MECHANICS AND DYNAMICS OF ORBITAL DRILLING OPERATIONS by ONUR MERT OZTURK B.Sc., Istanbul Technical University, Turkey, 2014 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2017 ยฉ Onur Mert Ozturk, 2017 ii Abstract Opening large number of holes takes considerable amount of time during manufacturing and assembly of aircrafts. Traditionally, tools having the same diameter of each hole have been used in drilling which take considerable amount of time for tool change and fixturing. Recently, orbital drilling technology has been introduced to open holes with a single set-up. The combined orbital motion around the hole and helical penetration in axial direction are either given by stationary computer numerically controlled (CNC) machines or hand held, portable heads that are attached to aircraft body with suction pads. Although the tool path and machine were developed, the process mechanics and dynamics have not been modeled to predict cutting forces, torque, power and chatter stability diagrams to identify most productive and safe cutting conditions. This thesis presents mathematical model to simulate the mechanics and dynamics of orbital drilling process. The mechanics of the process are modeled by identifying the chip thickness distribution along the peripheral and bottom cutting edges of the helical end mills used in orbital drilling, The pitch length of the path, tool and hole diameters, spindle speed, feed and material properties are used in the model which is experimentally proven by comparing predicted and measured cutting forces. The flexibilities of the orbital drilling head and tool are incorporated to the mechanics model to predict the dynamics of the system. It is shown that the additional delay contributed by orbital motion of the tool can be neglected, and the regenerative delay is dominated by the spindle speed. However, structural dynamic modes of the system need to be oriented along the tangential feed direction since it varies continuously along the orbital path. The chatter stability iii of the system has been developed in both frequency and semi-discrete time domains. The experimentally verified stability model considers spindle speed, tool and hole geometries, structural dynamics, material properties and orbital drilling pitch length. The proposed orbital drilling model allows optimal selection of spindle speed, feed, orbital speed, pitch length and tool diameter for a given work material without overloading the machine and chatter while achieving highest possible material removal rates. iv Lay Summary Drilling operations are widely used in aircraft industry to assemble light structural parts. Traditional drilling requires tools having the same diameter as the hole, hence significant number of drills are needed in production. Additionally, drills cannot be used to open holes with large diameters. The drilling must be followed by boring, which increases cost and production time. Orbital drilling is an emerging method to open holes with any diameter, which is larger than the tool diameter. Portable, computer controlled orbital drilling heads are clamped to aircraft body with suction pads. End mill with drilling edges at the bottom traverses along the hole circumference while helically penetrating into material along the hole axis. This thesis presents the mathematical model of the process to predict the chatter vibration free, most productive spindle speed, feed, and pitch length of the process as a function of tool geometry, work material properties and machineโs stiffness. v Preface This thesis presents a product of the research conducted by the author, Onur Mert รztรผrk, under the supervision of Professor Yusuf Altintas. This project has been completed in the Manufacturing Automation Laboratory at the University of British Columbia. I was the lead investigator of the thesis, responsible for conducting the experimental work, developing the algorithms and writing the manuscript. The thesis was proposed, supervised and edited by Professor Yusuf Altintas. vi Table of Contents Abstract .......................................................................................................................................... ii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Tables .............................................................................................................................. viii List of Figures ............................................................................................................................... ix Nomenclature ............................................................................................................................. xiii Acknowledgements ................................................................................................................... xvii Dedication ................................................................................................................................. xviii Chapter 1: Introduction ............................................................................................................... 1 Chapter 2: Literature Review ...................................................................................................... 3 2.1 Overview ..................................................................................................................... 3 2.2 Application and Experimental Investigation of Orbital Drilling Operations .............. 3 2.3 Mechanics and Dynamics of Orbital Drilling Operations .......................................... 6 Chapter 3: Mechanics of Orbital Drilling Operations .............................................................. 9 3.1 Introduction ................................................................................................................. 9 3.2 Geometric Modelling of Tool and Workpiece Intersection ...................................... 11 3.2.1 Modelling of the Radial Engagement ................................................................... 12 3.2.2 Comparison of the Engagement Models ............................................................... 15 vii 3.3 Mechanics of Orbital Drilling ................................................................................... 18 3.3.1 Identification of Cutting Force Coefficients at the Bottom Part of the End Mill . 18 3.4 Prediction of Cutting Forces ..................................................................................... 24 3.5 Simulations and Experimental Results ..................................................................... 27 Chapter 4: Chatter Stability of Orbital Drilling Operations .................................................. 33 4.1 Introduction ............................................................................................................... 33 4.2 Dynamic Cutting Forces in Orbital Drilling ............................................................. 34 4.3 Chatter Stability in Orbital Drilling .......................................................................... 37 4.3.1 Orbital Motion Effect on Time Delay ................................................................... 38 4.3.2 Chatter Stability in Frequency Domain Solution .................................................. 43 4.3.3 Chatter Stability in Semi-Discrete Time Domain Solution .................................. 51 4.4 Oriented Frequency Response Function ................................................................... 58 4.5 Simulations and Experimental Results ..................................................................... 59 Chapter 5: Conclusion ................................................................................................................ 70 5.1 Conclusions ............................................................................................................... 70 5.2 Future Research Directions ....................................................................................... 72 Bibliography .................................................................................................................................73 viii List of Tables Table 3.1 Axial feed rates and average forces of half plunge milling tests .................................. 22 Table 3.2 Identified cutting force coefficients for the bottom part of the tool and ....................... 23 Table 3.3 Specifications of the carbide end mill used for the experiments .................................. 28 Table 3.4 Cutting conditions used in orbital drilling tests. ........................................................... 28 Table 3.5 Prediction errors on average axial forces. ..................................................................... 32 Table 4.1 Modal parameters at the tool tip ................................................................................... 60 Table 4.2 Pitch length of tool path (๐๐) and spindle speed (๐) values for chatter tests .............. 61 Table 4.3 Modal parameters for semi-discrete time domain solution .......................................... 67 ix List of Figures Figure 2.1 Robotic orbital drilling unit. Taken from [10]. .............................................................. 5 Figure 2.2 a) Orbital drilling process b) Undeformed chip geometries during orbital drilling c) topography of the hole ground and the cross section of undeformed chip of the radial cut as a function of immersion angle. Taken from [11] ............................................................................... 6 Figure 3.1 Illustration of orbital drilling ......................................................................................... 9 Figure 3.2 Engagement zones in orbital drilling ........................................................................... 10 Figure 3.3 Comparison of radial engagements, left slot milling case, right orbital drilling case . 12 Figure 3.4 Graphic for the calculation radial engagement condition............................................ 13 Figure 3.5 Discretization of tool periphery along the axial direction of the tool .......................... 14 Figure 3.6 Tool motion on vertical (right) and horizontal (left) planes ........................................ 15 Figure 3.7 a) Virtual orbital drilling on ,MACHPRO , b) Radial engagement mapped on the end mill, c) Workpiece after end mill leaves the workpiece in zone2 ................................................. 16 Figure 3.8 Comparison of the radial engagement boundaries. Cutting conditions: Hole Diameter ๐ท๐ป = 30 mm, tool Diameter ๐ท๐ = 16 mm, pitch length of the tool path ๐๐ = 3 mm, spindle speed ๐ = 1000 rpm, orbital speed ๐๐ = 6 rev/min, ๐ ๐ก๐๐= 0.1319 mm/rev/tooth, ๐ ๐๐ฅ = 0.009 mm/rev/tooth. ................................................................................................................................ 17 Figure 3.9 Plunge milling geometry ............................................................................................. 19 Figure 3.10 Half plunge milling workpiece (bottom), CAD model of the workpiece (top) ......... 22 Figure 3.11 Average forces on each axis for half plunge milling ................................................. 23 Figure 3.12 Tangential and axial feed directions .......................................................................... 24 x Figure 3.13 Radial, tangential and axial cutting forces on the radial part of the tool at one discrete element .......................................................................................................................................... 26 Figure 3.14 Experimental setup for mechanics test ...................................................................... 27 Figure 3.15 Simulated maximum radial engagement boundaries for experiments ...................... 29 Figure 3.16 Measured cutting forces in x, y and z directions (Exp#1) ......................................... 30 Figure 3.17 Predicted cutting forces in x, y and z directions (Exp#1) .......................................... 30 Figure 3.18 Comparison of measured and predicted cutting forces (Exp#1) ............................... 31 Figure 3.19 Comparison of measured and predicted cutting forces (Exp#2) ............................... 