grinding forces in regular surface texture generation

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International Journal of Machine Tools & Manufacture 47 (2007) 2098–2110

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Grinding forces in regular surface texture generation

Piotr Stepien�

Department of Mechanical Engineering, Technical University of Koszalin, Rac!awicka 15-17, 75-620 Koszalin, Poland

Received 1 February 2007; received in revised form 4 April 2007; accepted 16 May 2007

Available online 23 May 2007

Abstract

Certain grinding operations need a specially shaped wheel for regular surface texture (RST) generation. In such cases, wheel nominal

active surface is reproduced on the ground surface in a special way. The simple version of the method consists in grinding with the wheel

having helical grooves which are deeper than the grinding depth. Pattern regularity depends in longer time on wheel wear. The grinding

force is thus one of the most important process indicators. A simulation model of grinding process, assuming random arrangement of

abrasive grains was developed and is presented in this paper. The model was verified by grinding force measurements. These

measurements showed specific features that were different from those characteristic of conventional grinding. Explanations of the

untypical effects observed at force–time series signals for the three basic types of surface patterns are provided.

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Keywords: Grinding; Simulation model; Force measurement; Regular surface texture

1. Introduction

Surfaces having local, regular groove cavities arranged ina regular way show many advantageous features, regardingmainly tribological effects. The main features of regularsurface texture (RST) are: reduction of fluid and boundaryfriction coefficients, absorption of small hard particlesfrom the lubricant, reduction of residual stress and shapedeviation, better leak-tightness of static and dynamiccouplings and better adherence of coating and adhesivebonds. RST may be generated by several methods: precisediamond turning [1], rolling [2], embossing [3,4], etching[5,6], vibrorolling [7], abrasive jet machining [8] and EDM[9]. Many recent papers [10–15] have demonstratedimpressive results using laser surface texturing. ‘‘Patterngrinding’’ with the wheel shaped in a special way was firstpresented in 1989 [16] as a simple, cheep and productivealternative to better known methods of RST generation.The practical application of ‘‘pattern grinding’’ for shapingceramic discs of gas lubricated, self-acting thrust bearingswas demonstrated [17] in 1994.

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The effect of coarse dressing on ground surfacetopography has generally been described [18–22] as agrinding fault. However, it is possible to generate RST by acontrolled grinding process with the wheel shaped in aparticular way [23,24]. The wheel circumference may have,for example, deep helical grooves of h depth which aredeeper than the grinding depth d. The vw:vs ratio should begreat enough to prevent any changes to the nominaldimensions of the resultant surface. Such a processgenerates regularly arranged grooves separated from eachother. A schematic view of the wheel surface reproductionis given in Fig. 1 for two variants of the method. The thirdvariant involves grinding with the wheel having a singlehelical groove in two reverse passes of the work material.Examples of the three basic types of RST are given inFig. 2.The grooves on the work material are generated without

sparking out in a single pass of the wheel by manyindividual grains. The roughness of the groove bottomsreaches high values (up to Ra ¼ 5 mm) while the roughnessof the nominal surface of the work material does notchange. The rough bottoms of the grooves increaselubricant and coating adherence. All types of RST,presented in Fig. 2, show better wear resistance as small,hard debris and contaminants coming form the lubricant

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Nomenclature

A cross-section area of an undeformed chipa undeformed chip thicknessc length of rectilinear segment of wheel cross

sectionCA mean number of cutting edges per wheel surface

unitCn, Ct coefficients of normal and tangential force

components for a single graind grinding (groove) depthfd dresser feed rate (pitch of helical groove)Fn, Ft normal and tangential grinding force compo-

nentsfn, ft specific normal and tangential grinding force

components per 1mm of the wheel heightFng, Ftg normal and tangential grinding force compo-

nents for a single grainFnm, Ftm normal and tangential grinding force compo-

nents for wheel moduleFns, Fts normal and tangential grinding force compo-

nents for wheel elementary sliceh height of a helical groove (dressing depth)H wheel heightHa wheel active heightl length of a grooveL longitudinal pitch of grooves shaped on work-

materiallc nominal length of the wheel-work material

contact zonelg mean directional distance between adjacent

cutting edgesm, u shape and scale parameters of the Weibull

distribution

n number of cutting edges generated at a singleelementary slice

nA number of active grains generated for a singleelementary slice

nn, nt exponents of normal and tangential forcecomponents for a single grain

nPA number of potentially active grains for a singleelementary slice

ns wheel rotational speedr radius of imaginable wheel, rolling without slip

along rolling line r ¼ R (vw/vs)R wheel radiusrD radius of diamond dresser tips number of elementary slices for one wheel

