), 70 mm over-hang length, 30 helix angle, and 4 flutes. A relatively long tool over-hang length was used in order to obtain a compliant t ool that could reasonably be modeled as single degree of freedom. Four measurements of the frequenc y response function of the tool (Figure 46) were made after running the spindle for 30 s econds at a specific spindle speed then completing a tap test in the xdirection, then running the spindle for another 30 seconds and taking a tap test in the ydirection, then removing the ho lder from the machine, and replacing and repeating the above procedur e for a different speed. This measurement procedure enabled the estimation of the vari ation of the modal parameters due to the spindle thermal effect and holder replacemen t effect. Figure 47 and Figure 48 show the frequency response measurement of the tool in the x and ydirections respectively. Table 11 lists the fitted tool modal parameters obtained by the peak amplitude method (see Milling Model section in Chapter 4) with thei r average and standard deviation, and Table 12 lists the correlation coefficient matrix The correlation coefficient between Mx and Kx, for example, is calculated according to 4 1 44 22 11ii xx iixxxx i MK xxxx iiMMKK r MMKK (5.23)

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85 The values shown in Table 12 indicate strong correlation coefficients for xx M Kr and y y M Kr. This is expected since the natural frequency of the tool is constant so that Kx or Ky depend entirely on Mx or My, respectively. Also, since the tool is symmetric the correlations for y x M Mr and y x K Kr are also expected to be high, which is the case for 0.86yxMMr As for the correlation coefficient, 0.75yxCCr some correlation is expected since the tool-holder interface damping is ideally symmetric for the round tool. The minimal correlation indicated between the damping and other modal parameters can be justified since there is no direct relationship between damping and mass or stiffness. The mean, standard deviation and the corr elation matrix are used to generate the random sample of input modal paramete rs in order to estimate uncertainty. Table 11. Tool modal parameters in x and y -directions. measurement stateM (kg)C (N.s/m)K (N/m) M (kg)C (N.s/m)K (N/m) static 0.0324.344.8E+060.0329.094.3E+06 5 krpm and replacement 0.0322.054.4E+060.0237.252.6E+06 10 krpm and replacement 0.0322.664.3E+060.0229.542.9E+06 20 krpm and replacement 0.0224.183.9E+060.0229.853.4E+06 mean 0.0323.314.4E+060.0231.433.3E+06 standard deviation 0.0020.9763.16E+050.0043.3686.60E+05 coefficient of variation (CV) 0.070.040.070.200.110.20 xy Table 12. Correlation coefficient matrix for modal parameters. Correlation CoefficientMxCxKxMyCyKy Mx 1.00 Cx 0.231.00 Kx 0.990.131.00 My 0.860.690.801.00 Cy -0.09-0.75-0.05-0.501.00 Ky 0.660.880.580.95-0.641.00

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86 Machine Spindle Instrumented Hammer Accelerometer Sensor interface Tool Holder Figure 46. Modal analysis test equipment t ypically used in machine tool structures. 0 500 1000 1500 2000 2500 3000 3500 4000 -2 -1 0 1 2 x 10-6 Real (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 -4 -3 -2 -1 0 x 10-6 Imag (m/N) Frequency (Hz) static 5 krpm and replacement 10 krpm and replacement 20 krpm and replacement Mean Figure 47. Frequency response function measurement of tool in x -direction. Four sets of measurements are made to estimate spindle thermal and holder replacement effects.

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87 0 500 1000 1500 2000 2500 3000 3500 4000 -2 -1 0 1 2 x 10-6 Real (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 -3 -2 -1 0 x 10-6 Imag (m/N)Freq. (Hz) static 5 krpm and replacement 10 krpm and replacement 20 krpm and replacement Mean Figure 48. Frequency response function measurement of tool in y -direction. Four sets of measurements are made to estimate spindle thermal and holder replacement effects. Stability Lobe Validation In this section the stability lobe diag ram for the 70 mm over-hang length compliant tool is verified. First the stability limit un certainty is predicted using Latin Hyper-Cube and Monte Carlo sampling, then a description of the experimental procedure is provided and results are discussed. Stability Lobe Uncertainty Due to the relatively high uncertainty in input parameters (Table 11), the sensitivity method cannot be used because of the nonlin earity of the axial depth limit to the respective parameters. The alternative random sampling methods (Latin Hyper-Cube and Monte Carlo) are used instead. The stability lobes are generated using TFEA. A random sample of size L=1000 is generated from the normal di stribution for each input parameter

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88 group (cutting force coefficients and m odal parameters) using their corresponding standard deviation, mean values and cova riance matrix. Latin Hyper-Cube sampling is used to generate the samples for the cutti ng force coefficients and modal parameters groups. Also, a random sample of the same size is generated for the radial immersion and spindle speed using Monte Carlo Simulation with no correlation assumed. This random sample of input parameters is used to generate the stability lobe diagram uncertainty intervals. Figure 49 shows the boxplot (a plot used to show variation and measures of central tendency for a sample) of axial dept h as a function of spindle speed. The grey boxes indicate the range of minimum a nd maximum axial depths, the black boxes indicate the lower 2.5 percen tile and upper 97.5 percentile (95 % confidence interval), while the two lines indicate the median and mean of the sample. It is seen that the median and mean lines do not match, which indicates the distribution is skewed. Examination of the histogram at selected spindle speeds (see Figure 50) validates this conclusion. At 10000 rpm the distribution appears close to normal, however, in checking the normality of the distribution at this speed (see Figure 51) we find that it is in fact not normal with a P-value of less than 0.005. This is illustrat ed by the deviation of the observations from the normal probability line.

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89 Omega (rpm x 10^3)b (mm) 20000 19000 18000 17000 16000 15000 14000 13000 12000 11000 10000 9 8 7 6 5 4 3 2 1 0 Figure 49. Boxplot of stability lobes bounda ry uncertainty. The minimum and maximum values are shown for each spindle speed (grey boxes), the mean and median of axial depth limit are indicated by the line and circles respectivel y, also shown is the 2.5 and 97.5 percen tiles (black boxes). Axial Depth Limit (mm)Number of Neutral Stability Observations 5 6 4 8 4 0 3 2 2 4 1 6 0 8 80 60 40 20 0 4 2 3 6 3 0 2 4 1 8 1 2 0 6 0 0 160 120 80 40 0 6 0 5 4 4 8 4 2 3 6 3 0 2 4 1 8 80 60 40 20 0 4 8 4 2 3 6 3 0 2 4 1 8 1 2 0 6 100 75 50 25 0 3 5 3 0 2 5 2 0 1 5 1 0 0 5 0 0 150 100 50 0 4 8 4 0 3 2 2 4 1 6 0 8 0 0 240 180 120 60 0 10000 12000 14000 16000 18000 20000 Mean3.054 StDev1.009 N1000 10000 Mean1.341 StDev0.6144 N1000 12000 Mean3.807 StDev0.8778 N1000 14000 Mean2.222 StDev0.8063 N1000 16000 Mean1.321 StDev0.5377 N1000 18000 Mean1.384 StDev0.7592 N1000 20000Histogram of Axial Depth Limit Distribution for Various Spindle Speeds Figure 50. Histograms of axial depth limit di stributions for various spindle speeds.

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90 Axial Depth Limit (mm)Percent 7 6 5 4 3 2 1 0 -1 99.99 99 95 80 50 20 5 1 0.01 Mean3.054 StDev1.009 N1000 AD3.835 P-Value<0.005Probability Plot for 10000 rpm Figure 51. Probability plot of axial depth limit distribution at 10000 rpm spindle speed. Experimental Procedure The stability lobes were verified experime ntally for 25% partial radial immersion down milling and 0.1 mm/tooth chip load. The same tool with modal tool parameters listed in Table 11 was used. A 7475 aluminum work-piece was mount ed (see Figure 52) to a Makino V55 vertical machining center tabl e. Cutting tests with different axial depths were conducted at a range of spindle speeds from 10000 rpm to 20000 rpm in 1000 rpm steps. The stability of each cutting oper ation was determined by recording the sound signal of the cut. The Fast Fourier Tran sform was used to transform the sound timedomain sound signal into the frequency domain. An analysis of the signal frequencies identified the chatter frequency, if one ex isted (i.e., significant content was seen at frequencies other than the runout and tooth passing frequencies) It was observed that the

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91 chatter frequency when it existed was always slightly higher than the tool natural frequency, as expected. Tool Machine Spindle Data Acquisition Stability test Workpiece x z y Figure 52. Schematic of stability tests for partial radial immers ion cutting conditions. Results The cutting test conditions are shown in Figure 53 with the boxplot of axial depth limit uncertainty. Also, the stab ility lobe boundary is overlaid using mean values of input parameters. In order to identify the stability of each cut, as noted previously, the sound signal was analyzed. In Figure 54 we can see some of these signals in the frequency domain. As noted previously, the chatter fre quency occurs near the natural frequency of the tool. The natural frequency of the tool is approximately 2000 Hz ( 1 / 2 fKM nyy meanmean, see Table 11). For 13000 rpm and a 1.52 mm axial depth, and 14000 rpm and 3.05 mm, for example, the chatter frequencies occur near 2100 and 2200 Hz, respectively. It should be noted that the chatter frequencies were difficult to

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92 identify when the tooth passing frequency or one of its harmonics are near the tool natural frequency. This is evident from the cutting test at 10000 rpm and 16000 rpm where there is high amplitude near the tool na tural frequency. In that case, examinations of the cut surface of the wor kpiece help in identifying chatter (due to the corresponding rough surface finish). In Figure 53, the stability of the cutting conditions agreed well with the median of stability prediction almost ev erywhere along the spindle spee d. However, near fractions of the tool natural frequency (60/1530000 rpmnnfNf ), poor agreement between prediction and experimental result is observed. This may be attributed to confidence in the modal fitting near the natural frequency of the tool 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 b (mm) (rpm x 103) Stable Unstable Marginal Stability lobe, mean parameters Figure 53. Stability lobe generated using mean values of input parameters with experimental results overlaid, also shown the boxplot co rresponding to each spindle speed used in the measurements.

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93 1800 1900 2000 2100 2200 2300 0 10 20 30 Magnitude20000 rpm 1800 1900 2000 2100 2200 2300 0 5 10 15 16000 rpm 1500 2000 2500 0 1 2 3 4 15000 rpmMagnitude 1500 2000 2500 0 5 10 15 20 25 14000 rpm 1800 1900 2000 2100 2200 2300 0 10 20 30 40 13000 rpm Frequency (Hz)Magnitude 1500 2000 2500 0 1 2 3 4 10000 rpm Frequency (Hz) 4.06 mm, stable 3.56 mm, stable 5.08 mm, stable 1.02 mm, stable 1.52 mm, chatter 2.03 mm, chatter 2.03 mm, stable 3.05 mm, chatter 2.54 mm, marginal 6.1 mm, stable 7.11 mm, stable 3.56 mm, stable 4.06 mm, stable 1.02 mm, stable 1.52 mm, chatter Figure 54. Fast Fourier Transform (FFT) of so und signals for selected stability tests. Pareto Front Validation This section begins with the calculation of the Pareto Front for a specific single degree of freedom tool considering conf idence levels in the axial depth limit, blim, (see Robust Optimization section in Chapter 4), after which the ex perimental procedure of the tests is described, followed by the results. Pareto Front Simulation Results The Pareto front for SLE and MRR is generated for the same material (7475 aluminum) and tool used in the stability tests (Table 11 and Table 12) and the same cutting conditions of radial immersion and chip load. Two cases are considered: 1) no uncertainty in the input parameters; and 2) uncertainty in input parameters or axial depth limit, blim, where an uncertainty of limcUub is used. The robust optimization algorithm (Chapter 4) is used to generate the Pareto front for the input parameters

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94 uncertainty case. Figure 55 and Figure 56 illustrate the Pareto front for the aforementioned cases. It can be seen that the uncertainty designs predict higher SLE for the same MRR compared to the mean value one. Also, in Figure 56 the knee in both Pareto Fronts indicate the design beyond which the SLE increases at a higher rate, which makes that knee a preferred design poin t. In considering Figure 56, the SLE difference between uncertainty and without uncertainty case s is larger at higher MRR than at lower MRR This is attributed to the fact that as higher MRR is required, the axial depth, b, approaches blim. Here, the predicted uncertainty in blim changes the design variables ( b or ) substantially to account for th e uncertainty. This penalizes the SLE for the uncertainty case and makes it significantly larg er than the no uncertainty case at higher MRR. However, at lower MRR the axial depth b is far from blim and is therefore less affected by uc(blim) 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 (x 103)b (mm)2 0 04 0 06 0 08 0 0 1 0 0 01 2 0 0 1 4 0 0 MRR (mm3/s) No uncertainty Uncertainty uc(b) Stability using Mean Parameters 5.4 m 85.7 m 46 m 3.8 m 2.1 m Figure 55. Stability boundary using mean values in the input parameters Pareto optimal designs are overlaid for two cases: mean values and uncertain input parameters.

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95 0 200 400 600 800 1000 1200 1400 0 10 20 30 40 50 60 70 80 90 MRR (mm3/s)perturbed average SLE (m) Uncertainty uc(b) Uncertainty not considered Figure 56. Pareto Front of perturbed average SLE and MRR The Pareto Front with uncertainty in axial depth is compar ed to the one with no uncertainty. Experimental Procedure and Results As a first step in conducting the surface lo cation error tests, the work-piece was machined to a specified dimension (nominally 40 mm width) usi ng shallow axial depth slotting cuts (see Figure 57). Careful atten tion was paid to minimizing positioning errors of the machine by feeding from the same di rection prior to cutting (i.e., minimize the influence of reversal errors). The cutting conditions of tw o mean value Pareto optimal designs were selected from Figure 56 for th e case of no uncertainty. The first point corresponds to the knee design poi nt in the figure and the second point corresponds to the maximum MRR for the case that uncertainty was not considered. For these two design conditions of axial depth and spindle speed, f our additional cuts were made for each case by varying the spindle speed around the sel ected design (see Table 13). The purpose of these extra cuts was to check the sensitivity of the stability and surface location error to

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96 spindle speed. The work-piece webs, shown in Figure 57, were milled from both sides. A coordinate measuring machine (CMM) was used to measure the base of the web (dimension a ) and the top portion (dimension b ). The measured surface location error was then taken to be 3.175 mm, 2 ab SLE where the commanded radial depth was 3.175 mm. Each dimension ( a and b ) was measured 15 times in order to evaluate the CMM machine measurement repeatability. The 15 measurements had a maximum standard deviation of2 m (the machine accuracy is estimated at < 5 m based on recent calibration tests). The measured SLE for the set of cutting tests is shown in Figure 58 and Figure 59. It should be noted here that all cuts were stable. Therefore, all SLE results are shown in the figures. The error of the reference dimension ( a=40 mm) is also shown in these figures. This would identify if there is a trend in the measured SLE due to the errors in the reference dimension. The standard deviation of the reference dimension is 4 m and 8 m for 4.45 mm and 2.12 mm axia l depth cuts, respectively. To illustrate the effect of the helix angle of the tool (30) on the SLE the CMM measurement was repeated for distances of {1, 2, 3, and 3.4} mm from the top surface of the work-piece web along the tool axis. Figure 61 shows that the SLE varies along the axial depth of the cut. This va riation corresponds with previous SLE studies [40] where similar variation of SLE was observed. Although the measured SLE does not agree well with the mean predicted value (Figure 62), there is some agreem ent in the trend of median of SLE (Figure 60) and the measured SLE (Figure 59). Also the measured SLE is within the uncertainty bounds on SLE (Figure 60). The disagreement betw een the measured and predicted SLE can be attributed to: 1) the milling model used in th e prediction assumed straight cutter teeth (the

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97 actual tool had a 30 helix angle) whic h would yield higher SLE ; 2) the cutting force coefficients used in the prediction were measured for 8900 rpm, while the SLE cuts were made for around 15000 rpm. At higher spindle speeds the cutting fo rce coefficients (cutting forces) tend toward lower values. Th ese two factors explai n the high prediction of SLE for the 1400 mm3/s case relative to the meas ured one. However, for 700 mm3/s case they fail to explain the difference. This may highlight model w eaknesses at this level of axial depth (2.12 mm). Tool Machine Spindle Surface Location Error Test Workpiece x z y 40 mm 3.175 mm 3.175 mm A B 2.12 or 4.45 mm Aluminum 7475 Figure 57. Surface location e rror experiment schematic.

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98 15.5 15.55 15.6 15.65 15.7 15.75 15.8 35 40 45 50 55 60 b = 2.12 mmSLE (m) 15.5 15.55 15.6 15.65 15.7 15.75 15.8 10 20 30 40 mError in reference dimension A (rpm x103) Predicted 1.2 m Figure 58. Measured surface location error of b=2.12 mm and the reference dimension (A) error. 14.75 14.8 14.85 14.9 14.95 15 15.05 0 20 40 SLE (m) b = 4.45 mm 14.75 14.8 14.85 14.9 14.95 15 15.05 5 10 15 20 Error in reference dimension Am (rpm x103) 2 mm 1 mm 2 mm 3 mm 3.4 mm Predicted 93.4 m Figure 59. Measured surface location error of b=4.45 mm and the reference dimension (A) error.

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99 14.753 14.803 14.853 14.903 14.953 20 40 60 80 100 120 140 SLE (m) (rpm x103) Measured SLE Figure 60. Boxplot of SLE uncertainty at spindle speeds for 4.45 mm axial depth case. The upper tail of the SLE uncertainty is not present due to the undefined SLE in the unstable region. 1 1.5 2 2.5 3 3.5 5 10 15 20 25 30 35 40 45 50 55 CMM measurement location (mm)SLE (m) 14753 rpm 14803 rpm 14853 rpm 14903 rpm 14953 rpm Figure 61. Measured surface location erro r of b=4.45 mm case. The CMM probe measurement is repeated along the axial depth of the tool with {1, 2, 3, and 3.4} mm from the top surface of the work-piece web.

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100 Table 13. Surface location error cutting conditi ons for two Pareto optimal designs with no uncertainty considered. Tool # (100 mm OAL) Cut No.b (mm) (rpm) MRR (mm3/s) SLE (m) 12.1215617 22.1215667 32.1215767 42.1215567 52.1215517 64.4514853 74.4514803 84.4514753 94.4514903 104.4514953 CRHEC500S4R30-KC610M 7001.2 93.4 1400 10 11 12 13 14 15 16 17 18 19 20 -20 0 20 40 60 80 100 120 X: 15.6 Y: 1.761 SLE ( m) (x 103 rpm) X: 14.8 Y: 93.42 b = 2.12 mm b = 4.45 mm Figure 62. Surface location error of prefe rred design conditions with no uncertainty considered in the optimization. Optimu m spindle speeds are indicated in the figure.

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101 Conclusions The uncertainty in axial depth limit, blim, and SLE indicates a non-normal distribution, although for conve nience a normal distribution was used to estimate the confidence levels of blim and SLE This non-normality predicates the use of different measures of uncertainty bounds in order to a ccount for a specific confidence interval. There is good agreement between the predic tion of stability and the experimental results. It is shown that th ere is a distinct grey region of neutral stability boundary (marginal stability) rather th an a black and white step change between stable and unstable zones as suggested by the single stability boundary typi cally indicated in stability lobe diagrams. Also, the uncertainty identified in the blim boxplot indicates that the distribution is skewed to higher blim values near the tool system natural frequency. This was also confirmed by the experimental results where higher axial depths were generally feasible. The measured surface location error of th e Pareto design points didnt show high sensitivity to spindle speed variation. Th is shows the validity of the optimization algorithm selection of a design that mitigates the effect of spindle speed on SLE. In this worst case scenario of uncertainty in modal parameters (thermal effects and dynamic variations due to tool removal and replacement), there was substantial uncertainty seen in blim and SLE. Reduction of uncertainty in the input parameters may be necessary to fully realize the benefits of high-speed milling. This may be done by conducting more tests to lessen the uncertainty bounds and/or completing modal tests of the tool-holder assembly each time it is removed and replaced. Although no experimental verification of robust optimum designs was done, it is interesting to note that the predicted robust optimum design corresponding to 700 mm3/s

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102 had approximately the same value of measured SLE (Figure 55 and Figure 56). This is due to the fact that the robust design select ed a design with spindl e speed that was 1000 rpm lower than the measured one (with no uncertainty considered). Also, in considering uncertainty (see Figure 55), the 1400 mm3/s was not realizable. This may be due to an overestimation of the uncertainty in the input parameters. As mentioned previously, this requires more testing in order to better esti mate the uncertainty of input parameters.

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103 CHAPTER 6 SUMMARY This chapter provides a summary of the work completed in this dissertation with a detailed procedure on how to implement the robust optimization algorithm. Then the limitations of the milling model are addressed with suggestions for future research. Robust Optimization Algorithm In order to implement the robust optimi zation algorithm for a specific compliant tool/work-piece system, the followi ng steps should be taken: 1. Measure the tool/work-piece frequency response and complete a modal fit to the measured response. The confidence levels in the fitted modal parameters are estimated by repeating the measurement at different thermal states of the machining spindle. In case this measurement cannot be repeated each time the tool/holder assembly is removed from the spindle, then several measurements should be made wherein the tool/holder assembly is rem oved from the spindle and replaced. This will account for the dynamic non-repeatability due to tool/holder replacement. The mean, standard deviation, and correla tion between the modal parameters are calculated. Equation Chapter 6 Section 1 2. Measure the cutting force coefficients fo r the tool/workpiece material. Chapter 5 gives the procedure used in estimating the mean values of these coefficients and details the regression analys is needed to estimate the mean values, standard deviation, and correlation between these coefficients. 3. The confidence levels in the spindle speed and radial depth can be either estimated from experience or machine manufacturer data. 4. These steps enable estimation of the mean values, confidence levels (standard deviation), and correlation in the input parameters to the milling model. Depending on the confidence levels in these paramete rs, an uncertainty prediction method can be used to estimate the confidence le vels in the stability boundary and SLE Two methods can be used: 1) the sensitivit y method; 2) the sampling methods (Monte Carlo and Latin Hyper-Cube). If the coeffici ent of variation in the input parameters (especially K or M ) is larger than 1%, then the sensitivity method cannot be used due to the non-linearity of axial depth limit to these parameters. Chapter 5 gives an

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104 example on how to use the sampling methods to assign confidence levels on the stability boundary and SLE. 5. The stability boundary confidence level, U obtained in step 4 is used in the robust optimization algorithm. The algorithm formulation is repeated here ,,, 3 : ,, 1 ... ,,,1,MRRifbfbfb SLESLESLEmin subjecttofbeforik gbUgbUgbU for a series of s ,MRRelected limits (e) on f (6.1) The spindle speed perturbed SLE average is used to account for SLE sensitivity to spindle speed. A typica l value for the perturbation is 50 rpm to 100 rpm In order to calculate the trade-off curve between SLE and MRR the optimization algorithm is run for a series of limits on the MRR objective. 6. The trade-off curve is used to select optimum cutting conditions that match the designer preferences. Typically a knee in the curve would indi cate a preferential design where the highest MRR can be achieved for a moderate SLE This completes the description of the selection of robust cutting conditions. Limitations and Future Research In this section, the limitations of this research are discussed as well as the recommendations for future research. Th e limitations and recommendations are as follows: 1. Further efforts should account for the pot ential variation of the cutting force coefficients as a function of spindle sp eed. This entails a significant amount of experimental testing. 2. The peak amplitude method used to obtai n the fitted modal parameters of the tool/workpiece system does not perform we ll near the system natural frequency. This makes the model predictions poor at regions where more accuracy is actually needed. 3. A weakness in the solution method ( TFEA ) was observed at shallow axial depths of cut (2 mm in our tests). Further testing at this condition is needed to verify this discrepancy and account for it.