31 Figure 3.20 Comparison of measured and predicted cutting forces (Exp#3) ............................... 32 Figure 4.1 Toolpath of orbital drilling process and simultaneous feed directions........................ 38 Figure 4.2 Vibration independent toolpath of orbital drilling process ......................................... 39 Figure 4.3 Variable time delay versus angular position of cutting tooth ...................................... 42 Figure 4.4 Dynamic radial chip thickness generation in orbital drilling ...................................... 43 Figure 4.5 Nyquist plot of the characteristic equation for stable and unstable process ................ 49 Figure 4.6 Predicted and discretized radial engagement profile for the case where Tool Dia.=16mm, Hole Dia.=30 mm, Pitch Length=5 mm, Discretization Angle: 6 deg. ................... 50 Figure 4.7 Discretized tool geometry(left) and a sample radial engagement(right) ..................... 51 Figure 4.8 Semi-discretization of tooth period ............................................................................. 53 Figure 4.9 Semidiscrete time domain stability lobes for orbital drilling process. Tool Dia.=16 [mm], Hole Dia.= 30 [mm], Helix Angle = 25 [deg], ๐พ๐ก๐, ๐ = 900 [MPa], ๐พ๐๐, ๐ = 270 [MPa], modal parameters taken from [12]. ๐ค๐๐ฅ1= 624 [Hz] , ๐๐ฅ1 = 0.052 , ๐๐ฅ1 = 8.2 x 107 [N/m], ๐ค๐๐ฅ2= 871 [Hz] , ๐๐ฅ2 = 0.033 , ๐๐ฅ2 = 16.1 x 107 [N/m], ๐ค๐๐ฅ3= 2311 [Hz] , ๐๐ฅ3 = 0.022, ๐๐ฅ3 = 8.7 x 107 [N/m], ๐ค๐๐ฅ4= 3052 [Hz] , ๐๐ฅ4 = 0.029 , ๐๐ฅ4 = 14.5 x 107 [N/m], ๐ค๐๐ฆ1= 692 [Hz] xi , ๐๐ฆ1 = 0.042, ๐๐ฆ1 = 7.6 x 107 [N/m], ๐ค๐๐ฆ2= 862 [Hz] , ๐๐ฆ2 = 0.052 , ๐๐ฆ2 = 13.5 x 107 [N/m], ๐ค๐๐ฆ3= 2289 [Hz], ๐๐ฆ3 = 0.019, ๐๐ฆ3 = 7.9 x 107 [N/m], ๐ค๐๐ฆ4= 3050 [Hz], ๐๐ฆ4 = 0.027 , ๐๐ฆ4 = 15.0 x 107 [N/m] ............................................................................................................. 57 Figure 4.10 Frequency response on local and global coordinates ................................................ 58 Figure 4.11 Experimental setup for chatter tests .......................................................................... 59 Figure 4.12 Two orthogonal direction FRFโs at the tool tip ......................................................... 60 Figure 4.13 FRF on global and local X coordinates of end mill when ๐ = 45 deg. ..................... 62 Figure 4.14 FRF on global and local Y coordinates of end mill when ๐ = 45 deg. ..................... 62 Figure 4.15 Stable orbital drilling: Measured sound signals and their FFT for ๐๐ = 3mm, ๐ = 8000 RPM, ๐ ๐ก๐๐ = 0.0275 mm/tooth, ๐ ๐๐ฅ = 0.002 mm/tooth, Tooth Passing Frequency=266.7 Hz .................................................................................................................................................. 63 Figure 4.16 Unstable orbital drilling: Measured sound signals and their FFT for ๐๐ = 6mm, ๐ = 9200 RPM, ๐ ๐ก๐๐ = 0.0275 mm/tooth, ๐ ๐๐ฅ = 0.0037 mm/tooth, Tooth Passing Frequency=306.7 Hz .................................................................................................................................................. 64 Figure 4.17 3D stability lobe for orbital drilling and chatter simulated chatter frequencies/Nyquist solution ........................................................................................................ 65 Figure 4.18 Comparison of Nyquist solution and experimentally measured chatter stability when end mill is at ๐ = 0 deg ................................................................................................................. 66 Figure 4.19 Stability limits during the rotation of the tool around the hole center for ๐ = 9200 RPM/ Nyquist solution ................................................................................................................. 66 Figure 4.20 3D stability lobe for orbital drilling /Semi-discrete time domain solution ............... 67 Figure 4.21 Comparison of predicted and experimentally measured chatter stability when end mill is at ๐ = 0 deg ........................................................................................................................ 68 xii Figure 4.22 Stability limits during the rotation of the tool around the hole center for ๐ = 9200 RPM .............................................................................................................................................. 69 xiii Nomenclature ๐๐ Pitch length of the tool path dz Differential axial depth of cut ๏จ ๏ฉD t Time varying directional matrix ๐ท๐ป Intended hole diameter PD Tool Path diameter ๐ท๐ Cutting tool diameter e Eccentricity axf Axial linear feed tanf Tangential linear feed ,btF , ,brF , ,baF Tangential, radial, axial forces on the bottom of the cutting tool ,bxF , y,bF , z,bF Forces on x, y and z directions on the bottom of the cutting tool ,rtF , ,rrF , ,raF Tangential, radial, axial forces on the periphery of the cutting tool ,rxF , y,rF , z,rF Forces on x, y and z directions on the periphery of the cutting tool x,bF , y,bF , z,bF Average forces on x, y and z directions on the bottom of the cutting tool J Tooth index ,b ,b ,b, ,tc rc acK K K Cutting force coefficients on the bottom of the cutting tool in tangential, radial, axial directions xiv ,b ,b ,b, ,te re aeK K K Edge coefficients on the bottom of the cutting tool in tangential, radial, axial directions ,r ,r ,r, ,tc rc acK K K Cutting force coefficients on the periphery of the cutting tool in tangential, radial, axial directions ,r ,r ,r, ,te re aeK K K Edge coefficients on the periphery of the cutting tool in tangential, radial, axial directions ๐ Spindle Speed [RPM] ๐๐ Orbital Speed [RPM] N Number of cutting tooth ๐๐ Number of natural modes q Differential physical displacement vector in time domain r Radial engagement length ๐ ๐ Cutting tool radius ๐ ๐ป Hole radius axs Axial feed rate tans Tangential feed rate PeripheryX , PeripheryY Periphery of the tool on xy plane ToolCenterX , ToolCenterY The position of the tool center on xy plane cw Instantaneous width of chip ๏ก Ramp angle of the tool path ๐ฝ Tool inspection arc radius xv h๏ข Helix angle of the cutter t๏ Time interval z๏ Vertical displacement of tool at each time interval ๐(๐ก) Rotation of the tool center around hole center [rad] ๏ธ Damping ratio ๏ด Tooth passing period 0๏ด Time varying delay xx๏ , yy๏ Direct FRFs on x and y direction xy๏ , yx๏ Cross FRFs โ ๐ ๐ก,๐, โ ๐๐ฅ,๐ Radial engagement boundaries pitch๏ฆ Pitch angle of the cutting tool , ( )j r k๏ฆ Instantaneous immersion angle ๏ท Spindle speed [rad/sec] c๏ท Chatter frequency n๏ท Natural frequency ๏ Orbital speed [rad/sec] CAD Computer aided design CFRP Carbon fiber reinforced polymers (CFRP) CFRP/Ti Carbon fiber reinforced polymers /Titanium CNC Computer numerical control FRF Frequency response function xvi STL Stereolithography TWE Tool-Workpiece Engagement 3D Three dimensional xvii Acknowledgements I would like to start with expressing my deepest gratitude to my research supervisor Dr. Yusuf Altintas for his guidance, patience and also belief in me during my studies. I would not have been able to finish this work without his research experience and his continuous support. It was a great privilege of mine working as an assistant under his supervision. I would like to accentuate my sincere heart-felt thanks to Dr. Doruk Merdol for his guidance and mentorship since the last year of my undergraduate study. Throughout my time in MAL, I had the chance to work with many brilliant people who showed me nothing but kindness. Among them, I would like to thank Alptunรง, Coลkun and Deniz for their sincere friendship. During my time in Vancouver I met a lot of special people, among them I would like to thank Kardelen รalฤฑkฤฑran for making my life better and more enjoyable. Besides, I also would like to thank Can Mahmut รelebi, my brother by heart, for always sharing my dreams and for always being a solid-funny support, even when we are miles away, since the first day of our high school years. Last but not least, I have been very fortunate to have my beloved familyโs unconditional support through my whole life. I would not be able to thank enough my mother Serap Kulalar, my grandmother Emel Kulalar, my uncle Doฤan Kulalar and also my grandfather ลinasi Kulalar, who passed away many years ago. Now, as a young man I see their sacrifices on another level which makes me nothing but grateful. My so-called success in life is just a small reflection of the love, kindness and compassion given by my family. Including this work, all of my past and future successes are dedicated to them. xviii Dedication To my mother, who taught me how to dream1 Chapter 1: Introduction Aircraft parts are mainly assembled by riveting or bolting them through holes opened with drilling operations. A typical mid-size aircraft may have over a million holes, hence the process is highly time consuming and costly. While holes with small diameter are opened through twist or indexed drills, and holes with large diameter have been opened combination of drilling and boring on large machine tools. Orbital drilling is a new technology which allows opening of medium to large diameter holes on the aircraft body directly during the assembly. The portable, computer numerical controlled (CNC) drilling head is attached to aircraft body through suction, and the drilling head follows an orbital โ helical motion to open and enlarge the hole by an end milling process. The operation has been found successful in drilling aluminum, titanium and composite parts in aerospace industry. In orbital drilling, an end mill moves along a three dimensional helical tool path by cutting at its periphery and bottom to open the hole. The cutting forces on both sections may excite the structural dynamics of the machine tool head during the operation. If the process parameters are not selected correctly, the flexibility of the orbital drilling head, spindle and end mill may experience severe chatter and large deflections which may destroy the costly aircraft parts during assembly. This thesis presents modeling the mechanics and dynamics of the orbital drilling process to predict the cutting forces, dimensional surface errors and chatter stability. The mathematical simulation model is intended for optimal selection of tool geometry, orbital and tangential feeds, and spindle speeds without causing chatter and overloading of the machine and cutting. . The thesis is organized as follows: 2 Chapter 2 summarizes the related background and the review of the literature in orbital drilling. In Chapter 3, the mechanics of orbital drilling operations is presented. A numerical method to identify tool-workpiece engagements (TWE) is introduced, and compared against the previous approaches [11]. The cutting force coefficients of the bottom part of the end mill are identified mechanistically, and the coefficient for the peripheral side are predicted from the orthogonal cutting parameters. The mechanics of orbital drilling have been experimentally validated by comparing the predicted and measured cutting forces. Chapter 4 is dedicated to the chatter stability of the orbital drilling operation. The regenerative time delay has two components: One is caused by the spindle speed while the other is contributed by the orbital โ helical feed motion of the tool. The mathematical modeling of the orbital โ helical motion has a negligible contribution to regenerative delay which is dominated by the spindle speed. The stability is reduced to regular end milling process but with feed direction dependent structural dynamics. Chatter stability of the operation is solved in semi discrete time domain and frequency domains. Nyquist stability criterion is used for frequency domain approach considering the radial engagement boundaries of the end mill. Three dimensional stability lobes are predicted as a position of the tool along the orbital path, spindle speed and pitch lengths of the tool path. The stability models are verified experimentally. The thesis is concluded in Chapter 5, summarizing the contributions of the thesis and future research directions. 