moduleT time of one wheel rotationTF time period of force periodicityTns nominal (maximal) contact time for wheel

elementary slicev* ratio between work material and grinding speedvs grinding speedvw work material speedy ¼ f(z) general equation of the helical groove profile in

axial wheel cross sectiona angular coordinate of cutting edges in polar

systemdH height of elementary wheel sliceDx, Dy coordinates of cutting edgesl parameter of exponential distribution (l ¼ 1/lg)rN(a) radius of nominal profile of grinding wheelj angle of wheel rotationo angular speed of grinding wheel

Fig. 1. Schematic views of a grinding with the wheel having single (a) and double (b) helical grooves for shaping regular patterns on a flat surface [17, 24].

P. Stepien / International Journal of Machine Tools & Manufacture 47 (2007) 2098–2110 2099

can be absorbed in the groove spaces. Hence, the lubricantfiltering system can be simplified to capture large pieces ofdebris only and is thus more productive. Types I and II of

RTS are preferred for cases where the lubricant is morecontaminated and open-surface structure facilitates thecirculation of the lubricant. Type III shows superior results

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Fig. 2. Three types of the grooved surface (top view photographs) obtained with the wheels having helical grooves (black areas represent fragments of flat

‘‘plateau’’ surface).

P. Stepien / International Journal of Machine Tools & Manufacture 47 (2007) 2098–21102100

in the reduction of the fluid friction coefficient, but closed‘‘canoe-like’’ grooves work as traps for the hard debris.

Wheel wear is a significant problem as it may affect thedimensions of individual grooves in long term machining.Theoretical analyses supported by simulation studies[23,24] show that active grains are not loaded uniformly.Grains located at the active edge of helical grooves are themost loaded ones, and they may be crushed even on firstcontact with the work material. The active edges becomemore rounded and the grains are loaded more uniformly.That mechanism of self-adaptive wheel wear providessatisfactory tool life. Experiments have shown that thewheel 99A-80-K-7-V02 having a once shaped single helicalgroove (fd ¼ 1.65mm, h ¼ 0.06mm) used for patterngrinding (Fig. 1a) with grooves depth d ¼ 0.015mm cangenerate at least 1.8m2 of RST without noticeable changesof groove dimensions. The productivity of the methodP ¼ Hvw is most impressive and reaches 0.0125m2 s�1 forthe H ¼ 25mm and vw ¼ 0.5m s�1.

Grinding with the entire wheel height at a depth d ofmore than 0.10mm would appear to be hazardous becauseof the risk of local burns and excessive grinding force.Discontinuities of the wheel surface cause grinding forceoscillations which may generate vibrations, which are of aquite different nature than for conventional grinding.These problems were analysed theoretically with thedeveloped model and other examined by force measure-ment. The findings are reported in this paper.

2. Model of the grinding process

Modelling of the grinding process is a complex matterand, although grinding kinematics is quite simple, the main

difficulties arise from random size, shape and arrangementof abrasive grains on the wheel surface. The noncircularwheel profile, caused by the deep helical groove, is the nextfactor complicating the model. Three procedures weredeveloped for the grinding force prediction: (1) simulationof grain arrangement, (2) calculation of the relative pathsof the active grains and the undeformed chips’ thicknessand (3) calculation of the grinding force components. Thethird procedure was developed for single grains, elementarywheel slices and the entire wheel.

2.1. Grinding wheel profile and arrangement of grains

For general description it was assumed that the profile ofthe dresser tip is given by the function y ¼ f (z), as shownfor the axial cross section in Fig. 3a. Function y ¼ f (z)determines the helical groove depth h, dresser feed rate fd,length c of the linear segment of the wheel profile andangles k1, k2 between the tangents to helical groove profileand the wheel nominal surface. The simple conversion z-aallows the determining of the wheel nominal radius rN(a)for 0pap2p. The nominal radius rN(a) is diversified in theplane perpendicular to wheel axis, as is shown in Fig. 3b,with points denoted in the same way as in Fig. 3a. Thepolar coordinate system rN(a) assumes the counter clock-wise sense of angle a (Fig. 3b). Knowing the grinding depthd, it is possible to indicate two points, C and E, which limitthe potential active part of the wheel circumference. Theentire wheel circumference was divided into four zones bycharacteristic points A, B, C, E of the wheel profile(Fig. 3b).To obtain an adequate c:fd ratio, the use of diamond

dressers with flat or rounded tips, for instance worn ones, is

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Fig. 3. Scheme of the wheel cross section (a, b), wheel segmentation (c) into elementary slices (d) and system (c) of coordinates Dx, Dy determining cutting

edges locations.


recommended. It is then possible to shape helical grooveswith a large dresser feed fd and a shallow depth h whichreduces dressing force and wheel radius decrement insubsequent wheel shapings. Characteristic angles a1, a2, a3,of the wheel profile (Fig. 3a), determine the sizes of thewheel cross-section zones.