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105 4. The solution method ( TFEA ) assumes straight cutter teet h while most cutters have a helix angle. This makes the model pred ictions more conservative and can over predict the SLE

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106 APPENDIX A TIME FINITE ELEMENT ANALYSIS Mechanical Model A schematic diagram of two degree of freedom milling process is shown in Figure 1 (repeated here). With the assumption of either a compliant structure or tool, a summation of forces gives the following equation of motion: Equation Chapter 1 Section 1 k y c y k x c x x y Figure 1. Schematic of 2-DOF milling tool 00 0() ()()() +, 00 0() ()()() mc kFt xtxtxt xx xx mc kFt ytytyt yy yy (A.1)

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107 where the terms mx,y, cx,y, kx,y and Fx,y are the modal mass, damping, spring stiffness, and cutting forces in the flexible directions of the system. The x and y cutting force components on the pth tooth are given by: () cos()sin() () (), () ()sin()cos() Ft tt Ft xppp tp gt p Ft Fttt np pp yp (A.2) where gp(t) acts as a switching function It is equal to one if the pth tooth is active and zero if it is not cutting [54] The tangential and normal cutting force components, Ftp(t) and Fnp(t), respectively, are considered to be the product of linearized cutting coefficients Kt and Kn, the nominal depth of cut b and the instantaneous chip thickness wp(t): () (), ()tp tte p np nneFt KK bwtb Ft KK (A.3) where wp(t) depends upon the feed per tooth, h the cutter rotation angle p, and regeneration in the compliant structure directions: ()sin()()()sin()()()cos(). wthtxtxttytytt p ppp (A.4) Here = 60/N is the tooth passing period, is the spindle speed given in rpm, h is chip load (used instead of c to differentiate it from cosp defined later) and N is the total number of cutting teeth. The angular position of the pth tooth for a cutter with evenly spaced teeth is p(t)=(2 /60)t+ p 2 / N The total cutting force equations are found by summing the forces on each cutting tooth in Eq. (A.2) and substituting Eqs. (A.3) and (A.4) into Eq. (A.2):

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108 2 2 () () 22 () ()() ()() 22tene teneKscKs KcKs n t h KsKc Ft KsKsc x n t gtb p Ft KscKsKcKsc y xtxt nn tt ytyt KsKscKscKc nn tt 1 N p (A.5) where s = sin p(t) and c = cos p(t). A more compact form fo r the equation of motion is realized by making the following substitutions: 22 ()() 22 1 KscKsKcKsc N nn tt tgt c p p KsKscKscKc nn tt K (A.6) 2 ()() 2 1tene teneKscKs N KcKs n t ftgth o p KsKc p KsKsc n t (A.7) Using Eqs. (A.5), (A.6) and (A.7) Eq.(A.1) can be written as: ()()()()()() X tXtXttbXtXt+ftb o MCKK c (A.8) where T X txtyt is the two-element position vector and M, C, and K are the 22 x mass, damping, and stiffness matrices of Eq.(A.1). Time Finite Element Analysis (TFEA) The dynamic behavior of the milling proce ss is governed by Eq.(A.8). Since this equation does not have a closed form solu tion, an approximate solution is sought to understand the behavior of the system. On e such approximation technique used for dynamic systems is TFEA [54]. An approximate discrete linear map is constructed using time finite elements in the cut to exact ma pping of free vibration out of the cut, where mapping is performed on displacement and ve locity components of vibration [54-57]. The formulation of the dynamic map for the multiple degree of freedom systems closely

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109 follows the discretization procedure outlined in references [54], but has been presented here for completeness. Free Vibration When the tool is not in contact with th e work-piece, the system is governed by the equation for free vibration. The cutt ing forces then become zero: ()()() XtXtXt MCK0, (A.9) and the exact solution for the free vibration can be written with a state transition matrix ,fccttt where ct is that the time the tool leaves the material and ft is the duration of free vibration. Exact mapp ing of displacement and veloci ty components can be written in terms of state transition matrix as: Xtt c Xt f c ttt cc f Xt c Xtt c f (A.10) Vibration during Cutting When the tool is in the cut, its motion is governed by a time delayed-differential equation. Since this equation does not have a closed form solution, an approximate solution for the tool displacement is assumed for the jth element of the nth tooth passage as a linear combination of polynomials [54]: 4 () 1n X tat jiij i (A.11) Here 1 1j ttnt j k k is the local time within the jth element of the nth period, the length of the kth element is tk and the trial functions i( j(t)) are cubic Hermite polynomials defined in Eq. (A.12),

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110 23 132 1 23 2 2 23 32 3 23 2 4jj j tt jj jjj t jj ttt jjj jj j tt jj jj t jj tt jj (A.12) Substitution of the assumed solution of Eq. (A.11) into the equation of motion (Eq. (A.8)) leads to a non-zero erro r. The error from the assume d solution is weighted by multiplying by a set of test functions and setting the integral of the weighted error to zero. Two test functions are ch osen to be a constant 1( j)=1 and 2( j)= j/tj-1/2 (linear). The integral is taken over the time for each element, tj=tc/E, thereby dividing the time in the cut tc into E elements. The resulting two equations are 44 i=1i=1 4 i=1 0 0 4 1 1 nn aa pp jiijjjiijj n ba cp t jjiijj j d j n ba cp jjiijj i bf op jj M+C KK K p=1,2 (A.13)

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111 where Kc( j) and f o j have been used in place of previously defined Kc(t) and f ot to explicitly show the dependence on local time. In Eq. (A.13), the index j refers to the number of elements and the index i refers to the corresponding Hermite polynomial. The displacement and velocity at tool entry into the cut are specified by the coefficients of the first two basis functions on the first element: 11 n a and 12 n a The relationship between the ini tial and final conditions during free vibration can be mapped through th e state transiti on matrix as ,1 3 11 12 4n n a a E a a E (A.14) where E is the total number of finite elements in the cut and the last term in Eq. (A.14) is the displacement and velocity of the element as it leaves the cut. For the remainder of the elements in the cut, a continuity constraint is imposed to set the position and velocity at the end of one element (13 a and 14 a for the 1st element) equal to the position and velocity at the beginning of the next element (21 a and 23 a for the 2nd element), see Figure 63.

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112 a 11 a 12 a 13 = a 21 a 14 = a 22 cutting zon e a 23 a 24 element 1 element 2 free vibration zone a ji j refers to number of elements i refers to Hermite coefficient Figure 63. Slotting cut with time in the cut divided into two elements. Transition matrix maps the position and velocity exactly in free vibration zone, while elements map them in cutting zone. Equations (A.13) and (A.14) can be arranged into a global matrix mapping the position and velocity of each tooth passage in terms of the previous one. Equation (A.13) maps the cutting zone approximately, while Eq. (A.14) maps the free vibration zone exactly. The following expression is for the case when number of elements E = 3

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113 1111 12 000000 21 1111 0000 1212 22 2222 0000 31 1212 3333 32 0000 1212 33 34 n aa a II a NNPP a a NNPP a NNPP a a ,1 0 0 12 1 21 2 22 1 31 2 32 1 33 2 34 n a C a C a C a C a C a C a (A.15) where the sub-matrices and elements of the sub-matrices for the jth element are j j NN NN jj 1314 1112 N = N= 12 NNNN 21222324 j j jj 1314 1112 = = 12 21222324 PP PP PP PPPP (A.16) 0 -b c -b c 0 0 t ijij j j Nd p pijj jij t j j Pd p pijijjj t j j Cbfd pop jjj MC KK K (A.17) Equation (A.15) describes a discrete dynamical system, or map, that can be written as A=B+ -1 or =Q+ -1 aaC n n aaD n n (A.18)

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114 Stability Prediction The eigenvalues of the transition matrix Q=A-1B are called characteristic multipliers (CMs) and take on a discrete ma pping analogy to the characteristic exponents that govern stability for continuous systems. The condition for stability is that the magnitudes of the CMs must be in a modulus of less than one for a given spindle speed () and depth of cut (b) for the milling process to be asymptotically stable. Surface Location Error Surface location error is defined as the e rror in the placement of the milling cutter teeth when the surface is generated. When the milling process is stable, the surface location error can be obtained by extracting th e position of the tool when the surface is generated. At steady state, the displacement a nd velocity coefficients are constant and are found from fixed points (* a n ) of the dynamic map: 1 aaa nn n (A.19) Substitution of Eq. (A.19) into Eq. (A.18) gives the fixed point map solution or steady state coefficient vector: -1 =-D a nIQ (A.20) Since Q and D can be computed for the milling parameters, the fixed point displacement solution can be found and used to specify surface location error as a function of machining process parameters.

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115 APPENDIX B MATLAB CODE Robust Optimization Code Main program % M. Kurdi (12/1/2004) % surface location error and MRR robust optimization clc; clear all; close all; pack; global Min_speed Max_speed Min_ depth Max_depth MRR_c band; warning off all; band =500; Min_speed = 10e3; Max_speed = 20e3; Min_depth = 1e-6; Max_depth = 18e-3; nteeth = 1; tic % % %%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% % % Finding the initial guess % %%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % % MRR_c MRR constraint in mm^3/s % Max_MRR = 400; hand = waitbar(0,'Please wait'); for MRR_c = 100:100:Max_MRR delete('MULTIPOINT OPTIMUMS.m') fid2 = fopen('MULTIPOINT OPTIMUMS.m','a'); speed_vec = 0:.1 :1.0; % spindle speed radial_vec = linspace(.03,1,11); % radial depth for i=1:length(speed_vec) % loop for spindle speed x0(2) = speed_vec(i); for j=1:length(radi al_vec) % loop for radial depth x0(3) = radial_vec(j); % figure % contour_plot; % obj_mrr; % hold on; a = x0(3) 25.4e-3; % radial depth of cut h = 0.1e-3; % feed per tooth % calculate initial depth, using MRR constraint and initial speed rpm = x0(2) (Max_speed Min_speed) + Min_speed;

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116 b = MRR_c / (a rpm h nteeth / 60 1e9); disp('axial depth(m) speed (rpm) radial immersion (m)'); design_point = [num2str(b),' ',num2str(rpm),' ',num2str(a)]; disp(design_point); x0(1) = (b Min_depth) / (Ma x_depth Min_depth); lb = [0 0 0.01]; ub = [1 1 1]; options = optimset('LargeSc ale','off','MaxFunEvals',50 );%,'Display','iter'); %fprintf(fid,'SLE_depth depth speed\n'); [x,fval,EXITFLAG] = fm incon(@obj,x0,[],[],[],[], lb,ub,@confun,options); [c,ceq] = confun(x); if EXITFLAG > 0 % solution found depth = x( 1) (Max_depth Mi n_depth) + Min_depth; rpm = x(2) (Max_speed Min_speed) + Min_speed; a = x(3); sle_exact = sle([rpm depth a]); % multipoint optimum file fprintf(fid2,'%e %e %e %e %e %e %e %e %e\n',depth,rpm,x(3),fva l,sle_exact,c(1),c(2),c(3),c(4)); end % end if loop end % end radial loop end % end spindle speed loop % finding minimu m of all solutions found fclose(fid2); fid2 = fopen('MULTIPOINT OPTIMUMS.m','r'); xx = fscanf(fid2,'%e %e %e %e %e %e %e %e %e\n',[9 inf]); xx = xx'; fclose(fid2); % find minimum value of sle_depth for the ra nge of speed initial % guesses and a particular MRR [minimum_sle,index]=min(xx(:,4)); fid3 = fopen('OPTIMUM POINTS.m','a'); % Optimum points file fprintf(fid3,'%e %e %e %e %e %e %e %e %e %e\n',MRR_c,xx(index,1),xx(index,2),xx( index,3),xx(index,4),xx(index,5),xx(ind ex,6),xx(index,7),xx(index,8),xx(index,9)); fclose(fid3); waitbar(MRR_c/Max_MRR,hand); end % end of MRR loop fclose(hand) Constraint Function % objective function for SLE / depth of cut function [c, ceq] = confun(x) global Min_speed Max_speed Min_ depth Max_depth MRR_c band;

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117 x1 = x(1) (Max_depth Min_depth) + Min_depth; x2 = x(2) (Max_speed Min_speed) + Min_speed; % MRR constraint on the first perturbed point c1 = confun1([x1 x2-band x(3)]); c2 = confun1([x1 x2 x(3)]); c3 = confun1([x1 x2+band x(3)]); % MRR constraint h = 0.1e-3; % m/tooth % b = x(1) ; % x(2) rpm % x(3) radial step a = x(3)*.0254; % radial depth in m nteeth = 1; MRR = a .* x1 h nteeth .* x2 / 60 1e9; c4 = MRR_c MRR; c = [c1 c2 c3 c4]; ceq = []; % objective function for SLE / depth of cut function c = confun1(x) % Input: % rpm speed (rpm) % E number of elements % Output: % CM eigen value for rpm and doc % b transition depth of cut (m) % b = x(1); rpm = x(2); E = 25; % number of elements Kt = 1295.9e6*(rpm/1000)^-0.2285; % N/m2 Kn = ((rpm/1000)^2*1.8485-54.604*(rpm/1000)+423.77)*1e6; Kte = ((rpm/1000)^2*-0.1335+3.2431*(rpm/1000)+27.216)*1e3; % N/m Kne = ((rpm/1000)^2*-0.0821+1.4447*(rpm/1000)+30.202)*1e3; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % CUT PROCESS DESCRI PTION GEOMETRY/IMMERSION/PROCESS PARAMETERS %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% h = 0.1e-3; % feed per tooth nteeth = 1; % number of teeth Diam = 1; % inches rstep = x(3); % radial immersion (inches) TRAVang = acos(1-rste p/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth

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118 rho = acos(1-rstep/(Diam/2))/(2*pi ); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'up'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % load SYSTEM_ID_1mode Kx = 4.4528e+006; Ky = 3.5542e+006; Mx = 0.4362; My = 0.3471; Cx = 82.5955; Cy = 89.8606; % zeta_x = .02996; zeta_y = 0.02576; % freq_x = 362.75; freq_y = 362.71; % Kx = 1.308e6; Ky = 1.194e6; % Mx = Kx/(freq_x*2*pi)^2; % My = Ky/(freq_y*2*pi)^2; % Cx = zeta_x 2 Mx 2*pi freq_x; % Cy = zeta_y 2 My 2*pi freq_y; M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); lmx=1; lmy=1; DOF=2; Mx=M(1,1); My=M(2,2); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut

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119 tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER ROTA TION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF-DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros(s ize(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos( 2*t0*omega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj-

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120 8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+ 2*omega*tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)-

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121 6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj

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122 )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)-

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123 15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^ 5-180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5;

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124 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)-

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125 10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome

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126 ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P 11(1,2)*ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P 12(1,2)*ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P 13(1,2)*ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P 14(1,2)*ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P 21(1,2)*ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P 22(1,2)*ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P 23(1,2)*ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P 24(1,2)*ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES

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127 %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; c = CM 1; return %save TFEA_STABSLE_LOW ss zz CM ee IMMERSION SLE % NOTES % ss spindle speeds % zz depth of cut % CM charistic multipliers or eigenvalues % SLEsurface location error Objective Function % % average surface location error % % Input: % x(1) depth % x(2) speed % x(3) radial function SLE_AVER = obj(x) global Min_speed Max_speed Min_ depth Max_depth MRR_c band; x1 = x(1) (Max_depth Min_depth) + Min_depth; x2 = x(2) (Max_speed Min_speed) + Min_speed; % MRR constraint on the first perturbed point sle1 = obj1([x1, x2-band ,x(3)]); sle2 = obj1([x1, x2, x(3)]); sle3 = obj1([x1, x2+band ,x(3)]); SLE_AVER = (sle1+sle2+sle3)/3;

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128 % objective function for SLE function SLE = obj1(x) % Input: % x(1) depth % x(2) speed % x(3) radial b = x(1); rpm = x(2); rstep = x(3); % radial immersion (inches) E = 25; % number of elements % adjust cutting coefficients to spindle speed Kt = 1295.9e6*(rpm/1000)^-0.2285; % N/m2 Kn = ((rpm/1000)^2*1.8485-54.604*(rpm/1000)+423.77)*1e6; Kte = ((rpm/1000)^2*-0.1335+3.2431*(rpm/1000)+27.216)*1e3; % N/m Kne = ((rpm/1000)^2*-0.0821+1.4447*(rpm/1000)+30.202)*1e3; %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%% % CUT PROCESS DESC RIPTION GEOMETRY/IMMERSION/PROCESS PARAMETERS %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% h = 0.1e-3; % feed per tooth nteeth = 1; % number of teeth Diam = 1; % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'up'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % load SYSTEM_ID_1mode Kx = 4.4528e+006; Ky = 3.5542e+006; Mx = 0.4362; My = 0.3471; Cx = 82.5955;

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129 Cy = 89.8606; % zeta_x = .02996; zeta_y = 0.02576; % freq_x = 362.75; freq_y = 362.71; % Kx = 1.308e6; Ky = 1.194e6; % Mx = Kx/(freq_x*2*pi)^2; % My = Ky/(freq_y*2*pi)^2; % Cx = zeta_x 2 Mx 2*pi freq_x; % Cy = zeta_y 2 My 2*pi freq_y; M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); lmx=1; lmy=1; DOF=2; Mx=M(1,1); My=M(2,2); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER ROTA TION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOFDOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%

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130 % STATE TRANSITION MATRIX %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros(s ize(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos(2*t0*o mega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2];

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131 P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega* tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)-

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132 6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)-

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133 6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*o mega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-

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134 10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)-

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135 120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*co s(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)-

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136 60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos( 2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5];

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137 P11 = [P11(1,1)*ones(lmx,1) P11(1,2) *ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P12(1,2) *ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P13(1,2) *ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P14(1,2) *ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P21(1,2) *ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P22(1,2) *ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P23(1,2) *ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P24(1,2) *ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop Q = A\B;

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138 [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = abs(sum(SLE_vec(locat))); else SLE = 100; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; return; Uncertainty Analysis Code Stability Uncertainty, Sensitivity Method % M. Kurdi (1/26/2005) % Function to find uncertainty in ax ial depth to change in cutting % coefficients, dynamic parameters and cutting process variables % Input: % b: depth of cut (m) % rpm: spindle speed % rstep: radial step (inches) % Kt % Kn % Kte % Kre % DELTA_Kt finite change in Kt % DELTA_b finite change in b % system_ID: Modal parameters % The derivative of Max eigen value is found for a miniscule perturbation % in input parameters, then its effect on the change of axial depth is % found. clear all; close all; clc;tic; % function uncer % percentage of uncertainty in cutting coefficients, dynamic parameters % and process parameters % tic; percent_Kcut = 0.05; % cutt ing coefficents uncertainty percent_Dyn = 0.005; % modal parameters uncertainty percent_rstep = 0.0001; % ra dial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % nominal values of process paramete rs and their calculated uncertainty rstep = 0.2;

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139 rpm_vec = 10000:200:30000; DELTA_rstep = percent_rstep*rstep; % cutting coefficient uncertainty Kt = 6e8; DELTA_Kt = percent_Kcut*Kt; Kn = .3*Kt; DELTA_Kn = percent_Kcut*Kn; Kte=0; DELTA_Kte = percent_Kcut*Kte; Kne=0; DELTA_Kne = percent_Kcut*Kne; % nominal values of dynamic parameters and their calcul ated uncertainty Kx = 4.4528e+006; Mx = 0.4362; Cx = 83; % Y direction parameters Ky = 3.5542e+006; My = 0.4362; Cy = 89.9; DELTA_Mx = Mx*percent_Dyn; DELTA_My = My*percent_Dyn; DELTA_Kx = Kx*percent_Dyn; DELTA_Ky = Ky*percent_Dyn; DELTA_Cx = Cx*percent_Dyn; DELTA_Cy = Cy*percent_Dyn; % to calculate the numerical deriva tive with respect to each input % variable set the miniscule change in each input % set miniscule change in input para meters to estimate the derivative step_percent = 0.002; dKt = step_percent*Kt; % N/m2 dKn = step_percent*Kn; % N/m2 dKte = step_percent*30; % N/m dKne = step_percent*30; % N/m drstep = step_percen t*rstep; % inch dKx = step_percent*Kx; % N/m dKy = step_percent*Ky; % N/m dCx = step_percent*Cx; % dCy = step_percent*Cy; dMx = step_percent*Mx; % Kg dMy = step_percent*My; % Kg h = waitbar(0,'Please wait...'); % computation here % for i=1:length(rpm_vec) waitbar(i/length(rpm_vec),h); rpm = rpm_vec(i); drpm = step_percent*rpm; % rpm

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140 DELTA_rpm = percent_rpm rpm; % Find depth of cut correspondi ng to stability bounda ry using nominal % settings of input parameters [b(i)] = bisection(rpm,rstep,Kt,K n,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % depth at boundary % Find numerical derivative of ma ximum eigenvalue with respect to input % parameters % perturb cutting coefficient Kt by dKt [b1] = bisection(rpm,rstep,Kt-dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt +dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_b_Kt(i) = (b2-b1)/dKt/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kn by dKn [b1] = bisection(rpm,rstep,Kt ,Kn-dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn+dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % % % derivative of eigen matrix w.r.t cutting coefficient Kt d_b_Kn(i) = (b2-b1)/dKn/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kte by dKte [b1] = bisection(rpm,rstep,Kt ,Kn,Kte-dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte+dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_b_Kte(i) = (b2-b1)/dKte/2; b1 =[]; b2 =[]; % perturb cutting coefficient Kne by dKne [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne-dKne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne+dKne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matrix w.r.t cutting coefficient Kne d_b_Kne(i) = (b2-b1)/dKne/2; b1 =[]; b2 =[]; % % perturb depth of cut rstep by drstep [b1] = bisection(rpm,rstep-drst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep+drst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rstep of cut d_b_rstep(i) = (b2-b1)/drstep/2; b1 =[]; b2 =[]; % perturb spindle speed by drpm [b1] = bisection(rpm-drpm,rst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm+drpm,rst ep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rpm d_b_rpm(i) = (b2-b1)/drpm/2; b1 =[]; b2 =[];

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141 % perturb Kx by dKx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx-dKx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx+dKx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Kx d_b_Kx(i) = (b2-b1)/dKx/2; b1 =[]; b2 =[]; % perturb Ky by dKy [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky-dKy,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky+dKy,Cy); % % derivative of eigen value w.r.t Ky d_b_Ky(i) = (b2-b1)/dKy/2; b1 =[]; b2 =[]; % perturb Cx by dCx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx-dCx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx+dCx,My,Ky,Cy); % derivative of eigen value w.r.t Cx d_b_Cx(i) = (b2-b1)/dCx/2; b1 =[]; b2 =[]; % perturb Cy by dCy [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy-dCy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy+dCy); % derivative of eigen value w.r.t Cy d_b_Cy(i) = (b2-b1)/dCy/2; b1 =[]; b2 =[]; % perturb Mx by dMx [b1] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx-dMx,Kx,Cx,My,Ky,Cy); [b2] = bisection(rpm,rstep,Kt ,Kn,Kte,Kne,Mx+dMx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Mx d_b_Mx(i) = (b2-b1)/dMx/2; b1 =[]; b2 =[]; % perturb My by dMy [b1] = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My-dMy,Ky,Cy); [b2] = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My+dMy,Ky,Cy); % % % derivative of eigen value w.r.t My d_b_My(i) = (b2-b1)/dMy/2; b1 =[]; b2 =[]; % DELTA_b(i) =( (DELTA_Kt d_b_Kt(i) )^2 + (DELTA_Kn d_b_Kn(i))^2 + ... (DELTA_Kne d_b_Kne (i))^2 + (DELTA_Kte d_b_Kte(i))^2 +... (DELTA_Kx d_b_K x(i))^2 + (DELTA_Mx d_b_Mx(i))^2 + ... (DELTA_Cx d_b_C x(i))^2 + (DELTA_Ky d_b_Ky(i))^2 +... (DELTA_My d_b_M y(i))^2 + (DELTA_Cy d_b_Cy(i))^2+... (DELTA_rstep d_b_rstep(i))^2 + (DELTA_rpm* d_b_rpm(i))^2)^0.5; end

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142 close(h); % d_b_rpm % % Find the uncertainty in depth of cut for a corresponding uncertainty in % % input paramters % % save uncer_march_10_smallconverror figure plot(rpm_vec*1/60/(sqrt( Ky/My)/2/pi),b*1000,'-g') % % set(gca,'fontname','times','fontsize',16); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('b (mm)','fontsize',14) % legend('Stability boundary nominal input','Stabili ty boundary uncertainty'); % axis([5 20 0 15]) Stability Uncertainty, Monte Carlo and Latin Hyper-Cube % % M. Kurdi (6/17/05) % 4 OAL TOOL % Program to complete LatinHyper and M onte simulation for TFEA stability lobes % clear all; close all; % function LatinHyper % tic; % chip_load=0.1e-3;% chip load % nteeth = 4; % Diam =0.5; % E=15; % N = 1000; % number of iterations % % % % AL 6061 % % percent_Kt = 7.13/100; % cutting coefficents uncertainty % % percent_Kn = 8.09/100; % % percent_Kte = 30.3/100; % % percent_Kne = 23.9/100; % % % 5 OAL TOOL UNCERTAINTIES % % percent_KX = 0.054; % modal parameters uncertainty % % percent_CX = .286; % % percent_MX =.045; % % % % percent_KY = 0.054; % modal parameters uncertainty % % percent_CY = .173; % % percent_MY =.055; % % 4 OAL TOOL UNCERTAINTIES due to thermal effect only

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143 % percent_MX = 0.074; % percent_CX = 0.042; % percent_KX = 0.073 ; % percent_MY = 0.2; % percent_CY = 0.107; % percent_KY = 0.2 ; % percent_rstep = 0.0005; % radial step uncertainty % percent_rpm = 0.005; % spindle speed uncertainty % % speed_min = str2num(input('Min_speed = ','s')); % % speed_max = str2num(input('Max_speed = ','s')); % % speed = speed_min:200:speed_max; % speed = 10000:100:20000; % h = waitbar(0,'Please wait...'); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%5 % % Cutting Coefficients %%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%% % % AL 6061 % % mean_Kt =7.06E+08; % N/m2 % % mean_Kn = 2.50E+08; % % mean_Kte = 1.29E+04; % N/m; % % mean_Kne = 6.57E+03; % % AL 7475 % mean_Kt = 690480868.527357; % mean_Kte = 12022.3004909002; % mean_Kn = 142535991.092323; % mean_Kne =11281.4601645315; % std_Kn=4009843*4.45; % N % std_Kne=310.909*4.45; % std_Kte=200.731*4.45; % std_Kt=2588583*4.45; % % std_Kt = percent_Kt*mean_Kt; % % std_Kn = percent_Kn*mean_Kn; % % std_Kte = percent_Kte*mean_Kte; % % std_Kne = percent_Kne*mean_Kne; % % Kne Kn Kte Kt % % AL 6061 % % SIGMA_K = [1.480E+07 -1.778E+11 -8.216E+06 9.871E+10; % % -1.778E+11 2.458E+15 9.871E+10 -1.365E+15; % % -8.216E+06 9.871E+10 9.163E+07 -1.101E+12; % % 9.871E+10 -1.365E+15 -1.101E+12 1.522E+16 % % ]; % % % AL 7475

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144 % SIGMA_K = [ 42157610.7365206 -506483170409.775 3598978.12573119 43238262783.6325; % -506483170409.775 7.00379474676691e+015 43238262783.6325 -597911116174549; % -3598978.12573128 43238262783.6335 17574719.1179838 211143357093.21; % 43238262783.6335 -597911116174562 -211143357093.21 2.91975098408051e+015]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%% % % Modal Parameters % % X %%%%%%%%%%%%%%%% % % 5 OAL TOOL % % mean_Kx = 2.64E+06; % % mean_Mx = 0.049; % % mean_Cx = 8.972; % % dynamic parameters for 4 OAL tool % mean_Mx = 0.027 ; % mean_Cx= 23.309; % mean_Kx= 4359275.000 ; % std_Cx = percent_CX*mean_Cx; % std_Kx = percent_KX*mean_Kx; % std_Mx = percent_MX*mean_Mx; % % Mx Cx Kx My Cy Ky 5 OAL % % SIGMA = [3.85E-06 4.03E-03 2.48E+02 2.40E-06 -3.18E-03 1.31E+02; % % 4.03E-03 5.27E+00 2.69E+05 4.08E-03 -3.32E+00 2.19E+05; % % 2.48E+02 2.69E+05 1.61E+10 1.67E+02 -2.07E+05 9.15E+09; % % 2.40E-06 4.08E-03 1.67E+02 4.23E-06 -1.88E-03 2.24E+02; % % -3.18E-03 -3.32E+00 -2.07E+05 -1.88E-03 2.71E+00 1.04E+05; % % 1.31E+02 2.19E+05 9.15E+09 2.24E+02 -1.04E+05 1.19E+10 % % ]; % % Mx Cx Kx My Cy Ky 4 OAL % SIGMA = [4.04188E-06 0.000450265 631.110625 7.25563E-06 -0.000584252 878.998125; % 0.000450265 0.953490935 38828.325 0.00283473 -2.467636648 567721.5525; % 631.110625 38828.325 1.00042E+11 1068.011875 -51720.94813 1.21332E+11; % 7.25563E-06 0.00283473 1068.011875 1.76519E-05 -0.007067488 2638.261875;

