Analytical Determination of the Distribution of Polishing Time over the Surface of Polished Tiles

Analytical Determination of the Distribution of Polishing Time over theSurface of Polished Tiles

Fabio J. P. Sousa,w,z Jan C. Aurich,y Walter L. Weingaertner,z and Orestes E. Alarconz

zUFSC — Laboratory of Materials, Department of Mechanical Engineering, Federal University of Santa Catarina,Campus Universitario Trindade, Florianopolis, SC 88040-900, Brazil

yDepartment of Mechanical and Process Engineering, Institute for Manufacturing Engineering and ProductionManagement, University of Kaiserslautern, D-67653 Kaiserslautern, Germany

Several kinds of glossiness pattern can be seen on the surfaceof porcelain stoneware tiles right after the polishing process,as a function of the kinematics performed by the polishingheads. For the newest generation of industrial polishingtrains, where a transverse oscillation is included, there is stilla great need for literature about the resulting patterns. Thispaper intends to find the spatial distribution of time underpolishing analytically using the kinematics equations involvedin the polishing process. The measured values of glossinesscollected from three polished tiles are also presented. Theimportance of adopting a good kinematics for the polishingprocess has been highlighted, and the equations developedherein are useful tools for further attempts at optimizing thepolishing process.

I. Introduction

IN the industrial polishing of porcelain stoneware tiles, theglossiness enhancement results from the abrasive contacts

carried out by a sequence of polishing heads with decreasingabrasive sizes. The set of polishing heads results in a polishingtrain, and according to the kinematics performed by these pol-ishing heads, several kinds of glossiness pattern can be seen onthe tile surface after the polishing process. One reason for this isprobably the differences between the time that each region in thetile surface remains under polishing.

For simple polishing trains, this polishing pattern is alreadywell known.1–4 However, for those polishing trains in which anextra motion of transverse oscillation is involved, more studiesabout the resulting polishing profile are still needed.

Such transverse oscillations are available in most modernpolishing machines, and because the glossiness of porcelainstoneware tile is considered to be the most important criterionto control the quality of such products,5 this paper intendsto determine such a polishing profile analytically using thekinematics equations involved in the polishing process. Allavailable motion involved in modern polishing trains can beseen in Fig. 1. Further details on the kinematics of a singleabrasive particle during the industrial polishing process can befound in Sousa et al.6

Each polishing head is formed by a horizontal spinning platein which six abrasive blocks, the fickerts, are coupled, maintain-ing a radial symmetry. Each fickert, in turn, is magnesium

oxychloride bonded, and contains innumerous silicon carbideparticles, which work as abrasive particles.1,7,8

The forward speed at which tiles are driven into the polishingtrain is represented by V (cm/s), while the transverse oscillationhas frequency f (s�1) and amplitude A (cm). The rotationW (rad/s) of the abrasive head causes no motion of point C,but it also causes a movement of the fickerts.

Thus, the trajectory of the active abrasive particles isgoverned by the preset movement of the polishing head,presented in Fig. 1. In view of this, certain regions undergo alonger duration of polishing than others, so that gradients ofglossiness are generated.

This can be easily understood considering the simple kine-matics at first, i.e., using no transverse oscillation. Owing to thelack of abrasive in the middle of the polishing heads, the centerof the tile undergoes less polishing than the adjacent area. Thisleads to the known profile detailed in Fig. 2, mentioned previ-ously by Hutchings et al.1,7

Such glossiness patterns are colloquially known as ‘‘polishingshadows’’ and they are usually sharp enough to be detected byoperators, who can therefore often ascertain the direction inwhich the tiles were polished.

To improve the glossiness distribution shown in Fig. 2(a),transverse oscillation motion was introduced in the newestgeneration of polishing trains. However, the problem of favoredpolishing areas was not completely solved. For such modernpolishing trains, the distribution of abrasive contact is expectedto vary as a function of the transverse oscillation motionin a certain zigzag overlapping. Therefore, unlike simple polish-ing trains, those polishing profiles might vary along the polish-ing direction, i.e., axis ı. A surface function results instead ofsingle profiles.