3 Chapter 2: Literature Review 2.1 Overview Hole machining is one of the most dominating in aerospace industry. While regular twist drilling operations have been widely studied and used, the orbital drilling is a new process and emerged during the last decade. Orbital drilling has shown a great capability of machining burr free and high quality holes in different types of aerospace materials. The mechanics and dynamics of twist drilling, where the drill rotates and moves only in axial direction, have been widely studied in the past. The prediction of drilling forces [20, 21], forced [22, 23] and self-excited chatter vibrations [24, 25, 26] have been widely published. However, the mechanics and dynamics of orbital drilling, where the tool follows an orbital and helical path to open and enlarge the hole, has not been studied widely to predict the cutting forces, chatter free cutting conditions and dimensional hole quality. The objective of this thesis is to predict the cutting forces and chatter stability diagrams to identify most productive and safe cutting conditions. This chapter presents the review of the orbital drilling literature. 2.2 Application and Experimental Investigation of Orbital Drilling Operations The major concern of the hole drilling processes is the dimensional quality of the hole while using tool geometry and cutting conditions which yield to highest productivity. Early studies on the orbital drilling process mostly focused on the applicability of the operation and its potential to replace conventional, twist drilling process. The first study by 4 Lindqvist et al. [1] from Navator Company of Sweden showed that the orbital drilling process is capable of eliminating delamination while machining holes on carbon sandwich structures for aerospace applications. They designed a novel portable, orbital drilling system attached to aircraft body to drill holes during the assembly, and demonstrated the applicability of the process with test trials. The academic investigations followed the introduction of orbital drilling concept. Kihlman et al. [2] compared the cutting forces in orbital drilling and conventional drilling process by conducting experiments. They showed that the orbital drilling leads to lower axial cutting forces than the conventional drilling process. Lindqvist and Kihlman [3] from Navator experimentally proved that the orbital drilling produces holes with higher surface quality and in comparison to conventional drilling. Furthermore, the work of Whinnem [4] on Boeing 787 demonstrated that the orbital drilling process can create holes without burr which leads to less scrap rate in producing costly aircraft parts. Iyer et al. [5] showed that H7 quality holes on AISI D2 hardened tool steel can be opened with orbital drilling with a surface finish of 0.3 ฮผm, experimentally. The orbital drilling allows the opening of holes with different diameter with the same tool diameter [6]. Furthermore, the same cutting tool can be used for finish machining of the hole, hence the number of post-processes (i.e. boring) for opening the borehole is reduced. The observed reduction in cutting forces led to the use of industrial robots or similar light weight but mobile structures for portable orbital drilling operations [7]. A portable orbital drilling unit on a robot is shown in Figure 2.1. 5 Figure 2.1 Robotic orbital drilling unit. Taken from [10]. Orbital drilling is highly used in machining of carbon fiber reinforced polymers (CFRP) and carbon fiber reinforced polymers/titanium (CFRP/Ti) alloy stacks in aerospace industry. Since the process results in lower drilling forces compared to conventional drilling, the process temperature is expected to be lower [9]. Orbital drilling also led to less low burr formation and reduced delamination in machining CFRP or CFRP/Ti relative to conventional drilling [6,8,9]. In a recent work [10], researchers investigated the effects of orbital drilling on the hole diameter variation and roundness of CFRP/Ti stacks. The outcome of their work showed that hole and tool diameter ratio has a greater impact on the roundness of the CFRP layer. All of the articles cited in this section are based on experimental trials without providing model based, analytical reasoning. 6 2.3 Mechanics and Dynamics of Orbital Drilling Operations The kinematics model of orbital drilling was first studied by Brinksmeier et al. [11], which led to the prediction of varying tool- workpiece engagements as a function of tool and hole diameters, the pitch length of helical - downward motion along the tool path. The model led to the prediction of chip geometry generated by cylindrical flat end mills as shown in Figure 2.2. Li and Liu [12] created a simulation model based on Z-mapping to simulate three-dimensional (3D) surface topography of the finished hole produced by orbital drilling process. Figure 2.2 a) Orbital drilling process b) Undeformed chip geometries during orbital drilling c) topography of the hole ground and the cross section of undeformed chip of the radial cut as a function of immersion angle. Taken from [11] Liu et al. [13] predicted orbital drilling forces as a function of spindle speed, helical feed, radial and axial cutting depth, and cutter geometry. Wang et al. [14] identified the cutting force coefficients mechanistically from a series of orbital drilling tests conducted on titanium. In a recent study, Meyer et al. [15] considered complex tool geometries in modeling orbital drilling 7 forces by considering the chip geometry and forces generated by the bottom and radial parts of the tool. The differences between the predicted and measured forces are attributed to the indentation of the bottom part of the end mill into the work material. The prediction of vibrations, hole surface quality and chatter free depth, speed and pitch length require the dynamic model of orbital drilling. The circular and thread milling operations have similarities with orbital drilling. They have two time delays which are functions of spindle speed and orbital motion. Kardes and Altintas presented a stability model of circular milling which resembles orbital milling but without axial motion [17]. They applied zero order solution of Altintas and Budak [27] and Finite Time model of Bayly et al. [28] in solving the stability. Thread milling has a path similar to orbital milling but with cutting edges similar to thread profile. Wan and Altintas [18] presented both mechanics and chatter stability models to predict the thread milling process. Dual time delay is considered in solving the stability of operations which are governed by coupled, time periodic, delayed differential equations. Li et al. [16] presented the only study to predict the chatter stability diagrams, i.e. critical depth of cut versus spindle speed, in orbital milling. They assumed the worst engagement conditions, i.e. slotting or peripheral milling, and applied Altintas and Budakโs zero order stability law to simplified orbital drilling process which does not consider varying engagement and dynamic flexibilities along the tool path. The TWEs are analytically calculated from the kinematics model proposed by Brinksmeier et al. [11]. The engagement conditions are used to describe the chip thickness distribution at the periphery and bottom edges of the helical end mill along the orbital tool path. The cutting forces are predicted as a function of chip thickness distribution. The effect of lateral 8 vibrations of the flexible tool on the regenerative, dynamic chip thickness is modeled as a function of tool and hole diameters, pitch length of the orbital motion, and feed. The frequency response functions (FRF) of tool structure are projected into the feed and normal directions along the orbital path. The chatter stability of the process is solved by applying the zero order stability law of Altintas and Budak [27] in frequency domain. The chatter stability is also solved in semi-discrete time domain proposed by Insperger and Stepan [19].The linear, semi discrete time model can also be used to predict the cutting forces and vibrations along the tool path [29]. In summary, this thesis presents integrated mechanics and dynamics models of orbital drilling process. The proposed models are experimentally proven. 9 Chapter 3: Mechanics of Orbital Drilling Operations 3.1 Introduction Orbital drilling is used to open large holes, especially common in aircraft fuselage manufacturing. The cutting tool follows a 3D helical toolpath with the combination of rotational and axial motions as shown in Figure 3.1. Figure 3.1 Illustration of orbital drilling This process can be considered as a combination of circular and plunge milling rather than drilling, since the cutting occurs both at the periphery and the bottom part of the tool. The 10 engagement conditions in orbital drilling can be separated into three different regions as shown in Figure 3.2. Figure 3.2 Engagement zones in orbital drilling 11 In this chapter an analytical engagement model from the literature is presented. A model to evaluate engagement conditions is introduced, and its results are compared against the values predicted by computationally costly, generic commercial software (Moduleworks). The kinematics and TWE models are used to evaluate the chip geometry generated during orbital drilling, which is needed for modeling the mechanics of the cutting process. The mechanics of orbital drilling process have been developed, and experimentally proven by comparing the predicted and experimentally measured cutting forces. 3.2 Geometric Modelling of Tool and Workpiece Intersection Engagement conditions during orbital drilling operation vary as the tool penetrates and leaves the work material, since both radial and axial edges of the cutting tool remove material from the workpiece. While the cutting tool moves along the 3D helical toolpath, axial part of the tool engages with the workpiece first followed by the radial edges (Figure 3.2.). The axial part of the tool has full contact with the workpiece material in regions 1 and 2. However, the radial engagement conditions change in region 1 where the cutting tool rotates 360 degrees around the center of the hole. The engagement area on the radial part of the tool increases as the tool plunges into the workpiece until the pitch length of the tool path is travelled. After tool finishes the first 360 degrees of rotation inside the workpiece, the engagement areas on both radial and axial cutting edges stay constant although the radial engagement boundaries, the entry โ ๐ ๐ก,๐ and exit โ ๐๐ฅ,๐ angles of the tool change along the longitudinal direction of the cutter. In region 3, the tool makes the last 360 degrees of rotation around the center of the hole on xy plane without a plunge motion which resembles circular milling. The engagement conditions vary significantly 12 in region 3. The radial engagement boundaries for slot milling and orbital drilling operations are shown in Figure 3.3 to highlight the differences. Figure 3.3 Comparison of radial engagements, left slot milling case, right orbital drilling case 3.2.1 Modelling of the Radial Engagement Brinksmeier et al. [11] presented the geometric engagement of cylindrical end mills zone 2 of orbital milling where radial engagement boundaries stay constant. The intended hole diameter ๐ท๐ป, cutting tool diameter ๐ท๐, and the pitch length of the tool path ๐๐ are used in the engagement model. 13 Figure 3.4 Graphic for the calculation radial engagement condition The eccentricity (๐) of the tool center from the hole center is, , ,2 2T HT HH TD DR Re R R๏ฝ ๏ฝ๏ฝ ๏ญ (3.1) The tool inspection arc angle ๐ฝ (Figure 3.4) is, ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ222 2cosarcsinsin cosT TT TR RR e R๏ช๏ข๏ช ๏ช๏ญ๏ฝ๏ซ ๏ซ (3.2) The cutting tool engagement boundaries change along the circumference of the tool (Figure 3.4). The depth of the radial chip along the tool circumference is given as: 1 1i ph a๏ข๏ฐ๏ฆ ๏ถ๏ฝ ๏ญ๏ง ๏ท๏จ ๏ธ (3.3) The radial engagement area along the circumference of the cylindrical end mill changes as the tool rotates around the center of the hole in zone 1 and 3, which is modeled here. 14 Figure 3.5 Discretization of tool periphery along the axial direction of the tool . Since the periphery of the end mill does not change along the axial direction of the cylindrical tool, its orbital motion can be projected on xy plane. If the rigid body motion of the tool is discretized at discrete time intervals ฮ๐ก, the position of the tool center can be found on xy plane as: ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉcossinToolCenterToolCenterX t e tY t e tt t๏ฑ๏ฑ๏ฑ๏ฝ ๏๏ฝ ๏๏ฝ ๏ ๏๏ (3.4) where ฮฉ is the orbital speed [rad/sec] and ๐(๐ก) is the rotation of the tool center around hole center in [rad]. Vertical displacement of the tool at each time interval is ฮ๐ง, axz t f๏ ๏ฝ ๏ ๏ (3.5) where ๐๐๐ฅ is the axial feed. Toolโs periphery at each time step can be evaluated as follows. ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉcossin, 0 2Periphery ToolCenter TPeriphery ToolCenter TX t X t RY t Y t Rwhere๏ช๏ช๏ช ๏ฐ๏ฝ ๏ซ ๏๏ฝ ๏ซ ๏๏ฃ ๏ฃ (3.6) 15 By considering the toolโs previous positions on both vertical and horizontal planes, as well as on its periphery, the radial engagement at each time step can be predicted with a line intersection algorithm. The intersection points represent the starting points of the radial engagement โ ๐ ๐ก,๐, along the toolโs axial direction. The engagement area along the tools feed direction is shown in Figure 3.6. Figure 3.6 Tool motion on vertical (right) and horizontal (left) planes 3.2.2 Comparison of the Engagement Models The proposed radial engagement of the tool with the workpiece during the orbital drilling is compared against commercial graphics system Moduleworks. In Moduleworks which is a module within UBCโs virtual machining system (MACHPRO), the TWEs are calculated at discrete points of the tool path. The engagement boundaries, the stereolithography (STL) file of the raw material geometry and NC tool path are used as inputs in calculating the engagements. Engagement information is extracted at user defined, constant 16 sampling distances along the tool path. A sample orbital drilling process (a), the radial engagement area of an end mill at a point where the tool is in zone 2 (b), and the corresponding hole shape (c) are shown in Figure 3.7. Figure 3.7 a) Virtual orbital drilling on ,MACHPRO , b) Radial engagement mapped on the end mill, c) Workpiece after end mill leaves the workpiece in zone2 The radial engagement on the end mill is projected to two dimensional plane where horizontal and vertical axes represent the immersion angle and the axial depth of chip, respectively. A comparison of engagement maps generated by Moduleworks in MACHPRO and proposed analytical models are shown in Figure 3.8 where the results are in good agreement. 17 Figure 3.8 Comparison of the radial engagement boundaries. Cutting conditions: Hole Diameter ๐ซ๐ฏ = 30 mm, tool Diameter ๐ซ๐ป = 16 mm, pitch length of the tool path ๐๐ = 3 mm, spindle speed ๐ = 1000 rpm, orbital speed ๐๐ = 6 rev/min, ๐๐๐๐= 0.1319 mm/rev/tooth, ๐๐๐ = 0.009 mm/rev/tooth. Moduleworks is based on the STL model of the part and tool, hence the resolution of STL mesh greatly affect the accuracy of evaluating engagement maps. Hence, Moduleworks cannot predict the engagements accurately, if the tool diameter is very small. The proposed engagement map solution is based on the analytical, geometric models of the cylindrical end mill and the hole diameter, hence it is independent of mesh size in STL models of the tool and the part. 0 20 40 60 80 100 120 140 160 18000.511.522.53Variation of Axial Depth of Chip- Hole Diameter 30mm, Tool Diameter 16mm, Pitch of Path 3mmAxial Depth of Chip [mm]Immersion Angle [Deg] Numerical SolutionAnalytical Solution[Brinksmeier et al.]MachPro Engagement Map18 3.3 Mechanics of Orbital Drilling Orbital drilling can be considered as a combination of plunge and circular milling operations. The cutting process at the bottom part of the end mill can be considered as plunge milling, whereas the process at the cylindrical surface resembles the circular milling process. The mechanics of the circular milling was investigated by Kardes and Altintas [17]. They considered the varying engagement conditions of the solid end mill and workpiece along the circular โ trochoidal tool path. Ko and Altintas [30] presented mechanics of plunge milling by utilizing the chip generated by the axially penetrating end mill as it rotates. In addition to the engagement geometry presented in the previous section, the mechanics model require the identification of cutting force coefficients, chip thickness and cutting force model which are presented below. 3.3.1 Identification of Cutting Force Coefficients at the Bottom Part of the End Mill The accuracy of the cutting force estimation significantly depends on the accuracy of cutting force coefficients. Unlike the geometry of cutting edges on the periphery of end mills, the manufacturers do not provide much information about the geometry of flutes at the bottom of the tool. Since it is also difficult to measure the bottom geometry on the standard tool microscopes, the cutting force coefficients need to be identified from the cutting experiments. During the orbital drilling process bottom part of the end mill plunges into the material like a drill, hence the 19 chip thickness distribution and the corresponding cutting force coefficients need to be identified from the geometry of cut shown in Figure 3.9. Figure 3.9 Plunge milling geometry The instantaneous width of chip ๐ค๐ is given by. ๏จ ๏ฉcos 90Tc TjR rw R๏ฆ๏ฌ ๏ผ๏ญ๏ฏ ๏ฏ๏ฝ ๏ญ ๏ญ ๏ฝ๏ญ๏ฏ ๏ฏ๏ฎ ๏พ (3.7) where ๐ is the radial engagement length. The tangential ( ,btF ), radial ( ,brF ) and axial ( ,baF ) forces caused by the plunge milling are expressed as a function of chip area ( ax ch w ) and cutting edge contact length ( cw ): ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ,b ,b ,br,b ,b ,ba,b ,b ,bt tc ax te crc ax re cac ax ae cF K h K wF K h K wF K h K w๏ฝ ๏ ๏ซ ๏๏ฝ ๏ ๏ซ ๏๏ฝ ๏ ๏ซ ๏ (3.8) 20 Where โ๐๐ฅ is chip thickness and equal to axial feed rate ax axh s๏ฝ , { ,b ,b ,b, ,tc rc acK K K } and {,b ,b ,b, ,te re aeK K K } are the cutting and edge force coefficients in tangential, radial and axial directions, respectively. The tools coordinate and global coordinate systems are coincided. The cutting forces on all teeth are transformed to x,y and z directions as: ๏ป ๏ฝ ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ,b ,b ,b ,b ,b1y,b ,b ,b ,b ,b1z,b ,b ,b1cos sinsin cosNx tc ax c te c j rc ax c re c jjNtc ax c te c j rc ax c re c jjNac ax ae cjF K h w K w K h w K wF K h w K w K h w K wF K h K w๏ฆ ๏ฆ๏ฆ ๏ฆ๏ฝ๏ฝ๏ฝ๏ฝ ๏ ๏ญ ๏ ๏ ๏ซ ๏ ๏ ๏ญ ๏ ๏ ๏ซ ๏ ๏๏ฝ ๏ ๏ ๏ซ ๏ ๏ ๏ญ ๏ ๏ ๏ซ ๏ ๏๏ฝ ๏ ๏ซ ๏๏ฅ๏ฅ๏ฅ (3.9) Instead of using orthogonal to oblique cutting transformation [31], the mechanistic prediction of cutting coefficients for the bottom edges of the tool is employed. The average cutting forces per tooth period are measured while the axial feed rate is varied during each experiment with constant immersion conditions. ๏จ ๏ฉ1exstavg jpitchF F d๏ฆ๏ฆ๏ฆ ๏ฆ๏ฆ๏ฝ ๏ฒ (3.10) Average forces on each axis can be evaluated as: ๏ป ๏ฝ ๏ป ๏ฝ๏ป ๏ฝ ๏ป ๏ฝ๏ป ๏ฝx,b 1 ,b ,b 2 ,b ,by,b 2 ,b ,b 1 ,b ,bz,b 3 ,b ,b2 22 22tc ax te rc ax retc ax te rc ax reac ax aeN NF K h K K h KN NF K h K K h KNF K h K๏ค ๏ค๏ฐ ๏ฐ๏ค ๏ค๏ฐ ๏ฐ๏ค๏ฐ๏ฝ ๏ญ ๏ ๏ซ ๏ญ ๏ ๏ซ๏ฝ ๏ ๏ซ ๏ญ ๏ ๏ซ๏ฝ ๏ ๏ซ (3.11) 21 Where ๐ฟ1, ๐ฟ2 and ๐ฟ3 are; ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ123coscos 90sincos 90cos 90exstexstexstTj TjTj TjTTjR rR dR rR dR rR d๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ค ๏ฆ ๏ฆ๏ฆ๏ค ๏ฆ ๏ฆ๏ฆ๏ค ๏ฆ๏ฆ๏ฆ ๏ถ๏ฆ ๏ถ๏ญ๏ง ๏ท๏ง ๏ท๏ฝ ๏ ๏ญ๏ง ๏ท๏ง ๏ท๏ญ๏จ ๏ธ๏จ ๏ธ๏ฆ ๏ถ๏ฆ ๏ถ๏ญ๏ง ๏ท๏ง ๏ท๏ฝ ๏ ๏ญ๏ง ๏ท๏ง ๏ท๏ญ๏จ ๏ธ๏จ ๏ธ๏ฆ ๏ถ๏ฆ ๏ถ๏ญ๏ง ๏ท๏ง ๏ท๏ฝ ๏ญ๏ง ๏ท๏ง ๏ท๏ญ๏จ ๏ธ๏จ ๏ธ๏ฒ๏ฒ๏ฒ (3.12) In order to obtain cutting coefficients associated with one tooth, intermittent plunge milling tests have been modeled with ๐ = ๐ ๐ , โ ๐๐ฅ,๐ = 180ยฐ and โ ๐ ๐ก,๐ = 0ยฐ. The corresponding average forces can be expressed as: ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝx,b ,b ,by,b ,b ,bz,b ,b ,b2Trc ax reTtc ax teTac ax aeNRF K h KNRF K h KNRF K h K๏ฐ๏ฐ๏ฝ ๏ญ ๏ ๏ซ๏ฝ ๏ ๏ซ๏ฝ ๏ ๏ซ (3.13) The cutting coefficients associated with the bottom part of the tool can be calculated by following: xc,b xe,b,b ,byc,b ye,b,b ,bzc,b ze,b,b ,b,,2 2,rc reT Ttc teT Tac aeT TF FK KNR NRF FK KNR NRF FK KNR NR๏ฐ ๏ฐ๏ฐ ๏ฐ๏ฝ ๏ญ ๏ฝ ๏ญ๏ฝ ๏ฝ ๏ฝ๏ฝ ๏ฝ (3.14) 22 Half plunge milling tests were carried at 5 different axial feed rates at 1000 rpm and the experiment parameters are given on the Table 3.1. Table 3.1 Axial feed rates and average forces of half plunge milling tests Exp. 1 Exp. 2 Exp. 3 Exp. 4 Exp. 5 Axial Feed Rate [mm/rev/tooth] 0.005 0.020 0.040 0.060 0.080 Average Forces on X axis [N] -1.16 -12.92 -24.82 -34.05 -41.55 Average Forces on Y axis [N] 67.35 144.6 245.6 349.4 442.2 Average Forces on Z axis [N] 105.4 136.8 196.4 248.1 297.6 Experiments were implemented on Aluminum 7050-T7451 and the workpiece and its computer aided design (CAD) model are shown in Figure 3.10. Figure 3.10 Half plunge milling workpiece (bottom), CAD model of the workpiece (top) 23 Figure 3.11 Average forces on each axis for half plunge milling The average forces are fitted to a line (Equation. 3.14) with a linear least squares, and the estimated cutting force coefficients are given in Table 3.2 Table 3.2 Identified cutting force coefficients for the bottom part of the tool and ๐ฒ๐๐,๐ [MPa] ๐ฒ๐๐,๐[N/mm] ๐ฒ๐๐,๐[MPa] ๐ฒ๐๐,๐[N/mm] ๐ฒ๐๐,๐[MPa] ๐ฒ๐๐,๐[N/mm] 104.63 0.20 986.04 8.62 326.44 11.22 24 3.4 Prediction of Cutting Forces The spindle speed ๐ and orbital travel speed ๐๐ are given in [rev/min] with corresponding angular speeds in [rad/s]: 22,60 60onn ๏ฐ๏ฐ๏ท ๏ฝ ๏ ๏ฝ (3.15) Figure 3.12 Tangential and axial feed directions Tangential linear feed (๐๐ก๐๐) and feed rate (๐ ๐ก๐๐) are evaluated in feed direction (๐ฅ๐) from Figure 3.12. tantan tan260H TD Dfs fNn๏ญ๏ฆ ๏ถ๏ฝ ๏๏ง ๏ท๏จ ๏ธ๏ฝ (3.16) The axial feed ๐๐๐ฅ and feed rate ๐ ๐๐ฅ along the axial (Z) direction are evaluated by considering the ramp angle of the tool path ๐ผ (Figure 3.12): 1tan P P H TpDD D Da๏ฐ๏ก ๏ญ๏ฆ ๏ถ๏ฝ ๏ฌ ๏ฝ ๏ญ๏ง ๏ท๏ง ๏ท๏จ ๏ธ (3.17) 25 Where ๐ท๐ is the diameter of the tool path. The axial linear feed (๐๐๐ฅ) and the axial feed rate per tooth ๐ ๐๐ฅ are given by: tantan tantan tan260tan tanH TaxaxD Df fs s fNn๏ก ๏ก๏ก ๏ก๏ญ๏ฆ ๏ถ๏ฝ ๏ฝ ๏๏ง ๏ท๏จ ๏ธ๏ฝ ๏ฝ (3.18) The instantaneous immersion angle of a tooth j at elevation z can be evaluated : , 0,tan( ) (j 1) hj r j r pitchTk zR๏ข๏ฆ ๏ฆ ๏ฆ๏ฝ ๏ซ ๏ญ ๏ซ (3.19) where โ ๐0,๐ is the immersion angle of the associated cutting flute at the bottom and ๐ฝโ is the helix angle of the cylindrical end mill with N number of flutes. The chip thickness within the engagement zone is defined by: ๏จ ๏ฉ๏จ ๏ฉ, tan ,sin ( )j j r j rh g k s k๏ฆ ๏ฆ๏ฝ ๏ (4.20) Where the following unit pulse function indicates whether the flute is in or out of material: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ, , ,,10st r j r ex rj rif z kg kelse๏ฆ ๏ฆ ๏ฆ๏ฆ๏ฌ ๏ฃ ๏ฃ ๏ผ๏ฏ ๏ฏ๏ฝ ๏ญ ๏ฝ๏ฏ ๏ฏ๏ฎ ๏พ (3.21) The cutting forces contributed by a cutting edge segment with a differential height (dz) are evaluated from Figure 3.13 as: ,r , ,,r , ,,r , ,tj tc r j te rrj rc r j re raj ac r j ae rdF K h dz K dzdF K h dz K dzdF K h dz K dz๏ฝ ๏ซ๏ฝ ๏ซ๏ฝ ๏ซ (3.22) 26 Figure 3.13 Radial, tangential and axial cutting forces on the radial part of the tool at one discrete element The total forces generated by the peripheral edges of the end mill can be calculated by summing the projected differential forces in the cartesian axes as: , , tj, rj, tj, rj,1y, , tj, rj, tj, rj,1z, , aj,1( , ) ( ) ( cos sin )sin ( sin cos )cos( , ) ( ) ( cos sin )cos ( sin cos )sin( , ) ( )Nx r j r r j r j r j r jjNr j r r j r j r j r jjNr j r rjF g F F F FF g F F F FF g F๏ฑ ๏ฆ ๏ฆ ๏ฆ ๏ฆ ๏ฑ ๏ฆ ๏ฆ ๏ฑ๏ฑ ๏ฆ ๏ฆ ๏ฆ ๏ฆ ๏ฑ ๏ฆ ๏ฆ ๏ฑ๏ฑ ๏ฆ ๏ฆ๏ฝ๏ฝ๏ฝ๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ซ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ญ ๏ซ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ซ ๏ป๏ฅ๏ฅ๏ฅ (3.23) The cutting forces contributed by the bottom edges of the end mill are evaluated when the tool plunges in the workpiece as: ,b tj,b rj,b tj,b rj,b1y,b tj,b rj,b tj,b rj,b1z,b aj,b1( , ) ( cos sin )sin ( sin cos )cos( , ) ( cos sin )cos ( sin cos )sin( , )Nx j j j jjNj j j jjNjF F F F FF F F F FF F๏ฑ ๏ฆ ๏ฆ ๏ฆ ๏ฑ ๏ฆ ๏ฆ ๏ฑ๏ฑ ๏ฆ ๏ฆ ๏ฆ ๏ฑ ๏ฆ ๏ฆ ๏ฑ๏ฑ ๏ฆ๏ฝ๏ฝ๏ฝ๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ซ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ญ ๏ซ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ซ ๏ป๏ฅ๏ฅ๏ฅ(3.24) 27 The total forces contributed by the peripheral and bottom edges of the tool are: x, , ,b, ,r ,b, , ,bTotal x r xy Total y yz Total z r zF F FF F FF F F๏ฝ ๏ซ๏ฝ ๏ซ๏ฝ ๏ซ (3.25) 3.5 Simulations and Experimental Results Series of orbital drilling tests have been carried out on Quaser UX600 5 axis machining center instrumented with a Kistler 9255B table top dynamometer as shown in Figure 3.14. Figure 3.14 Experimental setup for mechanics test The geometric properties of the helical end mill used in orbital drilling tests are shown in Table 3.3 and the cutting conditions are given in Table 3.4. 