Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8) New Mathematical Models o Axial Cutting Force and Torque in Drilling 2MoCr13 Stainless Steel MIHAIELA ILIESCU, AURELIAN VLASE Manuacturing Department POLITEHNICA University o Bucharest Splaiul Independentei no. 313 Street, District no. 6, zip code 642 ROMANIA Abstract: - Stainless steels represent materials which have known a continuous extend into various and interesting industrial ields application. Most o the time they needs machining and, one important procedure is drilling. The paper present aspects o the experimental research developed in the purpose o determining new and, more adequate, mathematical models o the cutting orce and torque, in drilling 2MoCr13 stainless steel. Graphs, as well as urther application o the obtained relationships are also, mentioned. Key-Words: - axial cutting orce, torque, drilling, stainless steel, mathematical model 1 Introduction There are, almost, 1 years since stainless steels have been discovered and, nowadays, their application ields are various and challenging. That is because o their important physical and mechanical characteristics, most o all, their high corrosion resistance to various chemical agents and, why not, because o their impressive good look [2]. Usually, ater obtaining, as rough material, machining is necessary, so as to obtain the shape, dimensions and surace roughness o the stainless steel part. These steels are very tough, with low thermal conductivity and, while machining, determines sever wear o the cutting tool, as well as, high value cutting orces [6]. Because o their high prices, researches on their machinability are necessary, in order to optimize the machining process, meaning, having high productivity and low costs o stainless steel parts. One important aspect o material s machinability is represented by the values o the cutting orce and torque, meaning the higher the values, the lower machinability is so, resulting a low energy eiciency use [3]. Speciic literature presents some relationships regarding variables o the machining process involving orce and torque but, when experimentally checking them, one can notice, relative high dierence (o the modeled ones) rom the real obtained values [5]. So, it has been considered useul to determine adequate models o some machining process parameters, regarding one widely used Romanian stainless steel 2MoCr13. Obtaining holes, in stainless steel parts, o various dimensions and precisions, is done by drilling. Cutting orce, specially axial one, and cutting torque, are important parameters (output variables) o the drilling process and, can be oten used or its optimization. Based on the above, this paper presents the experimental steps carried out in order to determine some mathematical models o axial cutting and torque in drilling 2MoCr13 stainless steel. 2 Research Methodology In order to experimentally determine a mathematical relationship o variables speciic to a machining process, there has to be mentioned, both the independent and the dependent ones [1]. Ater doing that, the dependence relation type must be settled and, correspondingly, the appropriate experiments design established. The mathematical relations, regarding axial cutting orce, in drilling stainless steel materials, presented by most o the articles and books dealing with this problem, are o the type: x F y F = C D a F [N] (1) F x M y M = C D a M [Nm] (2) M where: F is the axial component o the cutting orce; M the drilling torque; D the diameter o the drilling tool, [mm]; a cutting eed, o the drilling tool, [mm/rot]; x F, y F, x M, y M - polytropic exponents; C F, CM - constants. ISSN: 179-2769 21 ISBN: 978-96-474-19-2
Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8) I experiments were carried out, once the values o C F, C M, x F, y F, x M, y M known, or the same values o cutting tool s diameter and cutting eed but, or dierent values o cutting speed, dierent axial orce and torque values were obtained. So, one could think that the parameter not mentioned by relations (1) and (2), meaning cutting speed, should play an important role in drilling axial orce and torque prediction. As consequence o the above mentioned, this paper presents another mathematical relationship o the axial cutting orce and, respectively, o the torque, where one more independent variable appear, meaning the cutting speed v [mm/rot]. So, the new, original proposed mathematical models are: xf yf z F = C v F D a v [N] (3) xm ym z M = C M M D a v [Nm] (4) where: v is