外文翻譯---高速銑加工中軸轉(zhuǎn)矩的模糊控制

1、<p><b>　　附錄A：英文資料</b></p><p>　　Technical Briefs</p><p>　　Fuzzy Control of Spindle Torque in High-Speed Milling Processes</p><p>　　Rodolfo Haber-Guerra</p>&l

2、t;p>　　Instituto de Automática Industrial (CSIC),</p><p>　　kin. 22800 N-III,</p><p>　　La Proveda, 28500 Madrid, Spain</p><p>　　Steven Y. Liang</p><p>　　The Georg

3、e W. Woodruff School of Mechanical</p><p>　　Engineering,</p><p>　　Georgia Institute of Technology,</p><p>　　801 Ferst Drive, N.W.,</p><p>　　Atlanta, GA 30332-0405</p

4、><p>　　Jose R. Alique</p><p>　　Instituto de Automática Industrial (CSIC),</p><p>　　km. 22800 N-III,</p><p>　　La Proveda, 28500 Madrid, Spain</p><p>　　Rod

5、olfo Haber-Haber</p><p>　　Universidad de Oriente,</p><p>　　Ave. Americas s/n.,</p><p>　　Santiago de Cuba, 90400 Cuba</p><p>　　This paper presents the design and impleme

6、ntation of a two-input/two-output fuzzy logic-based torque control system embedded in an open architecture computer numerical control ( CNC) for optimizing the material removal rate in high-speed milling processes.The co

7、ntrol system adjusts the feed rate and spindle speed simultaneously as needed to regulate the cutting torque using the CNC's own resources. The control system consists of a two-input (i.e., torque error and change of

8、 error), two-output (i</p><p>　　Keywords:fuzzy control, torque, high-speed milling</p><p>　　Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFA

9、CTURING SCIENCE AND ENGINEERING. Manuscript received August 19,2005: final revision received February 14, 2006. Review conducted by C.J. Li.</p><p>　　Introduction</p><p>　　In order to improve ma

10、chining efficiency in a high-speed milling process through a higher material removal rate, this study focuses on the design and implementation of a two-input/two-output (TITO) fuzzy control system for spindle torque. The

11、 major issue to be dealt with is the new development and application of fuzzy logic (FL) using the CNC's own resources. No additional hardware overhead is required, since the control algorithm is embedded within the

12、kernel of a standard open control. Fuzzy l</p><p>　　This paper is organized as follows: In Sec.Ⅱwe present a brief study of a mechanistic model for predicting cutting force and spindle torque; in Sec. III we

13、 describe the design of the fuzzy controller to optimize the milling process; in Sec. IV we describe how the fuzzy controller can be embedded in open architecture CNCs, and we discuss the key design and programming stage

14、s; in Sec. V we review the experimental results and explore some of the comparative studies. Finally, we present conclusions </p><p>　　High-Speed Milling-Process Control Based on the Spindle Torque Signal<

15、;/p><p>　　The mechanistic model estimates cutting-force vectors and spindle torque on the basis of feed rate, spindle speed, and material constants. The mechanistic model of end milling implemented in this rese

16、arch is based on [1,2]. In order to make a simplified model, we used the following approximation [3,4]: If the feed-per-tooth value ft j is small with regard to the tool radius D/2, then for a tool fed in the positive X

17、direction, the instantaneous chip thickness hj is related to the rotation angle </p><p>　　hj(Φj) = fij sinΦj·sin κ (1)</p><p>　　Therefore, on the basis of t

18、he above approximation, and for roughing and semifinished cutting,it is not necessary to extract the instantaneous chip thickness from the volume information.Certainly, a correct value of the feed per tooth fij not only

19、increases tool performance but also improves the efficiency of the machining process. For a cylindrical end mill, the following conditions are defined for finding the general solution.</p><p><b>　　r(z)

20、 = </b></p><p><b>　　κ= 90º</b></p><p><b>　　(2)</b></p><p><b>　　Ψ= k0z</b></p><p>　　k0= (2 tanθ)/D</p><p>　　w

21、here Ψ(z) is the lag angle that appears due to the helix angle θ of the cutting tool. This angle is constant in the case of a cylindrical end mill, and it varies for a ball-end mill. The tangential, radial,and axial forc

