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VOI. 42 NO. 3 SCIENCE IN CHINA (Series E) June 1999 Chip-ejection interference in cutting processes of modern cutting tools * SHI Hanmin ( Ilifi E) (School of Mechanical -neering, Huazhong University of Science and Technology, Wuhan 430074, China) Received October 28, 1998 Abstrsct Based on the "principle of minimum energy", the basic characteristics of non-free cutting are studied; the phenomenon and the nature of chipejection interference emmuonly existing in the cutting process of modem cutting tools are explored. A "synthesis h o d of elementary cutting tools" is suggested for modeling the cutting procese of modem complex cutting tools. The general equation governing the chipejection motion is deduced. Real examples of non-free cutting are analyzed and the theoretically predicted results are supported by the experimental data or facts. The sufficient and necessary conditions for eliminating chip-ejection interference and for realizing free cutting are given; the idea and the technical appmach of "the principle of free cutting" are also discussed, and a feasible way for improving or optimiz- ing the cutting performance of modern cutting tools is, therefore, found. Keywords: metal cutting, principle d minimnm energy, ehipejdon interterence, synthesis method of dem- entaryeu#ing-. With the rapid advance in powder metallurgy, die manufacturing and NC grinding techniques, the modern cutting tools may be manufactured into a very complex shape, performing very complex non-free cutting. A real cutting tool may be regarded as a combination of a series of elementary cutting tools (ECTs) . As shown by refs. [ I , 21 and footnote 1) , the natural value of chip-ejection vector (including chip-ejection direction and chip-ejection velocity) of an ECT always minimizes the cutting power. This is the principle of minimum energy (PME) for ECTs. In a metal cutting process all the ECTs work concurrently, and each of them tries to eject its chips according to its own natural chip-ejection vector. All the chip-ejection vectors, each going his own way, will certainly interfere and conflict with each other; to keep the integrity of the common chip ejected by all the ECTs, they have to compromise with each other so as to coordinate the integral ejection motion of the whole chip. Researches also show that the natural law governing the integral chip-ejection motion is still PME, that is to say, among all the integral chip-ejection motions satisfy- ing the constraint conditions set by controlling parameters the realizable one must minimize the total cutting power of the cutting tool ( i , e . the sum of all the incremental cutting power consumed by all the ECTs) . A cutting process which possesses chip-ejection interference and thus needs chip-ejection com- promise is called non-free cutting. Nearly all the practical machining operations are of non-free cut- ting. In a non-free cutting process PME guarantees a minimum cutting power consumed by the entire * Project supported by the National Natural Science Foundation of China (Grant No. 59675058) . 1) Wang, J. L. , Shi, H. M . , Ni, J. et al. , The interference of chip ejection under non-free cutting, Int ME'94, l Win- h kr A d Meting of ASME ( PE-7) , Now. 6-1 1, Chicago , 1994. 276 SCIENCE IN CHINA (Seriea E) Vol. 42 tool, but it cannot make chip-ejection vectors of ECTs coincide respectively with their own natural . ones As a matter of fact, after chip-ejection compromise, each ECT, generally, has to make its con- cession and thus cannot eject its chip at its own natural chip-ejection vector. Therefore, the total cut- ting power of a cutting tool is normally larger than the sum of incremental cutting power consumed by a l the ECTs while each works separately. This shows a very strong nonlinear factor of non-free cut- l ting. This factor may intensify the chip deformation, accelerate the tool wear, deteriorate the surface finish and increase the cutting power consumed. Those harmful factors, in turn, shorten the tool life, leading to frequent tool changes and to excessive down time. Many problems arising in modem metal- cutting operations may be caused by this factor. In this paper an elementary cutting tool synthesis method (ECTSM) is suggested to explain, to predict and to optimize the cutting performance of complexly shaped cutting tools. Being distinct from the previous result^[^-^] , the author's method, on the basis of PME, has properly handled the chip ejection interference and the caused nonlinearity during the cutting process with complexly shaped toole. 