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5th International DAAAM Baltic Conference “INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOUR FORCE AND ENTREPRENEURS” 20–22 April 2006, Tallinn, Estonia A NEW APPROACH TO MATHEMATICAL MODELLING OF METAL CUTTING PROCESS CHARACTERISTICS Jaanson, A. least with a minimum number of passes. In resolving such a task, coefficients of stiffness of machine tool assemblies in cutting process and mathematical models of cutting process characteristics serve as initial data. To get a solution of practical importance the exactness of prediction values of process characteristics must be improved essentially. A way to tackle this problem is dealt with in this paper. Up to now no difference exists in achieving working accuracy either on a conventional machine tool or a modern precious CNC machine tool. Possibilities to raise productivity, to lower the cost of production and economize the energy consumed have not been made use of. Abstract: A review of mathematical methods used for composing empirical formulae and mathematical models of the metal cutting process is brought. The principle structure of process characteristics mathematical models in use and the main differences of a perspective model of a qualitatively higher level are given. Keywords: metal cutting, mathematical modelling. turning, 1. INTRODUCTION The development of metal machining may be divided into two periods: up to the introduction of modern computer controlled CNC machine tools, and afterwards. It can be observed based on the papers published till the end of 2005 that shifts depending on stiffness coefficients of machine tool assemblies have not been taken into account by the cutting data optimizing techniques. As machining accuracy is to be guaranteed, multiple passes (cuts) have been applied. There has been no change in the dimensional adjustment technique since modern CNC machine tools were introduced. The cutting tool is guided on a trajectory determined by nominal dimensions of the work, just like it has been done with conventional machine tools. In principle, it is possible to guide the cutting tool on a trajectory taking into account its dimensional wear and elastic shifts in the technological system. As a result, a work with no axial profile errors is achieved with a first pass or at 2. MATHEMATICAL METHODS USED TO DATE Metal cutting is probably the most widely used process of manufacturing. Research into this field of science began in the middle of the 19th century. In principle, the factors affecting cutting process, the mechanism of chip forming and the phenomena accompanying the process are known. The evergreen problem is how to raise machining efficiency, i.e. how to guarantee the required quality with higher productivity and a lower cost of production. Mathematical description of process characteristics is the first step to resolve the problem. F. Taylor [1] was the first investigator to compose an empirical tool life (T) formula. 133 number of well-known investigators in this field of science, e.g [2, 3, 4, 5], lies in the period from the 1940’s up to the 1960’s. The 2k type design of experiments for composing a tool life mathematical model was first used by an American scientist S. M. Wu in the year of 1964 [6]. (2k – number of experiments for determining estimations of coefficients of variables and their interactions). When using 2k designs, it is assumed that k factors are coded to the standardised levels ±1. Wu’s model: The principle form of Taylor’s polynomial: (1) T = Ca xp f y v n , where C – coefficient of proportionality; a p – depth of cut; f – feed rate; v – cutting speed; x,y,n – exponents. Taylor used his formula for optimizing cutting data in turning and achieved a multiple rise of productivity. The formula (1) has been used by a number of investigators as a pattern in composing empirical formulae for calculating metal cutting process characteristics (cutting force components, surface roughness indicators, cutting temperature). The values of the characteristics do not depend on cutting data only but also on the geometry of the cutting tool, properties of a workpiece and insert materials, and the environment. The influence of factors that are not variables of empirical formulae is taken into account by correction coefficients: C char = Ca xp f y v n K1 ...K n , (2) ŷ = bo + b1x1 + b2x2 + b3x3 + b4x1x2 + b5x1x3 + b6x2x3, (3) where ŷ – output of the model; x1…x3 – coded inputs of the model; bo – value of output if x1=x2=x3=0; b1…b6 – estimations of the model coefficients. Independent variables were the same as in formula (1). The estimations of model (3) coefficients were calculated by the method of least squares. Response surface methodology is often used for visual illustration of the dependence of output on input. Wu’s model allowed to raise the exactness of tool life prediction but did not resolve the problem exhaustively – by including essential factors into the model the number of experiments grows unacceptably. When using CNC machine tools chip breaking must be guaranteed. From the beginning of the use of CNC machine tools investigations related to this problem have been topical. Modern science and technology constantly require new materials of specific properties. As a rule, their machinability (easiness with which a work material is machined under a given set of cutting conditions) is worse. According to Sandvik [7] the three main parameters of machinability assessment