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Multiparameter representation of surface roughness
161
@‘ear,102 (1985) 161 - 176
MULTIPARAMETER REPRESENTATION OF SURFACE ROUGHNESS BOGDAN NOWICKI Warsaw Tee~njca~ Unjve~jty, (Received January 3,1984;
al. Ni~p~d~~~lo~ci 222, 00-663 macaw t~olu~d) accepted January 9,1986)
The parameters and functions used for the evaluation, analysis and modelling of roughness are reviewed in the present work. Since the estimation of the roughness performed with one parameter is ambiguous, a method of choosing parameters for the multiparameter estimation of roughness is proposed. On the basis of a comparative analysis of roughness features given by individual parameters, a selection procedure was carried out and a set of basic parameters proposed. Next, using the results of correlation analysis, a subset of four uncorrelated parameters was established. It is proposed that two-parameter characterizations should be based on parameters of this subset. This approach would considerably facilitate the exchange of licences, and of technical and scientific information.
Surface roughness is very important from the point of view of such fundamental problems as friction, contact deformation, heat and electric current conduction, tightness of contact joints and positional accuracy. For this reason snrFace roughne~ has been a subject of expe~mental and theoretical investigations for many decades. The real surface geometry is so complicated that a finite number of parameters cannot provide a full description. If the number of parameters used is increased, a more extensive and clearer description results. This has been one of the reasons for introducing new parameters for surface evaluation. The development of measurement methods and the trend towards more detailed description of surface roughness led to a change from subjective methods of visual evaluation via height p~~eters (R,, R, and R,,) to parameters describing peak shapes (A, and ic) as well as statistical distributions (tp, fh(y) and f,(y)). All these methods permitted the geometrical structure to be ordered with respect to its quality. After it was discovered that the roughness height is merely one estimator of surface quality, a statistical approach was taken. Researchers started using 0043.1648/%6/$3.30
@ EIsevier Sequo~j~i~ted
in Switzerland
162
such tools as the distribution of the bearing capacity and the distr$bution of ordinates. For abrasive wear this approach permitted much better evaluation of surface quality than was possible by other methods. Theoretical considerations and investigations of the contact deformations of microroughness have led to the formulation of parameters encompassing the shape of the roughness peaks and their pitch. In recent advanced invest~ations of the surface geometry the methods of correlation and spectrum analysis have been used. Up to now more than 30 parameters and functions describing the surface roughness are known. Such a great number results from the prerequisites already discussed and the historical development of national standards and measurement equipment. The parameters used for the estimation of roughness are listed in Table 1. A full list of standards and a bib~io~phy relating to Table I are included in ref. 1. Among these the following can be distinguished: height parameters, e.g. R,, % JLX, Rsk, R,, R,, R+,, and R; horizontal parameters, e.g. S,, h,, h,, n(O), na and s; parameters relating to the shape of the microroughness, eg. A,, A4, r,, rlj, 7L and 711;parameters associated with the shape, spatial extent and amplitude, e.g. lo, A and HSC!. Also used are statistical dis~butions which, in comparison with individu~ parameters, have properties generalizing the features of the geometric microstructure of the surface and are dependent on for example the shape and location of the peaks of the roughness. These include the following: the bearing capacity t, of the profile (evaluated as a percentage) and parameters of the bearing curve, e.g. Ir and Y;the d~tribution of ordinates of the profile, i.e. fh(y); the distribution fm(y) of peaks; the autocorrelation function &(A@; the spectral density function G(f). The numerical values of r, y, f’(y) and f,(y) are computed after their empirical distribution is determined and parameters indicating the degree of compliance of this distribution with the normal dist~bution (i.e. the skewness Sx and kurtosis b2) are calculated. It is evident from Table 1 that the estimation of the same features of the roughness is carried out with the use of parameters with very similar definitions. Anaiysing the theoretical works of Chusy and Witenberg [Z], Rudzit [ 31 and Whitehouse [ 4 ] on roughness description assuming a normal distribution, one can conclude that basic roughness parameters (R,=, A,, P and tP) may be determined on the basis of R,, n(O), m and R,(Ax). The relationships are given in Table 1. On the basis of these relationships we can determine the theoretical correlations between the appropriate roughness parameters. However, the roughness distributions for real surfaces are very different from the normal distribution. The preliminary verification of theoretical relationships made by the present author proves that the results obtained when these relationships are used give large errors and they may not be directly used for the choice of a multiparameter roughness representation. Therefore it is necessary to systematize the selection of a set of parameters which would fulfil the Profile
163
