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Combining Tool Wear and Dynamics in High-Speed Machining Performance Prediction

Permanent Link: http://ufdc.ufl.edu/UFE0041738/00001

Material Information

Title: Combining Tool Wear and Dynamics in High-Speed Machining Performance Prediction
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Karandikar, Jaydeep
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: force, milling, stability, wear
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A milling super diagram is described that incorporates limitations to milling productivity and part quality imposed by stability, surface location error (part errors due to forced vibrations), and tool wear. Combinations of axial depth of cut and spindle speed that offer stable cutting conditions with an acceptable, user-defined surface location error level are identified by a gray-scale color coding scheme. The effect of tool wear is incorporated by determining the variations in force model coefficients used for process dynamics prediction with tool wear. The force model coefficients increase as a function of the flank wear width (used as a measure of tool wear). The increase in force model coefficients is determined as a function of the volume of material removed. Using these coefficients, a super diagram is constructed for any user-defined volume of material removed using the selected cutter. Additionally, user beliefs about data and model accuracy are applied to identify safety margins relative to the deterministic boundaries in the diagrams. Experimental results are provided for an inserted (carbide) cutter used to machine 1018 steel. The wear behavior is incorporated as changes in the force model coefficients as a function of the volume of material removed at different operating parameters. The flank wear is also measured using an on-machine microscope (to avoid tool removal from the spindle) and correlated to the force model coefficients. Super diagrams are developed that correspond to the new and worn tool performance and experimental results are provided to verify changes in the process stability with tool wear. Once-per-revolution sampling is used to determine a variance parameter that is subsequently used to identify stable and unstable cuts. Test results are shown that compare the distribution in once-per-revolution sampled force data for new and worn inserts at various axial depths of cut. Synchronous (stable) behavior is characterized by a tight distribution, while asynchronous (unstable) behavior yields larger distributions. A normalized variance parameter is used to separate the two cases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jaydeep Karandikar.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041738:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041738/00001

Material Information

Title: Combining Tool Wear and Dynamics in High-Speed Machining Performance Prediction
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Karandikar, Jaydeep
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: force, milling, stability, wear
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A milling super diagram is described that incorporates limitations to milling productivity and part quality imposed by stability, surface location error (part errors due to forced vibrations), and tool wear. Combinations of axial depth of cut and spindle speed that offer stable cutting conditions with an acceptable, user-defined surface location error level are identified by a gray-scale color coding scheme. The effect of tool wear is incorporated by determining the variations in force model coefficients used for process dynamics prediction with tool wear. The force model coefficients increase as a function of the flank wear width (used as a measure of tool wear). The increase in force model coefficients is determined as a function of the volume of material removed. Using these coefficients, a super diagram is constructed for any user-defined volume of material removed using the selected cutter. Additionally, user beliefs about data and model accuracy are applied to identify safety margins relative to the deterministic boundaries in the diagrams. Experimental results are provided for an inserted (carbide) cutter used to machine 1018 steel. The wear behavior is incorporated as changes in the force model coefficients as a function of the volume of material removed at different operating parameters. The flank wear is also measured using an on-machine microscope (to avoid tool removal from the spindle) and correlated to the force model coefficients. Super diagrams are developed that correspond to the new and worn tool performance and experimental results are provided to verify changes in the process stability with tool wear. Once-per-revolution sampling is used to determine a variance parameter that is subsequently used to identify stable and unstable cuts. Test results are shown that compare the distribution in once-per-revolution sampled force data for new and worn inserts at various axial depths of cut. Synchronous (stable) behavior is characterized by a tight distribution, while asynchronous (unstable) behavior yields larger distributions. A normalized variance parameter is used to separate the two cases.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jaydeep Karandikar.
Thesis: Thesis (M.S.)--University of Florida, 2010.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041738:00001


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1 COMBINING TOOL WEAR AND DYNAMICS IN HIGHSPEED MACHINING PERFORMANCE PREDICTION By JAYDEEP MOHAN KARANDIKAR A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010

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2 2010 Jaydeep Mohan Karandikar

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3 To my family

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4 ACKNOWLEDGMENTS I would like to thank all members of my committee, Dr. Tony Schmitz, Dr. John Schueller and Dr. Hitom i Greenslet for their support and involvement with this project and taking the time to review this thesis A special thanks to my advisor, Dr. Tony Schmitz for giving me an opportunity to work in a great research environment. His advice and guidance helped a lot through my graduate studies. This work would not have been possible without support from Thomas Long and Srikanth Bontha of K ennametal, Inc. I would also like to thank all my lab mates at Machine Tool Research Center for their friendship and help, es pecially Raul Zapata. Last, but in no way the least, I would like to thank my family members for their constant encouragement and unwavering belief in me.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 8 ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION .................................................................................................... 13 Motivation and Research Objective ........................................................................ 13 Stability Lobes in Milling .......................................................................................... 14 Tool Wear ............................................................................................................... 15 Surface Location Error (SLE) .................................................................................. 17 2 LITERATURE REVIEW .......................................................................................... 19 3 SUPER DIAGRAM DESCRIPTION ........................................................................ 22 Concept .................................................................................................................. 22 Effect of Tool Wear ........................................................................................... 23 Incorporating Uncertainty ................................................................................. 23 Numerical Case Study ............................................................................................ 24 4 TOOL WEAR EXPERIMENTS ................................................................................ 31 Experimental Setup ................................................................................................ 31 Parameter Selection ......................................................................................... 32 Testing Procedure ............................................................................................ 33 Stability for Cutting Tests ........................................................................................ 34 Calculation of Force Coefficients ...................................................................... 34 Calculation of Stability Lobes ........................................................................... 36 Wear Tes t Results .................................................................................................. 37 5 STABILITY DIAGRAM VALIDATION ...................................................................... 45 Experimental Setup ................................................................................................ 45 Testing and Validation ............................................................................................ 47 6 CONCLUSIONS AND FUTURE WORK ................................................................. 62 Completed Work ..................................................................................................... 62 Future Work ............................................................................................................ 63

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6 APPENDIX A CALCULATI ON OF FORCE COEFFICIENTS ........................................................ 65 B SUPER DIAGRAM CODE ...................................................................................... 69 Stability Code .......................................................................................................... 81 SLE Code ............................................................................................................... 84 Fast Fourier Transform Code Used in SLE Calculations ........................................ 87 C TIME DOMAIN SIMULATION ................................................................................. 88 Code for Time Domain Simulation .......................................................................... 88 Code for Modal Fitting ............................................................................................. 92 D ONCE PER REVOLUTION AND VARIANCE CODE .............................................. 96 LIST OF REFERENCES ............................................................................................... 98 BIOGRAPHICAL S KETCH .......................................................................................... 102

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7 LIST OF TABLES Table page 3 1 Numerical case study parameters ...................................................................... 24 5 1 Force coefficients values for new and worn insert .............................................. 47 5 2 Normalized variance for new and worn inserts ................................................... 54

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8 LIST OF FIG URES Figure page 1 1 Typical stability lobes in milling identifying stable and unstable (chatter) zones .................................................................................................................. 15 1 2 Various forms of tool wear .................................................................................. 16 1 3 Increase in flank wear width with cutting time. .................................................... 17 1 4 Example of an undercut in down milling.. ........................................................... 18 3 1 FRF for the system dynamics parameters used for numerical study .................. 25 3 2 Super diagram with 50 m stability limit. ............................................................ 25 3 3 Variation in FWW assumed with volume removed at different speeds. .............. 26 3 4 Variation in Kt assumed with volume removed at different speeds ..................... 27 3 5 Variation in K n assumed with volume removed at different speeds .................... 27 3 6 Super diagram including tool wear effects (V = 20 cm3). .................................... 28 3 7 The safety limit is identified by testing the feasibility of the eight grid points surrounding {bi, i }.. .......................................................................................... 29 3 8 Super diagram including tool wear effects and the user defined safety margin .. 30 4 1 AFM measurement of carbide insert rake face ................................................... 31 4 2 Setup for tool wear experiments. ........................................................................ 32 4 3 Details of test block. ........................................................................................... 33 4 4 The tool path for machining workpiece. .............................................................. 34 4 5 Linear regression of mean x and y direction forces ............................................ 35 4 6 Tool point FRF of the system .............................................................................. 36 4 7 Stability lobe diagram for the system used for tool wear experiments ................ 36 4 8 Variation in FWW with volume removed ( = 2500 rpm). .................................. 37 4 9 Images of FWW at 60x magnification ................................................................. 38 4 1 0 Variation in Kt and Kn with volume removed ( = 2500 rpm) ............................ 38

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9 4 11 Variation in Kte and Kne with volume removed ( = 2500 rpm) ........................ 39 4 12 Variation in Kt and Kn with volume removed for various spindle speeds. ........... 40 4 13 Variation in Kt and Kn with FWW at various spindle speeds. ............................. 40 4 14 Variation in slope with spindle speed for the Kt and Kn versus volume removed lines (from figure 413). ........................................................................ 41 4 15 Variation in Kt and Kn with volume removed for vari ous feed per tooth values. ................................................................................................................ 43 4 16 Variation in Kt and Kn with normalized volume removed for various axial depths of cut. ...................................................................................................... 44 5 1 Long collet type holder for stability tests ............................................................. 45 5 2 Tool point FRF of the system. ............................................................................. 46 5 3 Change in spindle dynamics with spindle speed for the Mikron UCP Vario 600 used in this study as reported in [48]. .......................................................... 46 5 4 Stability lobes for new and worn insert. Note the stability limit is reduced for the worn insert due to higher cutting force coefficients. ...................................... 48 5 5 Force frequency spectrum for new insert at b = 1.6 mm. .................................... 48 5 6 Force frequency spectrum for worn insert at b = 1.6 mm. .................................. 49 5 7 Once per revolution samples for b= 1.6 mm and ..... 50 5 8 Once per revolution samples for b= 1.6 mm and insert ................................................................................................................... 51 5 9 Once per revolution plots for new and worn inserts at 0.8 mm, 1.6 mm, 2.2 mm and 3.0 mm. ................................................................................................. 52 5 1 0 Normalized variance in the onceper revolution samples with axial depth of cut for the new and worn inserts. ........................................................................ 54 5 11 Topography of the surface left by the new insert. ............................................... 55 5 12 Topography of the surface left by the worn insert. .............................................. 56 5 13 Variation in Kt and Kn with volume (normalized by the axial and radial depths of cut). ............................................................................................................... 57 5 14 Stability lobes generated at Vn = {0,101.5,203 and 304.5} mm. ......................... 58

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10 5 15 Variation in R with Vn ......................................................................................... 59 5 16 Once per .......................................... 59

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11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COMBINING TOOL WEAR AND DYNAMICS IN HIGHSPEED MACHINING PERFORMANCE PREDICTION By Jaydeep Mohan Karandikar May 2010 Chair: Tony L. Schmitz Major: Mechanical Engineering A milling super diagram is described that incorporates limitations to milling productivity and part quality imposed by stability, surface location error (part errors due to forced vibrations), and tool wear. Com binations of axial depth of cut and spindle speed that offer stable cutting conditions with an acceptable, user defined surface location error level are identified by a gray scale color coding scheme. The effect of tool wear is incorporated by determining the variations in force model coefficients used for process dynamics prediction with tool wear T he force model coefficients increase as a function of the flank wear width (used as a measure of tool wear). The increase in force model coefficients is determ ined as a function of the volume of material removed. Using these coefficients, a super diagram is constructed for any user defined volume of material removed using the selected cutter. Additionally, user beliefs about data and model accuracy are applied t o identify safety margins relative to the deterministic boundaries in the diagrams. Experimental results are provided for an inserted (carbide) cutter used to machine 1018 steel. The wear behavior is incorporated as changes in the force model coefficients as a function of the volume of material removed at different operating

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12 parameters The flank wear is also measured using an onmachine microscope (to avoid tool removal from the spindle) and correlated to the force model coefficients. Super diagrams are de veloped that correspond to the new and worn tool performance and experimental results are provided to verify changes in the process stability with tool wear. Once per revolution sampling is used to determine a variance parameter that is subsequently used t o identify stable and unstable cuts. Test results are shown that compare the distribution in once per revolution sampled force data for new and worn inserts at various axial depths of cut. Synchronous (stable) behavior is characterized by a tight distribut ion, while asynchronous (unstable) behavior yields larger distributions. A normalized variance parameter is used to separate the two cases.