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145 % -0.000584252 -2.467636648 -51720.94813 -0.007067488 11.34481701 1426512.396; % 878.998125 567721.5525 1.21332E+11 2638.261875 -1426512.396 4.36003E+11]; % % % Y %%%%%%% %%%%%%%%%%%% %%%%%%% % % 5 OAL TOOL % % mean_Ky = 2.26e+006; % % mean_Cy = 10.651; % % mean_My = 0.042; % % Y direction parameters 4 OAL TOOL % mean_Ky = 3301775.000; % mean_My = 0.021; % mean_Cy = 31.432; % std_My = percent_MY*mean_My; % std_Ky = percent_KY*mean_Ky; % std_Cy = percent_CY*mean_Cy; % % %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% % % % Radial step inches % % %%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%% % mean_rstep = 0.25*.5; % std_rstep = percent_rstep*mean_rstep; % % %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% % randn('state',0) % Mode = lhsnorm([mean_Mx mean_Cx mean_Kx mean_My mean_Cy mean_Ky],SIGMA,N); % % Mode(:,1) is Mx random vector % % Mode(:,2) is Cx random vector % % Mode(:,3) is Kx random vector % % Mode(:,4) is My random vector % % Mode(:,5) is Cy random vector % % Mode(:,6) is Ky random vector % Cut_Coeff = lhsnorm([mean_Kne mean _Kn mean_Kte mea n_Kt],SIGMA_K,N); % % Cut_Coeff(:,1) Kne % % Cut_Coeff(:,2) Kn % % Cut_Coeff(:,3) Kte % % Cut_Coeff(:,4) Kt % sample = randn(N, 2); % for j=1:length(speed) % waitbar(j/length(speed),h) % for i=1:N % % Unless otherwise specifi ed, all dimensions in m % % Define input parameters %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % Cutting coefficients % Kt = Cut_Coeff(i,4);

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146 % Kn = Cut_Coeff(i,2); % Kte = Cut_Coeff(i,3); % Kne = Cut_Coeff(i,1); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % milling parameters % % Spindle speed % mean_rpm = speed(j); % std_rpm = percent_rpm*mean_rpm; % rpm = mean_rpm + std_rpm*sample(i,1); % % rstep % rstep = mean_rstep + std_rstep*sample(i,2); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% % % Dynamic parameters % % X direction is feed direction % Kx =Mode(i,3); % Mx = Mode(i,1); % Cx = Mode(i,2); % % Y direction parameters % Ky = Mode(i,6); % My = Mode(i,4); % Cy = Mode(i,5); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % % Calculate axial depth corr esponding to input paramters % % that is on the stability boundaries % b(i,j) = bisection(rpm,rstep,Kt,Kn,Kte,Kne,Mx,Cx, Kx,My,Cy,Ky,chip_load,nteeth,Diam,E); % end % i end monte loop for one spindle speed % end % j end spindle speed range % std_dev = std(b) % b_mean = mean(b) % time=toc; % save Latin_AL7475 std_dev speed b_mean b time % close(h); % hold on; % h1 = plot(speed/1000,(mean (b)-2*std(b))*1000,'-r') % hold on; % h2 = plot(speed/1000,mean(b)*1000,'g-'); % hold on; % h3 = plot(speed/1000,(2*std (b)+mean(b)) *1000,'-r'); % legend([h1,h2,h3],'lower boundary','mean','upper boundary') figure plot(speed/1000,std(b)*2*1000) %

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147 % Function to stability lobe using bisection method. % Input: % rpm ; % rstep: radial immersion (inches) % Output: % b depth of cut (m) function [b] = bisection(rpm,rstep,Kt,Kn,Kte,Kne, Mx,Cx,Kx,My,Cy,Ky,h,nteeth,Diam,E) % E % h % feed per tooth % nteeth % number of teeth % Diam % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'down'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% % LOAD SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%% M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; b_r = [1e-10 100e-2]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% while (abs(b_r(1) b_r(2)) / b_r(1) > 1e-6) warning off MATLAB:singularMatrix; warning off MATLAB:nearlySingularMatrix;

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148 b = (b_r(1) + b_r(2)) / 2; speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth ; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER RO TATION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down' tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF -DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros (size(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos(2*t0*o mega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)-

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149 4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega* tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos(

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150 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos( 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)-

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151 2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)-

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152 60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)-

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153 180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^5-

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154 15*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega*tj*co s(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0*

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155 omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos( 2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj) -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P11(1, 2)*ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P12(1,2) *ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P13(1,2) *ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P14(1,2) *ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P21(1,2) *ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P22(1,2) *ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P23(1,2) *ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P24(1,2) *ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23;

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156 N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF,2* DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF )+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1); Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop size(A) Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = abs(sum(SLE_vec(locat))); else SLE = nan; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1; if CM < 1 b_r(1) = b; else b_r(2) = b; end

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157 end % while loop depth of cut Uncertainty SLE, Sensitivity Method % M. Kurdi (3/28/2005) % Function to find uncertainty in SLE to change in cutting % coefficients, dynamic parameters and cutting process variables % Input: % b: depth of cut (m) % rpm: spindle speed % rstep: radial step (inches) % Kt % Kn % Kte % Kre % DELTA_Kt finite change in Kt % DELTA_b finite change in b % system_ID: Modal parameters % The derivative of Max eigen value is found for a miniscule perturbation % in input parameters, then its effect on the change of axial depth is % found. clear all; close all; clc;tic; % function uncer % percentage of uncertainty in cutting coefficients, dynamic parameters % and process parameters % tic; percent_Kcut = 0.05; % cutt ing coefficents uncertainty percent_Dyn = 0.005; % modal parameters uncertainty percent_rstep = 0.0001; % ra dial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % nominal values of process paramete rs and their calculated uncertainty rstep = 0.2; b=1e-3; rpm_vec = 5500:50:5600; DELTA_rstep = percent_rstep*rstep; % cutting coefficient uncertainty Kt = 6e8; DELTA_Kt = percent_Kcut*Kt; Kn = .3*Kt; DELTA_Kn = percent_Kcut*Kn; Kte=0; DELTA_Kte = percent_Kcut*Kte; Kne=0; DELTA_Kne = percent_Kcut*Kne; % nominal values of dynamic parameters and their calcul ated uncertainty Kx = 4.4528e+006; Mx = 0.4362; Cx = 83;

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158 % Y direction parameters Ky = 3.5542e+006; My = 0.4362; Cy = 89.9; DELTA_Mx = Mx*percent_Dyn; DELTA_My = My*percent_Dyn; DELTA_Kx = Kx*percent_Dyn; DELTA_Ky = Ky*percent_Dyn; DELTA_Cx = Cx*percent_Dyn; DELTA_Cy = Cy*percent_Dyn; % to calculate the numerical deriva tive with respect to each input % variable set the miniscule change in each input % set miniscule change in input para meters to estimate the derivative step_percent = 0.002; dKt = step_percent*Kt; % N/m2 dKn = step_percent*Kn; % N/m2 dKte = step_percent*30; % N/m dKne = step_percent*30; % N/m drstep = step_percen t*rstep; % inch dKx = step_percent*Kx; % N/m dKy = step_percent*Ky; % N/m dCx = step_percent*Cx; % dCy = step_percent*Cy; dMx = step_percent*Mx; % Kg dMy = step_percent*My; % Kg h = waitbar(0,'Please wait...'); % computation here % for i=1:length(rpm_vec) waitbar(i/length(rpm_vec),h); rpm = rpm_vec(i); drpm = step_percent*rpm; % rpm DELTA_rpm = percent_rpm rpm; % Find depth of cut correspondi ng to stability bounda ry using nominal % settings of input parameters [sle(i)] = sle_f(b,rpm,rstep,Kt,K n,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % depth at boundary % Find numerical derivative of ma ximum eigenvalue with respect to input % parameters % perturb cutting coefficient Kt by dKt [sle1] = sle_f(b,rpm,rstep,Kt-dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt+dKt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_sle_Kt(i) = (sle2-sle1)/dKt/2; dsleKt_log(i) = d_sle_Kt(i)*Kt/(sle(i))*2; sle1 =[]; sle2 =[];

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159 % perturb cutting coefficient Kn by dKn [sle1] = sle_f(b,rpm,rstep,Kt,Kn-dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn+dKn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % % % derivative of eigen matrix w.r.t cutting coefficient Kt d_sle_Kn(i) = (sle2-sle1)/dKn/2; dsleKn_log(i) = d_sle_Kn(i)*Kn/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb cutting coefficient Kte by dKte [sle1] = sle_f(b,rpm,rstep,Kt ,Kn,Kte-dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte+dKte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kt d_sle_Kte(i) = (sle2-sle1)/dKte/2; dsleKte_log(i) = d_ sle_Kte(i)*Kte/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb cutting coefficient Kne by dKne [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne-dKne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne+dKne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen matr ix w.r.t cutting coefficient Kne d_sle_Kne(i) = (sle2-sle1)/dKne/2; dsleKne_log(i) = d_sle_Kne(i)*Kne/(sle(i))*2; sle1 =[]; sle2 =[]; % % perturb depth of cut rstep by drstep [sle1] = sle_f(b,rpm,rstep-drstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep+drstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rstep of cut d_sle_rstep(i) = (sle2-sle1)/drstep/2; dslerstep_log(i) = d_s le_rstep(i)*rstep/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb spindle speed by drpm [sle1] = sle_f(b,rpm-drpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm+drpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t rpm d_sle_rpm(i) = (sle2-sle1)/drpm/2; dslerpm_log(i) = d_sle_rpm(i)*rpm/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Kx by dKx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx-dKx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx+dKx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Kx d_sle_Kx(i) = (sle2-sle1)/dKx/2; dsleKx_log(i) = d_sle_Kx(i)*Kx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Ky by dKy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky-dKy,Cy);

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160 [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky+dKy,Cy); % derivative of eigen value w.r.t Ky d_sle_Ky(i) = (sle2-sle1)/dKy/2; dsleKy_log(i) = d_sle_Ky(i)*Ky/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Cx by dCx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx-dCx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx+dCx,My,Ky,Cy); % derivative of eigen value w.r.t Cx d_sle_Cx(i) = (sle2-sle1)/dCx/2; dsleCx_log(i) = d_sle_Cx(i)*Cx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Cy by dCy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy-dCy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My,Ky,Cy+dCy); % derivative of eigen value w.r.t Cy d_sle_Cy(i) = (sle2-sle1)/dCy/2; dsleCy_log(i) = d_sle_Cy(i)*Cy/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb Mx by dMx [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx-dMx,Kx,Cx,My,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx+dMx,Kx,Cx,My,Ky,Cy); % derivative of eigen value w.r.t Mx d_sle_Mx(i) = (sle2-sle1)/dMx/2; dsleMx_log(i) = d_ sle_Mx(i)*Mx/(sle(i))*2; sle1 =[]; sle2 =[]; % perturb My by dMy [sle1] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My-dMy,Ky,Cy); [sle2] = sle_f(b,rpm,rstep,Kt,Kn,Kte,Kne,Mx,Kx,Cx,My+dMy,Ky,Cy); % % derivative of eigen value w.r.t My d_sle_My(i) = (sle2-sle1)/dMy/2; dsleMy_log(i) = d_ sle_My(i)*My/(sle(i))*2; sle1 =[]; sle2 =[]; DELTA_sle(i) =( (DELTA_Kt d_sle_Kt(i ))^2 + (DELTA_Kn d_s le_Kn(i))^2 + ... (DELTA_Kne d_sle_Kn e(i))^2 + (DELTA_Kte d_sle_Kte(i))^2 +... (DELTA_Kx d_sle_Kx(i))^2 + (DELTA_Mx d_sle_Mx(i))^2 + ... (DELTA_Cx d_sle_ Cx(i))^2 + (DELTA_Ky d_sle_Ky(i))^2 +... (DELTA_My d_sle _My(i))^2 + (DELTA_Cy d_sle_Cy(i))^2+... (DELTA_rstep d_sle_r step(i))^2 + (DELTA_rpm* d_sle_rpm(i))^2)^0.5 end close(h); % % Find the uncertainty in depth of cut for a corresponding uncertainty in % % input paramters time_total=toc; % save uncer_march_30_sle % subplot(2,1,1)

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161 % plot(rpm_vec/1000,sle*1e6,'-g',rpm _vec/1000,(sle+2*DELTA_sle)*1e6,'k',rpm_vec/1000,(sle-2*DELTA_sle)*1e6,'-k') % set(gca,'fontname','times','fontsize',16); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('SLE (\mum)','fontsize',14) % legend('Stability boundary, nom inal input','\pm2u_c(SLE)'); % axis([5 20 -12 28]) % subplot(2,1,2) % plot(rpm_vec/1000,d_sle_Ky*Ky*1e6,'<',rpm_vec/1000,d_sle_My*My*1e6,'>',... % rpm_vec/1000,d_sle_Cy*Cy*1e6,'o',r pm_vec/1000,d_sle_rpm.*rpm_vec*1e6,'+',... % rpm_vec/1000,d_sle_rstep*rstep*1e6,'^',rpm_vec/1000,d_sle_Kt*Kt*1e6,'s',... % rpm_vec/1000, d_sle_Kn*Kn*1e6,'*'); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('x_i \partial(SLE)/\partial(x_i)'); % figure; % plot(rpm_vec/1000,dsleKy_log,'<',rpm_vec/1000,dsleMy_log,'>',... % rpm_vec/1000,dsleCy_log,'o',rpm_vec/1000,dslerpm_log,'+',... % rpm_vec/1000,dslerstep_log,'^',rpm_vec/1000,dsleKt_log,'s',... % rpm_vec/1000,dsleKn_log,'*'); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % xlabel('\Omega (x10^3 rpm)','fontsize',14) % ylabel('\partial(SLE)/\partial(x_i)x_i/SLE'); % figure; plot(rpm_vec/1000,abs(d_sle_Ky)*Ky*1e6,'.', rpm_vec/1000,abs(d_sle_My)*My*1e6,':',. .. % rpm_vec/1000,abs(d_sle_Cy)*Cy*1e6,'-',rpm_vec/1000,abs(d_sle_rpm).*rpm_vec*1e6,'-',... % rpm_vec/1000,abs(d_sle_rstep)*rstep*1e6,'^', rpm_vec/1000,abs(d_sle_Kt)*Kt*1e6,'s',... % rpm_vec/1000,abs(d_sle_Kn)*Kn*1e6); % legend('K_y','M_y' ,'C_y','\Omega','r_{step}','K_t','K_n'); % figure; % plot(rpm_vec/1000,abs(d_sle_Ky)*Ky./abs(d_sle_My)/My) % legend('K_y/M_y'); figure plot(rpm_vec/1000,DELTA_sle*1e6) xlabel('\Omega (x10^3 rpm)','fontsize',14) ylabel('u_c(SLE) (\mum)') figure plot(rpm_vec/1000,DELTA_sle*1e6,'-
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162 Uncertainty SLE, Monte Carlo and Latin Hype-Cube Sampling Methods % % M. Kurdi (6/17/05) % 4 OAL TOOL % Program to complete LatinHyper and Monte simulation for SLE clear all; % function LatinHyper tic; chip_load=0.1e-3;% chip load nteeth = 4; Diam =0.5; E=20; N = 1000; % number of iterations baxial=4.45e-3; % AL 6061 % percent_Kt = 7.13/100; % cutting coefficents uncertainty % percent_Kn = 8.09/100; % percent_Kte = 30.3/100; % percent_Kne = 23.9/100; % 5 OAL TOOL UNCERTAINTIES % percent_KX = 0.054; % modal parameters uncertainty % percent_CX = .286; % percent_MX =.045; % percent_KY = 0.054; % modal parameters uncertainty % percent_CY = .173; % percent_MY =.055; % 4 OAL TOOL UNCERTAINTIES due to thermal effect only percent_MX = 0.074; percent_CX = 0.042; percent_KX = 0.073 ; percent_MY = 0.2; percent_CY = 0.107; percent_KY = 0.2 ; percent_rstep = 0.0005; % radial step uncertainty percent_rpm = 0.005; % spindle speed uncertainty % speed_min = str2num(input('Min_speed = ','s')); % speed_max = str2num(input('Max_speed = ','s')); % speed = speed_min:200:speed_max; % speed = [ 14753 14803 14853 14903 14953]; 4.45 mm speed = [15517 15567 15617 15667 15767]; % 2.12 mm h = waitbar(0,'Please wait...'); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%5 % Cutting Coefficients %%%%%%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%

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163 % AL 6061 % mean_Kt =7.06E+08; % N/m2 % mean_Kn = 2.50E+08; % mean_Kte = 1.29E+04; % N/m; % mean_Kne = 6.57E+03; % AL 7475 mean_Kt = 690480868.527357; mean_Kte = 12022.3004909002; mean_Kn = 142535991.092323; mean_Kne =11281.4601645315; std_Kn=4009843*4.45; % N std_Kne=310.909*4.45; std_Kte=200.731*4.45; std_Kt=2588583*4.45; % std_Kt = percent_Kt*mean_Kt; % std_Kn = percent_Kn*mean_Kn; % std_Kte = percent_Kte*mean_Kte; % std_Kne = percent_Kne*mean_Kne; % Kne Kn Kte Kt % AL 6061 % SIGMA_K = [1.480E+07 -1.778E+11 -8.216E+06 9.871E+10; % -1.778E+11 2.458E+15 9.871E+10 -1.365E+15; % -8.216E+06 9.871E+10 9.163E+07 -1.101E+12; % 9.871E+10 -1.365E+15 -1.101E+12 1.522E+16 % ]; % AL 7475 SIGMA_K = [ 42157610.7365206 -506483170409.775 -3598978.12573119 43238262783.6325; -506483170409.775 7.00379474676691e+015 43238262783.6325 597911116174549; -3598978.12573128 43238262783.6335 17574719.1179838 211143357093.21; 43238262783.6335 -597911116174562 -211143357093.21 2.91975098408051e+015]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Modal Parameters % X %%%%%%%%%%%%%%%% % 5 OAL TOOL % mean_Kx = 2.64E+06; % mean_Mx = 0.049; % mean_Cx = 8.972; % dynamic parameters for 4 OAL tool mean_Mx = 0.027 ; mean_Cx= 23.309; mean_Kx= 4359275.000 ;

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164 std_Cx = percent_CX*mean_Cx; std_Kx = percent_KX*mean_Kx; std_Mx = percent_MX*mean_Mx; % Mx Cx Kx My Cy Ky 5 OAL % SIGMA = [3.85E-06 4.03E-03 2.48E+02 2.40E-06 -3.18E-03 1.31E+02; % 4.03E-03 5.27E+00 2.69E+05 4.08E-03 -3.32E+00 2.19E+05; % 2.48E+02 2.69E+05 1.61E+10 1.67E+02 -2.07E+05 9.15E+09; % 2.40E-06 4.08E-03 1.67E+02 4.23E-06 -1.88E-03 2.24E+02; % -3.18E-03 -3.32E+00 -2.07E+05 -1.88E-03 2.71E+00 1.04E+05; % 1.31E+02 2.19E+05 9.15E+09 2.24E+02 -1.04E+05 1.19E+10 % % ]; % Mx Cx Kx My Cy Ky 4 OAL SIGMA = [4.04188E-06 0.000450265 631.110625 7.25563E-06 -0.000584252 878.998125; 0.000450265 0.953490935 38828.325 0.00283473 -2.467636648 567721.5525; 631.110625 38828.325 1.00042E+11 1068.011875 -51720.94813 1.21332E+11; 7.25563E-06 0.00283473 1068.011875 1.76519E-05 -0.007067488 2638.261875; -0.000584252 -2.467636648 -51720.94813 -0.007067488 11.34481701 -1426512.396; 878.998125 567721.5525 1.21332E+11 2638.261875 -1426512.396 4.36003E+11]; % % Y %%%%%%%%%%% %%%%%%%%%%%%%%% % 5 OAL TOOL % mean_Ky = 2.26e+006; % mean_Cy = 10.651; % mean_My = 0.042; % Y direction parameters 4 OAL TOOL mean_Ky = 3301775.000; mean_My = 0.021; mean_Cy = 31.432; std_My = percent_MY*mean_My; std_Ky = percent_KY*mean_Ky; std_Cy = percent_CY*mean_Cy; % %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % % Radial step inches % %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% mean_rstep = 0.25*.5; std_rstep = percent_rstep*mean_rstep;

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165 % %%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%% randn('state',0) Mode = lhsnorm([mean_Mx mea n_Cx mean_Kx mean_My mean_Cy mean_Ky],SIGMA,N); % Mode(:,1) is Mx random vector % Mode(:,2) is Cx random vector % Mode(:,3) is Kx random vector % Mode(:,4) is My random vector % Mode(:,5) is Cy random vector % Mode(:,6) is Ky random vector Cut_Coeff = lhsnorm([mean_Kne mean_K n mean_Kte mean_K t],SIGMA_K,N); % Cut_Coeff(:,1) Kne % Cut_Coeff(:,2) Kn % Cut_Coeff(:,3) Kte % Cut_Coeff(:,4) Kt sample = randn(N, 2); for j=1:length(speed) waitbar(j/length(speed),h) for i=1:N % Unless otherwise specified, all dimensions in m % Define input parameters %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Cutting coefficients Kt = Cut_Coeff(i,4); Kn = Cut_Coeff(i,2); Kte = Cut_Coeff(i,3); Kne = Cut_Coeff(i,1); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % milling parameters % Spindle speed mean_rpm = speed(j); std_rpm = percent_rpm*mean_rpm; rpm = mean_rpm + std_rpm*sample(i,1); % rstep rstep = mean_rstep + std_rstep*sample(i,2); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% % Dynamic parameters % X direction is feed direction Kx =Mode(i,3); Mx = Mode(i,1); Cx = Mode(i,2); % Y direction parameters

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166 Ky = Mode(i,6); My = Mode(i,4); Cy = Mode(i,5); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% % Calculate axial depth corre sponding to input paramters % that is on the stability boundaries sle(i,j) = sle_f(baxial,rpm,rstep,Kt,Kn,Kte,Kne,Mx, Kx,Cx,My,Ky,Cy,chip_load,nteeth,Diam,E); end % i end monte loop for one spindle speed end % j end spindle speed range for i=1:length(speed) index = find(isnan(sle(:,i))==0); sle_mean(i) = mean(sle(index,i)); std_dev(i) = std(sle(index,i)); end time=toc; save Latin_AL7475SLE2p12 std_dev speed sle_mean sle time close(h); % hold on; % h1 = plot(speed/1000,(sle _mean-2*std_dev)*1e6,'-r') % hold on; % h2 = plot(speed/1000,sle_mean*1e6,'g-'); % hold on; % h3 = plot(speed/1000,(2*st d_dev+sle_mean)*1e6,'-r'); % legend([h1,h2,h3],'lower boundary','mean','upper boundary') % hold on; % for i=1:1000 % % plot(speed/1000,sle(i,:)*1e6,'.'); % end % % figure % % plot(speed/1000,std(sle)*2*1000) % % % % Input: % rpm ; % rstep: radial immersion (inches) % Output: % b depth of cut (m) function SLE = sle_f(b,rpm,rstep,Kt,Kn,Kt e,Kne,Mx,Kx,Cx,My,Ky,Cy,h,nteeth,Diam,E) % E=30; % % h = 0.1e-3; % feed per tooth % nteeth = 1; % number of teeth

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167 % Diam = 1; % inches TRAVang = acos(1-rs tep/(Diam/2)); % angular travel during cutting LAGang = 2*pi/nteeth; % separation angle for teeth rho = acos(1-rstep/(Diam/2))/(2*pi); % fraction of time in cut IMMERSION = rstep/Diam; opt = 'down'; if TRAVang>LAGang % MU LTIPLE TEETH ARE IN CONTACT teethNcontact = floor(TRAVang/LAGang) +1; else % SINGLE TOOTH IN CONTACT teethNcontact = 1; end %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % SYSTEM IDENTIFICATION MATRICES %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% M =[Mx zeros(size(Mx)); zeros(size(Mx)) My]; C =[Cx zeros(size(Mx)); zeros(size(Mx)) Cy]; K =[Kx zeros(size(Mx)); zeros(size(Mx)) Ky]; lmx = length(Mx(1,:)); lmy = length(My(1,:)); DOF = lmx+lmy; V = [ones(1,lmx) zeros(1 ,lmy); zeros(1,lmx) ones(1,lmy)]; A = zeros((E+1)*2*DOF,(E+1)*2*DOF); B = A; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % BEGIN LOOP CALCULATIONS OVER RPM vs DOC FIELD %%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% speed = rpm; omega = speed/60*(2*pi); % radians per second T = (2*pi)/omega/nteeth ; % tooth pass period TC = rho*T*nteeth; % time a single tooth spends in the cut tf = T-TC; % time for free vibs tj = TC/E; % time for each element %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% % SET CUTTER RO TATION ANGLE FOR UP/DOWN-MILLING %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% switch opt case 'up' t0mat = [0 tj*(1:(E-1))]; % upmilling locat = 2*DOF+lmx+1:3*DOF; case 'down'