In such a function, the effective time under abrasive contactsshould be provided for any point P in the tile surface, consid-ering the kinematic and geometric parameters adopted, just asshown in Fig. 3. For simplification, the rotational level of theabrasive head when compared with either transverse oscillationor forward motion was assumed here to be fast enough so thatthe polishing area regarding a given polishing head could beconsidered as a hollow circle. The lack of abrasive is limited byan inner radius r (cm), whereas the outer radius R (cm) definesthe reach of fickerts.

Both forward and transverse oscillation motion occur simul-taneously, and because the time spent in each oscillation cycle isby definition T5 1/f, a wave function results as the trajectory tobe followed by point C. The wave length l can be taken directlyas the forward displacement performed within period T by theproduction line, so that

l ¼ V

f(1)

Equation (1) above is true for each polishing head of thepolishing train, and the whole set accomplishes the same move-ment at the same time.

W. Lee—contributing editor

Supported by The National Council for Scientific and Technological Development —CNPq, an entity from the Brazilian Government.

wAuthor to whom correspondence should be addressed. e-mail: [email protected]

Manuscript No. 22839. Received February 22, 2007; approved July 6, 2007.

Journal

J. Am. Ceram. Soc., 90 [11] 3468–3477 (2007)

DOI: 10.1111/j.1551-2916.2007.01956.x

r 2007 The American Ceramic Society

3468

The center of the hollowed circle has coordinates (XC,YC).The values of YC vary according to the function fS, which isexplicit in Eq. (2)

YC ¼ fS ðXCÞ ¼A

2sin

2plXC

� �(2)

II. Limiting the Polished Area

The whole area polished by a single-polishing head must not besimply delimited by offsetting the trajectory of point C by dis-tance R; otherwise, the hatched area illustrated in Fig. 4 wouldnot be taken into account. For this reason, another function fB,corresponding to the boundaries of the whole polished area,needs to be determined.

The determination of function fB will be explained in the se-quence. Figure 5(a) introduces three points (t1, t2, and t3) thatbelong to the same function fB. Another function fC was includ-ed in the figure in order to represent the circle centered in C. Forsimplicity, only the outer circle, with radius R, was used forreasoning. As can be seen in Fig. 5(b), these points were coin-cident to circle fC while it was centered at points C1, C2, and C3,respectively.

In other words, the desired function fB is built by particularpoints of circle fC. These particular points have coordinates DXand DY related to the circle center. However, DX varies as thecircle moves along fS. Still, in Fig. 5(b), one might notice thateach one of points t1, t2, and t3 is placed in a different positionalong the circle. This fact in turn causes both components DXand DY also to vary. Such a variation can be then achievedconsidering the details presented in Figs. 6(a) and (b).

According to Fig. 6(a), function fB can be explicitly written as

fB Xð Þ ¼ YC � DY (3)

From Eq. (2) and Fig. 6(a), it follows that

fB XC � DXð Þ ¼ A

2sin

2pl

XC � DXð Þ� �

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � DX2p

(4)

It can be intuitively seen that as the circle center moves to-ward a given direction, the farthest point from point C will bealways the one tangent with the performed motion, thus yieldinga higher covered area. It follows from this that the instantaneousinclination of function fS at XC might be equal to that offunction fC.

In relation to a fixed point C, function fC can be written as

fC ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � X � XCð Þ2

q(5)

whose domain is {X; XC�RrXrXC1R} and its instantaneousinclination at an arbitrary point X, i.e., the derivative with X, is

f 0C ¼ �X � XCð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 � X � XCð Þ2q (6)

Fig. 2. Polishing profile and the typical glossiness pattern for simple polishing trains.

Fig. 1. Plan view of the available motion of a polishing head.