22、e differentials (dFt), (dFr), (dFa) act on an infinitesimal length dS of the cutting edge of the tool [4]</p><p>　　dFt = KtedS + Ktchj( Φ, κ)db</p><p>　　dFr = KredS + Krchj( Φ, κ)db

23、 (3)</p><p>　　dFa = KaedS + Kachj( Φ, κ)db</p><p>　　It is also considered that </p><p>　　db =

24、 (4) </p><p>　　Furthermore, the characteristics of a point on the cutting surface are identified using the properties of kinematic rigidity and the displacements between t

25、he tool and the workpiece. The constants or cutting coefficients (Ktc,Krc,Kac,Kte,Kre,Kae) can be found experimentally using cutting forces per tooth averaged for a specific type of tool and material [5].</p><

26、p>　　The total cutting forces as a function of Φ along the axial depth of cut for all the cutting edges that are in contact with the workpiece can be calculated as</p><p>　　N f N f z2 </p>

27、<p>　　Fx(Φ) =∑(Fx j(Φj(z)) =∑∫ [-dFr jsin Φj sin κj - dFi jcos Φj - dFa jsin Φj cos κj]dz</p><p>　　j=1 j=1 z1 </p><p>　　N f N f z2

28、 (5)</p><p>　　Fy(Φ) =∑(Fy j(Φj(z)) =∑∫ [-dFr jcos Φj sin κj - dFi jsin Φj - dFa jcos Φj cos κj]dz</p><p>　　j=1 j=1 z1 </p><p>　　

29、N f N f z2 </p><p>　　Fz(Φ) = ∑(Fz j(Φj(z)) =∑∫ [-dFr jcos κj - dFa jsin κj]dz</p><p>　　j=1 j=1 z1 </p><p>　　where z1 and z2 are the integration li

30、mits and Fx(Φ),Fy(Φ),Fz(Φ)are the resulting forces for each axis.</p><p>　　Cutting torque Tqe is estimated on the basis of the tangential force differential (dFt) and the tool diameter (D). The overall cutti

31、ng torque is</p><p>　　Tqe = Ft· D/2 (6)</p><p>　　Fuzzy Control of Spindle Torque in a High-Speed Milling Process</p><p>　　The manipulated (a

32、ction) variables we selected were the feed rate increment (△f as a percentage of the initial value programmed into the CNC) and the spindle speed increment (△s as a percentage of the initial value programmed into the CNC

33、). The three basic tasks known as fuzzification, decision making, and defuzzification were used. The error and output vectors were</p><p>　　eT=[KE·△TqKCE·△2Tq]</p><p>　　Fig.1 Fuzzy par

34、titions and membership functions for (a)△Tq,△2Tq,(b)△f, and (c) △s</p><p>　　u = GC·[△f △s ] (7)</p><p>　　where KE, KCE, and GC are scaling factors for

35、 inputs (error and change in error) and outputs (change in the feed rate and change in the spindle speed), respectively.</p><p>　　The torque values were acquired from an open architecture CNC. The reference

36、torque value Tr was estimated from the model described in Sec. 2. For each sampling period k, torque error and the change in torque error were calculated as</p><p>　　△Tq(k) = Tr - Tq(k)

37、 (8)</p><p>　　△2Tq(k) =△Tq(k)-△Tq(k -1) (9)</p><p>　　where △Tq is the torque error (in N m) and △2Tq is the change in torque error (in N m).&l

38、t;/p><p>　　The fuzzy partition of universes of discourse and the creation of the rule base were based on prior knowledge and experimental results. Figure I shows the resulting fuzzy partition. Seven fuzzy sets

39、were used for inputs and outputs: NB, negative big; NM,</p><p>　　negative medium; NS, negative small; ZE, zero; PS, positive small; PM, positive medium; and PB, positive big.</p><p>　　These memb

40、ership functions are essential to achieving good control performance. When trapezoidal membership functions are used, the resulting system is the sum of a global nonlinear controller (which is the static part) and a loca

41、l nonlinear PI controller</p><p>　　(which changes dynamically with regard to the input space) [6].</p><p>　　We considered a set of rules consisting of' linguistic statements linking each ant

42、ecedent with its respective consequent. The syntax followed the pattern below:</p><p>　　if △Tq is PB and △2Tq is PB, then △f PB and △s is NB</p><p>　　A total of 49 control rules for each output

43、 (△f and △s) were developed, summarized in Table 1. These fuzzy rules provide important principles and relevant information about the process. Under normal cutting conditions, the constant feed rate and spindle speed val

