# Stra ategies for Reducin ng Vibrat tions dur

Straategies for Reducin ng Vibrattions durring Milliing of Thin-walled Com mponentts J. Bertil B Waanner Licentiate Thhesis Stock kholm, Swedeen, 2012 ©J. Bertil Wanner, 2012 TRITA IIP- 12- 03 ISSN 1650-1888 ISBN 978-91-7501-322-0 KTH School of Industrial Engineering and Management Royal Institute of Technology SE-100 44 Stockholm Sweden Stockholm, Sweden, November 2012 Abstract Factors such as environmental requirements and fuel efficiency have pushed aerospace industry to develop reduced-weight engine designs and thereby light-weight and thin-walled components. As component wall thickness gets thinner and the mechanical structures weaker, the structure becomes more sensitive for vibrations during milling operations. Demands on cost efficiency increase and new ways of improving milling operations must follow. Historically, there have been two “schools” explaining vibrations in milling. One states that the entry angle in which the cutting insert hits the work piece is of greater importance than the exit angle. The other states that the way the cutter leaves the work piece is of greater importance than the cutter entry. In an effort to shed some light over this issue, a substantial amount of experiments were conducted. Evaluations were carried out using different tools, different tool-to-workpiece offset positions, and varying workpiece wall overhang. The resultant force, the force components, and system vibrations have been analyzed. The first part of this work shows the differences in force behavior for three tool-to-workpiece geometries while varying the wall overhang of the workpiece. The second part studies the force behavior during the exit phase for five different tool-to-workpiece offset positions while the overhang is held constant. The workpiece alloy throughout this work is Inconel 718. As a result of the project a spread sheet milling stability prediction model is developed and presented. It is based on available research in chatter theory and predicts the stability for a given set of variable input parameters. Keywords Milling, vibrations, chatter, stability, prediction, thin-wall, Inconel 718. i Acknowledgments I would like to express deep appreciation and gratitude to Prof. Mihai Nicolescu who has served as examiner and advisor for this work. I would like to thank Prof. Lars Pejryd and Assoc. Prof. Tomas Beno who have made this project possible and provided guidance and advice throughout. Dr. Mahdi Eynian has made himself available for many vital discussions and has assisted with mathematical issues and complex drawings and graphs. He has also performed the simulations presented in the published papers. Ulf Hulling and Edisa Sidzak have provided their time and expertise at the Production Technology Center to make the experiments possible during this project. Seco Tools is acknowledged for providing specially designed tools and inserts. This work is based on the research project “Vibrations during Milling of Thinwalled Aerospace Components” and supported by a grant from the Vinnova NFFP program and Volvo Aero. This support is gratefully acknowledged. The experiments were carried out at the Production Technology Center (PTC) at Trollhättan, Sweden. J. Bertil Wanner Stockholm, November, 2012 iii Publications The following papers have been published as a result of this thesis: Paper 1 B. Wanner, M. Eynian, T. Beno, L. Pejryd. Process Stability Strategies in Milling of Inconel 718. 4th Manufacturing Engineering Society International Conference (MESIC), Cadiz, Spain, 2011. Published in the American Institute of Physics Conference Proceedings 1431, 465-472 (2012). Paper 2 B. Wanner, M. Eynian, T. Beno, L. Pejryd. Milling Strategies for Thin-walled Components. Journal of Advanced Materials Research special edition “Advances in Materials Processing Technologies” Vol. 498, pp. 177-182 (2012). Paper 3 B. Wanner, M. Eynian, T. Beno, L. Pejryd. Cutter Exit Effects during Milling of Thin-walled Inconel 718. Journal of Advanced Materials Research special edition “Mechatronic Systems and Materials Application” Vol. 590, pp. 297-308 (2012). v Nomenclature alim Critical depth of cut [mm] ap Axial depth of cut [mm] ar Radial depth of cut [mm] c damping coefficient [Ns/m] f Feed rate [mm/revolution] fz Feed per tooth [mm/tooth] Fp Radial force, passive force (Fx) [N] F Resultant cutting force [N] Ff Axial force, feed force, (Fz) [N] l Length [mm] H Uncut chip thickness [mm] k Spring constant or stiffness [N/m] kc Specific cutting force [N/mm2] n Spindle rotational speed [rpm] r vc Nose radius [mm] vch Chip speed [m/min] m Mass [kg] Kt Tangential cutting constant Kr Radial cutting constant Kf Cutting coefficient in the feed direction Ω Angular speed [rad/sec.] Ф Immersion angle [degrees] ψ Effective exit angle [degrees] ωn Natural frequency [Hz] ωc Chatter frequency [Hz] Cutting speed in primary cutting direction [m/min] vii Acronyms CAD Computer Aided Design CNC Computer Numerical Control FEM Finite Element Method FFT Fast Fourier Transform MRR Material Removal Rate SLD Stability Lobe Diagram ix Table of Contents Abstract Acknowledgments Publications Nomenclature Acronyms Table of Content i iii v vii ix xi 1 Introduction 1.1 Background 1.2 Practical Example –Jet Engine Diffuser Case 1.3 Introduction to Milling Vibration Research 1.4 Fixtures and Tooling 1.5 Damping 1.6 Industrial and Scientific Issues 1.6.1 Industrial Problem 1.6.2 Scientific Approach 1.6.3 Research Questions 1.6.4 Research Scope and Limitations 1 1 2 2 7 8 8 8 9 9 10 2 Mechanics of Milling 2.1 Cutter Geometry Terminology on a Macro Level 2.1.1 Rake Angles 2.1.2 Lead Angle 2.1.3 Clearance Angle 2.1.4 Other Features 11 11 11 13 13 14 3 Milling System Vibrations 3.1 Introduction to Milling Vibrations 3.2 Free, Forced, and Chatter Vibrations 3.3 The Stability Lobe Diagram 3.4 The Dynamics of the Stability Lobes 3.5 Experimental Modal Analysis 17 17 17 20 22 23 xi 4 Modeling Work 4.1 General Simulation Modeling 4.2 Determination of the kc curve 4.3 The Milling Stability Prediction Model 25 25 25 27 5 Experimental Work 5.1 Definitions 5.2 Tools and Procedures 5.3 Dynamics of Stability Lobes Experiments 5.4 Tool-to-Workpiece Position Experiments 5.5 Cutting Geometry Sensitivity 5.5.1 Approach 5.5.2 Down Milling Position 5.5.3 Zero Rake Zero Exit Position 5.5.4 Zero Rake Zero Entry Position 5.5.5 Up Milling Position 5.5.6 Zero Offset Position 5.5.7 Force Profile versus Exit Angle 31 31 32 33 35 38 38 39 40 41 42 43 44 6 Conclusions 47 47 48 49 49 6.1 Research Questions – an Analysis 6.2 Vibration Prediction Modeling 6.3 Recommendations to Operators 6.4 The Diffuser Case – a Recap 7 Future Work 51 8 References 55 Appendix 1: Regenerative Stability Theory Appended Papers xii 1. Introduction 1.1 Background Taking 3 kg across the Atlantic requires about 1 kg of fuel! A European initiative “Clean Sky”1 has been established to support research in an effort to reduce aircraft emissions and fuel consumption. Volvo Aero is an Associate Member of Clean Sky (Sustainable & Green Engines or SAGE), a Joint Technology Initiative (JTI) that was initiated 2008. The goal of the project is to reach the environmental requirements for flights within Europe. One example of these requirements is to reduce carbon dioxide emissions to 50% of current levels by 2020. The project covers various advanced materials and process technologies for aircraft engine structures. To further increase pressure on the airline industry, the European Union has implemented legislatives regarding emissions trading starting 2012. In these, airlines are requested to purchase emission rights for operating aircraft. The Lufthansa Group’s2 aircraft consume on average 4.2 liters of fuel per 100 passenger kilometers. A major factor is their consistent modernization of the fleet. The Airbus A380, for example, consumes on average only 3.41 liters of fuel per 100 passenger kilometers. By 2016, a total of 160 new aircraft will be delivered to Lufthansa, thus ensuring further reduction in fuel consumption for their fleet. Because of programs like Clean Sky or SAGE, aerospace development has led to reduced-weight engine designs. Requirements for reduced emissions and fuel consumption have made aerospace components thinner at the same time as alloys more difficult to machine have been introduced. In addition, the demands in manufacturing have forced production speeds to increase in order to reduce production cost. One of the consequences of increased speeds is an increased risk for vibrations during the milling operation. This, in turn, may lead to poor surface finish, reduced dimensional accuracy, excessive tool wear or complete failure, and noisier work environment. It may also lead to rework and scrap components. 1 2 www.cleansky.eu www.lufthansa.com/responsibility 1 11.2 Practical Examp ple –Jet E Engine Difffuser Casse A typical component that exhibitss certain maachining diffficulties is th he jet engine difffuser case, Figure 1. It I is a com mponent locaated between n the compressorr inlet case an nd the combustion chambber case. It coonsists of an inner and outer ccase and a nu umber of hol low struts. FFigure 1 show ws the fuel injjector clamping m mounts whicch are typicaal locations fo for milling op perations. In nconel 718, a nickkel-base high temperature alloy, is usedd for the diffu user case. Figure 1: D Diffuser case. When macchining the diffuser case flanges at hhigh cutting speeds, vibraations may becom me uncontroollable, makiing the macchining process very diffficult. Vibrations can be so violent thatt the machinne-tool basee develops crracks. Because off chatter vibrrations (self-eexcited vibraations), noisee may reach levels harmful foor the operattor. Where ceramic insertts are used at a high mach hining rates, vibraations still preesent a challeenge for a co ntrolled proccess. This can n lead to tool failu ure. 11.3 Introd duction to o Milling V Vibration n Research h Concerns aabout millingg vibrations are nothing neew. As early as a 1907, Fred derick W. Taylor [1] described d machining vibrations ass the most ob bscure and deelicate of all the pproblems faccing the mach hinist. As shhown in man ny publication ns on machiningg, this obserrvation still holds true today. Therre will alwayys be vibrations in machinin ng systems. Whether W theyy are forced vibrations orr selfexcited vibbrations, theyy need to be controlled c inn order not too cause damaage to the component or the machining m sysstem. A simple vvibratory sysstem may bee representedd by a mass, a spring, and a a damper as depicted in Figure F 2. 2 Figure 2: Mass-sprin ng-damper sysstem. (Hooke’s law w) and awaay from equ uilibrium, (Newton’s second laaw), where m=mass, c= =damping coefficiient, and k=spring constaant or stiffneess. It is assu umed that the damper and th he spring haave no masss. Free vibraations with damping (cc≠0), free vibratioon without damping (c= =0), and unsstable (chatter) vibration ns can be visualizzed in Figuree 3. At equ uilibrium, Figure 3: Free vibrrations with damping d (topp), free vibrattions withoutt damping (middlle), and unstabble vibrationss (bottom). In a m milling situattion, the maass in Figuree 2 symbolizzes the inertia of the workpiiece/fixture system, thee spring syymbolizes th he stiffness of the workpiiece/fixture and/or a the to ool/spindle syystem, and th he damper syymbolizes the tottal milling syystem’s abilitty to reduce vibrations du uring and affter cutter engageement. 