November 2007 Porcelain Stoneware Tiles and the Glossiness Enhancement 3469

The instantaneous inclination of function fS at point XC is aconstant value K given by

K ¼ fS XCð Þ ¼ Apl

cos2plXC

� �(7)

The equivalence between instantaneous inclinations explainedabove implies f0S (XC)5 f0C. Moreover, in Eq. (6), the term(X–XC) can be replaced by �DX, so that

Apl

cos2pl

XCð Þ� �

¼ � DXð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � DXð Þ2

q (8)

which, after solving, yields

DX ¼ � KRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 þ 1p (9)

which is, in fact, the variable required to assess function fB bymeans of Eq. (4). It yields

fB XC � DXð Þ ¼ A

2sin

2pl

XCð Þ� �

� R

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

K2 þ 1

r(10)

In view of the signals, function fB can be finally divided intotwo functions f\B and f[B corresponding to the upper and lowerboundaries of the whole polished area. In sequence

f\B XC � DXð Þ ¼ A

2sin

2pl

XCð Þ� �

þ R


K2 þ 1

r(11)

f[B XC þ DXð Þ ¼ A

2sin

2pl

XCð Þ� �

� R


K2 þ 1

r(12)

As expected, one may notice in both equations above thatfunctions fB and fS tend to give the same result as the circle ra-dius R tends to 0. On the other hand, as the ratio A/l becomessmall, by either increasing the wavelength l or decreasing am-plitude A, the value of K tends to 0 and function fB simply be-comes two offsets of function fS expressed by fS7R.

Figure 7 contains an XY graphic in which the path of a givenpolishing head and its respective polishing boundaries are plot-ted. Radii of 23 and 11 cm were adopted, as well as the followingkinematics parameters: forward speed of 7.5 cm/s, and 12 cmand 0.2 s�1 as transverse oscillation amplitude and frequency.This condition was selected due to wide use in industrial pro-cesses. It must be mentioned that such values were obtained bythe expertise of employees, and that such an approach is used inmost floor tiles industries.9 The resulting wavelength of functionfS was 37.5 cm.

Because radius r stands for abrasive lacking, all areas insidethose inner boundaries will have their polishing process inter-rupted for some time, whereas no polishing interruptions are ex-pected for those areas between the outer and inner boundaries.

Thus, although outer polishing boundaries delimit the wholearea that will have at least one abrasive contact, the polishingover the tile surface might be far from uniform. A closer look atFig. 7 can provide some qualitative idea of the polishing profileto be expected for a set of polishing parameters including trans-verse oscillation motion. However, how long a given area willremain inside those polishing boundaries cannot be quantita-tively evaluated yet.

Similar to Fig. 2, which refers to a simple polishing machine,the quantitative evaluation of such a polishing profile consists indetermining the effective time under polishing for each positionover the tile surface. In theory, a given area may cross thosepolishing boundaries several times, depending on its location aswell as the geometric and kinematic parameters. However, onlythe cumulative time is of real interest, and one method by whichit can be assessed will be explained in sequence.

III. Determination of the Effective Time under Polishing

Let P (XP; YP) represent an arbitrary point on the surface of atile driven into a polishing train, while point O stands for the

Fig. 5. Geometrical relations between the sine function fS and theboundary function fB.

Fig. 6. Relationship between the instantaneous inclinations of fS and fC.

Fig. 3. Area under polishing for a given position at the polishing train.

Fig. 4. Limits of the area polished by a single polishing head.

3470 Journal of the American Ceramic Society—Sousa et al. Vol. 90, No. 11

center of a polishing head. Point O is also assumed to be theabsolute origin of the polishing train. Because points P and Oare in relative motion, the path to be performed by P can bedrawn as presented in Fig. 8(a). This path is another wave-likefunction named as fP.

Considering the dashed line in Fig. 8(a), function fP could bedirectly calculated if an increment YP as well as a phase delay XP

were computed in the function fS. For mathematical conve-nience, however, point P was assumed to be fixed. As shown inFig. 8(b), the same path of P inside the hollow circle is obtainedby considering function fS and point C (XC; YC) as the newcenter of the hollow circle, whose outer and inner circle func-tions were designed as fOC and fIC, respectively. Coordinates XC

and YC refer to point P as the relative origin, whereas (XP; YP)correspond to the real position of point P over the tile surface,regarding the origin of the polishing train. Both coordinates areeasily related to each other by (XC; YC)5 (–XP; –YP).