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168 tex = pi/omega; tent=tex-TC; % downmilling t0mat = [tent tent+tj*(1:(E-1))]; % downmilling locat = (E+1)*2*DOF -DOF-lmy+1:(E+1)*2*DOF-DOF; end %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% % STATE TRANSITION MATRIX %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%% G1 = [zeros(size(M)) M; eye(size(M)) zeros(size(M))]; G2 = [K C; zeros (size(M)) -eye(size(M))]; G = -G1\G2; PHI = expm(G*tf); %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% % N & P are used to create A & B which then become Q in..... a_n = Q a_n-1 + D %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% for e=1:E, t0 = t0mat(e); C1 = V'*[ -1/4*b*(-h*Kt*cos( 2*t0*omega+2*omega*tj)+2*h*Kn*omega*tjh*Kn*sin(2*t0*omega+2*omega*tj)+4*Kte*sin(t0*omega+omega*tj)4*Kne*cos(t0*omega+omega*tj)+h*Kt*co s(2*t0*omega)+h*Kn*sin(2*t0*omega)4*Kte*sin(t0*omega)+4*Kne*cos(t0*omega))/omega; 1/4*b*(2*h*Kt*omega*tjh*Kt*sin(2*t0*omega+2*omega*tj)+h*Kn*cos(2*t0*omega+2*omega*tj)4*Kte*sin(t0*omega+omega*tj)+4*Kne*cos (t0*omega+omega*tj)+h*Kt*sin(2*t0*ome ga)-h*Kn*cos(2*t0*omega)+4*Kte*sin(t0*om ega)-4*Kne*cos(t0*omega))/omega]; C2 = V'*[ 1/8*b*(h*Kt*sin(2*t0*omega+2*omega*tj)+h*Kt* cos(2*t0*omega+2*omega*tj)*omega*tj+h* Kn*cos(2*t0*omega+2*omega*tj)+h*Kn*s in(2*t0*omega+2*omega*tj)*omega*tj+8*K te*sin(t0*omega+omega*tj)-4*Kte *cos(t0*omega+omega*tj)*omega*tj8*Kne*cos(t0*omega+omega*tj)4*Kne*sin(t0*omega+omega*tj)*omega*tj+h*Kt*sin(2*t0*omega)h*Kn*cos(2*t0*omega)8*Kte*sin(t0*omega)+8*Kne*cos(t0*omega )+h*Kt*tj*cos(2*t0*omega)*omega+h*Kn* tj*sin(2*t0*omega)*omega-4*K te*tj*cos(t0*omega)*omega4*Kne*tj*sin(t0*omega)*omega)/tj/omega^2; 1/8*b*(h*Kt*cos(2*t0*omega+2*omega*tj)+ h*Kt*sin(2*t0*omega+2*omega*tj)*omeg a*tj+h*Kn*sin(2*t0*omega+2*omega*tj)h*Kn*cos(2*t0*omega+2*omega*tj)*omega*tj-

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169 8*Kte*sin(t0*omega+omega*tj)+4*Kte*cos (t0*omega+omega*tj)*omega*tj+8*Kne*co s(t0*omega+omega*tj)+4*Kne*sin (t0*omega+omega*tj)*omega*tjh*Kt*cos(2*t0*omega)-h*Kn*sin(2*t0*omega)+8*Kte*sin(t0*omega)8*Kne*cos(t0*omega)+h*Kt* tj*sin(2*t0*omega)*omegah*Kn*tj*cos(2*t0*omega)*omega+4*Kte*tj*co s(t0*omega)*omega+4*Kne*tj*sin(t0*o mega)*omega)/tj/omega^2]; P11 = [ 1/8*b*(3*Kt*sin(2*t0*omega+2*omega*tj)+3*K n*cos(2*t0*omega+2*omega*tj)+2*Kn*omeg a^4*tj^4+3*Kt*omega*tj*cos(2*t0*omega +2*omega*tj)+3*Kn*omega*tj*sin(2*t0*ome ga+2*omega*tj)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega)+2*Kt*cos(2*t0*omeg a)*omega^3*tj^3+2*Kn*sin(2*t0*omega) *omega^3*tj^3+3*Kt*omega*tj*cos(2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega))/o mega^4/tj^3, 1/8*b*(-3*Kt*cos(2*t0*omega+2*omega*tj)3*Kn*sin(2*t0*omega+2*omega*tj)+2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*ome ga*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(-3*Kt*cos(2*t0*omega+2*o mega*tj)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^43*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+3*Kn*omega*tj*cos(2*t0*omega+2*om ega*tj)+3*Kt*cos(2*t0*omega )+3*Kn*sin(2*t0*omega)2*Kt*sin(2*t0*omega)*omega^3*tj^ 3+2*Kn*cos(2*t0*omega)*omega^3*tj^33*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kt*omega*tj*cos(2* t0*omega+2*omega*tj)+3*Kn*omega*tj*s in(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+3*Kn*c os(2*t0*omega+2*omega*tj)+2*Kt*cos(2* t0*omega)*omega^3*tj^3+2*Kn*sin(2*t0*omega)*omega^3*tj^3+3*Kt*omega*tj*cos( 2*t0*omega)+3*Kn*omega*tj*sin(2*t0*omega)+3*Kt*sin(2*t0*omega)3*Kn*cos(2*t0*omega))/omega^4/tj^3]; P12 =[ 1/48*b*(6*Kn*omega*tj*sin(2*t0*omega+2*om ega*tj)+6*Kt*omega*tj*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^4*tj^49*Kt*sin(2*t0*omega+2*omega*tj)+9*K n*cos(2*t0*omega+2*omega*tj)+12*Kn*ome ga*tj*sin(2*t0*omega)+12*Kt*omega*tj*cos(2*t0*omega)6*Kt*omega^2*tj^2*sin(2*t0*omega)+6*Kn*om ega^2*tj^2*cos(2*t0*omega)+9*Kt*si n(2*t0*omega)-9*Kn*cos(2*t0*omega))/tj^2/omega^4, 1/48*b*(9*Kt*cos(2*t0*omega+2*omega* tj)+9*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^4*tj^4+6*Kt*omega*tj* sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega* tj)+12*Kt*omega*tj*sin(2*t0*omega)+6* Kn*omega^2*tj^2*sin(2*t0*omega)+ 6*Kt*omega^2*tj^2*cos(2*t0*omega)9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^4+6*Kt*omega* tj*sin(2*t0*omega+2*omega*tj)+9*Kt*cos(

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170 2*t0*omega+2*omega*tj)6*Kn*omega*tj*cos(2*t0*omega+2*omega *tj)+9*Kn*sin(2*t0*omega+2*omega*tj)+1 2*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga^2*tj^2*sin(2*t0*omega)+6*Kt*omega^ 2*tj^2*cos(2*t0*omega)-9*Kt*cos(2*t0*omega)-12*Kn*omega*tj*cos(2*t0*omega)9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(6*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega+ 2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)12*Kt*omega*tj*cos(2*t0*omega)12*Kn*omega*tj*sin(2*t0*omega)+6*K t*omega^2*tj^2*sin(2*t0*omega)6*Kn*omega^2*tj^2*cos(2*t0*om ega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega ))/tj^2/omega^4]; P13 =[ 1/8*b*(2*Kn*omega^4*tj^43*Kn*cos(2*t0*omega+2*omega*tj)+3*K t*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*cos(2*t0*o mega+2*omega*tj)2*Kn*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*cos(2*t0*omega+ 2*omega*tj)+3*Kn*cos(2*t0*omega)3*Kt*sin(2*t0*omega)-3*Kt*omega*tj*cos(2*t0*omega)3*Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kt*omega^4*tj^4+2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)2*Kn*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+3*Kt*cos(2*t0*omega+2*omega*t j)+3*Kn*sin(2*t0*omega+2*omega*tj)+3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj)-3*Kt*cos(2*t0*omega)3*Kn*sin(2*t0*omega)+3*Kt*omega*tj*sin(2*t0*omega)3*Kn*omega*tj*cos(2*t0*omega))/omega^4/tj^3; 1/8*b*(2*Kt*omega^4*tj^4+2*Kn*omega^3*tj^ 3*cos(2*t0*omega+2*omega*tj)+3*Kn *omega*tj*cos(2*t0*omega+2*omega*t j)-3*Kn*sin(2*t0*omega+2*omega*tj)2*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)3*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)3*Kt*cos(2*t0*omega+2*omega*tj)+3*Kn*s in(2*t0*omega)+3*Kt*cos(2*t0*omega)3*Kt*omega*tj*sin(2*t0*omega)+3*Kn*omega *tj*cos(2*t0*omega))/omega^4/tj^3, 1/8*b*(2*Kn*omega^4*tj^4+3*Kn*cos(2*t0*ome ga+2*omega*tj)+2*Kn*omega^3*tj^3 *sin(2*t0*omega+2*omega*tj)3*Kt*sin(2*t0*omega+2*omega*tj)+2*Kt *omega^3*tj^3*cos(2*t0*omega+2*omega*tj )+3*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+3*Kn*omega*tj*sin(2*t0*omega+2*o mega*tj)3*Kn*cos(2*t0*omega)+3*Kt*sin(2*t0*omeg a)+3*Kt*omega*tj*cos(2*t0*omega)+3* Kn*omega*tj*sin(2*t0*omega))/omega^4/tj^3]; P14 =[ -1/48*b*(6*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kn*omega*tj*sin(2*t0*omega+2*omega*tj)6*Kt*sin(2*t0*omega+2*ome ga*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)+2*Kn*omega^4*tj^49*Kn*cos(2*t0*omega+2*omega*tj)+9*K t*sin(2*t0*omega+2*omega*tj)6*Kn*omega*tj*sin(2*t0*omega)-

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171 6*Kt*omega*tj*cos(2*t0*omega)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega))/tj^2/omeg a^4, 1/48*b*(-2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4; 1/48*b*(2*Kt*omega^4*tj^412*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+12*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+6*Kt*cos(2*t0*omega+2*omega* tj)*omega^2*tj^2+6*Kn*sin(2*t0*omega+2 *omega*tj)*omega^2*tj^2-9*K t*cos(2*t0*omega+2*omega*tj)9*Kn*sin(2*t0*omega+2*omega*tj)6*Kt*omega*tj*sin(2*t0*omega)+6*Kn*ome ga*tj*cos(2*t0*omega)+9*Kt*cos(2*t0*o mega)+9*Kn*sin(2*t0*omega))/tj^2/omega^4, 1/48*b*(12*Kn*omega*tj*sin(2*t0*omega+2*omega* tj)-9*Kn*cos(2*t0*omega+2*omega*tj)2*Kn*omega^4*tj^4-6*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^212*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)+6*Kn*cos(2*t0*omega+2*omega*tj)*o mega^2*tj^2+9*Kt*sin(2*t0*omega+2*omega*tj)+9*Kn*cos(2*t0*omega)9*Kt*sin(2*t0*omega)-6*Kn*omega*tj*sin(2*t0*omega)6*Kt*omega*tj*cos(2*t0*omega))/tj^2/omega^4]; P21 =[ -1/80*b*(-15*Kt*cos(2*t0*o mega+2*omega*tj)*omega^2*tj^215*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^2+60*Kt*cos(2*t0*omega+2*omega *tj)+60*Kn*sin(2*t0*omega+ 2*omega*tj)+4*Kn*omega^5*tj^5+60*Kt*omega*tj*sin(2 *t0*omega)10*Kn*omega^3*tj^3*cos(2*t0*omega)+15*Kn* omega^2*tj^2*sin(2*t0*omega)+15*K t*omega^2*tj^2*cos(2*t0*omega)+1 0*Kt*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*cos(2*t0*omega)+10*Kn*sin(2*t0*omega)*omega^4*tj^4+10*Kt*co s(2*t0*omega)*omega^4*tj^4-60*Kn*sin(2*t0*omega)60*Kt*cos(2*t0*omega)+60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)60*Kn*omega*tj*cos(2*t0*omega+2*ome ga*tj))/omega^5/tj^4, -1/80*b*(60*Kn*cos(2*t0*omega)+60*Kt*sin(2*t0*omega)15*Kn*cos(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+15*Kt*sin(2*t0*omega+2*omega *tj)*omega^2*tj^2+60*Kn*omega* tj*sin(2*t0*omega+2*omega*tj)60*Kt*sin(2*t0*omega+2*omega*tj)+60*Kt *omega*tj*cos(2*t0*omega+2*omega*tj)+ 60*Kn*cos(2*t0*omega+2*omega*tj)+10*Kt*omega^3*tj^3*cos(2*t0*omega)15*Kt*omega^2*tj^2*sin(2*t0*omega)10*Kt*sin(2*t0*omega)*omega^4*tj^4+10*K n*cos(2*t0*omega)*omega^4*tj^4+15*K n*omega^2*tj^2*cos(2*t0*omega)+60*Kt*om ega*tj*cos(2*t0*omega)+10*Kn*omega^ 3*tj^3*sin(2*t0*omega)+60*Kn*omega*tj*si n(2*t0*omega)+4*Kt*omega^5*tj^5)/ome ga^5/tj^4; 1/80*b*(60*Kn*cos(2*t0*omega)60*Kt*sin(2*t0*omega)+15*Kn*cos(2*t 0*omega+2*omega*tj)*omega^2*tj^215*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^260*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*cos(2*t0*omega+2*omega *tj)-60*Kn*cos(2*t0*omega+2*omega*tj)-

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172 10*Kt*omega^3*tj^3*cos(2*t0*omega)+15*Kt* omega^2*tj^2*sin(2*t0*omega)+10*Kt *sin(2*t0*omega)*omega^4*tj^4-10*Kn*cos(2*t0*omega)*omega^4*tj^415*Kn*omega^2*tj^2*cos(2*t0*omega )-60*Kt*omega*tj*cos(2*t0*omega)10*Kn*omega^3*tj^3*sin(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)+4*Kt*omega^5*tj^5)/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*sin(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)10*Kn*sin(2*t0*omega)*omega^4*tj^4-10*K t*cos(2*t0*omega)*omega^4*tj^415*Kt*omega^2*tj^2*cos(2*t0*omega)10*Kt*omega^3*tj^3*sin(2*t0*omega)+10*Kn*omega^3*tj^3*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega)+60*K n*sin(2*t0*omega)+60*Kt*cos(2*t0*ome ga)60*Kt*cos(2*t0*omega+2*omega*tj)+4*Kn*o mega^5*tj^5+60*Kn*omega*tj*cos(2*t0 *omega+2*omega*tj)+15*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+15*Kn*sin (2*t0*omega+2*omega*tj)*omega^2*tj^2-60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega*tj))/omega^5/tj^4]; P22 = [ 1/480*b*(-2*Kn*omega^5*tj^5180*Kt*cos(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5, 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*Kn*co s(2*t0*omega+2*omega*tj)+2*Kt*o mega^5*tj^5+30*Kt*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(135*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+135*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)2*Kt*omega^5*tj^5+30*Kt*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^230*Kn*cos(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*cos(2*t0*omega)+ 120*Kn*omega^2*tj^2*cos(2*t0*omega)120*Kt*omega^2*tj^2*sin(2*t0*omega)30*Kn*omega^3*tj^3*sin(2*t0*omega)+ 225*Kt*omega*tj*cos(2*t0*omega)+225*Kn* omega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5, -1/480*b*(2*Kn*omega^5*tj^5-

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173 180*Kt*cos(2*t0*omega+2*omega*tj)-180*K n*sin(2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+135*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+30*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2+30*Kn*sin(2*t0*ome ga+2*omega*tj)*omega^2*tj^2225*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+225*Kn*omega*tj*cos(2*t0*omega)120*Kt*omega^2*tj^2*cos(2*t0*omega)+ 30*Kt*omega^3*tj^3*sin(2*t0*omega)30*Kn*omega^3*tj^3*cos(2*t0*omega)120*Kn*omega^2*tj^2*sin(2*t0*om ega))/tj^3/omega^5]; P23 = [ -1/80*b*(-4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4, -1/80*b*(60*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)-10*Kn*omega^ 3*tj^3*sin(2*t0*omega+2*omega*tj)4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4; -1/80*b*(-60*Kt*omega* tj*cos(2*t0*omega+2*omega*tj)60*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+60*Kt*sin(2*t0*omega+2*omega*tj)60*Kn*cos(2*t0*omega+2*omega*tj)10*Kt*omega^4*tj^4*sin(2*t0*omega+2*omega*tj)10*Kt*omega^3*tj^3*cos(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*cos(2*t0*o mega+2*omega*tj)10*Kn*omega^3*tj^3*sin(2*t0*omega+ 2*omega*tj)+4*Kt*omega^5*tj^515*Kt*sin(2*t0*omega+2*omega*tj)*omega ^2*tj^2+15*Kn*cos(2*t0*omega+2*omega *tj)*omega^2*tj^215*Kn*omega^2*tj^2*cos(2*t0*omega)+15*K t*omega^2*tj^2*sin(2*t0*omega)60*Kt*omega*tj*cos(2*t0*omega)60*Kn*omega*tj*sin(2*t0*omega)60*Kt*sin(2*t0*omega)+60*Kn*cos(2*t0*omega))/omega^5/tj^4, 1/80*b*(4*Kn*omega^5*tj^5-60*Kt*cos(2*t0*omega+2*omega*tj)60*Kn*sin(2*t0*omega+2*omega*tj)-

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174 60*Kt*omega*tj*sin(2*t0*omega+2*omega* tj)+60*Kn*omega*tj*cos(2*t0*omega+2*o mega*tj)+15*Kt*cos(2*t0*omega+2*omega *tj)*omega^2*tj^2+15*Kn*sin(2*t0*omega +2*omega*tj)*omega^2*tj^210*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+10*Kn*omega^4*tj^4*sin(2*t0*o mega+2*omega*tj)+10*Kt*omega^4*tj^ 4*cos(2*t0*omega+2*omega*tj)+10*Kn*omeg a^3*tj^3*cos(2*t0*omega+2*omega*tj)60*Kt*omega*tj*sin(2*t0*omega)+60*Kn*s in(2*t0*omega)+60*Kt*cos(2*t0*omega)+ 60*Kn*omega*tj*cos(2*t0*omega)-15*K t*omega^2*tj^2*cos(2*t0*omega)15*Kn*omega^2*tj^2*sin(2*t0*omega))/omega^5/tj^4]; P24 = [ 1/480*b*(2*Kn*omega^5*tj^5+180*Kt*cos( 2*t0*omega+2*omega*tj)+180*Kn*sin(2*t 0*omega+2*omega*tj)+225*Kt*omega*tj*sin(2*t0*omega+2*omega*tj)225*Kn*omega*tj*cos(2*t0*omega+2*omega*tj)120*Kt*cos(2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*sin(2*t0*omega+2*omega*tj)*omega^2*tj^230*Kt*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+30*Kn*omega^3*tj^3*cos(2*t0*o mega+2*omega*tj)+135*Kt*omega*tj*si n(2*t0*omega)-180*Kn*sin(2*t0*omega)180*Kt*cos(2*t0*omega)135*Kn*omega*tj*cos(2*t0*omega)+30*Kt *omega^2*tj^2*cos(2*t0*omega)+30*Kn*o mega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5, 1/480*b*(225*Kt*omega*tj*cos(2*t0*omega+ 2*omega*tj)+225*Kn*omega*tj*sin(2*t0 *omega+2*omega*tj)180*Kt*sin(2*t0*omega+2*omega*tj)+180*K n*cos(2*t0*omega+2*omega*tj)30*Kt*omega^3*tj^3*cos(2*t0*omega+2*omega*tj)30*Kn*omega^3*tj^3*sin(2*t0*omega+2*omeg a*tj)+2*Kt*omega^5*tj^5+120*Kt*sin( 2*t0*omega+2*omega*tj)*omega^2*tj^2120*Kn*cos(2*t0*omega+2*omega*tj)*omega^2* tj^2+30*Kn*omega^2*tj^2*cos(2*t0* omega)30*Kt*omega^2*tj^2*sin(2*t0*omega)+ 135*Kt*omega*tj*cos(2*t0*omega)+135*Kn*o mega*tj*sin(2*t0*omega)+180*Kt*sin(2*t0*omega)180*Kn*cos(2*t0*omega))/tj^3/omega^5; 1/480*b*(-225*Kt*omega*tj*cos(2*t0*omega+2*omega*tj)225*Kn*omega*tj*sin(2*t0*omega+2*omega *tj)+180*Kt*sin(2*t0*omega+2*omega*tj )180*Kn*cos(2*t0*omega+2*omega*tj)+30*Kt*omega^3*tj^3*cos(2*t0*omega+2*ome ga*tj)+30*Kn*omega^3*tj^3*sin(2*t0*om ega+2*omega*tj)+2*Kt*omega^5*tj^5120*Kt*sin(2*t0*omega+2*omega*tj)*omega^ 2*tj^2+120*Kn*cos(2*t0*omega+2*ome ga*tj)*omega^2*tj^230*Kn*omega^2*tj^2*cos(2*t0*omega)+30*K t*omega^2*tj^2*sin(2*t0*omega)135*Kt*omega*tj*cos(2*t0*omega)135*Kn*omega*tj*sin(2*t0*omega)180*Kt*sin(2*t0*omega)+180*Kn*cos(2*t0*omega))/tj^3/omega^5, 1/480*b*(2*Kn*omega^5*tj^5-180*Kt*c os(2*t0*omega+2*omega*tj)180*Kn*sin(2*t0*omega+2*omega*tj)225*Kt*omega*tj*sin(2*t0*omega+2*omega *tj)+225*Kn*omega*tj*cos(2*t0*omega+2 *omega*tj)+120*Kt*cos(2*t0*omega+2*omeg a*tj)*omega^2*tj^2+120*Kn*sin(2*t0*o mega+2*omega*tj)*omega^2*tj^2+30*Kt*omega^3*tj^3*sin(2*t0*omega+2*omega*tj)

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175 -30*Kn*omega^3*tj^3*cos( 2*t0*omega+2*omega*tj)135*Kt*omega*tj*sin(2*t0*omega)+180*Kn*s in(2*t0*omega)+180*Kt*cos(2*t0*omeg a)+135*Kn*omega*tj*cos(2*t0*omega)-30*Kt*omega^2*tj^2*cos(2*t0*omega)30*Kn*omega^2*tj^2*sin(2*t0*omega))/tj^3/omega^5]; P11 = [P11(1,1)*ones(lmx,1) P11(1,2) *ones(lmx,1); P11(2,1)*ones(lmx,1) P11(2,2)*ones(lmx,1)]*V; P12 = [P12(1,1)*ones(lmx,1) P 12(1,2)*ones(lmx,1); P12(2,1)*ones(lmx,1) P12(2,2)*ones(lmx,1)]*V; P13 = [P13(1,1)*ones(lmx,1) P 13(1,2)*ones(lmx,1); P13(2,1)*ones(lmx,1) P13(2,2)*ones(lmx,1)]*V; P14 = [P14(1,1)*ones(lmx,1) P 14(1,2)*ones(lmx,1); P14(2,1)*ones(lmx,1) P14(2,2)*ones(lmx,1)]*V; P21 = [P21(1,1)*ones(lmx,1) P 21(1,2)*ones(lmx,1); P21(2,1)*ones(lmx,1) P21(2,2)*ones(lmx,1)]*V; P22 = [P22(1,1)*ones(lmx,1) P 22(1,2)*ones(lmx,1); P22(2,1)*ones(lmx,1) P22(2,2)*ones(lmx,1)]*V; P23 = [P23(1,1)*ones(lmx,1) P 23(1,2)*ones(lmx,1); P23(2,1)*ones(lmx,1) P23(2,2)*ones(lmx,1)]*V; P24 = [P24(1,1)*ones(lmx,1) P 24(1,2)*ones(lmx,1); P24(2,1)*ones(lmx,1) P24(2,2)*ones(lmx,1)]*V; N11 = -C+1/2*K*tj+P11; N12 = -M+1/12*K*tj^2+P12; N13 = C+1/2*K*tj+P13; N14 = M-1/12*K*tj^2+P14; N21 = M/tj-1/10*K*tj+P21; N22 = 1/2*M-1/12*C*tj-1/120*K*tj^2+P22; N23 = -M/tj+1/10*K*tj+P23; N24 = 1/2*M+1/12*C*tj-1/120*K*tj^2+P24; N1 = [N11 N12; N21 N22]; N2 = [N13 N14; N23 N24]; P1 = [P11 P12; P21 P22]; P2 = [P13 P14; P23 P24]; %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%%% % BUILD GLOBAL MATRICES %%%%%%%%%%%%%%% %%%%%%%%%%%%%%% %%%%%%%%%%%%% %%%%%%%%%% A(1:2*DOF,1:2*DOF) = eye(2*DOF); A(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = N1; A(2*DOF*e+1:2*DOF*e+2 *DOF,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = N2; B(2*DOF*e+1:2*DOF*e+2*DOF ,2*DOF*(e-1)+1:2*DOF*(e-1)+2*DOF) = P1; B(2*DOF*e+1:2*DOF*e+2 *DOF,2*DOF*(e-1)+2*DOF+1:2*DOF*(e1)+2*DOF+2*DOF) = P2; B(1:2*(DOF),E*2*(DOF)+1:(E+1)*2*(DOF)) = PHI; Cvec(1:2*DOF,1) = zeros(2*DOF,1);

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176 Cvec(2*DOF*e+1:2*DOF*e+DOF,1) = C1; Cvec(2*DOF*e+DOF+1:2*DOF*e+2*DOF,1) = C2; end; % end # of elements loop ` Q = A\B; [vec,lam] = eig(Q); CM = max(abs(diag(lam))); D = A\Cvec; % Extract SLE coefficients if CM<1 SLE_vec = inv((eye(size(Q))-Q))*D; SLE = (sum(SLE_vec(locat))); else SLE = nan; end N1 = zeros(2*DOF,2*DOF); N2 = N1; P1 = N1; P2 = P1;

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185 BIOGRAPHICAL SKETCH Mohammad Kurdi was born and raised in the suburbs of Amman, Jordan. He finished his B.Sc degree in mechanical e ngineering in 1995 from the University of Jordan. He worked in his dads family opt ometric practice while attending graduate school at the University of Jordan, where he obtained an M.Sc. in mechanical engineering in 1999. After graduate school he joined R oyal Jordanian Airlines as an aircraft maintenance engineer where he worked for one year then he moved to Jordan Petroleum Refinery Co. and worked for 2 years as a development engineer In August 2002 he enrolled in the mechanical engineering gra duate program at the University of Florida where he obtained a Master of Science in December, 2003, and a Ph.D degree in August, 2005.


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ROBUST MULTICRITERIA OPTIMIZATION OF SURFACE LOCATION ERROR
AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER
UNCERTAINTY















By

MOHAMMAD H. KURDI


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Mohammad H. Kurdi


































To my Mom and Dad.















ACKNOWLEDGMENTS

I would like to thank my advisor Dr. Tony Schmitz for his advice and generous

financial support of my research. I would like to thank Dr. Haftka for his expert advice

and inspiring questions. I would like to thank Dr. Mann for introducing me to the field of

time finite elements. I would like to thank the committee members Dr. Schmitz, Dr.