following conditions: to ensure as complete as possible a description of the geometric features of the surface; to enable precise inte~re~tion; to be measurable by commonly available equipment; to fulfil requirements of economy, i.e. that the time and means required for measurements be compensated by, for example, high quality of machine parts or lower manufacturing costs; to be independent of each other. From the parameters relating to height the most useful features are It should be noted that the prevailing associated with Rsk, R, and R,,. measuring equipment is not fitted for measurements of &. This is why R, is commonly used in practice, in spite of its being less useful than R*. Horizontal parameters are represented by n(O), ry1and g. The remaining parameters in this group are their equivalents. The parameter g is rarely used in applied and basic works and moreover the equipment used for its measurement is frequently unav~ble. With respect to me~~ements, the present author suggests that the parameter m is to be preferred. The shape of the roughness peaks is described by r and y. The value of the crest radius has rather a large variance, and in numerical calculation it has considerable error associated with it. Opinions as to which of the variables r and y is more closely related to practical properties are divided. Nara [l] and Beck [5] propose that the profile shape should be the second parameter in the estimation of surface roughness, next in importance to the roughness height. Whitehouse [4] maintains that for frictional couplings the crest radius is more suitable for a description. The complex estimate A of the roughness is not widely used in practice; there is a lack of equipment suitable for its direct measurement and by definition it can be determined through other parameters. HSC has a definite relationship with m and a, and lo with A,. Moreover, this parameter is in infrequent use. Analysing the properties of the statistical distributions tp(y), fh(y) and f,(y), one can state [6] that tp(y) is the distribution function of the profile ordinate dist~bution fh(y), Because of this the joint use of both distributions is not necessary. From the standpoint of physical interpretation and feasibility of measurements the bearing capacity t, should be preferred. The distribution of the peaks of the profile has not been used so far, except in theoretical works [ 2,3]. Standard measuring equipment is unavailable and the relation between this distribution and the spatial distribution of the peaks is unknown. For these reasons the dist~bution of the peaks will be dropped from further analysis. The autocorrelation function R,(Ax) and the spectral power density Gy (f) permit us to obtain similar information about the geometric structure of the surface. The functions are related to each other by a Fourier transform. The main difference between them and the roughness parameters lies in the fact that the latter are used for inspection while the former are used for analysis of the surface roughness and in the search for the relationships between the state of machine tools, tools and machining parameters and the geometric structure of machined surfaces, or relations between the structure
164
9
8
7
5
Mean of maxi* mum height of profile
Profile solidity factor
k
Peak-to-valley height
+ R,+
. . . * RVs)
+ RP2 + ,.I + R,,,)
k = RvlRmax
% = +(R,,
Mean depth of valleys
R,
Formula
RP= $(Rp,
Graphical interpretation
Mean height of peaks
Definition
R,;Re
Equivalent designaCon
RQn
R
RP
ti0n
Item Fundamental designa-
TABLE 1 (continued)
Rl=RP,+R,
I
(continued)
Relationships with other parameters ~functians~
E
Fundamental designation
s,
n(O)
m
g
A,
Item
10
11
12
13
14
NP
NC
Aa; A,; &;A,
Equivalent designation
TABLE 1 (continued)
Mean slope of profile
Number of inflection points
Number of peaks in profile
Number of intersections of profile with mean line
Mean spacing of the profile irregularities
Definition Graphical interpretation
g = n,lL
m = n,/L
n(0) = n,/L
Formula
ly’l dx
A a iz: nn(O)R,
n(0) = 2/S,
Aa = 0.9&s,
l/2
(continued)
Relationships with other parameters (functions)
A,
r
P
I*
17
18
19
tan7
7,
16
15
Item Fundamental designation
rl
tan 0
8,
Relative length of profile
Curvature radius of profile
Radius of asperity
R.m.6. profile slope
Profile slope at the mean line
Equivalent Definition designation
TABLE 1 (continued)
Gmphical interpretation
=
O.lR,,,
or
d*y/dx*
(1 + (dy/dzc)2}3’2
l* = LJL
P’
I, LJO.O6R,,
I,
S’Y
valid for
CC*
r= -
Formula
la-
1
0))
1 ++A,*
(continued)
= ‘Z?r*E{R,}E{m)E{n(O)]
Efrl
E{tan 9) = 4E{R,}E{n(
Relationships with other parameters (functions)
5
b,v
HSC
23
24
m(h)
Bearing length curve
tp
22
High spot count
Parameters of equation of bearing length curve q = W/&d”
Relative approach
f
Appraach
Definition
21
Y,P
Equivalent designation
a
mental designation
Funda-
20
Item
TABLE 1 (continued) Graphical interpretation
E = a/R,,
Formula
q = bev
(continued)
Relationships with other parameters (functions)
fh(a)
f,(a)
R, (ax)
G(fo)
26
27
28
29
G(o), P(o), P(w), S(w)
R&M, ACF, A(P), A(x)
f,(Y)
P(Y ), ADF, APDF
Power spectral density
Autocorrelation function
Asperity height distribution; density of peaks
Amplitude density function
Dimensionless complex roughness ratio
A
A’
25
fh(Yh
Definition
Equivalent designation
Item
Fundamental designation
TABLE 1 (continued) Graphical interpretation
Gy(f)=
+Ax)dx
y2(x, f, An dx
1 lim -X Af-tO Af
y(x)y(x
= J LO
= -
1
R,(h)
A’ = R ,,,&b””
Formula
=2
I --
G,(f) +R,(k)
dx
(continued)
exp(-j2nfx)
Relationships with other parameters (functions)
5
Fundamental designation
SK
K
7
Item
30
31
32
P,P2,
b2
R.+r a, ci~“~, bl
Equivalent designation
TABLE 1 (continued)
Randomness coefficient
Kurtosis
Skewness
Definition Graphical interpretation
= p4/p22
EK=K-3
K
SK = 1.13/c(23’2
Formula Relationships with other parameters (functions)
171
and properties of tribological components. Both functions can be determined with the aid of very specialized equipment. From more than 30 parameters and functions now in use and frequently selected by the user at random, a set of parameters and fundamental functions should be chosen in such a way that the elements of this set enable various tasks to be carried out for which the whole range of parameters is currently used. It is assumed that parameters from outside the proposed set will be used if necessary. A reduction in the number of parameters used in production and research would improve the exchange of technical information and licences and would bring unification of equipment, reducing its cost and the cost of research work. The selection satisfying the first four demands can be made on the basis of data collected from the literature. Their statistical independence is discussed, however, in few published sources. On the basis of the preliminary selection of parameters and functions (Table 2) the proposed set would include the following: L
