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However, whenmaking surface texture measurements, z(x) will generally be determined overa discrete number of measurement points. In this case equation (8.3) shouldbe written asRa ¼N1XjZi jN i¼1(8.4)where N is the number of measured points in a sampling length. Theequations for the other profile parameters in this section, plus the arealparameters described in section 8.3, that involve an integral notation can beconverted to a summation notation in a similar manner.The derivation of Ra can be illustrated graphically as shown in Figure 8.6.The areas of the graph below the centre line within the sampling length areplaced above the centre line.
The Ra value is the mean height of the resultingprofile.The Ra value over one sampling length is the average roughness; therefore, the effect of a single non-typical peak or valley will have only a slightinfluence on the value. It is good practice to make assessments of Ra overa number of consecutive sampling lengths and to accept the average of thevalues obtained.
This will ensure that Ra is typical of the surface underinspection. It is important that measurements take place perpendicular tothe lay. The Ra value does not provide any information as to the shape of theirregularities on the surface. It is possible to obtain similar Ra values forsurfaces having very different structures (see section 8.3). For historicalreasons Ra is probably the most common of the all the surface textureSurface profile characterizationFIGURE 8.6 The derivation of Ra.parameters and is dominant on most engineering drawings when specifyingsurface texture. This should not deter one from considering other parametersthat may give more information regarding the functionality of a surface.8.2.7.2 The root mean square deviation of the assessedprofile, RqThe Rq parameter is defined as the root mean square value of the ordinatevalues, z(x), within the sampling length,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 l 2Rq ¼z ðxÞdx:(8.5)l 0The Rq parameter is another popular parameter along with Ra.
It iscommon to see it stated that Rq is always 11 % larger than Ra for a givensurface. However, this is only true of a sinusoidal surface, although Rq willalways be larger than Ra. The reason for the commonality of Ra and Rq ischiefly historical. Ra is easier to determine graphically from a recording of theprofile and was, therefore, adopted initially before automatic surface texturemeasuring instruments became generally available.
The Rq parameter is usedin optical applications where it is more directly related to the optical quality ofa surface. Also, Rq is directly related to the total spectral content of a surface.221222C H A P T ER 8 : Surface topography characterization8.2.7.3 Skewness of the assessed profile, RskSkewness is a measurement of the symmetry of the surface deviations aboutthe mean reference line and is the ratio of the mean cube value of the heightvalues and the cube of Rq within a sampling area,2Rsk ¼1 41Rq3 lðl03z3 x dx5:(8.6)The Rsk parameter describes the shape of the topography height distribution. For a surface with a random (or Gaussian) height distribution thathas symmetrical topography, the skewness is zero. The skewness is derivedfrom the amplitude distribution curve; it is the measure of the profilesymmetry about the mean line.
This parameter cannot distinguish whetherthe profile spikes are evenly distributed above or below the mean plane and isstrongly influenced by isolated peaks or isolated valleys. This parameterrepresents the degree of bias, either in the upward or downward direction, ofan amplitude distribution curve. A symmetrical profile gives an amplitudedistribution curve that is symmetrical about the centre line and an unsymmetrical profile results in a skewed curve.
The direction of the skew isdependent on whether the bulk of the material is above the mean line(negative skew) or below the mean line (positive skew). Figure 8.7 showsthree profiles with positive, zero and negative skewness. Use of thisparameter can distinguish between two surfaces having the same Ra value.FIGURE 8.7 Profiles with positive (top), zero (middle) and negative (bottom) values ofRsk (reprinted from ASME B46.1-1995, by permission of the American Society ofMechanical Engineers.
All rights reserved).Surface profile characterizationAs an example, a porous, sintered or cast iron surface will have a largevalue of skewness. A characteristic of a good bearing surface is that it shouldhave a negative skew, indicating the presence of comparatively few peaks thatcould wear away quickly and relatively deep valleys to retain lubricant traces.A surface with a positive skew is likely to have poor lubricant retentionbecause of the lack of deep valleys in which to retain lubricant traces.Surfaces with a positive skewness, such as turned surfaces, have high spikesthat protrude above the mean line. The Rsk parameter correlates well withload-carrying ability and porosity.8.2.7.4 Kurtosis of the assessed profile, RkuThe Rku parameter is a measure of the sharpness of the surface heightdistribution and is the ratio of the mean of the fourth power of the heightvalues and the fourth power of Rq within the sampling area,23ð1 41 l 4 5Rku ¼z x dx :Rq4 l 0(8.7)The Rku parameter characterizes the spread of the height distribution.
