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A self-balancing pneumatic wheatstone bridge for surface roughness measurement
Wear, 83 (1982)
37
37 - 47
A SELF-BALANCING PNEUMATIC WHEATSTONE SURFACE ROUGHNESS MEASUREMENT*
BRIDGE
FOR
L. H. TANNER Mechanical and Production BN2 4GJ (Gt. Britain)
Engineering
Department,
Brighton
Polytechnic,
Brighton
(Received May 27,1982)
Summary
The pneumatic conductance of the gap between a rough surface and a detector placed on it is roughly proportional to the 2.5 power of the roughness. To measure a roughness range of 4O:l it is therefore necessary to measure fluid resistances over a range of 104:1. This is made possible by the use of a pneumatic analogue of the Wheatstone bridge. The most important elements of the bridge are the sensor, a variable fluid resistor which can be calibrated and an extremely sensitive detector of pressure difference. If the detector gives an electrical output which is used to control the variable resistor, the bridge can be made self-balancing and will give an electrical readout of the roughness value. The design of a roughness meter of this type, its performance and potential for further development, and the dependence of the pneumatic roughness on the surface profile are discussed.
1. Introduction
The concept of using a pneumatic analogue of the Wheatstone bridge (Fig. 1) for roughness measurement has been implemented in three stages. In the first instruments [ 1, 21 the pressure source was a hand-operated bellows. The fixed resistors Ri and Rz were glass capillary tubes packed with stranded nylon thread to give a resistance of 2.6 X 10” Pa s mV3. The sensor (Fig. 2) was placed on the rough surface so that R4 was the resistance to air flow out from this sensor via the roughness elements. The variable resistor was formed by a lens rocked [1] or traversed [2] over a flat plate with a small hole. The gap between lens and hole is proportional to the square of the traverse distance from the datum position at which the lens is exactly *Paper presented at the Second International Conference on Metrology and Properties of Engineering Surfaces, Leicester Polytechnic, Leicester, Gt. Britain, April 14 - 16, 1982. 0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed
in The Netherlands
38
Fig. 1. Equivalent circuit of a pneumatic bridge: RI, Rz, Ri, fixed resistors; RJ, wirecontrolled resistor; W, S, water and solution sides of the detector; F, feedback circuit controlling R,; V, two-way valve. Fig. 2. Sensor and detail of rim shape.
over the hole. The conductance is roughly proportional to the 2.5 power of the gap and hence to the fifth power of the traverse distance. The traversing lens resistor has proved extremely stable and reliable and has been of great value throughout this research. The null detector was a small Perspex U-tube containing water with small particles in suspension. The channel joining the limbs was of small cross-sectional area, and a microscope was used to view particle motion in the channel. This gave an extremely sensitive way of detecting the pressure balance point when RJR3 = R2/R I. The micrometer reading was found [ 21 to vary linearly with RA1’3 where RA is the centre-line average roughness of the ground surfaces which were used for testing. The object of the second stage of development was to remove the need to observe particles through a microscope while manually balancing the bridge. If the detector can be made to give an electrical output and if this can then be used to control the fluid resistance R, then we have a feedback loop, the bridge becomes self-balancing and the electrical output can be used for a digital readout. The result [3] showed that this objective could be achieved using a fluid resistor controlled by the expansion of a heated wire. At this stage the system consisted of an array of separate elements (the power supply, the amplifier, the voltmeter, the wire-controlled resistor and the detector), and the RA value of the surface was roughly proportional to the square of the indicated voltage. The third stage has therefore been to incorporate all the elements in one unit and to add an electrical exponentiator and variable amplifier so as to give a readout indicating RA directly in microns. The design and performance of this instrument are described in the present paper, and the effect of surface profile on the relation between the pneumatic roughness RP and the RA value measured by profilometry is also discussed.