Figure 8(c) presents in a bold continuous line the pathsadopted by point P only during abrasive contacts. Becausecomponents S1 and S2 have the same direction as the polishingtrain, the cumulative time-point P that remains effectivelyunder polishing can then be taken as the time required by theproduction line to overcome a total distance ST, which iswritten as the sum of all components Si that may exist. Inview of this, distance ST can be considered as the effectivepolishing distance of point P. Distance ST will be a function ofboth coordinates of P (XP; YP), and also all other polishingparameters already detailed (V (cm/s), A (cm), f (s�1), R (cm),and r (cm)).

As a first attempt, distance ST could be achieved using theintersection points between a wave and a circle, that is, functionsfS and fOC. Thus, according to Fig. 8, functions that might becompared in order to achieve the desired intersection points aregiven by Eqs. (13) and (14). It should be remembered that thecumulative time P remains inside the inner radius r (fIC) must bereduced from the cumulative time it remains inside the outerradius R (fOC), since the former stands for abrasive lacking. For

convenience, only equations involving the outer radius R willcontinue to be expressed.

fOC Xð Þ ¼ YC �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � ðX � XCÞ2

q(13)

fS Xð Þ ¼ A

2sin

2plX

� �(14)

The equivalence fOC5 fS leads to

A

2sin

2plX

� �¼ YC �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � ðX � XCÞ2

q(15)

Equation (15) above is transcendental, so that solutionscannot be expressed in terms of common algebra. A numericalsolution could be used to bypass the problem, but in orderto try a more symbolic solution, another strategy was adoptedinstead.

As only the cumulative time inside the circle is of interest,regardless of the position of those intersection points, anattempt to find distance ST directly by means of area diagramswas then made.

As exemplified in Fig. 9, a small variation DYC leads to anarea variation DA that will tend to be proportional to interval Sas DYC tends to infinity. In other words, the interval S may beattained by the rate at which the internal area between wave andcircle functions changes with YC. Still noteworthy in Fig. 9 isthat in view of retaining the same resulting signal, those internalareas must be assessed by adopting either (fS–fC) or (fC–fS), ac-cording to the signal of YC.

Yet, according to the use of either outer radius R orinner radius r, the internal areas were, respectively, designedas AIntOC and AIntIC. The balance of area sketched in Fig. 10exemplifies the achievement of an arbitrary internal areawhere YC 40. The software Mathematica

s

version 5.2 wasused to carry out the integrations and differentiations needed.Constants generated by integration processes were not takeninto account.

The balance of area presented above can be mathematicallyexpressed according to Eqs. (16) and (17). In addition, fC canstill represent either the bottom half of circle, fOC[, or the tophalf of the circle, fOC\.

2AIntOC ¼Z XCþR

XC�RfS � fOC[j jdx

�Z XCþR

XC�RfS � fOC[ð Þdx

if YC � 0

(16)

Fig. 8. (a) Relative motion between a point at the tile surface and the origin of the polishing train. (b) Assuming point P as the relative origin.(c) Highlighting of the abrasive contact ranges.

Fig. 7. Path of the polishing head and its respective polishing boundaries.


2AIntOC ¼Z XCþR

XC�RfS � fOC\j jdx

�Z XCþR

XC�RfOC\ � fSð Þdx

if YC < 0

(17)

In a more compact way

2AIntOC ¼Z XCþR

XC�RfS � fOCj jdx� Afs � Af OC

(18)

AfOCis the area of a half circle with radiusR. According to the

half circle to be considered, it follows that

AfOC[ ¼ 2YCR� pR2

2forYC � 0 (19)

AfOC\ ¼ 2YCRþpR2

2forYC < 0 (20)

The value of area AfS in turn is then given by

AfS ¼Z

fSdx ¼Z XCþR

XC�R

A

2sin

2plX

� �� dX (21)

which yields

Afs ¼Al4p

� cos 2pðXC � RÞ

l

� �� cos 2p

ðXC þ RÞl

� �� (22)

Finally, area AIntOC can be written as

AIntOC ¼Z XCþR

XC�RfS � fOC[j j dx

2� YCRþ

pR2

4� Al

8p

� cos 2pðXC þ RÞ

l

� �� cos 2p

ðXC � RÞl

� ��

if YC � 0

(23)or

AIntOC ¼Z XCþR

XC�RfS � fOC\j j dx

2þ YCRþ

pR2

4þ Al

8p

� cos 2pðXC þ RÞ

l

� �� cos 2p

ðXC � RÞl

� ��

if YC < 0

(24)

The integral of term |fS–fOC|, present in the above equations,had to be retained in this symbolic form from known on,because any further development turned out to be unfeasible.

