Mathematical model for study and assessing of adhesive attrition process of coupling elements in sliding friction.

Ungur, Petru ; Pop, Petru ; Gordan, Mircea 等

1. INTRODUCTION

In specialty papers [Balasiu, 1990; Grama, 2007; Bowden, 1956; Ceausu, 1980], the wear is defined as a destroyed process of surfaces in contact of parts, in relative friction moving one to another, followed by a change of quality geometry and superficial layer properties of coupling materials.

The wear is a process with evolution in time and due to occurring of failure in working of gages, machine tools, installations, limited work precision [Kragelsky, 1978]. At friction coupling, as altering of contact surface coupling elements state in relative sliding moving has produced dislocation and removing of material from bodies surfaces by interaction of mechanical, thermal and chemical action [Pavlescu, 1983; Tudor, 1984]. The wear of friction coupling surfaces can be quantification by static characteristics:

* wear speed-[v.sub.w] , expressed loss of mass, altering volume or of a dimension in time, being reported of wear at function time-t;

* wear intensity-[I.sub.w], expressed by wear to unite sliding length, knowing as non-dimensional linear intensity wear.

The destroy wear due to excessive material removal from surfaces of friction coupling get to increase roughness and its destroyed. This evolution in time of wear it has known as Lorentz wear curve [Shrader et al, 2000].

This curve get by a plotting of wear function W=f(t), has an inflexion point at maxim wear in working, it's complicate used in practice, liked better a line equation of form [W.sub.m]=f(t) [Ceausu,1980].

The paper has presented the plotting of normal wear curve, accidental or damage wear curve and of wear line, following by a new mathematical model, derivates from them by using correlation of mathematical model and analytical approximation.

The paper's novel consists in new mathematical wear model defined.

2. ASPECTS ABOUT OF LORENTZ WEAR CURVE

After Kragelski, the normal wear of couple in friction has considered normal until the coupling and equipment became inadequate. Continuous operating of device or installation due to passing from normal wear to accidental or damage wear.

[FIGURE 1 OMITTED]

Normal wear of operating and damage wear of couplings in sliding friction has depicted in Fig.1, by well-known Lorentz curve W= f(t) [Georgescu,1984].

The limit of attrition or operating of installation or machine has determined by promptitude of wear process deployment of machine's components. The wear of elements in moving depends by: type of wear and moving, cycle moving, normal force-[F.sub.N], wear speed-[v.sub.w], loading-B, time of loading-[t.sub.B], temperature of working-T, etc. In tribo-technical system [Balasiu, 1990], friction and wear has determined by interaction between characteristic elements of friction state and wear device. A process of friction and wear has dependent of structure system-S and load force's group-B, defined as:

[F.sub.N] = f(B,S) (1)

W = f(B,S) (2)

Where, [F.sub.N] is similar with [F.sub.R]-friction force from coupling; B-load group; S-structure system and W-wear measurement. In Fig.2 has presented wear curve by loading time and in Fig 3 in detail of Lorentz's wear diagram in time. The normal wear is getting by the relation:

[W.sub.2] = W - [W.sub.1] (3)

Where, W-is admissible maxim wear value in working; [W.sub.1]-running in wear value; [W.sub.2]-normal wear value in working.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The wear intensity (Fig.3) has determined with relation:

tan [alpha] = [W.sub.2]/ [W.sub.1] (4)

,or:

tan [alpha] = W - [W.sub.1]/ t-[t.sub.1] (5)

Where: t-is maxim time in operating, [t.sub.1]-time in running in, [t.sub.2]-time in normal operating of equipment. Intensity of wear-tana depends of surfaces roughness, group of pressing forces-B, wear speed-[v.sub.w], material characteristics, lubrication, etc. In calculus, the wear curve W=f(t) is more less used, preferred by line [W.sub.m]=[alpha]-t, as:

t = W/tan [alpha] + ([t.sub.1] - [W.sub.1]/tan [alpha]) (6)

By analysing of Fig.3 and equation (6), it has resulted:

* if angle-[alpha] is small, that wear time-t will be great;

* term ([t.sub.1] - [W.sub.1/tan [alpha]] characterized the running in probe;

* if time of running in-[t.sub.1] has great value, and wear after finished running in-[W.sub.2] is small, that maxim time of operating-t will grow.