&, R,,,
n(O), m, r, 4,
t,, R&W
(1)
TABLE 2 List of roughness parameters, applications and measurement possibilities Parameters and functions describing the same roughness feature
Representa tive parameter (designation commonly used)
R,, Rskr (Rp) 4, R,, R, Rp, R, R, RIllax* 4, Ranax > $, n(O), NC SC,n(O) P g,‘s r(e), ta7, A,, As r, rl lo HSC 779r)20r 01,7)m $9
b,
m 4, r 7?20r
1)so
Measuring equipment available
Applicability in inspection
Application recommended by standards
Applicability in research work
Applicability in theore tical work and modelling of roughness
+ + + + + +
+ + -
+ + + + + +
+ + + + + +
+ + + + +
+ + + +
-
+ + +
+ +
+
-
+ + +
v -
Cal
fda) R&W, P,(W ‘%(f), Gy(~)
-
Ry@N -
+ -
-
+ + -
172
Parameters from the set (1) (excluding the autocorrelation function) can he determined for a given surface from its profile, and the majority of the parameters can be measured by currently manufactured laboratory profilemeasuring instruments. These parameters are defined by domestic or IS0 standards (K, R,, %,a,, n(0) and tP) and thus fulfil the condition of wide use. They represent the original or generalized features of the profile. However, few data are available in connection with the complex investigation of their statistical ~t~e~tionships. This fact induced the present author to carry out statistical investigations with the aim of determining the significance of relationships between these parameters. This investigation formed the basis for the choice of a narrower subset of statistically independent parameters. Investigations were conducted for specimens machined by plunge and face grinding, milling, turning and planing. The measurements were performed with a Rank Taylor Hobson profile inst~ment. The values of the correlation coefficients have been determined on the basis of the investigations that have been carried out for surfaces machined in industrial environments as well as for surfaces of roughness comparison specimens. In these investigations 20 specimens were used for each machining method. Measurements of typical parameters have been made for each surface and the other parameters have been determined from profilographs taken from the same places. Correlation coefficient calculations have been made with the help of a CDC computer using standard programs. The following types of calculations have been performed: calculations for individual machining methods; calculations for surfaces of random texture; calculations for surfaces of mixed texture; calculations for all surfaces taken as a whole. Table 3 illustrates the results of the investigations for all surfaces taken as a whole. The correlation coefficients for smaller groups of surfaces have higher values than those given in Table 3, with the exception of the case of TABLE
3
Matrix of structure
%
RZ
R max m
n(O) 7720 1)50 r
Y
correlation
1.000
0.963 1.000
coefficients
0.948 0.983 1.000
of roughness parameters
-0.775 -0.773 -0.733 1.000
-0.793 -0.779 -0.754 0.946 1.000
0.383 0.361 0.381 -0.298 -0.341 1.000
with random
0.2‘25 -0.192 -0.138 0.461 0.395 0.334 1.000
and mixed
0.392 0.399 0.417 0.203 0.246 0.242 0.050 1.000
0.528 0.601 0.656 0.385 0.402 0.188 0.002 0.041 1.000
173
milled surfaces, where the values of the correlation coefficients between m and n(0) as well as between R,, R, and R, and n(O) and m are low. On the basis of an analysis of the results presented in Table 3 the following conclusions can be formulated. (1) A strong correlation exists between the height parameters (R,, R, and R,,). The presence of all these parameters in standards should therefore be reconsidered. The same concern applies to R, and R, which are correlated with R,. (2) It is necessary for the normalizing committees to establish which parameters relating to the profile pitch will be obligatory. The parameters S, and h, have very close numerical values, and m and n(0) are strongly correlated. (3) In the joint analysis of surfaces with random and mixed types of roughness, a strong correlation is present between the height and the horizontal parameters. If the method of machining is known the amplitude parameter is quite sufficient. (4) It is proposed that, for the multiparameter characteristics of roughness used in research, parameters should be applied which show low statistical correlation, i.e. L
m, r, Aa, t20, t50
(2)
These represent both height (R,) and horizontal (m) parameters, making possible the description of the shape of the microroughness in terms of r and Aa. The bearing capacity tzo and t5,, carries generalized data about the probability density of the peaks and the geometry of the peaks. (5) The value of the expected wear or contact deformation can be estimated from an analysis of the working conditions of machine parts. Thus it is suggested that the bearing capacity parameter t2,, or t5,, which enables the estimation of the value of the true surface area during the contact between machine parts with better accuracy should be chosen. (6) In the inspection of functionally important surfaces, the choice of parameters should be based on the results of investigations of relationships between the parameters from the set (2) and useful properties of machined parts. Such a method would be helpful in choosing a set of adequate parameters for roughness estimation of various functions for the surfaces of machine parts, and it would be useful in formulating rational demands by, for example, tribologists. (7) It is deliberately arranged that each characteristic has one commonly shared parameter which meets the demands of common use, can be easily measured and about which a priori statements can be made concerning its significant relation to the useful properties of machine parts: this condition is fulfilled by R,. A choice of parameters for a two-parameter characteristic can be based on the analysis of theoretical relationships between useful properties of machine parts and those of surface layers or on the results of experimental investigations of these properties. In the first case particular attention should
174