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13 CHAPTER 1 INTRODUCTION Motivation and Research Objective High speed machining (HSM) has made significant technological advances in recent years. Improved spindle designs enable speeds up to 20000 rpm and higher in milling. High material removal rates (MRR) can be obtained by machining at higher axial depths of cut and spindle speeds. However, a limitation to machining at higher axial depths of cut is chatter or unstable cutting. Tlusty defined the mechanism for chatter or self excited vibrations as regeneration of surface waviness during material removal [1]. Subsequent work involved developing techniques like time domain simulations, frequency domain analyses or temporal finite element methods to predict the stability lobes in HSM [2 5]. Stability lobe diag rams separate stable cutting conditions from unstable or chatter conditions and are represented as a function of axial depth of cut and spindle speed. The models used to predict the stability lobes require the tool point frequency response function (FRF), tool geometry, cutting parameters and cutting force coefficients. Tool wear is also an important process limitation in machining. Taylor established empirical relationships between cutting parameters and tool wear, which is still used as the basis f or predicting tool life [6]. Tool wear is primarily dependent on cutting speed, which h as a strong influence on the temperatures in cutting. Higher maximum temperatures increase the severity of thermal stress cycling [7]. Tool condition monitoring (TCM) research has developed methods to estimate tool wear based on cutting force signals. Research has shown an increase in the cutting force and force coefficients due to tool wear [815]. However, the effect of tool wear on stability lobe

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14 diagrams has not been previously explored. The tool wear effects can be incorporated through the force model coefficients used to calculate the stability lobes. This research studies the role of tool wear as a process limitation through its effect on milling stability lobes. T he final objective is to present a milling super diagram which simultaneously incorporates stability, surface location error, tool wear and uncertainty. The super diagram will enable a user to select optimum cutting conditions to maximize MRR and reduce cost considering these process limitations. Stability L obes in Milling In milling, relative motion between a rotating cutter and workpiece is responsible for material removal. As the cutter is engaged, it experiences a cutting force whic h causes the tool to vibrate. The tool vibrations are imprinted on the workpiece leaving behind a wavy surface. The wavy surface left behind by one tooth is removed by the following tooth. Thus, surface regeneration occurs from one tooth to the next. The i nstantaneous chip thickness depends on the state of vibration of the current tooth and the surface left behind by the previous tooth and governs the cutting force. If the two surfaces are in phase, the chip thickness varies only according to cut geometry. This gives periodic cutting forces and tool vibrations and provides stable cutting conditions. However, an out of phase profile results in a variable chip thickness which affects the cutting force, and subsequently, the tool vibrations. The resulting vibrations again affect the chip thickness. This feedback mechanism may result in self excited vibrations or chatter in milling. Stability lobes separate stable operating points from unstable or chatter points. All operating points below the stability boundary are stable and the ones above are unstable. Fig ure 1.1 shows typical stability lobes in milling

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15 Fig ure 11 Typical stability lobes in milling identifying stable and unstable (chatter) zones The stability lobes are calculated at a certain radial depth of cut and feed per tooth. The user can select optimum operating conditions for spindle speed and axial depth of cut based on this diagram. Tool W ear Tool wear in machining is the loss of tool material due to interaction with the workpiece during cutting. Tool wear can ultimately result in catastrophic failure of the cutting edge. Tool wear is undesirable as it affects the cutting forces and quality of t he machined surface. Replacing a worn tool increase s tool change time, which increases the cost of the pr oduct. Tool wear also results in increased cutting forces and temperatures. The various mechanisms that can cause tool wear include mechanical microbreakages, abrasion, adhesion, diffusion and oxidation. Taylor established empirical relationships between tool life and cutting parameters [6]. The Taylor type tool life equation is given by T = Cvpft q, where T is tool life in

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16 minutes, v is surface speed of the cutter in mm/min ( given by 60 d v d is tool diameter in mm and spe e d in rpm ) and ft is the feed per tooth in mm/tooth. The constant C and exponents p and q can be calculated from the experiments where the time to reach a predetermined wear level is measured for various cutting conditions. The exponent p is usually hi gher than q, which indicates a greater dependence of tool life on the spindle speed than on feed per tooth. Tool wear features can be flank wear, crater wear and notch wear. Fig ure 1.2 shows the various forms of tool wear. Fig ure 1 2 Various forms of tool wear A ) Nose wear B ) Notch and Flank wear C ) Crater wear D ) Plastic/breakage [16]. Flank wear is the most common of the wear features and can be used to monitor tool wear. Flank wear is caused by abrasive wear of the main cutting edge against the workpiece and occurs on the tool flank face over the length equal to axial depth of cut in z ero helix end milling. Flank wear is expressed in terms of flank wear width (FWW). Flank wear increases with cutting time as shown in Fig. 1.3 [7]. The tool life is based on

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17 the time required for the maximum FWW to reach a preselected value. According to I SO, the permissible average value of FWW is 0.3 mm in the case of uniform wear or 0.6 mm maximum in case of irregular wear for cemented carbides Fig ure 1 3 Increase in flank wear width with cutting time. I) Initial rapid wear II) Uniform wear III. Final wear (catastrophic failure) [17]. The increase in flank wear width consists of three parts: I nitial rapid wear where FWW increases rapidly. U niform wear where FWW increases at a uniform rate. F inal accelerated wear leading to a catastrophic failure of the tool Surface Location Error (SLE) In milling the tool experiences a cutting force which causes it vibrate when it engages the workpiece Thus, the tool undergoes periodic forced vibrations, even under stable cutting conditions. These forced vibrati ons can cause surface location error s or geometric inaccuracies of the workpiece. The Surface L ocation E rror (SLE) is given by the difference between the location of the intended final surface and the machined surface. The position of the cutting edge as i t enters the cut in up milling or exits the cut in down milling as the tool vibrates determines the location of the machined surface. Depending on the state of the tool vibration as it leaves the final machined surface, it

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18 can be undercut or overcut. Figur e 12 shows an example of SLE in down milling. Because SLE is caused by forced vibration, its value depends on the spindle speed (excitation frequency) Figure 14 Example of an undercut in down milling. The difference between actual and intended surfac e is SLE. The feed of the tool is in x direction and cutting forces cause it vibrate in the y direction. The tool vibrations are exaggerated for depiction purposes. SLE can be predicted using the same inputs required for stability prediction including the tool point frequency response function, force model coefficients (dependent on the workpiece materials ), geometry of tool and cutting parameters A frequency domain approach cal culates SLE in the same axial depth of cut spindle speed domain as the frequency domain analysis [18]. The SLE graph can be shown a s contours of constant SLE values.

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19 CHAPTER 2 LITERATURE REVIEW The literature review presents prior research in: 1) tool condition monitoring using cutting force signals; and 2) techniques to predicting stability during machining. Both topics are critical for the scope of this thesis. Extensive research has been completed to better understand tool wear mechanisms and approaches to reduce tool wear. Taylor defined an empirical relation to calculate tool life by means of a tool life equation, which is still used as a basis for defining tool wear [6]. Subsequent research studied mechanisms of tool wear more closely for various materials and explored influencing factors. The effect o f various tool materials and coatings on tool wear was studied [1929]. Recently, efforts have been dedicated to tool conditioning monitoring (TCM). Multiple sensor systems can be used to monitor various signals during machining to determine inprocess tool wear status [30 31, 16]. Most TCM systems use indirect measurements of cutting performance to predict tool wear. In many cases cutting force is measured using a tablemounted or rotating spindle dynamometer [815, 3234]. The dynamometers measures three mutually perpendicular forces in the x (feed), y and z (axial) directions. Time series modeling of the x and y direction forces can then used to estimate the tool wear status. The increase in dynamic and static components of force signals has been found to increase with tool wear [9, 32]. Some studies measured the change in magnitude of cutting force harmonics in the frequency domain using the Fourier transform of the force signal [8, 15]. For example, a broken tooth can be detected by the corresponding in crease in the magnitude of the cutting force harmonics using a pre defined threshold

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20 value. A cutting force coefficient based method to monitor tool wear was developed which showed the force coefficients increase with tool wear as well [33 34] Chatter in high speed machining has been the subject of research studies for many years. Tlusty was among the first to develop an analytical method for predicting chatter in milling by means of stability lobes [1]. Tlusty assumed an average angle of the tooth in the cut and, therefore an average force direction. This eliminated the time dependence of cutting force direction in milling and simplified the analysis. Based on Tlustys analysis, stability lobes for any machinetool holder assembly can be calculated using an appropriate force model, geometric properties of the tool (number of teeth and helix angle) and parameters of the machining operations (radial depth of cut and feed per tooth). Altintas et al. defined an alternative technique where the time dependent coe fficients for the milling force equations are expanded into a Fourier series and the series is truncated to include only the average term. This technique also provides a closed form solution [35]. However, both the theories have been found to be invalid for low radial immersions (less than 20%) where the cutting forces resemble short duration impulses [36]. Research has shown that the instability for low radial immersion is due to quasi periodic chatter (referred to as flip bifurcation) which manifests it self as two tightly grouped clusters of points after once per revolution sampling. Therefore, techniques such as Poincare sectioning (once per revolution sampling) can be used to indentify instability [2 5, 37 43]. Other techniques like time domain analysi s, semi discretization method and temporal finite element analysis can also be used to predict stability [25]. More recently, a milling super diagram which combines stability with surface location

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21 error has been developed. The super diagram uses the freq uency domain analysis for predicting stability and surface location error [18,44].

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22 CHAPTER 3 SUPER DIAGRAM DESCRI PTION C oncept A milling stability lobe diagram gives information about stable and unstable (chatter) points over axial depth of cut spindle speed domain. A frequency domain approach can be used to calculate the deterministic stability boundaries [35] The method offers a convenient closed form solution to calculate the stability lobes. A frequency domain approach can be used to calculate surface location error ( SLE) in the same domain [18] The simulation results can be presented as contours of constant SLE values. Calculation of stability lobes and SLE requi re the same inputs such as tool point frequency response function, f orce model coefficients (dependent on tool and workpiece material) tool geometry (number of teeth, tool diameter and helix angle) and cutting parameters (radial dept h of cut and feed per tooth). The super diagram combines stability and SLE information in a single us er friendly format using a grey scale approach [46] The acceptable SLE threshold of SLE is user defined. To construct a super diagram, the user selects an axial depth of cut spindle speed domain, radial depth of cut, feed per tooth and limiting SLE value. The axial depth of cut spindle speed domain is discretized into a grid of points. The grid points are then penalized according to whether they are stable or unstable and if they fall within acceptable SLE limit or not. The penalties are given as fol lows No penalty is given to the points that are stable and within the user defined acceptable value of SLE; they are set as zero. The feasible zone is denoted by white on the super diagram. The points that are stable, but fall outside the acceptable SLE value are penalized by 1 (value 1). The SLE limited points are shown by a grey zone on the super diagram.

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23 The points that are unstable are penalized by 2 (value 2). The unstable points are represented by a black zone on the super diagram. Effect of T ool W e ar The milling cutting force model used for stability and SLE calculations in this work is given by equations 3 1 and 32 Ft = ktbh + kteb (3 1) Fn = kn bh + kneb (3 2) where, Ft is the tangential force component, kt is the tangential cutt ing force coefficient, b is the axial depth of cut, h is the instantaneous chip thickness (which depends on the feed per tooth and cutter angle), kte is the tangential edge coefficient, Fn is the normal force component, kn is the normal cutting force coeff icient, and kne is the normal edge coefficient [45]. The cutting force coefficients are used to calculate stability and SLE. It has been shown that the cutting force coefficients increase with tool wear [3334]. The cutting force model in equations 31 and 3 2 can be experimentally evaluated as a function of tool wear. Thus, by correlating the force model coefficients to wear status of the tool, tool wear effects can be included in the super diagram. Incorporating U ncertainty The super diagram provides inf ormation about stable/unstable cutting conditions and acceptable/unacceptable SLE values in a binary format. Both the stability and SLE predictions are deterministic. However, there is inherent uncertainty in their actual locations due to the distributions in the inputs that are used to calculate them. These input uncertainties are incorporated in the super diagram as a user defined safety limit, which is applied to the feasible (white) zone boundary. The user selects how close he/she is willing to operate to the feasible boundary. An additional penalty is then

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24 applied to the points that fall within the zone that violates the safety margin of the user. A light dark grey level is incorporated to indicate points that violate the safety margin. Dark gray now indicates the stable points where the SLE limit is exceeded, while light gray represents the previously feasible points which violate the safety margin. Numerical C ase S tudy To demonstrate the incorporation of tool wear and uncertainty into the super diagram a numerical case study is presented here. The parameters used for the case study are listed in Table 31 Table 3 1 Numerical case study parameters Parameter Value Units Stiffness 5 x 10 6 N/m Damping ratio 0.05 Natural fr equency 300 Hz Tool diameter 19.05 m m Helix angle 0 degrees Number of teeth 1 Tangential cutting coefficient 2 x 10 9 N/m 2 Normal cutting coefficient 0.667 x 10 9 N/m 2 Feed per tooth 0.06 mm/tooth Radial depth of cut 4.725 m m Figure 31 shows the FRF of the dynamic system calculated from the data in Table 3 1. It is a single degree of freedom system with natural frequency at 300 Hz. T he x and y direction dynamics were taken to be equal. Figure 32 shows the super diagram for the parameters listed in Table 31. The acceptable SLE limit was taken as 50 m. The super diagram was constructed using the frequency domain approaches of calculating stability and SLE. The feasible values of axial depth of cut and spindle speed are selected to be in the range 0 mm to 20 mm and 2000 rpm to 10000 rpm, respectively.