The wheel was divided (Fig. 3c) into so-called ‘‘elemen-tary slices’’ (Fig. 3d) of heights dH5H close to thedimensions of the interaction zone between the single grainand the work material. Knowing the number CA of thecutting edges observed on the wheel surface unit, theexpected number of cutting edges located at the circum-ference of a single elementary slice may be calculated asEsE2pRdHCA. The mean directional distance lg betweenadjacent cutting edges is the ratio between the entire length2pR of the circumference and the expected number Es ofthe cutting edges

lg ¼1

dHCA. (1)

A set of n cutting edges was generated for eachelementary slice, the locations of which were determinedby two random coordinates: circumferential Dx and radialDy (Fig. 3e). Incremental circumferential coordinates Dx

were generated as random values according to exponentialdistribution with the parameter l ¼ 1/lg. This assumed theuniform distribution of cutting edges in a circumferentialdirection. The number n of cutting edges in a singleelementary slice was a random value as the generation was

performed until SDxp2pR. Incremental coordinates Dx

were converted to incremental angular coordinates Da,determining the circumferential location of the cuttingedges in a polar system: Da ¼ Dx/R.The important assumption was made of the indepen-

dence of grain distribution in both radial and circumfer-ential directions. Opinions regarding the radial distributionof cutting edges were not coherent and at least twodistributions of coordinates Dy (Fig. 3e) have beenproposed: Gamma [25] and Weibull [26]. In the presentstudy, radial coordinates Dy were generated using Weibulldistribution, having a shape parameter m40 [�] and ascale parameter u40 [mm].Figs. 4a and 4b show examples of probability density

functions f(Dx) and f(Dy). Cumulative distribution func-tions F(Dx) and F(Dy) allow the generation of Dx and Dy

coordinates as they a have simple converse transformation.Replacement F(Dx) and F(Dy) with random number Rnd,taken from the range [0; 1), gives random incrementalcoordinates Dx and Dy, respectively

Dx ¼ �lg ln 1� Rnd½ �, (2a)

Dy ¼ �uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1� Rndð Þ

mp

. (2b)

Knowing the Dx and Dy coordinates of the cuttingedges, it was possible to obtain polar coordinatesr(a) ¼ rN(a)�Dy (Fig. 3e), where a was the consecutivesum of Da ¼ Dx /R. An example of n ¼ 392 cutting edgearrangements is given in Fig. 4c for a wheel having

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Fig. 4. Probability density functions of incremental cutting edges coordinates (a, b) and an example (c) of n ¼ 392 grain arrangement.


CA ¼ 6mm–2 cutting edges as was observed [21] by usingthe taper print method for the wheel 98A-60-K-6-V. Only apart of all grains, having radiuses r (a)XR�d, can contactthe work material. Because of the non-circular wheel crosssection and doh, the number nPA ¼ 248 of potentiallyactive grains is considerably smaller than n ¼ 392.

2.2. Grain paths

It was assumed that the wheel rotates counter clockwiseand that the wheel position is defined by angle j having thesame sense as the angle a defining any point of wheelcircumference. Grinding kinematics is relatively simpleowing to combination of wheel rotation and work feed.Any point of the wheel circumference moves over a cycloidpath defined in the system of coordinates 0xy (Fig. 3e)connected with the nominal surface of the work material.Cycloid paths are given as follows

for up�grinding :

x ¼ rj� rðaÞ sinðaþ jÞ;

y ¼ �rðaÞ cosðaþ jÞ þ g� R;

((3a)

for down�grinding :

x ¼ rj� rðaÞ sinðaþ jÞ;

y ¼ rðaÞ cosðaþ jÞ þ g� R;