be given to the assumptions concerning the formulae, i.e. whether the parameters from the set (2) were taken into account. The majority of relationships describing the contact of rough surfaces and related tribological phenomena were obtained under the assumptions of simplified models of roughness (spheres, lack of work hardening of the material etc.). The results obtained from these formulae, therefore, have considerable errors associated with them. For these reasons, with the current state of the art concerning the phenomena occurring in the contact area of rough surfaces, it is safer to determine the statistical relationships between R,, r, A, and t, and the useful properties of the surface layer in an experimental way. On the basis of these relationships one may narrow down the choice of roughness characteristics. So far two-parameter characteristics have been proposed by Whitehouse [4], Nara [l], McCool [7] and Spragg and Whitehouse [8]. However, these references do not furnish information about the set from which the characteristics were chosen. The solution to this problem may be obtained after determining the coefficient of correlation between all the parameters of the set (2) and given the use of machined parts. A method suitable for choosing a two-parameter characteristic for the contact deformation of rough surfaces is given below. It is useful to use an equation derived by Demkin [9] while choosing the form of the regression equation. This equation is valid for the elastic contact of a smooth surface with a rough surface with spherical peaks and a known peak distribution as a function of the distance of approach: j = eA,RaA,rA,AaA,t20A’mA’
(3)
The investigations were carried out on flat specimens. For each surface all parameters from the set (2) were determined. From analysis of the correlation coefficients it was established that the ratio of plastic and elastic deformations and the contact stiffness are strongly correlated with the following roughness parameters: R,, m, A,
(4)
As R, is correlated with m to an extent similar to that with j, the relation can be of an indirect type, and technical evaluation of the state of surfaces working under conditions of high pressure suggests that the following twoparameter characteristic should be used: R,, Aa
(5)
It is evident from the formulae for contact deformations that the roughness of the surface is represented by one amplitude parameter and the crest radius r. Therefore one may expect that at relatively large deformations (a soft specimen material measurement head with carbide inserts, considerable pressures) the more effective parameter will be the slope of the profile. This can be explained by the fact that the peaks become so deformed at small loads that the increase in surface resistance is associated with the slope
175 Standard inspection of roughness
R, l----1
I P I R, I I i-----I
Inspection of roughness of especially important surfaces Implementation
investigations
Routine investigations
F
Fundamental investigations
F
L_p_J P P
(4,)
Ra
(4)
R,
Ap,
R,
n(o)
l------1
tP Aa
L______: r
tp
f&&W
Fig. 1. Choices of parameters and techniques (F, photographs of the surface; P, profilograms) for the inspection and investigation of roughness. The box on the right hand side of the figure indicates that alternative parameters may be used.
of the profile, At this range of values of the approach the slope governs the increase in the real contact area. On the basis of the principles by which the multiparameter characteristics of roughness are to be established, it is possible to determine a hierarchical system for the estimation of surface roughness (Fig. 1). In basic research and advanced technologic~ invest~atio~ it is suggested that parameters from the set (1) should be used; these could eventually be complemented with specific parameters having a direct relation with the most significant useful properties according to surface profile traces and photo~ap~ (taken with a scanning microscope) of the surface. In routine ~vestigations the parameters from the set (2) should be used together with a profile trace and a photograph of the surface. While implementing new products two parameters are advised: R, and a parameter selected from the investigations of the first or the second kind, In production inspection, very high quality surfaces should be tested by checking two parameters, e.g. R, and A,. For stable and well- controlled production process conditions, the second of the parameters need not be inspected frequently. In other cases only one parameter is periodically controlled. The most important is the maintenance of correct production conditions and visual evaluation of the roughness, Both the inspection workers and other qualified and experienced workers can easily notice the essential changes in surface roughness.
The following conclusions can be drawn. There is an urgent need to establish from currently used parameters a small subset of parameters to be recommended for use in research work. The possibility of withdrawing strongly correlated parameters from the standards should be considered. Parameters describing the shape of the peaks should take their place. The application of two parameters in the inspection of surface rougbness should ensure constant high quality of machine parts.
176
References 1 I. Nara, Two dimensional representation of surface roughness, CZZU'Ann., (1) (1961). 2 A. ChusyandJu. Witenberg,Szerochowatost’PowierchnostejTeoretikowerojatnostnyj Podchod, Nauka, Moscow, 1975. 3 Ja. Rudzit, Mikrogeometria i Kontaktnoje WzaimodeistwiePowierchnostiej, ZINATNE, Riga, 1975. 4 D. Whitehouse, Approximate methods of assessment of surface topograph parameters,
Mat. Konf. Metrologii MP, Warsaw, 1976. 5 C. Beck, Statist&he KennegrSsaen der Profilwinkel a technischen Oberfhichen, Feingetitetechnik, (10) (1976). 6 B. Nowicki, Badanie mikrostruktury geometrycxnej powierzchni i metody jej oceny, Zest. Nauk. Politech. Wars%., 70 (1980). 7 J. McCool, Characterization of surface anisotropy, Wear, 49 (1) (1978) 19. 8 R. Spragg and D. Whitehouse, A new unified approach to surface metrology, Proc., Inst. Mech. Eng., London, (47) (1970 - 1971). 9 N. Demkin, Kontaktirowanie Szerochowatych Powierchnostej, Moscow, 1970. 10 ANSI Stand. B46, Part 1,1978 (American National Standards Institute, New York).
