Testing apparatus

A method of deriving the value of any one of the commonly used parameters for characterizing surfaces in surface measurement technology includes the steps of computing the autocorrelation function of a signal representing the surface, and selectively utilizing values of this autocorrelation function together with the value of the sampling interval used in generating the autocorrelation function, to derive the desired parameter values. This method considerably simplifies the information on a surface which is required, and enables micro-processor techniques to be used in the computation of the parameters.

The present invention relates to signal processing methods and apparatus 
for deriving the value of a parameter characterising a variable quantity 
by utilising autocorrelation function values of an input signal 
representing the variable quantity. In particular, but not exclusively, 
the invention relates to such methods and apparatus as applied to the 
evaluation of parameters characterising the surface topology of a 
component under test. 
Throughout the specification particular reference will be made to the 
evaluation of the surface of a component or workpiece. Such surface 
evaluation is conventionally undertaken by means of a surface testing 
instrument employing a contact stylus or pick-up drawn over a path along 
the surface under test to produce an electrical signal which, when 
amplified, represents the variations in height of the surface along the 
path. Optical transducers have also been used employing interferrometric 
principles. 
The electrical signal representing the surface contains all the information 
of interest to, for example, a manufacturer wishing to determine the 
results of a process on the surface, such as turning or grinding, but this 
information is not always readily apparent due to the complexity of the 
signal and the high degree of amplification which is necessary for display 
and to obtain the required sensitivity. Moreover, it is often required to 
be able to predict how the surface will behave in tribological situations, 
that is situations involving moving contact of the surface with another 
surface, for example to establish whether or not a suitable fit between 
two components has been acheived by the machining processes. For these, 
and other, purposes various parameters have been utilised for 
characterising the nature of the surface. Such parameters, include, for 
example, the centre line average, the high spot count, the peak density, 
mean peak height, mean curvature of peaks and mean slope of the signal. 
Such parameters are widely discussed in the literature of the art, and are 
well known to the man skilled in the art. For this reason they will not be 
defined in greater detail herein in their usual form. To produce an output 
signal representing any of the known parameters of the signal representing 
the surface (or, more generally, any variable quantity) it has generally 
been the practice to sample the electrical signal representing the surface 
and to perform operations on these samples in order to generate the 
required value. This sampling is carried out for each parameter it is 
desired to evaluate. Now, many of the known parameters are not independent 
from one another so that the process of deriving the values of the various 
parameters required for various differing purposes, has been unduly long 
and involved. 
It is therefore an object of the present invention to provide a method of 
analysing a signal representing a variable quantity, such as the height 
variation of a surface under test, which enables the derivation of various 
characterising parameters from a minimum of data extracted from said 
signal. 
It can be shown that the independent properties of a variable quantity such 
as the height of a surface are the height and the frequency of the 
asperities. In fact, for many tribological purposes, a random surface 
(such as one produced by grinding) can be described as a super position of 
a plurality of asperities having differing scales of size. Moreover, it 
can be shown that the contribution each scale of size may make to any one 
tribological parameter can be evaluated using sampled data theory. 
Additionally, it can be shown that information on the two independent 
properties of a variable quantity such as the height of a surface can be 
derived from an electrical signal representing the profile of the surface 
using only two functions, that is the amplitude probability density 
function (giving amplitude information) and the normalised autocorrelation 
function (giving spacing information). These two functions are independent 
and by using measures of these two independent functions any one of the 
multiplicity of parameters previously independently measured and computed 
separately can be derived. 
The amplitude probability density function of a surface gives information 
on the rate of change of the ratio of the material of the surface to air 
with height from the lowest valley to the highest peak of the surface. 
Mathematically it is possible to find the moments of the curve of the 
amplitude probability density function in much the same way as the centre 
of gravity of an object and its centre of gyration are found in mechanical 
engineering. These moments of the curve are termed, respectively, the 
variance, the skew, and the kurtosis going in ascending order. The square 
root of the variance is in fact the root mean square value of the surface 
and so is an estimate of the size, but the other two mentioned (and others 
of higher order) can be used to derive information on the shape. 
The autocorrelation function is useful in deriving the statistical 
properties pertaining to spatial features of a surface. As will be known 
to those skilled in the art the autocorrelation function is derived by 
obtaining a correlation function from a waveform with itself successively 
displaced by predetermined amounts called the sampling interval. 
Autocorrelation functions can probably best be obtained using digital 
techniques and have been employed in signal analysis methods used for a 
variety of applications (see, for example, U.S. Pat. No. 3,819,920 
relating to doppler radar signal analysis, and U.S. Pat. No. 3,331,955). 