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25 Figure 31 FRF for the system dynamics parameters used for numerical study As shown, three zones can be identified as follows W hite feasible zone stable and within acceptable SLE limit. G rey zone stable points higher than acceptable SLE limit. B lack unstable zone unstable points Fig ure 3 2 Super diagram with 50 m stability limit. Flank wear is a common wear feature. Tool life is often based on the cutting time required for the flank wear width (FWW) to reach a predetermined level. For the

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26 numerical study, a linear relationship of FWW with volume of material removed was assumed as shown in figure 33. Fi g ure 3 3 Variation in FWW assumed with volume removed at different speeds. The FWW tends to increase with cutting time or volume removed and is spindle speed, dependent. The end of tool life was defined as the time required to reach a FWW of 0.3 mm. As noted, the cutting forces and force coefficients tend to grow with FWW. The increase in cutting coefficients shown in equation 31, with volume removed, V, was assumed to be linear as shown in figures 34 and 35. The assumed linear relationship between t he coefficients Kt and Kn with volume removed at different spindle speeds is provided in equations 32 and 33, where the intercepts, c0,t and c0,n, are the coefficients for a new/unused tool (Table 31), and c1,t and c1,n are the speeddependent rates of increase in the force coefficients with V. Kt ( ,V) = c0,t + c1,t V (3 3) Kn ( ,V) = c0,n + c1,n V (3 4)

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27 Fig ure 3 4 Variation in Kt assumed with volume removed at different speeds Fig ure 3 5 Variation in K n assumed with volume removed at different speeds The slopes (c1,t and c1,n) were assumed to increase linearly with between 2000 rpm and 10000 rpm such that the coefficients doubled at 10000 rpm for V = 20 cm3 (where FWW = 0.3 mm; see Fig. 32) with no change at 2000 rpm for the same V. The assumption is based on the fact that FWW will be minimal at low speeds to remove 20

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28 cm3 and hence the force coefficients can be assumed to be constant. The edge coefficients are neglected in this numerical example without the loss of generality. To illustrate, at V = 20 cm3 for = 3000 rpm, the slope c1,t can be calculated by linear interpolation as follows 0, 1,30002000 100002000t tc c V = 1.25 x 107 N/m2/cm3 (3 5) Given this relationship between cutting coefficients, and V, the effect of tool wear can now be incorporated into the super diagram. T he volume to be removed must first be selected by the user. Then, new increased coefficients are calculated for each spindle speed as shown in equation 35. These coefficients are then used to determine the stability limit at each spindle speed. Also, SLE is calculated at each axial depth grid point for the given spindle speed using the increased value of the coefficients at that speed. The new super diagram for V = 20 cm3 is shown if figure 36. Figure 36. Super diagram including tool wear effects (V = 20 cm3).

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29 T he new super diagram takes into account the effect of tool wear by applying the higher coefficient value. Since the cutting coefficients grow with for a given volume to be removed, the stability limit decreases and the SLE infeasible zone grows while moving from left to right in the diagram. Next, the super diagram can be modified to incorporate the users uncertainty beliefs regarding the a ctual location of the deterministic boundaries. To carry out this task, the user defines safety limits for spindle speed, and axial depth of cut, b. These values give the distances from the boundaries that represent his/her 95% confidence level for actual feasible performance. For each f easible point in the { b } domain defined by the white zone, the penalty value of the surrounding eight points at distances and b from the test point are queried (see f ig ure 3 7 ). If any of these points are infeasible (with a penalty of 1 or 2) then the test point is penalized and also identified as infeasible as shown in figure 37. F igure 3 7 The safety limit is identified by testing the feasibility of the eight grid points surrounding {bi, i }. In this case, the test point is penalized ( 1) because the (black) point above it is unstable. A new gray scale is then implemented where the point values are: feasible (0, white), safety margin ( 1, light gray), SLE limit ( 2, dark gray), and unstable ( 3, black).

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30 Thus, the white feasible zone is further reduced on the application of safety margins by the user. Fig ure 3 8 shows a super diagram with uncertainties included as = 100 rpm and b = 0.5 mm Figure 38 Super diagram including tool wear effects and the user defined safety margin (V = 20 cm3, = 100 rpm, and b = 0.5 mm)

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31 CHAPTER 4 TOOL WEAR EXPERIMENT S Experimental S etup The setup for the tool wear experiments is described in this section. A 19 mm (3/4) diameter, single insert end mill (Kennametal KICR075SD260) was used for the tool wear experiments. An uncoated square carbide insert was used to achieve accelerated wear and reduce testing time. (Kennametal 107888126 C9 JC carbide insert; zero rake and helix angles, 15 deg relief angle, 9.53 mm square x 3.18 mm) The workpiece was 1018 steel (152.4 x 101.6 x 38.1 mm) with a Rockwell C hardness of 55. An atomic force microsc ope (AFM) was used to measure the topography of the carbide inserts. Figure 41 shows an example 50 m x 50 m measurement (256 line scans, no digital filtering) of the rake face. It is seen that there is a small chamfer with a 167 deg angle at the cutting edge. The roughness average, Ra, for the rake face was 310 nm. Fig ure 4 1 AFM measurement of carbide insert rake face Figure 42 shows the experimental set up. The workpiece was attached to a tablemounted cutting force dynamometer (Kistler 9257B).

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32 Figure 42 Setup for tool wear experiments. Parameter S election The parameters for the first test were selected using the tool life data for machining 1018 steel with uncoated carbide inserts reported in [34] and [47].The recommended speeds and feeds for machining 1018 steel with an uncoated carbide insert for a tool l ife of 45 minutes were = 2368 rpm and ft = 0.177 mm/tooth for a threetooth, 20 helix angle tool with 25.4 mm diameter, an axial depth of 5.08 mm and a radial depth of cut, a, of 25.4 mm (100% RI) [34]. As reported in [47], a tool life of 42 minutes was obtained by = 2500 rpm and ft = 0.06 mm/tooth using a single flute, 30 heli x angle tool with 12.7 mm diameter, an axial depth of 5.08 mm and a 50% radial immersion. Considering these two data sets, the parameters for the initial test was selected as = 2500 rpm, b = 3 mm, a = 4.7625 mm (25% RI) and ft = 0.06 mm/tooth. Based on t he selected parameters, the material removal rate (MRR) is given by equation 41 t tN abf MRR = 2143.125 mm3/min (4 1) workpiece dynamometer tool

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33 where Nt is the number of teeth (Nt = 1). Figure 4 3 shows details of the test block. A slot to a depth of 35 mm was made to facilitate partial radial immersion cutting. The effective volume of the block to be removed was equal to 80.96 mm x 101.6 mm x 33 mm = 2745 0 mm3. The time required to machine one block, based on the parameters selected, is equal to 126.6 minutes. Figure 43 Details of test block. Note the effective length that can be removed is equal to 80.96 mm. The figure shows one 3 mm layer removed from the block. A total of 17 passes can be made per layer using a = 4.7625 mm (25% RI). Testing P rocedure Fig ure 44 shows the tool path for machining the workpiece. The block was machined at the test parameters, while intermittently measuring forces at varying ft values. As seen from figure 44, the feed was in the x direction. The tool retracted after making a cut, returned to the x starting location, and then stepped over in the y direction by an amount equal to the radial immersion in preparation for the next cut.

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34 Figure 44 The tool path for machining workpiece. The tool was fed in the x direction while the radial depth of cut was commanded using th e y direction. The tool retracted after cutting returned to the starting x location, and was then incremented in the y direct ion by the radial depth of cut. All the cuts were downmilling. Stability for C utting T ests The stability lobes for the given spindleholder tool combination were calculated to ensure the wear tests are stable. The stability lobes were calculated by using t he method described in [35]. The cutting force coefficients for the tool workpiece material combination and the frequency response function of the system were measured and used as inputs for the stability lobe diagram development. Calculation of Force C oef ficients The cutting force coefficients for the given tool workpiece material combination were calculated using a linear regression on the mean x (feed) and y direction forces. The force was measured during cuts at 2500 rpm, 3.0 mm axial depth of cut and 25% RI over a range of feed per tooth values (0.04, 0.05, 0.06 and 0.07 mm/tooth). The slopes, a1x and a1y, and the intercepts, a0x and a0y, for the x and y direction data, x y z

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35 respectively, were used to calculate the cutting force coefficient given by equations 4 2 to 4 5 228(1.(22sin(2)sin(2))1.(cos(2)cos(2))) (().((22sin(2)sin(2))(cos(2)cos(2))))esse se t esse seay ax K Ntb (4 2) 228(1.(22sin(2)sin(2))1.(cos(2)cos(2))) (().((22sin(2)sin(2))(cos(2)cos(2))))esse se n esse seax ay K Ntb (4 3) (0.(in()sin())0.(cos()cos())) ().(1cos())eses te esaxs ay K Ntb (4 4) (0.(cos()cos())0.(sin()sin())) ().(1cos())eses ne esax ay K Ntb (4 5) where s e is the exit angle. The force coefficients were calc ulated as Kt = 2.1899 109 N/m2, Kn = 1.7327 109 N/m2, Kte = 54474 N/m Kne = 64471 N/m for a new insert. Figure 45 shows the linear regression of mean x and y forces used to calculate the cutting force coefficients [46,50]. Figure 45 Linear regression of mean x and y direction forces

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36 Calculation of S tability L obes The tool point FRF was measured by impact testing and the stability lobes were calculated for the system to ensure stable cutting tests. Figure 46 shows the x and y direct ion FRF of the tool spindleholder. Figure 47 shows the corresponding stability lobes for the dynamic system with 25% RI. A B Figure 46 Tool point FRF of the system: A) x direction B) y direction. Figure 47 Stability lobe diagram for the system us ed for tool wear experiments

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37 As seen from figure 44, the critical value of axial depth of cut is 17 mm. The axial depth of cut selected for wear tests was decided to be 3 mm. Thus, the wear tests for the selected parameters are stable at all cutting conditions. W ear T est R esults The first tool wear test was done at the parameters listed previously and was repeated three times. The cutting force coefficient regression was completed intermittently while wearing the tool. The feed per tooth values for the regression analysis were {0.03, 0.04, 0.05, 0.06 and 0.07 mm/tooth}. In addition to monitoring the cutting force, the insert wear status was also measured at the same intervals as the cutting force coeffi cients. To avoid removing the insert/tool from the spindle, a handheld microscope (60x magnification) was used to record the rake and flank surfaces. The calibrated digital images were used to identify the FWW. No crater wear was observed. Example FWW resu lts for 2500 rpm tests are provided in figure 48 (1 error bars), where the interval between measurements was 12 cm3. The test was terminated when the maximum FWW reached 0.7 mm. Figure 48. Variation in FWW with volume removed ( = 2500 rpm).

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38 Microsc ope images of the relief face for selected volumes of material removed are displayed in figure 49. The maximum FWW increases with the volume removed, V, as expected. Figure 49 Images of FWW at 60x magnification (from left to right, V = {50, 125, 200, and 275} cm3). The force coefficients were also calculated at each interval. Figures 410 and 411 shows the results. Figure 410. Variation in Kt and Kn with volume removed ( = 2500 rpm)

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39 Figure 411. Variation in Kte and Kne with volume removed ( = 2500 rpm) For the selected tool/material combination, Kt and Kn show an approximately linear increase while Kte and Kne show no clear trend. A linear fit provides a good appr oximation of the increase in Kt and Kn (R2 value is 0.877 and 0.853 for Kt and Kn, respectively). As a next step, the change in force coefficient behavior with spindle speed was evaluated by performing additional tests at {3750, 5000, 6250 and 7500} rpm. Figure 412 shows the results and the linear least square fits to the resul ts. The procedure was the same as described for the 2500 rpm testing. All parameters were unchanged, except for spindle speed. The Kte and Kne values again did not exhibit any significant trend at the additional spindle speeds As expected, the rates of Kt and Kn growth (i.e., the slopes) increase d with spindle speed. Interestingly, when plotted versus the corresponding FWW (measured with the handheld digital microscope), the five different spindle speed results collapse onto a single line as seen from figu re 4 13. Thus, the increase in Kt and Kn is dependent on FWW and independent of spindle speed. T his suggests that if the FWW

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40 were monitored, it could provide an inprocess approach to updating the force model coefficients based on the tool wear status. Figure 412. Variation in Kt and Kn with volume removed for various spindle speeds. Figure 413. Variation in Kt and Kn with FWW at various spindle speeds. To incorporate the variation in Kt and Kn with FWW at various spindle speeds, the slopes of the in dividual (Kt and Kn versus volume removed) lines in figure 412 are plotted against spindle speed. As seen from figure 414, the slopes increase linearly

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41 with spindle speed as was assumed in the numerical case study provided in Chapter 3. The error bars on slopes were obtained by performing a MonteCarlo simulation on the range of Kt and Kn values and calculating slopes for each random combination. F i gure 414. Variation in slope with spindle speed for the Kt and Kn ver sus volume removed lines (from f igure 4 13) Therefore, equations 3 3 and 34 which describe a linear relationship between Kt and Kn and the volume removed are applicable to this tool material pair. The slope of the lines in figure 414 are 7.1 103 (N/m2/cm3)/rpm and 9.1 103 (N/m2/cm3)/rpm and the intercepts are 1.3 107 N/m2/cm3 and 1.8 107 N/m2/cm3 for the Kt and Kn data, respectively. The negative intercept values are attributed to the linear fit with inherent experimental uncertainty. The terms c1,n and c1,n defined in equations 3 3 and 34 can be calculated at different speeds by multiplying these slopes (from the lines in figure 414) by the corresponding spindle speed adding the intercept shown in equation 4.6 and 4.7 The intercepts in equations 33 and 34 are taken as the st arting value (V = 0) of the force coefficients obtained using a new insert. Thus, equations 3 3 and 34 can be written for the tool material combination as shown in equations 46 and 47.