are: (i) tool life; (ii) surface finish; and (iii) cutting force. A number of new investigation methods (FEM – finite element method, neuron network method, Taguchi method) are in use for resolving where Cchar – arbitrary characteristic of the cutting process; K1…Kn – correction coefficients. To compose formulae like (1) and (2) the one-variable-at-a-time approach has been used. As interactions of many factors are essential, the correction coefficients are valid only as to the factors, by which they were determined. It is the reason, why the type (2) formulae are of a very low (in reality indeterminable) exactness. In workshops cutting data, geometry of the active part of cutting tools and other quantities needed are chosen as experience-based estimates. (Through the age of metal cutting the skill of operators has been looked at as an art in the sense of perfection). The “golden age” of classic papers of a great 134 number of earlier investigations beginning from the 1920’s chip root micrographs do not reflect the investigated process but a changed image of chip formation. In reality, chip formation process can be investigated with the help of chip root micrograhps gained by a quick stopping of cutting process. A device enabling an undisfigured chip root has been described in [24]. The criteria that make it possible to decide whether a micrograph reflects the process under investigation or a transition process are presented in [25]. An example of a correct investigation of chip formation is offered in [26]. Actually cutting process takes place within a technological system components of which are a machine tool, cutting tool, workpiece and in many cases also a jig. Astakhov’s theory of chip forming is not based on the micrographs but on arbitrary hand-drawn schemes and, therefore, leads to deeply erroneous results. So does Astakhov’s system concept in metal cutting [22, 23] as the cutting process cannot take place without a machine tool. The model inputs values are affected by the parameters of the machine tool assemblies. Reference systems are needed for mathematically describing the cutting tool and cutting process. ISO 3002-1:1982 consists of two reference systems: the tool-in-hand system for describing cutting tool and toolin-use system for modelling the cutting process. The standard was composed in a period when cutting blades were bronzed to the body of a cutting tool and the design geometry of the tool was guaranteed by grinding. The nose radius of such a tool was almost equal to zero. To-day inserts (cutting blades) are mechanically clamped to the cutting tool body and the nose radius varies in the limits of 0.4...10 mm. That is why the ISO 3002-1:1982 is not fully applicable. The structure of a mathematical model of cutting process characteristics in use is depicted in Fig. 1. problems of metal cutting. The opinion of the users of the methods is that a lot of problems can be resolved faster and cheaper than when making costly and time consuming experiments. Still the results brought in many papers make us arrive at a conclusion that the authors are not well informed as to the achievements of metal cutting theory. The results published in papers [8, 9, 10, 11] prove that with the neuron network and Taguchi methods trustworthy results are not attained which makes the methods unsuitable in solving metal cutting problems. In principle, FEM is suitable to resolve some problems of metal machining [12]. Though, in some cases it has also been used by researchers whose main profession is obviously not metal cutting. In such investigations, e.g. [13, 14, 15, 16 ], incorrect calculation designs or formulae can be found. As a rule, results of such investigations are useless for science and practice. Incorrect calculation designs can also be found in investigations not related to afore-mentioned calculation methods [17, 18, 19, 20 ], making the results published useless. Some investigators [21], presumably not mechanical engineers, are dealing with the problem of ensuring constant cutting force. Such a set of problem is meaningless from the position of resolving metal cutting problems. Astakhov et al. introduce in [22, 23] a “new system concept” in metal cutting. It can be read in [23], page 191, that cutting takes place within a cutting system that is defined as consisting of three components: the workpiece, the cutting tool and the chip. It is maintained in [22, 23]: “The uniqueness of the cutting process, among other closely related manufacturing processes, is identified by chip formation, which is considered to be caused by the bending stress in the deformation zone”. A similar mistake was made by E. Reuleaux at the beginning of the last century. Knowledge how the chip forms is essential in modelling the metal cutting process. In a 135 noise Inputs of basic model: v, a p , f , (M ) M Outputs of model (fixed also by experiments): Fx , Fy , Fz , T , Rz(DIN ) , Ra , θ Correction coefficients K1…Kn Figure 1. Inputs and outputs of cutting process characteristics mathematical models (formulae) by using ISO 3002-1:1982 tool-in-hand system. v – cutting speed; ap – depth of cut; f – feed rate; M – model; M – characteristic of workpiece material mechanical properties; K1…Kn – correction coefficients; Fx, Fy, Fz – components of cutting force; Rz(DIN), Ra – indicators of machined surface roughness; θ – cutting temperature. describe cutting process characteristics a polynomial model is used. Factors changing in the cutting