Whereas, for a random surface, the autocorrelation function falls from an 
initial value to zero in a distance known as the independence length as 
will be discussed in greater detail below, the presence of periodic 
components in a waveform will cause the autocorrelation function itself to 
be periodic, repeating with the same wavelength as the periodic component 
in the original wave. The autocorrelation function is therefore very 
useful in detecting periodicities in the presence of random elements 
since, if the autocorrelation function does not decay away entirely, its 
value after one or two cycles is the periodic component of the profile; 
this is so, due to the fact that all random elements must disappear within 
the independence length. This is useful for detecting, for example, 
grinding chatter in a manufactured surface since this introduces an 
element of waviness into the profile which is so small that it cannot be 
seen. 
One of the major problems in tribological situations is that of specifying 
the surface properties of mating surfaces. To control manufacture the 
surface parameter used need refer to only the surface in question whereas 
to predict tribological behaviour is must refer to both of two mating 
surfaces and it is the properties of the gap between the surfaces which 
are most likely to be significant. Unfortunately it is not usually 
possible merely to combine the surface parameters of the two mating 
surfaces (either by addition or subtraction) with one another in order to 
derive the properties of the gap. For example, the values of the parameter 
R.sub.a cannot be added to one another since the results would not be 
meaningful. Likewise, some of the more modern random process parameters 
such as the standard deviation of peak curvature would produce meaningless 
results if combined together. 
It is accordingly a further object of the present invention to provide some 
means of introducing into such processes a quantity akin to dimensional 
tolerances which can be summed from one component to another in order to 
assess whether functional requirements of the surface, such as state, will 
be satisfied. 
In accordance with one aspect of the present invention, there is provided a 
method of processing an input signal representing a variable quantity to 
derive the value of a selected parameter characterising that quantity, 
said method including the step of combining together, in predetermined 
manner, a data signal representing the value of a first origin-remote 
point of the autocorrelation function of the input signal with at least 
one further data signal selected from a signal representing the value of a 
second origin-remote point of said function, a signal indicative of the 
mean square value of the variable quantity, and a signal representing a 
sampling interval relating the said origin-remote point or points to the 
origin of the autocorrelation. 
By selectively combining two or more of the above four data signals 
derivable from the input signal it is possible to determine various of the 
known parameters used in characterising, for example, surface profiles. 
The precise manner in which these signals are combined will be given below 
for a number of such parameters. It will be noted that, as indicated 
above, the signals used provide a measure both of amplitude information 
(the mean square value signal which as previously mentioned corresponds to 
variance), and spacing information (the autocorrelation value signals). 
Preferably the two selected origin-remote points on the autocorrelation of 
the signal are spaced respectively, at the sampling interval and at twice 
the sampling interval from the origin. Likewise, the mean square value of 
the variable quantity is preferably obtained by deriving the origin value 
of the input signal. 
In order to give the results general applicability it is preferred that the 
signals representing the values of the origin-remote autocorrelation 
points are normalised. This normalisation can be effected by dividing the 
signals representing the values of the origin-remote autocorrelation 
points by the signal representing the mean square value of the variable 
quantity (the variance). As previously noted, this variance signal is 
preferably a signal representing the origin value of the autocorrelation. 
In its application to the evaluation of the surface of a component, the 
present invention preferably includes the steps of producing an input 
electrical signal representative of the variable quantity constituted by 
the height of the surface of said component along a path thereacross, 
processing said input signal to derive data signals representative of the 
value of the autocorrelation of said input signal at three selected points 
respectively residing at the origin of the autocorrelation, at a distance 
spaced from said origin by a predetermined sampling interval, and at a 
distance spaced from the said origin by twice said sampling interval, the 
origin-value data signal being indicative of the mean square value of said 
variable quantity, and selectively utilising the three 
autocorrelation-value data signals and a further data signal 
representative of the value of said sampling interval to derive an output 
signal representative of the value of a selected one of a plurality of 
surface topology parameters. 
Preferably, the electrical input signal is derived from a traverse of a 
transducer sensitive to surface height variations along a path on the 
surface. Such a transducer may be a stylus pick-up instrument drawn in 
contact with the surface of the component under test along the said path, 
or may be a non-contact type of instrument such as an optical transducer. 
Preferably, as a first stage in utilising the autocorrelation-value data 
signals, two normalised data signals are derived by dividing the values of 
the two origin-remote autocorrelation points by the origin value of the 
autocorrelation in order to normalise the origin-remote values. 