39
2. Elements
of the bridge
2.1. The pressure supply Air at about 5000 Pa gauge pressure obtained via a reducing valve from a compressed air supply is conveyed by a small-bore flexible tube to the point of entry to Ri and Rz. Here the pressure is reduced and controlled at 1400 Pa by a small weight on a hole 6 mm in diameter. This can be removed to give zero pressure when the instrument is not in use or for levelling the detector. 2.2. The fixed resistors R 1, R2 and Ri and valve V The fixed resistors are, as in the original instrument [ 11, glass capillary tubes packed with nylon thread. The size has been reduced to 0.5 mm bore and length 15 mm. The miniature two-way valve V is connected to the reference resistor R4) until the sensor is in position on the surface to be measured. 2.3. The electrically variable fluid resistor R3 This is the wire-controlled resistor described earlier [3] and shown in Fig. 3. Current through the wire causes it to expand, increasing the gap between nozzle and lever. The wire diameter is 75 pm and the Grashof number is lOA or less. In this range heat transfer is by conduction rather than by buoyant convection so the rise in wire temperature should be exactly proportional to the power dissipated. The wire resistance is expected to rise with temperature so the relation between the wire voltage V and the gap width h is expected to be of the form V2 = &(l + oh) where cy and fl are constants, and experiment confirms this [ 31. The moving magnet (Fig. 3) provides fine control for the wire tension and hence a zero setting for the gap width. Drift of zero can arise from changes in room temperature giving differential expansion, but this is readily corrected. Changes in sensitivity can arise from changes in air humidity which changes the air conductivity. To avoid this a very small flow of air from the supply is bled into the wire chamber to ensure that the air in the chamber has the same humidity as that flowing through R,.
NOZZLE
I
WIRE
1OOmm
I
Fig. 3. Wire-controlled resistor. The motion of the pallets for wire adjustment is shown by the arrows.
40
2.4. The detector This is as described earlier [ 3] but is smaller and is improved in detail. The chambers of the Perspex U-tube are 20 mm in diameter and 20 mm deep; one contains distilled water and the other contains dilute KC1 solution. Electrodes are placed at each end of the connecting channel, the small-bore part of which is 0.43 mm in diameter and 5 mm long. One of the electrodes is a dressmaker’s pin 0.6 mm in diameter which acts as a needle valve to permit adjustment of the fluid resistance of the channel. Excess pressure on one side or the other of the U-tube increases or decreases the concentration of solution in the channel, thus changing the electrical conductivity. A voltage of 1.0 V at 4 kHz is applied across the channel in series with a 56 ka resistor. The fraction developed across this resistor goes to a rectifying detector and amplifier giving a d.c. voltage output. For input amplitudes II less than u1 the d.c. output V is zero. As u rises from u1 to ua, V rises linearly from zero to 1.0 V, at which it remains if u rises further, thus preventing over-heating of the wire. To prevent oscillation and to reduce response time a differentiator and adder are included in the circuit to give proportional plus derivative feedback. A simplified analysis [3] has been given to show the effect of this on the system. Oscillation can still occur, particularly in the sensitive condition when the bridge arms are of equal resistance. Optimum conditions are obtained by setting the system into oscillation at this condition and then increasing the channel resistance by the needle valve until the oscillation is lightly damped. 2.5. The sensor The sensor (Fig. 2) is as described earlier [3]. The annulus is formed by lapping against two mating spherical surfaces, one convex and one concave, with a radius of curvature of 110 mm, to produce two approximately conical surfaces of slope l/25. A final brief lap on a flat surface gives the flat portion a width of 75 pm, ensuring perfect flatness of the rim and making it more durable. 2.6. Overall performance and the modified display The simplest analysis of the system would suggest that the resistances R, and R4 would be equal, and balance would be obtained, if the gap above the nozzle in R, were propo~ion~ to the gap between the sensor and the surface in R4 and hence to the surface roughness. Since the gap in Rs is roughly proportional to V2, we would expect V2 to be proportional to the surface roughness to be measured. The real situation is more complex in two respects. One is the nonlinearity arising from the change in wire resistance with temperature (Section 2.3). The other is that the different flow configurations of R, and R4 are likely to give differences in the resistance uersus gap relations and hence to give deviations from direct proportionality of one gap to the other for the condition of balance.