@‘ear,102 (1985) 161 - 176
MULTIPARAMETER REPRESENTATION OF SURFACE ROUGHNESS BOGDAN NOWICKI Warsaw Tee~njca~ Unjve~jty, (Received January 3,1984;
al. Ni~p~d~~~lo~ci 222, 00-663 macaw t~olu~d) accepted January 9,1986)
The parameters and functions used for the evaluation, analysis and modelling of roughness are reviewed in the present work. Since the estimation of the roughness performed with one parameter is ambiguous, a method of choosing parameters for the multiparameter estimation of roughness is proposed. On the basis of a comparative analysis of roughness features given by individual parameters, a selection procedure was carried out and a set of basic parameters proposed. Next, using the results of correlation analysis, a subset of four uncorrelated parameters was established. It is proposed that two-parameter characterizations should be based on parameters of this subset. This approach would considerably facilitate the exchange of licences, and of technical and scientific information.
Surface roughness is very important from the point of view of such fundamental problems as friction, contact deformation, heat and electric current conduction, tightness of contact joints and positional accuracy. For this reason snrFace roughne~ has been a subject of expe~mental and theoretical investigations for many decades. The real surface geometry is so complicated that a finite number of parameters cannot provide a full description. If the number of parameters used is increased, a more extensive and clearer description results. This has been one of the reasons for introducing new parameters for surface evaluation. The development of measurement methods and the trend towards more detailed description of surface roughness led to a change from subjective methods of visual evaluation via height p~~eters (R,, R, and R,,) to parameters describing peak shapes (A, and ic) as well as statistical distributions (tp, fh(y) and f,(y)). All these methods permitted the geometrical structure to be ordered with respect to its quality. After it was discovered that the roughness height is merely one estimator of surface quality, a statistical approach was taken. Researchers started using 0043.1648/%6/$3.30
@ EIsevier Sequo~j~i~ted
in Switzerland
162
such tools as the distribution of the bearing capacity and the distr$bution of ordinates. For abrasive wear this approach permitted much better evaluation of surface quality than was possible by other methods. Theoretical considerations and investigations of the contact deformations of microroughness have led to the formulation of parameters encompassing the shape of the roughness peaks and their pitch. In recent advanced invest~ations of the surface geometry the methods of correlation and spectrum analysis have been used. Up to now more than 30 parameters and functions describing the surface roughness are known. Such a great number results from the prerequisites already discussed and the historical development of national standards and measurement equipment. The parameters used for the estimation of roughness are listed in Table 1. A full list of standards and a bib~io~phy relating to Table I are included in ref. 1. Among these the following can be distinguished: height parameters, e.g. R,, % JLX, Rsk, R,, R,, R+,, and R; horizontal parameters, e.g. S,, h,, h,, n(O), na and s; parameters relating to the shape of the microroughness, eg. A,, A4, r,, rlj, 7L and 711;parameters associated with the shape, spatial extent and amplitude, e.g. lo, A and HSC!. Also used are statistical dis~butions which, in comparison with individu~ parameters, have properties generalizing the features of the geometric microstructure of the surface and are dependent on for example the shape and location of the peaks of the roughness. These include the following: the bearing capacity t, of the profile (evaluated as a percentage) and parameters of the bearing curve, e.g. Ir and Y;the d~tribution of ordinates of the profile, i.e. fh(y); the distribution fm(y) of peaks; the autocorrelation function &(A@; the spectral density function G(f). The numerical values of r, y, f’(y) and f,(y) are computed after their empirical distribution is determined and parameters indicating the degree of compliance of this distribution with the normal dist~bution (i.e. the skewness Sx and kurtosis b2) are calculated. It is evident from Table 1 that the estimation of the same features of the roughness is carried out with the use of parameters with very similar definitions. Anaiysing the theoretical works of Chusy and Witenberg [Z], Rudzit [ 31 and Whitehouse [ 4 ] on roughness description assuming a normal distribution, one can conclude that basic roughness parameters (R,=, A,, P and tP) may be determined on the basis of R,, n(O), m and R,(Ax). The relationships are given in Table 1. On the basis of these relationships we can determine the theoretical correlations between the appropriate roughness parameters. However, the roughness distributions for real surfaces are very different from the normal distribution. The preliminary verification of theoretical relationships made by the present author proves that the results obtained when these relationships are used give large errors and they may not be directly used for the choice of a multiparameter roughness representation. Therefore it is necessary to systematize the selection of a set of parameters which would fulfil the Profile
163