Given below are a number of relationships for deriving tribological 
parameters by selectively combining the data signals derived from the 
input signal representing the height variation of the surface under test. 
It will be appreciated that in fact these relationships can be used to 
process and analyse input signals representing variable quantities other 
than surface height. 
One of the parameters of interest in tribological assessments of surfaces 
is the high spot count, and in accordance with the principles of the 
present invention a signal representing the value of this parameter can be 
generated by combining the signal representing the sampling interval with 
the signal representing the normalised value of the autocorrelation at the 
sampling interval from the origin in the relationship (1/.pi.h) cos.sup.-1 
.rho..sub.1. where: .rho..sub.1 is the normalised value of the 
autocorrelation spaced at the sampling interval from the origin and, h is 
the sampling interval, that is the spacing of the digital readings taken 
in determining the autocorrelation of the input signal representing 
surface height variation. 
The density of peaks counted on a profile when digital techniques are used 
is also an important parameter, particularly in the steel industry where 
the peak count is one of the central parameters of interest. It can be 
shown, using the principles of the present invention that the peak density 
can be derived by combining signals representing the normalised values of 
the two origin-remote autocorrelation points with a signal representing 
the sampling interval in the relationship: 
##EQU1## 
where: .rho..sub.1 is the normalised value of the autocorrelation at a 
spacing equal to the sampling interval from the origin, 
.rho..sub.2 is the normalised value of the autocorrelation at a spacing 
equal to twice the sampling interval from the origin, 
h is the sampling interval. 
Of particular interest in tribology is the way in which the density of 
peaks changes from surface to surface, the way in which the mean peak 
height depends on the correlations between ordinates (that is the points 
at which samples of the value of the input signal representing surface 
height variation are taken) and the relationship between peak height and 
the curvature of the peak height. The mean peak height can be obtained, 
according to the principles of the present invention, from the probability 
density function using, results for the moments of a truncated trivariate 
normal distribution known in the art. For this, a peak is defined, using 
three ordinates, when the central ordinate is higher than the other two, 
and it can be shown from this that the parameter known as the "mean peak 
height" can be generated from a signal representing the normalised value 
of the autocorrelation at a distance equivalent to the sampling interval 
from the origin by combining this with a signal representing the square 
root of the mean square value of the surface height variation and with a 
signal representing the peak density as defined above, by the relation: 
##EQU2## 
where: .rho..sub.1 is the normalised value of the autocorrelation at a 
spacing equal to the sampling interval from the origin, 
.sigma. is the square root of the mean square value of the surface height 
variation, and 
D is the peak density as defined above. As mentioned above not only the 
mean peak height and peak density are of interest in tribology, but also 
the curvature of the peaks, especially when contact properties are being 
considered. Whether or not surface peaks deform elastically or plastically 
is determined by the plasticity index .psi. which is given by: 
##EQU3## 
where: E is the elastic modulus of the surface, 
H is the hardness, 
.sigma. is the root mean square (standard deviation) of the peak height 
distribution, and 
R is the radius of curvature of the peaks. 
Obviously, in all wear situations, R is a critical factor. From the 
probability density function of the distribution of the curvature of the 
peaks, which is obtained from the convolution of the joint probability 
density function of a bivariate normal distribution when both variables 
are truncated below at zero, it can be shown that the mean peak curvature 
can be derived from the signals representing the normalised values of the 
two origin-remote autocorrelation points by combining these with a signal 
representing the root mean square value of the surface height variation, a 
signal representing the peak density as defined above, and the signal 
representing the sampling interval in the relation: 
##EQU4## 
where: .rho..sub.1 is the normalised value of the autocorrelation at a 
spacing from the origin equal to the sampling interval, 
.rho..sub.2 is the normalised value of the autocorrelation at a spacing 
from the origin equal to twice the sampling interval, 
.sigma. is the root mean square value of the surface height variation, 
D is the peak density, and 
h is the sampling interval. 
The variance of the peak curvature can also be determined, and from a 
knowledge of this and of the mean peak curvature, the proportion of peaks 
on the surface which will elastically or plastically deform may be 
estimated. This is an important tribological feature which has so far been 
neglected. 
For verification of these results the special case when .rho..sub.1 
=.rho..sub.1.sup.2 may be considered (that is the case of an exponential 
autocorrelation function) in which case the expressions reduce to results 
previously obtained and verified and known in the art. 
It is possible to obtain similar results for the height and curvature of 
valleys, which can also be important in frictional studies involving the 
passage of fibres over rollers, or similar applications. 
