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W: “62. .- I~o .Jx” _ L , --—-:"""r (3)153?“ . .. gr: I; “at; fl§--' 17m llllllllllIllllllllllllllllllllllllllllllllllllllllllllllll LI 5 R A RY 293 02074 2056 2 00 0 Michigan State University This is to certify that the thesis entitled AN ANALYTICAL METHOD FOR PREDICTION OF CHATTER STABILITY IN A BORING OPERATION presented by Mukund Rajamony [yer has been accepted towards fulfillment of the requirements for MS Mechanical Engineering degree in F Major prof¥sor Date t5/, 2/00 0-7 639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE moo 12030009039659“ AN ANALYTICAL METHOD FOR PREDICTION OF CHATTER STABILITY IN A BORING OPERATION By Mukund Rajamony Iyer A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE Department of Mechanical Engineering 2000 Abstract AN ANALYTICAL METHOD FOR PREDICTION OF CHATTER STABILITY IN A BORING OPERATION By Mukund Rajamony Iyer An analytical method for the prediction of chatter stability in boring is developed. The dynamics of the boring process is obtained by combining a mechanistic cutting force model with the structural dynamics of the boring bar. Chip regeneration in the axial direction has been given importance in the dynamics of the process. Friction is not taken into account and the system is assumed to be linear. The chatter stability analysis is based on a delay-differential equation model. The solution of the first order characteristic system of this equation is used to compute the chatter stability lobes of the machining process. This model can be used to quickly compute the chatter stability of a multi-insert boring operation. The results obtained from this method are compared with time domain simulations and experiments. To my family iii Acknowledgements I would like to express my deepest appreciation to the following people for having made the completion of this thesis possible: To my advisor Dr. Brian Feeny, for his guidance and invaluable assistance, without which this thesis would not have been possible. To my other committee members, Dr. Steve W. Shaw and Dr. Dinesh Balagangadhar, for their valuable suggestions and comments. To Dr. Suresh Jayaram, Process Simulation and Analysis Tools Division, Caterpillar Inc., with whom this idea was formulated during the summer of 1999. His dedications to the subject and personal integrity have served as an example and an inspiration to me. To Dr. Robert Ivester and Dr. Matt Davies, NIST, for the help provided in setting up the experiment collecting data. Also to Dr. Tony Schmitz and Dr. Jon Pratt, for the fi'uitful discussions we had at NIST. To all my other instructors at MSU, for their skillfiil instruction in the classroom, but more importantly, for their fi'iendship and constant encouragement. To my friends, ‘Marcus’, ‘Vetti’, ‘Bill’ and ‘Tiger’, for having made my stay at MSU a memorable experience. To my family, for years of support, patience and understanding. And to GOD, for all the blessings showered upon me. Table of Contents List of tables .................................................................................... List of figures ................................................................................... Key to Symbols or Abbreviations ............................................................ 1. Introduction 1.1 Machining of Metals ............................................... 1.2 Machine Tool Vibrations .......................................... 1.3 High Speed Machining ............................................ 2. Literature Survey and Problem Description 2.1 General Machining Operations ................................... 2.2 Boring ................................................................ 2.3 Problem Description ............................................... 3. Theory 3. 1 Introduction ......................................................... 3.2 Cutting Force Model ............................................... 3.3 Structural Dynamics Model ....................................... 3.4 Formulation ......................................................... 3.4.1 Single Insert Boring : N=l ...................... 3.4.2 Twin Insert Boring : N=2 ....................... 3.5 Time Domain Simulations ........................................ 3.6 Delay Differential Equations ..................................... 3.7 Estimation of Cutting Coefficients .............................. 4. Results and Discussion 4.1 Measurements and Simulation .................................... 4.1.1 Single Insert Boring .............................. 4.1.2 Two Insert Boring ................................ 4.2 Measurements done at NIST ...................................... 4.2.1 Short boring Bar .................................. 4.2.2 Long boring Bar .................................. 4.3 Discussion ........................................................... 5. Conclusions 6. Suggestions for future work vii wu—I 12 15 l 8 23 26 28 29 30 33 34 37 39 41 43 45 63 Bibliography Appendix A Fig. 28 Section Y-Z of a single insert boring operation ....... 66 70 MPWN?‘ List of Tables Structural dynamics of the cutter ................................................... Calibration Constants ................................................................. Modal parameters of the Short bar .................................................. Cutting Coefficients .................................................................. Modal parameters of the Long bar .................................................. vii 40 48 52 57 :“P’Nt‘ 3" List of Figures A stability chart for a boring operation ............................................ Schematic of a boring operation ................................................... Single insert work piece interaction ............................................... Cutting Forces for Twin Insert Boring Bar at 4500 rpm and 0.75 mm Depth of Cut ...................................................................... Cutting Forces for Twin Insert Boring Bar at 4500 rpm and 0.85 mm Depth of Cut ......................................................................... Experimentally Measured Frequency response figures for the boring bar. . .. Stability Chart for a Single-insert Boring ......................................... Stability Chart for a Two-insert Boring ........................................... Experimental set up .................................................................. Magnitude of Transfer fimction Gxx(s) ............................................ Magnitude of Transfer function ny(s) ............................................ Magnitude of Transfer Function Gn(s) .............................................. Axial accelerations for stable depth of cut 0.381 mm ............................ Axial accelerations for unstable depth of cut 1.016 mm ........................ F FT analysis for the 1St second of stable cut operation .......................... F FT analysis for the 5th second of stable cut operation ......................... F FT analysis for the 1St second of unstable cut operation ...................... FFT armlysis for the 5th second of unstable cut operation ...................... Stability Lobes for the short boring bar ........................................... Chatter Signature of the short boring bar ......................................... Magnitude of Transfer function Gxx(s) ........................................... Magnitude of Transfer function ny(s) ............................................ Magnitude of Transfer function Gu(s) ............................................. Stability Lobes for the Long boring Bar ........................................... Chatter Signature of the Long Boring Bar ......................................... F FT analysis for the 3rd second of a stable cut operation ........................ FFT analysis for the 3rd second of an unstable cut operation ................... Section Y-Z of a single insert boring operation .................................. viii 18 20 35 35 40 41 42 45 46 47 47 49 49 50 51 51 52 53 54 55 56 56 57 58 59 59 57 4’1 $166 01 Key to Symbols or Abbreviations MEANING Axial Cutting Force Radial Cutting Force Tangential Cutting Force Displacement Transfer function in i-direction due to force in the j direction Axial Cutting force coefficient Radial Cutting force coefficient Tangential Cutting force coefficient Number of inserts Time period at a point between successive inserts Depth of cut Dynamic feed Static feed Revolutions per minute of the cutter Chip regeneration in axial direction Chip regeneration in radial direction Insert angle with respect to y—axis Angle between successive inserts Lead angle of the insert Chatter frequency UNITS N/m2 N/m2 N/m2 rad rad rad/s Chapter 1: Introduction Section 1.1: Machining of Metals Machining of metals has been studied extensively over the last hundred years, the focus primarily being on reduction of machining costs and a pragmatic approach to the manufacture of parts of acceptable dimensional accuracy and surface quality. Metal cutting phenomena were visualised and published as early as 1945 by Merchant [1]. Three important reasons define the necessity to develop a predictive theory for the metal cutting process [2]: o A predictive metal cutting theory would be beneficial to process planners by providing sufficient knowledge of process efficiency. 0 A predictive metal cutting theory would be beneficial to the tool designers, producers, and users, as it constitutes a proper basis for making right decisions. o A predictive metal cutting theory would benefit designers and users of machine tools by providing them with real process parameters such as cutting forces, heat generation rate and energy consumption. Boring, drilling, facing, milling, reaming and turning constitute the machine tool operations widely used in industry. Boring, drilling and reaming are generally the processes for producing internal cylindrical surfaces, i.e., holes of moderate accuracy in terms of position, roundness and straightness. These processes are typically done on engine lathes or milling machines, in addition to specialized tools on a transfer line. Drilled holes are often used for mechanical fasteners like bolts or rivets. Drilling a hole is a preliminary step for processes like tapping, boring or reaming. Internal cylinders of moderate accuracy are produced by drilling and high accuracy internal cylinders are produced by boring, in both cases the feed motion is axial. Turning and facing are lathe processes which involve using a single cutting edge with specified geometry in constant contact with workpiece to remove material. Relative motion is achieved by rotating the workpiece. Due to this relative motion, the tool either moves parallel or normal to the axis of rotation making lathes the best at producing surfaces of revolution. The sides of an external cylinder are produced by turning when the feed motion is parallel to the axis of rotation. The ends of a cylinder are generated by facing, where the feed motion is radial and normal to the axis of rotation. Milling machines are extremely productive as they employ multiple cutting edges as opposed to single cutting edges.The cutting speed is generated by rotating the cutter and moving the workpiece in a plane normal to the axis of spindle rotation. Milling processes are used to produce contour and planar surfaces. Common types ofmilling are face and end milling [3]. Section 1.2: Machine Tool Vibrations Due to inherent stiffness and damping properties of a tool and workpiece, vibrations during the cutting process are a common occurrence. Machine tool vibrations have been widely studied and a lot of progress has been made in order to suppress unwanted vibrations occurring during the operation. Machine tool vibrations are a very important problem in the manufacturing industry. Undesired vibration may lead to problems, viz., poor surface finish, tool wear and noise on the shop floor resulting in a loss of productivity. An operation displaying any or all of the symptoms above is said to chatter. Chatter is a nuisance to metal cutting and its effects are adverse. Chatter effects surface finish, dimensional