44、ues are set conservatively according to information in machining, cutting tool, and material handbooks. However, the feed rate values are manually adjusted in real time depending on the cutting parameters, in order to op

45、timize the machin</p><p>　　Table 1 Rule bases for manipulating (a) feed rate and (b) spindle speed</p><p>　　The sup-product compositional operator was selected for the compositional rule of inf

46、erence. Using the algebraic product operation, developing the fuzzy implication, and applying the maximum union operation, we obtained</p><p><b>　　49</b></p><p><b>　　(10)</b

47、></p><p><b>　　i=1</b></p><p><b>　　49</b></p><p><b>　　(11) </b></p><p>　　i=1 </p><p>　　The center-of-aver

48、age (COA) strategy was selected as the defuzzification strategy because of its suitable performance at steady state and its use as a standm-d defuzzification method in experimental and industrial fuzzy controllers. The c

49、risp controller outputs are obtained by dcfuzzilication</p><p><b>　　(12)</b></p><p><b>　　(13)</b></p><p>　　where △f (△s) is the crisp value of △f i (△si) for

50、 a given crisp input (△Tq,△2Tq ).</p><p>　　The output-scaling factor (GC) multiplied by the crisp control action (generated at each sampling instant) provides thc final actions that will be applied to the CN

51、C</p><p>　　f (k) =f (k - 1) + GC·△f (k)</p><p><b>　　(14)</b></p><p>　　s(k) = s (k- 1) + GC·△s (k)</p><p>　　Feed rate and spindle speed values were

52、 generated on line by the embedded controller and fed in with the set point for the torque Tr and measured value Tq from the internal torque signal provided by the open architecture CNC, as detailed in Sec. 4.</p>

53、<p>　　Open CNC and New Add-On Functions</p><p>　　This section briefly explains how the fuzzy controller is embedded in the open architecture CNC (see Fig. 2). Further details about software developmen

54、t and how to embed control and monitoring systems in an open CNC are provided in [7].</p><p>　　The application was developed on the basis of a Sinumerik 840D CNC [8]. First, the fuzzy controller was programm

55、ed in C/C+ +, and then it was compiled, and as a result a dynamic link library (DLL) was generated. The control kernel (NCK) was modified to enable tile real-time modifcation of the spindle speed and feed rate. A PC, the

56、 WINDOWS XP operating system, and</p><p>　　Visual C+ + were used to program the MMC. lntermtxlule communications between the MMC and the NCK were established through dynamic data exchange (DDE). DDE caused a

57、 delay that was considered in our work but was not relevant tor this case study. Finally, the user interface was programmed in Visual C+ +,for the sake of simplicity. The general outline of the control system is depicted

58、 in Fig. 3.</p><p>　　An internal data-acquisition system was developed and used to measure the internal torque signal. The sampling frequency was 500 Hz, defined by the servosystems' control cycle. The s

59、oftware consists of a data-acquisition module in the NCK that records the selected data into an internal buffer and a background task running on the MMC that receives the completed measurement and stores it in the hard

60、drive of the user-interlace PC.</p><p>　　Experimental Validation</p><p>　　Milling tests were carried out on the HSI000 Kondia highspeed milling machine, which was equipped with a Sinumerik 840D

61、 open CNC. A two-fluted end mill 12 mm in diameter was used as the tool for rough milling operations. All measurements were taken machining dry and supplying high-pressure air at the cutting zone. The workpiece material

62、was 220-HB F-1140 steel (DIN CK45, ASTM 1045). The maximum depth of cut was 0.5 mm. The nominal spindle speed and the nominal feed tale were set at f 0= 1600 mm</p><p>　　The reference torque value was deriv

63、ed from the model described in Sec. 2 and Eq. (6), using the following cutting coefficients (Kt c,Kr c, Ka c, Kt e, Kr e, Ka e)=(2178.23,879.65,798.44, 19.35,8.06,7.58). The torque value Tr=3.91 N m was set as the refere

64、nce torque. Sampling frequency was 500 Hz, the feed override range was 50-120%, and the spindle speed override range was 80-120 %.</p><p>　　Initially the controller was tuned by modifying the scaling factors

65、 for inputs and outputs (i.e., KE=1, KCE=0.5, and GC=1)although we did have to apply a "cut and trial" procedure as well.The performance of self-tuning strategies was analyzed in [9,10].However, constraints of

