Methods Applied for Determining Mathematical Models of Force and Torque in Widening 10TiMoNiCr175 Stainless Steel Holes

MIHAIELA ILIESCU, RUXANDRA BĂLAN

Manufacturing Department “POLITEHNICA” University of Bucharest

Splaiul Independentei no. 313 Street, District no. 6, zip code 060042 ROMANIA

iomi@clicknet.ro; http://www.imst.pub.ro Abstract: - Mathematical models of axial force and torque, when drilling or widening are important for scientific research and / or industrial application as, thus it would be possible to determine optimum process parameters values or, predict force and torque values, once process parameters set.. Widening holes in stainless steel is many times required and, given the fact that existing data on this topic are very poor, a study on 10TiMoNiCr175 has been done and results presented by this paper. Key-Words: - mathematical model, widening, force, torque, stainless steel 1 Introduction The corrosion resistance of iron-chromium alloys was first recognized in 1821 by French metallurgist Pierre Berthier, who noted their resistance against attack by some acids and suggested their use in cutlery. There are over 150 grades of stainless steel, of which fifteen are most commonly used. [7], [10]. Their application fields are more and more versatile as well as challenging, due to their important physical and mechanical characteristics, specially, their high corrosion resistance to various chemical agents and their impressive good look. Stainless steel alloy is milled into coils, sheets, plates, bars, wire, and tubing to be used in cookware, cutlery, hardware, surgical instruments, major appliances, industrial equipment (for example, in sugar refineries) and as an automotive and aerospace structural alloy and construction material in large buildings – see figure 1 . Stainless steel parts usually need machining to geometric precision required. These materials are very tough, with low thermal conductivity and, while machining they do involve sever wear of the cutting tool, as well as, high value cutting forces [6]. As they are expensive materials, detailed research on their machinability would be opportune, so that optimum process parameters values to be set and, thus, high productivity and low cost stainless steel parts could be obtained. Specially when drilling and / or widening are involved, one important aspect of material’s machinability consists in axial cutting force and torque values. So, the higher these values, the poorer machinability properties and the higher energy consumption [3].

[8]

[9] Fig. 1 Examples of stainless steel application

Specific literature presents relationships involving cutting force and torque but, when experimentally checking them, high difference of the modeled ones from the real experimental values can be noticed [2], [4], [6]. So, it has been considered appropriate the study presented by this paper, meaning to determine new adequate mathematical models of the axial force and torque in widening 10TiMoNiCr175 stainless steel holes.

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 306

2 Applied Methods Experimentally determining the mathematical model of parameters specific to a machining process, involves the definition of both independent and dependent variables associated to it [1]. Then, the appropriate experiments design has to be considered and, consequently, the fitted mathematical model should be obtained. The mathematical relations regarding axial cutting force in widening stainless steel material holes, that are mentioned by most of the articles and books dealing with this problem, do look like

[N] (1)

FFF zyf

xFz faDCF ⋅⋅⋅=

[Nm] (2) MMM zyf

xM faDCM ⋅⋅⋅=

where: Fz represents the axial cutting force; M – the torque; D – the diameter of the widening tool, [mm]; af – cutting feed, of the widening tool, [mm/rot]; f – cutting depth [mm] , , , - polytropic exponents; Fx Fy Mx My , - constants. FC MC

If experiments were carried out, once the values of , , , , , , and known, the same values of variables D, af and f experimented with different values of cutting speed, v, generated different axial force and torque values. So, one could think that the parameter not mentioned by relations (1) and (2), meaning cutting speed, should play an important role in axial force and torque values’ prediction.

FC MC Fx Fy Fz Mx My Mz

As consequence of the above mentioned, this paper presents new mathematical relations of axial cutting force and torque, where one more independent variable appear, meaning the cutting speed v [mm/rot]. So, the new, original proposed mathematical models are: [N] (3) vFFF wzy

fx

Fz vfaDCF ⋅⋅⋅⋅=

[Nm] (4) MMMM wzyf

xM vfaDCM ⋅⋅⋅⋅=

where: v is peripheral rotational speed of the drilling tool, usually mentioned as cutting speed [m/min]; , - polytropic exponents; Fw Mw For obtaining the constants and polytropic exponents’ values, relations (3) and (4) must be of linear type and, so, by logarithm their linear expressions are:

vwfz

ayDxCF

FF

fFFFz

lglglglglglg

++

+++= (5)

vwfz

ayDxCM

MM

fMMM

lglglglglglg

++

+++= (6)

So, the first method applied was that of solving a five linear equations system – as there were five constants to be determined (C, x, y, z and w). The second method applied dealt with experiments design and regression analysis – done with a specialized software, DOE KISS. [5] Due to limited license rights, there could only be studied the two most significant variables influencing force and torque (based on solving previously mentioned (5) and (6) relations). Thus, there were further studied only D and f parameters. The experiments design (Central Composite Design, CCD) is evidenced in table 1. Regression analysis performed by the software results in models like:

2

222

11

12210

faDa

fDafaDaaFz

⋅+⋅+

+⋅⋅+⋅+⋅+= (7)

2

222

11

12210

faDa

fDafaDaaM

⋅+⋅+

+⋅⋅+⋅+⋅+= (9)

The relationship of coded variables, and real

ones, is the following: jx

jz

2

2minmax

maxmin

zz

zzz

xj

j −

+−

= (10)

where: is the minimum experimental value minz - the maximum experimental value maxz

Table 1 Experiments Design

Run x1 x2

1. -1 -1 2. -1 +1 3. +1 -1 4. +1 +1 5. 0 0 6. 0 0 7. -1 0 8. +1 0 9. 0 -1

Cen

tral C

ompo

site

Des

ign

(CC

D)

10. 0 +1

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 307

3 Experiments and Results The experimental system designed and used in experiments, consists in the main components mentioned next – see figure 2.. → Drilling machine tool, coded GC032DM3. - electric motor of 3,5 kW power; - working plateau dimensions of 420 × 480 (mm);

- coupled to a data acquisition system, using the graphical programming LabVIEW software.

- main spindle with a no. 4 Morse cone. - available rotational speed range values of the drilling tool: 70 ÷ 1400 [rot/min], with 12 geometrical ratio levels variation; - possible cutting feed values: 0.12; 0.20; 0.32; 0.50 [mm/rot]. → Cutting tools made of Rp5 material with no. 62 Rockwell hardness .

→ Special rotational device, for adequate torques and forces measurements - individualized by an elastic sleeve with four resistive transducers, each inclined by 45° with respect to horizontal and vertical axes attached to it. → IEMI type electronic bridge

→ Studied material: 10TiMoNiCr175 (STAS 3583-87) → Cooling/lubricating fluid: 20% P emulsion. Experimental values obtained for the axial force and torque, in widening are shown in Table 2.

Table 2 Experimental results

Exp. No.

Initial hole diameter, D0 [mm]

Cutting tool

diameter, D [mm]

Cutting depth

f [mm]

Cutting feed

af [mm/rot]

Rotational speed

n [rot/min]

Cutting speed

v [m/min]

Axial force Fz [N]

Torque M [Nm]

10TiMoNiCr175 (stainless steel) 1 16 24 4 0.12 224 16.88 1163 20.91

2 12 16 2 0.20 355 17.83 513 11.41 3 16 24 4 0.32 224 16.88 1872 30.27 4 18 24 3 0.12 224 16.88 858 17.28 5 16 24 4 0.12 355 26.75 119/8 21.60

where: n is the rotational speed of the machine tool’s main spindle:

Dvn

π=

1000 [rot/min]

Fig. 2 Experimental stand - for study on widening 10TiMoNiCr175 stainless steel holes

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 308

The obtained experimental results were further “processed”, in order to solve the five equations system, required for model’s coefficients determination - see relations (1) and (12):

88.16lg4lg

12.0lg24lglg1163lg

FF

FFF

wzyxC

+⋅++⋅+⋅+=

83.17lg2lg

20.0lg16lglg513lg

FF

FFF

wzyxC

+⋅++⋅+⋅+=

88.16lg4lg

32.0lg24lglg1872lg

FF

FFF

wzyxC

+⋅++⋅+⋅+=

88.16lg3lg

12.0lg24lglg858lg

FF

FFF

wzyxC

+⋅++⋅+⋅+=

75.26lg4lg

12.0lg24lglg1198lg

FF

FFF

wzyxC

+⋅++⋅+⋅+=

88.16lg4lg

12.0lg24lglg91.20lg

MM

MMM

wzyxC

+⋅++⋅+⋅+=

83.17lg2lg

20.0lg16lglg41.11lg

MM

MMM

wzyxC

+⋅++⋅+⋅+=

88.16lg4lg

32.0lg24lglg27.30lg

MM

MMM

wzyxC

+⋅++⋅+⋅+=

88.16lg3lg

12.0lg24lglg28.17lg

MM

MMM

wzyxC

+⋅++⋅+⋅+=

75.26lg4lg

12.0lg24lglg60.21lg

MM

MMM

wzyxC

+⋅++⋅+⋅+=

Solving the equations system results in the real values of constant and , , , polytropic exponents [5].

MC Mx My Mz Mw

Thus, knowing that initial dependence relationships were exponential – see relations (5) and (6) - and that the ones used in equations systems are obtained from the first ones, by logarithm – see relations (11) and (12) - the mathematical models of the axial force and torque, in widening 10TiMoNiCr175 stainless steel holes are :

[N] (13) 065.056.1485.0832.05.44 vfaDF f ⋅⋅⋅⋅=

[Nm] (14) 070.0664.0377.0844.004.1 vfaDM f ⋅⋅⋅⋅=

Due to the fact that relation (13) and (14) were obtained by solving classical equations system – meaning five unknown parameters and five equations, an improvement of improve the obtained mathematical models was considered to be right.