3 The dynam mic state of a milling system can be desscribed by a frequency f resp ponse function, Φ . The real and im maginary parrts of the frrequency resp ponse function, F Figure 4, can be written ass where , Φ Eq. 1 Φ Eq. 2 is the stiffness, and d is the dam mping ratio. crosses over The real paart Φ o from poositive to negaative at the natural Φ frequency, . The imaginary part is at minim mum at the natural frequency, . If the curve c for thee imaginary ppart becomess very large in the negative ddirection, it indicates th hat the dam mping ratio nears n zero. In a machining system, the imaginary part p should aalways be negative. A po ositive o the imaginary part could suuggest an errror in the m easurement. The shape of maginary partss is dependen nt on the stiff ffness and thee damping ratio of real and im the system. Figure 4: R Real and imagginary part off transfer funct ction. Chatter th heory may bee used for determining thhe likelihood of chatter during d machiningg. Under certaain condition ns, the amplittude of vibrattion grows an nd the cutting sysstem becomees unstable. Self-excited chatter vibrrations in milling m develop ddue to dynaamic interacttion betweenn the cuttin ng tool and d the workpiece. This results in regenerattion of wavinness on the cu utting surface and modulation n of the chip thickness. Tobias et al. [2.3] and d Tlusty et al. a [4] presennted between n 1958 and 1967 basics of cchatter vibrattions on macchine tools bby explainingg the fundam mental mechanics of regeneratiion of chip thickness t andd calculating the uncondittional 4 stability boundary. Their theory was based on the physics of orthogonal cutting and the regenerative mechanism. If the values of the system parameters are such that the system is unstable, the smallest disturbance (such as a hard spot in the material) is sufficient to induce the system to leave steady state of motion and burst into oscillation (to chatter). Tobias analysis was concerned with the problem of stability, i.e., whether or not the machine will chatter under certain working conditions. His observation was that little use was made of the results by production engineers and machine tool designers of his time because “some of the recommendations based on theoretical considerations appeared to be contrary to practical experience”. He thus developed a regeneration theory that took into account that in a dynamic system, the chip thickness may vary independently of the machine tool feed rate. Shridhar et al. [5] presented in 1968 a detailed mathematical model of the dynamic milling process. They developed a comprehensive stability theory for milling that is based on the numerical integration of the milling equations for one period of the cutter’s revolution. Their computer algorithm permitted the determination of the stability boundaries in the space of controllable parameters associated with the cutting operation. It was able to handle six directions and any number of modes. Minis and Yanushevsky [6] proposed in 1990 a comprehensive analytical method and solved the two-dimensional milling problem by introducing the theory of periodic differential equations. They also improved the early analysis work of Shridhar et al. [5] by applying the theory of periodic differential equations on the milling dynamics equations. Although the algorithm depended on the numerical evaluation of the stability limits, it provided for comprehensive modeling in determining the stability limits for milling and described the aspects of milling dynamics. They concluded that if each tooth of the milling cutter remains in contact along the entire length of the arc being machined, then the dynamics of the milling system are well described by a set of linear differential equations with periodic coefficients. In the non-linear case where the cutting teeth loose contact with the workpiece at some point along the machined arc, the stability theory yields accurate results for most practical cases of milling. This is according to Minis and Yanushevsky due to the fact that prior to the onset of chatter, the nonlinearity occurs mainly at the very beginning of the machining arc. Altintas and Budak presented in 1995 an alternative method for analytical prediction of stability lobes in milling utilizing a transfer function [7]. They modeled the milling cutter and workpiece as multi degree-of-freedom 5 structures. The dynamic interaction in the cutting zone was modeled by including the variations in the cutter and workpiece dynamics in the axial direction. It was demonstrated with numerical examples that also for highly flexible workpieces, the accuracy of the predictions can increase. Their analysis resulted in analytical relations for the chatter frequency and chatter stability limit which were used to generate stability diagrams. Time-varying dynamic cutting force coefficients were approximated by their Fourier series components, and the chatter-free axial depth of cuts and spindle speeds were calculated directly from linear analytical expressions without any numeric iterations. The model can be used to determine the chatter free axial and radial depth of cuts without resorting to time domain simulations. Further development of their method and applications can be found in [8,9]. Quintana and Ciurana [10] recently made an extensive review of publications on chatter in machining processes. The article reviews the state of research on the chatter problem and defines the differences between free vibrations, forced vibrations, and self-excited vibrations. In addition, different forms of chatter such as frictional chatter, thermo-mechanical chatter, mode-coupling chatter, and regenerative chatter are explained. Further, the existing methods developed to ensure stable cutting are classified. Same authors [11] applied sound mapping methodology in order to determine the stability lobe diagram. During a milling operation, the three different types of mechanical vibrations (free vibrations, forced vibrations, and self-excited vibrations) propagate through air and generate a sound that intrinsically contains information about the process. The article presents the information in the form of a 3-D stability lobe diagram. Regarding further developments and improvements in the field, it suggests that models could become more sophisticated and accurate by a deeper consideration of process damping, part behavior, and changes in structure or system dynamics along the tool path. Advances in computers and sensors would undoubtedly play an important role in this field. A useful idea would be to identify the Stability Lobe Diagram, SLD, of a given system composed of a certain machine tool, tool holder, cutting tool and workpiece material system in the process planning. Finally, the article [10] mentions that some aspects in milling are still difficult to model, e.g., the spindle dynamic behavior variation at high rotational speeds, where the centrifugal force on the bearings, the gyroscopic effect, and thermal effects change the performance of the spindle. Studies by Pekelharing [12,13,14] show that interrupted cutting such as milling may yield excessive tool wear due to multiple tool entry and exit. 6 Initiallly, the entry shock s was blamed whereaas the exit waas considered d harmless or of m minor influeence [12]. The issue of eentry was coonsidered two-fold: A mechan nical impactt shock and d a thermal shock. Thee latter depeended on machin nability, coolling time, an nd cutting speeed and feed d. However, according a to Pekkelharing, thee exit can cau use immediatte and progreessive chippin ng. It can also caause severe bu urr build-up where the cuutter exits thee workpiece. Burrs are formedd because maaterial from th he exit face iss pushed asid de by each cuttter. Each tooth ttakes a little less than th he proper cutt and adds this to the bu urr. Over med but thee teeth take too certain n ranges of exxit angles, no o burr is form t much insteadd of too littlee. This “too much” is fouund at the end of the ch hip and is sometimes referred to as a “foot”” and is depiccted in Figure 5. This foot forming starts w where the furrthermost pro ogressed partt of the edge starts its exiit. It then spreadss over the wiidth of the cut. The foot is largest wh here the last exit takes place [[13]. The pap per states thaat the most efffective remedy against cu utter wear and faiilure is to prrevent foot fo ormation as m much as possible. This iss done by using ssmall diameteer cutters, sho ort cuts, and most importantly, to shiftt the path of the center of thee cutter awayy from the exxit face of thee workpiece whenever possiblle [14]. So according a to Pekelharing, the exit con nditions are of o greater importtance than initially expecteed. ns of process sttages. Figure 5: Definition 1.4 Fixxtures and d Tooling A com mmon solutioon to avoid vibration prroblems is too design fixttures that stabilizze the components. How wever, it is nnot always possible p or feeasible to design suitable macchining fixtu ures. Some fiixtures, for example, e mak ke certain parts oof the comp ponent inacccessible for m machining. Because B of their t size, weightt, complexityy, and cost, many m fixturess are only maade in single quantity. This leads to delaays while reeloading macchines as weell as an in ncrease in interm mediate parts storage. s Wheen such a fixtture fails, it may m cause prroduction interru uption and in ncur additionaal manufactuuring costs. Fiixtures are deesigned to 7 increase the stiffness of the workpiece. As the workpiece wall thickness becomes thinner, the workpiece itself becomes more flexible. Finding solutions to these issues becomes more and more urgent. 1.5 Damping Damping is a capacity of a vibrating system to transform a fraction of energy of the vibratory process during each cycle of vibration into another form of energy, mainly heat. This energy transformation leads to reducing intensity (amplitude) of a forced vibratory process or to a gradual decay and fading of a free vibration process. If a vibratory process is of a self-excited type (e.g., chatter vibrations in metal cutting operations), then the damping capacity of the system may partially suppress or completely prevent development of the self-excited vibrations [15]. Damping can also be described as an irreversible physical process that dissipates energy through the conversion of work into heat. In engineering structures it is present in several forms: internal hysteresis, friction via the rubbing action of surfaces or particles, viscous friction in fluids, radiation damping, electromagnetic damping etc. As demonstrated by Nicolescu et al. [16], damping plays a critical role in structural dynamics as it is the primary means by which resonant amplitudes are controlled thus enhancing durability, life cycle behavior and cost reduction. Damping is of great interest in vibration cases in machining operations. Layers of high damping capacity material may be applied at various places of the machining structure, such as the tool holder/tool interfaces. Although not covered in this work, it should not be neglected in importance. Rather, it should be considered carefully when developing further vibration control systems. 1.6 Industrial and Scientific Issues 1.6.1. Industrial Problem As higher material removal rates are required, ceramic inserts become an attractive solution. Using ceramic inserts, the cutting speed may be up to ten times faster than for carbide inserts. This would increase production rates with an equivalent amount. However, milling thin-walled components can induce vibrations which in turn can cause unfavorable machining stability. While large, complex, and costly fixtures have been the immediate solution to avoid vibrations, it is by no means the long-term or final solution. Vibration risks are still prevalent. The industry needs a robust methodology for assessing vibration 8 risks long before the machining operations take place. Stability effects should be assessed already during early stages in the process planning phase. 1.6.2. Scientific Approach As part of the preparation work a State-of-the-Art study was performed to investigate what work had already been done in this research area. A large number of experiments were conducted and the results observed and analyzed. Regarding insert material, it was decided to use carbide inserts instead of ceramic inserts in order to minimize tool failure during robust machining. The research presented in this thesis is divided up into several steps. First, the analysis will determine whether to approach the vibration issue through an examination of the cutter entry or the cutter exit. The path chosen depends on which of these two options that show the least amount of vibrations. Then, the vibration patterns will be examined using a variety of tool-to-workpiece offset positions. The plotting of the results will be used to acquire an understanding of how to better predict vibration risk during milling. 