28 Table 3.3 Specifications of the carbide end mill used for the experiments Tool Type Cutting Diameter Radial Rake Angle Axial Rake Angle Flute Helix Angle Max. Ramping Angle Orbital Drilling Capability 2- Fluted Solid Flat-End mill 16 mm 13.5ยฐ 13.5ยฐ 25ยฐ 15ยฐ Yes 3 experiments with different process parameters are experimentally verified and the parameters of each experiment are listed as follows. Table 3.4 Cutting conditions used in orbital drilling tests. Exp # Hole Diameter [mm] Spindle Speed [RPM] Orbital Speed [RPM] Pitch Length of the Tool Path [mm] Tangential Feed Rate [mm/rev/tooth] Axial Feed Rate [mm/rev/tooth] 1 25 1000 10 3 0.1414 0.015 2 25 1000 10 6 0.1414 0.030 3 30 1000 5 4.5 0.11 0.0113 The workpiece material was selected Aluminum 7050-T7451 with the cutting force coefficients ๐พ๐ก๐,๐ = 884.65 [MPa], ๐พ๐ก๐,๐ = 15.33 [N/mm], ๐พ๐๐,๐ = 214.64 [MPa], ๐พ๐๐,๐ = 35.36 [N/mm], ๐พ๐๐,๐ = 185.14 [MPa], ๐พ๐๐,๐ = 0 [N/mm]. Cutting coefficients related with the bottom part of the tool was given on the Table 3.2. When the cutting coefficients on both sides of the end mill are compared, the difference for both cutting and edge coefficients of radial, tangential and axial directions is obvious. The reason behind this difference is the cutting geometry on the periphery and on the axial part of the tool which are different. In addition to that, for 29 mechanistically calibrating the cutting force coefficients on the bottom part of the tool, cutting force coefficients are assumed to be constant throughout the cutter radius. However since the cutting speed is changing throughout the radius of the end mill, the cutting force coefficient values shouldnโt be different considering the orthogonal to oblique transformation approach. In order to present a more realistic solution the bottom geometry should be analyzed with an advanced tool microscope. The simulated maximum radial engagement conditions for each experimental case are shown in Figure 3.15. Figure 3.15 Simulated maximum radial engagement boundaries for experiments The predicted and experimentally measured cutting forces of experiment 1 in zone 2 of orbital drilling operation are given for one travel period of tool around the hole center in Figure 3.16 and Figure 3.17 respectively. Simulated and experimental results consider the cutting forces when tool starts cutting in zone 2 and rotates around the hole center 360 degrees which is equal to the pitch length of the path in vertical direction. A detailed comparison of the predicted and measured forces is given in Figure 3.18. 30 Figure 3.16 Measured cutting forces in x, y and z directions (Exp#1) Figure 3.17 Predicted cutting forces in x, y and z directions (Exp#1) 31 Figure 3.18 Comparison of measured and predicted cutting forces (Exp#1) The predicted and measured forces for the experiments 2 and 3 are shown in Figure 3.19 and Figure 3.20, respectively. Figure 3.19 Comparison of measured and predicted cutting forces (Exp#2) 32 Figure 3.20 Comparison of measured and predicted cutting forces (Exp#3) While the prediction accuracy is quite satisfactory in radial (X, Y) directions, there are slight deviations in the axial (Z) direction due to uncertainties in modeling the plunge cutting process. The prediction errors are shown on average axial forces in Table 3.5. Table 3.5 Prediction errors on average axial forces. Average Predicted Force [N] Average Measured Force [N] Average Error (%) Experiment #1 229 211 8.5 Experiment #2 281 273 2.9 Experiment #3 196 204 3.9 33 Chapter 4: Chatter Stability of Orbital Drilling Operations 4.1 Introduction Chatter is a self-excited vibration that is caused by the phase shift between two successive waves left on the workpiece material during the metal cutting operations. Chatter must be avoided to obtain good surface finish and high material removal rates while avoiding tool breakage and high dynamic loads on machine tool spindle. The acceptable quality of machined surface is determined by the static and dynamic deflections of the flexible tool which are imprinted on the hole surface. The mechanics of orbital drilling which leads to prediction of chip thickness and forces are presented in the previous chapter. This chapter extends the model to include the effects of vibrations on the chip thickness hence on the dynamic forces. The dynamic model allows the identification of chatter stability diagrams which lead to stable speeds and cutting engagement conditions which yield to good surface finish and high material removal rate. The model also allows the prediction of static deflection and forced vibration marks left on the hole surface. Henceforth, the chapter is organized as follows. First, the dynamic model of orbital drilling is modeled and the effect of orbital motion on time delay investigated. The chatter stability of the orbital drilling is solved both in frequency and semi-discrete time domain. The varying engagement conditions and position dependent structural dynamics of the tool are considered in the stability models. The models are experimentally verified. 34 4.2 Dynamic Cutting Forces in Orbital Drilling Dynamic cutting forces excite the machine tool structure in both radial (x,y) and axial (z) directions in global coordinate system. The dynamic displacements in the tangential and axial chip thicknesses are derived in terms of tools instantaneous angular position around hole (๐), instantaneous immersion angle at both radial and bottom part of the end mill โ ๐, and vibrations in global coordinates โ๐ฅ, โ๐ฆ, โ๐ง as follows: ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉtan , , ,, tan , , ,, sin cos sin,sin sin cossin coscos sinr j j r j r j rax j axr j r j r c j r c j rccch s x yh s zwhereh s x t y tx t x t y ty t x t y tz t z t๏ฆ ๏ฑ ๏ฆ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ๏ฆ ๏ฆ ๏ฆ ๏ฆ๏ฑ ๏ฑ๏ฑ ๏ฑ๏ฝ ๏ญ๏ ๏ญ ๏ซ ๏ ๏ญ๏ฝ ๏ซ ๏๏ฝ ๏ซ ๏ ๏ซ ๏๏ ๏ฝ ๏ญ๏ ๏ซ ๏๏ ๏ฝ ๏ญ๏ ๏ญ๏๏ ๏ฝ ๏ (4.1) ๏จ ๏ฉtan ,sin j rs ๏ฆ is the static and ๏จ ๏ฉ ๏จ ๏ฉ, ,cos sinj r j rx y๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ญ๏ ๏ญ ๏ซ๏ ๏ญ represents the dynamic part of the chip thickness in global coordinates. ๏จ ๏ฉ ( ) ( )x t x t x t ๏ด๏ ๏ฝ ๏ญ ๏ญ , ๏จ ๏ฉ ( ) ( )y t y t y t ๏ด๏ ๏ฝ ๏ญ ๏ญ and๏จ ๏ฉ ( ) ( )z t z t z t ๏ด๏ ๏ฝ ๏ญ ๏ญ denotes the difference between dynamic displacements at present and previous cuts in global coordinates. The static chip thickness does not affect the chatter stability when the cutting force coefficient is constant, hence it can be dropped from the stability formulation [32]. The tangential and axial chip thicknesses can be rewritten as follows: ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ, ,, cos sin,r j j r j rax jh x yh z๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ๏ฝ ๏ญ๏ ๏ญ ๏ซ๏ ๏ญ๏ฝ ๏ (4.2) 35 The cutting forces are evaluated in global coordinates ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ, , ,y, , ,, ,, , sin , cos, , cos , sin, ,j jj jjx r j tc r r j j rc r r j jr j tc r r j j rc r r j jz r j ac r r ja aa aaF K h K hF K h K hF K h๏ฆ ๏ฆ๏ฆ ๏ฆ๏ฆ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฝ ๏ญ ๏ญ ๏ญ๏ฝ ๏ญ ๏ซ ๏ญ๏ฝ (4.3) Where ๐พ๐ก๐,๐, ๐พ๐๐,๐, ๐พ๐๐,๐, are cutting force coefficients, โ ๐ instantaneous immersion angle of tooth j, ๐(โ ๐) is the instantaneous depth of cut, and โ๐(โ ๐, ๐) is radial chip thickness. Dynamic cutting forces at the bottom part of the end mill acting on tooth j is expressed ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ,b ,b ,by,b ,b ,b,b ,b, , sin , cos, , cos , sin, ,x j tc T ax j j rc T ax j jj tc T ax j j rc T ax j jz j ac T ax jF K R h K R hF K R h K R hF K R h๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฆ ๏ฑ ๏ฆ ๏ฑ๏ฝ ๏ญ ๏ญ ๏ญ๏ฝ ๏ญ ๏ซ ๏ญ๏ฝ (4.4) Where ๐พ๐ก๐,๐, ๐พ๐๐,๐, ๐พ๐๐,๐ are the cutting coefficients for the bottom part of the cuing edges, โ๐๐ฅ(โ ๐, ๐) is the axial chip thickness. 36 By considering the contributions of both radial and bottom part of the end mill, dynamic forces acting on tooth j can be as: ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ, , , , ,b ,b, , , , ,b ,b, ,yjzj122sin 2 1 cos 2 1 cos 2 sin 2 sin cos21 cos 2 sin 2 sin 2 1 cos 2 cos sin22 cos 2 sinjTtc r rc r tc r rc r tc rcjTtc r rc r tc r rc r tc rcjTac r ac rjxjaRK A K A K A K A K A K AaRK A K A K A K A K A K AaRK A K A KaFFF๏ฆ๏ฆ๏ฆ๏ฆ๏ฝ๏ซ ๏ซ ๏ญ ๏ญ ๏ญ๏ญ ๏ญ ๏ญ ๏ซ ๏ซ ๏ซ๏ญ๏ฌ ๏ผ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ญ ๏ฝ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ฎ ๏พ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ,bacx ty tz t๏ฉ ๏น๏ช ๏บ๏ช ๏บ๏๏ฌ ๏ผ๏ช ๏บ๏ฏ ๏ฏ๏ช ๏บ ๏๏ญ ๏ฝ๏ช ๏บ ๏ฏ ๏ฏ๏๏ช ๏บ ๏ฎ ๏พ๏ช ๏บ๏ช ๏บ๏ซ ๏ป (4.5) Where jA ๏ฆ ๏ฑ๏ฝ ๏ญ (4.6) Total dynamic cutting forces ๐น๐ฅ, ๐น๐ฆ and ๐น๐ง can be evaluated by summing the forces contributed by all the teeth which are engaged with the workpiece: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ,1,1,1jjjNxjxjNyjyjNzjzjF FF FF F๏ฆ ๏ฑ๏ฆ ๏ฑ๏ฆ ๏ฑ๏ฝ๏ฝ๏ฝ๏ฝ๏ฝ๏ฝ๏ฅ๏ฅ๏ฅ (4.7) 37 4.3 Chatter Stability in Orbital Drilling The chatter stability of orbital drilling process is solved in frequency domain and semi-discrete time domain with the following assumptions: ๏ท The radial chip load is always greater than the axial chip load due to the helical-orbital motion of the tool. Hence, it is assumed that the main source of the chatter is by the lateral deflections of the end mill and axial deflections are neglected in the chatter stability solution. ๏ท Since the core of the end mills are stronger than twist drills the torsional modes of the end mill is neglected. ๏ท Gyroscopic effects that might come from the orbital motion of the cutter are negligible due to slow orbital motion. Since the feed direction changes during the orbital motion, FRF of the machine on global coordinates is oriented onto the local coordinates of the end mill in lateral directions. As a result, the stability model is 3D and dependent on the position of the tool, spindle speed and axial pitch length of the tool path. Different than the approach presented by Li et al. [16], in frequency domain solution the exact engagement conditions are taken into account for solving the chatter stability of the operation. Besides, the presented semi-discrete time domain solution is the first attempt to study the dynamics of orbital drilling in time domain. 38 4.3.1 Orbital Motion Effect on Time Delay The stability of machining processes with by two different delay terms contributed by the spindle speed and feed motions has been studied before [17, 18]. Here, the effect of toolโs orbital helical motion on the regenerative delay is studied using an analysis model proposed by Long et al [33]. The spindle speed ๐ and orbital speed ๐๐ are expressed in [rad/sec] as: ๏ ๏ ๏ ๏22,60 60/ / orad sec rad sencn ๏ฐ๏ฐ๏ท ๏ฝ ๏ ๏ฝ (4.8) Which leads to tangential feed ๐๐ก๐๐[mm/sec] : tan2H TD Df๏ญ๏ฆ ๏ถ๏ฝ๏๏ง ๏ท๏จ ๏ธ (4.9) The motion of the end millโs tooth after one tooth period is shown in Figure 4.1 where end millโs center shifts from (๐1) to (๐2). The dashed and solid lines represent the tool paths of the first and the second tooth, respectively. ๐1 is the position of tool center when the first tooth cuts the workpiece at point B and ๐2 is the position of tool center when the subsequent tooth cuts the workpiece at point A. The kinematics of the motion leads to static chip thickness ๐ด๐ตฬ ฬ ฬ ฬ . Figure 4.1 Toolpath of orbital drilling process and simultaneous feed directions 39 In practice, the orbital speed ( on ) is significantly lower than the spindle speed ( n ), i.e. 100onn๏ณ . The speed ratio could change as a function of tool diameter, hole diameter and cutting conditions such as tangential and axial feed rates. Hence the angular orientation of the feed in one tooth period ( 060Nn๏ด ๏ฝ ) ( 0๏ด ๏๏ ) between two consecutive tool centers is extremely small: 0 00260 260n nNn N n๏ฐ ๏ฐ๏ค ๏ด๏ฝ ๏ ๏ฝ ๏ฝ (4.10) For a two fluted end mill ( 2N ๏ฝ ), orbital speed 0 12[ / min]n rev๏ฝ and spindle speed of1200 [ / min]n rev๏ฝ , the angular change in the direction of feed is 0.0314[rad] or 1.8 degrees which is negligible. As a result, it is assumed that the structural dynamics of the system in one tooth period are assumed to remain constant. Figure 4.2 Vibration independent toolpath of orbital drilling process 40 If the โ โฒ๐ and โ ๐ are assumed to be equal the constant time delay (tooth period) can be evaluated as: 060Nn๏ด ๏ฝ (4.11) If the vibrations of the tool are neglected, tool center shifts at amount of: ___1 2 0 tanO O f๏ด๏ป (4.12) By using the sine rule: ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ__0 tan' '/ 2 / 2sin / 2sin cosT Tjj j jf D D AB๏ด๏ฐ ๏ฆ๏ฆ ๏ฆ ๏ฆ๏ญ๏ฝ ๏ฝ๏ซ๏ญ (4.13) In practical orbital drilling applications 0 tanf๏ด is much smaller than the radius of the cutter hence the following assumption is made: ๏จ ๏ฉ' 'sin j j j j๏ฆ ๏ฆ ๏ฆ ๏ฆ๏ญ ๏ป ๏ญ (4.14) By using the definition of time delay: ๏จ ๏ฉ ๏จ ๏ฉ' ''0 02 / 260j j j jN nt๏ฆ ๏ฆ ๏ฆ ๏ฆ๏ฐ ๏ฐ๏ด ๏ด ๏ด ๏ท๏ท ๏ท ๏ท๏ญ ๏ญ๏ฝ ๏ญ ๏ฝ ๏ญ ๏ฝ ๏ญ ๏ฌ ๏ฝ (4.15) Where tโ is the time at which the previous cutter tooth reaches at the angular position โ โฒ๐ and t is the time when the current cutter tooth reaches at the angular position โ ๐. 