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42 9372.210(7.110.1.310)tKV (4 6) 9371.210(9.110.1.710)nKV (4 7) As noted earlier, since the Kte and Kne values did not show any significant trend, the mean values from figure 411 were applied i.e., Kte = 4.6 104 N/m and Kne = 3.9 104 N/m. Given this relationship, the super diagram that incorporates tool wear can then be developed at a user selected volume by calculating Kt and Kn for each spindle speed and using this value to calculate the speeddependent stability boundary and SLE as shown in the numerical example. All the previous testing was performed a t ft = 0.06 mm/tooth. This enables a super diagram to be developed at that value However, changing the ft value will change the SLE values with all other parameters remaining the same. Therefore, a similar set of experiments were completed at feed per tooth values of {0.03, 0.045, 0.075, and 0.09} mm/tooth. The tests were completed at 5000 rpm and other parameters remained the same as before. Figure 415 shows the variation in cutting force coefficients at different feed per tooth values. The wear rate i s higher and the volume of material that can be removed is lower for the smaller feed per tooth values. The wear rate trend suggests that strain hardening may be in effect. The thinner chips with increased hardness can cause accelerated wear. The reduced amount of material that can be removed could also be attributed to the increase in cutting time and the number of passes through the material required to remove the same volume.

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43 Figure 415. Variation in Kt and Kn with volume removed for various feed per tooth values. Finally, the variation in wear rate behavior with axial depth of cut was evaluated. The axial depths were {3, 4.5 and 6} mm, the spindle speed was 5000 rpm, the feed per tooth was 0.06 mm/tooth and the radial depth remained at 4.7 mm. Figur e 416 shows the results. Note that the Kt and Kn values are plotted against volume normalized by the axial depth of cut, Vn = V/b. This normalization was necessary because the independent variable, V, is a function of the dependent variable, b. As seen in the figure, the three tests sets collapse onto a single line for the usable tool life when plotted versus the normalized volume. The agreement between b values demonstrates that testing at a single axial depth is sufficient. The divergence at the highest Vn value for b = 6 mm is due to excessive FWW for that test (> 0.7 mm). It has also been suggested that variation in FWW is not observed at different radial depths of cut and differing number of teeth (assuming no runout) [7]. Similar results can be expec ted for tests with varying radial depth of cut or the number of teeth if Kt and Kn are again plotted against normalized volume.

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44 Figure 416. Variation in Kt and Kn with normalized volume removed for various axial depths of cut. By normalizing the volume removed by the axial and radial depths of cut and number of teeth, the required number of tests can be dramatically reduced. For a given tool workpiece combination, testing can be completed only at a selected axial depth of cut, radial depth of cut and num ber of teeth. The results can then be extended to other combinations by plotting the values of force coefficients against the normalized volume removed.

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45 CHAPTER 5 STABILITY DIAGRAM VALIDATION Experimental S etup The tool wear experiment al results showed a linear increase in cutting force coefficients Kt and Kn with volume removed due to progressive flank wear Th is increase in force coefficients causes the stability limit to decrease. Results also showed a spindle speed dependence on the rate of increase in the force coefficients. For a preselected volume to be removed (based on the workpiece geometry, for example) the rate of increase in Kt and Kn for the given material tool combination is given by equation 42 and 4 3 The stability diagram can be generated using the volume based force coefficients which are also spindle speed dependent. The stability diagram validation experiments were completed using new and worn inserts to show that the stability limit decreases with tool wear. The validation tests were carried out using 1018 steel and the same inserts as for to ol wear experiments. As seen in figure s 4 6 and 47, the stiff tool holder spindle combination yielded a critical stability limit of 17 mm. For the stability experiments, a long collet type holder was used in order to obtain a more flexible dynamic system and lower stability limit ( see figure 51). Fi gure 51 Long collet type holder for stability tests

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46 The workpiece was again attached to the tablemounted dynamometer to measure cutting force during the tests. For this setup, the workpiece was assumed to be rigid and only the tool dynamics were consid ered for the stability diagram calculations As a result, the changes in the mass of the part as the material was removed did not affect the dynamics. Figure 5 2 shows the tool point FRF s in the x and y directions, respectively. A B Figure 52 Tool point FRF of the system : A) x direction B) y direction. Figure 53 Change in spindle dynamics with spindle speed for the Mikron UCP Vario 600 used in this study as reported in [48]. Note that a different tool holder combination was used so the nat ural frequencies differ.

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47 However, the spindle dynamics change significantly with spindle speed for the particular machine used in this research (Mikron UCP Vario 600 with 20000 rpm, 14 kW Steptec spindle) [48]. As spindle speed increases the natural frequ ency reduces and the dynamic stiffness increases as seen in figure 53. Therefore, for the calculation of stability lobes, a 50% increase in the dynamic stiffness of the system at 5000 rpm was assumed relative to the FRF recorded at zero speed. Testing and Validation A new and worn insert (0. 5 mm FWW) were used for stability tests. The force coefficients for both the inserts were measured using the linear regression of the average x and y direction forces at varying ft values. These force coefficients are l isted in table 51 Table 51. Force coefficients values for new and worn insert Kt (N/m 2 ) Kn ( N/m 2 ) Kte ( N/m ) Kne ( N/m ) New insert 1.90 x 10 9 0.78 x 10 9 45500 46650 Worn insert 4.98 x 10 9 4.51 x 10 9 45500 25500 The values listed in table 5 1 are in close agreement with the values expected at FWW = 0.5 mm from figure 413. Stability diagrams were separately generated for both inserts using their respective force coefficient values. Figure 54 shows stability diagrams for the new and worn inserts. The stability lobes were generated at a = 19.05 mm (100% RI). Cutting tests were completed at 0.8 mm, 1.6 mm, 2.2 mm and 3 mm axial depth of cut using both inserts. The spindle speed was 5100 rpm. It can be safely assumed that there is no additional increase in the force coefficients during these tests due to small amount of material removed.

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48 Figure 54 Stability lobes for new and worn insert s. Note the stability limit is reduced for the worn insert due to higher cutting force coefficients. Figures 55 and 56 show the frequency spectrum of the force signal in the x and y directions for an axial depth of 1.6 mm with the new and worn inserts. A B Figure 55 Force frequency spectrum for new insert at b = 1.6 mm: A) x direction B) y direction. Figure 55 shows frequency content only at the tooth passing frequency (85 Hz) and its harmonics. The increased magnitude near 1400 Hz occurs due to the dynamometers lowest natural frequency

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49 A B Figure 56 Force frequency spectrum for worn insert at b = 1.6 mm: A) x direction B) y direction. Figure 56 shows increased magnitude at the tooth passing frequency and its harmonics relative to figure 55; this agrees with prior studies that demonstrate larger forces with tool wear [8,15]. Bec ause the forced vibrations are large, using the force frequency content to identify chatter (by content at frequencies other than the once per revolution, tooth passing frequency, and corresponding harmonics) is challenging in this case. Therefore, a onceper revolution force sampling strategy for the x (Fx) and y (Fy) directions was used to identify chatter. The once per revolution samples were obtained by sampling the force data at the commanded spindle rotating frequency [36]. The once per revolution sa mples for stable conditions (forced vibration only) are synchronous with spindle rotation and the behavior repeats each revolution. This produces a single small cluster of points in the plots. Unstable (chatter) behavior produces a more distributed set of points due to its asynchronous nature. In this case, it has been shown that the sampled tool displacement data collected during regenerative chatter is characterized by a ring of points which is characteristic of quasi periodic (or asynchronous) motion [3638]. Figure 57 shows the time domain simulation results for once per revolution sampling of tool displacements and forces (x and y directions) with

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50 b = 1.6 mm and [46, 49]. As seen from figure 57 the once per revolution samples are synchronous with spindle speed and produce a small cluster of points for both the displacement and the force plots. A B C D Figure 57 Once per revolution samples for b= 1.6 mm and A) time history of x and y displacements of the tool. The dark points represent the once per revolution samples. B) The x and y displacements are plotted against each other. The onceper revolution samples are again shown as dark points. Similar results are observed for: C) time history of Fx and Fy and D) Fx vs Fy. Figure 58 shows similar plots for an unstable cut at b = 1.6 mm and when using the worn insert (new cutting force coefficients). Note that while the onceper revolution plots for displacement show a distinctive ellipse, it appears as a line in the force plots.

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51 A B C D Figure 58 Once per revolution samples for b= 1.6 mm and worn insert: A) time history of x and y displacements of the tool. The once per revolution samples now vary considerably. B) The x and y displacements are plotted against each other. The once per revolution samples appear as an ellipse. C) Time history of Fx and Fy. D) Fx vs Fy. Note that the once per revolution samples collapse onto a straight line for the force data. The force data was measured using the tablemounted dynamometer. Once per revolution sampling was limited to the steady state portion of the force to remove the influence of the cut entry and exit transients on the results. Figure 59 shows the onceper revolution plots for new and worn insert tests.

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52 A B C D E F Figure 59. Once per revolution plots for new and worn inserts at 0.8 mm, 1.6 mm, 2.2 mm and 3.0 mm. A, C, E and G) N ew insert B, D, F and H) W orn insert.

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53 G H Figure 59 Continued. As seen from figure 59 the once per revolution scatter shows a marked increase for tests at b = 2.2 mm and b = 3.0 mm for new insert and from b = 1.6 mm onwar ds for the worn insert; as described previously, this increased scatter indicates instability. These results confirm to the stability diagram for the new and worn inserts shown in figure 54. Because the distribution of these points begin s to widen when ch atter occurs [36], t he scatter in the sample distribution can be quantified using the variance (the square of the standard deviation) in the sampl ed points. The statistical variance in the once per revolution data can therefore be used identify chatter. The variance for unstable cuts is typically at least an order of magnitude larger than the variance for stable cuts [44] The variance in the once per revolution sampled force data was normalized with the variance of the (unsampled) force data [48] 22 ,, 22 xrevyrev xyR (5 1) 2 x/y,rev is the variance in the once per revolution sampled force data, and the x and y subscripts refer to the x and y directions.

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54 Table 52 shows the R values for test cuts using the new and worn inserts. Figure 510 shows the increase in normalized variance with axial depth of cut for the new and worn inserts. Table 52 Normalized variance for new and worn inserts b (mm) R New Insert 0.8 0.0010 1.6 0.0010 2.2 0.0010 3.0 0.0099 Worn Insert 0.8 0.0038 1.6 0.0052 2.2 0.0039 3.0 0.0107 Figure 510. Normalized variance in the onceper revolution samples with axial depth of cut for the new and worn inserts. The R values are listed in table 52. The normalized variance values increases at b = 3.0 mm for new insert, which indicates chatter. The R values are larger for all instances when using the worn insert. The four to five times increase suggests that all the worn insert cuts were unstable, or marginally stable at best for the 0.8 mm, 1.6 mm and 2.2 mm tests.

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55 Sample surface profiles for w ith b = 1.6 mm for the new and worn inserts are provided in figure s 5 11 and 51 2 respectively The figures display the topography of the machined surface and were obtained using a scanning white light interferometer with a 10x magnification and 2. 5 mm by 1 mm field of view. A B C D Figure 511. Topography of the surface left by the new insert ; A) Topography of the surface shown in B) The image B) does not show any distinctive cutting marks indicative of chatter C ) is the color contoured image of stable cutting, with the line indicating the location at which the profile, D ), was obtained in the feed direction.

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56 A B C D Figure 512. Topography of the surface left by the worn insert ; A) Topography of the surface shown in B) B) shows wavy mar ks indicative of chatter. C) is the color contoured image, with the line indicating the location at which the profile, D ), was obtained. The average surface roughness for figure 511 D is 781.8 nm; this surface does not show any distinct chatter marks. Th e average surface roughness for the unstable result (worn tool) in figure 512 D is 4018.5 nm, which is five times the stable result obtained using the new insert. This shows that the stability limit decreases with increased tool wear. Additional tests were completed to verify the decrease in stability limit due to tool wear. All the conditions were same as the previous tests (

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57 and ft = 0.06 mm/tooth ) and the force was measured using the dynamometer. The increase in Kt and Kn with volume removed (see figure 412) was used to determine the new coefficient values and the corresponding stability limit. Since the stability testing was carried out at different axial and radial depths of cut than tool wear exper iments, the increase in Kt and Kn at 5000 rpm was plotted against volume normalized by both, i.e., Vn = V/(a*b). Figure 513 shows the increase in Kt and Kn at 5000 rpm with normalized volume Figure 513. Variation in Kt and Kn with volume (normalized by the axial and radial depths of cut). Note that this is the same plot as figure 412 for 5000 rpm, except for the normalized volume (Vn = V/(a*b)). From figure 513, the initial values (at Vn =0) for the cutting force coefficients are 2.0 109 N/m2 and 1. 1 109 N/m2, while the slopes of these lines are 3.49105 N/m2/cm and 4.30 109 N/m2/cm for Kt and Kn. For the additional stability evaluation, which was taken to indicate ch atter based on figure 510. The normalized volume removed per test cut was 10.1 cm. A series of stability lobes at the corresponding Vn

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58 were generated based on the increased coefficients (see figure 514); the new coefficients were selected using the inter cept and slope values from figure 513. Figure 514. Stability lobes generated at Vn = {0,101.5,203 and 304.5} mm. As seen from figure 514, the stability limit decreases with volume removed. Initial testing was performed at b = 2.5 mm. Figure 515 shows the variation in R with Vn. As seen in the figure, the R value reaches 0.0036 at a Vn of approximately 200 cm. The transition from stable to unstable behavior occurred later than predi cted by figure 514 (should have been unstable by Vn ~ 100 cm). The disagreement is attributed to uncertainty in the slopes from figure 513 as well as the force coefficients (see figures 412 and 414) When the threshold R value was reached (chatter), t he axial depth of cut was reduced to 2.0 mm and testing was continued. It is seen that the R value rapidly increased and quickly exceeded 0.004. This agrees with the stability limit predicted by figure 514. The axial depth was decreased to 1.5 mm at Vn ~ 300 cm by reducing b = 1.5 mm. The R value dropped, but remained higher than the chatter threshold value. Figure 516 shows the once per revolution force data for selected tests. V n = 0 cm V n = 101.5 cm V n = 203 cm V n = 304.5 cm

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59 Figure 515. Variation in R with Vn A B C D Figure 516. Once per G) corresponds to test numbers 1, 5, 9, 12, 16, 18 and 20 at b = 2.5 mm. Plots H J) corresponds to test numbers 21, 25 and 28 at b = 2.0 mm. Plots K) and L) correspond to test numbers 29 and 30 a t b = 1.5 mm. Each test included another Vn = 10.1 cm increment.