Because of complexity wear functions W= f(B,S) and W=f(t), these can be described more easy by quadratic complex functions. A mathematic model of wear as a parabolic, with a low slope has presented in following.

3. NEW MATHEMATICAL MODELS FOR ATTRITION

The wear workpieces depend by quality of cutting surface, materials used, lubrications, and of speed and specific pressure of parts in moving, etc. The attrition goes to altering of shape, volume and weight of parts, without modification of physics-chemical properties of material in friction, exhibited by adhesion, abrasion and erosion. After [Bowden &Tabor, 1956], the abrasion wear is characterising by micro-plastics deformations occurring and by cut of thin metallic layers from abrasive hard particles from surfaces.

Another variant of a wear function of quadratic equation (Fig.4) has given by:

W = 1/2p [t.sup.2] (7)

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Equation (7) is a symmetric parabola in report with axe-OW, with a flatten curve. Where: p-is parameter of wear parabola, t-durability of coupling in sliding friction.

The differentiation of function W= (1/2p)[t.sup.2] in point-to is:

dW = 1/[rho] dt (8)

In Fig.5, the size dt= [DELTA]t is differentiation of independent variable. The wear study has goal approximation of W= (1/2p)[t.sup.2] in proximity of point-A. Diagram from Fig.5 represents wear function, presumed differential in time interval and tangent at this curve in point A [[t.sub.0], f([t.sub.0])], curve-OA represents running in zone. The ordinate of proximity point-B [[t.sub.0] + [DELTA]t, f([t.sub.0] + [DELTA]t] is intersected by tangent in point-T. The portion of curve-AB corresponds of stability wear, and OB of normal operating period. The raising of function [bar.[A.sub.1]T] = [DELTA]W = f([t.sub.0] + [DELTA]t) - f([t.sub.0]), corresponded of abscissa-[DELTA]t growing, has decomposed in two terms [bar.[A.sub.1]T] and [bar.TB]. The raising of function [bar.[A.sub.1]T] is differentiation of wear function W(t), notated with dW. The size of dW= (1/p)dt is differentiation wear in point to.

4. CONCLUSIONS

The mathematical model proposed for analysis of adhesive mechanical wear reflected physical process of fast wear of friction couplings with high sliding speeds, used in modern machine tools with high speed machining.

At high speed machining, it has required a new mathematical model on which wear speed, volume of wear, intensity wear and its process has a parabolic evolution.

5. REFERENCES

Balasiu, D. (1990), Techniques of Investigation of Damage Process, Technical Editor, Bucharest

Bowden, F.P. & Tabor, D. (1956), Friction and Lubrification, Methuen et 6, London

Ceausu, I, et al. (1980), Organize and Leading of Activities of Maintenance and Reparation, Technical Editor, Bucharest

Georgescu, A. et al, (1984), Lubrification Practice in Industry, Vol. I-II, Technical Editor, Bucharest

Grama, L. (2007), Experimental Forecast in Construction of Machines, Methodology, Applications and Problems, Veritas Editor, Tirgu-Mures

Kragelsky, I.V. & Alisin, V.V. (1978), Friction Wear Lubrication. Tribology Handbook, Vol. I-II, Mir Publishers, Moscow

Pavelescu, D. (1983), Tribotechnica, Technical Editor, Bucharest

Schrader, G.F., Elshennawy, A.K. (2000), Manufacturing Processes & Materials, SME Editor, ISBN-087263-517-1, Dearborn, Michigan, USA

Tudor, A. (1984), Some Aspects about Reliability Indicators of Friction Couplings, Tribotechnica 84, Iasi 28-29 Sept.1984