accuracy, tool wear and machine life. Surface finish undulations are generally defined as chatter marks. Velocity variations on the tools due to imbalance in the drive system, servo instability or stick slip fi'iction, can result in periodical variations in the surface finish. However, forced vibrations and self-excited vibrations are the major sources of chatter. Forced vibrations can be attenuated by reducing the driving force and/or the dynamic compliance to permissible values. Many theories have been propounded to analyze self-excited chatter. Advancements in machining speed and machining of thermal alloys have resulted in the want of a better understanding of self-excited chatter [4]. Section 1.3: High Speed Machining The speed range of a mchining process is measured by the DN number. DN is defined as the product of the diameter in rmn and the spindle speed in rpm. DN is closely related to the surface cutting speed which is of the form v = nDN, (1) where D= is the cutter diameter, (m) N= is the rotational speed of the spindle, (revs/sec) Smith and Tlusty [5] showed that for a tool of diameter 25mm and dominant natural frequency 1000Hz, low range machining speed extends upto 2300 rpm, mid range between 2300 and 7500 rpm and high range extends beyond 7500 rpm. The upper limit of mid range machining is where the tooth passing fiequency is 1/4‘” of the natural frequency. There have been many instances in machine tool operations where large quantities of material had to be removed, especially in milling. Advances in tool materials have made it possible to reduce time by high-speed milling. Generally, high-speed machining ranges between spindle speeds of 7,500 and 50,000 rpm for the same tool described above. High speed machining leads to the employment of large axial and radial depths of cut paving the way for dramatic improvements in metal removal rates [6]. Mid range machining varies from speeds of 2300 rpm to 7500 rpm. It was shown that mid range machining did not have any process damping effects as in low speed machining [5]. Process damping will be discussed later. A stability clmrt shows the maximum depth of cut or feed a machining operation can undergo without chatter for a given spindle rpm. A typical stability chart is shown in Fig.1. The stability lobes shown here mark the boundary between stable and unstable cuts of operation, the area below the lobes representing stable operation zones and the area above, unstable operating zones. It can be seen from the figure that there are many stability lobes in the given range of speed and, given sufficient power, the operation can be suitably tuned to machine larger depths of cut or feed, thereby making the process more and more eflicient. Tlusty and Zaton [7] showed that the mid-range asymptotic stability limit was fairly constant and the machine operation could be improved by spindle speed variation and non-proportional tooth spacing. Our focus, in this work, lies on mid range and high speed machining. In high speed machining, the stable depth of cut depends very strongly on the spindle speed. It also displays a constant stability limit as in mid-range machining. It has been observed tlmt increases in metal removal rates can be obtained if the tooth passing fi'equency is a fiaction of the most dominant natural frequency. 1.15 1.1~ 1.05~ . ,1 i/i/ / 30003200940036003800400042004400460049005000 Speedtrpm) Deptbotcut (mm) 0 o o 26 8 .. p 8 .0 tn Fig. 1 A stability chart for a boring operation. The goal of the work done here is to develop an analysis tool to generate stability lobes for a boring operation quickly and reliably. A detailed statement of the problem is given in the following chapter. Chapter 2: Literature Survey and Problem Description Section 2.1: General Machining Operations Early attempts were made to examine the effect of flexible supports on the modes of vibration of a machine tool [8]. The goal of the analyses was to determine whether these supports would isolate a vibration without causing the machine tool to chatter. The force due the action of the tool on the workpiece was found to be a function of the vibration amplitude and tool speed among other factors. When chatter occurs, this force was such that the inherent damping of one of the modes was overcome and an oscillation was sustained at a frequency not far from the resonance frequency of the modes. The machine tool structure was assumed to be a collection of spring-mass systems, each having a specific resonance frequency. Findings showed that the chatter frequency was one of the frequencies whose peak was least damped. It was also shown that the tool wear primarily depended on the chatter velocity. From their studies, it was concluded that direct use of isolators would not help in vibration isolation and chatter simultaneously if the fiequencies causing chatter were low. Following this, a theory was developed to compute the stability lobes of a machine tool system [4]. However, this theory excluded the dynamics of the cutting process. Chatter was reasoned out to be caused primarily by the dynamic stifliiess of the system. Merritt also claimed that chatter would be a minor problem if the damping ratios of the most affecting modes was of the order of §=0.5. Merritt also suggested that this theory, developed for the turning process, could easily be extended towards other processes. A model [9] was proposed for metal cutting analysis on a milling machine. The model comprised of simple linear differential coefficients with perodic coeflicients. The model also took into consideration the rotation of induced forces. The equations were of first order and were very convenient for stability analysis. A regeneration factor, it, was introduced. The instantaneous chip thickness was now the sum of the nominal feed and the total deviation in the chip thickness. The deviation is computed fi'om the vibration of the workpiece and the surface generated at that point due to the passage of the previous tooth. From the model, it was concluded that chatter analysis was associated with the stability characteristics