A mathematical model for roughness in case of vibrorolling process.

Nagit, Gheorghe ; Fetecau, Catalin ; Negoescu, Florin 等

1. INTRODUCTION

In case of industrial processing, it is very important to know roughness parameter [R.sub.a] or [R.sub.z], which are currently determined by measurements.

The roughness parameter [R.sub.a] can be considered the main parameter of surface condition. Calculation of average deviation for [R.sub.a] can be done using the following relation (Nagit, 1997):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where: [L.sub.1] is the length of surface sector where the roughness is determined and y corresponds to surface profile, determined in XOY coordinate system. OX axis coincides with profile medium line.

2. DETERMINATION OF THE MATHEMATICAL MODEL

Starting from the geometrical similarity of turning process, where cutting tools with radius in the tool centre are used and from the mathematical model (2) of roughness (Braha et al., 2003) one will analyze figure no. 1.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where: r is the center tool radius, f is the work feed and m is a parameter defined as the distance from center point of the tool radius to medium line and has the following value (Nagit et al., 1998):

m = [square root of 4[r.sup.2] - [f.sup.2]/4] + 4[r.sup.2]/f arcsin f/2r (3)

In case of vibrorolling process there appears the sphericity radius ED having the value [d.sub.b]/2 and not radius of the tool point as in the case of the turning process.

[FIGURE 1 OMITTED]

In case of spherical diamond tool, radius will be [r.sub.v]. The feed f will be replaced with [L.sub.[alpha]], which is the minimal distance projection between elementary cell points L on feed direction.

[L[alpha]] = L x cos [alpha] (4)

or

[L.sub.[alpha]] = L x [square root of 1/1 + [tg.sup.2] [alpha]] (5)

where [alpha] is the network declination angle, determined by its tangent (Kvackaj, 2006):

tg [alpha] = 2 x e x i/[pi] x [d.sub.s] (6)

where: 2e represents the oscillation amplitude of deformation element;

i = [n.sub.cd]/n is synchronism rapport between double courses number in one minute of deformation element ([n.sub.cd]) and workpiece rotation speed;

[d.sub.s] is the workpiece diameter.

It is well known that in case of vibrorolling a micro-relief with a channel-network appears or a completely new micro-relief can be obtained, depending on the characteristics and properties of the work-piece that have to be improved (functional characteristics or surface quality (Nagit et al., 2003). [paragraph]

In case of completely new micro-relief, hexagonal or tetragonal elementary cells may be formed. Under these conditions, L represents the biggest distance between opposite corners of elementary resulted cells and takes the following values (Braha et al., 2003):

L = [square root of 2] x [f.sub.r] (7)

for micro-relief with tetragonal cells and

L = 2/[square root of 3] [f.sub.r] (8)

for hexagonal cells, [f.sub.[gamma]] is the tool progress distance in the direction of network cell inclination angle [gamma].

[f.sub.[gamma]] = [square root of ([pi] x [d.sub.s] x {i}/i).sup.2] + [f.sup.2] (9)

where {i} represents the fractional part of synchronism rapport p, and f is the work feed.

If these parameters are taken into account, [L.sub.[alpha]] can be expressed as follows:

[L.sub.[alpha]] = 2 x [pi] x [d.sub.s]/i x [square root of 1/3 [[pi].sup.2][d.sup.2.sub.s][{i}.sup.2] + [f.sup.2][i.sup.2]/[[pi].sup.2] x [d.sup.2.sub.s] + [(2ei).sup.2] (10)

for hexagonal cells and

[L.sub.[alpha]] = x [pi] x [d.sub.s]/i x [square root of 2 x [[pi].sup.2][d.sup.2.sub.s][{i}.sup.2] + [f.sup.2][i.sup.2]/[[pi].sup.2] x [d.sup.2.sub.s] + [(2ei).sup.2] (11)

for tetragonal cells, respectively.

Considering vibrorolling case, feed value f is replaced in relation number (1) by distance value [L.sub.[alpha]] and radius of the tool point with half of ball diameter [d.sub.b]/2; the relation of [R.sub.a] parameter thus becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Relation (12) represents the mathematical model for roughness parameter [R.sub.a] if taking into consideration a few parameters of vibrorolling behaviour (the pressing force is not taken into account; this parameter has influence on stamped channel depth and on remanent deformation of vibrorolled parts).