1.6.3. Research Questions There have been two “schools” on whether the cutter entry or the cutter exit is of greater importance. The research questions, therefore, are outlined to investigate this issue. The strategy is to investigate changes in structure or system dynamics along the tool path. It includes the dynamics explained by the stability lobe diagram as the cutter moves through the workpiece. The stability lobe diagram can be identified for a given system composed of a certain machine tool, tool holder, cutting tool, and workpiece geometry and material. In addition, micro features of the inserts can be studied more in depth. This thesis centers on the following four research questions: R1. How critical is the choice of offset between tool and workpiece during milling? R2. What effects do cutter entry and cutter exit have on system vibrations? R3. How does the effective exit angle affect vibrations during and after cutter exit? R4. What is the dynamic effect on the stability lobe diagram as the cutter moves through the workpiece? 9 1.6.4. Research Scope and Limitations The scope of this thesis is to investigate and analyze strategies for reducing vibrations during milling of thin-walled components. As an application of the industrial problem, an easy to use spread sheet milling stability prediction model is presented. The vibration studies conducted are limited to thin-walled Inconel 718 components. Inconel 718 was chosen as workpiece material because of its widespread usage in aerospace components and its difficult machining characteristics. The workpiece is assumed to be the most flexible part of the machining system. The tooling used has been limited to coated cemented carbide inserts mounted in two different 1-fluted milling tools. Cutter micro geometry effects are not considered. All cutting is done without cooling or cutting fluids. 10 2. M Mechanics of Millling Millingg mechanicss includes th he relationshhip between workpiece and tool surfacees, cutting and a insert angles, a and cutting forcces. The geo ometry is generallly significantly more com mplicated thaan for other cutting meth hods such as turn ning. 2.1 Cu utter Geom metry Term minologyy This iss a brief sum mmary of som me of the mosst frequently used cutter geometry termin nology. A general milling tool t is depicteed in Figure 6. 6 Figure 6: A generall milling tool serving as exaample for this section. It is a six-flute face miill with negatiive radial andd positive axiaal rake angles. 2. 1.1. Rakke Angles Radial Rake Angle The raadial rake anggle of a milling cutter is tthe angle bettween the rak ke face of the toooth and a rad dial line passiing through the cutting edge, e Figure 7. It may be posiitive, negativve, or zero and d controls thhe chip flow accordingly. a A positive radial rrake angle reqquires less force and creattes less heat than a zero orr negative radial rrake angle. However, H a po ositive rake aangle also putts greater streess on the cuttingg edge. A neggative radial rake angle ccutter starts the cut away from the edge off the rake face where the cutter c is stronnger. 11 Radial Rake angle a (showingg a negative raadial rake anggle). Figure 7: R Axial Rakee Angle For tools w with inserts, the axial rak ke angle is foormed between the axis of o the tool and th he cutting edgge, Figure 8. Just like the radial rake an ngle, the axial rake angle can bbe positive, negative, n or zero and conttrols the chip p flow accordingly. It influences the cuttingg forces and the strength of the cuttin ng edge. A po ositive axial rake angle allows for an easierr chip flow aas it lifts thee chip and cu urls it away from m the workpieece surface. A negative axxial rake anglee cutter bend ds the chip forwaard and down nward under pressure, cauusing potential chip evacu uation difficulties for soft alloyys. Figure 8: A Axial rake anggle (showing a positive axiaal rake angle). Helix Anglle and Helicaal Axial Rake Angle When the cutting edgee is formed along a a helix about the toool axis (as in the case of a ssolid carbide tool), the reesulting rake is called hellical rake anggle or helix anglee (see Figure 9). It is a fu unction of thhe tool radiuss and is steep per or 12 lower ttoward the ceenter of the tool. t The hellix angle is th herefore convverging to zero att the center of o the tool and larger towaards the perip phery. The helix angle for soliid tools correesponds to thee axial rake anngle for toolss with inserts. Figure 9: Helix anggle, flute, corner radius, andd tooth. 2. 1.2. Leaad Angle The leaad angle is deefined accord ding to Figurre 10. An increase in the lead l angle results in an increasse in the axiall force and a reduction in the radial forrce. From point oof view of vibrations, v th he magnitudee of the radiial force is important when iit is directed in the weakeest direction of the workp piece wall. A properly chosen n lead angle allows a the cuttter to enter aand exit the cut c more smoothly by reducin ng the shockk load on thee cutting edgge. This is because b the leead angle providees for a longeer entry and exit e phase forr cases wheree the axial rak ke angle is not eqqual to zero, Figure 10. The lead anngle is also important i fo or process dampin ng. Figure 10: Lead anggle. 2. 1.3. Clearance Anggle The cleearance anglee (both radiall and axial) iss the angle th he tool formss with the workpiiece and preevents the to ool from rubbbing againsst the workp piece (see Figure 11). It shou uld be large enough forr the tool to clear the workpiece. w Howevver, if it is tooo large, it will w weaken tthe cutter. Itt may be divvided into 13 primary an nd secondary clearance angles. The cleaarance angle can have an effect on the dam mping properties of the sysstem. Figure 11: Clearance an ngles (showingg the radial cle learance angless). 2.1.4.. Other Features F The radial runout or offfset of a millling cutter is a variation of o the cuttingg edge relative to the outer diaameter of thee tool. The axxial runout iss a variation of o the cutting edgge relative to the tool face, Figure 12. T The runout could c be a pro oduct of either ddesign or malfunctioning intolerances. i For examplee, the axial ru unout could be deesigned to function as a wiper. w Figure 12: Radial and axial a runout s of th he cutting to oth against which w the ch hip is The rake fface is that surface formed in the metal cuttting operatio on, Figure 133. 14 The laand is the part of the to ooth adjacennt to the cuttting edge. It I aids in avoidin ng interferencce between the tool itselff and the surfface of the workpiece, w Figure 13. The filllet is the curvved surface att the bottom of the flute. The shape off the fillet contribbutes to the strength s of th he tooth for solid cutters and defines the space where the chip flow ws, Figure 13.. d of the cylinder passing through the The outside diameeter is the diameter peripheeral cutting edges. It deffines the larggest slot that the tool can n cut in a single ppass, Figure 13. 1 The rooot diameter is the diameter of the cyllinder passing through th he bottom of the fillet. The larger root diameter a ttool has, thee greater its torsional strengtth, Figure 13. Figure 13: Tool rakke face, fillet, land, l flute, annd outer and root r diameterss. The cu utting edge is i the interseection of thee rake face of o the tooth with the leadingg edge of the land. The flu ute is a groovve on the periiphery of a cuutter that alloows for chip flow f away from th he cut. The teeeth are the cutting c pointss on a cuttingg tool. The number n of teeth is the same aas the numberr of flutes. The cu utter pitch orr density is deetermined byy the numberr of teeth in the t cutter body aand is depicteed in Figure 14. It rangess from coarsee to fine. Coaarse pitch allows for large feed d and chip th hickness, wheereas fine pittch may be used when 15 high efficieency and goood workpiece finish are reequired. Increeasing the nu umber of teeth w will increase the risk for regenerativee chatter while decreasing the number off teeth may induce vibrrations if thee system is working clo ose to resonance. Therefore, th he number of teeth on a ccutting tool should s be carefully consideredd. Staggered cutters c with unequal u spacinng (so-called differential pitch) p may be useed to reduce the risk for chatter c vibrattions in the syystem. The reason is that the chip load on n each cutter is varied, andd thus, the effective tooth h pass frequency is reduced. Differential pitch cutteers are thus used for sp pecific cutting appplications. Figure 14: Coarse, mediium, fine, andd differential ppitch. 16 3. Milling System Vibrations 3.1 Introduction to Milling Vibrations Vibrations may be divided into free vibrations, forced vibrations, and selfexcited vibrations (so-called chatter). Chatter may be divided into primary and secondary chatter [17]. Primary chatter is caused by friction between the tool and the workpiece, the thermodynamics of the cutting process, and from mode coupling. Secondary chatter or regenerative chatter results from a modulated chip thickness. 3.2 Free, Forced, and Chatter Vibrations Vibrations in milling arise from a flexible workpiece, a flexible machine tool, or both. Free vibrations occur when the mechanical system is displaced from its equilibrium and is allowed to vibrate freely. This could be for example as a result of a collision between the cutting tool and the workpiece. Forced vibrations appear due to external harmonic excitations, particularly when the cutting edge enters and exits the workpiece. They can also stem from unbalanced bearings or cutting tools. Chatter vibrations extract energy from the interaction between the cutting tool and the workpiece, and grow during the machining process to bring the system into instability. As thin-walled workpieces are flexible, chatter may occur. The reason is that structural modes of the machining structure are excited by the cutting forces. The first three modes of the workpiece used in this study are shown in Figure 15. 17 Figure 15: Mode 1, modde 2, and modde 3 of the woorkpiece used in i study. The wavy surface from m a specific cu utting tooth is removed by b the subseq quent tooth, also this leaving a wavy surfaace. The phasse shift betweeen the two waves w implies a ch hange in the chip thickneess as well as tthe cutting foorce, Figure 16. No viibrations No vibratioons Forced vibrations 0 In phase waves Constantt chip load No chatter Chattter vibration ns Out of phase waves Unstable chatter Figure 16:: Description of o no vibratioon, forced vibrrations, and chhatter vibrations as ness variation during d milling ng. visualized iin chip thickn is the ph hase shift anggle between the current aand previouss surface wavviness. Regenerativve chatter occcurs due to the differencce in vibratioon phase bettween the currentt cut and the previous cutt (relating to the chip thicckness). When n two vibration w waves are ou ut of phase, chatter occuurs. From reegenerative ch hatter theory, a reelationship is formed betw ween spindle speed and criitical depth of o cut. Here follow ws a brief sum mmary of ho ow to extract the stability lobe diagram m. For details, see Appendix 1,, Regenerativee Stability Theeory. A wavy surrface finish leeft by one too oth is removeed by the succceeding oscilllatory tooth. Thee resulting ch hip thickness will itself beecome oscillattory and produces 18 oscillattory cutting forces f havingg magnitudess proportionaal to the tim me-varying chip looad. In order to obtain thee stability lobbe diagram, ceertain parameeters such as the natural frequ uency, damping ratio, an d stiffness neeed to be dettermined. From tthese values, the real and imaginary paarts of the traansfer functio on may be calculaated and the dynamic cuttting coefficieents evaluateed. The stabiility lobes are calcculated as follows (also seee the schemattics in Figuree 17): Select a chaatter frequenccy ωc aroundd a dominant mode ωn. i. ii. Solve an eiggenvalue equ uation and callculate the critical depth of o cut alim. iii. Calculate the spindle sp peed n for eacch stability lobe. iv. Repeat thee procedure by b scanning the chatter frequencies f around all dominant modes m of the structure. Figure 17: Simplifieed schematics of procedure ffor graphing the t stability loobes. 