41
The matter was investigated experimentally [3] and it appeared that the two complicating factors roughly compensated one another and that V2 was at least approximately proportional to the surface roughness RA. At that time the voltage V was displayed, but it was suggested that display of V2 would be desirable. It is, however, almost as easy (by substituting a variable resistor for a fixed resistor) to display an adjustable power V” and, by adding a variable amplifier, to display FZV” where both 12 and n are adjustable. The extra degree of freedom makes it possible to adjust the calibration to ensure equality of pneumatic roughness R, and the profilometric RA at the maximum RA and at an intermediate value as well as at zero. This has now been done, and the display shows directly the surface roughness in microns. Adjustment of the index n effectively bends the calibration curve at the lower end. If any adjustment at the upper end is needed, advantage can be taken of the fact that at this end the resistances of the tubing connecting the detector to R, and R4 become comparable with Rs and R+ Ideally these series resistances should be equal, but deliberate inequalities can be used to bend the curves if required.
3. Effects of surface profile on the relation of pneumatic to profilometric roughness 3.1. Variation in resistance with sensor gap Consider a sensor held so that the minimum gap between it and a smooth flat surface is h. The way in which the fluid conductance varies with h depends on the form of the sensor. For a flat annulus (Fig. 4(a)) with negligible edge effects the conductance is proportional to h3. For the doublecone form (Fig. 4(d)) with a sharp edge it is proportional to h2. These can be regarded as extremes between which lie more practical forms, including the toroid (Fig. 4(b)) which has conductance proportional to h2.5 and the flattened double cone (Fig. 4(c)). This has conductance proportional to h3(1 + h/cd)-’ which approximates to the h2.5 law if h/d is not too far from unity. It therefore seems appropriate to assume an h2e5law in order to examine the effects of surface profile on pneumatic roughness. a
c
b
I
I
d
I
Fig. 4. Illustration of possible sensor designs.
42
3.2. Equivalent gap for a rough surface If the sensor is placed on a rough surface and the fluid resistance is the same as that for gap hE with a smooth surface, hE can be called the equivalent gap. It depends on the surface roughness RA, the probability distribution of the surface and the load on the sensor. Consider for example a surface with two-dimensional roughness with a gaussian distribution P(Z) =
1 u( 2n)“2
z* exp - 22 ( 1
where cr is the standard deviation and p is the probability corresponding to height z. Suppose that a sensor of the type of Fig. 4(c) has weight W and nominal contact area A = ndl and that the microhardness of the surface is M. Then the ratio of the actual to the nominal contact area is W _ kMA
a _= A
where k is a constant which Uppal et al. [4] have shown is about very light loads. Then the sensor will rest at height zi such that a -= A
l Jexpi(J(2?7)“2 *,
-$I
0.4 for
dz
This procedure enables zi to be estimated for any probability distribution if the load and hardness are known. For the sensor used on mild steel surfaces a/A was about 1.35 X lop3 corresponding to zi = 30. If each element of surface is regarded as an elementary flat plate with gap z 1 - z, then by using the 2.5 power law we should find that the surface conductance is proportional to 1 o(2n)“2
21 (2, --z)*.~ exp - gz dz J _m i 1
and that the equivalent gap is this quantity raised to the power 0.4. Similarly, for any other distribution the equivalent gap is *I (21 - z)*.“p(z) dz o.4 J 1 -cc I where z 1 is the point giving the same bearing area. Clearly this treatment is an approximation only, which is likely to be reasonably good so long as the surface slopes are not too large. For a gaussian surface with z1 = 30, the equivalent gap hE is 3.2350 which is equal to 4.054RA. If we use these surfaces as standards for calibrating the roughness meter by adjusting it to show RA for these surfaces,
43
then for any other surface the indicated pneumatic roughness RP will be h,/4.054. Hence we can calculate the ratio of RP to RA for the surface as
RP - = 0.247 2 HA RA 3.3. Effects of skewness and of truncation on RP and on Rp/RA While the gaussian distribution is the most commonly quoted, in practice few surfaces are exactly gaussian. Surfaces produced by a single process such as grinding often have distributions similar to gaussian but with more or less negative skewness. Those produced by double processes such as grinding followed by lapping or running in or, conversely, polishing followed by wear producing scratches often resemble truncated gaussian or skewed gaussian distributions. It is therefore desirable to examine the effects both of gradual change of skewness and of truncation. For gradual change of skewness, the logarithmic gaussian
which becomes gaussian as a0 tends to zero and has a skewness which increases with a0 is convenient. The distribution with u. = 0.3 and skewness -0.95 is compared with a gaussian having the same mean and RA in Fig. 5. Figure 6 shows the Rp/RA ratio uersus skewness which has a steady decrease, reaching 0.736 at a skewness of -1. The effect of truncation of the distributions is shown in Figs. 5, 7 and 8. This changes the value of zi in the definition of hx. Figure 5 shows Rp/RA against the truncation position for the gaussian and for the logarithmic gaussian with o. = 0.3. In both cases RP/RA falls as the truncation increases (z 1
Fig. 5. Gaussian and logarithmic gaussian probability distributions and the effect of truncation on R~/RA.