((3b)

where r ¼ R(vw/vs) ¼ Rv*.Paths y(x) are calculated for each potentially active

grain. Only a small quantity of nPA potentially active grainscan touch the work material. Grain activity depends onkinematical parameters: work feed vw and wheel speed vsand grain location in both circumferential and radialdirections. Knowing grain paths, it was possible to identifyactive grains by comparing sequential paths. Takingvs ¼ 25m s�1 and vw ¼ 0.5m s�1 for the wheel examplegiven in Fig. 4c, identification shows nA ¼ 68 active grains

(black points). This means that about 27% of potentiallyactive grains make contact the work material. The activegrains are located in three zones: 5 grains (zone 1), 62grains (zone 2) and 1 grain (zone 3).Fig. 5a shows the set of cycloids nA ¼ 68 active grain

paths, calculated according to Eq. (3a) for up grinding withthe cutting edges arrangement as in Fig. 4c. The length ofthe groove l ¼ 11.442mm is shorter than the longitudinalpitch of the grooves L ¼ 2pr ¼ 2pRv*

¼ 12.556mm. Thismeans that part of the flat nominal surface remainsuntouched by any grain. The rough bottom of the grooveis also clearly visible.The differences between the ordinates y(x) of two

successive paths determine the undeformed chip thicknessa(x) as shown in Fig. 5b. The first active grain enters andleaves the work material at the nominal surface (y ¼ 0) andhas a specific shape of a(x).The time T ¼ 2p/o of one full wheel rotation may be

expressed by using the wheel angular velocity o ¼ vs/R.A single wheel elementary slice touches the work materialbriefly because of the inactive zone 4 (Fig. 3a). Assumingthat active grains may by located in extreme positions Cand E (Fig. 3), nominal (maximal) contact time is equal toTns ¼ (2p+a2�a3)/o. For the given example (R ¼ 100mm,h ¼ 0.04mm, fd ¼ 2mm, rD ¼ 2mm, d ¼ 0.02mm, vs ¼

25m s�1) the following results were obtained: o ¼ 250 s�1,T ¼ 0.02513 s, Tns ¼ 0.01804 s.Each undeformed chip thickness a(x) was converted into

a time series a(t) by using exact the relations j(t) and x(j).Fig. 5c shows an example of the time series a(t)corresponding to undeformed chip thickness a(x) fromFig. 5b, for the location of grains shown in Fig. 4c. Incontrast to a(x), the time series a(t) appeared distinctpulsatory character of the work of elementary slice,although simultaneously contacts of different grains withthe work material were still possible.One wheel rotation consists of four phases. Grains

located in zone 1 (Fig. 3a) work in the indentation phase,

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Fig. 5. Active paths (a) of grains corresponding to Fig. 4c, undeformed chip thickness (b) converted into time series (c) for three phases of nominal contact

time Tns.

Fig. 6. Pictorial views of undeformed chips cut by first (a) and next (b, c)

active grains.


deepening the initial groove on the work material. The firstgrain of this phase (point C in Fig. 3) starts cutting atminimal depth while the last grain, located at point B(Fig. 3), shapes the left-side slope of the groove as thecycloid. The grains located in zone 2 work in the extending

phase in the same way as for conventional up grinding: theydo not deepen the groove, enlarging it in the work-feeddirection. The last grain of the extending phase, located atpoint A (Fig. 3), shapes the right-side groove slope. Forattainable work feed to grinding speed ratios v* ¼ vw/vs, thegrains located in zone 3 do not come into contact with thework material. They pass over the groove surface shapedby grains from zone 2. This is the exit phase. Zone 4 of thewheel circumference works in the so-called idle phase—grains do not make contact with the work material becauseof r(a)oR�d. In this phase the work material passes forsome distance leaving a part of a flat bearing surface.Generally, grains located in zone 1 cut chips thicker thangrains located in zone 2. However, the random arrange-ment of grains strongly affects their load and even grainslocated in zone 3 can touch the work material as is the casein the presented example. Three phases of one wheelrotation were marked in Fig. 5c.

2.3. Grinding force components

Force Fg acting on a single grain depends on theproperties of the work material, interaction conditions(lubrication, temperature, relative velocity) and the trans-

verse cross-section area A of the undeformed chip. Therelation A(a) between the transverse cross-section area A

and undeformed chip thickness a may be expressed asA ¼ Ca2, where C is a coefficient depending on grainshape. Fig. 6 shows three possible types of undeformedchip shape. The first active grain cuts chip removed from aflat surface (Fig. 6a) and the chip cross section is the sameas the grain profile. The shapes of the chip cross sectionscut by the next active grains are more complex, as theydepend on two successive active grain profiles. If the grain

ARTICLE IN PRESSP. Stepien / International Journal of Machine Tools & Manufacture 47 (2007) 2098–21102104

profile crosses a previous active grain profile (Fig. 6b),undeformed chip transverse cross section is asymmetrical(‘‘comma-like’’). The second shape (Fig. 6c) occurs whensequential active grain profiles do not cross and unde-formed chip shape is more symmetrical (‘‘boat-like’’). Thisis why the coefficient C should get different values for thefirst and for subsequent active grains.

