research article prediction model of coating growth rate

Research ArticlePrediction Model of Coating Growth Rate for Varied Dip-AngleSpraying Based on Gaussian Sum Model

Yong Zeng1 Yakun Zhang1 Junxue He2 Hai Zhou1 Chunwei Zhang1 and Lei Zheng1

1School of Mechanical Engineering Yancheng Institute of Technology Yancheng Jiangsu 224051 China2College of Electrical and Information Engineering Lanzhou University of Technology Lanzhou Gansu 730050 China

Correspondence should be addressed to Yong Zeng zengzhong188126com

Received 27 September 2016 Accepted 15 November 2016

Academic Editor Alessandro Gasparetto

Copyright copy 2016 Yong Zeng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In automatic spraying of spray painting robot in order to solve the problems of coating growth rate modeling for varied dip-angle spraying technology a prediction mode of coating growth rate using the Gaussian sum model is proposed Based on theGaussian summodel a theoretical model for coating growth rate with varied dip-angle spraying is established by using the theory ofdifferential geometryThe coating thickness of the sample points in the distribution range of the coating was obtained bymaking theexperiment of varied dip-angle spraying Based on the theoretical model the nonlinear least squaremethod is used to fit the coatingthickness of the sample points and the parameter values of the theoretical model are calculated By analyzing the variation law ofthe parameters with the spray dip-angle the prediction model of coating growth rate for varied dip-angle spraying is establishedExperiments have shown that the prediction model has good fitting precision it can satisfy the real-time requirement with varieddip-angle spraying trajectory planning in the offline programming system

1 Introduction

Trajectory planning of spray painting robot for complexfree surface has been a hot research topic both at homeand abroad One of the key difficulties in the research ofspraying trajectory planning is to establish a coating growthratemodel with high accuracy andwide applicability [1ndash3] Atpresent there are two main kinds of models for coatinggrowth rate a class of Gaussian distribution model [4 5] andCauchy distribution model [6] were proposed by Antonioand Freund both of them belong to an infinite range modeland this kind of model is only suitable for the spray gunperpendicular to the surface of the workpiece There is alsolimited range models for example piecewise function model[7] 120573 distributionmodel [8] analytical depositionmodel [9]parabolic model [10] ellipse dual-120573model [11] and Gaussiansummodel [12]Themathematical expressions of these mod-els are usually derived by means of mathematical analysis Atthe same time based on the mathematical expressions thefinal model of coating growth rate is approximated by thefitting of the experimental data These models can be widelyused in the case of complicated surface shape if these models

meet the error requirement In these models the Gaussiansum model has a wide range of application it can be appliedto many kinds of spray gun and its fitting precision is higher

In this paper a prediction model of coating growth ratefor varied dip-angle spraying based on the Gaussian summodel is established by the theory study and spraying experi-ment The model can be used to generate the coating growthrate model under a certain spray dip-angle quickly andaccurately

2 Theoretical Modeling of the Coating GrowthRate for Varied Dip-Angle Spraying

Spraying process is the coating produced by the spray gunthat atomize to the surface of the workpiece the process israther complicated Under the action of the spray gun and thepressure vessel the paint can realize the atomization and it issprayed from the spray gun nozzle the paint is formed in aconical or ellipse-conical shape in space In this paper thecoating growth rate model is established for varied dip-angle spraying based on a kind of spray gun the paint space

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9369047 7 pageshttpdxdoiorg10115520169369047

2 Mathematical Problems in Engineering

S

120572

c2

c1P1P2

HS120579S

120573 x

y

z

rS

O

H

(a) Schematic diagram of the dip-angle spraying calculation

120572

c1

c3

(b) Normal deflectionangles of 1198881ndash1198883

Figure 1 Schematic diagram of the coating growth rate modelcalculation for dip-angle spray painting

distribution of the spray gun is conical and the torch of thespray gun is a circular cross section and isotropic Underthe assumptions that the spray distance 119867 the spray flow119902V and the cone angle 120593 are constant when the spray gun isstill sprayed a mathematical model is used to describe thedistribution of spray coating on a flat plate The Gaussiansum model has the advantage of high fitting precision andnot affected by the shape of the coating distribution so it isassumed that the coating on the flat plate is distributed by theGaussian sum model its expression is