19 3.3 The Stability Lobe Diagrram Regenerativve stability theory t produ uces a stabiliity lobe diaggram that maay be used to predict and control c chattter. The stabbility lobe diagram d plotts the boundary bbetween stabble and unstaable regions aas a function n of spindle speed and depth of cut. Thee locations an nd shapes off the stabilityy lobes depen nd on many variaables, such as a material properties p annd tool-to-woorkpiece position. Each setupp and set off parameters gives a uniqque stability lobe diagram m. By applying su uch diagramss in milling or turning proocesses, the maximum m dep pth of cut may bbe optimized at the highest spindle sspeed used, thereby t increeasing material reemoval rates (MRR) and improving prroductivity. The T spindle speed can be direectly related to t the tooth-p pass frequenccy. As the nu umber of teeth h in a cutting toool increase, the tooth-p pass frequenncy increases. The work kpiece experiencess a greater number of cuts per unit ttime and thiss correspondss to a higher spin ndle speed. Stability loobe diagramss are created by intersectting a series of scallop-sh haped stability boorderlines [188]. These inteersections deefine the deep pest stable cu ut at a given spinddle speed and d form the lim mits for chat ter. Locally, for each lobee, it is stable below the lobe and a unstable above the lobbe. Since thee lobes intersect, a ne lobe coulld be above the neighboring lobe. Su uch a point locatted below on point musst be consid dered as unstable. Globaally, the relaationship bettween adjacent loobes needs too be considerred when dettermining staability. The upper u portions aabove the pooint of interrsection of two adjacent lobes coulld be trimmed ooff, connectin ng all the lob bes into chattter lines, Figure 18. All points p below the lines are stabble, whereas all points aboove the liness are unstablee. For increasing spindle speeeds, the stability lobes bbecome widerr, the intervening spaces betw ween consecu utive lobes greeater, and thee intersection n points higheer. Figure 18:: Stability lobbe diagram before b and aft fter connectingg lobes into ch hatter lines 20 Observving the lobees in Figure 19, the lobe to the far riight (lobe #0 0) has the maxim mum stable deepth of cut at a its intersecttion with thee lobe on its left (lobe #1). O On the other hand, h its righ ht branch haas no intersecction with oth her lobes, theorettically allowin ng for unlim mited depth oof cuts at veryy high spindle speeds. The loobes on the faar left move closer c togetheer the further left they are located. Also tthe intersectiion points move m downw ward eventu ually approacching the minim mum depth off cut, alim. Figure 19: Stabilityy lobe diagram m with upper aand lower bou undary limits ntire range off the stability lobe diagram m may be diviided into threee regions The en of stabbility: uncond ditionally staable, conditioonally stable,, and unconditionally unstable. nconditionallly stable regio on: The loweest points on the lobes (all the same The un value) represent alimm, the minimu um depth off cut. A lowerr horizontal borderline b may th hereby be draawn by conn necting the loowest points on all the lo obes. The region below this lower l border line is uncoonditionally stable, s indepeendent of spindlee speed or chatter c frequ uency. Manyy process en ngineers and machine operatoors prefer thiis region beccause it is chhatter free. However, H it allso means low prooductivity an nd low MRR. The un nconditionallly unstable reegion: An uppper border lin ne may also be b formed by fittiing a curve through t all th he intersectioon points of the lobes. Th he region above this upper border b line is i unconditioonally unstab ble. This imp plies that chatterr vibrations will w always occcur during m machining in this t region. The cconditionally stable regio on: The in--between reggion, i.e., th he region betweeen the lower and the upp per border linnes, is condittionally stable. In this 21 region, poiints are stablee when they are a below thee lobes and unstable u when n they are above tthe lobes. If this t region iss explored annd certain stab ble regions fo ound, then produ uctivity can be b increased without jeoppardizing maachined surfaace or tool integriity. When the spindle speeed approachees zero, the upper and lower border lines converge in nto a point of minimum m depth of ccut. The influ uence of diffferent parameterss on the stabiility lobe diaggram is descrribed more in n detail in Seection 4.3, The M Milling Stabiliity Prediction Model. 33.4 The Dynamics D of Stabiliity Lobes Bravo et all. [19] showeed that the naatural frequenncy of a thin n-walled work kpiece changes for each pass of o the cutter. An extensionn to this is th hat the shapees and locations oof the stabilitty lobes conttinually channge during a single cutterr pass through th he workpiecce material. As depictedd in Figure 20, they ch hange throughout the milling process as material m removval and cutterr contact variiation influence the natural freequency of th he workpiece . This is particularly noticceable during machining of th hin-walled co omponents si nce the mateerial removed d may constitute a considerablle portion of the starting sstock. The instantaneous cutter c direction aalso defines th he stiffness th he system exp eriences during the cut. Figure 20: The natural frequency f of the t workpiece changes as material is remooved. 22 Budak et al. [20] confirm that workpiece dynamics continuously change due to mass removal and variation of cutter contact. The article proposed an analytical method for modeling varying workpiece dynamics and its effects on process stability. The method is based on a finite element mesh used to obtain the frequency response function of the workpiece. It is updated by using the removed elements along the tool path as defined by the cutter location. The stability diagrams are then generated from the updated frequency response functions. Similarly, Shamoto et al. [21] reaffirms that chatter stability depends on the tool path relative to the dynamically most compliant direction. The article proposes a concept to optimize the tool path to avoid chatter vibrations in machining operations and claims that the optimum tool path can be defined based on given tool geometry and cutting conditions. 3.5 Experimental Modal Analysis By means of structural dynamic tests, the transfer function of an elastic structure may be identified. An impact hammer equipped with a piezoelectric force transducer is used to excite the machining structure. By means of a short impact by the hammer, an impulse is generated and a range of frequencies are excited that contain the natural modes of the system. The resulting vibrations are measured using an accelerometer. The frequency response function is measured and a stability lobe diagram may be extracted. In order to analyze the dynamics of machining systems, the interaction between the structural dynamics and the process dynamics must be analyzed. There is some inaccuracy associated with this method since the impulse is generated while the tool is stationary. The modal analysis is therefore done either on the workpiece/fixture structure or on the tool/machine structure. The results may be combined for a more accurate reading of the total system. Alternatively, if the machining forces and their directions are known, the milling tool may be statically placed against the workpiece exerting similar force magnitudes and directions. Then the hammer impulse can be recorded for the complete system. This may give a better indication of a system in motion than when only doing an impact test on the tool and the workpiece separately. 23 4. Modeling Work 4.1. General Simulation Modeling There are several tools currently available for predicting vibrations, including FEM and modal analysis. The aim has been to simulate or model vibration problems in milling of thin-walled components. Weck et al. [22] who suggested stability models where existing pocketing routines in a CAD/CAM system were corrected using a stability data bank during NC (Numerical Control) tool path generation. Bayly et al. [23] used FEM analysis localizing unstable zones within stable zones. Biermann et al. [24] published a simulation system consisting of an FE model of the workpiece coupled with a geometric milling simulation for computing regenerative workpiece vibrations during five-axis milling. FEM modeling is a relatively complex way to describe the process and requires a high level of understanding of vibration theory. An easy to use stability prediction model would therefore be of great value. A spread sheet with a simple user interface and scroll buttons for input parameters would be relatively easy to use. Therefore, building a prediction model based on a spread sheet would make it accessible to many engineers and operators. This chapter describes how a spread sheet prediction model could be built and applied to an elementary milling setup. 4.2. Determination of the Specific Cutting Force One way to extract the cutting forces of a certain alloy is by determining the specific cutting force, , from turning. This parameter is used to develop the milling prediction stability model in this work. The variation of depends on the workpiece material, tool geometry and coating, and cutting parameters. Also the tool wear influences because of removal of coating and changes in tool geometry during cutting. Feed is the parameter that has the greatest influence on the specific cutting force. The kc curve was extracted for Inconel 718 using a lathe equipped with a piezoelectric force sensor. The cutting speed was 50 m/min and the depth of cut 3.5mm. The cutting forces were measured with the force sensor and recorded into data files. Three measurements were taken and averaged for each 25 value of u uncut chip th hickness and d all measureements were done in ran ndom order. Thee value was w then extrracted for eaach averaged cutting forcce. 34 values of uncut chip thickness weere chosen bbetween 0.00 015 and 0.25mm which equaals the feed per p revolution. The cuurve ( as a function of uncut u chip thickn ness) was theen plotted acccording to Fiigure 21. Using a curve fitting f routine, th he equation foor the curvve was derivedd to be k 1139 9.7h . Eq. 3 m turning Incconel 718. Figure 21: The kc curvee extracted from 26 4. 3. The Milling Stability Prediction Model Figure 22: The geneeral layout off the Milling SStability Prediiction Model. The m milling stabilitty prediction model shoulld be able to predict the shape s and locations of the sttability lobess for a givenn dynamic system s and show s the frequen ncy responsee function. It should alsoo be a tool that can be used for visualizzation duringg class room instruction oor workshops. With this in mind, the miilling stabilityy prediction model was ddeveloped in the form off a spread sheet ggraphics reprresentation, Figure F 22. Itt is a tool th hat allows th he user to extractt a stability loobe diagram and a the frequuency responsse function fo or a set of workpiiece and tool parameters. The stabilityy lobe diagram m, Figure 23,, provides inform mation on how w to choose the t depth of cut for a givven spindle sp peed. The frequen ncy response function is used u to characcterize the dyynamics of th he system. Variable input parrameters inclu ude workpieece width, len ngth, height, density, Young’s Modulus, damping ratio, r numb er of teeth,, and start and exit immersion angles. The parameters are channged by means of scroll bars. b The stabilitty lobe diagrram and thee real and iimaginary paarts of the frequency f response function vary v accordin ngly. The wayy they vary ass input param meters are changeed is depicted d in Table 1. One applicattion of the model m is analyyzing how 27 to compen nsate the stabbility when a system channge occurs. For example, if the overhang increases byy 10%, the model can be used to determine what parameterss need to be changed and d by how muuch in order to t compensate for the changee. The modeel predictionss are based on the proccess described d in Append dix 1, Regenerativve Stability Theory, and Reference [17]. Princip pally, the ch hatter frequency iis scanned arround the nattural frequen cy of the worrkpiece. From m this the real an nd imaginaryy parts of the transfer fuunction are calculated toggether with the crritical depth of o cut and th he spindle speeed. The stab bility lobe diaagram and transfeer function will w change dyynamically toogether with all a parameterrs tied to the userr-changed parrameter. For example, if tthe workpiecce wall thickn ness is changed, th he graphs ch hange accordingly and so do affected parameters p su uch as workpiece stiffness and natural frequ uency. The curren nt model inccludes the ab bility to geneeralize basic workpiece shapes such as rou und and squaare bars. Lateer modeling ddevelopmentts may includ de the ability to suggest cuttin ng parameterss from a know wn location in i a given staability lobe diagraam. m as shown inn the Milling Stability S Pred diction Figure 23:: The stabilityy lobe diagram Model. 