Two main components (normal Fng and tangential Ftg)of force Fg acting on a single grain (Fig. 7) were calculatedby using the following formulas:

Fng ¼ CnAnn ; F tg ¼ CtAnt . (4a,b)

The values of coefficients C, Cn, Ct and exponentsnn ¼ 0.69, nt ¼ 0.75 were taken according to the results ofexperiments with single grit cutting [27]. The shape of thegrains and the interaction conditions were diversified bythe randomization of coefficients C, Cn, and Ct taken fromthe ranges [13.5; 16.5), [0.36; 0.44) and [0.108; 0.132),respectively. For the first active grains C, Cn, Ct were takenfrom the ranges [4.5; 5.5), [0.18; 0.22), and [0.081; 0.099),respectively. The randomisation assumed the uniformdistributions of C, Cn, Ct values. The applied values ofC, Cn, Ct relate hard steel grinding with alundum grains.

The components Fng and Ftg of force acting on eachactive grain were calculated as the time series Fng(t), Ftg(t)and added into the single time series Fns(t) and Fts(t) actingon the entire wheel slice. The equations

Fns tð Þ ¼XnAi¼1

Fngi tð Þ; F ts tð Þ ¼XnAi¼1

F tgi tð Þ, (5a,b)

considered simultaneously active grains. Results, corre-sponding to undeformed chip thickness a(x) from Fig. 5band grains location from Fig. 4c, were shown in Fig. 7.Local force components Fns(t) and Fts(t) were so great thatthe crush of some grains was very probable and the load ofactive grains in the next wheel rotation would be more

Fig. 7. Normal (a) and tangential (b) components of the

uniform. The mean values of Fns(t) and Fts(t) are in factrather small, as the force components remained zero forlarge parts of the time.Force components [Eqs. (5a) and (5b)] for subsequent

wheel slices were calculated in the same way, taking intoconsideration different grain arrangements. The cumula-tive effect of force components coming from subsequentwheel slices takes into account time shift t (Fig. 8). Therotational angle, corresponding to the same part of the nextwheel slice, is greater by dj, and the next wheel slice makescontact with the work material later by the time shift t

dj ¼ 2pdH

fd; t ¼

djo¼ 2p

dH

ofd. (6a,b)

Scheme (Fig. 8b) of the set of wheel slices, workingsimultaneously, shows a so-called ‘‘wheel module’’, con-sisting of s ¼ fd/dH wheel slices, each staggered by the timeshift t. The dimensions of the wheel module are: fd and2pR. The number ns of simultaneously working wheel slicesis an integer value changing in time by 1, while the meanvalue of ns is given as

ns ¼fd

dH

Tns

T. (7)

Force components Fnm(t), Ftm(t) acting on the entirewheel module were calculated for s ¼ fd/dH ¼ 20 wheelslices

Fnm tð Þ ¼Xs

j¼1

FnsðjÞ tþ j � 1ð Þt½ �, (8a)

F tm tð Þ ¼Xs

j¼1

F tsðjÞ tþ j � 1ð Þt½ �. (8b)

The force–time series Fnm(t), Ftm(t) are shown in Figs. 9aand 9b. The mean values of Fnm(t), Ftm(t) grow linearly(R2�0.95) vs. the number of wheel active slices, while the

force acting on a single wheel slice shown in Fig. 4c.

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Fig. 8. Angular shift (a) between two subsequent wheel slices and scheme (b) of the set of s ¼ fd/dH wheel slices staggered in time.

Fig. 9. Normal (a) and tangential (b) force components acting on the entire wheel module compound of s ¼ 20 wheel slices and the results (b, c) of the

force component compilations.


growth of the standard deviations of Fnm(t), Ftm(t) is non-linear (Figs. 9c and 9d). The decreasing ratios betweenmean values and standard deviations of Fnm(t), Ftm(t) showthe effect of force stabilization.