Haftka, Dr. Mann, Dr. Schuller and Dr. Akcali for their advice, time and effort.

In completing my research I was lucky to be a member of the Machine Tool

Research Center where I had the opportunity to work with intelligent and hard working

graduate students. I would like to thank all fellow members for their helpful suggestions

and interactions. Also, I would like to thank Ms. Christine Schmitz for taking the time to

edit the dissertation draft.

I would like to thank my wife Carolina for her continued support and

encouragement. I would like to thank my daughters Alanis and Alia for bringing laughter

and joy to my life. Finally, I would like to thank my mom and dad for their endless

encouragement and support.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ................................ .......... .. .... .. .... .............. viii

LIST OF FIGURES ......... ......................... ...... ........ ............ ix

N O M E N C L A T U R E .................................................. ................................................ xiv

ABSTRACT ........................................................... xvii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Justification of W ork ............................................... ..... ...... .............. .. 1
L literature R review ................. ..... .. .......................... ...... ........ .......... ...... .
O ptim ization in M achining......................................................... ............... 2
H igh-speed M killing O ptim ization...................................... ........................ 3
M ulti-objective O ptim ization ...................................................... ..... .......... 3
Stability and Surface Location Error ............... .............................................4
Scope of W ork ............................................................... ... .... ........ 6

2 MULTI-OBJECTIVE OPTIMIZATION .......................................... ............... 8

Fundamental Concepts in Multi-Objective Optimization ............................................8
Single and Multi-objective Optimization ............................................................8
Definition of Multi-Objective Optimization Problem.............. ...................10
Definition of Term s .............. ..................................................... 10
Pareto O ptim ality................................................................. ......... ......... 11
M ulti-objective Optimization M ethods ................................................................. 12
Methods with a Priori Articulation of Preferences using a Utility Function......13
Weighted global criteria method ............ .............................................14
W eighted sum m ethod............................................ ........... ............... 14
Exponential w weighted criterion ....................................... ............... 15
W eighted product m ethod ................................... ...................................... 15
C onjoint analysis ........ .................... ........................... ..........................15
Methods with a Priori Articulation of Preferences without using a Utility
F u n c tio n ..................................................................................................1 6


v









Lexicographic m ethod ...................................................................... 16
Goal program m ing m ethods......................................................................... 16
Methods for an a Posteriori Articulation of Preferences.............. .....................16
Bounded objective function method ............................ ...............17
Normal boundary intersection (NBI) method ...........................................17
N orm al constraint (N C) m ethod................................................................ 18
Hom otopy m ethod .................. .............. ..................... ..... 18
Choice of Optimization Method ............... .. ..... .........................18

3 MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM ...........................20

In tro d u ctio n ........................................................................................................... 2 0
M killing Problem .................................20
Milling ModelEquation Chapter 3 Section 1.................. ............................ 20
Solution M ethod ....................................................... ................. 21
Problem Specifics ............. .................... ........ ... ...................... .. 22
Stability B boundary ............................................ .. ......... .. .... ..... ...... .. 22
Surface location error and stability boundary: C1 discontinuity ..................23
T F E A conv erg en ce ............................................................ .....................24
Optimization Method.................... .... ......... .. ................................. 26
Particle Swarm Optimization Technique.................................. ............... 26
Sequential Quadratic Programming (SQP) ............................... ................27
Problem Form ulation .................. ........................... .... ... ..... ............ 27
P problem Statem ent........... ............................................................ .. .... .. ... .. 27
Tradeoff M ethod .................. ....................................... .......... .... 28
R obu st O ptim ization ......... ........................................................ ...... .... ..... 29
Problem solution ...................... ................ .................. ........ 29
R eform ulation of problem .................................... ..................................... 32
B i-objective space ............ .......... .. .. .... .... ...................................... 37
Selection of spindle speed perturbation bandwidth.............................. 38
Case Studies....................... ..................41
Radial im m version (a)... ........ .................................... .... ........ .......... 41
C h ip lo a d (c ) ............................................................................. 4 2
D isc u ssio n ............................................................................................................. 4 6

4 UNCERTAINTY ANALYSIS ............................................................................47

M killing M odel ................................. .. ............. .............. ........... 48
Stability and Surface Location Error Analysis..................................................50
Bi-section Method Convergence Criterion........................................................ 50
Number of Elem ents .................. ............................. ......... ........ .... 50
N um erical Sensitivity Analysis ............................................................................ 51
Truncation Error ......... ...... .. .......... ................. ............. .... 51
Step S ize ................................................................... 52
C ase Studies................................................................................................................52
Stability Sensitivity A nalysis........................................................... ............... 58
Surface Location Error Sensitivity Analysis ................................... .................61









Uncertainty of Stability Boundary and Surface Location Error .............................63
Input Parameters Correlation Effect ................................ ................63
M onte C arlo Sim ulation ........................................................... .....................64
Sensitivity M ethod................................ ........ .......... .......... ...... ........ 65
Latin Hyper-Cube Sampling M ethod ...................................... ............... 68
Robust Optimization under Uncertainty ................ ..............................................69
D discussion ..................................... .................. ............... ........... 70

5 EXPERIMENTAL RESULTS ............................................................................72

C cutting F orce C oefficients............................................................... .....................72
M illin g F o rc e s ............................................................................................... 7 2
Experim ental Procedure ....................................... .............................. 74
Covariance Matrix (Linear Multi-Response Model)...............................................80
Com pliant Tool M odal Param eters....................................... .......................... 84
Stability Lobe V alidation ............................................................................. 87
Stability Lobe U uncertainty ............................................................................ 87
E xperim ental P procedure ........................................................... .....................90
R e su lts ......................................................................... 9 1
Pareto Front V alidation .................................................... .............93
Pareto Front Sim ulation R esults................................... .................................... 93
Experim ental Procedure and R esults................................................................ 95
C onclu sion s ......................................................................10 1

6 SUM M ARY ............... ............... ......... ................... .......... 103

Robust Optim ization A lgorithm ........................................ .......................... 103
Lim stations and Future Research ........................................................... ... .......... 104

APPENDIX

A TIME FINITE ELEMENT ANALYSIS ...................................... ............... 106

B MATLAB CODE.................. ....................... ......... 115

LIST OF REFEREN CES ........................................................... .. ............... 177

BIOGRAPHICAL SKETCH ............................................................. ............... 185
















LIST OF TABLES


Table p

1 Classification of solutions ......................................................... ............... 13

2 Cutting conditions and modal parameters for the tool used in optimization
sim u latio n s ............................................................................................................... 3 0

3 Cutting conditions, modal parameters and cutting force coefficients used in bi-
objective space sim ulations ......................................................... .............. 37

4 Cutting conditions, modal parameters and cutting force coefficients used in
radial im m version case study .............................................................................. 41

5 Milling cutting conditions, modal parameters and cutting force coefficients used
in chip load study case ...................... .. ........................ .... .... ........... 43

6 Cutting force coefficients, modal parameters and cutting conditions of milling
p ro c e s s ........................................................................... 4 9

7 Cutting coefficients for 1 insert endmill for slotting cutting tests..........................77

8 Up milling cutting coefficients for 12% radial immersion ....................................78

9 Estimated cutting force coefficients and their correlation matrix for 7475
aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30
degree h elix angle................................................ ................. 83

10 Tool modal parameters in x and y-directions. .............. ...... .................. 85

11 Correlation coefficient matrix for modal parameters........................ ...............85

12 Surface location error cutting conditions for two Pareto optimal designs with no
u n certainty con sidered ............................................. ......................................... 100















LIST OF FIGURES


Figure pge

1 (a) Typical Pareto front in the criteria space (b) Design variables xl and x2, and
constraint in the design space .................................. ............... ............... 9

2 Pareto optimality and domination relation. ....................................................... 13

3 Schem atic of 2-DOF milling tool................................ ........................ ......... 20

4 Surface location error and its absolute .......................................... ............... 24

5 A typical stability boundary. ............................................ ............................ 24

6 Convergence of stability constraint for 5% radial immersion and different
spindle speeds for an 18 mm axial depth. ..................................... ............... 25

7 Schematic of milling cutting conditions and various types of milling operations...28

8 Stability, IfSE and fJ contours with optimum points overlaid ............................ 31

9 A typical optimum point found; optimum point sensitivity with respect to
spindle speed is apparent...... ....................................................... ...... .........3 1

10 Perturbed average of fsLE validation as optimization criterion that avoids
spindle speed sensitive fsLE .........................................................33

11 Stability, IfS, and f, contours with optimum Pareto front points found using
PSO and SQP (average perturbed spindle speed formulation)............................ 34

12 Pareto front showing optimum points found using three optimization
algorithms/formulations; the same trends are apparent................ .............. ....35

13 Variations in the eigenvalues, surface location error, and removal rate for PSO
and SQP optima, where f, is the objective for both.............................................36

14 Average surface location error contours for 300 rpm bandwidth perturbation,
stability boundary and material removal rate (see Table 3)..................................38

15 F easib le dom ain .................................................. ................ 39









16 Contour lines corresponding to constant spindle speed in feasible region of bi-
objectiv e sp ace ..................................................... ................ 3 9

17 Average surface location error contours for 100 and 300 rpm band width,
stability boundary and material removal rate contours .........................................40

18 Pareto front for spindle speed and axial depth as design variables with radial
immersion 0.508 mm, compared to the case where radial immersion is added as
a third design variable. ........................................ ........................ 43

19 Pareto front using chip load as a third design variable compared to spindle speed
and axial depth as design variables. .............................................. ............... 44

20 Stability, perturbed average If,,E and f, contours with optimum Pareto front
points found using 100 rpm and 400 rpm bandwidth ...........................................45

21 Schem atic of 2-D m killing m odel. ................. ........................... ........................49

22 The effect of error limit in the bisection method on numerical noise in the
sensitivity calculation ................................. ............ ................. .. ..... 53

23 Sensitivity of SLE with respect to Kx. ..... ............ .. ......... ..................54

24 Comparison between 2nd and 4th order central difference formulas.........................55

25 The logarithmic derivative of axial depth with respect to input parameters versus
step size percentage ....................... ........ ............ ................... .. ......56

26 The variation of axial depth blum with respect to a 10% change in nominal input
p aram eters. ........................................................ ................. 57

27 The variation of b,,m with respect to a 10% change in Kt and K,. The sensitivity
of blum with respect to each parameter is superimposed. Linearity of blm,(Xi) can
be ob served (see T able 6) ................................................................. ............... 57

28 Sensitivity of axial depth blim to changes in modal mass M and modal stiffness
K in the x and y-directions (see Table 6) .... ........... ........ ......................... 58

29 Sensitivity of axial depth blum to changes in modal damping C in the x and y-
d irectio n s ............................................................................. 5 9

30 Sensitivity of axial depth blum to changes in spindle speed. The spindle speed
sensitivity is compared here to the modal mass and stiffness in y-direction...........60

31 Sensitivity of axial depth blum to changes in force cutting coefficients in the
tangential Kt and normal directions K........................................... ...............60









32 Sensitivity of surface location error SLE to changes in modal parameters in y-
d irectio n ............................................... ................. .........................6 1

33 Sensitivity of SLE to cutting force coefficients.................................................. 62

34 Sensitivity of SLE to spindle speed and radial depth of cu ................................62

35 Confidence in stability boundary due to input parameters uncertainties using
M onte C arlo sim ulation .......................... ..................... ................. ............... 65

36 Uncertainty boundary in axial depth limit using two standard deviation
confidence interval. ....................... ...................... ................... .. .....66

37 Uncertainty in axial depth using sensitivity and Monte Carlo methods. ...............67

38 Surface location error uncertainty with two standard deviation confidence
interval on the nominal SLE ........... .. ......... ............................ 68

39 Example simulation of cutting forces to facilitate proper selection of
dy n am o m eter ...................................................... ................ 7 5

40 W ork-piece, dynamometer and tool setup.............................................. .......... 76

41 Cutting coefficient in tangential direction (Kt) ....... ... ....................................... 77

42 Cutting coefficient in normal direction (K,) .......... .............. ........................78

43 Simulated and measured forces for 0.12 mm/tooth chip load and 1000 rpm..........79

44 Simulated and measured cutting forces for 0.2mm/tooth chip load, b=0.4 mm
and 5000 rpm .................................................... ................. 79

45 Simulated and measured forces at 20 krpm and b=0.4 mm for slotting ...................80

46 Modal analysis test equipment typically used in machine tool structures. ..............86

47 Frequency response function measurement of tool in x-direction ...........................86

48 Frequency response function measurement of tool in y-direction .........................87

49 Boxplot of stability lobes boundary uncertainty ............................................... 89

50 Histograms of axial depth limit distributions for various spindle speeds ...............89

51 Probability plot of axial depth limit distribution at 10000 rpm spindle speed.........90

52 Schematic of stability tests for partial radial immersion cutting conditions. ...........91









53 Stability lobe generated using mean values of input parameters with
experimental results overlaid, also shown the boxplot corresponding to each
spindle speed used in the measurements. ...................................... ...............92

54 Fast Fourier Transform (FFT) of sound signals for selected stability tests. ............93

55 Stability boundary using mean values in the input parameters Pareto optimal
designs are overlaid for two cases: mean values and uncertain input parameters. ..94

56 Pareto Front of perturbed average SLE and MRR. The Pareto Front with
uncertainty in axial depth is compared to the one with no uncertainty....................95

57 Surface location error experiment schematic. .................................. ............... 97

58 Measured surface location error of b=2.12 mm and the reference dimension (A)
error. .................................................................................9 8

59 Measured surface location error of b=4.45 mm and the reference dimension (A)
error. .................................................................................9 8

60 Boxplot of SLE uncertainty at spindle speeds for 4.45 mm axial depth case...........99

61 Measured surface location error of b=4.45 mm case .....................................99

62 Surface location error of preferred design conditions with no uncertainty
considered in the optimization. Optimum spindle speeds are indicated in the
figure. ............................................................................ 100

63 Slotting cut with time in the cut divided into two elements................................112















NOMENCLATURE


A slotting transformation matrix
Cx modal damping in x-direction
Cy modal damping in y-direction
E number of elements
F(x) vector of objective functions
F utopia point
F average cutting force
Fx average cutting force in x-direction
Fx average normal cutting force in x-direction
Fxe average normal edge cutting force in x-direction
FY average cutting force in y-direction
F average tangential cutting force in y-direction
Fy average tangential edge cutting force in y-direction
IQ Identity matrix of size Q
K, cutting force coefficients matrix defined in Appendix A
K, tangential cutting force coefficient
Kn normal cutting force coefficient
K, edge tangential cutting force coefficient
K edge normal cutting foce coefficient
Kx modal stiffness in x-direction
Ky modal stiffness in y-direction
L sample size
Mx modal mass in x-direction
My modal mass in y-direction
MRR material removal rate
N number of teeth on the cutting tool
PSO particle swarm optimization









Q number of experimental runs
R cutter radius
R2 adjusted coefficient of determination
SLE surface location error
SQP sequential quadratic programming
U used for utility function
U, expanded uncertainty
X feasible design space
X(t) two-element position vector
X, milling model ith input parameter
IY vector of observations in ith response
Z feasible criterion space
Z, Q x p, matrix of rankp,
a radial depth of cut
b axial depth of cut
bi set of goals for objective functions
b,, maximum stable axial depth
c chip load
d deviation from the goals
e MRR constraint limit

QA cutting force coefficients vector defined in Appendix A
fSE surface location error objective function
f material removal rate objective
g, set of inequality constraints
g, absolute value of maximum characteristic multiplier
h, set of equality constraints
h step size used to estimate numerical derivative
p parameter in exponential weighted criterion
p, rank of Z, matrix
r number of response variables
1r correlation coefficient between x and y
uC combined standard uncertainty
Sector of weights (preferences)
x x-direction











x vector of design variables
y y-direction
a parameter used in homotopy method
, vector of unknown constant parameters
/ unbiased estimate of /
3 spindle speed perturbation (half of bandwidth)
E absolute error limit
E, random error vector associated with ith response
A variance-covariance matrix
A system characteristic multipliers
radial depth angle
S radial depth angle at start of cut
ex radial depth angle at end of cut
Q spindle speed
Y covariance matrix
0o- i, j element of covariance matrix Y
Oyj estimate of o-
r tooth passing period
1 damping factor















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ROBUST MULTICRITERIA OPTIMIZATION OF SURFACE LOCATION ERROR
AND MATERIAL REMOVAL RATE IN HIGH-SPEED MILLING UNDER
UNCERTAINTY

By

Mohammad H. Kurdi

August 2005

Chair: Tony L. Schmitz
Cochair: Raphael T. Haftka
Major Department: Mechanical and Aerospace Engineering

High-speed milling (HSM) provides an efficient method for accurate discrete part

fabrication. However, successful implementation requires the selection of appropriate

operating parameters. Balancing the multiple process requirements, including high

material removal rate, maximum part accuracy, chatter avoidance, and adequate surface

finish, to arrive at an optimum solution is difficult without the aid of an optimization

framework.

Despite the attractive gain in productivity that HSM offers, full realization of the

benefits is dependent on the proper selection of cutting parameters. Parameters selected

must achieve the required productivity while maintaining an acceptable accuracy. Milling

models are used to aid in the proper selection of these cutting parameters. They provide

information on whether a cutting condition is stable and/or predict the surface accuracy.

However, this selection is rather tedious, costly and time consuming and might not even









provide an optimum solution. Parameters are selected based on experience until a point is

found that provide the productivity and surface accuracy required. Difficulties

encountered in this selection process include sensitivity of surface accuracy to cutting

parameters, uncertainties in several parameters in the milling model and the

computational effort needed to account for stability and surface accuracy. Therefore,

balancing the multiple requirements, including high material removal rate, minimum

surface location error and chatter avoidance, to arrive at an optimum solution is difficult

without the aid of optimization techniques.

In this dissertation a robust optimization algorithm that accounts for the inherent

process uncertainty and surface location error sensitivity is developed. Two optimization

criteria are considered, namely, surface location error and material removal rate under the

stability constraint. The trade off curve of surface location error versus material removal

rate is calculated for the mean values of input parameters, as well as for a confidence

level in the stability boundary. An experimental validation of the robust optimization

algorithm is also conducted, including an experimental validation of the variation of the

cutting forces as a function of spindle speed. The confidence level in the axial depth limit

and surface location error prediction is found using two methods: 1) sensitivity analysis;

and 2) sampling methods. The sensitivity study highlights the most significant factors

affecting process stability and surface location error. The effect of input parameters

correlation is included in the confidence level predictions using Monte Carlo and Latin

Hyper-Cube sampling methods.


xviii














CHAPTER 1
INTRODUCTION

Justification of Work

Intense competition in manufacturing places a continuous demand on developing

cost-effective manufacturing processes with acceptable dimensional accuracy. High-

speed milling, HSM, offers these benefits provided appropriate operating parameters are

selected. Some typical applications include, but are not limited to, orthopedic surgery [1],

end milling (pocketing) of airframe panels [2] and ball end milling of stamping dies [3, 4]

in automotive manufacturing.

Despite the attractive gain in productivity that HSM offers, full realization of the

benefits is dependent on the proper selection of cutting parameters. Parameters selected

must achieve the required productivity while maintaining an acceptable accuracy. Milling

models are used to aid in the proper selection of these cutting parameters. They give us

information on whether a cutting condition is stable and/or they predict the surface

accuracy. However, this selection is rather tedious, costly and time consuming and might

not even provide an optimum solution. Parameters are selected based on experience until

a point is found that provides the productivity and surface accuracy required. Difficulties

encountered in this selection process include sensitivity of surface accuracy to cutting

parameters, uncertainties in several parameters in the milling model and the

computational effort needed to account for stability and surface accuracy. Therefore,

balancing the multiple requirements, including high material removal rate, f ,









minimum surface location error fSLE and chatter avoidance, to arrive at an optimum

solution, is difficult without the aid of optimization techniques.

Literature Review

The literature review proceeds with a summary of previous implementations of

optimization methods in machining, with particular attention to high-speed milling and

multi-objective optimization. Also, a review of milling models for stability and surface

location error is provided.

Optimization in Machining

Previous research in machining process optimization [5] has focused on

mathematical modeling approaches to determine optimal cutting parameters with regard

to various objective functions. Three main objectives have been recognized: 1) maximum

production rate or minimum cycle time [6-9]; 2) minimum cost [10-21]; and 3) maximum

profit [12, 22], or a combined criterion based on a weighted sum of these [23, 24].

The machining optimization problem can be formulated using deterministic and

probabilistic approaches [11, 25]. Several optimization techniques were used to handle

both formulations. For the deterministic approach they include linear and nonlinear

programming techniques [9, 15, 26, 27], while for the probabilistic approach chance-

constrained programming can be used [17, 28]. Other optimization techniques used in

machining include graphical optimization [12, 22], polynomial geometric programming

[6, 18-20, 29, 30], geometric programming [10] based on quadratic posylognomials

(QPL) [31], goal programming with linear [32, 33] and nonlinear [34] goals, fuzzy

optimization [35], and global search methods such as particle swarm optimization [21]

and simulated annealing [16].









The machining optimization literature can also be classified according to different

constraints and design variables handled. Several authors [7, 14] considered cost

optimization for single-pass milling and turning [10, 17, 19, 20, 29]. The range of

constraints considered are machine tool constraints, such as cutting speed and feed rate,

tool dynamics constraints such as cutting force, power and stability, and product

constraints such as surface roughness. In reference, [17] some of the constraints

considered are of probabilistic nature. Also, multi-pass peripheral and end milling to

maximize production rate are considered [8] under a range of constraints with relevance

to rough milling such as the machine tool limiting power, torque, feed force and feed-

speed boundaries while in another work. In addition to the previous constraints, arbor

rigidity and deflection are used [6].

High-speed Milling Optimization

Few references are found on optimization of high-speed milling. The concept of

adaptive learning (polynomial network) [16] is used to construct a machining model.

Simulated annealing was then used to minimize production cost for rough high-speed

machining operations for three cutting condition parameters namely cutting speed, chip

load and axial depth of cut. A similar study was done for low speed milling [21] where an

artificial neural network was used to build the machining model. However, particle

swarm optimization was used to optimize production cost under machine, tool and

product constraints.

Multi-objective Optimization

Multi-objective optimization (MOO) addresses the issue of competing objectives

using concepts first introduced by Edgeworth [36] then expanded and developed by

Pareto [37], the French-Italian economist who established an optimality concept in the









field of economics based on multiple objectives. A Pareto front [38] is generated that

allows designers to trade off one objective against another.

In the area of machining, Jha [24] studied two objective function optimization

based on cost and rate of production where example constraints were machine power,

cutting speed limitations, depth of cut, and table feed. The two objectives were combined

using weights. Koulams [28] studied single-pass machining considering the influence of

tool chatter failure where a tool failure probability function effect was added as a penalty

cost function to the objective function.

Stability and Surface Location Error

As explained earlier, the full exploitation of HSM demands mathematical models to

predict stability and surface location error. An unstable milling process is caused by a

phenomenon called chatter. Among the first to describe chatter is Taylor, [39] who

described chatter as "the most obscure and delicate of all problems facing the machinist."

Chatter [40] is a self-excited vibration that occurs if the chip width is too large with

respect to the dynamic stiffness of the system. It causes undulations in the machined

surface (poor surface finish) and could result in tool breakage. Extensive work has been

done to generate stability boundaries or lobes. The lobes define a region below which

chatter is nonexistent. Two approaches are used to generate these lobes: 1) analytical [41]

with a continuous cutting model or with an interrupted cutting model [42]; and 2) time

domain simulation [43, 44].

Surface location error is defined as the error in the placement of the milling cutter

teeth when the surface is generated. This error depends on the interaction of work-

piece/tool dynamic stiffness and the cutting forces. The correct prediction of this error

depends on correct prediction of the cutting forces and resulting deflections. Mechanistic









models can be used to estimate these forces. The cutting force is found by summing the

forces acting on incremental sections of a helical cutting edge [45, 46], then the surface

location error is computed based on the static stiffness of the tool [47]. However, the

effect of the deflection of the cutter on the cutting forces is not included. In an

improvement of the previous model, the static deflection is fed back to correct the cutting

forces [48, 49]. A more realistic regenerative force model [50] considered the effect of

undulations in the surface generated by previous tooth passage on the next tooth passage.

In this model the dynamic deflection of the tool imprints waviness on the generated

surface. Using time domain simulation, surface location error, cutting forces and stability

lobes are predicted. An improvement on this model considered [42, 51-53] interrupted

cutting as a factor influencing the stability lobes and surface location error. A newly

developed method uses time finite element analysis (TFEA) to model the governing time

delayed differential equation [54-58]. Regenerative cutting forces and dynamic deflection

of the tool are all implicitly included in the governing differential equation. The

advantage of this method is that it concurrently provides surface location error and

stability information on the milling process in a semi-analytical manner. In this method

the governing differential equation is modeled by dividing the time in the cut into a

number of elements, where displacement and velocity continuity are enforced between

each element. A discrete linear map is formed by mapping the time in the cut to free

vibration. The eigenvalues of the discrete map determine the stability boundaries,

whereas fixed points of the dynamic map determine surface location error (fsLE)









Scope of Work

The purpose of this dissertation is to use optimization as a tool to efficiently

determine preferred and robust operating conditions in HSM, considering multiple

objectives. Although known optimization methods and machining models will be applied,

there are a number of innovative aspects of this research. First, proper formulation of the

objective functions to account for practical application of the preferred conditions is

necessary. The formulation should account for uncertainty in the milling model and

sensitivity of objectives) to process variables. Uncertainty has not previously been

considered. Second, two objectives are simultaneously optimized: surface location error

fSLE and material removal rate, f, Stability and side bounds of design variables are

considered as constraints. Prior research has focused only on the empirical tool life, not

the unavoidable milling dynamics and the inherent limitations they impose. The tradeoff

curve (Pareto front) [38, 59] of fA, and fsLE is generated based on nominal

experimental model parameters. Experimental case studies are conducted to verify the

validity of the Pareto front. The uncertainty in the milling model is addressed using

Monte Carlo simulation and/or sensitivity analysis, where a confidence interval is applied

to the stability limit. The uncertainty of different input parameters such as cutting force

coefficients, tool/work-piece dynamic parameters and milling process parameters are

considered in the uncertainty prediction. This uncertainty is used in the selection of a

robust design that would allow a venue for the practical application of the stability lobe

theory at the shop floor.