r
t
L_--_I.
0
-0.5
SKEWNESS
-1.0
Fig. 6. Rp/RA vs.skewness
for the logarithmic
Fig. 7. Rp vs. the logarithm mic gaussian distributions.
of the bearing
Fig. 8. Variation in Rp with RA arising rithmic gaussian distributions.
from
gaussian
area ratio
distributions.
for truncated
progressive
truncation
gaussian
and logarith-
of gaussian
and loga-
falling), but reaches a minimum when the t~ncation is around the modal point of the distribution and then rises if zi falls further. Figure 7 shows, for the same distributions, RP uersus the logarithm to the base 10 of the bearing area ratio, i.e. the proportion of the total area that has been rendered flat by the t~ncation. A notable feature is the large difference between the slopes of the two curves when the bearing area ratio is small. This is relevant to the effect of loading of the sensor. For the gaussian, a change in load by a factor of 10 at low loads changes RP by about 22%, while for the skewed d~~ibu~on the corresponding change would be only 10%. This suggests that measurement of RP under varying sensor loads could be valuable for study of surface compliance. Figure 8 shows how RP varies with RA as the truncation point changes. For the gaussian surface with zl = 30, RP = RA. If z1 is reduced, for example by lapping against a flat surface, RP is at first rapidly reduced with very little change in RA. With greater truncation RA does fall. As already shown in Fig. 5, R~/RA falls to a minimum of 0.44 when z1 = 0 but rises back through unity when the roughness has been reduced to 0.067 of its original value.
45
The corresponding curve for the skewed distribution starts at RP = 0.743 (the RP/RA ratio corresponds to a0 = 0.3 in Fig. 6) and behaves similarly except that the fall in RP arising from truncation is smaller. Despite the lower initial RP value, the minimum RP/RA is slightly higher at 0.47 and the original RP/RA value is regained when the roughness has fallen to 0.17 of the original value. It has been assumed that the roughness meter is calibrated using gaussian surfaces. If ordinary ground surfaces are used, they may have slightly skewed distributions lying between those shown in Fig. 5. The effects of lapping would then be less drastic than those for the gaussian in Fig. 8 and the minimum Rp/RA value reached by truncation would be higher. If a truly gaussian profile were then tested it would appear to have RP/RA > 1. Figure 8 suggests that pneumatic roughness is very sensitive to processes of wear or running in which flatten the tops of the distribution, and the rapid decrease in RP may give a better indication of the resulting improvement of the tribological properties of surfaces than would be obtained by profilometry . An interesting example of a profile with a low RP/RA value is a gramophone record. This has right-angled V-shaped grooves alternating with flats of about the same width; the calculated RP/RA value is about 0.4, and measurements give RP = 5.4 pm and RA = 13.8 pm. 4. Prospects Further
for further
development
developments
seem possible along the following
lines.
4.1. Use of smaller sensors The present sensor diameter of 8.55 mm makes it very sensitive to surface waviness and some reduction is desirable. Fortunately the conductance is more sensitive to h (being proportional to h’.‘), and hence a reduction in the diameter by a factor of 5 would involve a gap reduction in Rs of only a factor of 1.9. With a very small detector, more sophisticated means of handling might be required. 4.2. Use of slit sensors A slit, rather than an annular sensor, would give a more exact comparison with profilometry. The more important advantage, however, would be the possibility of using a slit sensor on cylindrical surfaces, and preliminary work has suggested that this is worthy of further study. 4.3. Study of surface profile effects Some of the effects to be expected for two-dimensional roughness have been discussed in Section 3, and while some experimental confirmation has been obtained further work is needed. A study of the effects of anisotropy is a further obvious need.