3. EXPERIMENTAL VALIDATION

For the experimental validation, a vibrorolling device attached on universal lathe SN 400x100 was used.

The test pieces were realized by five different steels with a chemical composition as follows:

OLC35-0.34%C, 0.6%Mn, 0.2%Si, 0.3%Cr, 0.3%Ni, 0.3%Cu. OLC45-0.44%C, 0.6%Mn, 0.2%Si, 0.3%Cr, 0.3%Ni, 0.3%Cu. OLC15-0.16%C, 0.6%Mn, 0.2%Si, 0.3%Cr, 0.3%Ni, 0.3%Cu. OL60-0.4%C, 0.6%Mn, 0.37%Si.

18MnCr10-0.2%C, 0.2%Si, 1%Mn, 1%Cr.

The test pieces were machined in advance by turning in the same working conditions at diameter: [d.sub.s] = 58 mm. The roughness value after this operation was: [R.sub.a] = 1.5-1.7 [micro]m.

[FIGURE 2 OMITTED]

The input parameters of vibrorolling process were: [n.sub.cd] = 470 cd/min, n = 100 rot/min, 2e = 1.6 mm, db = 15.85 mm, F = 400 N. The feed f was modified so as to obtain a completely new micro-relief.

The results obtained are presented in table number 1 and a graphical representation may be analyzed in figure number 2. Constants: n=100 rot/min; [n.sub.cd]=470 cd/min; 2e=1,6mm; [d.sub.b]=15.85 mm; F=400N; [d.sub.s]=58mm.

4. CONCLUSION

Experimental results and mathematical model respect the same tendency of roughness variation. For small values of feed, the model fits better for steel with low hardness, while for big values of feed, hard steels correspond better to the mathematical model (fig 2.).[paragraph]

These unconformities apear because pressing force is not taken into account. The mathematical model must be revised with a coefficient which describes the dependencies between material hardness and pressing force.

5. ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support offered to researchers of this paper by Romanian Ministry of Education, Research and Youth, through CEEX Grant no. 62/2006--BIOMEC.

6. REFERENCES

Braha, V., Nagit, Gh. & Negoescu, F. (2003). Cold plastic deformation technology (in Romanian), Technical and scientific publishing house CERMI, ISBN 973-667-025-2, Iasi-Romania

Kvackaj, I. (2006). Development of bake hardening effect by plastic deformation and annealing conditions. Availble from:http://public.carnet.hr/metalurg/Metalurgija/2006_vol _45/No_1/MET_45_1_051_055_Kvackaj.pdf Accesed: 2008-05-16

Nagit, Gh., (1997). Theoretical and experimental contributions on vibrorolling process (in Romanian), PhD thesis, Technical University Gheorghe Asachi of Iasi

Nagit, Gh., Braha, V., Slatineanu, L. & Musca, G. (1998). Experimental researches concerning pressure force influence and lubrication on roughness at vibrorolling. Technical Institut's Bulletin of Iasi, Vol. XLIV (2003), pp. 391-396, ISBN 1011-2855

Nagit, Gh., Dodun, O., Negoescu, F. & Ignatescu, E.(2003). Cold superficial plastic deformation-a solution to replace heat treatment in small laboratory, Proceedings of Baia Mare-North University, Vol. XVII, pp. 227-231, ISSN 1224-3264, Baia Mare-Romania

Tab. 1. Experimental and theoretical values of roughness for
external vibrorolled cylindrical surfaces.

[paragraph] f(mm/rot) 0,024 0,029 0,048 0,053 0,065

OLC45 0,26 0,3 0,38 0,4 0,44
OLC35 0,23 0,28 0,33 0,36 0,41
OL60 0,2 0,22 0,26 0,29 0,33
18MnCr10 0,18 0,19 0,23 0,27 0,31
OLC15 0,16 0,18 0,21 0,25 0,29
Model(12) 0,021 0,033 0,086 0,11 0,16