If the parameter of interest is the mean slope of the surface height 
variation, this can be derived from a signal representing the normalised 
value of the autocorrelation at twice the sampling interval from the 
origin by combining this signal with a signal representing the root mean 
square value of the surface height variation and a signal representing the 
sampling interval in the relation: 
##EQU5## 
where: .rho..sub.2 is the normalised value of the autocorrelation at a 
spacing from the origin equal to twice the sampling interval, 
.sigma. is the root mean square value of the surface height variation, and 
h is the sampling interval. 
Many other parameters can be determined by selectively using only the 
values of the two origin-remote points on the autocorrelation, the 
sampling interval and the root mean square of the surface height 
variation. For example the root mean square curvature of the surface 
height variation can be determined by the relation: 
##EQU6## 
where: .rho..sub.1 is the normalised value of the autocorrelation at a 
spacing corresponding to the sampling interval from the origin, 
.rho..sub.2 is the normalised value of the autocorrelation at a spacing 
corresponding to twice the sampling interval from the origin, 
.sigma. is the root mean square value of the surface height variation, and 
h is the sampling interval. 
Likewise the correlation between curvature and peak height can be derived 
from the relation: 
##EQU7## 
where .rho..sub.1 and .rho..sub.2 have the significance previously 
assigned to them. The standard deviation of the peak distribution is given 
by the relation: 
##EQU8## 
where K is a scaling factor dependent on the value of the root mean square 
of the surface height variation. If .sigma.=1 then K=1. 
The RMS value of peak curvature is given by the relation: 
##EQU9## 
where: D is the peak density as defined above and .rho..sub.1, 
.rho..sub.2, and h have the previously assigned significance. 
Other parameters, such as the correlation between a peak of given curvature 
at given height can also be determined. This latter is given by the 
relation: 
##EQU10## 
The advantage of using signals representing the normalised value of the 
autocorrelation function at points related to the sampling interval at 
which samples are taken in computing the autocorrelation function are that 
in determining the properties of a gap between two mating surfaces the 
values of the autocorrelation function can simply be summed to derive 
equivalent autocorrelation values of the gap between the two surfaces. The 
resultant signals can then be selectively combined with a signal 
representing the root mean square value of the gap between the two 
surfaces and/or a signal representing the sampling interval used in 
determining the autocorrelations to provide the value of a selected 
tribological parameter of the gap between the surfaces. 
In accordance with a further aspect of the present invention, there is 
provided apparatus for processing an input signal representing a variable 
quantity to derive the value of a selected parameter, characterising that 
quantity, said apparatus comprising: 
a correlator arranged to receive said input signal and to generate three 
data signals respectively representative of the value of the 
autocorrelation of the input signal at the origin of the autocorrelation, 
at a distance spaced from said origin by a predetermined sampling 
interval, and at a distance spaced from said origin by twice said sampling 
interval, the correlator being further arranged to generate a fourth data 
signal representative of said sampling interval, and signal combining 
means connected to receive said four data signals and arranged to 
selectively combine them to derive an output signal representive of the 
value of a selected one of a plurality of parameters.

Referring first to FIG. 1 there is shown a waveform A representing the 
profile of a surface, such as the signal generated by a surface testing 
instrument. In order to determine the autocorrelation function for this 
waveform it is sampled at a plurality of points to measure the value of 
the signal in terms of its displacement from the mean line in order to 
derive values thereof at a predetermined spacing interval h, such values 
being indicated y.sub.1, y.sub.2, y.sub.3 . . . y.sub.n. The same profile 
signal is then superimposed on the original as shown in curve B and the 
same process repeated. Each sampled value (y.sub.i) of the profile signal 
is termed on ordinate. The ordinates of the two signals A and B are then 
each multiplied by the corresponding ordinate directly aligned with it, 
and all of these products are then summed. Since the two signals are 
exactly in phase all the products will be positive. The products are then 
divided by the total number of products and the resultant N.sub.1 is 
plotted as the ordinate at the origin on a new graph F. The profile signal 
B is then displaced to the right by a distance equal to the sampling 
interval and the newly aligned ordinates are multiplied as before. This 
time each ordinate of signal B is multiplied with that ordinate which 
would have been spaced one position to the left of it in signal A in the 
original position. Again, the products are summed and divided by the total 
number of products. This time some of the products will be negative since, 
in the region of the intersection with the mean line, some positive 
ordinates will be multiplied with some negative ordinates. The value (say 
N.sub.2) of the resultant obtained by dividing the summed products is 
plotted on curve F as the second ordinate shifted to the right of the 
origin by the sampling interval. Because some of the products were 
negative the value of N.sub.2 will be less than the value of N.sub.1. This 
process is repeated each time displacing the signal B one sampling 
interval to the right. 