Simulation procedures were repeated for the next fourwheel modules differentiated only by arrangements of thegrains. The sums of five force–time series, representingHa ¼ 10mm of active height of the wheel, were thuscalculated. The cumulative effect of 100 wheel slices showsthe very regular (linear) nature of the growth of meanforce components. Specific mean force componentsfn ¼ 11.13Nmm�1 and ft ¼ 4.51Nmm�1 acting on 1mmof the wheel height were determined.

Individual input data (h, fd, rD, lg, d, m, u, vw, vs) werechanged in many repetitions of simulation procedure,giving different results for the specific mean forcecomponents fn and ft. The most significant input data (h,

fd, rD, lg, d, vw, vs) were indicated by multiple non-linearregressions. The parameters m and u of the radialdistribution of grains were less significant than other inputparameters and were neglected in the final models. More-over, m and u were not technological parameters. Eqs. (9a)and (9b) show the form of regression functions

f n ¼ Anhenhfenfd renr

D lenlg dendvenw

w venss ½Nmm�1�, (9a)

f t ¼ Athethf

etfd retr

D letlg detdvetw

w vetss ½Nmm�1�. (9b)

Estimated values of coefficients and exponents, standarderrors, p-values, as well as goodness-of-fit parameters aregiven in Table 1The signs of exponents (Table 1) show the effect of each

input data on specific mean force components. The effectsof depth h, pitch fd of helical groove and radius rD of

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Fig. 10. Fixing unit (a): force gauge (1), base (2), workpiece holder (3), initial load screw (4), nut (5), workpiece view (b) and an example (c) of the

workpiece profile taken in A–A or B–B.

Table 1

Results of non-linear estimation of specific mean force component models (estimated values regard hard-steel grinding with alundum wheel)

Specific normal force component fn ¼ Anhenh fenfd renr

D lenlg dendvenw

w venss

Input data Specific tangential force component ft ¼ Atheth f

etfd retr

D letlg detdvetw

w vetss

Explained variation: 0.9781 R ¼ 0.9890 Explained variation: 0.9732 R ¼ 0.9865

Estimated value Std. error t (df ¼ 26) p-value Estimated value Std. error t (df ¼ 26) p-value

An 8.850 3.228 2.741 1.1E-02 3.046 1.274 2.392 2.4E-02

enh �0.198 0.055 �3.602 1.3E-03 h (mm) �0.144 0.064 �2.263 3.2E-02

enf 0.647 0.074 8.743 3.2E-09 fd (mm) 0.624 0.084 7.425 7.0E-08

enr �0.279 0.057 �4.872 4.7E-05 rD (mm) �0.293 0.065 �4.533 1.2E-04

enl 0.106 0.040 2.660 1.3E-02 lg (mm) 0.178 0.046 3.879 6.4E-04

end 1.346 0.077 17.523 6.7E-16 d (mm) 1.360 0.087 15.601 1.0E-14

enw 1.040 0.071 14.559 5.2E-14 vw (m s�1) 1.090 0.081 13.491 3.0E-13

ens �0.825 0.067 �12.368 2.1E-12 vs (m s�1) �0.836 0.076 �10.963 3.0E-11


dressing tool may be explained indirectly by using Fig. 3a.By increasing h and rD, angle a1 decreases. The length c ofwheel cross section and the ratio Tns:T decrease. Smalleractive parts of wheel circumference cause smaller specificmean force components. Increasing fd enlarges length c andgives opposite effect. The effect of the distance lg betweenadjacent cutting edges is not very distinct. By increasing lgthe number nA of active grains decreases but theundeformed chip thickness increases and the grinding forceincreases slightly. The effects of grinding depth d, work andgrinding speeds (vw, vs) do not need explanations.

3. Grinding force measurements

Grinding force components (normal Fn, tangential Ft

and axial Fa) were measured by using KISTLER-9602AQ01 piezoelectric force gauge, KEITHLEY-KPCI-1800HC interface and Test Point v. 5.0 software. To avoidelectromagnetic interference the force measurement systemwas battery powered. Each force component was sampledwith 10 kHz frequency and recorded on a PC. Force gauge

was mounted in the fixing unit (Fig. 10a) and the initialload was applied according to producer guidelines.The presented results of force measurements relate to

pattern grinding with the 200*32*76/99A-80-K-7-V wheelhaving helical grooves (hE0.05mm, fdE1.7mm) shaped witha single point diamond dresser (rDE1.1mm). The wheel speedvs ¼ 25.4–24.9ms�1, changed with the decreasing wheelradius R, caused by subsequent dressing. Hard steel work-pieces (70 HRA) were used in experiments. The roughness ofworkpieces (RaE0.5mm) was obtained by precise initialgrinding and flatness deviations were less than 0.007mm.Elastic deflections of the complex set of wheel-work-