1 The general law of chip-ejection motion during non-free cutting 1 .1 The dividing and specification of ECTs A cutting tool may be divided into a se- ries of ECTs along its cutting edge as shown in fig. 1 . All the parameters of the ith ECT, . such as the radius vector pi ( i e . the vector from the origin to the ith ECT) , the unit vec- tor bi in the direction of elementary cutting edge, the unit vector ai in the rake and nor- mal to the edge (see fig. 2 ) , the length of in- cremental edge A bi , the main geometrical an- gles: Yoi , A,;, and the cutting parametem: > a,, V;; i = 1, 2 , -.., n , may be obtained Y through a geometrical and kinetic analysis on the cutting tool. Fig. 1. Integral chipejection motion and the ECT dividing. The main cutting force generated by the unit length of the ith ECT Iroi Iloi(yoi, Ari, aci, V;), i = 1 , 2 , -.., n = (1 and the natural chip-ejection vector Uoi = U o i ( ~ o iASi, aCi, Vi), i = 1 , 2 , ---, , n (2) may be determined by means of experiments or theoretical analyses on ECTs (there are some 0 t h factors, such as the materials of the work-piece and the tool, the cooling condition etc . , also affecting the main cutting force and the natural chip-ejection vector, but they are equal for all the ECTs on a cutting tool, i. e . they are constants, so those factors are not explicitly expressed in the above two 1) See footnote on page 275. No. 3 CHIP-EJECTION INTERFERENCE IN CUTPING PROCESSES 277 Natural chip-ejection vector Uoi can be resolved into components parallel with and normal to the cutting edge, respectively (see fig. 2 ) , Uo, = Uoi bi , Uoh = Uoi ai. (3) According to PME for ECTS"~211) , the pow- er consumed by the entire cutting tool may be ex- pressed as i " A = CAA~ ,=I = 2 l z o i v i ~ b i (+l ,=1 I: + P~c:) 9 (4) where ti and 5, are the variation ratios of U with respect to U, in the directions parallel and normal U, to the cutting edge, respectively (see fig. 2) . gi = AU,/UOi = ( u , - U0,)/Uoi, Ei = AUin/Uoi = ( Uh - Uok)/Uoi. (5) And the coefficients pl and p2 indicate the strength of the influence of ti and Ci on cutting t power, respectively. Fig. 2 . Chip-ejection vectors. . U , Natural chip-ejection vec- 1.2 The motion of the common chip tor; U , real chipejection vector; U', induced chipejection Once formed, the common chip as a whole is vector. assumed to undergo a kind of rigid body motion (ignoring the elastic motion) . As everybody knows a rigid body motion may be composed by a transla- tion T and a rotation d2 as shown in fig. 1, and the real chip-ejection vector of the ith ECT is ui = T + sa x pi, i = I , 2 , n, .-., (6) which may be resolved into Uir = Ui bi , Uin = Ui ai . (7) 1.3 The PME for the whole cutting tool . Substituting eqs (5 ) ( ) into (4) yields -7 A = p I,V.Abi{l + P l [ ( T + KJ x Pi - uoi) bi]' + P2 [(T+ x Pi - Uoi) ai]'} 9 i=l ui o o ui where T and 51 are two state parameters (vectors) which should be determined on the basis of PME . Therefore, letting a A / a T = 0 and a A / a n = 0 leads to - - u: p i x u: = p i x ULi, = U'&; where Upi and ULi are the induced chip-ejection vectors and the induced natural chip-ejection vec- tors, respectively (see fig. 2, the subscript " i" has been ignored in the figure) , The vinculum in eq . (9) refers to the weighted average, i .e . 278 SCIENCE IN CHINA (Seriee E) Val. 42 i.1 and the ith weight is the cutting power consumed by the ith ECT. The normalized weights are, I a pair of Trnand l2, does minimize cutting power A , then eq. (9) must be satisfied, and the e- f quation, on the other hand, provides a way of calculating Tm and . This equation summarizes the arn general rule of determining state parameters on the basis of PME. Substituting T , and a, obtained into eq. ( 8 ) yields the real minimized cutting power Amin. 2 A case study on non-free cutting A lot of experimental data and facts given by the author'" 21 ') has verified the validity of the general law described by eq. ( 9 ) under some special cases. The general solutions of this equation and the ecumenical method for calculating Tm l2, are beyond the scope of this paper. Here, only and an example of forming turning with circular edged tools (fig. 3) is given so as to show how to analyze a non-free cutting process by means of ECTSM on the basis of PME. Fig. 3 . Circular edged tools used for experiments and analysis, plunge turning, work-piece : bronze, tool: HSS , 7, = A, = O" . (a) Convex, without chip splitting slots ; ( b) convex, with chip splitting slots ; ( c) concave. 