of linear differential equations with periodic coefficients. Sridhar also concluded that stability methods, based on frequency analysis, could not be used to study chatter in milling operations. Sridhar et a1. [10] followed up with their previous theory and came up with a stability algorithm for the general milling process. The algorithm involved the solution of a transcendental equation by employing existing graphical solution methods. Sridhar used a linear theory, which, in the past, was found to sufficiently accurate. The algorithm presented an elegant technique for solving for more realistic models. Tlusty and Zaton [7] used a time domain simulation approach to develop a better understanding of milling stability. Many previous assumptions, such as uniform orientation of all the cutter teeth, were abolished. The theory showed that the gains in stability were actually smaller than they were previously thought to be. However, time domain simulations are very tedious and take a long time to predict a stability chart. Smith and Tlusty [6] showed that the optimum speed of operating a milling operation would be in the region of its natural fi‘equency since the regeneration of waviness on the surface, which causes chatter stability, inhibited the development of forced vibrations. Subsequent work by Smith and Tlusty [1 1] resulted in the development of a control system to adjust the spindle speed and the feed rate automatically to achieve stable milling. Other algorithms employed to study stability were Peak to Peak Diagrams of forces, deflections or surface finishes [12]. A new method for the prediction of chatter in milling was propounded by Minis and Yanushevsky [13]. The dynamics of the milling process were described by a set of differential equations with time varying periodic coefficients and time delays. The resulting characteristic equation was of infinite order and its truncated version was used to determine the limit of stability by employing standard techniques of control theory. Excellent agreement with previous experimental results was the validity of the proposed stability method. Based on transfer functions of the structure of the cutter workpiece contact zone, and the static force coefficients, a new analytic method was presented to predict the stability lobes in the milling operation by Altintas and Budak [14]. This analysis showed that frequency domain simulations could be performed on the milling operations. The method was shown to yield the stability diagram much faster, and of the same accuracy or better, compared to conventional time domain methods, used in the past. The method was based on the formulation of dynamic milling with regeneration in the chip thickness. The theory was robust and was able to incorporate time varying modal parameters into its algorithm. Tlusty [15] suggested that in the case of high speed machining operations, tool dynamics could be effectively manipulated to get the best out of the spindle. Chatter was also found due to the varying flexibility of the workpiece during machining. High speed machining became a very economically significant process. Smith and Tlusty [5] discussed how attempts to make practical use of high speed machining led to developments in tool materials, spindle and machine design, chatter avoidance and 10 structural dynamics of machine tools to name a few. Chen, Ulsoy and Koren [16] presented a computational stability analysis for turning. The solution scheme was simple since the characteristic equation simplified to the solution of a single variable. It was also suggested that this method could be employed for other machining processes, as long as the system equations were expressed as a set of linear time-invariant difference difi‘erential equations. Rao and Shin [17] presented a model of the dynamic cutting force process for the three-dimensional or oblique turning operation. The methodology involved linking the mechanistic force model to the tool-workpiece vibration model. Cross coupling between radial and axial states was paid particular attention to as this inclusion was believed to be important in predicting the unstable-stable chatter phenomenon occurring due to non- linearity in the process. More recently, Bayly, Halley and Young [18] came up with a quasi-static model for reaming. This model neglected inertial and damping forces, but included regeneration and rubbing effects. The eigenvalue solution was found to closely resemble the tool behavior seen in practice. The same authors also came up with another simulation of radial chatter in drilling and reaming [19]. ll Section 2.2: Boring Boring bars are metal cutting tools that are used to bore deep precise holes in a variety of manufacturing operations. These cutting holes are used alter a hole has already been made from a drilling operation. The boring bar is characterized by a large length-to- diameter (L/D) ratio. The boring bar is clamped at one end by a tool holder, and has a cutting insert at the free end. The cutting insert at the free end is used to cut the metal in the bore of the workpiece to a close tolerance. The cutting operation is achieved either by keeping the boring bar stationary and moving the workpiece or vice versa. A boring bar is characterized by a low transverse dynamic stiflhess. Hence, it is susceptible to excessive mechanical vibrations in the transverse direction. Merritt [4] had earlier shown metal cutting processes, including boring, as a closed loop feedback system. This system can be explained from the fact that the motion of the tool in the previous tool pass affects the force acting on the cutting tool in its current pass. Until recently, most of the machine tool operations were developed for milling and turning, as discussed before, and suitably modified for boring. Few theories were built specifically for the boring process. One early study by Anshofl’ [20] was in the field of chatter suppression. Anshofl’ concluded that there were two ways to suppress chatter. One was to fit an absorber on the 12 boring head. The second method was to utilize certain bar materials