66、the open CNC (e.g., for real-time computation) and the characteristics of high-speed milling processes(i.e., stringent real-time requirements) severely restricted the implementation of a self-tuning algorithm.</p>

67、<p>　　We verified the effectiveness of two fuzzy controllers. The first was a two-input/one-output (TISO) fuzzy controller where the spindle speed was set as a constant and the feed rate increment was adjusted acco

68、rdingly. The second was a TITO controller with real-time modilication of both feed rate and spindle speed. We did not include linear controllers, because, according to previous study, fuzzy controllers yield better resul

69、ts than linear control loops for this type of case study [11]. The issue</p><p>　　Control system behavior was evaluated by assessing accuracy and oscillations. Various performance indices, such as integral a

70、bsolute errors (IAE), integral square errors (ISE), and integral of time per absolute errors (ITAE), were calculated in order to assess the inner loop control's performance. The cycle time tmcch and the productivity

71、improvement in machining operations Eff were also calculated. The results are summarized in Table 2. Finally, surface roughness was computed according to the R</p><p>　　The results are shown in Fig. 5. The b

72、ehavior of the torque signal for all cases, including the CNC working alone, is depicted in Fig. 5(a). Control signals corresponding to feed rate and spindle</p><p>　　speed are shown in Fig. 5(b). The TISO c

73、ontroller is represented as a gray line. The TITO controller is represented as a solid line.The T1TO fuzzy controller outperformed the others, as shown in Table 2.</p><p>　　The decrease in the cycle time Eff

74、 was close to 10%, which clearly shows progress in productivity. Moreover, the IAE, ISE,and ITAE performance indices indicated more accurate behavior,which corroborated the advantage of using these two control variables

75、. Finally, the roughness values were in the 0.34-0.79μm range (N4-N6), in accordance with ISO 1320:1992.</p><p>　　Final Remarks</p><p>　　This paper introduces a two-input/two-output fuzzy contro

76、ller to regulate torque for the optimization of high-speed milling processes. The main advantages of the approach include a two-input/two-output fuzzy controller embedded in an open architecture CNC to deal with nonlinea

77、r and time-variant milling-process behavior. The results of the fuzzy control strategy show higher machining efficiency in actual industrial tests. The influence of the proposed control system on useful tool life, the ap

78、pea</p><p>　　Acknowledgment</p><p>　　This work was supported in part by "Ramón y Cajal" Fellow Research Programme and DP12005-04298 COREMAV project of the Spanish Ministry of Educ

79、ation and Science. The authors wish to express their gratitude to Dr. Angel Alique and Dr. Salvador Ros for their assistance in providing useful comments and suggestions during the preparation of this paper. Finally, the

80、 authors would like to thank anonymous referees for their helpful suggestions and comments.</p><p>　　References</p><p>　　[I] Engin, S., and AItimas. Y., 2001, "Mechanics and Dynamics of Gen

81、eral Milling Cutters Part 1: Helical End Mills," Int. J. Math. Tools Manuf. 41, pp.2195-2212.</p><p>　　[2] Haber, R. E., Jiménez, J. E., Coronado, J. L., and Jiménez. A., 2004. "Cutting F

82、orce Model for a High-Speed Machining Processf,"Rev. Metal, Madrid,40(4), pp. 247-258.</p><p>　　[3] Roth, D., Ismail, F., and Bedi. S.. 2003, "Mechanistic Mod

83、el of the Milling Process Using an Adaptive Depth Buffer." Compul.-Aided Des. 35, pp. 1287-1303.</p><p>　　[4] Martellotti, M., 1945. "An Analysis of the Milling Process, PartⅡ--Down Milling,"

84、; Trans. ASME, 67(1), pp. 233-251.</p><p>　　[5] Budak, E., Ahintas, Y., and Armamgo, E. J. A., 1996, "Prediction of Milling Force Coefficients fi'om Orthogonal Cutting Data," ASME J. Eng. lnd..