28 Table 1: The effectss in the stabillity lobe diagrram due to an n input pparameter. Lo obe Workpiece aliim Effect intersection natural Param frequency meter Wall th hickness Compoonent length Overalll height Densitty N/C Dampiing ratio Number of teeth Young’s Modullus Entry aangle Exit an ngle increase in a variable Workpiece Stiffness + + N/C + - N/C N/C N/C N/C N/C + + N/C N/C N/C N/C The arrrows represeent the direcction in whicch the pointss in the stab bility lobe diagram m move durring an increease of a varriable input parameter. Regarding R workpiiece natural frequency and a stiffnesss, “+” mean ns an increase, “-“ a decreasse, and “N/C C” no changee. The positioon and shapee of the stabiility lobes are parrticularly senssitive to component wall thickness and d wall height, more so than too wall length h. Also when n the numberr of teeth vaaries, the stab bility lobe diagram m displays significant changes. Reegarding thee alim and the lobe intersections, they divide the staability lobe ddiagram into the regions described in Secttion 3.3, Thhe Stability Lobe L Diagram m. The millin ng stability prediction p model is an easy way w to visuaalize how theese regions behave b under varying millingg conditions. 29 5. Exxperimen ntal Worrk 5. 1. Definitions This seection clarifiees some of th he definitionns used in this work. Thee effective exit an ngle ψ is defi fined as the angle a betweeen the rake face f and the exit face, Figure 24. This angle is depend dent on the ggeometry of the cutting tooth t and the toool offset from m the exit face of the workppiece. Figure 24: Effectivee exit angle iss defined in a and b. The zero rake zero exit tool a depicted inn (c) and (d), respectively. and thee zero rake zerro entry tool are The exxit phase is defined as the t part of tthe cut where the chip thickness (cuttin ng load) contiinually decreaases. In proceesses such as down millingg, the exit 31 phase includes the main part of the cutting process, including the in-process region depicted in Figure 5. The zero rake zero exit tool position, on the other hand, has a zero degree effective exit angle and therefore an instantaneous exit phase. 5.2. Tools, Experimentals, and Procedures Two different tools were used; a standard on-the-market tool and a zero rake custom-made tool. The standard tool was a single coated cemented carbide cutting insert mounted on a 25mm diameter three-flute milling cutter. The milling tool had an axial rake angle of 8.0 degrees, a radial rake angle of -7.6 degrees, and a nose radius of rε=0.8mm. The feed was set at fZ=0.08mm/tooth. The zero rake tool was a single coated cemented carbide cutting insert mounted on a 25mm diameter two-flute milling tool. The insert was configured for zero degree radial and axial rake angles. It had a nose radius of rε=0.8mm and the feed was set at fZ=0.08mm/tooth. A three-axis milling machine was utilized for the experiments. The cutting speed was fixed at vc=50m/min corresponding to a spindle speed of n=637rpm. The axial depth of cut was set at ap=0.5mm. There were no cutting or cooling fluids used during any of the experiments. Using a three-component piezoelectric force sensor mounted under the workpiece fixture, the cutting forces were measured and recorded into data files. From these, the force components and the resultant cutting forces were extracted. The data was analyzed and plotted using numerical software. The workpiece was Inconel 718 plate with a cross section of 5mm x 40mm and was mounted in a fixture as shown in Figure 25. The workpiece overhang was varied between 5mm and 40mm. The cutting insert was replaced for each machining run. By altering the offset position between the tool and the workpiece, down milling, zero offset milling, and up milling can be obtained. These three offset positions were chosen for the standard tool. The zero offset position was defined as head-on face milling. The effective exit angles for these positions were 82.4 (down milling), 3.9 (zero offset milling), and -44.5 (up milling) degrees respectively. In addition, two offset positions were chosen for the zero rake tool; a zero entry angle position corresponding to a 23.5-degree effective exit angle, and a zero exit angle position corresponding to a 0-degree effective exit angle. The five offset positions are depicted in Figure 26. 32 n machining ffixture Figure 25: Workpieece installed in Figure 26: Top vieew of the offsett positions usedd in this workk 5. 3. Dyynamics off Stabilityy Lobes - Experimen E nts As is illlustrated in Figure F 27, th he likelihood to find stablee regions is dependent d on how w fast the spin ndle is rotatin ng. The daata for the graaphs were derived from onne pass of a milling m cutterr during a face milling operatiion for the zeero offset geoometry. The workpiece w waas Inconel 718 wiith a cross section of 5mm m x 40mm annd a wall oveerhang of 40mm. The 33 axial depth h of cut was 0.5mm, the spindle speedd 250rpm, an nd the uncutt chip thickness 00.08mm. Three poin nts from the data d file weree analyzed: onne at the begginning, one at a the data, middle, an nd one at the end of the cu ut. From the FFT of the accelerometer a the naturaal frequenciess were found d to be 19990Hz, 2004H Hz, and 202 24Hz, respectivelyy. As noted in Section 3.4, The Dyynamics of Sttability Lobess, the change in natural freqquency is atttributed to material rem moval and cutter c contact varriation. The stability lobee diagrams w were derived for f each value and then overlaaid in the graaphs shown. This T indicatess a change in natural frequ uency by about 1.7% from staart of cut to end e of cut. At lower sppindle speeds, Figure 27 left graph, tthe three curvves show a sp pread making it very difficullt to define a region aboove alim that would stay stable s throughout the cut. Altthough the stability lobe diagram givees some indiccation of the critiical depth off cut when the t system gooes from stab ble to unstab ble, it does not provide mu uch useful information i on how too choose milling m parameterss. At higher spindle speeeds, Figure 27 right graaph, the reggions between n the stability lobbes are markkedly more well-defined w annd, thereforee, the diagram m will give a clearrer indication n on stable reggions above a lim. The stability lobe diaggram, therefore, is significanttly more useeful for deterrmining milling parameteers at high spindle speeds than n at low spindle speeds. Figure 27: 7: Stability Loobe Diagram at the beginnning (1990H Hz), at the middle m (2004Hz), and at the end e (2024Hz) z) of cut for loow spindle speed (left) and d high spindle speeed (right). 34 5.4. Tool-to-Workpiece Position Experiments This part of the research focused on three common offset positions using the standard milling insert, namely down milling, zero offset milling, and up milling. The vibration behavior was investigated during cutter entry, in-process milling, and cutter exit for a thin-walled Inconel 718 component. These offset positions were chosen to exemplify the impact the milling geometry has on the resultant cutting force and on the onset of vibrations. The manner in which the component overhang affects the overall stability of the system depends greatly on the offset position. The resultant forces from the measurements taken during the first three offset positions are shown in Figure 28. For a component wall overhang of 5mm, the depth of cut ap=0.5mm does not cause any instability in the system. As the wall overhang increases, a higher degree of instability is observed, especially for zero offset and up milling. The highest levels of vibrations are seen for the zero offset geometry. Figure 28 also shows that onset of vibrations occurs at different component wall overhang for the three offset geometries. For down milling, vibrations starts at a wall overhang of 25mm, for the zero offset case at a wall overhang of 20mm, and for up milling at a wall overhang of 30mm. These are critical points where the dynamic system changes from chatter-free vibrations to chatter vibrations. The three geometries show significant variations in resultant force amplitudes. Greater forces are observed for zero offset than for down milling or up milling. Up to a component wall overhang of 30mm, down milling and up milling require more or less the same amount of resultant force to achieve cutting. For wall overhangs of 35 mm and above, the down milling plots of the resultant force exhibits minor vibrations, whereas the up milling plots exhibits more severe vibrations. 35 Figure 28 : Down milllling (left), zeero offset millling (middle) e), and up milling m (from bottom to top) wall height h (right) at 55, 10, 15, 20,, 25, 30, 35, and 40mm (f overhang. Figure 28 indicates thaat the offset position p has ggreater influeence on vibraations than the am mount of waall height oveerhang of thee workpiece. This is espeecially true for sm mall amounts of workpiecee overhang. FFor large amoounts of overhang, both the zeero offset and d the up millling geometriies show sign nificant vibrations. The plot su uggests that the t down miilling positionn is preferablle in order to keep vibrations to a minimu um. Because of the dynaamic circumsstances durin ng the cutting prrocess, it is not obvious that chatteer vibrations can be preecisely predicted using existin ng methods. As noted bby Surmann et al. [25], even chatter-freee milling proocesses can produce p a higgh surface loocation error since chatter-freee does not necessarily mean vibrattion-free. Su uch circumsttances include fleexible deform mations of thee workpiece before cuttin ng takes placee and burr formaation during cutter exit. The T results oof this study can be used as an indicator fo for how to avooid chatter viibrations. For down m milling, maxximum force is i observed dduring cutter entry as the cutter c is travelingg in a relativeely flexible direction d of thhe workpiecee. The cutterr exit, on the oth her hand, occcurs as the cu utter is travelling in the sttiffest direction of 36 the woorkpiece. Forr up milling,, we have thhe opposite scenario. s For the zero offset ggeometry, the cutter is traveling in thhe predominaantly flexible direction througghout the cu ut, includingg entry and exit. From Figure 28 it i can be concluded that dow wn milling (ccharacterizedd by a smooth cutter exit) exhibits significcantly less vibbrations than n up milling (characterizeed by a smoo oth cutter entry) for same com mponent overrhang. A genneralization off this is that it is more importtant to have a smooth cutter c exit thhan a smooth cutter en ntry. This supporrts the findin ngs of Peckelharing [12,113,14] mentioned in Secction 1.3, Review w of Milling Vibration V Reseaarch History. In ordeer to determine whether the vibrationns displayed were w forced vibrations v or chattter vibration ns, the force measurement m ts from a ran ndom cut was overlaid with th he force meassurements fro om the subseequent cut. Iff the two are in phase, forced vibrations arre inherent in n the processs, whereas if they are out of phase, chatterr vibrations prevail. p Figuree 29 shows exxamples of th hese two casess. Figure 29: Two subsequent forcee measuremennts: at left, in n phase depictting forced ight, out of phhase depicting chatter vibrattions. vibratioons, and at rig If the ddepth of cut ap is held con nstant, the crritical depth of cut alim deecreases as the component ovverhang increeases, Figure 30. For a dynamic system with constan nt depth of cut c and increeasing compoonent overhaang, chatter vibrations v will eveentually be in ntroduced. Figure 30: The effeects on alim as the wall heighht overhang iss increased froom 35mm mer impact tessts. (left) too 40mm (righht). The diagraams are deriveed from hamm 37 5.5. Cutting Geometry Sensitivity 5.5.1 Approach In this part of the research, a zero rake milling insert is used in addition to the standard cutting insert described above. Besides the three standard tool positions, two offset positions are defined as the zero rake zero entry position corresponding to an effective exit angle of 23.5 degrees and the zero rake zero exit position at an effective exit angle of zero degree. The resultant force and its components are presented for each of the five cases. At the end of the section, the relationship between the force profile and the exit angle is analyzed. 