The probability characteristics of surface topography parameters described by the composition of the deterministic component and the homogeneous random normal field were analysed. Formulae for the calculation of the mathematical expectation of the Ras parameter and the evaluation of its variance are given

Haghi, A. K.

Acta Polytechnica

English

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionNominally flat surfaces are widely used in practice. Thesecan be mathematically described by the composition of thedeterministic and random components of irregularities in thegiven Cartesian coordinate system:     h x y S x y x y, , ,   (1)where S(x, y) is the deterministic function of the surface (x, y)coordinates and (x, y) is the homogeneous random normalfield. The parameters of surface irregularities measured overthe whole surface (the topographic parameters) characterizemore positively the functional properties of the surface thanthe h profile parameters. Surface deviations are functions oftwo coordinates (x, y) and therefore the profile evaluationgives incomplete information about the surface.2 Surface topography parametermeasurementFor surface topography parameter measurement it is nec-essary to determine the actual value of the parameter andto know the accuracy of the measurements. In this casethe analogue mean value is taken for the actual value ofthe parameter, and[s] the measurement error is determinedby the systematic and random components. The series ofthe surface topographic parameters can be represented asan averaging operator of the generalized transformationG{h(x, y)} of the surface coordinates on the given rectangulararea of the surface L1 × L2 with sides L1 and L2 [1]:  PL LG h x y x ysLL 11 20021, d d (2)Since there is a random component on the measured sur-face, the topographic parameter measured is a random value,which is characterized by the mathematical expectation E(Ps)and the variance D(Ps). Therefore, one of the problems inmeasuring the topographic parameter is the determinationof its probability characteristics, i.e., the mathematical expec-tation E(Ps) and the variance D(Ps). It is known that themathematical expectation of the parameters given by equa-tion (2) can be derived by integration of the mathematicalexpectation E(G) of the transformation G{h(x, y)} in equation(2) by the x and y variables. As an example of the applicationE(G) of the transformation G{h(x, y)} in equation (2) bythe x and y variables. As an example of the application of mea-suring methods of the topographic parameters the PsRasparameter is used. This is the arithmetic mean deviation ofthe surface coordinates of the mean plane.    E RL LE h x y x yLLas d d 11 20021, (3)where  h x y, is the absolute value of the h(x, y) surface coor-dinate. This expression can also be extended for the h(x, y)surface:         E h x yS x yS x yS x y, exp,,, 	 221 2 22 21 2(4)where     z t tz 21 2 20exp d is the Laplace function.The generalized transformation G{h(x, y)} is the randomfield which has the correlation function [3] defined as         K x x y y G h G h f h h h hE G h G hG 1 2 1 2 1 2 1 2 1 21, , , ,  d d  2 ,(5)where  h h x y1 1 1 , ,  h h x y2 2 2 , are the coordinates of thesurface at the x y1 1, and  x y2 2, points,  f h h1 2, distributiondensity expands into a series in terms of Hermite polynomi-als. The correlation function (5) can be represented as [3]:       K x x y y C x y C x ynG n nn1 2 1 2 1 1 1 1 1 21 21, , , , ,,!        (6)    C x y G h H hS h Shn 1 1 1 2212 2, exp 	 n d (7)1 2 1 x x2 2 1 y y is the r.m.s. deviation of the random component (x, y) and   1 2, is the correlation coefficient of the random compo-nents (x, y) ,  h h x y , ,  S S x y , . For the generalizedtransformation     G h x y h x y , , , which determines theRas parameter, the coefficients Cn in equation (7) are writtenas follows after the transformation:35Acta Polytechnica Vol. 42  No. 2/2002A Mathematical Approach for Evaluationof Surface Topography ParametersA. K. HaghiThe probability characteristics of surface topography parameters described by the composition of the deterministic component and thehomogeneous random normal field were analysed. Formulae for the calculation of the mathematical expectation of the Ras parameter andthe evaluation of its variance are given.Keywords: mathematical approach, surface topography, deterministic component.      