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it is the height of the highest point of the profile fromthe mean line; see Figure 8.3. This parameter is often referred to as anextreme-value parameter and as such can be unrepresentative of the surfaceas its numerical value may vary so much from sample to sample. It is possibleto average over several consecutive sampling lengths and this will reduce thevariation, but the value is often still numerically too large to be useful in mostcases. However, this parameter will succeed in finding unusual conditionssuch as a sharp spike or burr on the surface that may be indicative of poormaterial or poor processing.8.2.6.2 Maximum profile valley depth, RvThis is the largest profile valley depth within the sampling length, i.e.
it is thedepth of the lowest point on the profile from the mean line and is an extremevalue parameter with the same disadvantages as the maximum profile peakheight (see Figure 8.4).8.2.6.3 Maximum height of the profile, RzThis is the sum of the height of the largest profile peak height, Rp, and thelargest profile valley depth, Rv, within a sampling length.FIGURE 8.3 Maximum profile peak height, example of roughness profile.Surface profile characterizationFIGURE 8.4 Maximum profile valley depth, example of roughness profile.8.2.6.4 Mean height of the profile elements, RcThis is the mean value of the profile element heights within a samplinglength.
This parameter requires height and spacing discrimination asdescribed earlier. If these values are not specified then the default heightdiscrimination used shall be 10 % of Rz. The default spacing discriminationis 1 % of the sampling length. Both of these conditions must be met. It isextremely rare to see this parameter used in practice and it can be difficult tointerpret. It is described here for completeness and, until it is seen on anengineering drawing, should probably be ignored (it is, however, used in theGerman automotive industry).8.2.6.5 Total height of the surface, RtThis is the sum of the height of the largest profile peak height and the largestprofile valley depth within the evaluation length (see Figure 8.5). Thisparameter is defined over the evaluation length rather than the samplinglength and as such it has no averaging effect.
Therefore, scratches orcontamination on the surface can strongly affect Rt.8.2.7 Amplitude parameters (average of ordinates)8.2.7.1 Arithmetical mean deviation of the assessed profile, RaThe Ra parameter is the arithmetic mean of the absolute ordinate values,z(x), within the sampling length, l,Ra ¼1lðl0jzðxÞjdx:(8.3)219220C H A P T ER 8 : Surface topography characterizationFIGURE 8.5 Height of profile elements, example of roughness profile.Note that equation (8.3) is for a continuous z(x) function. 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.
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