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60 E F G H I J Figure 516. Continued.

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61 K L Figure 516. Continued. Note the increased scatter of the onceper revolution data as testing continues from test numbers 1 to 20. Also, the scatter decreases as b is reduced from 2.5 mm to 2.0 mm at test number 20, which is reflected in the reduced R value. Similar results are obtained for further testing at 2.0 mm and 1.5 mm as shown. These test results further confirm the reduction in stabi lity limit with tool wears (modeled here as the increase in force coefficients Kt and Kn).

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62 CHAPTER 6 CONCLUSIONS AND FUTURE WORK Completed Work Tool wear is an important process limitation in milling. In this work, the effect of tool wear on cutting force was determined experimentally. By representing the force variation using a force model with appropriate cutting force coefficients, tool wear could be incorporated as a process limitation together with stability and SLE in the milling super diagram. Therefore, a new comprehensive milling super diagram can be developed that provides information at the process planning stage for stability, surface location error, tool wear, and uncertainty in a user friendly graphical format. The gray scale color schem e identifies: 1) stable combinations of axial depth of cut and spindle speed that offer both stable cutting conditions and an acceptable, user defined surface location error level within a user selected safety margin (white); 2) stable cutting conditions t hat meet the deterministic SLE limit but are not within the safety margin (light gray); 3) stable cutting conditions that do not meet the surface location error limit (dark gray); and 4) unstable cutting conditions (black). A numerical case study was prese nted to describe the diagram development for user specified values of SLE, volume of material removed, and safety margins. Tests were completed to establish the variation in cutting force coefficients with tool wear as a function of spindle speed and volum e removed for a zero rake/zero helix angle, 15 deg relief angle, square, uncoated carbide insert used to machine 1018 steel. The single insert was mounted in a 19 mm diameter steel tool body. For the selected insert material pair, it was observed that the cutting coefficients, which relate the tangential and normal force components to chip area, increased linearly with volume

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63 removed and the corresponding slope increased linearly with spindle speed. However, the edge (plowing) coefficients, that relate the forces to chip width only, showed no appreciable trend with tool wear. Tests also showed the increases in force coefficients with volume are independent of axial, b, and radial, a, depths of cut. The volume removed can therefore be normalized by both b and a and the results can be extended to any other combination. Using this force model, a comprehensive super diagram can be developed for the user selected volume to be removed by incorporating tool wear through an appropriate increase in the force coefficie nts. Finally, stability predictions were validated by two sets of experiments. In the first experiments, stability lobes were calculated for new and worn inserts based on the corresponding force coefficient values and cutting tests were completed to validate the stability limits. In the second experiments, the increase in force coefficients with volume from tool wear experiments were used to determine the stability lobes at different volumes. In both cases, the normalized statistical variance in the onceper revolution sampled force signal was used to identify chatter. The tests showed good agreement with the predicted stability limits. Future Work This study investigated the effect of tool wear on cutting force coefficients and the corresponding influence on process stability. Tests showed a linear increase in cutting force coefficients with tool wear, while edge coefficients showed no particular trend for the given tool material combination. Also, the tests showed that the increase in force coefficients wa s closely related to flank wear width (FWW). Similar experiments can be carried out for different tool material combinations to determine if the cutting force coefficients show similar trends in both their growth and relationship to FWW with

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64 volume removed. Two options are possible: 1) the FWW can be used to update the force coefficients for a worn tool without requiring the knowledge of volume removed; or 2) the cutting force can be monitored and used to identify the wear state and corresponding force coe fficients. Also, in the tool wear tests performed here, no crater wear was observed. Future experiments can be completed to study the effect of different forms of tool wear on force coefficients. The increase in force coefficients causes not only the stabi lity limit to decrease, but also a proportional increase in the SLE as shown in the numerical study. Tests can be completed to determine the effect of tool wear on SLE. In the stability tests, a once per revolution sampling strategy and data analysis was used instead of the analyzing the frequency spectrum of the cutting force signal. The frequency spectrum analysis was not an effective tool for detecting chatter for worn insert due to increased magnitude of forces and energy in the tooth passing frequency harmonics. Future work can be completed to explore effective analysis techniques for identifying chatter with worn tools. The super diagram can be improved by the addition of surface roughness as a process limitation. Surface roughness depends strongly on the feed per tooth. Therefore, the selection of feed per tooth for the super diagram can be based on the surface roughness. Additionally, uncertainty can be better incorporated by using the probability distribution functions of the input parameters and pr opagating them through the milling process model, rather than the safety margin approach applied here.

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65 APPENDIX A CALCULATION OF FORCE COEFFICIENTS % Jaydeep Karandikar % Coefficient testing close all clear all clc % Remeber that of the 1 and 2 versions of this test the signs have to be % reversed from one another. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Feed Per tooth DATA % %%%%%%%%%%%%%%%%%%%%%%%%%%%% FT = [0.03 0.04 0.05 0.06 0.07]*1e3; % chip load m/tooth X_sensitivity=1; Y_sensitivity=1; mean_FX = zeros(5,5); mean_FY = mean_FX; Nt = 1; a = 4.7625e3; b = 3e3; d = 19.05e3; % teeth diameter, m gamma = 0; % helix angle, deg phis = 120*pi/180; phie = 180*pi/180; % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %%%%M0 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tic FILENAME = 'C:\ Study\ Dynamics of Prod machinary\ Tests\ Test19\ 0measure_6250_0.03_3_25%_down_pass1.pcs'; low_limit=5; high_limit=11; [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; index=find(Time>=low_limit & Time<=high_limit); % id_noise = find(Time<=0.5); % remove_noise_avg_X = mean(Fx(id_noise)); % remove_noise_avg_Y = mean(Fy(id_noise));

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66 % F_X = Fx(index)remove_noise_avg_X ; % Force in X for feed rate i % F_Y = Fy(index)remove_noise_avg_Y ; % Force in Y for feed rate i % % % for cnt1 = 1:250 % for cnt2 = 1:1000 % F_X(cnt2+(cnt11)*2000) = 0; % end % end F_X =Fx(index); F_Y = Fy(index); FX_mean1 = mean(F_X); FY_mean1 = mean(F_Y); FILENAME = 'C:\ Study\ Dynamics of Prod machinary\ Tests\ Test19\ 0measure_6250_0.04_3_25%_down_pass1.pcs'; low_limit=2; high_limit=7; [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; index=(Time>=low_limit & Time<=high_limit); F_X = Fx(index); F_Y = Fy(index); FX_mean2 = mean(F_X); FY_mean2 = mean(F_Y); FILENAME = 'C:\ Study\ Dynamics of Prod machinary\ Tests\ Test19\ 0measure_6250_0.05_3_25%_down_pass1.pcs'; low_limit=2; high_limit=7; [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; index=(Time>=low_limit & Time<=high_limit); F_X = Fx(index); F_Y = Fy(index); FX_mean3 = mean(F_X); FY_mean3 = mean(F_Y); FILENAME = 'C:\ Study\ Dynamics of Prod machinery\ machinary\ Tests\ Test19\ 0measure_6250_0.06_3_25%_down_pass2.pcs'; low_limit =3; high_limit=8;

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67 [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; index1=(Time>=low_limit & Time<=high_limit); F_X = Fx(index1); F_Y = Fy(index1); FX_mean4 = mean(F_X); FY_mean4 = mean(F_Y); FILENAME = 'C:\ Study\ Dynamics of Prod machinary\ Tests\ Test19\ 0measure_6250_0.07_3_25%_down_pass2.pcs'; low_limit =2; high_limit=7; [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; index1=(Time>=low_limit & Time<=high_limit); F_X = Fx(index1); F_Y = Fy(index1); FX_mean5 = mean(F_X); FY_mean5 = mean(F_Y); mean_FX(1,:) = [FX_mean1 FX_mean2 FX_mean3 FX_mean4 FX_mean5]; mean_FY(1,:) = [FY_mean1 FY_mean2 FY_mean3 FY_mean4 FY_mean5]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Linear regression using the meanforce data points p1(1,:) = polyfit(FT,mean_FX(1,:),1); p2(1,:) = polyfit(FT,mean_FY(1,:),1); a0x(1) = p1(1,2); a1x(1) = p1(1,1); a0y(1) = p2(1,2); a1y(1) = p2(1,1); % rx2(1) = (sum((mean_FX(1,:) mean(mean_FX(1,:))).^2) sum((mean_FX(1,:) a0x(1) a1x(1)*FT).^2))/(sum((mean_FX(1,:) mean(mean_FX(1,:))).^2)); ry2(1) = (sum((mean_FY(1,:) mean(mean_FY(1,:))).^2) sum((mean_FY(1,:) a0y(1) a1y(1)*FT).^2))/(sum((mean_FY(1,:) mean(mean_FY(1,:))).^2)); %calculate coefficients kte_fit(1) = (pi*(a0x(1)*(sin(phie)sin(phis))a0y(1)*(cos(phie)cos(phis))))/(Nt*b*(1cos(phiephis))); % Eq. 4.7.24 (modify this line based on your solution)

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68 kne_fit(1) = (pi*(a0x(1)*(cos(phie)cos(phis))+a0y(1)*(sin(phie)sin(phis))))/(Nt*b*(1cos(phiephis))); % Eq. 4.7.24 (modify this line based on your solution) kt_fit(1) = 8*pi*(a1y(1)*(2*phie2*phis+sin(2*phis)sin(2*phie))+ a1x(1)*(cos(2*phis)cos(2*phie))) / ((Nt*b)*((2*phie2*phis+sin(2*phis)sin(2*phie))^2 + (cos(2*phis)cos(2*phie))^2)); kn_fit(1) = 8*pi*(a1x(1)*(2*phie2*phis+sin(2*phis)sin(2*phie))a1y(1)*(cos(2*phis)cos(2*phie))) / ((Nt*b)*((2*phie2*phis+sin(2*phis)sin(2*phie))^2 + (cos(2*phis)cos(2*phie))^2)); figure(1) plot(FT, mean_FX(1,:), 'bo', FT, mean_FY(1,:), 'bs') hold on set(gca,'FontSize', 14) xlabel('f_t (m/tooth)') ylabel('Mean Force (N)') legend('x', 'y') %best fit lines plot(FT, a0x(1)+a1x(1)*FT, 'b:', FT, a0y(1)+a1y(1)*FT, 'b--' ) hold on % % T oc

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69 APPENDIX B SUPER DIAGRAM CODE % Ral E. Zapata % Call program for super diagram data collection % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This program calls the stability_alt_trim program to create stability % lobes and test a series of operating points to see if they are above or % below the stability limit (unstable or stable). % Afterwards the program inputs these same points into a surface location % error calculation to obtain the contours of levels of error. % These are then combined to form a diagram delimiting the areas that are % and are not of use according to the user selected bounds of surface % location error. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all clear all clc % Inputs. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Variables % %%%%%%%%%%%%% % Max Real Values 0.75mm cuts Ktc_vec = 2000e6; % tangential cutting force coefficient, N/m^2 Krc_vec = 666e6; % radial cutting force coefficient, N/m^2 Kte_vec = 0; % tangential edge constant, N/m Kre_vec = 0; % radial edge constant, N/m % Tool description N = 1; % teeth, integer beta = 0; % helix angle, deg d = 19.05*1e3; % teeth diameter, m % Stability lobes Num_lobes = 40; % number of stability lobes to be calculated. ss = 2000:10:20000; % spindle speed, rpm % Machining specifications for test cases ft = .06e3; % feed per tooth (m/tooth) RDOC = .25*d; % Radial depth of cut. rho = RDOC/d; % Radial Immersion as a fractional quantity not a percent % Milling direction used to determine entry and exit angle. %1 = upmilling and 2 = downmilling

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70 dir = 2; ADM = 0; SLE_MAX = 20; %%%%%%%%%%%%%%%%%%%%%%% % FRF %%%%%%%%%%%%%%%%%%%%%%% % MODAL FIT (I created) % n_teeth = 1; % # of teeth on the tool SS_best1 = 18000; % "best speed" arbitrary value for the peak around the first lobe. F_best1 = SS_best1/60*n_teeth; % % Tool description ky = [5e6]; % N/m zetay = [0.05]; % damping coefficient wny = [F_best1]*2*pi; % rad/s my = ky./(wny.^2); % kg cy = 2*zetay.*(my.*ky).^0.5; % Ns/m % Define ydirection FRF (could read in measurement from TXF) fmeas = 0:1:5000; % Hz w = fmeas*2*pi; % rad/s FRFy = (wny(1)^2/ky(1))./(wny(1)^2 w.^2 + i*2*zetay(1)*wny(1).*w); % this for loop adds up any other modes of vibration to the FRF if they % exist in the formulation above. for cnt = 2:length(ky) FRFy = FRFy + (wny(cnt)^2/ky(cnt))./(wny(cnt)^2 w.^2 + i*2*zetay(cnt)*wny(cnt).*w); end FRFx = FRFy; % for this test I am assumming the X and Y dynamics are the same. figure subplot(211) plot(fmeas,real(FRFx),'b') subplot(212) plot(fmeas,imag(FRFx),'b') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % .TXF FILE CALL % Assumptions: % 1. Files already have the hammer and sensor calibration factor in them, but they are presented anyway just in case. % 2. G consists of 6 columns the first of which is the FRF that we want, so it will be extracted. % 3. f consists if the frequencies used in the G vector.