following conditions: to ensure as complete as possible a description of the geometric features of the surface; to enable precise inte~re~tion; to be measurable by commonly available equipment; to fulfil requirements of economy, i.e. that the time and means required for measurements be compensated by, for example, high quality of machine parts or lower manufacturing costs; to be independent of each other. From the parameters relating to height the most useful features are It should be noted that the prevailing associated with Rsk, R, and R,,. measuring equipment is not fitted for measurements of &. This is why R, is commonly used in practice, in spite of its being less useful than R*. Horizontal parameters are represented by n(O), ry1and g. The remaining parameters in this group are their equivalents. The parameter g is rarely used in applied and basic works and moreover the equipment used for its measurement is frequently unav~ble. With respect to me~~ements, the present author suggests that the parameter m is to be preferred. The shape of the roughness peaks is described by r and y. The value of the crest radius has rather a large variance, and in numerical calculation it has considerable error associated with it. Opinions as to which of the variables r and y is more closely related to practical properties are divided. Nara [l] and Beck [5] propose that the profile shape should be the second parameter in the estimation of surface roughness, next in importance to the roughness height. Whitehouse [4] maintains that for frictional couplings the crest radius is more suitable for a description. The complex estimate A of the roughness is not widely used in practice; there is a lack of equipment suitable for its direct measurement and by definition it can be determined through other parameters. HSC has a definite relationship with m and a, and lo with A,. Moreover, this parameter is in infrequent use. Analysing the properties of the statistical distributions tp(y), fh(y) and f,(y), one can state [6] that tp(y) is the distribution function of the profile ordinate dist~bution fh(y), Because of this the joint use of both distributions is not necessary. From the standpoint of physical interpretation and feasibility of measurements the bearing capacity t, should be preferred. The distribution of the peaks of the profile has not been used so far, except in theoretical works [ 2,3]. Standard measuring equipment is unavailable and the relation between this distribution and the spatial distribution of the peaks is unknown. For these reasons the dist~bution of the peaks will be dropped from further analysis. The autocorrelation function R,(Ax) and the spectral power density Gy (f) permit us to obtain similar information about the geometric structure of the surface. The functions are related to each other by a Fourier transform. The main difference between them and the roughness parameters lies in the fact that the latter are used for inspection while the former are used for analysis of the surface roughness and in the search for the relationships between the state of machine tools, tools and machining parameters and the geometric structure of machined surfaces, or relations between the structure
164
9
8
7
5
Mean of maxi* mum height of profile
Profile solidity factor
k
Peak-to-valley height
+ R,+
. . . * RVs)
+ RP2 + ,.I + R,,,)
k = RvlRmax
% = +(R,,
Mean depth of valleys
R,
Formula
RP= $(Rp,
Graphical interpretation
Mean height of peaks
Definition
R,;Re
Equivalent designaCon
RQn
R
RP
ti0n
Item Fundamental designa-
TABLE 1 (continued)
Rl=RP,+R,
I
(continued)
Relationships with other parameters ~functians~
E
Fundamental designation
s,
n(O)
m
g
A,
Item
10
11
12
13
14
NP
NC
Aa; A,; &;A,
Equivalent designation
TABLE 1 (continued)
Mean slope of profile
Number of inflection points
Number of peaks in profile
Number of intersections of profile with mean line
Mean spacing of the profile irregularities
Definition Graphical interpretation
g = n,lL
m = n,/L
n(0) = n,/L
Formula
ly’l dx
A a iz: nn(O)R,
n(0) = 2/S,
Aa = 0.9&s,
l/2
(continued)
Relationships with other parameters (functions)
A,
r
P
I*
17
18
19
tan7
7,
16
15
Item Fundamental designation
rl
tan 0
8,
Relative length of profile
Curvature radius of profile
Radius of asperity
R.m.6. profile slope
Profile slope at the mean line
Equivalent Definition designation
TABLE 1 (continued)
Gmphical interpretation
=
O.lR,,,
or
d*y/dx*
(1 + (dy/dzc)2}3’2
l* = LJL
P’
I, LJO.O6R,,
I,
S’Y
valid for
CC*
r= -
Formula
la-
1
0))
1 ++A,*
(continued)
= ‘Z?r*E{R,}E{m)E{n(O)]
Efrl
E{tan 9) = 4E{R,}E{n(
Relationships with other parameters (functions)
5
b,v
HSC
23
24
m(h)
Bearing length curve
tp
22
High spot count
Parameters of equation of bearing length curve q = W/&d”
Relative approach
f
Appraach
Definition
21
Y,P
Equivalent designation
a
mental designation
Funda-
20
Item
TABLE 1 (continued) Graphical interpretation
E = a/R,,
Formula
q = bev
(continued)
Relationships with other parameters (functions)
fh(a)
f,(a)
R, (ax)
G(fo)
26
27
28
29
G(o), P(o), P(w), S(w)
R&M, ACF, A(P), A(x)
f,(Y)
P(Y ), ADF, APDF
Power spectral density
Autocorrelation function
Asperity height distribution; density of peaks
Amplitude density function
Dimensionless complex roughness ratio
A
A’
25
fh(Yh
Definition
Equivalent designation
Item
Fundamental designation
TABLE 1 (continued) Graphical interpretation
Gy(f)=
+Ax)dx
y2(x, f, An dx
1 lim -X Af-tO Af
y(x)y(x
= J LO
= -
1
R,(h)
A’ = R ,,,&b””
Formula
=2
I --
G,(f) +R,(k)
dx
(continued)
exp(-j2nfx)
Relationships with other parameters (functions)
5
Fundamental designation
SK
K
7
Item
30
31
32
P,P2,
b2
R.+r a, ci~“~, bl
Equivalent designation
TABLE 1 (continued)
Randomness coefficient
Kurtosis
Skewness
Definition Graphical interpretation
= p4/p22
EK=K-3
K
SK = 1.13/c(23’2
Formula Relationships with other parameters (functions)
171
and properties of tribological components. Both functions can be determined with the aid of very specialized equipment. From more than 30 parameters and functions now in use and frequently selected by the user at random, a set of parameters and fundamental functions should be chosen in such a way that the elements of this set enable various tasks to be carried out for which the whole range of parameters is currently used. It is assumed that parameters from outside the proposed set will be used if necessary. A reduction in the number of parameters used in production and research would improve the exchange of technical information and licences and would bring unification of equipment, reducing its cost and the cost of research work. The selection satisfying the first four demands can be made on the basis of data collected from the literature. Their statistical independence is discussed, however, in few published sources. On the basis of the preliminary selection of parameters and functions (Table 2) the proposed set would include the following: L
&, R,,,
n(O), m, r, 4,
t,, R&W