119891 (119903119878) =

119873sum119894=1

120596119894 exp[minus (119903119878 minus 119903119894)2

21205902119894 ] 10038161003816100381610038161199031198781003816100381610038161003816 le 1198770 10038161003816100381610038161199031198781003816100381610038161003816 gt 119877

(1)

where 120596119894 119903119894 and 120590119894 are the unknown parameters 119894 =1 2 119873 119877 is the radius of spray painting When 119873tends to infinity the function can approximate any form ofcoating distribution [13] but its complexity has increaseddramatically so generally take119873 = 3

In consideration of the dip-angle spraying assume thatthe angle between the axis of the spray gun and the planenormal direction is 120572 as the spray dip-angle Using the areaamplification theorem of differential geometry assumingthat the small round regions are 1198881 and 1198882 by sprayingrespectively in the reference planes P1 and P2 which areperpendicular to the spray direction of the spray gun asshown in Figure 1(a) the areas of 1198881 and 1198882 are 119878P1 and 119878P2respectively and the corresponding coating thicknesses are1198911 and 1198912119867119878 is the distance from any point 119878 on the plane tothe nozzle along the spray gun axis Therefore

119878P1119878P2 =11986721198781198672 (2)

So the coating thicknesses of 1198911 and 1198912 are satisfied11989111198912 =

11986721198672119878 (3)

Suppose there are two plane regions at the same cone anglerespectively which are 1198881 and 1198883 as shown in Figure 1(b)where 1198881 is perpendicular to the spray direction the anglebetween 1198883 and the spray direction is 120572 so the coatingthickness of 1198883 can be expressed as

1198913 = 1198911 cos120572 (4)

Combining formulas (2)ndash(4) the theoreticalmodel of coatinggrowth rate which takes into account the spray dip-angle as avariable can be expressed as

119891 (119909 119910 120572) = 119891 (119903119878) ( 119867119867119878)2

cos120572 (5)

where

119867119878 = 119867 + 119909 sin120572119903119878 = 119867radic119909

2cos2120572 + 1199102119867 + 119909 sin120572

(6)

119909 and 119910 are the horizontal and vertical coordinates of anypoint 119878 within the coating range

3 Fitting Method of Coating GrowthRate Model

To establish the coating growth rate model under the pre-scribed dip-angle spraying unknown parameters in formula(5) need to be solved further Coating thickness data of thesample point is measured in the coating distribution range bydoing the spraying experiment of the dip-angle Fitting thesedata with the theoretical model by the least square methodunknown parameters can be obtained

The rectangular coordinate system 119883119874119884 is establishedwith the intersection point 119874 of the spray gun axis and thespraying plane as the coordinate origin Since the coating isformed on the plane after the dip-angle spraying the coatingcovering range shape is elliptical Here a rectangle is used tocontain the ellipse the length and width of the rectangle arethe long and short axis of the ellipse Isometric parallel linesparallel to the119883-axis and the 119884-axis are respectively done inthe rectangular box the intersection point between parallellines is the sampling point of coating thickness as shownin Figure 2 119872 sampling points in the measurement rangeof coating thickness are measured optimization function isestablished by using nonlinear least square method as shownin the following

119864 (1205961 1205962 1205963 1199031 1199032 1199033 1205901 1205902 1205903)= min

119872sum119894=1

[119879 (119909119894 119910119894) minus 119891 (119909119894 119910119894)]2 (7)

Mathematical Problems in Engineering 3

Sample point

O

Y

X

Figure 2 Sample points of coating thickness

where 119864 is the sum of squares of the difference between thecoating thickness of the sampling point and the theoreticalcalculation value 119879(119909119894 119910119894) is the coating thickness of thesampling point 119891(119909119894 119910119894) is the theoretical value calculated byformula (5)