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71 % % Y direction % Fmin = 220; % Fmax = 5000; % % % [FILENAME,PATH] = uigetfile('*.txf'); % FILENAME = [PATH FILENAME]; % [f G] = Txfnew(FILENAME); % % index = find(f>Fmin & fFmin & f
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72 end end clear count % SS_test = [5000 5000 5000 7500 7500 7500 9000 15000 15000 18000 18000 18000 18000]; % test point spindle speeds % ADOC_test = [.5 1 2 .5 1 2 1 .5 2 .5 1 2 3]/2; % test point axial depths %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculations and procedures/functions %%%%%%%%%%%%%%%%%%%%%% % "MEAN" FMC data calculations Ktc = Ktc_vec; Krc = Krc_vec; Kte = Kte_vec; Kre = Kre_vec; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DETERMINING ENTRY AND EXIT ANGLE % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if rho <= 0.5 % Entry and Exit angles (radians) if dir == 1 phi_in = zeros(1,length(rho)); % entry angle (upmilling) phi_out = acos(1rho*2); % exit angle (upmilling) elseif dir == 2 phi_in = piacos(1rho*2); % entry angle (downmilling) phi_out = pi*ones(1,length(rho)); % exit angle (downmilling) else fprintf('The direction you have chosen is incorrect please run again and select "up = 1" or "down = 2" for the dir variable. \ n'); return end elseif rho > 0.5 & rho<1 if dir == 1 phi_in = zeros(1,length(rho)); %phi_out = phi_in + pi/2 + acos(1(rho0.5)*2); phi_out = phi_in + pi/2 + asin(2*rho1); elseif dir == 2 phi_out = pi*ones(1,length(rho)); phi_in = phi_out pi/2 asin(2*rho1); else

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73 fprintf('The direction you have chosen is incorrect please run again and select "up = 1" or "down = 2" for the dir variable. \ n'); return end elseif rho == 1 phi_in = zeros(1,length(rho)); phi_out = pi*ones(1,length(rho)); elseif rho > 1 || rho <=0; fprintf(' Please review the radial immersion of the cut, the current value is illogical. \ n'); return end phistart = phi_in*180/pi; % starting angle, deg phiexit = phi_out*180/pi; % exit end, deg %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DTR = pi/180; % conversion Degrees to radians %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Stability %%%%%%%%%%%%%%%%% % This should output the stability conditions and operating conditions % selected for testing in a table or matrix form, along with the vectors % required to create a stability boundary plot (spindle speed and axial depth). tic [Combination_mat,SS_final,Blim_final] = stability_alt_trim(Ktc, Krc, N, phistart*DTR, phiexit*DTR, ss, Num_lobes,ADOC_test,SS_test, fmeas, FRFx, FRFy); figure plot(SS_final,Blim_final,'b');%,SS_test,ADOC_test,'r*'); xlabel('Spindle speed (rpm)') ylabel(' Axial depth (mm)') title(num2str(RDOC)) axis([Inf Inf 0 20]) stab = Combination_mat(:,3); toc %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % SLE %%%%%%%%%%%%%%%%% quant1 = length(stab); SLE_data_mat = zeros(quant1,4); counter = 0; tic for cnt = 1:length(ADOC)

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74 [e] = sle_SSvec_capable(fmeas, FRFy, Ktc, Krc, Kte, Kre, N, beta, d, phistart, phiexit, spinspeed, ADOC(cnt)*1e3, ft, ADM); sle = e*1e6; for cnt1 = 1:length(sle) counter = counter+1; if abs(real(sle(cnt1)))<=SLE_MAX %acceptable SLE_mult = 0; elseif abs(real(sle(cnt1)))>SLE_MAX %unacceptable SLE_mult = 1; end SLE_data_mat(counter,:) = [spinspeed(cnt1) ADOC(cnt) real(sle(cnt1)) SLE_mult]; end end SLE = SLE_data_mat(:,4); index = find(SLE_data_mat(:,2)>= 2.006 & SLE_data_mat(:,2)<= 2.007); figure plot(SLE_data_mat(index,1),SLE_data_mat(index,3)); xlabel('Spindle speed (rpm)') ylabel('SLE (mm)') % ADOC_sle = 2.0068; % for slecnt = 1:length(spinspeed) % [e] = sle_SSvec_capable(fmeas, FRFy, Ktc, Krc, Kte, Kre, N, beta, d, phistart, phiexit, spinspeed(slecnt), ADOC_sle, ft, ADM); % sle_aodc(slecnt) = e*1e6; % end % % figure % plot(spinspeed,sle_adoc); % xlabel('Spindle speed (rpm)') % ylabel('SLE (mm)') % toc %%%%%%%%%%%%%%%%%%%%%%%%% % Data for Diagram %%%%%%%%%%%%%%%%%%%%%%%%% Total = stab+SLE;%life; Total_mat = zeros(length(ADOC),length(spinspeed)); for cnt = 1:length(ADOC) Total_mat(cnt,:) = Total(1+(cnt1)*length(spinspeed):cnt*length(spinspeed)).'; end

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75 % % figure % %contourf(spinspeed,ADOC,Total_mat) % [c, h] = contour(spinspeed,ADOC,Total_mat,[0:3]); % xlabel('Spindle speed (rpm)') % ylabel('Axial Depth (mm)') % clabel(c,h) % figure contourf(spinspeed,ADOC,Total_mat,[0:3]); xlabel('Spindle speed (rpm)') ylabel('Axial Depth (mm)') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% %MRR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% for cnt2 = 1:length(SS_test) MRR1(cnt2) = RDOC*ft*N*SS_test(cnt2)*ADOC_test(cnt2); end %MRR_max = max(MRR) for cnt3 = 1:length(SS_test) if Total(cnt3) == 1 | Total(cnt3) == 2 | Total(cnt3) == 3 | Total(cnt3) == 4 MRR(cnt3) = 0; elseif Total(cnt3) == 0 MRR(cnt3) = MRR1(cnt3); end end MRR_max = max(MRR); a=0; b=0; for cnt4 = 1:length(SS_test) Max_MRR = RDOC*ft*N*SS_test(cnt4)*ADOC_test(cnt4); if Max_MRR == MRR_max Max_Speed = SS_test(cnt4); Max_ADOC = ADOC_test(cnt4); b= cnt4; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% %Safety limits to Parameters %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% SS_tol = 100; % Safety limit for Spindle speed ADOC_tol = 0.5; %Safety limit for Axial Depth of Cut

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76 ds = spinspeed(2)spinspeed(1); da = ADOC(2)ADOC(1); SS_factor = ceil(SS_tol/ds); % grid steps ADOC_factor = ceil(ADOC_tol/da); % grid steps MRR_safety = zeros(1,length(SS_test)); for cnt5 = 1:length(ADOC) counter = cnt51; for cnt6 = 1:length(spinspeed) if cnt5<=ADOC_factor if cnt6<=SS_factor if Total_mat(cnt5+ADOC_factor,cnt6+SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6 )==0&Total_mat(cnt5,cnt6+SS_factor)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6>=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5+ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6=ADOC_factor if cnt5>=length(ADOC)ADOC_factor if cnt6<=SS_factor if Total_mat(cnt5ADOC_factor,cnt6+SS_factor)==0&Total_mat(cnt5,cnt6+SS_factor)==0&Total_mat(cn t5ADOC_factor,cnt6)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6>=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 %check for surroundind 9 data points MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6
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77 if Total_mat(cnt5ADOC_factor,cnt6+SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5,cnt6+SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 %check for surroundind 9 data points MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end end end elseif cnt5=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5+ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6)==0&Total_mat(cnt5ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 %check for surroundind 9 data points MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6
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78 [row,col] = size(Total_mat); MRR_mat = zeros(row,col); for cnt = 1:row MRR_mat(cnt,:) = MRR_safety(1+(cnt1)*col:cnt*col); end % % figure % contourf(spinspeed,ADOC,MRR_mat) id_MRR = find(MRR_mat>0); Saferegion = MRR_mat; Saferegion(id_MRR) = 1; % % figure % contourf(spinspeed,ADOC,Saferegion) % xlabel('Spindle speed (rpm)') % ylabel('Axial depth (mm)') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Repeat with zero limits for comparison SS_tol = 0; %Safety limit for Spindle speed ADOC_tol = 0; %Safety limit for Axial Depth of Cut ds = spinspeed(2)spinspeed(1); da = ADOC(2)ADOC(1); SS_factor = ceil(SS_tol/ds); % grid steps ADOC_factor = ceil(ADOC_tol/da); % grid steps MRR_safety = zeros(1,length(SS_test)); for cnt5 = 1:length(ADOC) counter = cnt51; for cnt6 = 1:length(spinspeed) if cnt5<=ADOC_factor if cnt6<=SS_factor if Total_mat(cnt5+ADOC_factor,cnt6+SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6 )==0&Total_mat(cnt5,cnt6+SS_factor)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6>=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5+ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end

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79 elseif cnt6=ADOC_factor if cnt5>=length(ADOC)ADOC_factor if cnt6<=SS_factor if Total_mat(cnt5ADOC_factor,cnt6+SS_factor)==0&Total_mat(cnt5,cnt6+SS_factor)==0&Total_mat(cn t5ADOC_factor,cnt6)==0 MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6>=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 %check for surroundind 9 data points MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end elseif cnt6=SS_factor if cnt6>=length(spinspeed)SS_factor if Total_mat(cnt5+ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5ADOC_factor,cnt6SS_factor)==0&Total_mat(cnt5+ADOC_factor,cnt6)==0&Total_mat(cnt5ADOC_factor,cnt6)==0&Total_mat(cnt5,cnt6SS_factor)==0 %check for surroundind 9 data points MRR_safety(cnt6+counter*length(spinspeed)) = MRR(cnt6+counter*length(spinspeed)); end

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80 elseif cnt60); Saferegion1 = MRR_mat; Saferegion1(id_MRR1) = 1; % % figure % contourf(spinspeed,ADOC,Saferegion1) % xlabel('Spindle speed (rpm)') % ylabel('Axial depth (mm)') Comparison = Saferegion1Saferegion; figure contourf(spinspeed,ADOC,Comparison) xlabel('Spindle speed (rpm)') ylabel('Axial depth (mm)')

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81 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % figure % %contourf(spinspeed,ADOC,Total_mat) % [c, h] = contour(spinspeed,ADOC,Total_mat,[0:3]); % xlabel('Spindle speed (rpm)') % ylabel('Axial Depth (mm)') save numerical_study_19_2000_20.mat SLE_data_mat Combination_mat spinspeed ADOC MRR Comparison Stability Code % Ral E. Zapata % Altintas Stability Lobes Trimmed with search algorithm for stable and % unstable individual points. % October 2006 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [Combination_mat,SS_final,Blim_final] = stability_alt_trim(kt, kn, N, phis, phie, spin_speed, Num_lobes, ADOC_vec, SS_vec, f, Gxx, Gyy) % This function provides the full set off tested points along with the % spindle speed and Blim vectors for the stability lobe diagram they were % compared to. The last column on Combination_mat is the stability % multiplier where 1 means that the parameter set is stable and zero means % the parameter set is unstable. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Calculations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% w_temp = f*2*pi; w = zeros(1,length(w_temp)); for cnt = 1:length(w_temp) w(1,cnt) = w_temp(cnt); end % Enter in the Cutting Coefficient Information Ktc = kt; Krc = kn; % calculate cutting coefficients for monte carlo run Kta=Ktc; Kra=Krc/Ktc; % -----------------------------------------Altintas Code alphaxx=0.5*((cos(2*phie)2*Kra*phie+Kra*sin(2*phie))(cos(2*phis)2*Kra*phis+Kra*sin(2*phis))); alphaxy=0.5*((sin(2*phie)2*phie+Kra*cos(2*phie))( sin(2*phis)2*phis+Kra*cos(2*phis))); alphayx=0.5*((sin(2*phie)+2*phie+Kra*cos(2*phie))( sin(2*phis)+2*phis+Kra*cos(2*phis)));