(1)
TABLE 2 List of roughness parameters, applications and measurement possibilities Parameters and functions describing the same roughness feature
Representa tive parameter (designation commonly used)
R,, Rskr (Rp) 4, R,, R, Rp, R, R, RIllax* 4, Ranax > $, n(O), NC SC,n(O) P g,‘s r(e), ta7, A,, As r, rl lo HSC 779r)20r 01,7)m $9
b,
m 4, r 7?20r
1)so
Measuring equipment available
Applicability in inspection
Application recommended by standards
Applicability in research work
Applicability in theore tical work and modelling of roughness
+ + + + + +
+ + -
+ + + + + +
+ + + + + +
+ + + + +
+ + + +
-
+ + +
+ +
+
-
+ + +
v -
Cal
fda) R&W, P,(W ‘%(f), Gy(~)
-
Ry@N -
+ -
-
+ + -
172
Parameters from the set (1) (excluding the autocorrelation function) can he determined for a given surface from its profile, and the majority of the parameters can be measured by currently manufactured laboratory profilemeasuring instruments. These parameters are defined by domestic or IS0 standards (K, R,, %,a,, n(0) and tP) and thus fulfil the condition of wide use. They represent the original or generalized features of the profile. However, few data are available in connection with the complex investigation of their statistical ~t~e~tionships. This fact induced the present author to carry out statistical investigations with the aim of determining the significance of relationships between these parameters. This investigation formed the basis for the choice of a narrower subset of statistically independent parameters. Investigations were conducted for specimens machined by plunge and face grinding, milling, turning and planing. The measurements were performed with a Rank Taylor Hobson profile inst~ment. The values of the correlation coefficients have been determined on the basis of the investigations that have been carried out for surfaces machined in industrial environments as well as for surfaces of roughness comparison specimens. In these investigations 20 specimens were used for each machining method. Measurements of typical parameters have been made for each surface and the other parameters have been determined from profilographs taken from the same places. Correlation coefficient calculations have been made with the help of a CDC computer using standard programs. The following types of calculations have been performed: calculations for individual machining methods; calculations for surfaces of random texture; calculations for surfaces of mixed texture; calculations for all surfaces taken as a whole. Table 3 illustrates the results of the investigations for all surfaces taken as a whole. The correlation coefficients for smaller groups of surfaces have higher values than those given in Table 3, with the exception of the case of TABLE
3
Matrix of structure
%
RZ
R max m
n(O) 7720 1)50 r
Y
correlation
1.000
0.963 1.000
coefficients
0.948 0.983 1.000
of roughness parameters
-0.775 -0.773 -0.733 1.000
-0.793 -0.779 -0.754 0.946 1.000
0.383 0.361 0.381 -0.298 -0.341 1.000
with random
0.2‘25 -0.192 -0.138 0.461 0.395 0.334 1.000
and mixed
0.392 0.399 0.417 0.203 0.246 0.242 0.050 1.000
0.528 0.601 0.656 0.385 0.402 0.188 0.002 0.041 1.000
173
milled surfaces, where the values of the correlation coefficients between m and n(0) as well as between R,, R, and R, and n(O) and m are low. On the basis of an analysis of the results presented in Table 3 the following conclusions can be formulated. (1) A strong correlation exists between the height parameters (R,, R, and R,,). The presence of all these parameters in standards should therefore be reconsidered. The same concern applies to R, and R, which are correlated with R,. (2) It is necessary for the normalizing committees to establish which parameters relating to the profile pitch will be obligatory. The parameters S, and h, have very close numerical values, and m and n(0) are strongly correlated. (3) In the joint analysis of surfaces with random and mixed types of roughness, a strong correlation is present between the height and the horizontal parameters. If the method of machining is known the amplitude parameter is quite sufficient. (4) It is proposed that, for the multiparameter characteristics of roughness used in research, parameters should be applied which show low statistical correlation, i.e. L
m, r, Aa, t20, t50
(2)
These represent both height (R,) and horizontal (m) parameters, making possible the description of the shape of the microroughness in terms of r and Aa. The bearing capacity tzo and t5,, carries generalized data about the probability density of the peaks and the geometry of the peaks. (5) The value of the expected wear or contact deformation can be estimated from an analysis of the working conditions of machine parts. Thus it is suggested that the bearing capacity parameter t2,, or t5,, which enables the estimation of the value of the true surface area during the contact between machine parts with better accuracy should be chosen. (6) In the inspection of functionally important surfaces, the choice of parameters should be based on the results of investigations of relationships between the parameters from the set (2) and useful properties of machined parts. Such a method would be helpful in choosing a set of adequate parameters for roughness estimation of various functions for the surfaces of machine parts, and it would be useful in formulating rational demands by, for example, tribologists. (7) It is deliberately arranged that each characteristic has one commonly shared parameter which meets the demands of common use, can be easily measured and about which a priori statements can be made concerning its significant relation to the useful properties of machine parts: this condition is fulfilled by R,. A choice of parameters for a two-parameter characteristic can be based on the analysis of theoretical relationships between useful properties of machine parts and those of surface layers or on the results of experimental investigations of these properties. In the first case particular attention should
174