of high damping capacity such as manganese copper. Chatter free machining was reported if the longitudinal flats were machined on the bar and a suitable position was selected for the cutting tool in the boring head. The depth of the cut was analyzed to be sensitive to changes in the angle of the tool with respect to the plane of the flats (lead angle) [21]. Parker [22] showed that boring could be well modeled if the penetration rate effect (or the dependence of the forces on the velocity of the tool) was considered. The force due to this effect was proportional to the process damping constant. However, recent research has revealed that process damping, talked about previously, assumes an important role only in low speed machining and has very little or no effect in mid-range and high speed machining. Parker concluded that tangential vibrations were an important consideration at certain head angles as their exclusion caused serious problems to the simulation. Rivin [23] suggested a structural optimization approach to improve the dynamic stability of the boring bars with long overhangs. Little effort was made to simulate stability charts of boring operations. Kapoor et al. [24] developed an elegant mathematical model based on dynamic equations in which chip regeneration was an important parameter. The model involved 13 development of a transfer function, which made it not only possible for predicting boring bar chatter but also predicting the generation of irregularities on the surface. The formulation eliminated chip regeneration from the feedback path in the closed loop system, thereby greatly simplifying the simulation. Zhang and Kapoor [25] followed this method with particular attention to tangential vibrations. The simulations yielded good results with some discrepancies in the stability lobes. Tewani et al. [26] suggested the boring bar to be extremely susceptible to chatter owing to its cantilever type structure which reduced its dynamic stifliiess in the transverse directions. He suggested that chatter fiee boring bars are typically of the overhang ratio 4.5 and less. Recently, Li, Ulsoy and Endres [27] showed that there exist differences between stationary and rotating bores. The work was primarily focussed on regenerative chatter due to tool rotation. Stability lobes varied for stationary and rotating bores. Qualitative explanations were given without mathematical proof to show the discrepancies between exact and approximate solutions, especially at low spindle speeds. The regeneration term (the increment or decrement in material removal rate due to the difference between the current and previous insert depths) affects the stability limit. The proposed analysis is for high speed and mid range machining. Since the feed rate is much slower than the speed range of the boring process, fiill chip regeneration in the axial direction of the boring bar is assumed, i.e, [F]. 14 Several analytical methods have been used in the past to compute the stability limits [4, 13, 28]. It is possible to linearize the non-linear cutting force model and use the linear model to compute the chatter stability limits. However, these linear models are only accurate close to the region of linearization and will introduce errors if extrapolated over wide ranges [27, 29]. The assumption of no nose radius implies that the analytical results can be applied to tools where the depth of cut is significantly larger than the nose radius. Analytical models which include the effect of nose radius and depth of cut variations have been overcome in recent work by Ozdoganlar et aL [28]. These methods can be incorporated into the present analytical method to make the model more comprehensive. The analytical model proposed in this work provides a quick means to estimate the stability limits. Emphasis is laid on vibrations in the feed or the axial direction. If a detailed analysis is required, then the more accurate time domain simulation can be performed. Section 2.3: Problem Description This work aims to develop a simple analytic method for the prediction of chatter limits in a boring operation. The eigen-value method developed by Altintas [14] for milling has been extended towards the boring operation. The boring bar is assumed to be stationary while the workpiece rotates. Certain assumptions have been made in view of previous experiences in analysing similar machine tool operations. 15 0 Properties in the X, Y and Z directions have been considered and incorporated into the equations. Though the tool is considered stiffer in the axial directions, properties have been included as they have an effect on the analysis. Torsion effects have been neglected. o The cutting forces are assumed to be linearly dependent on the depth of cut and the feed, thereby linearly proportional to the chip area. 0 Cutting coefficients are treated to be constant for the range of spindle speed considered. The spindle speed we are concerned with is between 3,000 and 15,000 rpm. Hence, coeflicients related to penetration rate and velocity variation have been removed to make the analysis simple. 0 The structure is assumed to be linearly and proportionally damped. o The tool lead angle has been taken into account, but no nose radius has been used. However the theory can be easily extended to incorporate nose radius changes. 