7


The dissertation organization proceeds as follows: Chapter 2 gives a general

description of multi-objective optimization; Chapter 3 describes Pareto front generation

formulation of the optimization problem, optimization methods and case studies; Chapter

4 provides the uncertainty analysis of stability and surface location error; Chapter 5

describes the robust optimization algorithm and presents some practical case studies to

verify stability lobes and selected design points on the Pareto front. Chapter 6

summarized the results and outlines future work in this area.


















CHAPTER 2
MULTI-OBJECTIVE OPTIMIZATION

Fundamental Concepts in Multi-Objective Optimization

Optimization is an engineering discipline where extreme values of design criteria

are sought. However, quite often there are multiple conflicting criteria that need to be

handled. Satisfying one of these criteria comes at the expense of another. Multi-objective

optimization deals with such conflicting objectives. It provides a mathematical

framework to arrive at an optimal design state which accommodates the various criteria

demanded by the application.

This chapter begins with a comparison of single- and multiple-objective

optimization. Next, the definition of the multi-objective optimization problem and terms

are explained. Then, a summary of multi-objective optimization methods is presented.

Finally, reasons are given for the choice of the multi-objective optimization method.

Single and Multi-objective Optimization

In single objective optimization one is faced with the problem of finding the

optimum of the objective function. For example considering the decision making

involved in an investment (Figure 1). There are several possible designs in the feasible

domain (A, B and 1-6). These designs are mapped from the design space Figure 1 (b) into

the criteria space Figure 1 (a). In the design space there are two design variables xl

(spindle speed) and x2 (axial depth) where the feasible domain is limited by the










constraint. If we are only concerned about profit with no regard to risk (profit is our

single objective), then point B would correspond to the maximum profit optimum design.

A risk averse investor would choose risk as an objective function. The optimum design

for the risk objective would correspond to point A. Depending on the objective function,

constraints, and design variables, different techniques are used to solve for the single-

objective optimum. However, in multi-objective optimization, a vector of objectives

needs to be optimized. For the investment example, two objectives are considered. In this

two objective case there is no unique optimum, rather a set of optimum solutions is

found. In Figure 1, for instance, points A, B and 5-6 are all candidate solutions.

Depending on the decision maker's risk aversion, a single solution can be chosen from

that set.



B
90% -. B constraint

design space
Feasible domain
3
*4 6 x2

2 t
S 5 B
4 0
criteria space

10%- A 1 A Feasible domain

$1000 Profit $5000 x

(a) (b)
Figure 1. (a) Typical Pareto front in the criteria space (b) Design variables x] and x2, and
constraint in the design space.


The similarity between single- and multi-objective optimization makes it possible

to use the same optimization algorithms as for the single-objective case. The only









required modification is to transform the multi-objective problem into a single one. This

may be accomplished in a number of ways, such as introducing a vector of preferences,

v, to get a single objective as a weighted sum, or by solving one of the objectives for a

different set of limits on the other objectives [60-62]. In any case, a set of optimal

solutions are found rather than a single one. It is worth noting, however, that when the

objective functions are non-conflicting, the optimal set reduces to a single solution rather

than a set. This can be related to the commodity example. For instance, if we want to

maximize both cost and quality, then solution B is the only one.

Definition of Multi-Objective Optimization Problem

The mathematical representation of the multi-objective optimization problem is

formulated as follows:

Minimize F(X) = (k), F (5),...,Fk()1]
subject to gJ (,)<0, j = 1,2,..., (2.1)
h, ()= 0, =1,2,...,e

where subscript k denotes the number of objective functions F, m is the number of

inequality constraints and e is the number of equality constraints; and x e E" is the

vector of design variables, where n is the number of independent design variables.

Definition of Terms

The feasible design space (inference space), X, is defined as the set of design

variables that satisfy the constraint set, or

{gx,()<0, j=1,2,...,m; andh(, )=0, =l1,2,...,e (2.2)









Thefeasible criterion space, Z, (often called the cost space or attainable set) is

defined as the set of cost functions F () such that V maps to a point in the feasible

design space X or (F ) Y G X .

The preferences refer to the decision maker's opinion in terms of points in the

criterion space. The preferences can be set apriori (before solution set is obtained) or a

posteriori (after solution set is obtained).

The preference function is an abstract function of points in the criterion space

which perfectly satisfies the decision maker's preferences.

The utilityfunction is an amalgamation of individual utility functions of each

objective that approximates the preference function, which typically cannot be expressed

in mathematical form. The formation of a utility function requires insight into the

physical aspects of each objective. This may require finding the Pareto front (explained

next) in order to properly formulate the utility function.

A utopia point is a point F0 e Zk that satisfies F = minimum{F () i e X} for

each i = 1,2..., k.

Pareto Optimality

The multi-objective optimization problem has more than one global optimum. The

predominant concept in defining an optimal point is that of Pareto optimality [37] which

is defined as follows: a point, x*" X, is Pareto optimal if there does not exist another

point, x e X, such that F(x)
That is the set of Pareto optimal points dominates any other optimal set. This can be

defined by the domination relation [60], where a vector x, dominates a vector x2 if: l is









at least as good as j2 for all the objectives, and 2, is strictly better than j2 for at least

one objective. To better understand the domination relation, or Pareto optimality, an

example is provided [63] (Figure 2). A two-objective problem of maximizingfi and

minimizing2 is addressed. Table 1 presents the set of solutions, classified with respect to

each other. A solution P is designated as +,- or = depending on whether it is better, worse

or equal to a solution Q for the corresponding objective. For example, comparing

solutions A and B, we find that solution A is worse forfi (maximizingfi) compared to B,

therefore it is designated as (-) for objectivefi. Also, comparing objective2 we find that

solution A is worse than B (-). Now for a solution to belong to the non-dominated set it

must be as good as the other solutions for both objectives and it must be strictly better for

at least one objective. Considering solution A in Figure 2 we see it is worse than all other

solutions (dominated); solution B is also worse than C for both objectives (dominated).

Solution C is not dominated by point E (couple (+,-) at the intersection of the row E and

the column C) and it does not dominate point E (couple (-,+) at the intersection of the row

C and the column E), therefore points C and E are non-dominated. Solution D is worse

than C for both objectives therefore solution D is dominated.

Multi-objective Optimization Methods

As explained earlier the solution to a multi-objective optimization problem is a

Pareto optimal set that gives a tradeoff between the different objective functions

considered. Depending on the decision maker's preferences, a solution is selected from

that set. Therefore multi-objective optimization methods can be categorized according to

how the designer articulates his preferences (by order or by importance of objectives).

This includes three cases: apriori, aposteriori, and progressive articulation of






13


preferences. A brief overview of the methods used is outlined. For a detailed description

of the methods the reader is referred to reference 64.



Table 1. Classification of solutions


Solutions A B C D E
A (-,-) (-,-) (-,-) (-, -)
B (+,+) (-,-) (-,=) (-,=)
c (+,+) (+,+) (+,+) (-,+)
D (++) (+,=) (-,=)
E (+,+) (+,=) (+,-) (+,=)


B D


10 12 14 16


Figure 2. Pareto optimality and domination relation.

Methods with a Priori Articulation of Preferences using a Utility Function

In these methods, the decision maker's preferences are incorporated as parameters

in terms of a utility function apriori. Typically these parameters can be coefficients,

exponents, constraint limits, etc. These parameters determine the tradeoff of objectives

before implementation of the optimization method. The optimum solution found would










reflect the tradeoff made a priori. Depending on whether the solution found turned out to

satisfy the preferences or not, the decision maker can re-adjust the parameters to get a

better solution. However the beauty of these methods is that they do not require doing a

multi-objective optimization problem since the a priori preferences and utility function

reduce the optimization to a single one.

Weighted global criteria method

In this method, all objective functions are combined to form a single utility

function. The weighted global criterion is a type of utility function U in which

parameters are used to model preferences. The simplest form of a general utility function

can be defined as

k
U= w,, (x))P, F, (x)> 0Vi, or (2.3)
i=1


U [w (x)]P, F (x)> Vi, (2.4)
i=1

where i5 is the vector of weights set by the decision maker such that 4i > 0 and

w = 1. The difference between the two above formulations is related to conditions


required for Pareto optimality. Complete discussion can be found in reference [64].


Weighted sum method

This is a special case of the weighted global criteria method in which the exponent

P is equal to one; that is,

k
U Zw, (x). (2.5)
i=1

The method is easy to implement and guarantees finding the Pareto optimal set,

provided the objective function space is convex. However, a uniformly distributed set of









weights does not necessarily find a uniformly distributed Pareto optimal set, which makes

it difficult to obtain a Pareto solution in a desired region of the objective space.

Exponential weighted criterion

It is defined as follows:


U (eP' -1)ePF(x), (2.6)
i=1

where the argument of the summation represents an individual utility function for F( (x).

Weighted product method

To avoid transforming objective functions with similar significance and different

order of magnitude, one may consider the following formulation [65]:

k w
U= J[F (x)] (2.7)
1=1

where w, are weights indicating the relative significance of the objective functions.

Conjoint analysis

This method [66, 67] uses a concept borrowed from marketing, where a product is

characterized by a set of attributes, with each attribute having a set of levels. An

aggregated utility function is developed by direct interaction with the customer/designer;

the designer is asked to rate, rank order, or choose a set of product bundles. In

engineering design studies, we can assume that people will choose their most preferred

product alternative. Conjoint analysis takes these sets of attributes and converts them into

a utility function that specifies the preferences that the customer has for all of the

product's attributes and attribute levels. The advantage of this method is that it

automatically takes into account marginal diminishing utility (i.e., no cost is expended in

a design that does not really have practical utility).









Methods with a Priori Articulation of Preferences without using a Utility Function

Lexicographic method

Here the objective functions are arranged in a descending order of importance [68].

The highest preference objective is optimized with no regard to the other objectives, and

then a single objective problem is solved consecutively (in order of preference of

objectives) for a set of limits on the optimums of previously solved for objectives. This

can be defined as

Minimize F (x)

subject to F (x) 1, (2.8)
i = 1,2,...,k

where i represents the function's position in the preferred sequence and FI (xi)

represents the optimum of the jth objective function found in the jth iteration.

Goal programming methods

Here, goals bi are specified for each objective function FJ (.) [69]. Then the total

k
deviation from the goals, \d is optimized, where d is the deviation from the goal
J=1

bJ for thejth objective.

Methods for an a Posteriori Articulation of Preferences

The inability of the decision maker to set preferences a priori in terms of a utility

function makes it necessary to generate a Pareto optimal set after which an aposteriori

articulation of preferences is made; such methods are sometimes referred to as cafeteria

or generate-first-choose-later. These methods however require the generation of the

Pareto optimal set which may be prohibitively time consuming. It is worth noting that

repeatedly solving the weighted sum approaches presented earlier can be used to find the









entire Pareto optimal solution for convex criteria space; however, these methods fail to

provide an even distribution of points that can accurately represent the Pareto optimal set.

Bounded objective function method

In this method [70], the single most important objective function, Fg (.), is

minimized, while all other objective functions are added as constraints with lower and

upper bounds such that / I, F (2) < c,, i = 1,2,...,k,i # s. A variation of this method is

thee constraint [71] or trade-off method in which the lower bound 1 is excluded and

the Pareto optimal set is obtained using a systematic variation of S,. This method is

particularly useful in finding the Pareto optimal solution for convex or non-convex

objective spaces alike. However, choice of the constraint vector s must lie within the

minimum and maximum of the objective function considered; otherwise, no feasible

solution will be found. Also the distribution of the Pareto optimal solution will usually be

non-uniform for the objective functions) minimized.

Normal boundary intersection (NBI) method

This method provides a means for obtaining an even distribution of Pareto optimal

points for a consistent variation in parameter vector of weights [72, 73], even with a non-

convex Pareto optimal set (a deficiency found in weighted sum method). For each

parameter weight the NBI problem is solved to find an optimum point that intersects the

criteria feasible space boundary, however, for non-convex problems, some of the

solutions found can be non Pareto optimal. Details of the method can be found in the

references.









Normal constraint (NC) method

This method uses normalized objective functions with a Pareto filter to eliminate

non-Pareto optimal solutions [74]. The individual minima of the normalized objective

functions are used to construct the vertices of the utopia hyper-plane. A sample of evenly

distributed points on the utopia hyper-plane is found from a linear combination of the

vertices with consistently varied weights in criterion space. Each Pareto optimal point is

found by solving a separate normalized single-objective function with additional

inequality constraints for the remaining normalized objective functions.

Homotopy method

In this method the convex combination of bi-objective functions (1- a)f + af, is

optimized for an initial value of the parameters Then homotopy curve tracking methods

are used to generate the Pareto optimal solution curve for a e [0,1] whenever the curve is

smooth [75, 76] or even non-smooth [62, 77] at points corresponding to changes in the

set of active constraints.

Choice of Optimization Method

The ease of implementation of the c -constraint method [71] for a bi-objective

problem makes it a good candidate method. In this method, one of the objectives is

optimized for systematic variation of limits (e, ,...,, ) on the second objective. A

uniform distribution of the Pareto optimal set can be found for the constrained objective.

There is no limitation on the convexity or non-convexity of the objective space in finding

the Pareto optimal set. However, choice of the constraint set of limits (e, e..., ) must

lie within the minimum and maximum of the objective function considered; otherwise, no

feasible solution would be found. In our case the material removal rate (fm) and






19


surface location error ( fLE) are the bi-objective criteria. The material removal rate

objective would be a better choice for the constrained objective, since the set of limits

(E, ..., ) of f, constraint can be more easily constructed according to designer's

preference, whereas that would be difficult for the fsLE objective.














CHAPTER 3
MILLING MULTI-OBJECTIVE OPTIMIZATION PROBLEM

Introduction

In this chapter, a description of the milling problem and solution method used to

solve the mathematical model is presented. Two optimization methods of interest are

briefly described. These methods are then applied to the multi-objective optimization

problem and a discussion of results is provided.

Milling Problem

Milling Model

The schematic for a two degree-of-freedom (2-DOF) milling process is shown in

Figure 3 (repeated here). With the assumption of either a compliant tool or a structure, a

summation of forces gives the following equation of motion:


Figure 3. Schematic of 2-DOF milling tool









mX 0 1x(t) c 0 xt) kX 0 x(t)= iXQ) (3.1)
0 my y(t) 0 Cy y(t) 0 ky y(t) Fy(t) I


where the terms mx, cx, kx and my, cy, k are the modal mass, viscous damping, and

stiffness terms, and F, and Fy are the cutting forces in the x and y directions, respectively.

A compact form of the milling process can be found by considering the chip thickness

variation and forces on each tooth (a detailed derivation is provided in references [54-58]

and Appendix A):

MX(t) + C(t) + KX(t) = Kc (t)b (t) (t r))+ (t) b (3.2)


where X(t)=[x(t) y(t)]T is the two-element position vector and M, C, and K are the

2x2 modal mass, damping, and stiffness matrices, Kc and f0 (function of the cutting

force coefficients) are defined in Appendix A, b is the axial depth of cut, r = 60/(NQ) is

the tooth passing period in seconds, Q is the spindle speed given in rev/min (rpm), and N

is the number of teeth on the cutting tool. As shown in Eq. (3.2), the milling model is

dependent on modal parameters of the tool/work-piece combination and the cutting force

coefficients.

Solution Method

As described in Chapter 1, a solution of Eq. (3.2) can be completed using numerical

time-domain simulation [43, 44, 50] or the semi-analytical TFEA [54-58]. Compared to

the first approach, TFEA can obtain rapid process performance calculations of surface

location error, fL,, and stability. The computational efficiency of TFEA compared to

conventional time-domain simulation methods makes it the most attractive candidate for

use in the optimization formulation. In this method a discrete linear map is generated that









relates the vibration while the tool is in the cut to free vibration out of the cut. Stability of

the milling process can be determined using the eigenvalues of the dynamic map, while

surface location error (see Appendix A) is found from the fixed points of the dynamic

map. Details can be found in references [54-58]. An added advantage of TFEA is that it

provides a clear and distinct definition of stability boundaries (i.e., eigenvalues of the

milling equation with an absolute value greater than one identify unstable conditions).

Problem Specifics

In this section, the calculation of the stability boundary is analyzed, the continuity

of surface location error and stability boundary is addressed, TFEA convergence is

described, and sensitivity of the milling model to cutting force coefficients is defined.

Stability Boundary

In order to find the axial depth limit, bl,,, of neutral stability at corresponding input

parameters, the bi-section method is used in the TFEA algorithm to solve for bm,, at

which the maximum characteristic multiplier is equal to one (stability limit)

gA =max < 1 (3.3)

where I is the eigenvalues of the dynamic map. An absolute error is used as a criterion

for convergence

b, < (3.4)
b

where E corresponds to the error tolerance and b is the root corresponding to

max = 1 at iteration i. The value of E is set based on the numerical accuracy required

in the calculation of b,,,. A value of E = le 3 can be adequate for the calculation of bl,,.









Surface location error and stability boundary: C1 discontinuity

Correct use of an optimization method depends on its limitations. Gradient-based

methods, for example, depend on C1 continuity (the first derivative of the function is

continuous) of the objective functions ( f and IfSL ) and stability constraint (Eq. (3.3)).

The objective f, is defined analytically in Eq. (3.6), where it is clear that it is C1

continuous. However, the IfSL and stability (g,) functions are only found numerically

using TFEA. A graphical description of both functions provides some insight into the

continuity of these functions. Figure 4 depicts the variation of fSL and IfS,, as a

function of spindle speed for a typical set of cutting parameters. Although fIS is C1

continuous in the region where it is defined (stable region), IfSLE is C1 discontinuous.

This can be easily verified analytically by considering the functions f(x) = x and

f (x) = Ix =x { for x > 0 and -x for x <0}. The absolute function is clearly C1 discontinuous

at x = 0. The same argument can be made for the near-zero IfSL range shown in Figure 4.

In Figure 5 the variation of stability function g, versus spindle speed shows lobe peaks

where C1 (slope) discontinuity of g, is also observed. C1 discontinuity makes

convergence of gradient-based optimization algorithms near the discontinuity rather

difficult. This requires the use of multiple initial guesses in order to converge to even a

local optimum.



















2
1
0

10 12 14 16 18 20
S ( x31epr

Figure 4. Surface location error and its absolute. Discontinuity of the absolute surface
location error is apparent in the lower insert.


3.5

3

2.5

25

1.5




0.5
1 1.1 1.2 1.3 1.4 1.5
W(x 1 arpm)
Figure 5. A typical stability boundary. The cusps where C' discontinuity in the stability
boundary are depicted.



TFEA convergence

The convergence of TFEA depends on the cutting parameters. A higher number of

elements must be used when convergence is not achieved. Either If, or the stability


boundary g, can be used to check for convergence. A typical procedure to test for the


convergence of finite element meshes is to compare the change in the estimated value (g


or ISE I) as the number of elements is increased (mesh refinement). In Figure 6, the









dependence of convergence on the spindle speed is shown for a randomly selected cutting

condition of 5% radial immersion (percentage of radial depth of cut to tool diameter) and

18 mm axial depth. As seen in Figure 6, the flawed convergence for a small number of

elements (=1) would give the impression of a sufficient number of elements. However,

further increasing the number of elements (=12) shows poor convergence for the low

speed. This can be due to the fact that as the spindle speed decreases, the time in the cut

increases, which requires a higher number of elements to achieve convergence. The fact

that the optimization algorithm will pick milling parameters within the design space

makes it necessary to choose a rather high number of elements to ensure convergence

anywhere in the design space. However, a penalty in computational time is incurred.

40 50
500 (rpm)
1000 40
20 30
30

10

-20
0 10 20 30 40 2 4 6
No. of elements No. of elements
4
10 3 x10

8
2 /

S 4 \

2
0 0
20 25 30 35 40 10 15 20
No. of elements No. of elements
Figure 6. Convergence of stability constraint for 5% radial immersion and different spindle
speeds for an 18 mm axial depth. We can see that convergence at lower speeds
require substantially more number of elements.










Optimization Method

Optimization methods can be categorized according to the searching method used

to find the optimum [78]. They are either direct where only the values of the objective

function and constraints are used to guide the search strategy, or gradient-based, where

first and/or second order derivatives guide the search process. Particle swarm

optimization (PSO) and sequential quadratic programming (SQP) will be used to test the

feasibility of both methods, respectively, for the problem at hand.

Particle Swarm Optimization Technique

Particle swarm optimization is an evolutionary computation technique developed

by Kennedy and Eberhart [79, 80]. It can be used for solving single or multi-objective

optimization problems. To find the optimum solution, a swarm of particles explores the

feasible design space. Each particle keeps track of its own personal best (pbest) fitness

and the global best (gbest) fitness achieved during design space exploration. The velocity

of each particle is updated toward its pbest and the gbest positions. Acceleration is

weighted by a random term, with separate random numbers being generated for

acceleration toward pbest and gbest.

In order to accommodate constraints, Xiaohui et al. [79] presented a modified

particle swarm optimization algorithm, where PSO is started with a group of feasible

solutions and a feasibility function is used to check if the newly explored solutions satisfy

all the constraints. All the particles keep only those feasible solutions in their memory

while discarding infeasible ones.









Sequential Quadratic Programming (SQP)

The basic idea of this method is that it transforms the nonlinear optimization

problem into a quadratic sub-problem around the initial guess. The nonlinear objective

function and constraints are transformed into their quadratic and linear approximations.

The quadratic problem is then solved iteratively and the step size is found by minimizing

a descent function along the search direction. Standard optimization algorithms may be

used to solve the quadratic sub-problem.

Usually SQP leads to identification of only local optima. In order to better

converge to the global optimum, a number of initial guesses is used to scan the design

space and the optimum of these local optima is close to the global optimum.

Problem Formulation

In this section, the multi-objective optimization problem is defined and then a

description of the tradeoff method is given. The problem solution is then presented in the

order it has been addressed in the robust optimization section. Finally, discussion of the

simulation results is provided.

Problem Statement

The problem of minimizing surface location error If, I and maximizing material

removal rate f, is stated as follows:

min[ fsE (a, b, c, N, ),- fI f(a,b, c, N, ()],
(3.5)
subject to: g (b,Q) = max i(a,b,N,Q) <1

where g, is the stability constraint obtained from the dynamic map eigenvalues, fs, is

found from the fixed points, and the mean f, is given as:

fm = abcNQ, (3.6)









where a, b, c, N and 2 are radial depth of cut, axial depth of cut, feed per tooth (chip

load), number of teeth, and spindle speed, respectively (Figure 7). From Eq. (3.5) it can

be seen that only the stability constraint is not a function of the feed per tooth. In Eq.(3.5)

, fAE and f, are explicitly stated as a function of cutting conditions (a, b, c, N and ( ).

This reflects the relative ease by which these conditions can be adjusted to achieve

optimality of the objectives.




Radial depth
N=2 of cut, a Axi
Axial
depth,
Chip load,c b


Y Ost=O L y Pst=0 "y


x ex x
a R
I ex= 7 | ex I
Slotting Up milling Down milling
Figure 7. Schematic of milling cutting conditions and various types of milling operations.


Tradeoff Method

To address the multi-objective problem the constraint method is used, where the

two-objective problem is transformed into a single objective problem of minimizing one

objective with a set of different limits on the second objective. Each time the single

objective problem is solved, the second objective is constrained to a specific value until a

sufficient set of optimum points are found. These are used to generate the Pareto front

[38] of the two objectives. In the case that f,, is chosen as the objective function to be

minimized then Eq. (3.5) is transformed to:









min fE{ (a, b, c, N,),
subject to: f, (a,b,c,N, ) ) (3.7)
gp (a,b,N,Q ) = maxl (a,b,N, Q) < 1,
for a series of k selected limits (e) on f,,.

Where the cutting conditions: a, b, c, N and n are the design variables. On the other

hand if f, is chosen as the objective function to be maximized, then Eq. (3.5) is

transformed to:

min f (a,b,c, N, ),
subject to : If, (a,b,c,N, Q) e,, for i = 1 ... k
(3.8)
gp (a,b,N, )= maxl(a,b,N, ) < 1,
for a series of k selected limits (e) on I f I.

It should be noted that applying Eq. (3.7) using the SQP method is more

straightforward than Eq. (3.8). The reason is that in order to use a number of initial

guesses along the Ifs, contour in Eq. (3.8), the axial depth corresponding to that IfL

needs to be found, whereas in Eq. (3.7) the axial depth can be explicitly expressed found

as shown in Eq. (3.6).

Robust Optimization

Problem solution

In the first iteration of the problem, only axial depth (b) and spindle speed (n2) are

considered as design variables. Other cutting conditions are held fixed (Table 2) for a

down milling cut. Modal parameters for a single degree-of-freedom tool with one

dynamic mode in x andy directions are used (Table 2). The nominal values of the

tangential (Kt) and normal (K,) cutting force coefficients are 550 N/mm2 and 200 N/mm2,

respectively. The SQP method is used to find the Pareto front using the formulation in

Eq. (3.7). Here Ifs I is minimized for a set of limits on f As mentioned earlier, the









SQP method is a local search method that is highly dependent on C1 continuity of the

objective function and constraints. To obtain a global optimum, a number of initial

guesses are used along each f, constraint limit. A set of optimum points are obtained

for these initial guesses. The minimum of these optimum points is nominated as a global

optimum. The number of initial guesses is increased and another run of the optimization

simulation is made to check the validity of that global optimum.



Table 2. Cutting conditions and modal parameters for Tool used in optimization
simulations


M (kg) C (Ns/m) K (N/m)
0.056 0 3.94 0 1.52x106 0
0 0.061 0 3.86 0 1.67x106
Tool diameter (mm) c (mm) a (mm) N
19.05 0.178 0.76 2
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
550 x 106 200 x 106 0 0


In this formulation, the minimum f,, I points were found to favor spindle speeds

where the tooth passing frequency is equal to an integer fraction of the system's natural

frequency (Figure 8), which corresponds to the most flexible mode (these are the

traditionally-selected 'best' speeds which are located near the lobe peaks in stability lobe

diagrams). Because ,,L can undergo large changes in value for small perturbations in 2

at these optimum points, the formulation provided in Eqs. (3.7) and (3.8) leads to optima

which are highly sensitive to spindle speed variation (Figure 8) To show the sensitivity

of these optimum points, a typical optimum point is superimposed on a graph of fs,E Ivs.