2.1 ECTs ' nonlinear synthesis method From the experimental data an empirical formula of the main cutting force generated by a unit length of cutting edge is fitted as I = 1 5 9 3 ~ ~ 0 , . ~+( 0.76d2), ' 1 (13) which is the basis for further analysis, where 9 is the forced deviation in chip-ejection angle. If each ECT ejects chips in its natural chip-ejection direction, i .e . in the radius direction, then B = 0'. The elementary cutting force generated by an edge A1 would have been AF, = 1 5 9 3 a t . 6 1 ~ l , (14) 1) See footnote on page 275. No. 3 CHIP-EJECTION INT&UERENCE IN CWllNG PROCESSES 279 and the cutting force of the whole tool (see fig. 3(a) ) would be calculated by simply superposing the elementary forces of all the ECTs, F, = IAF, = r 5 9 3 ( f ~ i n a ) ~Rda = 3 6 4 7 ~ P . ~ l , l 0 .~l where f is plunge feed rate, fsina = a,, a is the orientation angle of the ECT and R = 2 mm is the radius of the circular edge. As a matter of fact, because of symmetry of the circular edge the common chip as a whole can only be ejected in the longitudinal direction on the tool shank. Because of this, the deviation angle of the real chip-ejection direction from the natural one is 0 = x/2 - a . Substituting this into eq. (13) and integrating lead to the real cutting force of the tool, This shows an increase of 43% in the main cutting force caused by chip-ejection interference under non-free cutting. Let us see fig. 3(b) . The two slots divide the circular edge into three segments, each of which ejects its chips independently. First, let us see the central segment, which is also a circular edged tool. Following eq . ( 16) , the cutting force of this central segment can be expressed as F = J A F . = r 593(f ~ i n a ) O . ~ ' [ l 0 . 7 6 ( ~ / 2 - a)'] ~ d = 1 4 1 3 . 5 ~ f O . ~ ' . (17) i l + a u/3 Second, let us see the right segment of the edge, whose cutting force can be expressed as The parameter in the equation is the common chip-ejection angle of this segment (fig. 3 ( b ) ) , which is a state parameter and should be determined by PME. Letting a F : / a n = 0 yields Substituting it into eq. (18) and integrating lead to F: = 1 077Rj"~~l. Note that the elementary free-cutting force of the ith ECT can be expressed as d F,, = 1 593 Rda and dp = d F Z o / r dF , is the normalized weighting coefficients. Thus the above e- (fsina )0.61 0 quation may be rewritten as r u/3 This is the embodiment of the weighted average relation expressed by eq. (9) in this particular case, and the relation shows that the ECT with larger cutting force has stronger influence on chip-ejection compromising. Finally, let us see the left segment of the edge. Because of symmetry the cutting force gener- ated by this segment of the edge is equal to that of the right segment, i .e . = F: , and the total cutting force of the entire tool is F,, = F: + F : + F > 2F: + F", = 3 904Rf61. (20) A comparison of eq. (20) with eq. (16) shows that the chip splitting slots have reduced the main 280 SCIENCE IN CHINA (Series E) Vol. 42 cutting force by one fourth. 2.2 A comparison of experiments with the theory The main cutting forces of the tools shown in fig. 3 were theoretically predicted and experimen- tally measured (the results are given in table 1 ) . Table 1 The main cutting force of circular edae tools f/mmarev-' 0.019 0.022 0.026 0.032 0.039 Without chip splitting slots Experimental Convex 915 971 1 137 1 303 1 331 Concave 887 957 1 109 1 317 1 331 Predicted 929 1 002 1 125 1 277 1 429 With chip splitting slots Experimental Convex 702 735 832 930 988 Concave 684 721 777 943 961 Meted 696 750 843 957 1 010 3 The degree of M o r n confinement and free cutting 3.1 Coefficient of non-free cutting Coefficient of non-free cutting (CNFC) of a cutting process @ is defined as = F ~ / ~ A F , , (21) where F, stands for the real cutting force (reckoning in chip-ejection interference), and A F , is the sum of the incremental forces generated by all the ECTs each working separately (without interfer- ence) . CNFC is defined to indicate the degree of freedom confinement of a cutting process. 45 a 1, and the bigger the coefficient is, the severer the chip-ejection interference is. If and only if "@ = 1" then the cutting process is free. 3.2 The sacient and necessary condition for realizing free cutting heo or em^'^. A cutting process is h e , i . e . 