C x y S x xS x y S x1 1 11 21 21 1222, , exp, 	   1 11 22, y    C x yS x yHS x yn 1 11 2 222122, exp,, 	 !n     1 1 1 1 1   S x yHS x y, ,n 1(8)where Hn(z) are Hermite polynomials. The variance of theparameter PS is determined by integration of the correlationfunction  K x x y yG 1 2 1 2, , , of the generalized transformationG{h(x, y)} using variables x1, x2, y1, y2:   D PL LK x x y y x x y yLLLS G d d d d 112 22 1 2 1 2 1 2 1 2000211, , , . (9)Calculation of the integral in equation (9) involves consid-erable difficulties; therefore, in the general case, formula (9)in not suitable for calculations. However, having  S x y, " and  S x y, #  offers simplified evaluations of the variance.Thus, having  S x y, "  from equation (8),C1 $ Cn $ 0 n  %2 3, , (10)The correlation function of the transformation G{h(x, y)}from equation (6) is reported approximately as   KG      1 2 1 2, ,$ . (11)By using  S x y, #  from equation (8) we obtainC1 0$ C Hn n$ 	 201 22 .The correlation function of the transformation G{h(x, y)}from equation (6) is    K CnG nnn   1 22 1 22,,!$ . (12)In both cases the correlation function  KG  1 2, is a func-tion of only two variables 1 and 2. Thus, for the above-men-tioned approximations, D(PS) can be calculated by using theformula:   D PL L L LKGLS d d 	 	4 1 11 211221 2 1 202    ,01L . (13)The following notation is now used:  SKKK GGGd d10 0 1 2 1 200,,    (14)with L L SK1 2! " Gequation (13) may be written approxi-mately:   D PL LK SKS G G$40 01 2, . (15) KG 0 0, is the variance of G{h(x, y)}. It is not possible to carryout analogue measurements of the topographic parameterPS. Therefore the analogue-discrete and discrete methods arethe only ones that can be used to measure the topographicparameters. With analogue-discrete measurements the aver-aging operator is:   ,PN LG h i y yLiNS d  1 1210121& (16)where &1 is the sampling in the sampling interval between theprofiles and N1 is the number of profiles. The evaluation ofthe parameter in equation (2) in the general case is thereforeshifted together with the task of determining the parameterprobability characteristics, i.e., for its mathematical expecta-tion and variance it is necessary to determine the shift in thevalue of the evaluation of equation (16) with respect to that ofequation (2). We can regard the probability characteristics ofthe evaluation (16) for that particular transformation as    G h i y h i y& &1 1, , . (17)Ras represents the analogue-discrete evaluation of the top-ographic parameter Ras. The mathematical expectation ofthe transformation (17) will be determined by the expressionin equation (4) with    S x y S i y, , &1 . The mathematicalexpectation of the Ras parameter is identically:    E R N L E h i y yLiN ,as d  1 11 210121& . (18)The integrals in equation (18) are not taken into account,but if the deterministic component  S i y&1, is linearizedwith a small error using the y argument on the terminal num-ber of intervals the length of which is 	2 then it is possibleto obtain an exact expression for the evaluation  E Ras of thetopographic parameter Ras. For this, we expand the deter-ministic component  S i y&1, in its Taylor series by using they argument up to the linear terms at the point j = 1, 2, …, n:       S i y S i jS i jyy& & && &&1 1 21 22, ,,$  . (19)By substituting equation (19) into equation (18) and tak-ing into account equation (4), we obtain:  E RN LjjjniNexpas  ''  	 ~~  1 121 21111 2221&&     ij ijij ijij ijyyy 	221 222dy(20)where    ijijS i jyS i jS i jyj~~ & && && &&1 21 21 22,,,.