process must be chosen as model inputs. The characteristics of the process are made up of two parts, differently depending upon the inputs. The intensity of wear of various cutting wedge parts has been substituted for tool life as one of the model outputs. A different approach to the problem to be resolved makes it possible to compile a systematic mathematical model of the cutting process, where interactions of the model inputs and variation of the process characteristic values in time are considered. The possibility to raise the prediction exactness of the cutting force components values up to 20 times has been proved experimentally. In the middle of the 20th century when intensive investigations in metal cutting were done computer engineering was not yet at a level to allow solutions of today. To all appearance, the users of the metal cutting theory of the present day must be so accustomed to the simplifications in use that no attempt has been made to find better solutions, or perhaps they have not been found. Modern instrumentation allowing the experimental results to be directly forwarded to a computer for data processing makes it possible to compile a model for any material in a very short time (about 2 to 4 days). A structure of the new model is shown in Fig. 2 3. A POSSIBILITY TO PREDICT PROCESS CHARACTERISTICS ON A QUALITATIVELY HIGHER LEVEL In the monograph “Fundamentals of Mathematical Modelling of the Metal Cutting Process” [27] (in Estonian) reasons of inaccuracy of metal cutting process mathematical models are analysed. It is emphasized that most of the reasons are subjective and therefore avoidable. The metal cutting process and technological system are treated as a unitary system. In [27] the ISO 3002-1:1982 tool-in-hand system has been adapted to modern cutting tools. Two new reference systems have been introduced. An insert reference system is envisaged to describe the insert. The tool-in-use system has been replaced by the tool-in-process system making a more precise determination of the face and clearance angles in different points of the cutting wedge possible. Coordinate systems are made use of to calculate parameters of the cutting process starting from the design geometry of the insert, insert bases orientation on the cutting tool body and kinematics of the cutting tool. The formal geometrical inputs of the models can thus be replaced with these harmonised with the physical essence of the process, and at the same time the number of inputs can be minimized. To 136 Noise Model outputs fixed on experiments and got by data processing Fγx , Fαx , Fγy , Fαy , Fγz , Fαz , dhα dt , dbγ /dt dβ n2 dt , Rz(DIN ) , Rz (DIN )cal , Inputs of the model Rz (DIN ) proc , Ra , Racal , Ra proc , θ γ cp1 , γ cp 2 , bγ , auc , buc , v M Outputs of the model Fγx , Fγy , Fγz , pn, pt, dhα dt , dbγ dt , dβ n2 dt , θ , Rz(DIN)proc, Raproc Figure 2. γ cp1 , γ cp 2 – tool rake of the first and second face in cutting process; bγ – width of the first face; auc – weighted average of uncut chip thickness; buc – width of uncut chip; v – cutting speed; Fγx , Fγy , Fγz – components of the chip generating force; Fαx , Fαy , Fαz – components of the force acting on the flank; dhα / dt – intensity of flank wear; dbγ / dt – intensity of the first face land wear; dβ n 2 / dt – intensity of decreasing of the normal wedge angle of the second face; Ra – arithmetical mean deviation of assessed profile; Rz (DIN ) – ten-point height of irregularities; Ra proc , Rz ( DIN ) proc – components of Ra and Rz (DIN ) depending on cutting process; Ra cal , Rz ( DIN ) cal – components of Ra and Rz (DIN ) depending of the nose radius of the cutting edge and feed rate; θ – cutting temperature; p n , pt – normal and tangential pressure on the flank of cutting tool. 4. CONCLUSION sions) employed for turning. For taking into use an applied research of the new model is needed. The new model is a real-time model with the help of which a number of technical problems unsolved so far with the conventional models can now be resolved. Compiling programs for CNC machine tools makes it possible to achieve the required quality with a minimum number of cuts, and actually optimise the cutting data. So far the relation of the feed and speed alone has been optimised. The monograph also provides principles for building up a CNC machine tool control program. In [27] turning of outer surfaces has only been treated. Still, the cutting process mathematical model can be usable for the majority of machining methods by correspondingly adapting the mathematical apparatus (about 700 mathematical expres 5. REFERENCES 1. Taylor, F. M. On the Art of Cutting Metals. Trans. ASME, 1907, 28. 2. Merchant, M. E. 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A system concept in metal cutting. J. Mater. Process. Technol., 1998, 79, 189–199. 24. Hastings, W.F. A new quick-stop device and grid techniques for metal cutting research. Ann. CIRP, 1967, XV, 109–116. 25. Jaanson, A. About quick-stop devices for metal cutting research. Proc. TPI, 1971, 306, 3–13 (in Russian). 26. Komanduri, R., Brown R.H. On the mechanics of chip segmentation in machining. J. of Engineering for Industry, 1981, 1, 33–51. 27. Jaanson, A. Fundamentals of Mathematical Modelling of the Metal Cutting Process. TUT Press, Tallinn, 2005 (in Estonian). 6. CORRESPONDING ADDRESS Arvo Jaanson Tallinn University of Technology Department of Machinery 138 Ehitajate tee 5, 19086 Tallinn, Estonia Arvo.Jaanson @ ttu.ee 139