85、 118,pp. 216-224.</p><p>　　[6] Ying, H., 1999, "Analytical Structure of the Typical Fuzzy Comrnllers Employing Trapezoidal Input Fuzzy Sets and Nonlinear Control Rulesf Inf. Sci.(N,Y.), 116(2-4), pp. 17

86、7-203.</p><p>　　[7] Haber, R. E., Alique, A., Alique, I. R,, Hem:iodcz, J., and Uribe-Elxabarria.R., 2003, "Embedded Fuzzy Control System for Machining Processes. Resuhs of a Case-Study,"Comput Ind

87、., 50, pp. 353-366.</p><p>　　[8] Sinumerik 840d. OEM,package NCK, software release 4, User's Manual, Siemens AG, 1999,</p><p>　　[9] Haber, R. E., Haber, R. H., Alique, A.. and Ros, S., 2002,

88、 "Application of Knowledge-Based Systems for Supervision and Control of Machining Processes," in Handbook of Software Engineering and Kmm'ledge Engineering 2,S. K. Chang, ed., World Scientific. Singapere, p

89、p. 673-710.</p><p>　　[10] Haber, R. E., Haber-Haber, R.. and Alique, A., 2000, "HierarchicaJ Fuzzy Control of the Milling Process with a Self-Tuning Algorithm." in Proceedings of the IEEE Internati

90、onal symposium on httelligent Control. Patras, Greecc,pp. 115-120.</p><p>　　[11] Jiménez, J. E., Haber, R. E., and Alique, J. R.. 2004. "A MIMO Fuzzy Control System for High Speed Machining Process

91、es. Results of a Case Study," in Proceedings of the IEEE Conference on Fuzzy Systems, Budapest, Hungary,pp. 901-905.</p><p>　　[12] Haber, R. E., Schmiu-Braess, G., Haber-Haber. R., Aliquc. A., and Aliqu

92、e, J.R., 2003, "Using Circle Criteria for Verifying Asymplotic Slability in PI-Like Fuzzy Control Systems. An Application to the Milling Process," IEE Proc.:Control Theory Appl. 150(6), pp. 619-627.</p>

93、<p>　　附錄B：英文資料翻譯</p><p>　　高速銑加工中軸轉(zhuǎn)矩的模糊控制</p><p>　　這篇論文介紹了把一個(gè)內(nèi)含兩輸入/兩輸出基于邏輯轉(zhuǎn)矩的模糊控制系統(tǒng)的開環(huán)數(shù)控系統(tǒng)用于使材料切除速率最優(yōu)的高速銑加工的設(shè)計(jì)和執(zhí)行。這個(gè)控制系統(tǒng)同時(shí)調(diào)整了流入速度和軸轉(zhuǎn)速當(dāng)做需要使用數(shù)控系統(tǒng)自身的資源調(diào)整切割扭矩。這個(gè)控制系統(tǒng)內(nèi)含一個(gè)標(biāo)準(zhǔn)開放控制內(nèi)核，由一個(gè)兩輸入（也就是，轉(zhuǎn)矩誤差和

94、轉(zhuǎn)變誤差），兩輸出（流入速度和軸速增量）模糊控制器組成。兩個(gè)途徑被試驗(yàn)過，并且使用單獨(dú)的性能測(cè)量來評(píng)定他們的性能。這兩種途徑是分別用一個(gè)兩輸入/兩輸出模糊控制器和一個(gè)單輸出（也就是，只有流入速度修正）模糊控制器。結(jié)果證明被提議的控制策略比其它策略提供更好的精確度和加工周期，因此增加了金屬切除速度。</p><p>　　關(guān)鍵詞：模糊控制，轉(zhuǎn)矩，高速銑床</p><p><b>　　

95、緒論</b></p><p>　　為了以更高的材料切除速度來提高高速銑加工的加工效率，這個(gè)研究集中在軸轉(zhuǎn)矩的一個(gè)兩輸入/兩輸出的模糊控制系統(tǒng)的設(shè)計(jì)和執(zhí)行。處理的主要問題是新的發(fā)展和在使用數(shù)控特有資源的同時(shí)模糊邏輯的應(yīng)用。不需要額外的硬件，因?yàn)檫@個(gè)控制運(yùn)算法則是內(nèi)含標(biāo)準(zhǔn)開放控制內(nèi)核的。模糊邏輯是從所有可利用的技術(shù)中選出的，因?yàn)樗C明了在對(duì)控制和工業(yè)工程方面做為一個(gè)非常實(shí)用的最優(yōu)化工具是有用的。盡我們所知