In this study the optimization algorithm of Levenberg-Marquardt is used to solve the unknown parameter valuesits basic steps are as follows

(1) Determine the initial values of 119901 120583 V and 120576(2) Calculate error matrix corresponding Jacobi matrix119869 and module 119865 of error matrix(3) Calculate Hessian matrix and gradient 119892(4) Calculate step size119898 and determine whether 119865 is less

than convergence value 120576 if yes then the iterationends otherwise go to the next step

(5) Calculate 119901 = 119901 + 119898 and gain ratio 120588(6) Determine whether 120588 is greater than 0 and update

iteration value(7) Determine whether 119865 is less than convergence value120576 if yes then the iteration ends otherwise go to the

step (4)

The above algorithm is used to obtain a series of unknownparameter values of the corresponding coating growth ratemodel of spray dip-angle by using the finite number ofvaried dip-angle spraying experiments With spray dip-angleas variable the value of these parameters is analyzed with thechange of spray dip-angle Finally a predictionmodel of coat-ing growth rate for varied dip-angle spraying is established

4 Application of Prediction Model of CoatingGrowth Rate for Varied Dip-Angle Spraying

In order to ensure the coating uniformity on the inside-corner surface and reduce the waste of paint on the outside-corner surface the continuous varied dip-angle spray trajec-tory planning method can be used to design spray trajectoryon the inside-corner or outside-corner surface as shown inFigure 3 If the continuous varied dip-angle spray trajectoryparameters were optimized on the inside and outside cornerssurface the coating thickness model is needed to establishafter the spray gun along the trajectory of the continuousvaried dip-angle spraying In theory based on the predictionmodel of coating growth rate the dynamic coating thickness

model can be obtained by integrating it under a certain spray-ing dip-angle

Assume the trajectory length of the continuous varieddip-angle spraying is 119889 spray gun to achieve the maximumdip-angle is 120572th In order to reduce the complexity of theproblem the trajectory of the continuous varied dip-anglespraying is divided into 119894 segments by the discretemethod thespray dip-angle of each segment is (1119894)120572th (2119894)120572th 120572threspectively and the walking speeds of the spray gun areV1 V2 and V119894 respectively At this time the coating thick-ness model of each small segment after the spray painting onthe point 119878 can be expressed as follows

119879119878 (119909 119910 119895) = 1V119895 int(119895119894)119889

((119895minus1)119894)119889119891(119909 119910 119895119894 120572th 120574119895)119889119911

119895 isin [1 119894] 119895 is a positive integer(8)

where

120574119895 = 119867radic1199102 + (119909 minus 119911)2 cos2 ((119895119894) 120572th)

119867 + (119909 minus 119911) sin ((119895119894) 120572th) (9)

Based on formula (8) the dynamic coating thickness modelof the whole continuous varied dip-angle spray trajectory canbe obtained by the superposition expression is as follows

119879119878 (119909 119910) = 119894sum119895=1

119879119878 (119909 119910 119895) (10)

From the above formula the dynamic coating thicknessmodel is established by the superposition of coating thicknessmodels of constant dip-angle spraying According to the pre-diction model of coating growth rate for varied dip-anglespraying the coating thickness model of constant dip-anglespraying can be established quickly and meet the demand ofthe offline programming system for rapid modeling of theconstant dip-angle spraying after continuous varied dip-anglespraying this is also an important reason for the establish-ment of the prediction model of coating growth rate

5 Experimental Verification

Theexperimental platform is composed of an automatic spraygun an air compressor and a bracket of spray gun withadjustable dip-angle and heightThe workpiece to be sprayedis a 460 lowast 400mm steel plate Adjust the spray height 119867 tobe 200mm and the cone angle 120593 to be 28∘ Spraying time isset to 1 s the coating thickness of fixed point is measured bythickness gauge after spraying is finished and the coating iscompletely dried