As reasoned previously, the effective polishing interval ST foran arbitrary polished point will be given as follows, including thesubtraction regarding the lack of abrasive in the center of thepolishing head:

ST ¼dAIntOC

dYC� dAIntIC

dYC(25)

which is detailed below

ST ¼d

dYC

Z XCþR


2

� �� d

dYC

�Z XCþr

XC�rfS � fIC[j j dx

2

� �� R� rð Þ

if YC � 0

(26)

Fig. 9. Effective polishing distance given by means of area variation.

Fig. 10. Determining the common areas between functions fS and fOC.


ST ¼d

dYC

Z XCþR


2

� �� d

dYC

�Z XCþr

XC�rfS � fIC\j j dx

2

� �þ R� rð Þ

if YC < 0

(27)

Nevertheless, the above equations still need a complement inorder to avoid errors like those exemplified in Figs. 11(a) and (b),indicated by e. Because the dashed lines in the circle functions ofFig. 11 were not considered so far, no intersection between thewave and circle could be detected in those sides of the circle.

Unlike the polishing effective interval, errors e are related toan external area, designed as AextOC. Thus, considering a similarbalance of areas sketched in Fig. 10

2AExtOC ¼Z XCþR

XC�RfS � fOC\j jdx

�Z XCþR

XC�RfOC\ � fSð Þdx

if YC � 0

(28)

2AIntOC ¼Z XCþR

XC�RfS � fOC[j jdx

�Z XCþR

XC�RfS � fOC[ð Þdx

if YC < 0

(29)

Analogous to the polishing interval, error e is then given by

e ¼ dAExtOC

dYCþ dAExtIC

dYC(30)

which yields

e ¼ d

dYC

Z XCþR


2

� �þ d

dYC

�Z XCþr


2

� �� R� rð Þ

if YC � 0

(31)

e ¼ d

dYC

Z XCþR


2

� �þ d

dYC

�Z XCþr


2

� �þ R� rð Þ

if YC < 0

(32)

Error e can then be subtracted from the effective polishinginterval, which finally yields

ST ¼d

dYC

Z XCþR


2

� �

� d

dYC

Z XCþr


2

� �

� d

dYC

Z XCþR


2

� �

þ d

dYC

Z XCþr


2

� �þ R� rð Þ if YC � 0

(33)and

ST ¼d

dYC

Z XCþR


2

� �

� d

dYC

Z XCþr


2

� �

� d

dYC

Z XCþR


2

� �

þ d

dYC

Z XCþr


2

� �� R� rð Þ if YC < 0

(34)

Once ST is determined, the effective polishing time EPT

regarding point P can finally be found by considering theforward speed of the polishing train, so that

EPT ðXP;YPÞ ¼ST

V(35)

The polishing profiles attained considering some typical set ofpolishing conditions are presented in the following section. Itmust be remembered that the above assumptions describe theeffect of a single polishing head. Because all polishing headsoscillate together in a coherent way, further polishing heads canbe promptly included by considering additional functions fSi

.

IV. Simulation and Checking

Different polishing conditions were introduced into Eq. (35) inorder to obtain values of effective polishing time. The results ofall simulation were calculated adopting Dx5 0.01 cm, by nu-merical integrations, and Dy5 0.001 cm by differentiations. Thedomain admitted was limited to the surface of three adjacentsquare tiles and surface graphics were adopted for presenting theresults. The following polishing conditions were common for allthe simulations carried out: oscillation amplitudeA5 12 cm andouter and inner diameter of the polishing head R5 23 cmand r5 11 cm. The tile nominal dimensions were 45� 45.

Fig. 11. Errors resulting from using only one side of the circle functions.


Nevertheless, for simulation purposes, each tile was divided into24 lines and 36 rows, resulting in 864 identical small rectangles,1.75 cm� 1.25 cm each.