45 = 1 , i and only i the natural chip-ejectbn f f vectors of all the ECTs fonn a rigid body motion , i . e . Uoi = To + no x p i , i = 1 , 2 , ..., n , (22) where Toand noare two constant vectors and in this case , the real chip motion is composed of transha- tion To = Toand rotation = Do(proof ignored ) . Equation (22) has two special cases : To = 0 and no= 0. The theorem crs lf ae i a prevalent misunderstanding: only when the cutting tool is a straight line edged one can it perform free cutting. This, unfortunately, is not true. As a matter of fact, the theo- rem says that a cutting process is free has no inevitable connection with whether or not the cutting edge is a straight line. 3 .3 The principle of free cutting As revealed by the research, for arbitrarily edged cutting tool the principle of free cutting lies in NO. 3 CHIP-EJECTION INTERFERENCE IN CUTITNG PROCESSES 28 1 properly designing the tool face with a purpose of dredging the chip-ejection so as to eliminate chip e- jection interference from all the segments of the working edge. As verified by experiments free-cutting technique notably improves cutting condition, reduces the cutting force and the power consumed, per- fects chip control and elongates the tool life (Chinese patent ZL 96 2 35153 .9)[61 . 4 Conclusions 1) Chip-ejection interference and compromise. Modem cutting tools may be regarded as a com- bination of a series of ECTs and all the ECTs are working concurrently. In the process each ECT tries to eject its chips according to its own natural chip-ejection vector, and this will certainly result in in- terference among chip-ejections. The interfering chip ejections have to compromise with each other so as to keep the integrity of the common chip. The compromised common chip motion must minimize the cutting power consumed by the whole cutter, which is the PME for cutting processes with complexly shaped tools. A great deal of experiments and analysis have verified or support this principle. 2 ) Non-free cutting. Whenever chip ejection compromise is necessary due to existing of interfer- ence the cutting process is called non-free cutting. During a non-free cutting process ECTs normally cannot eject chips at their natural chip ejection vectors and this factor increases the cutting force and intensifies the chip deformation, causing many real problems in today ' s machining operations. 3) General equation governing chip ejection motion. With an assumption of chip's rigid body motion and based on PME, the general equation governing the translation and the rotation components of the chip ejection motion is deduced. From the equation a law is concluded: a state parameter, such as chip ejection angle, relating to integral chip motion is equal to the average of the same param- eter relating to each ECT weighted by its free cutting force. By means of the law, state parameters of a non-free cutting process are conveniently determined, and the influence of various factors on state parameters is promptly analyzed. The law is supported by experiments. 4) The conditions for realizing free cutting. A coefficient of non-free cutting is defined to indicate the degree of freedom confinement of a cutting process. The conditions for realizing free cutting are that the natural chip ejection vectors UOi i = 1 , 2 , ... , n ) of all the ECTs form a rigid body motion. ( 5) The principle of free cutting. Properly designing the geometry of the tool rake may eliminate chip ejection interference and realize free cutting so as to increase the efficiency and to improve the quality of machining operations. Acknowledgement Wang Xibing and Lu Tao carried out relevant experiments and computer analysis. References 1 Shi, H. M . , Wang, J . L . , A model for non-free-cutting, Int . J. Mach. Took M...fnct, 1994, 35( 11) : 1507. 2 Shi. H. M . , Lu. T., Wang, X. B . , On the nature of minimum energy of metal-cutting process, Proc. of Int . Wohhop on Aut . Manufc. Sci . & Tech. , Jm20-21 , Shanghai , China , 87 - 92, Shanghai : Shanghai Sci . & Tech. Literature Pub- lishing House, 1996. . 3 Stephenson, D. A . @pin, I . S. , Calculation of main cutting edge forces and torque for drills with arbitrary point geome- . tries, Int J . Mach. Twb MMyfaet. , 1992, 32(4) : 521. 4 Watson, A . R ., Drilling model for cutting lip and chisel edge and comparison of experimental and predicted results-I-IV , . Int . 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