38 5. 5.2 Dow wn Millingg Position Duringg down milliing, Figure 31, the cutterr exit phase iss mainly or altogether a in the least flexible direction. The T exit phas e starts from m a relatively low force level aand exhibits a long smoo oth decline. IIn addition, the chip thickness is reduced throughou ut the cut. According tto the definiition of Section 5.1, Definittions, the exitt phase thus includes thee entire cuttin ng process affter cutter entry. Because of a smooth cutter exit, vibraations are con nsiderably lesss than for the oth her offset possitions studieed and vibratiions are in efffect absent. Since the cutter exits in the least flexiblee direction off the workpiiece, the cuttting force requireed is also sign nificantly lesss. There is nno after-ringin ng visible in the force profile except from free vibration ns. Figure 31: Force prrofile during down d millingg at 40mm ovverhang. Resulltant force and off ffset position (top) (t and forcee components (bottom). Thhe effective exxit angle is 82.4 de degrees. 39 5.5.3 Zero Rake R Zero Exit E Positionn For the Zeero rake zeroo exit case, the t exit phas e starts from m a relatively high force level and the exiit takes placee abruptly. A hump (a su udden increaase in force and vvibrations) iss clearly visib ble during th e post-exit phase p althouggh the vibrations rapidly decaay at the end d of the hum mp. Significaant vibrationss and forces are ggenerated beccause the cuttter exits in thhe most flexib ble direction of o the workpiece,, Figure 32. It exhibits basically no ex exit phase and d has the shortest exit of the offset positioons considereed in this stuudy. The resu ultant force profile p shows a hu ump behaviorr which is also clearly visibble in the y and a z compon nents. It is, howevver, completeely absent in the x componnent. Figure 322: Force profi file for the zeero rake zeroo exit tool att 40mm overrhang. ffset position (top) and foorce components (bottom). The Resultant fforce and off effective exiit angle is zeroo degree. 40 5. 5.4 Zerro Rake Zerro Entry Poosition By inccreasing the exit angle byy 23.5 degreees, the vibrrations and the t forces decreasse to the leveel shown in Figure 33 (zeero rake zeroo entry positiion). The exit ph hase starts froom a low to moderate m leveel. Less vibrattion is observved at the same ttime as less force f is required for the ccutting proceess. Also, the post-exit after-riinging almostt completely disappears. Figure 33: Force profile p for thee zero rake zzero entry toool at 40mm overhang. Resultaant force andd offset positiion (top) andd force compponents (botttom). The effectivve exit angle iss 23.5 degrees. 41 5.5.5 Up Mi lling Position Up millingg, Figure 34 exhibits the opposite o behhavior from down d milling. The exit phase starts from a moderately high forcee level and the t chip thicckness increases th hroughout th he cut. The cutter c directiion changes to t an all the more flexible diirection of the workpiiece and, ttherefore, viibrations inccrease throughout the cut. Cu utting forces are also muchh larger than n for down milling m since the ccutter exits paartly in the most m flexible direction. It can be seen from the graphss that the y and z com mponents sttart to dom minate over the t x component after the cu utter exit ph hase. The afteer-ringing ob bserved, therefore, mainly stem ms from the cutting forcee in these twoo directions. A hump is clearly c seen in thee resultant foorce graph bu ut is not as nnoticeable in the graphs fo or the force compponents. Altthough up milling m causess more vibraations than down d milling, it exhibits less vibration v thaan the zero offfset and the zero rake zero exit positions. Up millingg causes sevvere burr foormation du uring cutter exit. Characterisstic only for up milling is i that the viibrations con ntinue on thrrough the post exxit phase. Figure 34:: Force profile during up milling at 40m mm overhang. Resultant R forcce and offset positioon (top) and force f componeents (bottom).. The effectivee exit angle is -44.5 degrees. 42 5. 5.6 Zerro Offset Po osition The strrongest vibraations are seen n for the zeroo offset posittion with thee standard tool, F Figure 35. The T exit phasse starts at hhigh force leevels and theen decays rapidlyy down to a minimum after which a large-sized d hump appeears. The vibratioons continuee through thee hump but disappear almost compleetely after the hump. It generaates after-ringging similar iin nature to the zero rakee zero exit positioon. The cutterr exits near th he most flexibble direction of the workp piece and, therefoore, strong viibrations are present duriing and afterr cutter exit. Also the cuttingg forces are considerablyy larger throoughout the cut compareed to up millingg and down milling. In this tool possition, the ch hip thicknesss remains relativeely constant throughout the t cut. The after-ringingg is characterrized by a large-siized hump having h approxximately the same span ass the tool engagement region of the force profile. This hump is barrely noticeablle in the x co omponent but cleearly visible in n the y and z components . Figure 35: Force prrofile during zero offset miilling at 40m mm overhang. Resultant nd force compponents (botto tom). The effe fective exit force aand offset posiition (top) an angle iss 3.9 degrees. 43 5.5.7 Force Profile P versu us Exit Anggle The force profile behaavior in the region r follow wing the cuttter exit (the afterringing and the hump)) may be cau used by severral factors. In n addition to o free vibrations and built-up p edge, the beeat phenomennon, Figure 36, may influ uence the force bbehavior. In the t beat phen nomenon, thee response is composed of preexisting ch hatter vibratioons at frequency ωc and ffree vibration n of the work kpiece at frequenccy ωn. The reesponse could d be written aas: sin sin 2 cos sin Eq. 4 Here the ccosine factorr is an envelo ope for the ssine wave. Iff the two staarting frequenciess are close to each other, the t frequencyy of the cosin ne on the righ ht side is too slow w to be percceived as a pitch. p Insteadd it is perceiived as a perriodic variation oof the sine in the expresssion with a frequency of o , i.ee. the average of the two freqquencies. Thee successive vvalues of maxxima and miinima form a waave whose freequency equaals the differeence between n the two staarting frequenciess. b producedd (1400Hz vibbration with 58Hz 5 beat) Figure 36: Example of beat At an effecctive exit anglle of 82.4 deggrees (down m milling positiion), the optimum vibration-ffree region is found, Figu ure 37. This is no doubt what experieenced machine ooperators woould expect. However, seeemingly favvorable ampllitude versus effecctive exit anggle scenario iss seen at 23.55 degrees, corrresponding to t the zero rake zero entry position. p This merits furtther studies to investigatte the sensitivity of exit angle choice and to t what degreee a model coould be developed to predictt vibration in i the proccess. In ordeer to predicct vibration risk, geometricaal factors havve to be tak ken into acccount. The model shoulld be capable off handling th hese phenomeena. To furthher develop this, cutter micro m geometry sshould be con nsidered. Forr the five offs fset positions studied, exceept in 44 the casse of down milling, m a hump is seen inn the force profile p after the t cutter exit. T This hump is most noticeaable for the zeero offset possition for thee standard tool an nd for the zerro exit positio on for the zeero rake tool. When usingg the zero rake toool, it was nooted that a change c from zero to 23.5 5 degrees in exit e angle significcantly reducees the size off the hump aand the amou unt of vibratiions. It is notewoorthy that the t hump amplitude a foollows the maximum amplitude a througghout the eff ffective exit angle a range. The conclu usion is that the exit directioon should bee chosen in the least flexibble direction n possible and d that the effectivve exit angle should be as large as posssible (so that the tool-to-w workpiece positioon is as close to t the down milling m positiion as possiblle). Figure 37: Maxim mum amplitudde during exxit phase andd hump ampplitude vs. effectivve exit angle After ccutter exit, th he workpiece vibrates at a certain freq quency and am mplitude. These vvibrations deepend on facttors such as thhe exit angle and force levvel at start of exitt phase. As the t effective exit angle neears zero deggrees, these vibrations v become stronger. In ndependentlyy of whether tthe standard cutter or the zero rake cutter is used, the most m favorable cutter exitt is in the leaast flexible dirrection of the woorkpiece. By choosing c an offset o positio n such as dow wn milling, vibrations v are avooided almost altogether, evven at relativvely large worrkpiece overh hang. The force reequired for th he cut is also reduced. In addition, burrr formation is kept to a minimum. It wass determined that exit forrces should be avoided in the most flexiblee direction off the work piiece and thatt down millin ng is more ro obust and less proone to generaate vibrationss than up millling and zeroo offset millin ng. In the case of down millling, it prod duces a cleann surface with w virtually no burr ormation. formattion. Up millling, on the other hand,, produces severe burr fo Thin-w walled compoonents are paarticularly proone to burr formation f at high exit 45 forces and large chip thickness since the material tends to be plastically deformed rather than cut. Burr formation mechanisms are explained in [26] and reviews on burr minimization techniques are found in [27]. 46 6. Conclusions 6.1. Research Questions - an Analysis The research questions raised in the introduction give rise to queries much greater than the scope for this thesis. The answers are therefore given in regards the limitations of the scope. R1. How critical is the choice of offset between tool and workpiece during milling? The results from the experiments indicate that by changing the offset location of the tool in relation to the workpiece, the amount of vibrations in the system may change significantly. This is particularly noticeable when the offset location is changed from zero offset to down milling. Generally, for small component overhangs, the vibration risk is limited. However, as the component overhang increases, the choice of offset position becomes crucial. Down milling is more robust and less prone to generate vibrations than up milling and the zero offset geometry. It also displays the smallest force amplitude during the exit phase and no hump in the post exit phase. In addition, it produces the least amount of burr formation. R2. What effects do the cutter entry and cutter exit have on system vibrations? Down milling (characterized by a smooth cutter exit) exhibits significantly less vibration than up milling (characterized by a smooth cutter entry) for same component overhang. A generalization of this is that it is more important to have a smooth cutter exit than a smooth cutter entry in order to avoid vibrations. Entry and exit forces should be avoided in the most flexible direction of the workpiece. R3. How does the effective exit angle affect vibrations during and after cutter exit? The tool position should be chosen so that the cutter exits in the least flexible direction possible for the workpiece. As was observed for the standard cutting tool, down milling clearly is more favorable than up milling regarding 47 vibrations, force requirements, and burr formation. The effective exit angle, therefore, should be chosen so that the cutter exits in the least flexible direction possible at the same time as the tool-to-workpiece position is as close to the down milling position as possible. The region around the zero degree effective exit angle is very sensitive. Even small changes in the processing conditions, such as the effective exit angle, may result in substantial changes in vibrations. R4. What is the dynamic effect on the stability lobe diagram as the cutter moves through the workpiece? The shapes and locations of stability lobes change as the cutter passes through the workpiece material. They change throughout the milling process as material removal and cutter contact variation influence the natural frequency of the workpiece, Figure 19. This is particularly noticeable during machining of thin-walled components since the material removed may constitute a considerable portion of the starting stock. The instantaneous cutter direction also defines the stiffness the system experiences during the cut. The usefulness of the stability lobe diagram is dependent on how fast the spindle rotates, Figure 27. Under the same milling situation, it is easier to define stable regions above alim during milling at high spindle speeds than at low spindle speeds. 