This process is repeated continuously, each time shifting the lower profile 
further to the right and plotting the value of N in a corresponding 
position on the graph F. If the profile is random, as in the example 
shown, the value of N will fall eventually to a low value or zero, and 
will not rise again to any significant extent irrespective of how far to 
the right the lower profile is shifted and how long the computing process 
is continued. This is because, eventually, as the displacement gets 
larger, any one feature on the lower profile bears no relationship to the 
feature directly above it on the upper profile. Hence there is an equal 
probability of the products of ordinates being positive or negative, that 
is the product sign can be equally probably obtained by the product of two 
positive values, a negative and a positive value, a positive and a 
negative value, or a negative and a negative value and this being the case 
the resultant sum is obviously zero. The distance through which the line B 
has to be displaced in order to achieve this condition on the graph of N 
is called the independence length of the profile. Normalisation of the 
graph F is achieved by dividing the value of each ordinate by the value of 
the first ordinate so that the curve decays from a value of one unit to 
zero. This is illustrated in curve G which then represents the 
autocorrelation function of the curve represented in line A. It should be 
noted that the physical dimension of the abscissa of the autocorrelation 
function is one of length, the units being the same as those of the 
horizontal axis of the profile graph. This means that the curve of the 
autocorrelation function can be compared directly, in the length 
direction, with the profile itself. This is an important idea in the 
development of the present invention. If a similar computation is carried 
out on a periodic wave, as shown in FIG. 2, the normalised autocorrelation 
function g will also be periodic because the condition for a maximum value 
of unity arises every time the displacement of the signal B is an integral 
number of wavelengths of the periodicity. The autocorrelation function 
thus repeats with the same wavelength as the original wave; it never 
decays away as the displacement is increased unlike the autocorrelation 
function of a random profile, which does. It will be appreciated, 
therefore, how the autocorrelation function can be used to detect 
periodicities in the presence of random elements. If the autocorrelation 
function does not decay to zero, the periodic component thereof represents 
the periodic component of the profile. All random elements must disappear 
within the independence length so that even if, for example, grinding 
chatter is introducing an element of waviness into a profile which is too 
small to be detected by other techniques, it can be sensed using the 
autocorrelation function and appropriate remedial action on the machine 
taken. 
As can be seen with reference to FIG. 3 the two values of the 
autocorrelation function of interest in the present invention, that is the 
value of the autocorrelation function at a distance equal to the sampling 
interval and to twice the sampling interval from the origin, can be 
determined for two surfaces, and the corresponding values of the 
autocorrelation function of the combined signals, which represent the gap 
between the two surfaces, can be derived simply by summing the 
corresponding values of the two autocorrelation functions to derive what 
is, in effect, an autocorrelation function of the gap. From this the 
parameters of interest can be derived directly using the expressions given 
above so that the behaviour of the two components in tribological 
situations can be assessed. 
FIG. 4 illustrates a block diagram showing how the parameters discussed 
above could be computed. The legends to the boxes in the diagram 
illustrate the processes performed therein. Dedicated apparatus formed in 
accordance with this block diagram would be difficult to produce, however, 
and it is considered more suitable for the techniques to be undertaken 
using a microprocessor programmed to perform appropriate routines to 
generate the parameters indicated, as shown schematically in FIG. 5. This 
figure shows a pick-up 50 of the stylus type the analogue output signal of 
which is fed to an analogue-to-digital converter 51. The digitised signals 
are then fed to a correlator and store 52 which has three outputs on which 
are fed out signals representing, respectively, the said two values of the 
autocorrelation function computed from the incoming digital data, and 
(also computed from the incoming digital data) the root mean square value 
of the profile height, which latter constitutes the variable quantity. 
These three output signals are then fed to a processor 53 which computes 
from the incoming signals the value of the required parameter from a 
programme selected by an operator from among a number of such programmes 
previously written into the processor for computing, for example, one of 
the relations set out above, or some other relation for another parameter 
calculated in the same way as those outlined herein to involve only the 
three signals given. The functions of the analogue-to-digital converter 
51, the correlator and store 52, and the processor 53 could all be 
performed by one microprocessor such as, for example, a Motorola 6800. The 
external equipment of the microprocessor could include a set of push 
buttons for selecting, in the simplest possible manner, the programme to 
be performed by the processor 53 to generate the value of a selected 
parameter.