2 in Figure 9. It is seen that the optimum point is located in a high ,,L slope region.












IF i


i*I
Ii

ii;

!I I

iii I
!i
!! i'
II i!
*uJ 1


I"i
i


i





!.F-
I
r
i I



I 3
I
i jj9


5 10 15 20 25 30
Q (x 103 rpm)


Figure 8. Stability, IfS, L and fJ contours with optimum points overlaid. The figure
shows that optimum points occur in regions sensitive to spindle speed variation
(Table 2).


Surface location error
-- Spindle speed sensitive optimum







4'


4 6 8 10 12 14
Q2 (x 13 rpm)
Figure 9. A typical optimum point found; optimum point sensitivity with respect to
spindle speed is apparent (Table 2).









Reformulation of problem

The optimization problem was redefined in order to avoid convergence to spindle

speed-sensitive optima. Two approaches were applied: 1) an additional constraint was

added to the IfS slope; and 2) the f,, objective was redefined as the average of three

perturbed spindle speeds. The latter proved to be more robust than the former. This is due

to the difficulty in setting the value of the If I slope constraint apriori. The spindle

speed perturbed form of the problem transforms Eqs. (3.7) and (3.8) to


in fLE (b, Q +3) + fsLE (b, Q) + fsLE (b, Q )|
3
subject to: f (b, ) < e,, for i = 1... k (3.9)
(g, (b, 6)n g (b,i)n g, (b,+ )} < 1,
for a series of selected limits (e) on f,

and

min f, (b, ),
S rSLE (b, Q + 3) + \fSLE (b, 4) + SLE ](b,Q1 S)
subject to: fSLE (bQ+-) Q) l fSLE' Q- 5 C, for i = 1...k
3
(g, (b, 6)n g, (b, )n g, (b, + )} < 1,
for a series of selected limits (s) on average perturbed Ifs ,

(3.10)

where 5is the spindle speed perturbation selected by the designer (a typical value for our

analyses was 50 rpm). A study of spindle speed perturbation selection is provided in the

next section.

The validity of the perturbed IfsL, average as a convergence criteria can be seen in

Figure 10. In this figure the perturbed average Ifs, is plotted with fsE where points A

and B correspond to highly and moderately spindle speed-sensitive fs,L respectively.











The average perturbed If, I at point A (high slope point) is shown to be higher than at


point B. Therefore, using the perturbed average |, I as an objective function criteria can


avoid convergence to spindle speed sensitive fE (such as IfsE region near point A).






Perturbed average fSLE
IfsLEI
---------------I-----
1.2







5" 0.8

around point A relatively
more sensitive fSLE| region
A
0.6



0.4



0.2
10.5 11 11.5 12 12.5 13 13.5 14
C (x 103 rpm)

Figure 10. Perturbed average of fSE validation as optimization criterion that avoids

spindle speed sensitive fSLE Shown in the figure are points A (close to steep

slope region of fSLE ) and B (close to moderate slope region of fSLE ), the

perturbed average of fsLE near A is higher than at B. Therefore, using the

perturbed average as an optimum criterion is valid.


The SQP method is used to solve Eqs. (3.9) and (3.10). In case Eq. (3.9) is

implemented then initial guesses of 2 and b (design variables) are made along the f,


contour. In the other case (Eq. (3.10)) the initial guesses of 2 and b are made along If,,










contour. The number of initial guesses along the constraint is made such that convergence

is towards a global optimum. The initial guesses for the spindle speed are increased in

625 rpm increments for the corresponding spindle speed range considered. Also, the PSO

method is used to solve Eq. (3.8). When using PSO, the optimum points do not tend to

converge to spindle speed sensitive optimums. Therefore, there is no need to solve the

reformulated form of the problem in PSO. This leads to a fewer number of evaluations of


IfsLE I and is a computationally more efficient optimization method.

A comparison of the three optimization schemes is shown in Figure 11 and Figure

12. Figure 11 shows the optima for each approach superimposed on the corresponding

stability lobe diagram. In Figure 12, the Pareto fronts for the three methods are shown.

The optimum points found using the two SQP formulations closely agree with the PSO

method (Figure 12).


5 eL bI\\\ b' iJr3ly" r
I':.. p \. \ SQP SLE objective
"" \ SQPMRRobjectve
4.5 0- PSO MRR objective








14 16 1 20 22 24 26 2 30
35
7





10

so -- 7. 71.-.


14 16 18 20 22 24 26 28 $0
0 (x 1 rpm)

Figure 11. Stability, IfS| and f, contours with optimum Pareto front points found
using PSO and SQP (average perturbed spindle speed formulation). The figure
shows that optimum points are not in regions sensitive to spindle speed (Table 2).









20
S SQP SLE objective ? /
18 SQP MRR objective
1- PSO MRR objective
16

14,

12

10



f 6-

4-

2
0

50 100 150 200 250 300 350 400 450
fM,, mm /s)
Figure 12. Pareto front showing optimum points found using three optimization
algorithms/formulations; the same trends are apparent. However, the SQP
methods required additional computational time (Table 2).
Although the PSO points show the same trend, some improvement in the fitness is

still possible relative to the SQP results. Because the PSO search inherently avoided

optimum points that are spindle speed insensitive, there is no need to use average

perturbedfsLE as with SQP, which leads to a decreased number of fLE evaluations in

PSO. However, narrow optimum points may go undetected when using PSO.

As noted, when comparing the Pareto fronts in Figure 12, it is seen that the PSO

approach did not converge to the same fitness as SQP method. A check of the optimum

points which correspond to a value of If,,LE I = 4 rim, for example, shows that PSO

converged to 100 mm3/s, while SQP converged to 150 mm3/s. To better understand this











result, the design space was divided between the two design vectors, b and Q, for SQP


and PSO using a factor, a, that was normalized between 0 and 1. The PSO and SQP


optimums were normalized to a = 0 and 1, respectively. Next, the stability constraint


(g,), f., and IfS E were plotted against that ratio. In Figure 13 it is seen that


discontinuities exist in the fs, I constraint and the first derivative of the eigenvalue


constraint within this region. Although PSO is not significantly affected by a


discontinuity in the derivative constraint, it can be affected by a discontinuity of the fsA,


constraint, where the discontinuity tends to narrow the search region of the swarm.





1.1 PSO optimum at 4tm -
O SQP optimum 41tm
o


0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
150
0 PSO optimum at 4pm
S100 O SQP optimum 41m

50 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a
160
0 PSO optimum at 41tm
140 SQP optimum 41tm

I 120 -
1200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
a

Figure 13. Variations in the eigenvalues, surface location error, and removal rate for PSO
and SQP optima, where f, is the objective for both. The discontinuities in the surface
location error cause PSO to not converge on the SQP optimum.









Bi-objective space

In this section, the bi-objective domain (the feasible space of the objective

functions) of average perturbed If, I and f, for the set of input parameters listed in

Table 3 for an up milling case is provided. Figure 14 shows the objective contours in the

design space of spindle speed (n2) and axial depth (b). The respective bi-objective space

is shown in Figure 15 and Figure 16. In Figure 15 the contours of constant axial depth are

shown, while the contours of constant spindle speeds are shown in Figure 16. These

figures give an idea of the feasible design and bi-objective space. It can be seen that the

bi-objective feasible space can be non-convex (not all points on a straight line connecting

two points in the feasible domain belong to that domain). This makes the choice of using

the tradeoff method as a multi-objective optimization approach a suitable one, since this

method can handle both convex and non-convex problems. A good observation can be

made from Figure 15, where it can be seen that for the high f, region with high b

values, the relative sensitivity of If, increases compared to the lower f, region.



Table 3. Cutting conditions, modal parameters and cutting force coefficients used in bi-
objective space simulations


M (kg) C (Ns/m) K (N/m)
0.44 0 83 0 4.45x106 0
0 0.35 0 90 0 3.55x106
Tool diameter (mm) c (mm) a (mm) N
25.4 0.1 21.8 1
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
700 x 106 20 x 106 46 x 10 33 x 10









Selection of spindle speed perturbation bandwidth

In Figure 10, it was shown that the average perturbation of IfS provided an

adequate optimization criteria. However, the choice of the spindle speed perturbation step

size or bandwidth, 2,, depends on the designer preference. Any spindle speed

perturbation in If, would avoid convergence to sensitive If, optima. Depending on

the machining center spindle drive accuracy, the perturbation bandwidth can be set

accordingly. The average perturbed If,, contours of 100 and 300 rpm bandwidth are

shown in Figure 17 (use Table 3 parameters). The high slope region of average If, I in

the 100 rpm bandwidth case is replaced by higher values of average If, I, making the

optimization formulation favor insensitive spindle speed If




C.) Stability boundary
5.5 200 f mm /S
5 2C00 perturbed average |fsLE| m
4.5 00

%07 /h h :ei \ d,
S 3 100 \ n SLE :l: ul.


2.5 \ 200
\900 1000 4 C
2 \

1.5 700 0
1 z/ 35 "

10 12 14 16 18 20
S(x 10 rpm)
Figure 14. Average surface location error contours for 300 rpm bandwidth perturbation,
stability boundary and material removal rate (see Table 3).














160

,140

-120
Cl
S100

80 / Feasible domain

B 60

o 40

20
20 TPareto front

500 1000 1500 2000 2500
fMRR(mm3/s)
Figure 15. Feasible domain. Contour lines corresponding to constant axial depth in the
stable region in the bi-objective space (see Table 3).


120

100 -

80

60-

40-

20-

1-.2 ,, pK0'.....)
500 1000 1500 2000
fMR (mm )
Figure 16. Contour lines corresponding to constant spindle
objective space (see Table 3).


19

18

17

16

15

14

13

12

11


2500


speed in feasible region of bi-









100 rpm bandwidth


i
3

2


1
10


12 14 16 18
Q (x 103 rpm)


300 rpm bandwidth


12 14 16 18
Q (x 103 rpm)


Figure 17. Average surface location error contours for 100 and 300 rpm band width,
stability boundary and material removal rate contours (see Table 3).


5


4

E3


2


1










Case Studies

As opposed to the previous analysis of two design variables (2 and b), two cases

of an added third design variable were analyzed. The first one was for radial immersion

(a) and the second one was for chip load (c). These cases are compared to the two design

variable case.

Radial immersion (a)

Previous simulations considered spindle speed and axial depth of cut as design

variables. Another simulation was completed using radial immersion as a third design

variable for an up milling cut. It was compared to a two design variable case where radial

immersion was held constant at 0.508 mm in a 25.4 mm tool (Table 4). Figure 18 shows

the Pareto front for these two cases. It is seen that adding radial immersion as a third

design variable improved the value of perturbed average If,, with respect to the constant

radial immersion case. The optimum radial immersion found was 0.58 mm for all

optimum points up to 500 mm3/s. In both simulations the same spindle speed perturbation

(6 = 170 rpm) was used. As seen in Figure 18, a better calculation of the Pareto front

(smoother than Figure 12) is found by using small increments in the spindle speed (each

100 rpm) initial guesses. However, the If,, found in Figure 18 appear to be

unrealistically small which may warrant further analysis.



Table 4. Cutting conditions, modal parameters and cutting force coefficients used in
radial immersion case study
M (kg) C (Ns/m) K (N/m)
0.25 0 34.4 0 1.30 x106 0
0 0.23 0 27.0 0 1.20 x106
Tool diameter (mm) c (mm) a (mm) N
25.4 0.1 0.508 2
Kt(N/m) (N/m)) Kte (N/) K(Nm) Kne (N/m)
700 x 106 210 x 106 0 0









Chip load (c)

To study the effect of chip load on surface location error, it is added as a third

design variable in addition to spindle speed and axial depth. The parameters used in this

study are listed in Table 5 for a down milling case. For the two design variable case (0.1

mm/tooth chip load), the Pareto optimal points are found for two different bandwidths,

100 rpm and 400 rpm, respectively (Figure 20). It is noted that 2 of the optimum points

is almost constant up to 700 mm3/s ( 31,325 rpm) where it changes to another almost

constant 2 (29,500 rpm) for the higher f, range. Also the effect of bandwidth size

does not show significant effect on the optimum points found. Figure 19 shows the Pareto

front for a constant chip load of 0.1 mm/tooth compared to the three design variable case,

where the chip load (3rd design variable) side constraints are from 0.01 mm/tooth to 0.2

mm/tooth. An improvement in the average perturbed If,,E can be seen. It should be noted

here that for the latter case, 2 is also found to be constant (31,325 rpm same as two

design variable case) while the chip load increased from 0.16 to 0.2 mm/tooth. The effect

of adding the chip load is therefore seen as an improvement in the fitness of average

perturbed If,, objective, where further improvement is possible while eliminating the

need to switch to a lower speed (29,500 rpm) where the If, S error is much higher. This

explains the agreement between the two design variables case and three design variable

case in the f, range below 600 mm3/s. When higher f, is needed the two design

variable case fails to account for the f, constraint at the same spindle speed. However

the three design variable case (with chip load) can accommodate this by increasing the

chip load while keeping the spindle speed unchanged. This makes the I| in the three

design variable case substantially lower.















0.02


S0.015


S 0.01


0 100 200 300
fR (mm3s)


400 500 600


Figure 18. Pareto front for spindle speed and axial depth as design variables with radial
immersion 0.508 mm, compared to the case where radial immersion is added as a
third design variable. The optimum radial immersion for the latter case is 0.58
mm up to 500 mm3/s (see Table 4).


Table 5. Milling cutting conditions, modal parameters and cutting force coefficients used
in chip load study case


M (kg) C (Ns/m) K (N/m)
0.027 0 7 0 1.0x106 0
0 0.03 0 2 0 1.6x106
Tool diameter (mm) c (mm) a (mm) N
12.7 0.1 0.635 2
Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
600 x 106 180 x 106 0 0












chip load constant
chip load 3rd design


c=0.16 to 0.2 mm/tooth


200


400


Figure 19. Pareto front using chip load as a third design variable compared to spindle
speed and axial depth as design variables. For the three design variable case, an
improvement in the average surface location error can be seen (see Table 5).


600


f (mm 3s
MRR


800


1000


~c~s~


^-^-"





























16 -0- Optimum points 100 rpm bandwidth



14



12











6
4lO












10 15 20 25 30 35 40
2 (rpm)



Figure 20. Stability, perturbed average If,,, and f, contours with optimum Pareto front points found using 100 rpm and 400 rpm

bandwidth. This case study shows the difficulty in selecting optimum points based on experience (Table 5).









Discussion

The formulations provided in Eqs. (3.9) and (3.10) proved adequate in finding the

Pareto optimal set insensitive to spindle speed variation, provided an appropriate number

of initial guesses is made. Also, the Eq. (3.9) formulation is easier to apply using the SQP

method, where the initial guesses are made along the f, contour.

The generation of the Pareto front for the multi-design variable case can be rather

time-consuming. However, if the designer is given that freedom of choice, it might be a

necessity. For example, the effect of adding chip load or radial immersion as a third

design variable gave a substantial improvement in the surface location error in

comparison to the two design variable case. This is counterintuitive to using a lower

value of c or a as means of reducing the surface location error.

The effect of spindle speed perturbation bandwidth on the sensitivity of optimum

points is rather complex. Qualitatively, in Figure 17 it is shown that increasing the

bandwidth from 100 rpm to 300 rpm had the same effect of increasing the value of IfS,,L

near the sensitive region. Further investigation is needed to establish a quantitative

relation between bandwidth and sensitivity of optimum points.














CHAPTER 4
UNCERTAINTY ANALYSIS

In Chapter 3, optimization was used to find preferable designs for two objectives:

material removal rate (MRR) and surface location error [48, 81, 82] (SLE), with a Pareto

front, or tradeoff curve, found for the two competing objectives. Although the milling

model used in the optimization algorithm is deterministic (time finite element analysis),

uncertainties in the input parameters to the model limit the confidence in the optimum

predictions. These input parameters include cutting force coefficients (material- and

process-dependent), tool modal parameters, and cutting conditions. By accounting for

these uncertainties it is possible to arrive at a robust optimum operating condition.

In previous studies [83-85], uncertainty in the milling process was handled from a

control perspective. The uncertainty in the cutting force was accommodated using a

control system. The force controller was designed to compensate for known process

effects and accounted for the force-feed nonlinearity inherent in metal cutting operations.

In this study, the uncertainties in the milling model are estimated using sensitivity

analysis and Monte Carlo simulation. This enables selection of a preferred design that

takes into account the inherent uncertainty in the model a priori.

This chapter begins with a description of the milling model and continues with a

discussion of stability lobes and surface location error analysis with regard to their

numerical accuracy. Sensitivity analysis is discussed in the next section. Then, case

studies for the numerical accuracy of the sensitivities of the maximum stable axial depth,

blm, and SLE are presented for a typical two degree-of-freedom tool. This enables us to









carry out the stability lobe and surface location error sensitivity analysis in the next two

sections. Sensitivity is used to determine the effect of input parameters on b,,m and SLE.

This enables the determination of which parameters) is the highest contributor to

stability enhancement and SLE reduction. The uncertainties in bm, and SLE predictions

are then calculated using two methods 1) the Monte Carlo simulation; and 2) the use of

numerical derivatives of the system characteristic multipliers to determine sensitivities.

The uncertainty in axial depth effects a reduction in the MRR, and the SLE uncertainty

provides bounds on SLE mean expected value. This allows robust optimization that takes

into consideration both performance and uncertainty.

Milling Model

A schematic of a two degree-of-freedom milling tool is shown in Figure 21. The

tool/work-piece dynamics and cutting forces are used to formulate the governing delay

differential equation for the system. Solution of the delay differential equation is found

using time finite element analysis (TFEA) [54-56]. This method provides the means for

predicting the milling process stability and quality (SLE). However, the uncertainty in the

input parameters to the solution method places an uncertainty on the stability and SLE

prediction. These parameters are divided into two groups; 1) uncertainty from lack of

knowledge of the tool modal matrices, K, C and M, and the cutting force coefficients

(mechanistic force model); and 2) uncertainty in other machining parameters, such as

spindle speed, chip load and radial depth. To estimate the parameters in the former,

modal testing is used to measure the dynamic parameters while cutting tests are

completed to estimate the cutting force coefficients. In the modal parameter estimation

the peak amplitude method is used to fit the measured frequency response function. In

this method [86, 87], the peak of the magnitude of the frequency response function










corresponds to the natural frequency. From this the half power frequencies are used to

estimate the damping ratio. Table 6 lists the mean modal values for 25.4 mm diameter

endmill having a 12 helix angle with 114 mm overhang length and the corresponding

cutting force coefficients for 6061 aluminum (assuming a mechanistic force model, see

Chapter 5). The cutting conditions are also listed in the table. These parameters will be

used in the simulations in this chapter for a down milling cut.


Feed
----------------


Figure 21. Schematic of 2-D milling model. Surface location error (SLE) due to phasing
between cutting force and tool displacement is also shown.


Table 6. Cutting force coefficients, modal parameters and cutting conditions of milling
process.


M 1. K(. I x10 6) C(N.s/m) i
x 0.44 4.45 83 0.030
Y 0.44 3.55 90.9 0.03
K,(. m2 x106) K ,(. 2 x106) K,,( mx103) K,(. mx103)
600 180 6 12
Tool diameter (mm) radial depth,a (mm) chip load, c (mm/tooth) N
25.4 0.508 0.1 1









Stability and Surface Location Error Analysis

The stability lobes are used to represent the stable space of axial depth (b) and

spindle speed 12 of a milling process. In TFEA [54-57], a discrete map is used to match

the tool-free vibration while out of the cut, with the tool vibration in the cut. The system

characteristic multipliers (A ) of the map provide the stable cutting zone where max A

is less than one.

TFEA provides a field of max A in the design space of b and Q. The limit of

stability, blm can be found using root-finding numerical techniques. Here we use the bi-

section root-finding method. The convergence criterion of the bi-section method should

account for the amplification of numerical noise induced by sensitivity estimation. It

should be noted that the number of elements affects the accuracy of the estimation.

For calculation of SLE in TFEA, the numerical noise is only due to the number of

elements. In this section we will discuss the effect of both the convergence criterion and

the number of elements on the sensitivity estimation of bl.m and SLE.

Bi-section Method Convergence Criterion

As described in Chapter 3 the axial depth limit, blim, was calculated using the bi-

section method (Eq. (3.4)). Although a relatively large value of E can be adequate for the

calculation of the stability lobes, a tighter limit is needed to calculate the sensitivities.

This is attributed to amplification of numerical noise in the derivative calculation. This

comparison is made in the Case Studies section.

Number of Elements

The accuracy of TFEA prediction of stability and SLE is highly dependent on the

number of elements used. The effect of the number of elements is even more apparent









when calculating the sensitivity of the prediction, where a higher number of elements is

needed to eliminate numerical noise from the sensitivity calculation.

Numerical Sensitivity Analysis

The sensitivity of axial depth to input parameters (Sb / axi) is cumbersome to

compute analytically using the TFEA method; therefore, a numerical derivative is used

by implementing a small perturbation.

Factors which affect accurate calculation of sensitivity to inputs include: 1) central

difference truncation error; and 2) step size selection. Therefore, a balance needs to be

achieved in determining the sensitivity that provides a stable estimate of the sensitivity

while maintaining computational efficiency. In the following, we describe these factors

and their consideration in the calculation of stability and SLE sensitivities.

Truncation Error

The central difference method is used in the sensitivity calculation. The formula for

this method is

Bb b1- b
b +0(h2), (4.1)
OXA 2h

where h denotes the step size in input parameter X,, b b (X, + h) b = b(X, h) and

O(h2) is the 2nd order truncation error. A higher order formula with 4th order truncation

error O(h4) can also be used. However, as shown in Eq. (4.2), it is two times more

computationally expensive than Eq. (4.1),

b -b2 +8b, -8b +b
X, 12h ) (4.2)
8 X 12h









In order to help decide whether the higher truncation error formula need be applied

(Eq. (4.2)), the sensitivity of b,,m with respect to modal stiffness Kx is calculated as a

function of step size h. This comparison is made in the Case Studies section.

Step Size

The step size, h, in Eqs. ((4.1) and (4.2)) should be carefully chosen. This is

especially important when there is numerical noise in the calculated bum, due to the

convergence criterion (Eq. (1)). The step size should be large enough to be out of the

numerical noise range, however, not so large that the non-linear variation in the output

(b,,m or SLE) takes effect. The following section illustrates this idea.

Case Studies


In this section, numerical estimations of the sensitivity are made based on different

variations of convergence criterion, number of elements, sensitivity analysis formula (Eq.

(4.1) and Eq. (4.2)), and step size. The comparisons are made for a 10 krpm spindle

speed, 10 elements and e =3x10-4 unless otherwise noted. The logarithmic derivative can

be used in making these comparisons by evaluating the percentage of change in an output

(axial depth, b) due to a percentage change in the input, X,. It is expressed as

aln((b) _X Qb
(4.3)
aln(X) b c8X

To illustrate the effect of convergence criterion, the logarithmic derivative of blum

with respect to Mx (the X direction modal mass) is calculated for two error limits as a

function of step size percentage (/oh = AX, IX, x 100), see Figure 22. It can be seen that a

tighter error limit nearly eliminates the numerical noise in the derivative calculation.










The effect of the number of elements on SLE sensitivity is illustrated in Figure 23,

where the SLE sensitivity with respect to Kx is calculated. The OSLE / Kx is used to

illustrate the effect of the number of elements because it is known that the SLE does not

depend on the Kx stiffness (tool feeding direction being the x-axis). Therefore

8SLE / 8Kx = 0, which would amplify and illustrate more clearly the effect of the number


of elements on the sensitivity estimation. The higher number of elements provides a

larger stable region of sensitivity. It should be noted that the 2nd order finite difference

method is used in this sensitivity comparison and the bi-section convergence criterion is

not applicable here since SLE is found from fixed points of the dynamic map (see Eq.

(A.18) in Appendix A) when the cutting conditions provide a stable cut.





2nd order central difference = 3e-4
-0.2 E=10 = 3e-10

-0.4

-0.6

-0.8 -







-1.6

-1.8

2 i I I I
0 0.1 0.2 0.3 0.4 0.5
%h
Figure 22. The effect of error limit in the bisection method on numerical noise in the
sensitivity calculation (see Table 6).







54


-23
x10
14
1-- E=10
12 ---- E=30 -
--E=50
10

8

6

84-'i












%h
-2



-4-

-6 I I I I
0 0.1 012 03 0 .4 0.5



Figure 23. Sensitivity of SLE with respect to Kx. The higher number of elements, E,
provides more stable sensitivity estimation. The second order finite difference
formula is used here (see Table 6).



Figure 24 shows the effect of the central difference truncation error. A finite step


size percentage is needed to reach a stable value of the derivative for both formulas. It


can be seen that Eq. (4.2) gives a wider range of step sizes at which the sensitivity


calculation is stable. However, the improved stability range, or reduction in numerical


noise, is not significant to sacrifice computational efficiency for its usage.









-9
x 10
2.5,
2nd Order central difference
4th order central difference
2 Convergence limit e= 3e-4
E=10


1.5



O1 i i i



0.5




0 0.1 0.2 0.3 0.4 0.5
%h
Figure 24. Comparison between 2nd and 4th order central difference formulas. The 4th
order formula shows a wider stable region for step size, but higher computation
time (see Table 6).


The importance of step size selection can be illustrated by Figure 25, which shows

the logarithmic derivative of axial depth with respect to input parameters versus step size

percentage. It can be seen that the step size should be chosen high enough to be out of the

numerical noise range but not so high so that the non-linear variation is included (in this

range of %h only D is non-linear). The figure also indicates the relative sensitivity of

axial depth to each input parameter, spindle speed having the largest effect followed by

modal mass and stiffness.













1-










K2 C- -M K- Kn


-4 i\
^ ~~ ---














sensiiviesi i ver step size o see Table 6).








s = 3 x 10-4 E=10, and the 2nd order finite difference approximation give correct

calculation of sensitivity, the variations of b to modal parameters and cutting coefficients

are plotted in Figure 26 and Figure 27, respectively. Also, the slope predicted using Eq.
-3-

























(4.1) with h 0.2% is superimposed on the same plot. The suitable selection ofh is


indicated by the tangency of the predicted slope to the functional variation. On the other

hand, it can be seen that when the variation is linear, the linear approximation can be

accurate for a large variation of the input parameter.
accurate for a large variation of the input parameter.