Adjust the spray dip-angle and in turn 0∘ 10∘ 20∘ 30∘ and40∘ static spraying experiment with continuous spraying 3times in each dip-angle take a coating covering the best of theexperimental sample piece to measure the coating thicknessand thickness gaugeMC-2010A is used as shown in Figure 4The coating thickness of each sampling point was measured 3times and average value was obtained as the coating thicknessof the sampling point


Table 1 Parameter values of different spraying dip-angle for Gaussian sum model

120572 (∘) 1205961 1199031 1205901 1205962 1199032 1205902 1205963 1199033 12059030 1006 24 135 minus452 274 71 913 273 7610 1025 21 122 minus816 276 67 1265 272 8120 1001 21 144 minus1857 274 73 2214 277 7830 1010 22 135 minus3028 273 75 3424 269 7840 1007 26 133 minus4068 274 71 4460 274 74

d

d

Inside-corner surface

Outside-corner surface

120572th

120572th

Figure 3 Sketch of continuous varied dip-angle spraying

Figure 4 Sample piece and measuring tool

After obtaining the coating thickness of the samplingpoint for each dip-angle spraying experiment the unknownparameter values of the model were obtained for each spraydip-angle by using formula (7) as shown in Table 1

By parameter values in Table 1 1205962 and 1205963 have obviouschange with the spray dip-angle the change law can beobtained by the curve fitting toolbox in MATLAB 2010 andthe expressions are as follows

1205962 (120572) = 3952 cos2 (120572) minus 5446 cos (120572) + 14471205963 (120572) = minus3598 cos2 (120572) + 4841 cos (120572) minus 1151 (11)

In addition to 1205962 and 1205963 the remaining 7 groups of param-eters with the change of the dip-angle are not obvious Thereasons of the difference between these parameter values withthe dip-angle are mainly caused by the measurement error ofcoating thickness and the calibration error of spray height andspray dip-angle It is assumed that these parameter values are

not related to the change of the dip-angle The average valuesof each parameter are taken here as follows

1205961 = 1011199031 = 231205901 = 1341199032 = 2741205902 = 711199033 = 2731205903 = 77

(12)

The prediction model of coating growth rate for varied dip-angle spraying is established by formulas (11) and (12) intoformula (5) In order to verify the validity of the predictionmodel under the premise of ensuring that spray height spraytime and cone angle are the same spray experiments were


0 20 40 60 80 1000

102030405060708090

100

Gaussian sum modelOriginal point

minus20minus40minus60

x (mm)

T(120583

m)

(a) 119910 = 0


0102030405060708090

100

0 20 40 60minus20minus40minus60

y (mm)

T(120583

m)

(b) 119909 = 0


0102030405060708090

100

0 20 40 60 80minus20minus40minus60

x (mm)

T(120583

m)

(c) 119910 = 20


0102030405060708090

100


y (mm)

T(120583

m)

(d) 119909 = 20

Figure 5 Fitting of coating thickness when 120572 = 25∘

done when the spray dip-angle is 25∘ and 45∘ respectivelyThe coating thicknesses of the sample point were measuredat 119909 = 0 119910 = 0 119909 = 20mm and 119910 = 20mm The coatingthicknesses of the sampling point were compared with theprediction model where Figure 5 shows the case when 120572 =25∘ and Figure 6 shows the case when 120572 = 45∘

From Figures 5 and 6 the prediction model can be usedto well simulate the coating growth rate with varied dip-anglespraying however with the increase of spray dip-angle thefitting precision of the prediction model has declined