41 Varying time delay is derived as: ๏จ ๏ฉ๏จ ๏ฉtan/ 2 cos / 2 cos2T TjH TT jT jD DD DN D fN D๏ฐ ๏ฐ๏ด ๏ฆ๏ท ๏ฆ๏ท ๏ฆ๏ฝ ๏ฝ๏ฆ ๏ถ๏ฆ ๏ญ ๏ถ๏ซ ๏ฆ ๏ถ๏ซ ๏๏ง ๏ท๏ง ๏ท๏ง ๏ท๏จ ๏ธ๏จ ๏ธ๏จ ๏ธ (4.16) As it can be seen from the Equation 4.16 the effect of orbital motion on time delay depends on the hole diameter HD , tool diameter TD , spindle speed ๏ท [rad/sec] and orbital speed ๏ [rad/sec]. Time delays are evaluated as a function of orbital speeds, with fixed tool and hole diameters at the angular position of the each tooth in Figure 4.3. The following parameters are used in the analysis: spindle speed n = 8000 rpm, tool diameter TD = 16 mm, hole diameter HD = 30 mm, number of the teeth N = 2. The following orbital speeds and tangential feed rates are used: i. on = 20 rpm, tans = 0.055 mm/tooth/rev ii. on = 40 rpm, tans = 0.110 mm/tooth/rev iii. on = 80 rpm, tans = 0.220 mm/tooth/rev iv. on = 120 rpm, tans = 0.330 mm/tooth/rev In practical orbital drilling operations, the tangential feed rate (๐ ๐ก๐๐) is often less than 0.25 mm/tooth/rev. As the orbital speed increases, the variable delay changes. For the case where the orbital speed is on = 120 rpm, the variation in the delay amplitude is expected to be less than 1.5% of constant time delay ( 0๏ด ). 42 Figure 4.3 Variable time delay versus angular position of cutting tooth The results given in Figure 4.3 demonstrates the real time delay remains within 1.5%๏ฑ of the tooth period at extreme orbital speeds, hence the effect of orbital speed on the delay term used in chatter stability is negligible. 43 4.3.2 Chatter Stability in Frequency Domain Solution If the cutting with the bottom part is neglected the dynamics of the orbital drilling at a frozen point along the path is similar to the milling. The regeneration of radial chip thickness is shown in Figure 4.4 and can be expressed as [32]: Figure 4.4 Dynamic radial chip thickness generation in orbital drilling ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ, , ,sin cosr j r j r j rh x y๏ฆ ๏ฆ ๏ฆ๏ฝ ๏ ๏ซ๏ (4.17) The dynamic cutting forces in x and y directions are: ,12xx xyxtc ry yx yya aF xaKF a a y๏ฉ ๏น๏ฌ ๏ผ ๏๏ฌ ๏ผ๏ฏ ๏ฏ๏ฝ ๏ช ๏บ๏ญ ๏ฝ ๏ญ ๏ฝ๏๏ฏ ๏ฏ ๏ฎ ๏พ๏ช ๏บ๏ฎ ๏พ ๏ซ ๏ป (4.18) 44 where the time varying directional dynamic milling coefficients are represented by ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ1,01,01,01,0sin 2 1 cos 21 cos 2 sin 21 cos 2 sin 2sin 2 1 cos 2Nxx r rjNxy r rjNyx r rjNyy r rja Ka Ka Ka K๏ฆ ๏ฆ๏ฆ ๏ฆ๏ฆ ๏ฆ๏ฆ ๏ฆ๏ญ๏ฝ๏ญ๏ฝ๏ญ๏ฝ๏ญ๏ฝ๏ฉ ๏น๏ฝ ๏ญ ๏ซ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ซ ๏ซ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ญ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ซ๏ซ ๏ป๏ฅ๏ฅ๏ฅ๏ฅ (4.19) , , tc,/r r rc r rK K K๏ฝ . The dynamic cutting forces in time domain can be expressed in vector form as: ๏จ ๏ฉ๏ป ๏ฝ ๏จ ๏ฉ ๏ป ๏ฝ,1( )2tc rF t K a D t t๏ฝ ๏๏ฉ ๏น๏ซ ๏ป (4.20) Where ๏จ ๏ฉD t is the time varying directional matrix. Altintas and Budak [27] introduced frequency domain linear stability law to predict critically stable depth of cut and spindle speeds as a function of machineโs dynamics, tool geometry and work material properties. They transformed the time varying, periodic directional matrix into frequency domain: ๏จ ๏ฉ ๏ ๏๏ ๏ ๏จ ๏ฉ01ir trrir trD t D eD D t e dt๏ด๏ด๏ท๏ด๏ท๏ด๏ฅ๏ฝ๏ญ๏ฅ๏ฝ๏ฉ ๏น๏ซ ๏ป๏ฝ ๏ฉ ๏น๏ซ ๏ป๏ฅ๏ฒ (4.21) Where ๐ is the tooth passing period. Budak and Altintas [34] proposed two approaches: Multi-Frequency solution where the harmonics of periodic coefficients are considered; and Zero Order method where the harmonics are neglected to eliminate time dependency of directional 45 coefficient. The Zero Order method is used here by taking only the mean value of directional coefficients as: ๏ ๏ ๏จ ๏ฉ001D D t dt๏ด๏ด๏ฝ ๏ฉ ๏น๏ซ ๏ป๏ฒ (4.22) Which is valid only when the tool is in engagement boundaries (,st r๏ฆ ,,ex r๏ฆ ) ๏ ๏ ๏จ ๏ฉ,,01ex rst rpitchD D d๏ฆ๏ฆ๏ฆ ๏ฆ๏ฆ๏ฝ ๏ฉ ๏น๏ซ ๏ป๏ฒ (4.23) Where pitch๏ฆ is the pitch angle of the cutter. The resulting average directional milling coefficients become: ,,,,,,,,, ,,,, ,1cos 2 2 sin 221sin 2 2 cos 221sin 2 2 cos 221cos 2 2 sin 22ex rst rex rst rex rst rex rst rxx r r r rxy r ryx r ryy r r r ra K Ka Ka Ka K K๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ๏ฆ ๏ฆ ๏ฆ๏ฆ ๏ฆ ๏ฆ๏ฆ ๏ฆ ๏ฆ๏ฆ ๏ฆ ๏ฆ๏ฉ ๏น๏ฝ ๏ญ ๏ซ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ซ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ซ ๏ซ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ญ ๏ญ ๏ญ๏ซ ๏ป (4.24) The dynamic cutting forces in Equation 4.20 can be expressed as: ๏จ ๏ฉ๏ป ๏ฝ ๏ ๏ ๏จ ๏ฉ๏ป ๏ฝ, 012tc rF i K a D i๏ท ๏ท๏ฝ ๏ (4.25) 46 The lateral directions ๏จ ๏ฉi๏ท๏ can be expressed from the frequency response of the machine tool ๏จ ๏ฉi๏ท๏ and the dynamic cutting forces ๏จ ๏ฉF i๏ท as. ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ11xiixx i F ie iy i Fy iwherex i x ii ey i y ix i F iiy i Fy i๏ท๏ด๏ท๏ด๏ท ๏ท๏ท๏ท ๏ท๏ท ๏ท๏ท๏ท ๏ท๏ท ๏ท๏ท๏ท ๏ท๏ญ๏ญ๏๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฝ ๏ญ ๏๏ฉ ๏น๏ญ ๏ฝ ๏ญ ๏ฝ๏ซ ๏ป๏๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฎ ๏พ ๏ฎ ๏พ๏๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ ๏ฝ ๏ฝ ๏ญ๏ญ ๏ฝ ๏ญ ๏ฝ๏๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฎ ๏พ ๏ฎ ๏พ๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฝ ๏๏ฉ ๏น๏ญ ๏ฝ ๏ญ ๏ฝ๏ซ ๏ป๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฎ ๏พ ๏ฎ ๏พ (4.26) Where ie ๏ท๏ด๏ญ is the delay between two successive teeth. The frequency response matrix includes both direct ( ๏จ ๏ฉxx i๏ท๏ , ๏จ ๏ฉyy i๏ท๏ ) and cross ( ๏จ ๏ฉxy i๏ท๏ , ๏จ ๏ฉyx i๏ท๏ ) FRFs. However machine tools usually have orthogonal structures, hence cross FRFs can be neglected: ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ00xx xy xxyyyx yyi i iiii i๏ท ๏ท ๏ท๏ท๏ท๏ท ๏ท๏ฉ๏ ๏ ๏น ๏ฉ๏ ๏น๏ ๏ฝ ๏ฝ๏ฉ ๏น ๏ช ๏บ ๏ช ๏บ๏ซ ๏ป ๏๏ ๏ ๏ช ๏บ๏ช ๏บ ๏ซ ๏ป๏ซ ๏ป (4.27) The dynamic cutting forces can be reduced to following: ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ, 0, 011211 02c c cc cx xi i itc r cyx i itc r cyF Fe K a D e i eF FyFe I K a D e iF๏ท๏ด ๏ท๏ด ๏ท ๏ด๏ท ๏ด ๏ท ๏ด๏ท๏ท๏ญ๏ญ๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ๏ฉ ๏น๏ฝ ๏ญ ๏๏ญ ๏ฝ ๏ญ ๏ฝ๏ซ ๏ป๏ฏ ๏ฏ ๏ฎ ๏พ๏ฎ ๏พ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฉ ๏น๏ฉ ๏น๏ญ ๏ญ ๏ ๏ฝ๏ญ ๏ฝ ๏ซ ๏ป๏ช ๏บ๏ซ ๏ป๏ฏ ๏ฏ๏ฎ ๏พ (4.28) The characteristic equation of the system becomes ๏จ ๏ฉ๏ ๏ ๏จ ๏ฉ, 01det 1 02citc r cI K a e D i๏ท๏ด ๏ท๏ญ๏ฉ ๏น๏ฉ ๏น๏ญ ๏ญ ๏ ๏ฝ๏ซ ๏ป๏ช ๏บ๏ซ ๏ป (4.29) 47 Or in a short form as: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏ ๏ ๏จ ๏ฉ0,0 0det 014ccitc rc cI iwhereNK a ei D i๏ท๏ด๏ฌ ๏ท๏ฌ๏ฐ๏ท ๏ท๏ญ๏ฉ ๏น๏ฉ ๏น๏ซ ๏ ๏ฝ๏ซ ๏ป๏ซ ๏ป๏ฝ ๏ญ ๏ญ๏ฉ ๏น ๏ฉ ๏น๏ ๏ฝ ๏๏ซ ๏ป ๏ซ ๏ป (4.30) Which leads to the following quadratic function: ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ20 1011 0xx c yy c xx yy xy yxxx xx c yy yy ca awherea i i a a a aa a i a i๏ฌ ๏ฌ๏ท ๏ท๏ท ๏ท๏ซ ๏ซ ๏ฝ๏ฝ ๏ ๏ ๏ญ๏ฝ ๏ ๏ซ ๏ (4.31) Then the eigenvalue ๏ฌ is obtained as 21 1 0Im042Ra a aia๏ฌ ๏ฌ ๏ฌ๏ญ ๏ซ ๏ญ๏ฝ ๏ฝ ๏ซ (4.32) The eigenvalue of the system has both real and imaginary part R Ii๏ฌ ๏ฌ ๏ฌ๏ฝ ๏ซ . Using the cos sincic ce๏ท๏ด ๏ท ๏ด ๏ท ๏ด๏ญ ๏ฝ ๏ญ transformation and the eigenvalue given in Equation 4.32 the critical depth of cut a at the chatter frequency c๏ท can be calculated as follows: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉIm Im,1 cos sin 1 cos sin41 cos 1 cosR c c c R ctc r c ca iNK๏ฌ ๏ท ๏ด ๏ฌ ๏ท ๏ด ๏ฌ ๏ท ๏ด ๏ฌ ๏ท ๏ด๏ฐ๏ท ๏ด ๏ท ๏ด๏ฉ ๏น๏ญ ๏ซ ๏ญ ๏ซ๏ฝ ๏ญ ๏ซ๏ช ๏บ๏ญ ๏ญ๏ช ๏บ๏ซ ๏ป (4.33) 48 Since the depth of cut is a real number, the imaginary part of the Equation 4.33 vanishes ๏จ ๏ฉIm 1 cos sin 0c R c๏ฌ ๏ท ๏ด ๏ฌ ๏ท ๏ด๏ญ ๏ซ ๏ฝ . By substituting sin1 coscIR c๏ท ๏ด๏ฌ๏ซ๏ฌ ๏ท ๏ด๏ฝ ๏ฝ๏ญ (4.34) Chatter free axial depth of cut is reduced to: 2,2[1 ]tc raNK๏ฐ๏ซ๏ฝ ๏ญ ๏ซ (4.35) The phase distance between the inner and outer cut surface is: 2 2 2c k k๏ท ๏ด ๏ฐ ๏น ๏ฐ ๏ฅ ๏ฐ๏ฝ ๏ญ ๏ซ ๏ฝ ๏ซ (4.36) Where k is the integer number of waves generated within one tooth period (๏ด ) and 2๏ฅ ๏ฐ ๏น๏ฝ ๏ญ is the phase shift between the inner and outer waves. The spindle speed n (rev/min) can be is found as: 60nN๏ด๏ฝ (4.37) Equations 4.35 and 4.37 lead to analytical prediction of critically stable depth of cuts and spindle speeds in regular milling operations. However as the radial engagement boundaries in orbital drilling process vary throughout the tool diameter as shown in Chapter 3, which brings additional delays and varying directional coefficients depending on the cutting conditions and speed. As a result, the frequency domain, analytical solution presented by Altintas and Budak [27] may not always lead to accurate solutions for the stability of orbital drilling operations. By considering the spindle speed, Nyquist stability criterion is used in this thesis as illustrated in Figure 4.5[35][36]. 49 Figure 4.5 Nyquist plot of the characteristic equation for stable and unstable process The Nyquist stability criterion sweeps around the most flexible modes of the structure and looks clock-wise encirclements of the complex plane origin as illustrated in Figure 4.5. If the Nyquist plot encircles the origin, it means unstable process. The stability lobes are searched at different pitch lengths of tool path (๐๐) and spindle speed ( n ) pairs. The radial engagement profile changes at each different pitch length, hence it is required to calculate the most severe radial engagement in second zone, first. Then the calculated engagement area is discretized into segments as shown in Figure 4.6. 50 Figure 4.6 Predicted and discretized radial engagement profile for the case where Tool Dia.=16mm, Hole Dia.=30 mm, Pitch Length=5 mm, Discretization Angle: 6 deg. After discretizing the radial engagement into different segments, the characteristic equation of each segment can be written as follows: ๏จ ๏ฉ ๏จ ๏ฉ, 0,1det 1 02citc r i i cI K a e D i๏ท๏ด ๏ท๏ญ๏ฉ ๏น๏ฉ ๏น ๏ฉ ๏น๏ญ ๏ญ ๏ ๏ฝ๏ซ ๏ป๏ซ ๏ป๏ช ๏บ๏ซ ๏ป (4.38) Where ๏ ๏ , ,, ,0, 0ex r ist r iiD D๏ฆ๏ฆ๏ฉ ๏น ๏ฝ๏ซ ๏ปis the coefficient matrix and ๐๐ is the height of each segment. The characteristic equation for each pitch length (๐๐) and spindle speed ( n ) pair can then be expressed as : ๏จ ๏ฉ๏ ๏ ๏จ ๏ฉ,0,11det 1 02citc r T cKT i iiI K e D iwhereD D a๏ท๏ด ๏ท๏ญ๏ฝ๏ฉ ๏น๏ฉ ๏น๏ญ ๏ญ ๏ ๏ฝ๏ซ ๏ป๏ช ๏บ๏ซ ๏ป๏ฉ ๏น๏ฝ ๏ซ ๏ป๏ฅ (4.39) 51 The characteristic equation of each case is assessed by the Nyquist stability criterion and then stability lobes are generated for different pitch lengths. 4.3.3 Chatter Stability in Semi-Discrete Time Domain Solution Stability of the milling was predicted by extending semi-discretization method proposed by Stepan et al.