6.2. Vibration Prediction Modeling In addition to these research questions, this work has discussed some aspects of modeling. In order to better predict the vibration risk for a component during early stages in the process planning phase, an easy to use milling stability prediction model has been developed. It predicts the shape and locations of the stability lobes and the frequency response functions for a given set of milling parameters. It also shows dynamic changes in parameters such as cutting forces, chip thickness, and workpiece natural frequency and stiffness. It is a tool that can be used for visualization during class room instruction or workshops. It is in the form of a spread sheet graphics representation. The workpiece geometry discussed within this thesis has been a rectangular block, but the model is generalizable for other basic workpiece shapes such as round and square bars. 48 6.3. Recommendations to Operators Process engineers and operators may consider that vibration risks are reduced in the following cases: A smooth cutter exit Small component overhangs Offset position close to the down milling position Cutter exit in the least flexible direction possible for the workpiece In addition, the following needs to be considered: The stability changes during cutter pass Considering these guidelines will assure a more robust process that is less prone to generate vibrations. Large force amplitudes during the exit phase would be avoided and the post-exit hump eliminated. In addition, burr formation would be minimized. 6.4. The Diffuser Case – a Recap In Section 1.2, Practical Example – Jet Engine Diffuser Case, it was explained that some milling difficulties had been experienced for this type of component. From what has been observed in this work, the following milling strategies could be recommended: When milling around and into the holes of the fuel injector mounting clamps or flange bosses, down milling should be used. In addition to tool-to-workpiece position, also tool diameter and pitch have to be taken into account. Although it is preferable for the cutter entry to be smooth, it is essential for the cutter exit to be as smooth as possible. This will inhibit vibrations in the system and call for a cleaner surface condition and tighter dimensional accuracy. It would also reduce burr formation and subsequent surface reworking. The axial rake angle should be chosen in such a way as to provide a smooth cutting surface and a limited heat input into the material. The lead angle can be increased so that the cutter exit is smooth and adequate amount of material is removed for each cut. 49 7. Future Work For a more complete picture of the vibration issues encountered during milling of thin-walled components, some specific aspects have to be analyzed in-depth. Some suggestions for future work are provided here. The Licentiate-andBeyond Research Circle in Figure 38 depicts some avenues for continued research. As opposed to a real-life traffic circle where only one choice can be done once inside the circle, the research circle allows for some simultaneous work. The milling stability prediction model, for example, can be developed in parallel with the effective exit angle sensitivity research. The sensitivity of the effective exit angle needs to be considered for a complete realization of the behavior of the force and vibration response. This requires an analysis of the micro geometry of the cutting process. Also the other regions of the cutting process needs to be considered, including entry, inprocess, and post-process. This will give insight into what options the process planners and operators have when choosing machining parameters According to Figure 37, there is a region from about -25 to +25 degrees effective exit angle where vibrations are especially noticeable in the machining system. This region could be examined with smaller angular increments using a variety of cutting tools, on both macro and micro levels. This could be done by utilizing a turning holder mounted in a milling holder. Inserts with various macro and micro geometries could then be used for a complete study of the vibration behavior. On the macro level, lead angle could be varied, and on micro level, chip breaker and edge radius could be varied. As the lead angle is increased, the exit path and exit time are extended. This, in turn, will influence the cutting forces and vibrations on the system. This would determine which areas are unsuitable for milling and which areas are feasible although not optimum. The dynamics of the stability lobe diagram could be investigated further taking into account the wall thickness of the workpiece material. Also the direction of vibrations developed during the cut could be taken into account. Changes in material properties such as Young’s Modulus, density, and the damping ratio affect the natural frequency of the workpiece and this, in turn, 51 will affect the shapes of the stability lobes. Also how the stability lobes change with spindle speed as these properties change would be an area of interest. The variation in chip thickness as a function of offset could be taken into account when analyzing the dynamics of the machining system (see Figure 26). Also, the chip thickness varies together with the component stiffness during the cut as depicted in Appendix 1, Figure 1. Damping layers may be applied at various places of the machining structure, such as the tool/insert, a rotating force sensor/tool, and the tool holder/tool interfaces. Such layers would play a critical role in structural dynamics and resonant amplitude control thus enhancing durability, life cycle and cost reduction. Laser Vibrometry could be used to determine the stability of a machining system in real time during the actual machining. The cutting parameters are then varied instantaneously through a control system as the dynamics of the system changes. A comparison with Euler buckling could be done to determine the maximum overhang a workpiece may have without risk for chatter vibrations. The study would include a cutting parameter matrix for a complete analysis of the phenomenon. The Milling Stability Prediction Model presented in this thesis does not suggest cutting parameters from a known location in a given stability lobe diagram. The development of this feature will make the model highly useful during component redesign. The model may also be expanded to include more advanced geometries such as workpieces with hollows. It could also include the dynamics of the stability lobes as noted in Sections 3.3, The Dynamics of Stability Lobes and Section 5.3, Dynamics of Stability Lobes Experiments. By reducing system vibrations, ceramic inserts may be used without failing and the noise may be reduced to acceptable levels in the workshop. Ceramic inserts can be said to be a common goal for all the various avenues out of the research circle. 52 Figure 38: The Liceentiate-and-B Beyond Researcch Circle. 53 8. References 1. F.W. Taylor. On the Art of Cutting Metals. ASME 1907. 2. S.A. Tobias and W. Fishwick, Theory of Regenerative Machine Tool Chatter. The Engineer, London, 1958. 3. S.A. Tobias, Machine Tool Vibration. Blackie and Sons Ltd., 1965. 4. F. Koenigsberger and J. Tlusty, Machine Tool Structures Vol. 1: Stability Against Chatter. Pergamon Press, 1967. 5. R. Shridhar, R.E. Hohn and G.W. Long, A Stability Algorithm for the General Milling Process. Contribution to Machine Tool Chatter Research-7. Transactions of the ASME Journal of Engineering for Industry 90:330-334, 1968. 6. I. Minis, R. Yanushevsky, R. Tembo and R. Hocken, Analysis of Linear and Nonlinear Chatter in Milling. Annals of the CIRP 39:459-462, 1990. 7. Y. Altintas and E. Budak, Analytical Prediction of Stability Lobes in Milling, Annals of the CIRP, Vol. 44, No. 1, pp. 357-362, 1995. 8. E. Budak and Y. Altintas, Analytical Prediction of Chatter Stability in Milling – Part I: General Formulation. Journal of Dynamic Systems, Measurement and Control 120:22-30, 1998. 9. E. Budak and Y. Altintas, Analytical Prediction of Chatter Stability in Milling – Part II: Application of the General Formulation to Common Milling Systems. Journal of Dynamic Systems, Measurement and Control 120:31-36, 1998. 10. G. Quintana and J. Ciurana, Chatter in Machining Processes: A Review. International Journal of Machine Tools and Manufacture. International Journal of Machine tools and manufacture, Vol 51, No 5 pp 363–376, 2011. 55 11. G. Quintana, J. Ciurana, I. Ferrer and C. Rodriguez. Sound Mapping for Identification of Stability Lobe Diagrams in Milling Processes. International Journal of Machine Tools and Manufacture, Vol 49, 203-211, 2009. 12. A.J. Pekelharing. The Exit Failure in Interrupted Cutting. Annals of the CIRP, Vol. 27, No. 1, pp. 5-10, 1978. 13. A.J. Pekelharing. The Exit Failure of Cemented Carbide Face Milling Cutters, Part I – Fundamentals and Phenomenae. Annals of the CIRP, Vol. 33, No. 1, pp. 47-50, 1984. 14. A.J. Pekelharing. The Exit Failure of Cemented Carbide Face Milling Cutters, Part II – Testing of Commercial Cutters. Annals of the CIRP, Vol. 33, No. 1, pp. 51-54, 1984. 15. E.L. Rivin. Handbook on Stiffness & Damping in Mechanical Design. ASME 2010. 16. C.M. Nicolescu, V. Chiroiu, T. Nadea & O. Marin. On the Modeling the Damping Across the Length Scales by using Tzitzeica Surfaces. 17. Y. Altintas. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge University Press, 2000. 18. J. Yue. Creating a Stability Lobe Diagram. Proceedings of the 2006 IJME Intertech Conference. 19. U. Bravo, O. Altuzarra, L.N. Lopez de la Calle, J.A. Sanchez & F.J. Campa. Stability Limits of Milling Considering the Flexibility of the Workpiece and the Machine. International Journal of Machine Tools& Manufacture, 45, 1669-1680, 2005. 56 20. E. Budak, L.T. Tunc, S. Alan & N. Özgüven. Prediction of Workpiece Dynamics and its Effects on Chatter Stability in Milling. Annals of the CIRP, Vol. 61, pp. 339-342, 2012. 21. E. Shamoto, S. Fujimaki, B. Sencer, N. Suzuki, T. Kato & R. Hino. A Novel Tool Path-Posture Optimization Concept to Avoid Chatter Vibration in Machining - Proposed Concept and its Verification in Turning. Annals of the CIRP, Vol. 61, pp. 331334, 2012. 22. M. Weck, Y. Altintas & C. Beer. CAD Assisted Chatter Free NC Tool Path Generation in Milling. International Journal of Machine Tool Design in Research. 1994, 34(6):879-89. 23. P.V. Bayly, J.E. Halley, B.P. Mann, M.A. Davies. Stability of Interrupted Cutting by Temporal Finite Element Analysis. Journal of Manufacturing Science and Engineering. 2003, 125:220-225. 24. D. Biermann, P. Kerstinga & T. Surmann. A general approach to simulating workpiece vibrations during five-axis milling of turbine blades. Annals of the CIRP, Vol. 59, pp. 125-128, 2010. 25. T. Surmann & D. Biermann. The Effect of Tool Vibrations on the Flank Surface Created by Peripheral Milling. Annals of the CIRP, Vol. 57, pp. 375-378, 2008. 26. M.C. Avila and D.A. Dornfeld. On the Face Milling Burr Formation Mechanism and Minimization Strategies at High Tool Engagement. Consortium on Deburring and Edge Finishing, Laboratory for Manufacturing and Sustainability, UC Berkeley, 2004. 27. S. Tripathi and D.A. Dornfeld. Review of Geometric Solutions for Milling Burr Prediction and Minimization. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture April 1, 2006 220: 459-466. 57 Appendix 1 Regeneration Stability Theory This information is based on the theory presented in Y. Altintas. Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design. Cambridge University Press, 2000. The general dynamic chip thickness is described by: Τ t Τ Eq. 1 where y(t) is the modulation of the surface of the workpiece during the current pass of the milling cutter at time t and y(t-T) is the modulation of the surface of the workpiece during the previous pass at one spindle revolution period (T) before t. Τ Eq. 2 [mm] is the intended chip thickness (equal to the feed rate of the machine) [N/mm2] is the cutting coefficient in the feed direction [kg] is the mass is the damping coefficient [N/m] is the stiffness [mm] is the depth of cut The following Laplace definitions will prove useful: Eq. 3 0 1 Eq. 4 0 0 Eq. 5 Τ Eq. 6 By definition, therefore, the dynamic chip thickness in the Laplace domain becomes Eq. 7 or 1 Eq. 8 The dynamic cutting force in the Laplace domain thus becomes Eq. 9 Define Φ as the transfer function of the workpiece structure (single degree of freedom). The current vibration is Φ Eq. 10 Φ Eq. 11 0 0 0 Eq. 12 Set the initial conditions 0 0 0 Eq. 13 ⟹ When the damping is constant at Eq. 14 0, the system natural frequency is Eq. 15 or Eq. 16 2 Let the damping ratio be ζ Eq. 17 Generally for mechanical structural systems, ζ is less than 1. The following is then valid: 2 2 2 ⟹ Eq. 18 Φ 2 Eq. 19 The transfer function of the workpiece structure becomes Φ Eq. 20 ∴ Φ Eq. 21 Combining Eq. 8 and Eq. 11 gives: 1 1 1 Φ Φ Eq. 22 Eq. 23 Eq. 24 3 The stability of the closed-loop transfer function is determined by the roots (s) of the characteristic equation, i.e.: 1 1 Φ 0 Eq. 25 Let the root of the characteristic equation be Eq. 26 If the real part 0, the time domain solution will have an exponential term with positive power, i.e. | | . The chatter vibrations will grow indefinitely and the system will be unstable. If the real part 0, the time domain solution will have an exponential term with negative power, i.e. | | . The vibrations will be suppressed by time and the system will be stable with chatter vibration free vibrations. If the real part 0, then and the system is critically stable. The workpiece oscillates with constant vibration amplitude at chatter frequency . The characteristic equation of the dynamic cutting process has additional terms beyond the structures transfer function. Therefore, the chatter vibration frequency does not equal the natural frequency of the structure. However, the chatter vibration frequency is still close to the natural mode of the structure. ⟹1 1 Φ 0 Eq. 27 is the maximum axial depth of cut for chatter vibration-free where machining. The transfer function may be divided into real and imaginary parts, i.e. Φ Eq. 28 This is not to be confused with the dynamic chip thickness in the Laplace domain, H(s). Generally: Eq. 29 Eq. 30 4 Therefore, 1 1 Τ Τ Τ Τ 1 Eq. 31 0 Τ Τ Eq. 32 Τ Τ 0 Eq. 33 Collecting all real and imaginary terms separate gives: 1 1 Τ Τ 1 Τ Τ 0 Eq. 34 Both the real and imaginary parts must be equal to zero. For imaginary part equal to zero: Τ 1 Τ Τ Τ 0 Eq. 35 1 Eq. 36 tan Eq. 37 where ψ is the phase shift of the structure’s transfer function. Using the trigonometric identities cos 2 cos sin 2 2 sin cos and substituting 2 Τ sin Eq. 38 Eq. 39 Τ, we get cos Τ⁄2 sin 5 Τ⁄2 Eq. 40 sin Τ Τ⁄2 cos 2 sin ⁄ tan Τ⁄2 Eq. 41 ⁄ ⁄ Eq. 42 ⁄ From cos sin cos Τ⁄2 1 Eq. 43 it becomes 1 Τ⁄2 sin Eq. 44 Inserting into Eq. 42 yields: tan 2 sin sin Τ⁄2 cos Τ⁄2 sin Τ⁄2 Τ⁄2 Τ⁄2 cos Τ⁄2 Τ⁄2 2sin 2 sin Τ⁄2 Τ⁄2 cos sin cot Τ⁄2 Eq. 45 Using the trigonometric identities cot tan Eq. 46 tan tan Eq. 47 tan tan Eq. 48 6 It becomes tan tan tan ⁄2 2 ⁄2 2 ⁄2 tan tan ⁄2 tan ⁄2 ⁄2 2 2 Eq. 49 Eq. 50 3 Eq. 51 tan Eq. 52 The spindle speed n and chatter vibration frequency have a relationship that affects the dynamic chip thickness. Assume that the chatter vibration frequency is or . The number of vibration waves left on the surface of the workpiece is: Τ Eq. 53 is the fractional wave generated. The is the integer number of waves and angle represents the phase angle between the inner and outer modulations. For 0 or 1 the chip thickness will remain constant despite the presence of vibrations. For other values of , the chip thickness changes continuously. 2 Τ 2π Eq. 54 2π Eq. 55 7 tan ⁄2 tan Eq. 56 ⁄2 3 2 2 2 3 3 Eq. 57 Eq. 58 2 2 Eq. 59 2 Eq. 60 The spindle period is Τ and the spindle speed By setting the real part of the characteristic equation to zero, we may now derive the critical axial depth of cut: 1 1 Τ Τ 0 Eq. 61 or Eq. 62 ⁄ From Eq. 61 we have that ⁄ The right hand side denominator thus becomes 1 Τ Τ Τ Τ 1 1 Τ sin 1 Τ Τ 1 1 Τ Τ sin 1 Τ Τ 8 1 2 cos 1 Τ cos 1 Τ Τ 2 cos 1 Τ 1 Τ 2 1 1 2 sin Τ Τ Τ Eq. 63 Therefore: Eq. 64 must be a positive number. The solution Because it is a physical quantity, is therefore only valid for negative values of the real part of the transfer function and chatter vibrations may occur at any frequency satisfying is chosen at the minimum value of , no this condition. When chatter is generated no matter spindle speed. The harder the work material, the larger the cutting constant , with a reduction in the axial depth of cut as a consequence. Another factor reducing the axial depth of cut is the flexibility in the machine tool or work piece structure. This also reduces the productivity. By means of the following five steps, we may plot the stability lobe diagram: For known transfer function Φ of the structure at cutting point and Kf cutting constant, 1. Select a chatter frequency at the negative real part of the transfer function. 2. Calculate the phase angle of the transfer function at from tan (Eq. 37) 9 3. Calculate the critical depth of cut from (Eq. 64) 4. Calculate the spindle speed for each stability lobe number k=0,1,2…. Eq. 65 From Τ Eq. 66 5. Repeat the procedure by scanning the chatter frequencies around the natural frequency of the structure 10 Dynaamic Milling Modeel Figure 1: Regenerattion Assume two orthoggonal Degreess of Freedom , N teeth, and zero degreee helix angle D Define dynam mic displacem ments as x andd y where x is i the feed dirrection and y iis the normal direction. A coorrdinate transfo formation in the t radial (chhip thickness)) direction givves: sin cos Eq. 67 is th he instantaneeous angular immersion i off tooth j. If Ω is the spindle angular a speed d in rad/s, theen Ω Eq. 68 The resulting chip thickness t sin Eq. 69 , 11 is the feed rate per tooth and cutter. sin , is the static part and Define , is the dynamic displacements of the is the dynamic part. , as a step function 1, 0, Since sin Eq. 70 is staic, it does not contribute to the dynamic chip load. sin cos Eq. 71 , sin cos sin sin Δ sin , and structure. , cos cos Δ cos Eq. 72 represent the dynamic displacements of the cutter Define as the tangential and radial cutting forces acting on tooth j. They are proportional to the axial depth of cut and to the chip thickness . Eq. 73 Eq. 74 and are constants. 12 Resolving cutting forces into x,y directions gives cos sin sin cos Eq. 75 The cutting forces contributed by all teeth will be ∑ Eq. 76 ∑ , where is the pitch angle. cos sin cos Δ sin Δ cos Δ sin Δ sin Δ cos sin cos sin cos Δ cos Δ cos Δ sin cos sin sin Δ cos sin sin Δ cos Eq. 77 Using sin /2 cos /2 Eq. 78 13 With and sin /2 sin cos 2 sin cos , it yields: sin 1 sin 2 cos cos cos 2 cos 2 Eq. 79 1 2 cos 2 cos 2 1 cos 2 2 2 cos cos sin sin sin 2 Eq. 80 sin cos sin Δ sin cos Δ cos Δ sin Δ sin Δ Δ 2 cos 2 sin 1 1 2 1 sin sin Δ cos Δ cos sin Δ cos sin sin cos Δ cos sin cos cos cos sin cos Δ cos sin Δ cos cos sin cos sin 1 1 cos 2 2 1 1 2 1 cos 2 2 2 cos 2 14 cos cos sin Eq. 81 sin sin sin 2 Eq. 82 cos sin cos cos 1 2 cos 2 sin sin 1 Eq. 83 sin 2 1 cos 2 Δ sin 2 Δ 1 cos 2 yields Δ cos 2 cos 2 cos 2 sin 2 and 2 cos 2 1 sin 2 2 Inserting into 2 1 Eq. 84 1 cos 2 sin 2 Δy sin 2 cos 2 Eq. 85 Define time-varying dimensional dynamic milling force coefficients: ∑ sin 2 ∑ 1 ∑ 1 ∑ sin 2 1 cos 2 cos 2 sin 2 cos 1 15 Eq. 86 Eq. 87 sin Eq. 88 cos 2 Eq. 89 Resulting expressions in matrix form: Δ Δ Eq. 90 The angular position of parameters changes with time and angular velocity. In time domain in matrix form: Δ Eq. 91 For milling, the direction of the force is not constant but varies with time. is periodic at tooth passing frequency Ω 2 / or tooth period . Eq. 92 Expansion into Fourier series gives: ∑ Eq. 93 Eq. 94 is the number of harmonics of the tooth passing frequency . depends on the immersion conditions and on the number of teeth. For the most simplistic approximation, 0. Eq. 95 16 is valid only between the entry and exit angle angles, i.e., where 1. Also, Ω Ω Eq. 96 at becomes equal to the average value of 0 . Eq. 97 are integrated functions defined as: cos 2 2 sin 2 2 sin 2 2 cos 2 2 sin 2 Eq. 98 cos 2 Eq. 99 cos 2 Eq. 100 sin 2 Average directional factors are dependent on on width of cut , . Eq. 101 (redial cutting constant) and The dynamic milling expression: Δ (Eq. 92) is reduced to: Δ Eq. 102 17 is a directional cutting coefficient matrix (time invariant but immersion dependent). Average cutting force per tooth period is independent of the helix angle. Therefore, is valid also for helical end mills. The transfer function matrix: and and and Eq. 103 are the direct transfer functions in the x and y direction are the cross transfer functions. is the vibration vector at the present time at previous tooth period . and In frequency domain using harmonic functions, the vibrations at the chatter frequency will be: Eq. 104 Eq. 105 Substituting Δ gives Δ 1 Eq. 106 is the phase delay between the vibrations at successive tooth periods T. 18 Substituting Δ into the dynamic milling equation Δ (Eq. 103) 1 Eq. 107 yields This has a non-trivial solution if its determinant is zero: 1 0 Eq. 108 This is the characteristic equation of the closed-loop dynamic milling system. To simplify, define the oriented transfer function matrix: Eq. 109 The eigenvalue for the characteristic equation is: Λ 1 Eq. 110 Resulting characteristic equation becomes: Λ 0 Eq. 111 Its eigenvalue can be solved for a given chatter frequency , static cutting , , factors , radial immersion , and transfer function of the structure. 19 Consider two orthogonal degrees of freedom in the feed (x) and normal (y) 0 . directions Then the characteristic equation becomes a quadratic function. Λ Λ 1 0 Eq. 112 Eq. 113 Eq. 114 The eigenvalue becomes: 4 Λ Eq. 115 For the plane of cut (x,y), the characteristic equation is quadratic. The transfer functions are complex: Λ Λ Λ Eq. 116 cos Substituting the eigenvalue and Λ 1 sin into gives the critical depth of cut at chatter frequency : Eq. 117 20 is a real number, the imaginary part is zero. Since Λ 1 Λ sin cos 0 Eq. 118 into the real part of Substituting , the final expression for chatter free axial depth of cut is 1 Eq. 119 / tan / tan /2 /2 Eq. 120 tan is the phase shift of the eigenvalue. 2 is the phase shift between inner and outer modulations. 2 where is an integer number of full vibration waves (lobes) imprinted on cut arc. 2 Eq. 121 The spindle speed is given by Eq. 122 The transfer functions are identified and the dynamic cutting coefficients are evaluated. Then, the stability lobes are calculated as follows: i. ii. Select a chatter frequency from transfer functions around a dominant mode. Solve the eigenvalue equation Λ Λ 1 0 (Eq. 113) 21 iii. Calculate the critical depth of cut 1 iv. (Eq. 120) Calculate the spindle speed for each stability lobe, k=0, 1, 2, …. (Eq. 123) v. Repeat the procedure by scanning the chatter frequencies around all dominant modes of the structure evident on the transfer function. For a thin-walled component, flexibility is assumed only in the y-direction (see Figure 1). For such a case, 0 Eq. 123 becomes 00 0 Eq. 124 Solve: Λ Λ 1 0 (Eq. 113) 0 Λ Eq. 125 Eq. 126 Therefore: Λ Λ Λ Eq. 127 22 Using Eq. 128 yields 1 Λ 1 Eq. 129 Λ Eq. 130 Λ cos 2 2 sin 2 Eq. 131 and that denote start and exit of cut are derived The immersion angles from the tool to workpiece position. and are constants and are the slopes of the force curves for and respectively. 23

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