piece-fixed unit, workpiece flatness deviations and mount-ing errors caused an oblique position of the nominalworkpiece surface relative to wheel axis and vw vector.Hence, individual grooves formed on workpieces haddifferentiated depths. The actual grinding depth d wasnot uniform for the whole workpiece and this was observedat surface profiles registered with HOMMELWERKET2000 in two planes A–A and B–B. Fig. 10c shows anexample of a workpiece profile with local burrs (max.

ARTICLE IN PRESSP. Stepien / International Journal of Machine Tools & Manufacture 47 (2007) 2098–2110 2107

2.5 mm in height) visible at groove sides. The mean value ofthe groove depths in A–A and B–B profiles was taken asthe grinding depth d.

Workpiece feeds vw were calculated by dividing work-piece diameter (32mm) (Fig. 10b), enlarged with twolengths lcE(2Rd)0.5 of wheel–workpiece contact zone, bytotal contact time, known from registered force signals.Because of the rounded ends of the workpiece, thewheel–workpiece contact started with Ha ¼ 0 then increas-ing to Ha ¼ 18mm finally increasing to at the Ha ¼ 18mmat a distance of 4.72mm, equal to the sum of lc and2.77mm, which was the rise of the arc F32mm (Fig. 10b).This was the entry time. The same time distance wassubtracted from the end of the signal as the exit time.

3.1. Grinding force in type I RST generation

A typical example of grinding force componentsregistered for generating I type of RST is shown inFig. 11. Total contact time was 102.3m s�1 and for a giventotal length of grinding 32mm+2lcE35.9mm, workpiecefeed vw ¼ 0.351m s�1 was calculated.

Measured force components were compared with themodels [Eqs. (9a) and 9(b)] based on simulation results.Mean force components, for the time range correspondingto the entire workpiece with (18mm), are shown in Fig. 11.Specific mean force components fn ¼ 8.39Nmm�1 andft ¼ 3.39Nmm�1 were obtained as the ratio between meanforces and Ha ¼ 18mm. Taking h ¼ 50 mm, fd ¼ 1.7mm,rD ¼ 1.1mm, lg ¼ 1.25mm, d ¼ 19 mm, vw ¼ 0.351m s�1,vs ¼ 25.4m s�1 into Eqs. (9a) and (9b), specific mean forcecomponents fn ¼ 7.04Nmm�1 and ft ¼ 2.87Nmm�1 wereobtained as the model results. The relative differences wereabout 15% and should be assumed as small enough forforce prediction. The simulation model developed wasverified not only by comparison of the force value levels.Comparison of Figs. 9a, 9b and 11 show an acceptableconsistency of frequency characteristics of force–time seriesas well, which is more convincing. The considerable

Fig. 11. An example of the grinding force c

difference between time sampling intervals Dt should benoted. For simulated force–time series Dt ¼ 5.138 s�7,while for measured force–time series Dt ¼ 1 s�4, which isalmost 200 times greater. Simulated force–time series thusneed extensive smoothing for comparison.Taking into account the considerable active height of the

wheel (Ha ¼ 18mm), the grinding force should be evalu-ated as small one, compared to the conventional grindingforces. There are two causes supporting such a conclusion:the greater values of undeformed chip parameters anddecreased number of active grains. Such a conclusion iscompatible with known recommendations concerninggrinding energy reduction by using grooved wheels. Forcecomponents do not show oscillations emanating fromwheel discontinuity if active wheel height Ha is greatenough: Ha4fd. Axial force component Fa oscillatesaround zero, which is typical for the grinding withoutcross feed. However, the workpiece should be pushed in anaxial direction because of the helical groove formed on thewheel circumference, as was supposed.

3.2. Grinding force in type II RST generation

The second type of RST generation needs two reversalpasses of the workpiece (Fig. 2). In general, the second passmay be performed with a different wheel and differentparameters (R, fd, vs, vw, d). However, most often the samewheel is used in reverse pass and only the work feed differsslightly (vw16¼vw2). The first pass, described above, does notneed further explanations. Of more interest is the secondpass, when the wheel, having a single helical groove forms asecond set of grooves inclined at the opposite angle. Thegrinding force is thus smaller and distinct oscillations canbe seen (Fig. 12) for the second grinding pass.To explain the nature of force oscillations the grooved

surface was presented as a composition of equal modules,denoted in Fig. 13a by a broad line. Each module containsone convex ‘‘island’’ (white area) surrounded by groovebottoms (grey areas). The length Lm of the module depends

omponents for type I surface grooving.