[37,38]. If the machines structural flexibilities are assumed to be orthogonal, the FRF of the lateral directions can be represented as follows: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ2 222 22 222 2/22/22nx x nxxx x nx nx xxx nx nxny y nyyy y ny ny yy ny nyks x t x t x t F tks sks y t y t y t F tkys s๏ท ๏ท๏ธ ๏ท ๏ท๏ธ ๏ท ๏ท๏ท ๏ท๏ธ ๏ท ๏ท๏ธ ๏ท ๏ท๏ ๏ฝ ๏ฎ ๏ซ ๏ซ ๏ฝ๏ซ ๏ซ๏ ๏ฝ ๏ฎ ๏ซ ๏ซ ๏ฝ๏ซ ๏ซ (4.40) Figure 4.7 Discretized tool geometry(left) and a sample radial engagement(right) 52 The end mill is discretized into differential segments with uniform height of dz as it is shown in Figure 4.7. The dynamic forces inn x and y directions at each segment can be expressed as: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ,12xx xyxtc ry yx yya aF t x t x tdzKF t y t y ta a๏ด๏ด๏ฉ ๏น ๏ฆ ๏ถ๏ฌ ๏ผ ๏ญ๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฝ ๏ญ๏ง ๏ท๏ช ๏บ๏ญ ๏ฝ ๏ญ ๏ฝ ๏ญ ๏ฝ๏ง ๏ท๏ญ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ฏ ๏ฏ ๏ช ๏บ ๏ฎ ๏พ ๏ฎ ๏พ๏ฎ ๏พ ๏จ ๏ธ๏ซ ๏ป (4.41) which is valid if the cutting flute at a specific segment is in cut, i.e. within engagement boundaries. The dynamics of the cutting at each differential segment is expressed as, ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ222, 222/ 012 0 /x nx nxy ny nyny x xx xytc ryx yyny yx t x t x ty t y t y tk a a x t x tdzKy t y ta ak๏ธ ๏ท ๏ท๏ธ ๏ท ๏ท๏ท ๏ด๏ด๏ท๏ฌ ๏ผ๏ซ ๏ซ๏ฏ ๏ฏ๏ญ ๏ฝ๏ซ ๏ซ๏ฏ ๏ฏ๏ฎ ๏พ๏ฉ ๏น ๏ฉ ๏น ๏ฆ ๏ถ๏ญ๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ช ๏บ๏ฝ ๏ญ๏ง ๏ท๏ช ๏บ ๏ญ ๏ฝ ๏ญ ๏ฝ๏ง ๏ท๏ญ๏ช ๏บ ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ช ๏บ ๏ฎ ๏พ ๏ฎ ๏พ๏จ ๏ธ๏ซ ๏ป๏ซ ๏ป (4.42) Or in matrix as follows: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ222, 22 0 01 00 20 1 0/ 012 0 /x nx nxy ny nyny x xx xytc ryx yyny yx x xy y yk a a x t x tdzKy t y ta ak๏ธ ๏ท ๏ท๏ธ ๏ท ๏ท๏ท ๏ด๏ด๏ท๏ฉ ๏น๏ฉ ๏น๏ฉ ๏น ๏ฌ ๏ผ ๏ฌ ๏ผ ๏ฌ ๏ผ๏ช ๏บ๏ซ ๏ซ๏ญ ๏ฝ ๏ญ ๏ฝ ๏ญ ๏ฝ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ซ ๏ป ๏ฎ ๏พ ๏ฎ ๏พ ๏ฎ ๏พ๏ซ ๏ป ๏ซ ๏ป๏ฉ ๏น ๏ฉ ๏น ๏ฆ ๏ถ๏ญ๏ฌ ๏ผ ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ช ๏บ๏ฝ ๏ญ๏ง ๏ท๏ช ๏บ ๏ญ ๏ฝ ๏ญ ๏ฝ๏ง ๏ท๏ญ๏ช ๏บ ๏ฏ ๏ฏ ๏ฏ ๏ฏ๏ช ๏บ ๏ฎ ๏พ ๏ฎ ๏พ๏จ ๏ธ๏ซ ๏ป๏ซ ๏ป (4.43) The two sets of second order delayed differential equations are reduced to state space form as: ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ1 2 1 21 2 1 2, ,, ,dx tx x t x x xdtdy ty y t y y ydt๏ฝ ๏ฝ ๏ฝ๏ฝ ๏ฝ ๏ฝ (4.44) First order representation of the system becomes ๏ป ๏ฝ ๏ ๏๏ป ๏ฝ ๏ ๏ ๏จ ๏ฉ๏ป ๏ฝ๏จ ๏ฉ๏ป ๏ฝiq L q R q tq t๏ด๏ด๏ฝ ๏ซ ๏ญ๏ญ (4.45) 53 where ๏ป ๏ฝ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏ ๏ ๏ ๏ ๏ ๏๏ ๏๏ ๏ ๏ ๏๏ ๏ ๏จ ๏ฉ ๏ ๏ ๏ ๏๏ ๏๏ ๏ ๏ ๏๏ ๏ ๏จ ๏ฉ ๏ ๏,2 21222 2 2 21111, ,2/ 0 2 0 0, ,0 2 00 /0 0 0,0xx xytc ryx yyny x x nx nxy ny nyny yx xx ta t a ty tq D t dzKx t a t a ty tkM CkIL RM D tM D t C๏ค๏ท ๏ธ ๏ท ๏ท๏ธ ๏ท ๏ท๏ท๏ค๏ค๏ญ๏ญ๏ญ๏ญ๏ฌ ๏ผ๏ฏ ๏ฏ ๏ฉ ๏น๏ฏ ๏ฏ๏ฝ ๏ฝ ๏ฝ๏ฉ ๏น ๏ช ๏บ๏ญ ๏ฝ ๏ซ ๏ป๏ช ๏บ๏ฏ ๏ฏ ๏ซ ๏ป๏ฏ ๏ฏ๏ฎ ๏พ๏ฉ ๏น ๏ฉ ๏น๏ญ๏ฉ ๏น๏ช ๏บ ๏ช ๏บ๏ฝ ๏ฝ ๏ ๏ฝ๏ช ๏บ๏ญ๏ช ๏บ ๏ช ๏บ๏ซ ๏ป ๏ซ ๏ป๏ซ ๏ป๏ฉ ๏น ๏ฉ ๏น๏ช ๏บ ๏ช๏ฝ ๏ฝ๏ช ๏บ ๏ญ ๏ฉ ๏น๏ช๏ญ ๏ญ ๏๏ฉ ๏น ๏ซ ๏ป๏ซ๏ซ ๏ป๏ซ ๏ป๏บ๏บ๏ป (4.46) The tooth period ๏ด is discretized into small time intervals t๏ as shown in Figure 4.8. Current state at time it is represented by ๏ป ๏ฝiq where the state a tooth period before is represented by๏จ ๏ฉ๏ป ๏ฝiq t ๏ด๏ญ . Figure 4.8 Semi-discretization of tooth period 54 If the time interval t๏ is very small, the states ๏ป ๏ฝq can be defined as the average of two consecutive states as given as: ๏จ ๏ฉ๏ป ๏ฝ๏จ ๏ฉ๏ป ๏ฝ ๏จ ๏ฉ๏ป ๏ฝ ๏ป ๏ฝ ๏ป ๏ฝ12 2i i i m i mq t t q t q qq t๏ด ๏ด๏ด ๏ญ ๏ซ ๏ญ๏ญ ๏ซ๏ ๏ซ ๏ญ ๏ญ๏ญ ๏ป ๏ฝ (4.47) The dynamics of the system can be rewritten at discrete time intervals as: ๏ป ๏ฝ ๏ ๏๏ป ๏ฝ ๏ ๏ ๏ป ๏ฝ ๏ป ๏ฝ๏จ ๏ฉ112i i i i i m i mq L q R q q๏ญ ๏ซ ๏ญ๏ฝ ๏ซ ๏ญ (4.48) The differential Equation 4.48 has homogeneous ๏จ ๏ฉ๏ป ๏ฝ,i hq t and particular ๏จ ๏ฉ๏ป ๏ฝ,piq t solutions. The details of the semi-discrete solution can be found in [32]. ๏จ ๏ฉ๏ป ๏ฝ ๏จ ๏ฉ๏ป ๏ฝ ๏จ ๏ฉ๏ป ๏ฝ, ,pi i h iq t q t q t๏ฝ ๏ซ (4.49) The state solution at time 1it t ๏ซ๏ฝ is given as: ๏ป ๏ฝ ๏ ๏ ๏ป ๏ฝ ๏ ๏ ๏ ๏๏จ ๏ฉ๏ ๏ ๏ ๏ ๏ป ๏ฝ ๏ป ๏ฝ๏จ ๏ฉ11 112iL ti i i i i i m i mq e q L t I L R q q๏ญ๏๏ซ ๏ญ ๏ซ ๏ญ๏ฝ ๏ซ ๏ ๏ญ ๏ซ (4.50) Where 1i it t t๏ซ๏ ๏ฝ ๏ญ and 1it t ๏ซ๏ฝ . Equation 4.50 requires the previous value of the state๏ป ๏ฝiq and values of the state a delay before ๏ป ๏ฝ ๏ป ๏ฝ๏จ ๏ฉ1 1,i m i mq q๏ญ ๏ซ ๏ญ ๏ซ . Equation 4.50 can be expressed as follows: ๏ป ๏ฝ ๏ ๏๏ป ๏ฝ1i i iz B z๏ซ ๏ฝ (4.51) 55 where ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ๏ป ๏ฝ๏จ ๏ฉ๏ ๏๏ ๏ ๏ ๏ ๏ ๏ ๏ ๏๏จ ๏ฉ๏ ๏ ๏ ๏ ๏ ๏ ๏ ๏๏จ ๏ฉ๏ ๏ ๏ ๏๏ ๏ ๏ ๏ ๏ ๏ ๏ ๏๏ ๏ ๏ ๏๏ ๏ ๏ ๏ ๏ ๏๏ ๏ ๏ ๏๏ ๏ ๏ ๏ ๏ ๏ ๏ ๏111 11 14 1 4 11 102 20 0 000 00 0 0 0i i iiiii mi m m xL t L t L ti i i iim x mqqzqqe e I L R e I L RIBII๏ญ๏ญ ๏ซ๏ญ ๏ซ๏ญ ๏ญ๏ ๏ ๏๏ซ ๏ซ๏ฌ ๏ผ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ฝ ๏ญ ๏ฝ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ฏ ๏ฏ๏ฎ ๏พ๏ฉ ๏น๏ญ ๏ญ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ฝ ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ซ ๏ป (4.52) The stability of the process can be assessed for one tooth period ๐ with ๐ number of discrete time intervals as: ๏ป ๏ฝ ๏ ๏๏ป ๏ฝ ๏ ๏ ๏ ๏ ๏ ๏ ๏ป ๏ฝ2 1..i m i imz z B B B z๏ซ ๏ฝ ๏ ๏ฝ (4.53) According to Floquetโs linear periodic system theory, the system is defined as unstable if any of the eigenvalues of the transition matrix ๏ ๏๏ have a modulus greater than one, critically stable if the modulus is equal to one, and stable if the moduluses of all the eigenvalues are less than one. The stability lobes are obtained by checking different pitch length spindle speed pairs as carried out in Nyquist method. However, semi-discrete method considers time varying directional coefficients which are averaged and assumed to be time independent when Nyquist approach is used for the stability. 56 The equations given above are valid when one dominant mode is considered for each direction. However when there are ๐๐ number of dominant modes for each direction,๏ ๏1M๏ญ, ๏ ๏C and ๏ ๏๏ matrices should be expanded as follows: ๏ ๏๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏ ๏๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ๏ ๏1 11 11 11 1221222 22 2/ 0 00 0 00 0 // 0 00 0 00 0 /2 0 00 0 00 0 22 0 00 0 00 0 2m mN Nm mm mN Nm mm mm mN Nm mm mN Nm mm mnx xN x Nnx xny yN x Nny yN x Nx nxN x Nx nxy nyN x Ny nyN x NkkMkkC๏ท๏ท๏ท๏ท๏ธ ๏ท๏ธ ๏ท๏ธ ๏ท๏ธ ๏ท๏ญ๏ฉ ๏น๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ฝ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ซ ๏ป๏ญ๏ฉ ๏น๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ญ๏ช ๏บ๏ฝ๏ช ๏บ๏ญ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ญ๏ช ๏บ๏ซ ๏ป๏ ๏ฝ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏ ๏๏จ ๏ฉ ๏จ ๏ฉ๏จ ๏ฉ ๏จ ๏ฉ1122222 20 00 0 00 00 00 0 00 0m mNmm mNmm mnxN x NnxnyN x NnyN x N๏ท๏ท๏ท๏ท๏ฉ ๏น๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ช ๏บ๏ซ ๏ป (4.54) The procedure for the rest of the solution is identical to what is given for the one dominant mode case. 57 Results of stability solution is given for a sample case in Figure 4.9 when the end mill is at ๐ = 0 [deg.] Figure 4.9 Semidiscrete time domain stability lobes for orbital drilling process. Tool Dia.=16 [mm], Hole Dia.= 30 [mm], Helix Angle = 25 [deg], ๐ฒ๐๐,๐ = 900 [MPa], ๐ฒ๐๐,๐ = 270 [MPa], modal parameters taken from [39]. ๐๐๐๐= 624 [Hz] , ๐๐๐ = 0.052 , ๐๐๐ = 8.2 x ๐๐๐ [N/m], ๐๐๐๐= 871 [Hz] , ๐๐๐ = 0.033 , ๐๐๐ = 16.1 x ๐๐๐ [N/m], ๐๐๐๐= 2311 [Hz] , ๐๐๐ = 0.022, ๐๐๐ = 8.7 x ๐๐๐ [N/m], ๐๐๐๐= 3052 [Hz] , ๐๐๐ = 0.029 , ๐๐๐ = 14.5 x ๐๐๐ [N/m], ๐๐๐๐= 692 [Hz] , ๐๐๐ = 0.042, ๐๐๐ = 7.6 x ๐๐๐ [N/m], ๐๐๐๐= 862 [Hz] , ๐๐๐ = 0.052 , ๐๐๐ = 13.5 x ๐๐๐ [N/m], ๐๐๐๐= 2289 [Hz], ๐๐๐ = 0.019, ๐๐๐ = 7.9 x ๐๐๐ [N/m], ๐๐๐๐= 3050 [Hz], ๐๐๐ = 0.027 , ๐๐๐ = 15.0 x ๐๐๐ [N/m] The multiple mode solution considers the flexibilities of all modes of the end mill. 58 4.4 Oriented Frequency Response Function The methods that presented so far assess the stability of the system at one point however during orbital drilling operation the feed direction of the end mill changes as the tool moves around the hole center as it is shown in Figure 4.10. Figure 4.10 Frequency response on local and global coordinates Koenigsberger and Tlusty [40] defined the frequency response function between cutting forces and the resultant vibration directions (Xโ,Yโ) as oriented FRFs. Hence, the FRFs on global coordinates ๏จ ๏ฉ,xx yy๏ ๏ has to be projected to the feed and normal directions as the tool follows orbital drilling path. The projected frequency responses along instantaneous feed and normal direction ๏จ ๏ฉ' ',xx yy๏ ๏ of the end mill can be evaluated as: 2 2'2 2yy'cos (90 ) cos ( )cos (180 ) cos (90 )xx xx yyxx yy๏ฑ ๏ฑ๏ฑ ๏ฑ๏ ๏ฝ ๏ ๏ซ ๏ซ๏๏ ๏ฝ๏ ๏ซ ๏ซ๏ ๏ซ (4.55) 59 4.5 Simulations and Experimental Results A set of orbital drilling experiments have been carried out a range of hole pitch lengths and spindle speeds listed in Table 4.2. An end mill with radius ๐ ๐ = 8 [mm], pitch angle ๐ฝโ = 25 [deg] and ๐ = 2 teeth has been used for opening holes with ๐ท๐ป = 30 [mm] diameter. The workpiece material is Al7050-T7451 with the experimentally identified cutting coefficients ๐พ๐ก๐,๐ = 752.4 [MPa] and ๐พ๐๐,๐ = 178.3 [MPa]. The presence of chatter has been detected from the spectrum of sound and hole surface. If the spectrum peaks close to one of the natural frequencies of the machine, with peaks spindle/tooth passing frequencies away on the left and right, the presence of chatter is assumed. The presence of forced vibrations is decided when the spectrum is dominated by the spindle and/or tooth passing frequencies and their harmonics. Figure 4.11 Experimental setup for chatter tests Modal parameters of the structures in x and y directions are given in Table 4.1. FRFโs on the tool tip are given in Figure 4.12. 60 Table 4.1 Modal parameters at the tool tip Measurement Direction Natural Frequency [Hz] Damping Ratio ๐ Stiffness ๐ [N/m] X 376.6, 503.9, 1008.1, 1241.5, 1584.7, 1821.1, 2357.8, 3554.9, 4574.7 0.0251, 0.0927, 0.0351, 0.0245, 0.0461, 0.0277, 0.0769, 0.0291, 0.0214 2.31e7, 1.50e7, 5.19e7, 5.49e7, 2.59e7, 4.26e8, 2.58e7, 3.41e8, 2.58e8 Y 443.4, 522.6, 996.1, 1138.9, 1344.1, 1608.0, 2369.3, 3818.8, 4586.8 0.0797, 0.0657, 0.0729, 0.0687, 0.0500, 0.0320, 0.0589, 0.0366, 0.0243 1.57e7, 1.42e7, 4.25e7, 8.81e7, 9.62e7, 4.28e7, 3.07e7, 3.81e8, 2.76e8 Figure 4.12 Two orthogonal direction FRFโs at the tool tip 61 Table 4.2 Pitch length of tool path (๐๐) and spindle speed (๐) values for chatter tests Experiments Spindle Speed [RPM] Orbital Speed [RPM] Pitch of Tool Path [mm] Tangential Feed of the Tool [mm/tooth] Axial Feed of the Tool [mm/tooth] # 1. 8000 10 3 0.0275 0.002 # 2. 8000 10 5 0.0275 0.003 # 3. 8000 10 6 0.0275 0.0037 # 4. 8400 10.5 3 0.0275 0.002 # 5. 8400 10.5 4 0.0275 0.0025 # 6. 8400 10.5 5 0.0275 0.003 # 7. 8400 10.5 6 0.0275 0.0037 # 8. 8800 11 3 0.0275 0.002 # 9. 8800 11 4 0.0275 0.0025 # 10. 8800 11 5 0.0275 0.003 # 11. 8800 11 6 0.0275 0.0037 # 12. 9200 11.5 2.5 0.0275 0.0016 # 13. 9200 11.5 4 0.0275 0.0025 # 14. 9200 11.5 5 0.0275 0.003 # 15. 9200 11.5 6 0.0275 0.0037 FRFs have been oriented as the feed direction changes along the orbital tool path. Sample oriented FRFs for feeds with 45 deg๏ฑ ๏ฝ are given for X and Y directions in Figure 4.13 and Figure 4.14 respectively. 62 Figure 4.13 FRF on global and local X coordinates of end mill when ๐ฝ = 45 deg. Figure 4.14 FRF on global and local Y coordinates of end mill when ๐ฝ = 45 deg. 63 Sound measurements and the corresponding spectrums are shown along the orbital drilling path starting at 0๏ฑ ๏ฝ degree and completing the full hole at 360๏ฑ ๏ฝ degrees. A stable orbital drilling process carried out at the spindle speed of 8000 rev/min is shown in Figure 4.15. The spectrum is dominated by the spindle frequency (133.33Hz) due to run out, tooth passing frequency (266.7 Hz) and their harmonics which indicate forced vibrations hence the stability. The visually observed surface finish was also smooth. Sample, unstable orbital drilling test results are shown in Figure 4.16 where the system chatters close to 522 Hz natural mode. The spectrums are spread to the right and left with integer number (k) of tooth passing frequencies away (i.e. c Tk๏ท ๏ท๏ฑ ) which indicates the presence of chatter. The hole surface was also observed to be rough containing chatter marks. Figure 4.15 Stable orbital drilling: Measured sound signals and their FFT for ๐๐ = 3mm, ๐ = 8000 RPM, ๐๐๐๐ = 0.0275 mm/tooth, ๐๐๐ = 0.002 mm/tooth, Tooth Passing Frequency=266.7 Hz 0 60 120 180 240 300 360-0.015-0.01-0.00500.0050.010.015Angular position of the cutter center [Deg.]Sound [-]0 500 100000.511.522.533.5x 10-3Amplitude of the signal0 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-3Frequency [Hz]0 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-3P2P2P1P3P3P4P4P5P5P6P6266.4 Hz 267.2 Hz 267.2 Hz 267.2 Hz 267.2 Hz 267.2 HzP164 Figure 4.16 Unstable orbital drilling: Measured sound signals and their FFT for ๐๐ = 6mm, ๐ = 9200 RPM, ๐๐๐๐ = 0.0275 mm/tooth, ๐๐๐ = 0.0037 mm/tooth, Tooth Passing Frequency=306.7 Hz The predicted stability charts have been experimentally validated by recording the sound from a microphone attached close to the cuting zone. The assesment of stable and unstable process was made when the end mill starts cutting in zone 2, where the radial engagement reaches its maximum. Frequency domain solution requires FRF of the structure in feed and normal directions whereas the semi-discrete time domain solution requires the modal parameters of the system in each direction. Regarding the feed direction change during the orbital drilling process, calculation of modal parameters in both directions at each predefined point is necessary for solving the 3D chatter stability of the operation in time domain. Besides, if higher number of the modes of the structure is included in the time domain solution, the solution starts being time consuming since the matrixes corresponding with the modal parameters start expanding. 