6 Conclusion

In this paper in order to solve the modeling problem ofcoating growth rate for varied dip-angle spraying technologyin the offline programming system of spray painting robotthe Gaussian sum model is used to establish the predictionmodel of coating growth rate for varied dip-angle sprayingExperiment shows that the Gaussian sum model also has ahigh fitting precision in describing the coating growth ratefor varied dip-angle spraying the prediction model can be

used to reflect the coating growth rate quickly and accuratelywith varied dip-angle spraying and itmeets the requirementsof offline programming system for modeling real-time per-formance and practicality There are some advantages in thevaried dip-angle spraying technology compared with the ver-tical spraying technology For example the varied dip-anglespraying technology can be a better way to ensure the coatinguniformity on the inside-corner surface and reduce the wasteof paint on the outside-corner surface Based on the predic-tionmodel established in this paper when spraying trajectoryplanning the surface of workpiece with inside and outside-corner feature a varied dip-angle spraying trajectory opti-mization model can be established to optimize the coatingthickness uniformity in the corners of the surface so as toachieve the purpose of the best spraying effect for the surfaceof workpiece with corner feature

Competing Interests

The authors declare that they have no competing interests


0 20 40 60 80 100 120 1400

102030405060708090

100


minus20minus40minus60x (mm)

T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100


0 20 40 60 80 100 120minus20minus40minus60

x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments

The project was supported by the National Natural ScienceFoundation of China (Grant no 51405418) and the NaturalScience Foundation of the Jiangsu Higher Education Institu-tions of China (Grant no 14KJB460031)

References

[1] H P Chen and N Xi ldquoAutomated tool trajectory planningof industrial robots for painting composite surfacesrdquo TheInternational Journal of Advanced Manufacturing Technologyvol 35 no 7 pp 680ndash696 2008

[2] H P Chen T Fuhlbrigge and X Z Li ldquoA review of CAD-basedrobot path planning for spray paintingrdquo Industrial Robot vol36 no 1 pp 45ndash50 2009

[3] H P Chen N Xi S K Masood Y Chen and J Dahl ldquoDevelo-pment of automated chopper gun trajectory planning for sprayformingrdquo Industrial Robot vol 31 no 3 pp 297ndash307 2004

[4] M Hyotyiemi ldquoMinor moves-global results robot trajectoryplanningrdquo in Proceedings of the IEEE International Conferenceon Tools for Artificial Intelligence pp 16ndash22 Herndon Va USANovember 1990

[5] E Freund D Rokossa and J Rossmann ldquoProcess-orientedapproach to an efficient off-line programming of industrialrobotsrdquo in Proceedings of the 24th Annual Conference of the IEEEIndustrial Electronics Society pp 208ndash213 Los Alamitos CalifUSA 1998

[6] J KAntonio R Ramabhadran andT L Ling ldquoA framework fortrajectory planning for automated spray coatingrdquo InternationalJournal of Robotics and Automation vol 12 no 4 pp 124ndash1341997

[7] C Feng and Z-Q Sun ldquoModels of spray gun and simulation inrobotic spray paintingrdquo Robot vol 25 no 4 pp 353ndash358 2003

[8] T Balkan andM A S Arikan ldquoModeling of paint flow rate fluxfor circular paint sprays by using experimental paint thicknessdistributionrdquoMechanics Research Communications vol 26 no5 pp 609ndash617 1999

[9] D C Conner A Greenfield P N Atkar A A Rizzi and HChoset ldquoPaint deposition modeling for trajectory planning onautomotive surfacesrdquo IEEE Transactions on Automation Scienceand Engineering vol 2 no 4 pp 381ndash391 2005

[10] T Balkan and M A S Arikan ldquoSurface and process modelingand off-line programming for robotic spray painting of curvedsurfacerdquo Journal of Robotic Systems vol 17 no 9 pp 479ndash4942000


[11] Y G Zhang Y M Huang and F Gao ldquoNewmodel for air spraygun of robotic spray-paintingrdquo Journal of Mechanical Engineer-ing vol 42 no 11 pp 226ndash233 2006

[12] B Zhou Z H Shao Z D Meng and X Z Dai ldquoGaussiansum based coat growth rate modeling of spray painting robotsrdquoJournal of Huazhong University of Science and Technology(Natural Science Edition) vol 41 supplement 1 pp 463ndash4662013

[13] C E Rasmussen and C K I Williams Gaussian Processes forMachine Learning MIT Press Cambridge UK 2006