Asurface with a Gaussian height distribution has a kurtosis value of three.Unlike Rsk this parameter can not only detect whether the profile spikes areevenly distributed but also provides a measure of the spikiness of the area. Aspiky surface will have a high kurtosis value and a bumpy surface will havea low kurtosis value. Figure 8.8 shows two profiles with low and high valuesFIGURE 8.8 Profiles with low (top) and high (bottom) values of Rku (reprinted fromASME B46.1-1995, by permission of the American Society of Mechanical Engineers. Allrights reserved).223224C H A P T ER 8 : Surface topography characterizationof Rku. This is a useful parameter in predicting component performance withrespect to wear and lubrication retention.
Note that kurtosis cannot differentiate between a peak and a valley.8.2.8 Spacing parameters8.2.8.1 Mean width of the profile elements, RSmThe RSm parameter is the mean value of the profile element widths withina sampling length (see Figure 8.9). In other words, this parameter is theaverage value of the length of the mean line section containing a profile peakand adjacent valley.
This parameter requires height and spacing discrimination. If these values are not specified then the default height discrimination used is 10 % of Rz. The default spacing discrimination is 1 % of thesampling length and both of these conditions must be met.8.2.9 Curves and related parametersThe profile parameters described so far have resulted in a single number(often with a unit) that describes some aspect of the surface. Curves andrelated parameters give much more information about the surface fromwhich, often, functional information can be gained [7]. All curves and relatedparameters are defined over the evaluation length rather than the samplinglength.8.2.9.1 Material ratio of the profileThe material ratio of the profile is the ratio of the bearing length to theevaluation length.
It is represented as a percentage. The bearing length is theFIGURE 8.9 Width of profile elements.Surface profile characterizationsum of the section lengths obtained by cutting the profile with a line (slicelevel) drawn parallel to the mean line at a given level. The ratio is assumed tobe 0 % if the slice level is at the highest peak, and 100 % if it is at the deepestvalley. Parameter Rmr(c) determines the percentage of each bearing lengthratio of a single slice level or nineteen slice levels that are drawn at equalintervals within Rt respectively.8.2.9.2 Material ratio curveThe material ratio curve (formally known as the Abbot-Firestone or bearingratio curve) is the curve representing the material ratio of the profile asa function of level.
By plotting the bearing ratio at a range of depths in theprofile, the way in which the bearing ratio varies with depth can be easily seenand provides a means of distinguishing different shapes present on theprofile. The definition of the bearing area fraction is the sum of the lengths ofindividual plateaux at a particular height, normalized by the total assessmentlength, and is the parameter designated Rmr (see Figure 8.10).
Values of Rmrare sometimes specified on drawings; however, such specifications can leadto large ambiguities if the bearing area curve is referred to the highest andlowest points on the profile.Many mating surfaces requiring tribological functions are usuallyproduced with a sequence of machining operations. Usually the first operation establishes the general shape of the surface with a relatively coarsefinish, and further operations refine this finish to produce the propertiesrequired by the design. This sequence of operations will remove the peaks ofthe original process but the deep valleys will be left untouched. This processleads to a type of surface texture that is referred to as a stratified surface. Theheight distributions will be negatively skewed, therefore making it difficultfor a single average parameter such as Ra to represent the surface effectivelyfor specification and quality-control purposes.
A honed surface is a goodexample of a stratified surface.FIGURE 8.10 Material ratio curve.225226C H A P T ER 8 : Surface topography characterization8.2.9.3 Profile section height difference, RdcThe profile section height difference is the vertical distance between twosection levels of given material ratio.8.2.9.4 Relative material ratio, RmrThe relative material ratio is the material ratio determined at a profile sectionlevel Rdc, and related to a reference, C0, where C1 ¼ C0 Rdc and C0 ¼C(Rmr0). Rmr refers to the bearing ratio at a specified height (seeFigure 8.11).
A way of specifying the height is to move over a certainpercentage (the reference percentage) on the bearing ratio curve and then tomove down a certain depth (the slice depth). The bearing ratio at the resultingpoint is Rmr. The purpose of the reference percentage is to eliminate spuriouspeaks from consideration – these peaks tend to wear off in early part use. Theslice depth then corresponds to an allowable roughness or to a reasonableamount of wear.8.2.9.5 Profile height amplitude curveThe profile height amplitude curve is defined as the sample probabilitydensity function of the ordinate, z(x), within the evaluation length. Theamplitude distribution curve is a probability function that gives the probability that a profile of the surface has a certain height, at a certain position.The curve has the characteristic bell shape like many probability distributions (see Figure 8.12).
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