K
y
C
y
M
y
Sensitivity Prediction


5
-c


/7
^ -'-

//8r
^^*-Q^^


100
AX/X


Figure 26. The variation of axial depth b1lm with respect to a 10% change in nominal input
parameters. The sensitivity of blm with respect to each parameter is superimposed.
Linearity and non-linearity of bin(X,) can be observed (see Table 6).


4.8


Sensitivity Prediction


90 95 100 105 110
AXl/Xi (x 100)

Figure 27. The variation of blm with respect to a 10% change in Kt and K,. The sensitivity
of bl1; with respect to each parameter is superimposed. Linearity of bl,(Xi) can be
observed (see Table 6).







58


Stability Sensitivity Analysis

In this section, calculations of the sensitivity of bm,, to the input parameters are

provided. The parameters used in the sensitivity calculations are provided in Table 7. In

Figure 28 a comparison between the sensitivities of stiffness, K, and modal mass, M, are

compared in the x (feed) and y-directions of the tool. As can be seen in the figure, the

sensitivities in the x and y-directions are comparable in magnitude; however, the

sensitivity in they-direction is inaccurate near discontinuities in the system characteristic

multipliers. This will be explained in the Uncertainty section with a graphic depicting

these discontinuities.




Table 7. Parameters used in sensitivity analysis.


h (%) E C
0.2 10 2

20

15 -

10 -

X- 5'




-5

-10 -

-15

-20
5


central difference
nd order


Figure 28. Sensitivity of axial depth blim to changes in modal mass M and modal
stiffness K in the x and y-directions (see Table 6).


8

3x10 '


bh


S(rpm x103)
- Ky -


Mx My


I
:I` n



'
iI










In Figure 29, the effect of damping on the stability is shown to be minimal

compared to the modal stiffness and mass. This is a somewhat counter-intuitive result,

but can be explained by regeneration (undulations in the cut surface experienced by the

tooth in the current cut that are caused by the tooth vibration in the previous cut), which

is a primary physical phenomenon that causes instability. The modal mass and stiffness

have a great effect on the system's natural frequency, which has a significant effect on

regeneration. This also explains the result shown in Figure 30, where the sensitivity of

axial depth bl,m to a change in spindle speed is significant and comparable to modal mass

and stiffness. The effect of cutting force coefficients is shown in Figure 31, where the

tangential cutting force coefficient, Kt, has more effect on the axial depth limit than the

normal direction coefficient, K,.



30
K
C
X
20 C


10 -


0_ ------


-10 \ t


-20- C1 discontinuities in bhmM


-30
5 10 15 20
0 (x 103)
Figure 29. Sensitivity of axial depth bl,m to changes in modal damping C in the x and y-
directions. The damping sensitivity is compared to modal stiffness sensitivity in
the x-direction (see Table 6).





























-150


K ]C1 discontinuities in blm
-K
y


-200 _____ M
y


-250
5 10 15 20
Q (rpm x03)
Figure 30. Sensitivity of axial depth bl,m to changes in spindle speed. The spindle speed
sensitivity is compared here to the modal mass and stiffness in y-direction (see
Table 6).


U.4
0.4-----------------------------------------------
--... Kt
0.2 K

0-


-0.2

S-0.4


%
-0.6

S-0.8 Jk h I. ..



-1.2


-1.4
5 10 15 20

Q (rpm x03)
Figure 31. Sensitivity of axial depth bl,m to changes in force cutting coefficients in the
tangential Kt and normal directions K,. Higher sensitivity can be seen for Kt (see
Table 6).










Surface Location Error Sensitivity Analysis

The sensitivity of surface location error, SLE, to changes in input parameters is

examined here. The parameters listed in Table 6 are used with b=l mm and down milling

case. In Figure 32, the sensitivity of SLE to changes in modal parameters in they-

direction is shown. Again, it can be seen that changes in Ky and My contribute more than

Cy to a change in SLE. In Figure 33, the effect of cutting force coefficients is shown,

where it is observed that the highest contributors to SLE sensitivity are K, and Kte. Also,

in Figure 34, SLE sensitivity to spindle speed and radial depth, rtep, is shown. Substantial

sensitivity to spindle speed can be seen. This is due to the dependence of SLE on the

relationship between the tool point frequency response and the selected spindle speed. As

the spindle speed changes, it tracks different parts of the response.



250
K
Y
lM Y
200- C



150



100 l



50



5 10 15 20
Q (rpm x 103)
Figure 32. Sensitivity of surface location error SLE to changes in modal parameters in y-
direction (see Table 6).













15


K
ne


Q (rpm x103)
Figure 33. Sensitivity of SLE to cutting force coefficients (see Table 6).


500


400


step


300


200


100


Q (x 103)
Figure 34. Sensitivity of SLE to spindle speed and radial depth of cut (see Table 6).









Uncertainty of Stability Boundary and Surface Location Error

Input Parameters Correlation Effect

The correlation between the input parameters can have significant effect on the

prediction of uncertainty. Neglecting the correlation can result in erroneous estimation of

the uncertainty, especially when the input parameters are highly correlated. Inclusion of

the covariance matrix between parameters is necessary in this case. The input parameters

can be classified into three groups: dynamic modal parameters of the tool (work-piece

assumed rigid), cutting force coefficients and machining parameters (e.g., radial step and

spindle speed). In Chapter 5, estimation of the correlation between parameters of the first

two groups is explained and used in the uncertainty prediction.

The combined standard uncertainty uc can be found using sensitivities of output

(b,,m or SLE) to input parameters. For the case of axial depth limit, uc is given as [88]:


1m m 1m 1m1 OA
uc (bhm)2 = u (X) +2ZZ y '1imbu( XX), (4.4)


where u(X,) refers to the standard uncertainty in the input parameter X1, u(X,,Xj) is the

estimated covariance between parameters X, and X,,. and m is the number of input

parameters. The degree of the correlation between X, and X, is characterized by the

correlation coefficient

u X(,,X,)
r (X,, X)= u (4.5)
(x,) (x,)

In the Monte Carlo and Latin Hype-Cube sampling methods (described next), the

multivariate normal distribution can be used to estimate the confidence level, in which

case the covariance matrix between parameters controls the random sampling procedure.









Monte Carlo Simulation

The combined standard uncertainty, uc, of the stability boundary (blm) and surface

location error (SLE) can be predicted using Monte Carlo simulation. In this method, a

random sample of size n is selected from the population of each input parameter. A

normal distribution of the input parameters is assumed. In the sample n, the nominal

value and standard deviation of each input parameter are used to generate the sample.

The axial depth limit and surface location error are then calculated using TFEA for each

point in the sample. The standard deviation of the predicted bi,, and SLE is then

calculated from sample output for the range of spindle speeds of interest. It should be

noted here that in doing so, no correlation between the input parameters is assumed,

which is a common, and sometimes erroneous, approach.

To illustrate the effect of uncertainty in the input parameters on stability boundary

uncertainty, standard uncertainties of 5%, 0.5%, 0.001% and 0.5% are assigned to

nominal values of the cutting force coefficients, modal parameters, radial step, and

spindle speed, respectively. The values of the standard uncertainties assigned correspond

to practical variation in the parameters involved. The parameters are assumed to be

uncorrelated here. A sample size of 1000 is used. The stability boundary uncertainty is

found, as shown in Figure 35, for one standard deviation interval around the neutral

stability boundary.










mean one std
mean
mean + one std


5 10 15 20
2 (rpm x03)
Figure 35. Confidence in stability boundary due to input parameters uncertainties using
Monte Carlo simulation (see Table 6).


Sensitivity Method


The combined standard uncertainty uc in axial depth limit while neglecting

correlation between input parameters can be obtained from Eq. (4.4) as


I m b bh
uc (blhm)= lml(X),
vI \^ 8X, )


(4.6)


where u(X,) refers to the standard uncertainty in the input parameter X, (same used for

Monte Carlo method), and m is the number of input parameters. Although this relation

assumes no correlation between input parameters it should be noted that cutting force









coefficients (Kt, Kn, Kte, Kne) and modal parameters (K, C, M) may be correlated in

practice.

The same standard uncertainty is assumed in the input parameters as in previous

sections and the confidence level in axial depth limit is calculated for an interval of + 2

uc(bjzm). Figure 36 shows the close agreement found using the two methods. However, it

should be noted that the sensitivity method can be inaccurate near points where the

function (b,,m) is C1 discontinuous. Figure 37 shows the direct correspondence between

the inaccurate sensitivity and C1 discontinuity inA The C1 discontinuity in blum leads to

inaccurate estimation of uc(bizm) (see Eq. (4.6)).



20
Sensitivity
18 -- Monte Carlo
Nominal
16

14

12





10




5 10 15 20
Q (x 103)
Figure 36. Uncertainty boundary in axial depth limit using two standard deviation
confidence interval. Uncertainty is calculated using sensitivity method and Monte
Carlo method (see Table 6).
Carlo method (see Table 6).










Derivative method
S2 -- Monte Carlo Simulation


0


1
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



0-



5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1








Q (x 103)
Figure 37. Uncertainty in axial depth using sensitivity and Monte Carlo methods.
Inaccuracies in the sensitivity method can be seen near C1 discontinuity in the real
and imaginary part of system characteristic multipliers (see Table 6).


It should be noted here that predicting the uncertainty by Eq. (4.6) uses a linear

approximation. The standard uncertainties assumed earlier are small where the linear

approximation is still valid. However, if the uncertainties in the input parameters are

large, then that linear approximation is no longer valid. In this case, simple random

sampling methods (such as Monte Carlo simulation) are more appropriate.

The surface location error uncertainty is found similarly using both methods.

However, as shown earlier (see Figure 32 and Figure 34), the SLE sensitivities are

accurate and do not depend on the characteristic multipliers' continuity. Since the SLE is

only defined for stable cutting conditions (see Eq. (A. 18) in Appendix A) and explains








the close prediction of uncertainty in SLE using sensitivity and Monte Carlo methods

(Figure 38).


30


I \
o~ l\
5 "'

0 -'

5 \ i

0-


/



I\
!i \l
i ':


[I ,;

I~


Monte Carlo method
Sensitivity method


Q2 (x 103)
Figure 38. Surface location error uncertainty with two standard deviation confidence
interval on the nominal SLE. Close agreement is observed (see Table 6).

Latin Hyper-Cube Sampling Method
This method was originally proposed as a variance reduction technique [89] in

which the estimated variance is asymptotically lower than with simple random sampling

(Monte Carlo method) [90, 91]. That is, for a sample size L, this method gives a lower

estimate of the output variance than is possible with the Monte Carlo method. The basic

idea of this method is that each value (or range of values) of a variable is represented in

the sample, no matter which value turns out to be the most important. In this way, the









sampling distribution is divided into a number of strata with a random selection inside

each stratum. The Latin Hyper-Cube method will be used in Chapter 5 for predicting the

standard combined uncertainty of the stability and surface location error cutting tests in

that chapter.

Robust Optimization under Uncertainty

In order to account for uncertainty in the axial depth stability limit, the safety factor

design analogy is used here. The deterministic optimization algorithm implemented in

Chapter 3 (Eq. (3.9)), repeated here, can be modified to account for the axial depth

uncertainty.




min fSLE,, (b, +) + fSLE (b, ) + fSLE (b, 3)
3
subject to: f, (b, Q) < e,, for i = 1 ... k (4.7)
{g, (b, ) ng, (b, )n g, (b, + 6)} <1,
for a series of selected limits (e) on fe,



Therefore, the axial depth b used in the stability constraint is set equal to an uncertainty

inflated value. That is, b is replaced by b+ Ue, where Ue = kuc (b) is the expanded

uncertainty, k is a factor that estimates the uncertainty confidence interval and uc(b) is

the combined standard uncertainty in the axial depth. Therefore Eq. (4.7) becomes



min fsLE, (b, +) + fSLE (b, Q) + fSLE (b, )|
3
subject to: f, (b,Q) e,, for i = 1...k (4.8)
{g, (b+Ue, ) g,) (b+Ue,)n g, (b+Ue,,+6)}< 1,
for a series of selected limits (e) on f,,,









Discussion

In this chapter, the sensitivities of axial depth limit and surface location error to

model input uncertainties were studied. Numerical estimation of the sensitivities can be

challenging, where several factors contribute to the accuracy of the estimation. Step size

is one of the significant factors that affect the accuracy of the estimation.

The sensitivity analysis aids in identifying the relative contribution of the milling

model input parameters to the sensitivity of either axial depth limit or surface location

error. For the case of axial depth, the spindle speed, followed by modal stiffness and

mass, is the most significant contributor. In the case of cutting force coefficients, the

tangential cutting force coefficient is found to contribute more to the sensitivity than the

normal cutting force coefficient. As for the surface location error sensitivity, the same

trend can be observed. However, for the cutting force coefficients, the edge tangential

cutting force coefficient significantly contributes to the SLE.

The uncertainty in axial depth and surface location error was predicted using two

methods: the sensitivity method and the Monte Carlo simulation approach. Comparable

agreement is shown. However, the sensitivity method is more efficient computationally.

For example, in the case of SLE uncertainty prediction, Monte Carlo simulation required

9.34 hours, while the sensitivity method needed only 0.26 hours (36 times more

efficient). It is noted that for the uc(SLE) case, when the milling parameters are well into

the stable region, the accuracy of the sensitivity method is not sacrificed at the cost of

efficiency as is the case for uc(b) at discontinuities in the characteristic multipliers.

Finally, the optimization algorithm introduced in Chapter 3 was modified to

account for confidence levels in the axial depth limit. This allows robust optimization to






71


account for inherent uncertainty in the mean values of the input parameters. In Chapter 5

an implementation of this algorithm is demonstrated.













CHAPTER 5
EXPERIMENTAL RESULTS

The milling model accuracy depends on reliable determination of cutting force

coefficients and tool or work-piece modal parameters. These values are found

experimentally and their uncertainties contribute to the uncertainty of the model

prediction. In this chapter, the experimental procedure used to determine these

parameters is described and then the optimization algorithm is executed using the

experimentally determined input parameters to find the Pareto optimal points. Another set

of experiments is completed to validate/invalidate these optimal points. Using the

optimization algorithm, the strength and weakness of the mathematical model or solution

method can be obtained.

Cutting Force Coefficients

Milling Forces

The average milling forces during one tooth period in the x and y-directions are [92,

93],


=Nb [K,cos(2) -K [20-sin(20)]]+ N[-K, sin(0)+K, cos(0)]l
'(5.1)
S Nbc [K, [2-sin(2)] + Kcos(20)] Kcos() + Ksin )
< 8ffcos(O+ sin(


where K, and K are the tangential and normal edge cutting force coefficients,

respectively. In slotting tests (see Figure 7), the entry and exit angles of the cutter are









, = 0 and 0ex = .i, respectively. The average forces per tooth period for this case are

found to be:

Nb Nb
F= K c- Kne
4 KZ (5.2)
SNb Nb
F = Kc+- Ke
4 Z

Equation (5.2) can be written as a function of chip load (c) as:

FF = FcC + F, (q = x, y,z) (5.3)

The experimental procedure consists of completing multiple cutting tests at varying

chip loads and recording the cutting forces. For each chip load increment, the average

cutting forces in the x and y-directions are measured, and then a linear regression of the

data points is made to extract the cutting coefficients using Eqs. (5.2) and (5.3):

4F orF
Kt Kte y
Nb Nb (54)
4F(5.4)
K K ne
Nb Nb

For radial immersions less than the cutter diameter, the entry and exit angles differ

from the slotting case. For up-milling (see Figure 7) the entry and exit angles of the cutter


are b, =0 and x = cos 1- a- where a is the radial depth of cut. Substituting in Eq.


(5.1) gives:



F Nb= K, cos(2 )- 1]- K [2q -sin(2 )+-sin( (2J)] Nb sin )+ Ke[cos(ex)- 1]
S(5.5)
(5.5)


Factoring Eq. (5.5) in terms of chip load c gives:








Fx = Fxcc + Fxe (5.6)

F Nb [K, [cos(2x)-1] -K [2x sin (20)]]
(5.7)
F Nb _Kt, sin ( x) + Kn [cos(e)- ]].

Similarly, the following equations are obtained for the y-direction.

Nb[K, [2 sin(2qJe) +Kn [cos(2x)-l ]] [K cos(Oex)-1]+Ksin(ex)]

(5.8)

F= Fyc +F (5.9)


=c Nb [K, [2, -sin(2ex) + K [cos(25ex-1]]
(5.10)
Fy = [ K cos (ex) 1+ Ke sin ( )]

Writing Eqs. (5.7) and (5.10) in matrix form to solve for the cutting coefficients the

final equation can be expressed as shown in Eq. (5.11).



Fx; Nb[ cos(20e)-l -2 x + sin(2 ex) 0 0 K,
FC= 8-r 20 -sin(2 ex) cos(2,)-l 0 0 K,
&Fo 0 Nb[ -sin(0,) cos(ex)- KK,
SFye 0 0 2 1-cos(o) -sin( x) K,
(5.11)

The same procedure can be used to solve for the cutting coefficients in the down-

milling case (Figure 7).

Experimental Procedure
Proper selection of a suitable dynamometer to measure the dynamic cutting forces

is important. Some of the factors that need to be addressed are the calibration range of the










dynamometer and its dynamic response. Simulation of the cutting forces helps in

addressing the issue of cutting force magnitude range. Using time-domain simulation of

the cutting forces and approximate cutting coefficient values, an estimate of the typical

range of cutting forces can be found. Euler integration is used to solve for the tool

displacement during the cut in the 2nd order differential equation (Eq. (3.2)) and find the

corresponding cutting forces in the x and y-directions. An example is shown in Figure 39.

It is seen that a dynamometer with the 0 kN to 5 kN range is acceptable, although the

force levels are relatively small compared to the full scale value. A Kistler 9257A

dynamometer with 5 kN range was available for these tests. One requirement for this

dynamometer is that the cutting force is applied to the dynamometer not more than 25

mm above the top surface of the dynamometer.



50


0-

-50


-100
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)

100


50-




-50
0 0.1 0.2 0.3 0.4 0.5
Time (s)
Figure 39. Example simulation of cutting forces to facilitate proper selection of
dynamometer.









A 25 mm thick 6061-T6 aluminum work-piece was sized to 100 mm x 85 mm, then

faced and drilled to fit the dynamometer hole pattern as shown in Figure 40. Slotting

cutting tests were made for a 25.4 mm diameter end mill with a 145 mm overhang and a

single 12 helix insert for chip load range of 0.1-0.24 mm/tooth in 0.02 mm/tooth steps.

The cutting forces in x and y-directions were measured for each chip load using an axial

depth of 0.4 mm. Two sets of measurements were made for a 1000 rpm spindle speed. To

address the influence of spindle speed on cutting coefficients, the above two sets were

repeated for {5000, 10000, 15000 and 20000} rpm. The average value of the measured

cutting forces was inserted into Eq. (5.4) to solve for the cutting coefficients. Average

cutting coefficients of the two sets of measurements at each spindle speed are listed in

Table 8.


Figure 40. Work-piece, dynamometer and tool setup










A regression analysis of the cutting force coefficients as a function of spindle speed

was carried out. For Kt and K,, a linear regression with logarithmic transformation of

spindle speed indicates a statistically significant relation with a P-Value of less than

0.007. Figure 41 and Figure 42 show the trend line for this regression for both Kt and K,,

respectively. For the edge cutting force coefficients K,e and Kte the regression doesn't

indicate a significant statistical relation between K,e or Kte and spindle speed. The P-

Value for the slope of the regression was 0.39 and 0.55, respectively.



Table 8. Cutting coefficients for 1 insert endmill for slotting cutting tests


Q Kt Kn Kte Kne
(krpm) (N/mm2 (N/mm2) (N/mm) (N/mm)
1 1321 379 28 32
5 832 183 47 34
10 841 62 37 38
15 655 34 52 33
20 670 65 37 26


1400

1200

1000

800

600

400

200

0


0 0.5


1
logio(Q (rpm) x103)


Figure 41. Cutting coefficient in tangential direction (Kt)










400

350

300
y = -268.84x + 369.14
250- R2 adj = 0.93

200

150

100

50

0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
3
logo( (rpm) x10)
Figure 42. Cutting coefficient in normal direction (K,)


A similar set of measurements were made using partial radial immersion (up

milling) for a 15000 rpm spindle speed. Equation (5.11) was used to find the cutting

coefficients in this case. The results are provided in Table 9.




Table 9. Up milling cutting coefficients for 12% radial immersion


Kt(N/mm2) Kn( N/mm2) Kte(N/mm) Kne(N/mm)
833 431 6 8



To verify the fit, the cutting coefficients obtained were used in a time-domain

simulation of the cutting forces. The measured forces were then overlaid on the simulated

forces. Figure 43 shows a case for 0.12 mm/tooth chip load and 1000 rpm. Also Figure

44 and Figure 45 show the fit for higher spindle speeds of 5000 and 20000 rpm,

respectively.


















0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time secss)


Simulated force
Measured force


0


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Time secss)
Figure 43. Simulated and measured forces for 0.12 mm/tooth chip


-- Simulated force
Measured force


0 0.02 0.04 0.06 0.08
Time secss)


0.4

load and 1000 rpm.


0.1 0.12


Simulated force
Measured force


0.02 0.04 0.06 0.08
Time secss)


0.1 0.12


Figure 44. Simulated and measured cutting forces for 0.2mm/tooth chip load, b=0.4 mm
and 5000 rpm.


40
20
0
-20
-40
-60
-80
0


I









100
80g o / Simulated force




20 '
0
-20
1.01 1.015 1.02 1.025 1.03 1.035
Time secss)

50- Simulated force
Measured force




-50

1.01 1.015 1.02 1.025 1.03 1.035
Time secss)
Figure 45. Simulated and measured forces at 20 krpm and b=0.4 mm for slotting.


Covariance Matrix (Linear Multi-Response Model)

The regression analysis performed in the previous section is a single response

analysis. However, the measured responses are the forces in both the x and y-directions

during a single measurement (dynamometer). Obviously this is a multi-response

measurement. Therefore analysis of the data should take into consideration the

multivariate nature of the data. The interrelationship existing between the variables could

render univariate investigations meaningless. The development for a multi-response

model follows the description in [94]. If we let Q be the number of experimental runs and

r be the number of response variables measured for each setting (two in our case, i.e., Fx

and Fy) of a group of variables (chip load only in our case). The ith response model can be

written in vector form as









Y =Z Z,+, i=1,2,...,r (5.12)

where Y, is an Q x vector of observations in the ith response, Z, is an Q x p, matrix of

rank p, (for the simple linear model p, = 2), f/ is a p, x 1 vector of unknown constant

parameters, and E, is an Q x 1 random error vector associated with ith response. The

assumption of simple linear regression apply here, that is E(E) = 0 and Var (es) = cIg.

However, the covariance matrix between the responses is not zero,

Var(s,)= ,, i = 1,2,...,r
(5.13)
Cov (E,,e = ,, i, j = 1,2,..,r;i i j

We denote the r xr covariance matrix whose (i,j) th element is o-j (i, j = 1,2,..., r)

by For the case of two responses, Eq. (5.12) can be written in matrix form as:


r Zo 0i 01 E
=] x2 8 + ]xl (5.14)
Y2 0 Z2 02 -2
Q xl Q x2_ 1 Q x 1


where
Z,=Z2 = 1 c (5.15)

where c represents the chip load vector (see Eq. (5.3)) and the left hand side vector of Eq.

(5.14) represents the observed average cutting forces in the x and y-directions. From Eq.

(5.13) it can be seen that E has the following variance-covariance matrix,

A =Var (s)= E o0 (5.16)

where 0 is a symbol for the direct (or Kronecker) product of matrices. The direct

product of two matrices E and I. both of size r xr gives an r2 xr2 matrix which is








partitioned as o-JI where -,, is the (i,j)he element of matrix E. The best linear unbiased

estimate of ,f is given by [95]

f =(Z'A 'Z) Z'A' 1 (5.17)

where Y is the left hand side of Eq. (5.14). The variance-covariance matrix of the

estimated vector / is

Var ()=(Z'A 'Z) (5.18)

Since E is usually unknown, it is estimated using the following equation [95]


Y' I,-Z, Z Z,) Z;Z I,-Z Z;'Z Z; ^Y
--j -- (5.19)
i,j = 1,2,...,r

It should be noted that j is computed from the residual vectors which result from

ordinary least-squares fit of the ith andjth single response models to their respective data

sets. Using this estimate for E in Eq.(5.19), an estimate of the variance of / can be

obtained. The cutting force coefficients are determined using a linear transformation

[K]= [A][] (5.20)

where the matrix A for slotting (see Eq. (5.4)) is
0 0 0
Nb
4 0
0 0 0
A Nb (5.21)
0 0 z 0
Nb
0 0 0 4
Nb









Therefore the variance-covariance matrix of cutting force coefficients can be found

as

Var(K})= [A]'Var(f)[A]. (5.22)


Using the procedure outlined above, the cutting force coefficients and their

corresponding correlation matrices are calculated and listed in Table 10 for cutting tests

carried out according to the same procedure described earlier, noting that the correlation

matrix is obtained directly from the covariance matrix (see Eq. (4.5)) As indicated in

Table 10, a high correlation between K, and Kte, and Kn and Kne is found. This high

correlation is justified since both of the corresponding cutting coefficients (K, and Ke or

K, and Kne) are obtained from the same regression fit and cutting force direction.

However, a small correlation between the x and y-directions of the forces is found

(between K, and Kn or Kne) which may be due to experimental error.



Table 10. Estimated cutting force coefficients and their correlation matrix for 7475
aluminum and a 12.7 mm diameter solid carbide endmill with 4 teeth and 30
degree helix angle.


Kt (N/m2) Kn (N/m2) Kte (N/m) Kne (N/m)
Mean 8.41E+08 2.53E+08 1.27E+04 1.01E+04
Standard deviation 2.19E+07 2.66E+07 1.70E+03 2.07E+03
Coefficient of variation 0.03 0.11 0.13 0.20
P Value 2.E-08 8.E-05 3.E-04 3.E-03
Correlation Coeff. Matrix Kne Kn Kte Kt
Kne 1.00
Kn -0.93 1.00
Kte -0.13 0.12 1.00
Kt 0.12 -0.13 -0.93 1.00