peripheral rotational speed o the drilling tool, usually mentioned as cutting speed [m/min]; z v, z M - polytropic exponents; For obtaining the constants and polytropic exponents values, relations (3) and (4) must be o linear type and, so, by logarithm their linear expressions are: lg F = lg CF lg D + yf lg a + zf lg v (5) lg M = lg CM lg D + ym lg a + zm lg v (6) 3 Mathematical Models Obtaining inal ormula o the new mathematical models implies, irst, experiments and, ater that, constants and polytropic exponents determination. There has been used a cooling/lubricating luid, 2% P emulsion. Cutting tools were helix drilling ones, made o Rp5 material and having Rockwell hardness no. 62. The edge angle was 2 χ =14 and the diameter values considered were: Φ 1 = 8 ; Φ 2 = 12 ; Φ 3 = 14 [mm] As or the drilling experimental conditions, they were according to R137/2-69 Standard, type A. For adequate measuring o axial cutting orces and torques, in drilling, there has been deigned and manuactured a special rotational device. Its most important element is represented by the elastic sleeve, on which there were attached our resistive transducers, each inclined by 45 with respect to horizontal and vertical axes. The exit cables o this device were connected to an IEMI type electronic bridge which, was coupled to a data acquisition system, using the graphical programming LabVIEW sotware. see Figure 1. The studied material was 2MoCr13, its chemical structure being presented in Table 1, while its mechanical characteristics are mentioned by Table 2 Table 1 Chemical Structure. C Mo Ni Cr Mn.2 3.82.24 13.2.68 Si S P.3.17.35 Table 2 Tensile Strength, R m [N/mm 2 ] Mechanical Characteristics. Flow Strength, R 2 [N/mm 2 ] Relative Elongation δ Hardness, HB 92 67 1.7 24 3.1 Experiments There were carried out experiments under specially designed conditions. So, the machine tool was a drilling machine, coded GC 32DM 3. whose electric motor had 3,5 kw power. The working table dimensions were 42 48 (mm) and the main spindle had a no. 4 Morse cone. Possible rotational speed range values o the drilling tool were 7 14 [rot/min], with 12 geometrical ratio levels variation and possible cutting eed values were.12;.2;.32;.5 [mm/rot]. Fig.1 LabVIEW graphical program ISSN: 179-2769 211 ISBN: 978-96-474-19-2
Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8) Images o the designed stand, taken while experimenting are shown in igure 2. An example o the obtained graphic, or all drilling orce s components, as well as or the power involved by the process is presented in Figure 3. Fig.2 Experimental stand Fig.3 LabVIEW data acquisition drilling orce s components graphics ISSN: 179-2769 212 ISBN: 978-96-474-19-2
Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8) Exp. No. Cutting Tool Diameter, D [mm] Cutting Feed a [mm/rot] Table 3 Experimental results Rotational Speed n [rot/min] Cutting Speed v [m/min] Axial Cutting Force, Drilling Torque 1 8.12 56 14.7 2282 4.97 2 8.2 56 14.7 2976 6.68 3 8.12 9 22.61 1998 4.43 4 12.12 56 21.1 3522 9.54 5 12.2 56 21.1 452 12.58 6 14.12 9 39.56 3565 1.71 where: n is the rotational speed o the machine tool s main spindle v n = 1 [rot/min] πd Experimental values obtained or the axial orce and torque, in drilling 2MoCr13 stainless steel are shown in Table 3. 2.2 Obtaining Mathematical Models Based on the experimental results, and on research methodology mentioned, the equation systems necessary or models determination are as ollows: lg 2282 = lg CF lg8 + yf lg.12 + zv lg14.7 lg 2976 = lg CF lg8 + yf lg.2 + zv lg14.7 lg1998 = lg CF lg8 + yf lg.12 + zv lg 22.61 lg 3522 = lg CF lg12 + yf lg.12 + zv lg 21.1 (7) lg 4.97 = lgcm lg8 + ym lg.12 + zm lg14.7 lg 6.68 = lgcm lg8 + ym lg.2 + zm lg14.7 lg 4.43 = lgcm lg8 + ym lg.12 + zm lg 22.61 lg 9.54 = lgcm lg12 + ym lg.12 + zm lg 21.1 (8) By solving the equations systems, the values o constants, C F, C M and polytropic exponents, x F, y F, x M, y M are obtained [5]. Knowing that the initial dependence relationships were exponential ones, and the ones used in solving are obtained rom the irst ones, by logarithm, the inal mathematical models o the axial cutting orce and torque, in drilling 2MoCr13 stainless steel are: 1.35,52.28 F = 87D a v [N] (9) 1.85,58.24 =.684D a v M [Nm] (1) Graphs o the axial orce and torque variances, on some o the considered variables are shown in Figure 4 and, respectively, Figure 5. 8 7 6 5 4 3 2 1 25 2 15 1,5,1,15,2,25,3,35 5 v = 2 m/min a [mm/rot] v = 2 m/min,5,1,15,2,25,3,35 a [mm/rot] Fig.4 Axial orce, F, and torque variation, M, on cutting eed, a ISSN: 179-2769 213 ISBN: 978-96-474-19-2
Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8) 6 5 a =.12 m/rot Figures 6 and 7 show the obtained results meaning constants and coeicients values, standard errors o the coeicients, standard error, R 2 coeicient (determination coeicient), values o t and F tests, etc. 4 3 2 v = 14,7 [m/min] v = 21,1 [m/min] Based on this regression analysis, the mathematical models or the axial component o e drilling orce, F and or the drilling torque, M are as shown by relations (11) and (12): 1 v = 39,56 [m/min] 5 1 15 D [mm] 16 14 v = 14,7 [m/min] 12 1 8 a =.12 m/rot 6 v = 39,56 [m/min] 4 2 5 1 15 D [mm] v = 21,1 [m/min] Fig.6 SPC KISS regression analysis or axial orce, F model Fig.5 Axial orce, F, and torque variation, M, on cutting tool diameter, D As relations (9) and (1) were obtained by solving classical equation systems our unknown parameters and our equations, one can think o trying to improve the obtained mathematical models. That is i, changing one o the equations in systems (7) and, respectively, (8), by considering another experimental results rom Table 3 experiment number 5 or, 6, dierent values or the constants and polytropic exponents should be obtained. So, considering all the values rom Table 3, and using a specialized sotware SPC KISS, regression analysis has been carried out [4]. Fig.7 SPC KISS regression analysis or torque, M model ISSN: 179-2769 214 ISBN: 978-96-474-19-2
Proceedings o the 1th WSEAS International Conerence on MATHEMATICAL and COMPUTATIONAL METHODS in SCIENCE and ENGINEERING (MACMESE'8),2,18,16,14,12,1,8,6,4,2,5,1,15,2,25,3,35 16 14 12 1 8 6 4 2.58,5,1,15,2,25,3,35 1.399.6 F = 648D a v [N] (11).8 1.555 a [mm/rot].6 v = 2 m/min v = 2 m/min a [mm/rot] Fig.8 Axial orce, F, and torque variation, M, on cutting eed, a M =.893D a v [Nm] (12) Graphs o the axial orce and torque variances, are shown in Figure 8. 4 Conclusion Analyzing the mathematical models obtained, can be noticed that the ones obtained by SPC KISS sotware do not it the experimentally obtained results. Even the regression analysis proved to be adequate, or real, when checking with the observations resulting rom experiments, high dierences occurred. Perhaps the data, were not enough or, not adequately established or the implied regression sotware. So, there will be considered right, only the mathematical models given by relation (9) or axial drilling orce and, by relation (1) or drilling torque. From these models, one can notice that the higher inluence on the dependent variable (F, or M) is that o cutting tool diameter, D, meaning, the larger the drilled hole, the higher the orce and moments values. It can, also, be noticed out o the variation graphs plotted in Figure 5. The lower inluence, on the same studied dependent variables is that o the cutting speed, v, but, it is a reveres inluence the higher values o v, the lower values o F and M. Both mathematical models, or axial drilling orce and drilling torque, are, somewhat, correlated, meaning it resulted the same, similar, inluences o the independent variables studied (D, s, v). Once determined, the considered models were urther checked, by more experiments or dierent values o the parameters. All the experimentally obtained results were in good concordance with the mathematically predicted values. Further research should be developed so as, to implement the obtained results mathematical models, into an automated optimization system.o the manuacturing process. Reerences: [1] Iliescu M., Vlădăreanu L, Statistic Models o Surace Roughness MET 4 Metallized Coating in Grinding Manuacturing System, 12 th WSEAS International Conerence on Systems, pag. 777-782, ISSN 179-2769, Greece, July, 28 [2] Grigoriu M, Gheorghiu L., Energy Eiciency Pumping Systems Improvement Method, Energetica Journal, no. 2/28, pag. 74-78, ISSN 1453-236, Bucharest, 28 [2] Dumitrescu C. I.,, Grigoriu M, National Energy Market Highliths, Energetica Journal, no. 4/28, pag. 19-113, ISSN 1453-236, Bucharest, 28 [3] Montgomery D, Runger G, Applied Statistics and Probability or Engineers, John Wiley & Sons, Inc., 23 [4] Ghionea A., Vlase A, Ghionea I, Wear Evaluation o Cutting Edge and Cutting Speed in Drilling Process o Some Manganese Steel, Proceedings o 18 th International DAAAM Symposium ISSN 1726-9679, Austria, 27 [5] Vlase I, Contribution to Determining o Some Indexes or Quantiying the Stainless Reractory Steels Machinability, Doctoral Thesis, 22. ISSN: 179-2769 215 ISBN: 978-96-474-19-2