96、，這個(gè)途徑的主要優(yōu)勢(shì)是它包括了：1）內(nèi)含在開環(huán)數(shù)控的一個(gè)模糊控制器用來處理生產(chǎn)環(huán)境；2）實(shí)現(xiàn)時(shí)間要求的一個(gè)簡(jiǎn)單計(jì)算的程序；3）傳感器成本范圍（開環(huán)數(shù)控提供轉(zhuǎn)矩信號(hào)）、配線、或者與數(shù)控系統(tǒng)同步?jīng)]有限制。</p><p>　　這篇論文組織起來如下：在第二部分我們介紹關(guān)于一個(gè)機(jī)械論模型的簡(jiǎn)短研究來預(yù)言切削力和軸轉(zhuǎn)矩；在第三部分我們描述使銑加工最優(yōu)化的模糊控制器的設(shè)計(jì)；在第四部分我們描述模糊控制器如何才能被嵌入到開環(huán)數(shù)控

97、里，并且討論關(guān)鍵設(shè)計(jì)和規(guī)劃發(fā)展的進(jìn)程；在第五部分我們回顧實(shí)驗(yàn)結(jié)果并且探測(cè)一些比較的研究。最后，我們?cè)诘诹糠纸榻B結(jié)論。</p><p>　　基于軸轉(zhuǎn)矩信號(hào)的高速銑加工控制</p><p>　　這個(gè)機(jī)械論模型評(píng)估了切削力向量和以流入速度、軸轉(zhuǎn)速、材料數(shù)量為基礎(chǔ)的軸轉(zhuǎn)矩。這個(gè)研究中最終銑執(zhí)行的機(jī)械論的模型是基于文獻(xiàn)[1，2]。為了做一個(gè)簡(jiǎn)單的模型，我們使用接下來的文獻(xiàn)[3，4]：如果每齒流入值

98、ft j小于刀具半徑D/2，那么對(duì)于流入X正方向，即時(shí)碎片厚度hj涉及到刀具的旋轉(zhuǎn)角。當(dāng)切斷刀具旋轉(zhuǎn)，碎片厚度改變被看作一個(gè)關(guān)于切斷刀具半徑角(Φ) 和 軸向角(κ)的函數(shù)。</p><p>　　hj(Φj) = fij sinΦj·sin κ (1)</p><p>　　因此，在以近似值為基礎(chǔ)上，對(duì)于粗磨和半完成切斷，沒有必要從大量信

99、息獲得即時(shí)的碎片厚度。當(dāng)然，每齒流入值ft j的校正值不僅增加了刀具性能，而且提高了機(jī)加工的效率。對(duì)于一個(gè)圓柱端銑刀，發(fā)現(xiàn)常規(guī)解決方法需要以下條件。</p><p><b>　　r(z) = </b></p><p><b>　　κ= 90º</b></p><p><b>　　(2)</b>

100、;</p><p><b>　　Ψ= k0z</b></p><p>　　k0= (2 tanθ)/D</p><p>　　滯后角Ψ(z)的出現(xiàn)是由于切削刀具的螺旋角θ。在這個(gè)圓柱端銑刀的例子中，這個(gè)角是不變的，而它在球銑刀中是變化的。切線的、半徑的、軸向的力的微分(dFt), (dFr), (dFa)作用于刀具切削刃的一個(gè)無限小的長(zhǎng)度dS [

101、4]。</p><p>　　dFt = Kt edS + Kt chj( Φ, κ)db</p><p>　　dFr = Kr edS + Kr chj( Φ, κ)db (3)</p><p>　　dFa = Ka edS + Ka chj( Φ, κ)db</p><p><b>　

102、　也可以寫成</b></p><p>　　db = (4) </p><p>　　此外，切斷表面的分?jǐn)?shù)特征相當(dāng)于刀具上運(yùn)動(dòng)學(xué)硬度和工件上的位移。這些常數(shù)或切斷系數(shù)(Kt c,Kr c,Ka c,Kt e,Kr e,Ka e)可以用試驗(yàn)方法建立，就是用同一種種型的刀

103、具和材料的平均每齒的切削力[5]。</p><p>　　總切削力可以被看作關(guān)于和工件有聯(lián)系的所有切削刃順著軸向的切削深度Φ的一個(gè)函數(shù)，它可以這樣被計(jì)算出</p><p>　　N f N f z2 </p><p>　　Fx(Φ) =∑(Fx j(Φj(z)) =∑∫ [-dFr jsin Φj sin κj - dFi jcos Φj - dF