0 The inserts are spaced at equal angles and the inserts are equally distant fi'om the center of the boring bar. 0 Extreme cases of boring resulting in no contact between tool and workpiece have been neglected in the simulation. The work is organized as follows: The Theory section describes the method of analysis. Following the theory are the results obtained using the formulation developed in this work. The formulation is compared with a time domain simulation. Models are also verified with experimental results. Application of the theory is discussed in the 16 Discussion part. A brief summary follows the discussion. Scope for possible future work is also discussed at the end. 17 Chapter 3: Theory Section 3.1: Introduction A boring bar as described previously, is a tool with a large length-to-dhmeter (L/D) ratio. L/D ratios vary typically fiom 2 to 9. A simple diagram of a boring bar is shown in Fig. 2. Boring Bar Z Fig. 2 Schematic of a boring operation 18 The above figure is a model of a 2 insert boring bar. The boring bar is assumed to be irrotational and feeding in the negative Z direction. For the insert shown, f, and f, are the radial and tangential forces experienced by the tool during the cutting process. In addition to this, the tool also experiences an axial force fax in the positive Z-direction. The radial and tangential forces, shown in fig. 2, are on the X-Y plane. The workpiece is assumed to rotate in the problem of our concern. If the bar is assumed as a perfectly uniform beam free at the loaded end and clamped at the other, the static stifl‘ness for the bending motion is given by K = -—, (1) where, E: modulus of elasticity, N/m2 1= transverse moment of inertia in the x direction, In“, and L= length of the beam, m Clearly from Eq. 1, it is evident that the longer the boring bar, the more weaker it is in the transverse directions. This weakness in transverse stifiiiess is a primary cause for Chatter. 19 A A Workpiece V v V Static depth of cut, d. 4’: D Direction of feed, f; Y Z Fig. 3 Single insert workpiece interaction The movement of the boring bar in the negative Z-direction is called the feed as shown in Fig. 3. A detailed diagram of a Y-Z section of a single insert boring operation is shown in the Appendix A. Feed is typically expressed in mm/rev or mm/sec. The instantaneous feed rate is obtained as the sum of the static feed rate, the projection of the current vibration in the direction of the feed, and the projection of the previous vibration (imprinted on the workpiece surface) in the direction of the feed. The static feed rate, f; is defined as the feed rate with which the operation begins. Hence, the instantaneous feed for the i’h-cutting insert can be represented by the following equation: f.=f..+AZ+Artan(¢1), (2) where, 20 Ar = 1‘ Sill(¢.)+ yCOS(¢.) -x. Sin(¢.- -¢.,...) — yo Sin(¢.- -¢,,,,), (3) and, Az=z—,uzo (4) x, y, and z are the deflections of the cutter at time t and x0, yo, and 20 are the deflections of the cutter at time t-T, i.e., one revolution before. T defines the period of revolution of the workpiece, the inverse of which multiplied by 60, gives the workpiece rpm. At the time t- T, the workpiece was at an angle ¢i-¢,-,,c to the Y-axis. It must be noted that the angle, ¢1, is the angle of the current insert with respect to the Y-axis and the angle, ¢,- - all“, is that of the previous cutting insert. This is different from the turning and milling operations where the angle, ¢.-, is used for both [14]. Since there exists a possrbility of the tool overlapping part of the surface previously cut, a regeneration parameter, ,u is introduced in Eq.4. In the case of spiral cutting as observed in a threading process, the previous tool position has no relation to the present position. Hence it is assumed to be 0. The feed of the system is slow as compared to the machining speed concerned in our analysis. Owing to this, a large portion of the tool tip machines the same surface in the axial direction for a number of revolutions of the workpiece. Therefore, the regeneration factor ,u is assumed as 1. The 21 regeneration direction of the boring bar is in the axial direction since the tool is moving in the Z-direction. The radial position of the tool tip with respect to the workpiece in the radial direction is defined as the instantaneous depth of cut. For this analysis, the depth of cut is a parameter to be found. When looking at linear stability, we consider either (a) purely linear systems or, (b) small deviations fiom equilibrium, which would not involve the machine tool leaving the cut. Hence, the extreme case where the tool insert exits cut, i.e., the tool does not touch the surface of the workpiece (a clear case of chatter) has been left out. Time domain analysis can be used to predict exactly and speeds and the feed where the tool inserts exits cut. This phenomenon is regarded as tool run out. The instantaneous depth of the cut for the 1"” insert is represented by d,. A boring process typically moves in the feed direction, hence, the dynamics are assumed to be primarily in the feed direction. Dynamics in the depth of cut direction have been omitted since our main focus of interest lies in the axial direction, which is also the feed direction. Fig.3 shows a typical case of tool dynamics indicating a static depth of cut, the direction of feed and the lead angle. Eq. (2) also shows the effect of lead angle on the feed. The vibrations in the X and Y directions affect the feed in the Z-axis [30]. The lead angle is defined as the angle made by the tool as shown in Fig. 3. Lead angles are typically from —20 to +20 degrees for boring bars [31]. The lead angle is shown in Fig.3. 22 Section 3.2: Cutting Force Model We assume that each insert i is oriented at angle ¢.- to the Y axis. The inserts on the tool are spaced equally apart. The forces on a single insert is given by K! f. = K. A. (5) K01 Eq. 5 takes into account only the dynamic chip area. The dynamic chip area is defined as below: Ac = fidr _ fed: (6) The dynamic chip area for the ith insert is defined as the difference between the instantaneous chip area and the static chip area. The static chip area does not influence the dynamics of the system and therefore is removed fiom the analysis. The static chip area does not result in tool chatter. The depth of cut is a parameter to be found. K,, K, and Kax are cutting coefficients in the tangential, radial and axial directions respectively. Cutting coefficients are found by running experiments in the static cutting region and evaluating forces that occur in the tangential, radial and axial directions. The forces are then divided by the chip area of the cutting process to obtain cutting coeflicients. Hence the involvement of the static feed occurs only the evaluation of the 23 cutting coeflicients. This evaluation of the cutting coeflicients is valid only in the stable zone of operation, where vibrations to the tool are damped out and stabilized. Cutting forces are approximated as linear functions of static feed and the static depth of cut only for mid-range and high speed machining processes. For low speed machining processes, the forces are not a true linear function of the feed and the depth of cut. Therefore, prediction of cutting coefficients is difficult to make. This analysis assumes linearized cutting coeficients. More on cutting coefficients is discussed in section 3.7. Using Eqs. 2, 3, 4 and 6 into Eq. 5, we obtain f, K,dtan(¢,) K,d Ar f. = Krd tan((15,) Krd [A2] (7) f... ,. Kmdtamd) Kmd ,. a a C The matrix C contains the cutting coeflicients. The forces experienced by each tool insert can be conveniently transformed in the global X, Y and Z directions by the following expression: F. -cos(6.) —sin(0.