ARTICLE IN PRESS

Fig. 12. An example of the grinding force components for type II surface grooving.

Fig. 13. Scheme (a) explaining grinding force oscillations (b) during the second pass of type II RST generation and periodogram (c) of the original Fn(t)

signal.


on the pitches L1 and L2 of grooves (see Fig. 1a) shaped inboth passes. The relationships can be expressed by usingthe work feed harmonic mean vwHM.

Lm ¼ 2L1L2

L1 þ L2¼

2

ns

vw1vw2

vw1 þ vw2¼

vwHM

ns. (10)

According to the scheme shown in Fig. 13a, coincidenceof convex ‘‘islands’’ corresponds to the lower grindingforce (broken arrows) while the coincidence of groovescorresponds to the higher force (full arrows). The distancebetween successive peaks and valleys of oscillating grindingforce is exactly half the module length Lm. The time periodTF of force periodicity can be calculated according toEq. (10)

TF ¼Lm

2vwHM¼

1

2ns. (11)

Eq. (11) shows that the force oscillation period dependsonly on wheel rotational speed and this allows actual wheelrotational speed to be determined by measurement of TF atthe second pass of the II type of RST generation. Theexplanation of the force oscillations presented was con-firmed by time series analyses of normal force componentshown in Fig. 12b. In the first attempt (Fig. 13b) period TF

was determined for the smoothed (50-point moving average)force signal Fn(t) as TFE0.0122 s. Harmonic analysis of theoriginal signal Fn(t) showed the same results (Fig. 13c).Actual wheel rotational speed was nsE40.96 r.p.s.

3.3. Grinding force in type III RST generation

An example of the grinding force components, registeredat the III type of RST generation (Fig. 14a), shows similaroscillations, but the nature of that phenomena is different.

ARTICLE IN PRESS

Fig. 14. An example (a) of the grinding force components for type III surface grooving, cross sections (b) of the wheel and top view (c) of the surface

formed as odd and even rows of the grooves.


Crossing of two helical grooves causes wheel roundnessdeviations. The coincidence of helical groove bottoms takesplace at two opposite points of the wheel cross section,denoted in Fig. 14b by points A and B where the wheelradius is smallest, as described by Yokogawa [23]. In aperpendicular direction the wheel radius reaches maximalvalue and only two opposite parts of the wheel can makecontact with the work material (doh). This takes place twotimes for one wheel revolution when two rows of groovesare formed: odd ‘‘O’’ and even ‘‘E’’, each at the distance L

(Figs. 1b and 14c). The time period between sequentialpeaks of grinding force is half the time of a single wheelrotation: TF ¼ 1/(2ns), the same as for the II type of RST[Eq. (11)], but the mechanism of force oscillation is different.

4. Conclusions

(1)
Pattern grinding with the wheels having helical groovesis an untypical surface finishing method. Grindingforces show several untypical features which areimportant for the wheel wear prediction and vibration.
(2)
The model of the grinding process with the wheelhaving a single helical groove and random arrangementof grains clearly explained the qualitative relationshipsbetween the input parameters of the process (h, fd, rD,lg, d, vw, vs) and the grinding force. Increasing h, rD andvs results in a lighter grinding force, while increasing fd,lg, d and vw causes the opposite effect. Quantitativedescription was obtained by analyses of the simulationresults based on the model.
(3)
The grinding force during the second grinding pass withthe wheel having a single helical groove (II type ofRST) shows oscillations caused by grinding over a setof inclined linear grooves shaped in the first pass.
(4)
The grinding force during grinding with the wheelhaving two crossed helical grooves (III type of RST)shows oscillations caused by the not circular wheelcross section which has two active parts.
(5)
The frequency of force oscillations during II and IIItype surface texturing is twice as great as the wheel
rotational frequency and does not depend on the workfeed. Such oscillations do not cause any extra vibra-tions as the frequency is too low.

(6)
The ratio between normal Fn and tangential Ft forcecomponents is about 2.5:1. Axial force components Fa
are many times smaller, even for grinding with thewheel having a single helical groove.

Acknowledgement

The author would like to thank Dr. Mariusz Kasprzykfor his technical assistance in the configuration of forcemeasurement systems.

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