So, for the new set of experiments, the values of the independent variables af and v were set to 0.20 (mm/rot) and, respectively, 17.83 {m/min). The other two variables, meaning D and f were varied according to the CCD – DOE KISS software. It should be mentioned that, for each experience, replicates number equaled five. The obtained experimental results for axial force, Fz and torque, M are presented in table 3 Regression analysis results are presented in figure 3.

(11)

Some aspects, like the ones that follow, should be evidenced, meaning: → a factor is considered to have significant influence on the output as long as the P (2 Tail) value is less. or equal, to 0.05; → when using the software, there are considered the coded values for the variables studied → the relationship of coded and real variables values (x1, x2 and D, f) is defined by relation (13) – see, also relation (10) .

3

151

−=

Dx ; (13) 32 −= fx

→ the software enables getting the marginal means plot to strengthen which of the two independent variables considered do higher influence the output – see figure 4

(12)

Table 3 Experiments Results – for DOE KISS

Axial force, Fz [N] Exp. no. 1 2 3 4 5 6

403.

61

840.

11

566.

19

1177

.19

787.

68

789.

01

Exp. no. 7 8 9 10

620.

09

868.

79

513.

33

791.

31

Torque, M [Nm]

Exp. no. 1 2 3 4 5 6

20.0

8

31.8

3

28.2

8

44.8

1

33.5

2

35.0

1

Exp. no. 7 8 9 10

CC

D

26.2

9

37.0

2

25.6

1

40.5

7

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 309

- for axial force, Fz

- for torque, M Fig. 3 DOE KISS -regression analyses results

- for axial force, Fz - - for torque, M -

Fig. 4 Marginal means plots

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 310

So, based on all the above mentioned, mathematical models of axial force and torque in widening 10TiMoNiCr175 stainless steel holes, obtained by regression analysis with DOE KISS software are:

[N] 22 96.5991.354.14

56.34424.11911.483

fDfD

fDFz

⋅−⋅+⋅⋅+

+⋅+⋅−=

(14)

[Nm] 22 74.024.040.0

64.581.729.110fDfD

fDM⋅−⋅−⋅⋅+

+⋅+⋅+−=

(15) 4 Conclusion Stainless steels application fields are more and more versatile as well as challenging, due to their important physical and mechanical characteristics, Stainless steel parts usually need machining to geometric precision required. This paper presents two methods used for determining new adequate mathematical models of the axial force and torque in widening 10TiMoNiCr175 stainless steel holes. The first method, consisted in solving a five linear equations system. Obtained models proofed that the factors significantly influencing axial force and torque values are the tool diameter, D and cutting depth, f. The second method, involved design of experiments and regression analysis. Using a specialized software (DOE KISS), there were obtained polynomial type models – that evidenced the influence of all studied variables combination, on axial force and torque values. Further research involves other process parameters and materials to be studied, as well as the application of obtained models on real time control of the machining processes.

References: 1] Iliescu M., Vlădăreanu L, Statistic Models of Surface Roughness MET 4 Metallized Coating in Grinding Manufacturing System, 12th WSEAS International Conference on Systems, pg. 777- 782, ISSN 1790-2769, Greece, July, 2008 [2] Iliescu M, Vlase A., New Statistical Models of Cutting Tool Wear and Cutting Speed in Turning 20MoCr130 Stainless Steel, 10th WSEAS MACMESE'08, pg. 204-209, ISSN 1790-2769, Bucharest, Romania, November 7-9, 2008 [3] Iliescu M, Vlase A, Maturin S, Processing Transducers’ Signals when Determining Mathematical Models of Force and Torque in Drilling 2MoNiCr175 Steel, 8th WSEAS International Conference ISPRA '09, pg. 45-48, ISSN 1790-5117, Cambridge, UK, February 21- 23, 2009 [4] Hsu-Hwa Chang, Chih-Hsien Chen, Use of esponse Surface Methodology and Exponential Desirability Functions to Paper Feeder Design, WSEAS TRANSACTIONS on Applied and Theoretical Mechanics Issue 9, Volume2, September 2007, ISSN: 1991-8747

[5] Montgomery D, Runger G, Applied Statistics and Probability for Engineers, John Wiley & Sons, Inc., 2003 [6] Vlase I, Contribution to Determining of Some Indexes for Quantifying the Stainless Refractory Steels Machinability, Doctoral Thesis, 2002.. [7] http://en.wikipedia.org/wiki/Stainless_steel accessed on March, 2011. [8] http://www.brass-screws.com/stainless-steel- machined-parts.htm, accessed on May, 2011. [9] http://www.euro-inox.org/pdf/build/envelope/ accessed on May, 2011. [10] http://www.ssbearings.com/enpro5.asp accessed on April, 2011.

Recent Researches in Computational Techniques, Non-Linear Systems and Control

ISBN: 978-1-61804-011-4 311