Firstly, in order to check the adequacy of simulations, thevalues of effective polishing intervals for 10 different points weremeasured directly by a CAD software, AutoCad

s

version 14.Both the measured and simulated results are listed in Table I.Coordinates of the checking points were randomly chosen andincluded in the table.

Small differences were found between simulated and measuredresults, as shown in the last column of Table I. The average errordue to the simulation was estimated to be 0.32%, within astandard deviation of 0.47%. Once the model was tested and itsaccuracy had been established, six other different polishing con-ditions were simulated: V52.5 cm/s adopting f50.2, 0.4, and 0.6s�1, and V57.5 cm/s adopting f50.4, 0.6 s�1, and no oscillation.These kinematic parameters yielded the following wavelengths l,respectively: 12.5, 6.25, and 4.17 cm, and 18.75, 12.5, and, in thecase of null transverse oscillation, l tends to infinity.

Beyond this, the glossiness from three real polished tiles wasmeasured all over the surface using a glossmeter (model IG-320,Horiba Ltd., Kyoto, Japan). In this case, a pixel size of 2.5 cmwas adopted and the results were then compared with thesimulated values.

Figure 12 presents the effective polishing time, hereafternamed as EPT, simulated using the same kinematic conditionused in the model checking. In addition to a surface graphic, asketch of the polishing limits and also a gray-scale graphiccontaining a plan view of the polishing train were included inthe figure, so that most polished areas could be clearly indicated.

As expected, the simulation reveals that those most polishedareas take place where no polishing interruptions due to abra-sive lacking in the middle of the polishing head do occur. Limitsindicating these unaffected areas are sharply defined so thatvariations of polishing time may occur abruptly. Apart fromthis, a complete agreement was found between the polishinglimits and both surface and gray-scale graphics. It must beremembered that equations were developed independently.

When no transverse oscillation is available, e.g., simplepolishing machines, a constant well-known polishing profile isexpected to result, just as found in the literature.1,7 Such a resultis shown in Fig. 13, and it contrasts to the result of using f5 0.4s�1, also presented in the figure. The forward speed in both caseswas maintained as 7.5 cm/s.

In sequence, the same values of transverse oscillation fre-quency were then simulated considering this time V5 2.5 cm/s.The results can be found in Fig. 14.

Apart from the higher values in the scale of EPT found forV5 2.5 cm/s, which results directly from the increase of timerequired to overcome each tile, it can be seen in the simulationsthat the distribution of EPT becomes less biased as the wave-length l becomes smaller. This tendency was also seen in theresults with f5 0.6 s�1, found in Fig. 15.

A slightly better result regarding the distribution of EPT wasobtained by reducing l from 6.25 to 4.17 cm. Moreover, asobserved by comparing Figs. 14(a) and 15(b), a similar distribu-tion of EPT was found for a different kinematic condition thatretained the same ratio V/f.

Apart from the simulated results, Fig. 16(a) presents the finalglossiness of three adjacent polished tiles measured with aglossmeter, mark Horiba, model IG-320. The tiles were pro-vided by the Brazilian company Ceramica Portobello S/A(Tijucas, Brazil). The kinematics used in this case were V5 12cm/s, A5 19 cm, and f5 0.3 s�1. For comparison purposes,Fig. 16(b) shows the corresponding simulated results of EPT. Apixel size of 2.5 cm� 2.5 cm was adopted in both graphics.

Although several polishing heads account for the final glossi-ness, only five adjacent polishing heads are shown in Fig. 16(b).This was based on the fact that within such stages, thetile glossiness usually increases from 20 glossiness units (gu) toalmost 80 gu.1,10 Each further polishing head, i.e., functionfSi

, differs from the previous one by a phase advancement of58 cm, corresponding to the distance between two adjacentpolishing heads.

A coarse zigzag pattern of glossiness was revealed by theexperimental results. On the other hand, the likeness betweenmeasured and simulated patterns was found to be very little.This fact confirms that not only how long a given region remainsunder polishing is mandatory for defining the final glossiness but

Fig. 12. Distribution of EPT over the tile surfaces, considering V5 7.5 cm/s and f5 0.2 s�1.