-) o f. Fy = sin(0,.) —cos(6,.) 0 f, (8) F, , 0 O —1 fat ' 4 > Tr 24 Tr, in the above equation is the transformation matrix. In order to compute the chatter stability limits, it is assumd that the system is vibrating at the chatter fiequency, at The chatter frequency is simulated over a range of frequencies. Altintas [14] used a frequency range of 1/2 the lowest modal frequency to 2 times the highest modal frequency of the tool in his analysis. It was argued in his analysis that the chatter frequency is found in the range of the modal frequencies of the tool of operation. Our analysis intends to do the same. The system is also assumed to be harmonic, i.e., the deflections at the cutter at time t—T are obtained by multiplying e'm’T to the deflections at the cutter at time t. T is defined as the tooth passing period, the inverse of which is called the tooth passing frequency. This fi'equency is also equal to the ms of the workpiece multiplied by the number of inserts or teeth, N on the tool. Writing in matrix form we have, x0 x y. = y e”“”' (9) Z Z This expression represents synchronous periodic motion. Later we will find conditions that allow for periodic motion. This will represent the transition between stable (exponentially decaying) and unstable (exponentially growing) regimes. Eq. 7, along with Eqs. 3 and 4, shows the dependence of the axial, radial and tangential forces on an insert tip to be dependent on the X, Y and Z vibrations of the boring bar at time t and the x,y,z vibrations of the insert at a previous time t-T. The x,y,z vibrations of the insert are the same as the surface undulation on the workpiece. The term i a) is represented also by the Laplace transform variable 5. 25 Eqs. 3 and 4 can be rewritten into a single matrix equation using Eq. 10 as: V x Ar sin( 0,. ) — e'"”7 sin( 0, — 6m) cos(6l, ) — em"? cos(0, — 0m.) 0 = -151 y (10) A2 0 0 1 — e z 4 a P Note the restriction of periodic motion is incorporated into P. Section 3.3: Structural Dynamics Model The vibrations in the X, Y and Z directions can be computed fiom the forces by using the transfer function matrix. In the Laplace transfer frequency domain, the expression governing this relation is given by: x(s> G..(s) 0,6) G..(s> P16) )(S) = 0,.(8) ny(S) 0,.(3) F,(S) (11) 26) 0.6) 0,6) cats) E6) 4 5 0(5) The forces F,(s), Fy(s) and F,(s) represent the Laplace transform of the forces exerted on the cutting tool due to the interaction of all the inserts with the workpiece. In 26 mathematical terms, the forces in the time domain are computed by summing up the individual forces exerted on each insert as shown below in Eq. 12. 4; N F. F, =2 F, (12) F; i=1 F151. Eq. 11 has a transfer function matrix G(s). Element Gg-(s) is the transfer function between the deflection in the 1‘” direction due to the force in the 1"" direction. There could be a possrbility of more than one mode existing in a particular direction influenced by a force in another direction. Merritt [4] showed the importance of using transfer functions in machining operations. Element G,-,-(s) can be written mathematically as numberofmodes Gym: 2 [ “m" ], (13) n=l S2 + 2§nwn(S) + a): The mass (m), damping ratio (3.) and fi'equency (6a.) are the n'” modal parameters of the dominant modes in the 14" direction influenced by the force in the 1" direction. Elements Gm ny and Gzz are known as the direct transfer functions. The rest are known as cross transfer functions. It has been observed that direct transfer functions are more influential in the transfer fimction matrix and the cross transfer functions are smaller in magnitude compared to the direct transfer ftmctions. Hence, for simplicity, the cross transfer 27 functions are eliminated. Eq. 11 is rewritten as -X(S) (ia(5) 0 0 12(5) y(S) = 0 ny(S) 0 . Fy(S) z(s) 0 0 G: (s) F,(s) Section 3.4 Formulation Eqs. 7, 8, 10 and 12 are combined in Eq. 14 to give x N x y =[GlZZlTrICIP y z ”I z (14) (15) The problem has two parameters, the depth of cut d and the tooth-passing period T. The preceding formulation is restricted to synchronous harmonic motion. Matrix P incorporates this restriction. Indeed, such a response representing the transition between stable and unstable cutting can occur under specific circumstances, for which Eq. 15 has a non-trivial solution. We can determine these circumstances by solving for d and T. The non-trivial solution to Eq. 15 is possible if the determinant l1 - [GlngrICIP1= o 28 (16) Eq. 16 is a clmracteristic equation used to solve for two variables, T and d. The fact that d, the depth of cut, is a real number is made use of in finding the solution for T, by setting the imaginary parts of d to zero. Section 3.4.1: Single Insert Boring: N=1 We assume the insert to be aligned along the Y-axis, i.e., 61:09 Also since there is only one insert, 6,..5360". Each element of Eq. 16, Tr, C, P and G can be determined separately and put back into Eq. 1 6 yielding the characteristic equation 1+ Kmd(l - (“”7 )0: + K,d tan(¢,)va (1 — e""”) = 0 (17) Eq. 17 does not involve Gn(s), i.e., the transfer function in the X-direction related to the Force in the X-directon. It was earlier assumed that the insert was located on the Y- axis. Gn(s) plays a role in Eq. 17 if the insert is located on the X-axis. Substituting A=1—e""” (18) into Eq. 1 7, the depth of out can be solved as 1 1 d = —. 