0 60 120 180 240 300 360-0.03-0.02-0.0100.010.020.03Angular position of the cutter center [Deg.]Sound [-]0 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-3Frequency [Hz]0 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-30 500 100000.511.522.533.5x 10-3Amplitude of the SignalP1 P2 P3 P4 P5 P6P2 P3 P4 P5 P6P1wc+wt=824.2 Hzwt=307 Hzwc=519.5 Hz wc=521.9 Hzwc=521.1 Hz wc=520.3 Hzwc+wt=833.6 Hzwc=529.7 Hzwc=518 Hz65 Therefore, it is decided to solve the stability of the system in frequency domain in order to observe the dominant modes around the experiment points. The chatter stability diagram (Figure 4.17) is obtained in Frequency Domain as functions of spindle speed, angular position of the tool and pitch length of the tool path. Figure 4.17 3D stability lobe for orbital drilling and chatter simulated chatter frequencies/Nyquist solution 66 Results of the experiments and simulation can be better compared on a two dimensional graph when the tool is at 0 deg๏ฑ ๏ฝ and when n=9200 rpm as shown in Figure 4.18 and Figure 4.19. Figure 4.18 Comparison of Nyquist solution and experimentally measured chatter stability when end mill is at ๐ฝ = 0 deg Figure 4.19 Stability limits during the rotation of the tool around the hole center for ๐ = 9200 RPM/ Nyquist solution 67 Frequency domain solution and the experimental results are in great agreement. Furthermore, from the frequency domain solution it is observed that when the spindle speed changes between 7500 rpm and 9500 rpm the dominant natural frequencies of the structure are around 500 Hz and 1600 Hz. Thatโs why time domain simulation considers the modal parameters given on Table 4.3. Table 4.3 Modal parameters for semi-discrete time domain solution Direction Natural Frequency [Hz] Damping Ratio ๐ Stiffness ๐ [N/m] X 503.9, 1584.8 0.0927, 0.0461 1.50e7, 2.59e7 Y 522.6, 1608 0.0657, 0.0320 1.42e7, 4.28e7 The chatter stability diagram (Figure 4.20) is obtained in Semi-Discrete Time Domain as functions of spindle speed, angular position of the tool and pitch length of the tool path. Figure 4.20 3D stability lobe for orbital drilling /Semi-discrete time domain solution 68 Time domain and frequency domain solutions can be better compared on a two dimensional graph when the tool is at 0 deg๏ฑ ๏ฝ as shown in Figure 4.21. Figure 4.21 Comparison of predicted and experimentally measured chatter stability when end mill is at ๐ฝ = 0 deg The frequency and time domain results have small discrepancies and they show consistency with the experimental results. The radial engagement profile is discretized into segments and average directional coefficient matrix for each segment is calculated with the zero order frequency domain method. However as the engagement area gets smaller, the averaging of directional coefficients lead to errors. On the other hand, semi discrete time domain solution considers the cutters exact rotation within the engagement, hence time varying directional coefficients are considered, hence leads to more accurate solutions. 69 Frequency and time domain simulation results are computed at each 20 degrees rotation of the end mill around hole center with 50 rpm intervals. Both solution algorithms are implemented on Matlab R2016a platform. Nyquist solution that considers all of the natural frequencies in both lateral directions takes seventy six minutes whereas Nyquist solution that considers only two modes, takes thirty two minutes. Semi-discrete time domain solution that considers two modes takes three hundred and one minutes. Figure 4.22 Stability limits during the rotation of the tool around the hole center for ๐ = 9200 RPM 70 Chapter 5: Conclusion 5.1 Conclusions The mechanics and dynamics of orbital drilling operations have been modeled in this thesis. The model is able to predict the cutting forces, chatter stability and dimensional hole quality for cylindrical helical end mills with drilling edges at their bottom. The mechanics model uses orthogonal to oblique transformation at the helical flutes at the periphery of the end mill. The orthogonal model is based on the plastic shearing, and uses shear yield stress, shear angle and average Coulomb friction coefficient of the work-tool material couple. Due to indentation and severe ploughing, the orthogonal to oblique cutting model has shown to be unsuitable for the penetration of the bottom flutes into the material during downward motion, i.e. drilling with the end mill. The cutting force coefficients have been mechanistically identified from cutting tests, and used in modeling the contribution of the cutting forces by the cutting edges at the bottom face of the end mill. The kinematic model of the orbital drilling motion has been modeled as a function of tool and hole geometries, spindle speed, tangential and orbital feed velocities. The distribution of chip thickness along the bottom and periphery of the end mill have been modeled, and used in predicting the vibration free (i.e. static) cutting forces. The model has led to accurate prediction of cutting forces quite well in lateral directions, but not well in the axial direction where the forces are dominated by ploughing due to plunging into the material. 71 The chatter stability model has been reduced to orbital feed independent, regular milling system by orienting the structural flexibilities in the feed and normal directions. The stability has been solved as a function of pitch length of the orbital motion in frequency domain using Nyquist Stability Criteria, and in Semi Discrete Time domain proposed by Insperger and Stepan. The predicted stability lobes have been experimentally proven on a mill turn machine tool. The contributions of the thesis can be summarized as follows: ๏ท A general force model of orbital drilling for cylindrical flat end mills is developed. A numerical model for calculating the TWE is introduced. ๏ท The dynamic force model of orbital drilling that considers the vibration on lateral (x, y) and axial (z) directions is presented. ๏ท The effect of orbital motion on time delay is analyzed and found to be negligible. ๏ท Since the feed direction of the end mill is changing during orbital drilling, FRF on global coordinates projected along instantaneous feed and normal direction. Hence the stability charts are 3 dimensional considering the end mills position around the hole center, spindle speed and pitch length of the tool path. ๏ท In frequency domain, Nyquist stability criterion is utilized in order to obtain accurate stable pitch lengths since the radial engagement conditions are not constant throughout the end mill diameter. The proposed mathematical model has the following impact on the orbital drilling operations used in industry: ๏ท The model allows the chatter free spindle speed, orbital feed and pitch length of the operation that leads to highest material removal rates. 72 ๏ท The prediction of cutting forces, hence the torque and power allows the selection of tool diameter and cutting conditions which do not violate the toolโs breakage and machine toolโs spindleโs torque-power limits. 5.2 Future Research Directions Future research directions can be pursued in following topics: ๏ท The cutting coefficients related with the bottom part of the end mill are obtained by using the mechanistic approach in this thesis. However, the process is dominated by indentation mechanics which can be modeled based on slip line field or other plasticity methods. ๏ท For high speed orbital drilling applications (i.e. robotic orbital drilling), the gyroscopic effects that might come from the orbital motion of the cutter could be analyzed. ๏ท Orbital drilling is a highly used process in aerospace industry especially on difficult to cut materials at low spindle speeds. The modeling of process damping that considers the contact between the flank face of the cutting edge and cut surface is needed. ๏ท Optimization of the orbital drilling process which considers structural dynamics of the machine, work material properties, Computer Numerical Control (CNC) servo drive bandwidth, tool breakage and dimensional hole surface errors can be highly useful tool for manufacturing industry. 73 Bibliography [1] Lindqvist, Richard, Ingvar Eriksson, and Mathias Wolf. Orbital drilling of sandwich constructions for space applications. 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Mechanics and dynamics of orbital drilling operations Ozturk, Onur Mert 2017
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Title | Mechanics and dynamics of orbital drilling operations |
Creator |
Ozturk, Onur Mert |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | Opening large number of holes takes considerable amount of time during manufacturing and assembly of aircrafts. Traditionally, tools having the same diameter of each hole have been used in drilling which take considerable amount of time for tool change and fixturing. Recently, orbital drilling technology has been introduced to open holes with a single set-up. The combined orbital motion around the hole and helical penetration in axial direction are either given by stationary computer numerically controlled (CNC) machines or hand held, portable heads that are attached to aircraft body with suction pads. Although the tool path and machine were developed, the process mechanics and dynamics have not been modeled to predict cutting forces, torque, power and chatter stability diagrams to identify most productive and safe cutting conditions. This thesis presents mathematical model to simulate the mechanics and dynamics of orbital drilling process. The mechanics of the process are modeled by identifying the chip thickness distribution along the peripheral and bottom cutting edges of the helical end mills used in orbital drilling, The pitch length of the path, tool and hole diameters, spindle speed, feed and material properties are used in the model which is experimentally proven by comparing predicted and measured cutting forces. The flexibilities of the orbital drilling head and tool are incorporated to the mechanics model to predict the dynamics of the system. It is shown that the additional delay contributed by orbital motion of the tool can be neglected, and the regenerative delay is dominated by the spindle speed. However, structural dynamic modes of the system need to be oriented along the tangential feed direction since it varies continuously along the orbital path. The chatter stability of the system has been developed in both frequency and semi-discrete time domains. The experimentally verified stability model considers spindle speed, tool and hole geometries, structural dynamics, material properties and orbital drilling pitch length. The proposed orbital drilling model allows optimal selection of spindle speed, feed, orbital speed, pitch length and tool diameter for a given work material without overloading the machine and chatter while achieving highest possible material removal rates. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-05-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0347429 |
URI | http://hdl.handle.net/2429/61610 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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