Table I. Comparison between Simulated and MeasuredValues for Distance ST

Coordinates of checked points

XP (cm) YP(cm) ST by CAD (cm) ST simulated (cm) Error (%)

64.375 �4.375 21.88 21.87 0.05106.875 �18.375 22.13 22.09 0.1943.125 �2.625 26.25 26.10 0.5749.375 16.625 22.06 22.12 �0.2595.625 11.375 35.03 35.28 �0.7126.875 �6.125 20.94 20.96 �0.1076.875 �2.625 28.06 28.08 �0.0869.375 9.625 42.25 41.82 1.0329.375 20.125 19.90 19.88 0.0966.875 0.875 23.49 23.51 �0.09


also the corresponding polishing condition. The contact pres-sure during polishing is known to change with the scratchingradius,3,10 as well as with time, due to wear of the fickert.1

Nevertheless, it must be remembered that no simulation of thepolishing process is intended in this work, but to develop usefultools for further optimizations.

Figure 16(a) still reveals an asymmetry regarding the glossi-ness pattern from all the three polished tiles. A similar asymme-try can also be noticed in experimental results elsewhere.3,4 Thiscan be explained by the introduction of an undesired inclinationof the support device of the tile, inside the polishing train.

Finally, the standard deviation of the distribution of EPT wasthen calculated, aiming to quantify the homogeneity for each

simulation above. The results are shown in Fig. 17 according tothe corresponding l value. The distribution of final glossiness isalso included. In view of scale differences, the coefficient ofvariation was adopted instead of the standard variation.

The coefficient of variation regarding the glossiness distribu-tion was found not to differ that much from those regarding thedistributions of EPT, despite the difference between graphicsfrom Figs. 16(a) and (b). The difference would have been evensmaller had the support of the tile been less inclined.

The points corresponding to both kinematic conditions withl5 12.5 cm were coincident. This indicates that different kine-matic conditions can be equivalent regarding the distribution ofEPT, as long as the same ratio V/f is adopted. Moreover, the

Fig. 14. Distribution of EPT over the tile surface for V5 2.5 cm/s. (a) f5 0.2 s�1, (b) f5 0.4 s�1.

Fig. 13. Distribution of EPT over the tile surface for V5 7.5 cm/s. (a) f5 0.0 s�1, (b) f5 0.4 s�1.


graphic confirms the trend in yielding higher homogeneity as ldecreases.

One might also note that the distribution for l5N wasfound to be better than for l5 37.5 cm. It follows that theavailability of the transverse oscillation does not necessarily leadto better polishing results than those produced in simple polish-ing machines, but it is rather a question of the kinematicadopted. A better glossiness distribution could then be expectedusing smaller values of l. However, Fig. 17 still reveals that nosignificant improvement should be expected by reducing from12.5 to 4.17 cm.

The importance of the distribution of EPT relies on the factthat, referring to the glossiness criterion, neither coarse-polishedareas nor overpolished areas are desired. The former limits thefinal quality of the polished tile, whereas the latter simplyrepresents extra costs, due to the consumption of energy,abrasive, polishing tools, and also waste generation. Besides,after a limit defined by the microstructure of the tile surface,2,8,10

no glossiness profit is achieved from oversupplying abrasivecontacts.

Reducing values of l imply either a reduction in V or anincrease in f. However, an increase in the transverse oscillationfrequency demands a higher acceleration of the whole set ofpolishing heads, which causes the energy consumption to in-crease. In addition, as the mechanical solicitations in the equip-ment increase, the lifetime of the polishing train may decrease.

On the other hand, reductions of V decrease the productivityto the same extent. Therefore, further researches on the finaleffect of all the features mentioned above, as well as on theminimum number of abrasive contacts, are needed so that anoptimization of the polishing process could be planned.

V. Conclusions

An attempt at determining the effective time polishing EPT foreach region over an industrial polishing train was carried outconsidering the transverse oscillation motion. In some areas, thepolishing process is interrupted due to lack of abrasives in themiddle of the polishing head, so that notable differences regard-ing the polishing time were seen even between adjacent areasover the tile surface.

Fig. 16. (a) Measured values of glossiness and (b) simulated values of EPT for three adjacent polished tiles.