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82 alphayy=0.5*((cos(2*phie)2*Kra*phieKra*sin(2*phie))( cos(2*phis)2*Kra*phisKra*sin(2*phis))); % Calculate and Sort Eigenvalues eigen1 = zeros(1,length(w)); eigen2 = zeros(1,length(w)); for cnt=1:length(w) A=[alphaxx*Gxx(cnt) alphaxy*Gyy(cnt);alphayx*Gxx(cnt) alphayy*Gyy(cnt)]; E=eig(A); temp=E(1); eigen1(cnt)=temp; temp=E(2); eigen2(cnt)=temp; if (cnt>1) dot_prod1=real(eigen2(cnt))*real(eigen2(cnt1))+imag(eigen2(cnt))*imag(eigen2(cnt1)); dot_prod2=real(eigen2(cnt))*real(eigen1(cnt1))+imag(eigen2(cnt))*imag(eigen1(cnt1)); if(dot_prod2>dot_prod1) temp=eigen2(cnt); eigen2(cnt)=eigen1(cnt); eigen1(cnt)=temp; end end end eigen1=eigen1'; eigen2=eigen2'; % Calculate alim values for each eigenvalue alim1=(2*pi/N/Kta)./((real(eigen1)).^2+(imag(eigen1)).^2).*(real(eigen1).*(1+ (imag(eigen1)./real(eigen1)).^2)); alim2=(2*pi/N/Kta)./((real(eigen2)).^2+(imag(eigen2)).^2).*(real(eigen2).*(1+ (imag(eigen2)./real(eigen2)).^2)); % Choose positive Values of alim1 [index1]=find(alim1>0); alim1=alim1(index1); alim1=alim1*1000; w1=w(index1).'; psi1=atan2(imag(eigen1),real(eigen1)); psi1=psi1(index1); epsilon1=(pi2*psi1); % Note that this is a column vector %Choose positive Values of alim2 [index2]=find(alim2>0); alim2=alim2(index2); alim2=alim2*1000; w2=w(index2).'; psi2=atan2(imag(eigen2),real(eigen2)); psi2=psi2(index2); epsilon2=pi2*psi2; % Note that this is a column vector

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83 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Matrix Formulation N1_matrix = zeros(length(epsilon1),Num_lobes); N2_matrix = zeros(length(epsilon2),Num_lobes); for cnt1 = 1:Num_lobes N1_matrix(:,cnt1) = (60/N)*w1./(epsilon1+2*(cnt11)*pi); N2_matrix(:,cnt1) = (60/N)*w2./(epsilon2+2*(cnt11)*pi); end SS_final = zeros(1,length(spin_speed)); Blim_final = SS_final; for cnt2=1:length(spin_speed) SS_final(cnt2)=spin_speed(cnt2); % Initiallize Search Matrices blim1_temp = zeros(length(epsilon1),Num_lobes); blim1_temp(1,:)=1e20; blim2_temp = zeros(length(epsilon2),Num_lobes); blim2_temp(1,:)=1e20; % Populate search matrices by linearly interpolating the desired SS points. for cnt3 = 1:Num_lobes for cnt4=2:length(epsilon1) if(spin_speed(cnt2)>N1_matrix(cnt4,cnt3) & spin_speed(cnt2)N1_matrix(cnt41,cnt3)) blim1_temp(cnt4,cnt3)=alim1(cnt41)+(alim1(cnt4)alim1(cnt41))*(spin_speed(cnt2)N1_matrix(cnt41,cnt3))/(N1_matrix(cnt4,cnt3)N1_matrix(cnt41,cnt3)); else blim1_temp(cnt4,cnt3)=1e20; end end for cnt4=2:length(epsilon2) if(spin_speed(cnt2)>N2_matrix(cnt4,cnt3) & spin_speed(cnt2)N2_matrix(cnt41,cnt3)) blim2_temp(cnt4,cnt3)=alim2(cnt41)+(alim2(cnt4)alim2(cnt41))*(spin_speed(cnt2)N2_matrix(cnt41,cnt3))/(N2_matrix(cnt4,cnt3)N2_matrix(cnt41,cnt3)); else blim2_temp(cnt4,cnt3)=1e20;

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84 end end end Blim1_min = min(min(blim1_temp)); % Find minimum boundary for alim1 lobes at this SS Blim2_min = min(min(blim2_temp)); % Find minimum boundary for alim2 lobes at this SS Blim_final(cnt2) = min([Blim1_min Blim2_min]); % Find global minimum boundary for this SS end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Test parameters stability check Counter = 0; Combination_mat = zeros(length(ADOC_vec),3); %for cnt50 = 1:length(ADOC_vec) for cnt51 = 1:length(SS_vec) cnt50 = cnt51; Counter = Counter+1; index_low = max(find(SS_final < SS_vec(cnt51))); index_high= min(find(SS_final >= SS_vec(cnt51))); SS_left = SS_final(index_low); SS_right = SS_final(index_high); B_lim_left = Blim_final(index_low); B_lim_right = Blim_final(index_high); Blim_value = B_lim_right+(B_lim_leftB_lim_right)*(SS_vec(cnt51)SS_right)/(SS_leftSS_right); if Blim_value > ADOC_vec(cnt50) %stable STAB_mult = 0; elseif Blim_value <= ADOC_vec(cnt50); %unstable STAB_mult = 2; end Combination_mat(Counter,:) = [ADOC_vec(cnt50) SS_vec(cnt51) STAB_mult]; end %end SLE Code function [e] = sle_SSvec_capable(fmeas, FRFmeas, kt, kn, kte, kne, m, beta, d, phistart, phiexit, omega, b, ft, ADM) % T. Schmitz (10/20/05)

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85 % This is a program to find the SLE in helical peripheral end milling using a frequency domain approach. % Regeneration is not considered. % Consider only up and downmilling cases if phistart == 0 flag = 0 + 2*ADM*tan(beta)/d; % upmilling else flag = 180 + 2*ADM*tan(beta)/d; % downmilling end DTR = pi/180; % degrees to radians conversion % Simulation specifications n = 11; steps = 2^n; % steps for one cutter revolution, int dt = 60./(steps*omega); % integration time step, s dphi = 360/steps; % angular steps size between time steps, deg if beta == 0 % straight teeth db = b; % discretized axial depth, m else % nonzero helix angle db = d*(dphi*DTR)/2/tan(beta*DTR); end steps_axial = round(b/db); % number of steps along tool axis tooth_angle = 0:360/m:(360360/m); % angles of m cutter teeth starting from zero, deg % Initialize vectors teeth = round(tooth_angle/dphi) + 1; phi = linspace(0, (steps1)*dphi, steps); Force_y = zeros(1, steps); %************************** MAIN PROGRAM (Force)****************************** for cnt1 = 1:steps % time steps, s for cnt2 = 1:m teeth(cnt2) = teeth(cnt2) + 1; % index teeth pointer one position (rotate cutter by dphi) if teeth(cnt2) > steps teeth(cnt2) = 1; end end Fy = 0; for cnt3 = 1:m % sum forces over all teeth, N for cnt4 = 1:steps_axial % sum forces along axial depth of helical endmill, N phi_counter = teeth(cnt3) (cnt41); if phi_counter < 1 % helix has wrapped through phi = 0 deg phi_counter = phi_counter + steps; end phia = phi(phi_counter); % angle for given axial disk, deg

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86 if (phia >= phistart) && (phia <= phiexit) % verify that tooth angle is in specified range for current disk, deg h = ft*sin(phia*DTR); % chip thickness, m ftan = kt*db*h + kte*db; frad = kn*db*h + kne*db; else % tooth angle is outside range bounded by radial immersion ftan = 0; frad = 0; end Fy = Fy frad*cos(phia*DTR) + ftan*sin(phia*DTR); % N end % cnt4 loop end % cnt3 loop Force_y(cnt1) = Fy; end % cnt1 loop %************************** END OF MAIN PROGRAM ****************************** e = zeros(1,length(dt)); for cnt_dt = 1:length(dt) % Compute FFT of Fourierbased ydirection force [FY, freq] = spec(Force_y', 1/dt(cnt_dt)); % compute FFT (in spec.m, there should be no multiplication by T) FY = FY/(2^n); % correct magnitude to N % Define ydirection FRF on proper frequency vector, freq index = find(freq >= min(fmeas) & freq <= max(fmeas)); freqtemp = freq(index); FRFytemp2 = interp1(fmeas, FRFmeas, freqtemp, 'spline'); FRFytemp1 = ones(1, (index(1)1))*FRFytemp2(1); FRFytemp3 = ones(1, (length(freq)index(length(index))))*FRFytemp2(length(FRFytemp2)); FRFy = [FRFytemp1 FRFytemp2' FRFytemp3]'; Yf = FY.*FRFy; % F X/F = X Yf_dc = Yf(1); % DC component Yf(1) = 0; % DC extraction y = real(ifft(Yf*(2^n))); % convert to timedomain y = y+Yf_dc; % DC insertion to create full signal t = 0:2*dt(cnt_dt):(length(y)1)*2*dt(cnt_dt); % new time vector after inverse FFT % Use automatic method to sample y tfirst = (flag*DTR)/(omega(cnt_dt)/60*2*pi); % time for first SLE point, s

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87 index = find(t > tfirst); first_point = index(1) 1; % first point for SLE index = first_point:round(60/(omega(cnt_dt)*m)/(2*dt(cnt_dt))):length(t); y_sampled = y(index); % sampled position vector -SLE values e(cnt_dt) = y_sampled(length(y_sampled)); % record SLE, m end Fast Fourier Transform C ode U sed in SLE C alculations % Computes the fft X of signal x and the corresponding frequency vector f given % the sampling frequency fs. % % [X,f]=spec(x,fs) % % [X,f]=spec(x,fs,'whole') returns values around the whole unit circle function [X,f]=spec(x,fs,whole) T=1/fs; N=length(x); %X=T*fft(x); X=fft(x); f=[0:fs/N:(1 1/(2*N))*fs]'; if nargin == 2 X=X(1:N/2+1,:); f=f(1:N/2+1,:); end

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88 APPENDIX C TIME DOMAIN SIMULATI ON Code for Time Domain Simulation % University of Florida % T. Schmitz clc close all clear all %Define cutting force coefficients % Ks = 2000e6; % N/m^2 % beta = 60; % deg kt = 4.9883e9;%Ks*sin(beta*pi/180); kn = 4.5145e9;%Ks*cos(beta*pi/180); kte = 45500;%55600; kne = 25500; % Define modal parameters for x direction load MODAL_FIT_short2.mat k_x wn_x zeta_x k_y wn_y zeta_y ky = 1.5*k_y; zetay = zeta_y; wny = wn_y; kx = 1.5*k_x; zetax = zeta_x; wnx = wn_x; my = ky./(wny.^2); % kg cy = 2*zetay.*(my.*ky).^0.5; % Ns/m x_modes = length(ky); % number of modes in x direction % Define modal parameters for y direction mx = kx./(wnx.^2); % kg cx = 2*zetax.*(mx.*kx).^0.5; % Ns/m y_modes = length(kx); % number of modes in y direction % Define cutting parameters Nt = 1; d = 19.05e3; % teeth diameter, m gamma = 0; % helix angle, deg phis = 0; % deg phie = 180; % deg omega = 5100; % rpm b = 1.6e3; % m ft = 0.06e3; % m steps_rev = 360; % Calculate actual steps_rev to ensure integer value % for ratio (steps_rev/Nt) temp = round(steps_rev/Nt);

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89 steps_rev = temp*Nt; dt = 60/(steps_rev*omega); % s dphi = 360/steps_rev; % deg if gamma == 0 db = b; else % discretized axial depth, m db = d*(dphi*pi/180)/2/tan(gamma*pi/180); end % number of steps along tool axis steps_axial = round(b/db); rev = 200; steps = rev*steps_rev; % Initialize vectors for cnt = 1:Nt teeth(cnt) = (cnt1)*steps_rev/Nt + 1; end for cnt = 1:steps_rev phi(cnt) = (cnt 1)*dphi; end surf = zeros(steps_axial, steps_rev); % Euler integration initial conditions x = 0; y = 0; dp = zeros(1, x_modes); p = zeros(1, x_modes); % xdirection modal displacements, m dq = zeros(1, y_modes); q = zeros(1, y_modes); % ydirection modal displacements, m % Function to keep track of simulation progress handle = waitbar(0, 'Please wait... simulation in progress.'); for cnt1 = 1:steps waitbar(cnt1/steps, handle) for cnt2 = 1:Nt teeth(cnt2) = teeth(cnt2) + 1; if teeth(cnt2) > steps_rev teeth(cnt2) = 1; end end Fx = 0; Fy = 0; for cnt3 = 1:Nt for cnt4 = 1:steps_axial phi_counter = teeth(cnt3) (cnt41); if phi_counter < 1 % helix has wrapped through phi = 0 deg phi_counter = phi_counter + steps_rev; end