(21)The integrals in both components of expression (20) canbe tabulated [4]. Then after transformation we obtain:36Acta Polytechnica Vol. 42  No. 2/2002 E R N L CA CACDjniNijij ij ijijas  	  1 1 121421 2 111    (A Cij(22) (   	 B CBCD B CAijij ij ijij ijijij  1421 2exp    C ABC Bijijijij2 21 22 2 expwhereAij ij ij  &1 B jij ij ij   1 1&C 12 12Dijij2 212The   z and exp( )z2 and the standard calculation pro-gram for the   z and exp( )z2 functions can be used tocalculate E R(  )as on the computer. Formula (22) shows that themathematical expectation of the parameter Ras depends onthe characteristic of the   S i y&1 , the deterministic compo-nent S i y( , )&1 and the random component 2. With thedefinite relationship between the deterministic and the ran-dom components, an approximate evaluation of E R(  )as canbe obtained.Thus, having  S x y, #  and linearizing equation (4) weobtain:   E R N L S i y yLiN ,as d$  1 11 210121& . (23)By using  S x y, #  and linearizing equation (4):      E h x yS x y,,$ 	 2212 21 2 . (24)©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 37Acta Polytechnica Vol. 42  No. 2/20025,06 mm 5,06 mmμm2000 1 2 3 4 mm090 μmFig. 1: Representation of surface roughness measurement2,9 mm2,5 mm001 2 mm2010μmFig. 2: Representation of surface roughness measurement38 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 42  No. 2/2002Figs. 3 and 4: A 3D topographical image©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 39Acta Polytechnica Vol. 42  No. 2/2002Fig.5:Representationof3DroughnesstestinalargescaleThen    E RL LS x y x yLL ,as $	  2 12121 21 220021 d d . (25)To analyse the E R(  )as dependence on S  and to deter-mine what relationship with S  is needed in order to makeformulae (23) and (25) applicable to the determination of themathematical expectation E R(  )as , calculations were carriedout using formulae (22), (23) and (25). The mathematicalmodels of the composition surface and the experimentalcalculation of the parameterR hijjNiN ias   111(26)where hij is the deviation of the h (x, y) surface from the meanplane at the discrete points. The random component wasmodelled with the help of the random number generator.The deterministic component was designated by formula(19).3 Discussion If the deterministic component is a piecewise linear func-tion and the random component is a homogeneous normalfield, then the mathematical expectation of parameter Rasincreases with increasing S . IfS   1, when E R(  )as is calculated the influence of the de-terministic component can be ignored and the error is lessthan 10 %. When S   3, formula (25) can be used to calculate E R(  )aswith an error of less than 10 %. If S   4, E R(  )as can be evaluated with an error of lessthan 10 % by using the deterministic component (for-mula 23). Formula (22) holds for the whole range of S  changes. Full agreement with the theoretical expression can bedemonstrated by the experimental calculations using for-mula (26). Some optical profilometers based on these principles areshown in figures (1–6) to measure the statistical parametersof rough surfaces. The contrast is found to be related to sur-face roughness when the length of coherence of the light[are] is comparable in magnitude.4 ConclusionThe theoretical dependence of the mathematical expecta-tion of parameter Ras has been determined for the case whenthe deterministic component S(x, y) can be linearized on theseparate sections. The dependence of the evaluation bias inparameter Ras on the relationship between the random anddeterministic components has been determined.References[1] Lukyan, V.: Measurement of the surface topography. ICTE,2000, p. 103–154.[2] Lewis, J.: Fundamentals of Microgeometry. RPI, 2001,p. 32–65.[3] Zahedi, M.: Mesure Sans Contact de la Topographie d’uneSurface. IUT de Belfort, 2000, p. 43–78.[4] Galat, T.: Optical Surface Roughness Determination. Ap-plied Optic, 2000.Dr. A. K. Haghie-mail: haghi@kadous.gu.ac.irGuilan UniversityP.O. Box 3756Rasht, IRAN40 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 42  No. 2/2002Fig. 6: Microtopograph of a rough surface

A Mathematical Approach for Evaluation of Surface Topography Parameters

A. K. Haghi

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A Mathematical Approach for Evaluation of Surface Topography Parameters

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