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


S

120572

c2

c1P1P2

HS120579S

120573 x

y

z

rS

O

H

(a) Schematic diagram of the dip-angle spraying calculation

120572

c1

c3

(b) Normal deflectionangles of 1198881ndash1198883

Figure 1 Schematic diagram of the coating growth rate modelcalculation for dip-angle spray painting

distribution of the spray gun is conical and the torch of thespray gun is a circular cross section and isotropic Underthe assumptions that the spray distance 119867 the spray flow119902V and the cone angle 120593 are constant when the spray gun isstill sprayed a mathematical model is used to describe thedistribution of spray coating on a flat plate The Gaussiansum model has the advantage of high fitting precision andnot affected by the shape of the coating distribution so it isassumed that the coating on the flat plate is distributed by theGaussian sum model its expression is

119891 (119903119878) =

119873sum119894=1

120596119894 exp[minus (119903119878 minus 119903119894)2

21205902119894 ] 10038161003816100381610038161199031198781003816100381610038161003816 le 1198770 10038161003816100381610038161199031198781003816100381610038161003816 gt 119877

(1)

where 120596119894 119903119894 and 120590119894 are the unknown parameters 119894 =1 2 119873 119877 is the radius of spray painting When 119873tends to infinity the function can approximate any form ofcoating distribution [13] but its complexity has increaseddramatically so generally take119873 = 3

In consideration of the dip-angle spraying assume thatthe angle between the axis of the spray gun and the planenormal direction is 120572 as the spray dip-angle Using the areaamplification theorem of differential geometry assumingthat the small round regions are 1198881 and 1198882 by sprayingrespectively in the reference planes P1 and P2 which areperpendicular to the spray direction of the spray gun asshown in Figure 1(a) the areas of 1198881 and 1198882 are 119878P1 and 119878P2respectively and the corresponding coating thicknesses are1198911 and 1198912119867119878 is the distance from any point 119878 on the plane tothe nozzle along the spray gun axis Therefore

119878P1119878P2 =11986721198781198672 (2)

So the coating thicknesses of 1198911 and 1198912 are satisfied11989111198912 =

11986721198672119878 (3)

Suppose there are two plane regions at the same cone anglerespectively which are 1198881 and 1198883 as shown in Figure 1(b)where 1198881 is perpendicular to the spray direction the anglebetween 1198883 and the spray direction is 120572 so the coatingthickness of 1198883 can be expressed as

1198913 = 1198911 cos120572 (4)

Combining formulas (2)ndash(4) the theoreticalmodel of coatinggrowth rate which takes into account the spray dip-angle as avariable can be expressed as

119891 (119909 119910 120572) = 119891 (119903119878) ( 119867119867119878)2

cos120572 (5)

where

119867119878 = 119867 + 119909 sin120572119903119878 = 119867radic119909

2cos2120572 + 1199102119867 + 119909 sin120572

(6)

119909 and 119910 are the horizontal and vertical coordinates of anypoint 119878 within the coating range

3 Fitting Method of Coating GrowthRate Model

To establish the coating growth rate model under the pre-scribed dip-angle spraying unknown parameters in formula(5) need to be solved further Coating thickness data of thesample point is measured in the coating distribution range bydoing the spraying experiment of the dip-angle Fitting thesedata with the theoretical model by the least square methodunknown parameters can be obtained

The rectangular coordinate system 119883119874119884 is establishedwith the intersection point 119874 of the spray gun axis and thespraying plane as the coordinate origin Since the coating isformed on the plane after the dip-angle spraying the coatingcovering range shape is elliptical Here a rectangle is used tocontain the ellipse the length and width of the rectangle arethe long and short axis of the ellipse Isometric parallel linesparallel to the119883-axis and the 119884-axis are respectively done inthe rectangular box the intersection point between parallellines is the sampling point of coating thickness as shownin Figure 2 119872 sampling points in the measurement rangeof coating thickness are measured optimization function isestablished by using nonlinear least square method as shownin the following