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90 phia = phi(phi_counter); % angle for given axial disk, deg if (phia >= phis) & (phia <= phie) n = x*sin(phia*pi/180) y*cos(phia*pi/180); % m h = ft*sin(phia*pi/180) + surf(cnt4, phi_counter) n; % m if h < 0 Ft = 0; Fn = 0; surf(cnt4, phi_counter) = surf(cnt4, phi_counter) + ft*sin(phia*pi/180); else Ft = kt*db*h + kte*db; Fn = kn*db*h + kne*db; surf(cnt4, phi_counter) = n; end else Ft = 0; Fn = 0; end Fx = Fx + Ft*cos(phia*pi/180) + Fn*sin(phia*pi/180); Fy = Fy + Ft*sin(phia*pi/180) Fn*cos(phia*pi/180); end end Forcex(cnt1) = Fx; Forcey(cnt1) = Fy; % Numerical integration for position x = 0; y = 0; % x direction for cnt5 = 1:x_modes ddp = (Forcex(cnt1) cx(cnt5)*dp(cnt5) kx(cnt5)*p(cnt5))/mx(cnt5); dp(cnt5) = dp(cnt5) + ddp*dt; p(cnt5) = p(cnt5) + dp(cnt5)*dt; x = x + p(cnt5); % m end ax(cnt1) = ddp; % y direction for cnt5 = 1:y_modes ddq = (Forcey(cnt1) cy(cnt5)*dq(cnt5) ky(cnt5)*q(cnt5))/my(cnt5); dq(cnt5) = dq(cnt5) + ddq*dt; q(cnt5) = q(cnt5) + dq(cnt5)*dt; y = y + q(cnt5); % m end xpos(cnt1) = x; ypos(cnt1) = y; ay(cnt1) = ddq; end close(handle); % close progress bar time = ((1:steps)1)*dt; % s

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91 figure(1) subplot(211) plot(time, Forcex, 'b') hold on xlim([0 max(time)]) set(gca,'FontSize', 14) ylabel('F_x (N)') subplot(212) plot(time, xpos*1e6) xlim([0 max(time)]) set(gca,'FontSize', 14) xlabel('t (s)') ylabel('x (\ mum)') figure(2) subplot(211) plot(time, Forcey, 'b') hold on xlim([0 max(time)]) set(gca,'FontSize', 14) ylabel('F_y (N)') subplot(212) plot(time, ypos*1e6) xlim([0 max(time)]) set(gca,'FontSize', 14) xlabel('t (s)') ylabel('y (\ mum)') F = (Forcex.^2 + Forcey.^2).^0.5; % N figure(3) plot(time, F) %axis([0.09 0.1 0 1500]) set(gca,'FontSize', 14) xlabel('t (s)') ylabel('F (N)') low_limit =2.3e3; high_limit=low_limit+200*60/omega; index1=(time>=low_limit & time<=high_limit); Forcex1 = Forcex(index1); Forcey1 = Forcey(index1); time = time(index1); x1 = xpos(index1); y1 = ypos(index1); fx2 = Forcex1(1:steps_rev:length(Forcex1)); time2 = time(1:steps_rev:length(time)); fy2 = Forcey1(1:steps_rev:length(Forcey1)); x2 = x1(1:steps_rev:length(x1)); y2 = y1(1:steps_rev:length(y1)); figure(9) subplot(211)

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92 plot(time, Forcex1,'y') hold on plot(time2,fx2,'k.') set(gca,'FontSize', 14) xlabel('Time (sec)') ylabel('F_x (N)') subplot(212) plot(time, Forcey1,'y') hold on plot(time2,fy2,'k.') set(gca,'FontSize', 14) xlabel('Time (sec)') ylabel('F_y (N)') figure(10) plot(Forcex1,Forcey1,'y.') hold on plot(fx2,fy2,'k.') set(gca,'FontSize', 14) xlabel('F_x (N)') ylabel('F_y (N)') figure(11) subplot(211) plot(time, x1*1e6,'y') hold on plot(time2,x2*1e6,'k.') set(gca,'FontSize', 14) xlabel('Time (sec)') ylabel('x ( \ mum)') subplot(212) plot(time, y1,'y') hold on plot(time2,y2,'k.') set(gca,'FontSize', 14) xlabel('Time (sec)') ylabel('y (\ mum)') figure(12) plot(x1,y1,'y.') hold on plot(x2,y2,'k.') set(gca,'FontSize', 14) xlabel('x (\ mum)') ylabel('y (\ mum)') % % Code for Modal Fitting % Ral Zapata % Peak Picking modal fit % Based off Scott Duncan's code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all; clear all; clc;

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93 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % FRF's % %%%%%%%%% % % % % Midlength tool % % % load G_middlex.mat G_mass f; % % % FRF_xdir = G_mass; % % % freq1 = f; % % % clear G_mass f % % % % % % load G_middley.mat G_mass f; % % % FRF_ydir = G_mass; % % % clear G_mass f % Short tool load FRFx.mat FRFx fmeas; FRF_xdir = FRFx; freq1 = fmeas; clear FRFx fmeas load FRFy.mat FRFy fmeas; FRF_ydir = FRFy; clear FRFy fmeas %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % X DIRECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % FIGURES FOR PEAK PICKING % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plots of the unprocessed holder/spindle data figure(1) subplot(211) plot(freq1,real(FRF_xdir)) title('Unprocessed Frequency Response Function, Holder/Spindle, Real: X') xlabel('Frequency (hz)') ylabel('Real (m/N)') subplot(212) plot(freq1,imag(FRF_xdir)) title('Unprocessed Frequency Response Function, Holder/Spindle, Imaginary: X') xlabel('Frequency (hz)') ylabel('Imaginary (m/N)') % Choose number of modes number_modes = input('How many modes do you wish to fit?'); % Select the points for peak picking imag_min_x = ginput(number_modes); real_max_x = ginput(number_modes); real_min_x = ginput(number_modes);

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94 % Calculate the values for the initial guess parameters for nonlinear least squares fit for n = 1:number_modes fn_x(n) = imag_min_x(n,1); wn_x(n) = fn_x(n)*2*pi; zeta_x(n) = abs((real_max_x(n,1)real_min_x(n,1))/(2*imag_min_x(n,1))); k_x(n) = 1/(2*imag_min_x(n,2)*zeta_x(n)); m_x(n) = k_x(n)/wn_x(n)^2; c_x(n) = zeta_x(n)*2*sqrt(k_x(n)*m_x(n)); end f = 1:10000; w = f*2*pi; G_measured_x = 0; for n = 1:number_modes G_measured_x = ones(size(w))./(k_x(n)*ones(size(w))m_x(n)*w.^2+i*c_x(n)*w)+G_measured_x; end % Write m, c, and k back to main gui program figure(2) subplot(211) plot(freq1,real(FRF_xdir),f,real(G_measured_x)) title('Actual Versus Fit FRF for Holder/Spindle, Real: X') xlabel('Frequency (hz)') ylabel('Real (m/N)') legend('Actual', 'Fit') subplot(212) plot(freq1,imag(FRF_xdir),f,imag(G_measured_x)) title('Actual Versus Fit FRF for Holder/Spindle, Imaginary: X') xlabel('Frequency (hz)') ylabel('Imaginary (m/N)') legend('Actual', 'Fit') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Y DIRECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % FIGURES FOR PEAK PICKING % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Plots of the unprocessed holder/spindle data figure(3) subplot(211) plot(freq1,real(FRF_ydir)) title('Unprocessed Frequency Response Function, Holder/Spindle, Real: Y') xlabel('Frequency (hz)') ylabel('Real (m/N)') subplot(212) plot(freq1,imag(FRF_ydir)) title('Unprocessed Frequency Response Function, Holder/Spindle, Imaginary: Y')

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95 xlabel('Frequency (hz)') ylabel('Imaginary (m/N)') % Choose number of modes number_modes = input('How many modes do you wish to fit?'); % Select the points for peak picking imag_min_y = ginput(number_modes); real_max_y = ginput(number_modes); real_min_y = ginput(number_modes); % Calculate the values for the initial guess parameters for nonlinear least squares fit for n = 1:number_modes fn_y(n) = imag_min_y(n,1); wn_y(n) = fn_y(n)*2*pi; zeta_y(n) = abs((real_max_y(n,1)real_min_y(n,1))/(2*imag_min_y(n,1))); k_y(n) = 1/(2*imag_min_y(n,2)*zeta_y(n)); m_y(n) = k_y(n)/wn_y(n)^2; c_y(n) = zeta_y(n)*2*sqrt(k_y(n)*m_y(n)); end f = 1:10000; w = f*2*pi; G_measured_y = 0; for n = 1:number_modes G_measured_y = ones(size(w))./(k_y(n)*ones(size(w))m_y(n)*w.^2+i*c_y(n)*w)+G_measured_y; end % Write m, c, and k back to main gui program figure(4) subplot(211) plot(freq1,real(FRF_ydir),f,real(G_measured_y)) title('Actual Versus Fit FRF for Holder/Spindle, Real: Y') xlabel('Frequency (hz)') ylabel('Real (m/N)') legend('Actual', 'Fit') subplot(212) plot(freq1,imag(FRF_ydir),f,imag(G_measured_y)) title('Actual Versus Fit FRF for Holder/Spindle, Imaginary: Y') xlabel('Frequency (hz)') ylabel('Imaginary (m/N)') legend('Actual', 'Fit') % SAVING DATA save MODAL_FIT_short2.mat k_x wn_x zeta_x k_y wn_y zeta_y

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96 APPENDIX D ONCE PER REVOL U TION AND VARIANCE CO DE % Jaydeep Karandikar % Coefficient testing close all clear all clc % Remeber that of the 1 and 2 versions of this test the signs have to be % reversed from one another. % % %%%%%%%%%%%%%%%%%%%%%%%%%%%% % Feed Per tooth DATA %%%%%%%%%%%%%%%%%%%%%%%%%%%% FT = [0.01 0.02 0.03 0.04 0.05]*1e3; % chip load m/tooth X_sensitivity=1; Y_sensitivity=1; Nt = 1; a = 19.05e3; b = 3e3; d = 19.05e3; % teeth diameter, m gamma = 0; % helix angle, deg phis = 0*pi/180; phie = 180*pi/180; omega = 5100; % rpm tic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %M0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% handle = waitbar(0, 'Please wait... simulation in progress.'); for cnt=1:181 waitbar(cnt/181, handle) FILENAME = 'C:\ Study\ Dynamics of Prod machinary\ Tests\ Stability_11_15_Final\ worn_tool_5100rpm_0.4adoc_100ri_0pt06ft .pcs'; [Signal,Time] = pcscopenew(FILENAME); Signal = Signal; Fx=Signal(:,1)*X_sensitivity; Fy=Signal(:,2)*Y_sensitivity; time_rev = 1/(omega/60); sample_pts = 1080; incr = time_rev/sample_pts; low_limit =2.0161+(cnt1)*(5.8823e3/180);%1.9607e4;%floor(1.0085/time_rev)*time_rev; high_limit=low_limit+15;

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97 index1=(Time>=low_limit & Time<=high_limit); FX = Fx(index1); FY = Fy(index1); Time = Time(index1); time_rev = 1/(omega/60); sample_pts = 1080; incr = time_rev/sample_pts; dt = low_limit:incr:high_limit; % F_X = interp1(Time,FX,dt); F_Y = interp1(Time,FY,dt); fx = F_X(sample_pts:sample_pts:length(F_X)); time = dt(sample_pts:sample_pts:length(dt)); fy = F_Y(sample_pts:sample_pts:length(F_Y)); Vx_rev(cnt) = var(fx); Vy_rev(cnt) = var(fy); % Vx(cnt) = var(FX); % Vy(cnt) = var(FY); % R(cnt) = (Vx_rev(cnt)^2+Vy_rev(cnt)^2)/(Vx(cnt)+Vy(cnt)); end toc Vxrev = mean(Vx_rev) Vyrev = mean(Vy_rev) Vx = var(FX) Vy = var(FY) R = (Vxrev^2+Vyrev^2)/(Vx^2+Vy^2) close(handle); % close progress bar figure subplot(211) plot(Time, FX,'b') hold on plot(time,fx,'k.') xlabel('Time (sec)', 'FontSize', 14) ylabel('F_x (N)', 'FontSize', 14) xlim([low_limit high_limit]) ylim([1000 1300]) set(gca,'FontSize', 14) subplot(212) plot(Time, FY, 'b') hold on plot(time,fy,'k.') xlabel('Time (sec)', 'FontSize', 14) ylabel('F_y (N)', 'FontSize', 14) ylim([500 1300]) xlim([low_limit high_limit]) set(gca,'FontSize', 14)

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102 BIOGRAPHICAL SKETCH The author was born in the city of Mumbai ( formerly known as Bombay), India in 1985. After spending his early years in Mumbai he along with his family, moved to the small town of Pune, India popularly known as the Oxford of the East. After finishing his high school education in 2003, he pursued his under graduate degree in m echanical e ngineering at the University of Pune, India. After four grueling years of studies, project assignments and voluntary community service activities, the author obtained his b achelors degree in m echanical e ngineering with distinction. He secured a job with Forbes Marshall India Pvt Ltd. a Company engaged in the manufacturing of control valves. After working with Forbes as a design engineer for a year, the author quit his job in order to pursue m asters in m echanical e ngineering from University of Florida, Gainesville. The author plans to pursue PhD under the guidance of his advisor Dr. Tony L. Schmitz at the Machine Tool Research Center.