2 Re(K,,G,, + K, tan(¢,)G,,,) (19) 29 and T can be solved from the expression, Cot(%7:) = 2d Im(Ka,.Gz + K, tan(¢,)GW) (20) The spindle speed is then obtained by using the expression (2 = —, (21) N being equal to l in this case. Section 3.4.2: Twin insert boring, N=2 In the case of twin insert boring, Eq. 16 involves a summation of the interaction between both the inserts. The inserts are placed such that 0,=0°, 62:1800and 0m=180°. Eq. 16 is simplified to obtain a determinant, 1 2K,dtan(¢,)ny(l +e"‘"") o 0 1+ 2K,d1an(¢,)ny(1+e""”) 0 =0, (22) o 0 1+ 2Kdez(1—e"‘”7) Once again, Gn(s) is absent because the inserts are aligned along the Y-axis. The analysis therefore shows that transfer function in the X-direction do not have an effect on 30 the solution of our system. However, it is also observed that the modal parameters in the x and y directions are very similar and therefore, will affect the solution in a similar way. Two solutions arise fi'om the solution of the determinant shown in Eq. 22. The first solution is solely dependent on the axial cutting coeflicient and the axial modal parameters. The second solution involves the lead angle A and modal parameters in the y direction. Each of these solutions for the depth of cut d have a corresponding spindle speed .0. The lowest depth of cut corresponding to a speed is considered for the stability plot. Solution 1: Equ 22 is satisfied if 1 + 2K,d tan(¢,)ny (I + e""”) = 0 (23) Eq. 23 can be solved, bearing in mind that d can only be real. The solution for d would then be d_1__[_l__] (24, tan(¢,) 4.Re(GW) and the solution of T can be obtained from, tan(5’2—T-] = 2d Im(-2K,d tan(¢, )ny) (25) 31 The spindle speed can be computed fiom the time T by employing Eq. 21. Solution 2: Eqn. 22 is also satisfied if 1+ 2Kdez(1— e-W’ ) = 0 fi'om which, the solution for d is, 1 4.19,. Re(G,_,) ’ and T can be obtained by solving, cot[%T—) = 2d1m(2Ka,G,,) (26) (27) (28) Again, the spindle speed, (2, is derived from T using Eq. 22. The data obtained fi'om Eqs. 26 thru 29 are charted and the lowest positive depth of cut for a particular spindle speed is selected as a stability limit. Eqs. 19 and 27 show the strong dependence of the axial parameters on the depth of cut. The dependence of the stability boundaries on axial parameters is the main thrust of this work 32 Multi-insert solutions can also be obtained from the analytic method. However experimental data for chatter stability of multi insert boring bars is yet to be found to validate the theory. Section 3.5: Time Domain Simulations Simulation of machining operations for a certain number of revolutions at a given range of speeds and depths of cut is known as time domain simulations. Time domain simulations are very accurate. However, these simulations must be performed for each value of depth of cut and speed. Hence, the time taken to plot a stability chart is large. Analytic methods are preferred to time domain simulations when time is a constraint. The time domain simulation algorithm is a general algorithm that can also be used for boring processes. In operations such as milling, facing and turning, there are entry and exit angles, i.e., angles where the insert enters and exits cut every time the tool bar rotates. In the case of boring, the insert is always removing material and actually does not enter and exit the cut at any time. Extreme cases of insert run-out due to excessive vibration have been neglected, as these cases are a sure example of chatter. The time step or the angular increment of the tool rotation depends on the speed of operation. Jayaram, 1997 [32] showed that the angular increment should satisfy the inequality n/(mn’IO) 2 60*0im/(Q*360). This means, 01... S (Pb/(20%;). The smallest angular increment required for Q = 1000 rpm and fi,=500 Hz is 0.6°. Hence, a value of 05° was used for these simulations. 33 The algorithm was used to work on 10 passes or 10 revolutions of the workpiece over the tool. Therefore, for example, an operation for 5000 rpm was simulated for 2 ms. Forces in the X, Y and Z directions were plotted for the simulated time and the system was said to chatter if the forces were observed to grow. This phenomenon of growth in forces could be readily observed in a single insert boring bar. In a multi insert boring bar, due to the symmetry of the forces in the simulation, the resultant forces acting on the structure is minimal, causing no excitation of the structure. This phenomenon could be observed in the force plot of a time domain simulation as shown in Figs. 4 and 5. An artificial spike is introduced which simulates a hard spot or a poor workpiece material at a particular place. Then the force plot is observed after the introduction of the spike. If this spike causes the vibrations to increase with time (become unstable) then this condition is termed as chatter. The depth of cut was incremented in steps of 0.05 mm until chatter was detected. The criterion for the detection of chatter is to determine if any of the cutting forces, j}, fy or jg, or the dynamic feed rate increases with time after the first few revolutions. Section 3.6: Delay Differential Equations Let us examine a simple case of a single insert boring. The insert is at angle 0=0° with respect to the Y-axis. The forces then acting on the insert are 34 .fr = Krds tan(¢l) Krds 2 fax Kurds tan(¢l) Kurds 0 (29) f. K161381104) K161. y ] Fore. (N) 0 an 0.01 o.“ 0.00 o. 1 Q12 Tm (Inc) Fig. 4 Cutting Forces for Twin Insert Boring Bar at 4500 rpm and 0.75 mm Depth of Cut. 1m- . A . , . . . .1 Putnam) O -mr r i j I 0 one 0.04 0.“ 1' 0.1 0.12 Thu (see) Fig. 5 Cutting Forces for Twin Insert Boring Bar at 4500 rpm and 0.85 mm Depth of Cut. 35 The forces in the radial, tangential and axial directins can be transformed to the global X,Y and Z coordinates. With the aid of the modal parameters of the tool obtained fi'om impact hammer tests, the forces in the X,Y and Z directions can be used to compute the dynamics of the tool. Therefore, F. m, x+ c, x+ kxx __ j: Fy =