119864 (1205961 1205962 1205963 1199031 1199032 1199033 1205901 1205902 1205903)= min

119872sum119894=1

[119879 (119909119894 119910119894) minus 119891 (119909119894 119910119894)]2 (7)


Sample point

O

Y

X








step (4)






119879119878 (119909 119910 119895) = 1V119895 int(119895119894)119889

((119895minus1)119894)119889119891(119909 119910 119895119894 120572th 120574119895)119889119911


where


119867 + (119909 minus 119911) sin ((119895119894) 120572th) (9)


119879119878 (119909 119910) = 119894sum119895=1

119879119878 (119909 119910 119895) (10)








d

d



120572th

120572th








1205961 = 1011199031 = 231205901 = 1341199032 = 2741205902 = 711199033 = 2731205903 = 77

(12)



0 20 40 60 80 1000

102030405060708090

100



x (mm)

T(120583

m)

(a) 119910 = 0


0102030405060708090

100


y (mm)

T(120583

m)

(b) 119909 = 0


0102030405060708090

100


x (mm)

T(120583

m)

(c) 119910 = 20


0102030405060708090

100


y (mm)

T(120583

m)

(d) 119909 = 20




6 Conclusion



Competing Interests



0 20 40 60 80 100 120 1400

102030405060708090

100



T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100



x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sample point

O

Y

X








step (4)






119879119878 (119909 119910 119895) = 1V119895 int(119895119894)119889

((119895minus1)119894)119889119891(119909 119910 119895119894 120572th 120574119895)119889119911


where


119867 + (119909 minus 119911) sin ((119895119894) 120572th) (9)


119879119878 (119909 119910) = 119894sum119895=1

119879119878 (119909 119910 119895) (10)








d

d



120572th

120572th








1205961 = 1011199031 = 231205901 = 1341199032 = 2741205902 = 711199033 = 2731205903 = 77

(12)



0 20 40 60 80 1000

102030405060708090

100



x (mm)

T(120583

m)

(a) 119910 = 0


0102030405060708090

100


y (mm)

T(120583

m)

(b) 119909 = 0


0102030405060708090

100


x (mm)

T(120583

m)

(c) 119910 = 20


0102030405060708090

100


y (mm)

T(120583

m)

(d) 119909 = 20




6 Conclusion



Competing Interests



0 20 40 60 80 100 120 1400

102030405060708090

100



T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100



x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












d

d



120572th

120572th








1205961 = 1011199031 = 231205901 = 1341199032 = 2741205902 = 711199033 = 2731205903 = 77

(12)



0 20 40 60 80 1000

102030405060708090

100



x (mm)

T(120583

m)

(a) 119910 = 0


0102030405060708090

100


y (mm)

T(120583

m)

(b) 119909 = 0


0102030405060708090

100


x (mm)

T(120583

m)

(c) 119910 = 20


0102030405060708090

100


y (mm)

T(120583

m)

(d) 119909 = 20




6 Conclusion



Competing Interests



0 20 40 60 80 100 120 1400

102030405060708090

100



T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100



x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 1000

102030405060708090

100



x (mm)

T(120583

m)

(a) 119910 = 0


0102030405060708090

100


y (mm)

T(120583

m)

(b) 119909 = 0


0102030405060708090

100


x (mm)

T(120583

m)

(c) 119910 = 20


0102030405060708090

100


y (mm)

T(120583

m)

(d) 119909 = 20




6 Conclusion



Competing Interests



0 20 40 60 80 100 120 1400

102030405060708090

100



T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100



x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 20 40 60 80 100 120 1400

102030405060708090

100



T(120583

m)

(a) 119910 = 0

0102030405060708090

100



y (mm)

T(120583

m)

(b) 119909 = 0

0102030405060708090

100



x (mm)

T(120583

m)

(c) 119910 = 20

0102030405060708090

100



y (mm)

T(120583

m)

(d) 119909 = 20


Acknowledgments


References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article prediction model of coating growth rate

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