be given to the assumptions concerning the formulae, i.e. whether the parameters from the set (2) were taken into account. The majority of relationships describing the contact of rough surfaces and related tribological phenomena were obtained under the assumptions of simplified models of roughness (spheres, lack of work hardening of the material etc.). The results obtained from these formulae, therefore, have considerable errors associated with them. For these reasons, with the current state of the art concerning the phenomena occurring in the contact area of rough surfaces, it is safer to determine the statistical relationships between R,, r, A, and t, and the useful properties of the surface layer in an experimental way. On the basis of these relationships one may narrow down the choice of roughness characteristics. So far two-parameter characteristics have been proposed by Whitehouse [4], Nara [l], McCool [7] and Spragg and Whitehouse [8]. However, these references do not furnish information about the set from which the characteristics were chosen. The solution to this problem may be obtained after determining the coefficient of correlation between all the parameters of the set (2) and given the use of machined parts. A method suitable for choosing a two-parameter characteristic for the contact deformation of rough surfaces is given below. It is useful to use an equation derived by Demkin [9] while choosing the form of the regression equation. This equation is valid for the elastic contact of a smooth surface with a rough surface with spherical peaks and a known peak distribution as a function of the distance of approach: j = eA,RaA,rA,AaA,t20A’mA’
(3)
The investigations were carried out on flat specimens. For each surface all parameters from the set (2) were determined. From analysis of the correlation coefficients it was established that the ratio of plastic and elastic deformations and the contact stiffness are strongly correlated with the following roughness parameters: R,, m, A,
(4)
As R, is correlated with m to an extent similar to that with j, the relation can be of an indirect type, and technical evaluation of the state of surfaces working under conditions of high pressure suggests that the following twoparameter characteristic should be used: R,, Aa
(5)
It is evident from the formulae for contact deformations that the roughness of the surface is represented by one amplitude parameter and the crest radius r. Therefore one may expect that at relatively large deformations (a soft specimen material measurement head with carbide inserts, considerable pressures) the more effective parameter will be the slope of the profile. This can be explained by the fact that the peaks become so deformed at small loads that the increase in surface resistance is associated with the slope
175 Standard inspection of roughness
R, l----1
I P I R, I I i-----I
Inspection of roughness of especially important surfaces Implementation
investigations
Routine investigations
F
Fundamental investigations
F
L_p_J P P
(4,)
Ra
(4)
R,
Ap,
R,
n(o)
l------1
tP Aa
L______: r
tp
f&&W
Fig. 1. Choices of parameters and techniques (F, photographs of the surface; P, profilograms) for the inspection and investigation of roughness. The box on the right hand side of the figure indicates that alternative parameters may be used.
of the profile, At this range of values of the approach the slope governs the increase in the real contact area. On the basis of the principles by which the multiparameter characteristics of roughness are to be established, it is possible to determine a hierarchical system for the estimation of surface roughness (Fig. 1). In basic research and advanced technologic~ invest~atio~ it is suggested that parameters from the set (1) should be used; these could eventually be complemented with specific parameters having a direct relation with the most significant useful properties according to surface profile traces and photo~ap~ (taken with a scanning microscope) of the surface. In routine ~vestigations the parameters from the set (2) should be used together with a profile trace and a photograph of the surface. While implementing new products two parameters are advised: R, and a parameter selected from the investigations of the first or the second kind, In production inspection, very high quality surfaces should be tested by checking two parameters, e.g. R, and A,. For stable and well- controlled production process conditions, the second of the parameters need not be inspected frequently. In other cases only one parameter is periodically controlled. The most important is the maintenance of correct production conditions and visual evaluation of the roughness, Both the inspection workers and other qualified and experienced workers can easily notice the essential changes in surface roughness.
The following conclusions can be drawn. There is an urgent need to establish from currently used parameters a small subset of parameters to be recommended for use in research work. The possibility of withdrawing strongly correlated parameters from the standards should be considered. Parameters describing the shape of the peaks should take their place. The application of two parameters in the inspection of surface rougbness should ensure constant high quality of machine parts.
176
References 1 I. Nara, Two dimensional representation of surface roughness, CZZU'Ann., (1) (1961). 2 A. ChusyandJu. Witenberg,Szerochowatost’PowierchnostejTeoretikowerojatnostnyj Podchod, Nauka, Moscow, 1975. 3 Ja. Rudzit, Mikrogeometria i Kontaktnoje WzaimodeistwiePowierchnostiej, ZINATNE, Riga, 1975. 4 D. Whitehouse, Approximate methods of assessment of surface topograph parameters,
Mat. Konf. Metrologii MP, Warsaw, 1976. 5 C. Beck, Statist&he KennegrSsaen der Profilwinkel a technischen Oberfhichen, Feingetitetechnik, (10) (1976). 6 B. Nowicki, Badanie mikrostruktury geometrycxnej powierzchni i metody jej oceny, Zest. Nauk. Politech. Wars%., 70 (1980). 7 J. McCool, Characterization of surface anisotropy, Wear, 49 (1) (1978) 19. 8 R. Spragg and D. Whitehouse, A new unified approach to surface metrology, Proc., Inst. Mech. Eng., London, (47) (1970 - 1971). 9 N. Demkin, Kontaktirowanie Szerochowatych Powierchnostej, Moscow, 1970. 10 ANSI Stand. B46, Part 1,1978 (American National Standards Institute, New York).