Fig. 15. Distribution of EPT over the tile surface for f5 0.6 s�1. (a) V5 2.5 cm/s, (b) V5 7.5 cm/s.

Fig. 17. Correlation between l and the coefficient of variation of thedistribution of EPT.


A geometrical approach was used to bypass the transcendentalequation involved in the analytical determination of the distribu-tion of EPT over the tile surface, so that it could be explicitlyfurnished as a function of the polishing parameters. The accuracyof the resulting model was checked and several polishing condi-tions were simulated and presented in surface graphics.

The simulations revealed the importance of the ratio V/frather than individual values of each kinematic parameter. Thequality of the distribution of EPT, which may lead to betterdistributions of glossiness, was found to increase meaningfullyas the ration V/f was reduced from 37.5 to 12.5 cm, and onlyslightly improvements were observed after further reductions.

Although the transverse oscillation can be seen as a usefuldevice to reduce the biased glossiness enhancement typicallyattained by simple polishing machines, a wrong choice ofkinematic parameters can lead to results even worse than thosewhere no transverse motion is available. In addition, becausetwo other parameters are introduced, transverse oscillationfrequency and amplitude, the polishing kinematics becomesmore complicated to optimizing by trial and error.

Acknowledgment

The authors would like to formally thank the companies Ceramic Portobelloand Ceusa for providing all the facilities required to carry out this work.

References

1I. M. Hutchings, K. Adachi, Y. Xu, E. Sanchez, M. J. Ibanez, and M. F.Quereda, ‘‘Analysis and Laboratory Simulation of an Industrial Polishing Processfor Porcelain Ceramic Tiles,’’ J. Eur. Ceram. Soc., 3151–6 (2005).

2I. M. Hutchings, Y. Xu, E. Sanchez, M. J. Ibanez, and M. F. Quereda,‘‘Development of Surface Finish During the Polishing of Porcelain Ceramic Tiles,’’J. Mater. Sci, 40, 37–42 (2005).

3V. Cantavella, E. Sanchez, M. J. Ibanez, M. J. Orts, and J. Garcia-Ten,‘‘Grinding Work Simulation in Industrial Porcelain Tile Polishing,’’ Key EngMater, 1467–70 (2004).

4V. Cantavella, E. Sanchez, J. Garcia-Ten, M. J. Ibanez, J. Sanchez, C. Soler,J. Sales, F. Mulet, and S.Mor,Modelizacion de la Operacion Industrial de Pulido deBaldosas Ceramicas (Modelling of the Industrial Polishing Operation of CeramicTiles), pp. 111–22. Castelon, Spain, 2006.

5C. Y. Wang, T. C. Kuang, Z. Qin, and X. Wei, ‘‘How Abrasive MachiningAffects Surface Characteristics of Vitreous Ceramic Tile,’’ Am. Ceram. Soc. Bull.,9201–8 (2003).

6F. J. P. Sousa, J. C. Aurich, W. L. Weingaertner, and O. E. Alarcon,‘‘Kinematics of a Single Abrasive Particle During the Industrial Polishing Processof Porcelain Stoneware Tiles,’’ J. Eur. Ceram. Soc., 27, 3183–90 (2007).

7I. M. Hutchings, K. Adachi, Y. Xu, E. Sanches, and M. J. IbanezQUALICER,Laboratory Simulation of the Industrial Ceramic Tile Polishing Process, pp. 19–30.Castelon, Spain, 2004.

8L. Esposito, A. Tucci, and D. Naldi, ‘‘The Reability of Polished PorcelainStoneware Tiles,’’ J. Eur. Ceram. Soc., 785–93 (2005).

9A. Tucci and L. EspositoQUALICER, Polishing of Porcelain Stoneware Tile:Surface Aspects, pp. 127–36. Castelon, Spain, 2000.

10E. Sanchez, J. Garcia-Ten, M. J. Ibanez, M. J. Orts, and V. Cantavella,‘‘Polishing Porcelain Tile. Part 1: Wear Mechanism,’’ Am. Ceram. Soc. Bull, 50–4(2002). &


Analytical Determination of the Distribution of Polishing Time over the Surface of Polished Tiles

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