46
4.4. Study of effects of sensor load It was suggested in Section 3, and experiment confirms, that large changes of sensor load produce relatively small changes of conductance and hence of the apparent RP value. A study of the optimum load which would give the most consistent results without damaging the sensor or the surface is required, however. As already suggested, a sensor with a variable load might give a useful indication of the compliance of the surface.
5. Postcript:
design improvements
Following the acceptance in 1981 of this paper for the Conference, a number of design improvements have been made which increase by an order of magnitude the speed of response of the instrument. First, analysis showed that a better combination of response speed and rigidity would be obtained by using a thinner wire that pulled directly rather than at the 1:2.5 inclination shown in Fig. 3. The wire diameter has been reduced from 75 to 15 pm. The wire now controls a needle whose point passes through the nozzle of resistor R, and rocks a “seesaw” type of lever, the other end of which forms the gap over another nozzle which now forms the resistor Ri. Thus as R, falls, RI rises. This enables the ratio R3/R1 to vary over a wider range without involving very large values for either resistor. Thus the range and response speed are both improved. The range improvement is such that removing the sensor from the surface (thus reducing R4 to a very low value) no longer gives any ill effects. It is now unnecessary to switch between R4 and R/ when placing the sensor on, or removing it from, the surface. With the detector described in Section 2.4, the use of water, which has high and variable surface tension, made it necessary to use fairly large chambers for the U-tube. This has now been replaced by a smaller U-tube with chambers 6.3 mm in diameter which uses silicone oil. A rolling sphere 0.6 mm in diameter whose position variation alters the transmission of IR light from an emitter on one side to a detector on the other is now located in the connecting channel. The viscosity (100 mm* s-l) of the silicone oil was chosen to provide the correct damping. The reduction by a factor of about 30 in the chamber volume which this new detector has provided is an essential complement to the reduction of the wire diameter in improving the response speed of the instrument. The system now oscillates at 50 Hz compared with 4 Hz for the previous design. A stable reading is now obtained after a delay of 0.25 - 1.0 s, rising to 1.5 s only for the very smoothest surfaces. A situation where the improvement in response speed is particularly valuable arises in the use of a slit sensor with cylindrical surfaces. It is necessary to traverse the slit to obtain the minimum wire voltage corresponding to the minimum gap position, and it now seems likely that it will be possible to do this automatically and reasonably quickly.
47
References 1 L. H. Tanner, A pneumatic Wheat&one bridge for surface roughness measurement, J. Phys. E, 12 (1979) 957 - 960. 2 L. H. Tanner, An improved pneumatic Wheatstone bridge for roughness measurement, J. Phys. E, 13 (1980) 593 - 594. 3 L. H. Tanner, A self-balancing pneumatic potentiometer and Wheatstone bridge with electrical readout. Applications to surface roughness measurement, pneumatic gauging and measurement of pressure difference ratios, Precis. Eng., 3 (4) (1981) 201 - 207. 4 A. H. Uppal, S. D. Prober-t and T. R. Thomas, The real area of contact between a rough and a flat surface, Wear, 22 (1972) 163 - 183.
37
37 - 47
A SELF-BALANCING PNEUMATIC WHEATSTONE SURFACE ROUGHNESS MEASUREMENT*
BRIDGE
FOR
L. H. TANNER Mechanical and Production BN2 4GJ (Gt. Britain)
Engineering
Department,
Brighton
Polytechnic,
Brighton
(Received May 27,1982)
Summary
The pneumatic conductance of the gap between a rough surface and a detector placed on it is roughly proportional to the 2.5 power of the roughness. To measure a roughness range of 4O:l it is therefore necessary to measure fluid resistances over a range of 104:1. This is made possible by the use of a pneumatic analogue of the Wheatstone bridge. The most important elements of the bridge are the sensor, a variable fluid resistor which can be calibrated and an extremely sensitive detector of pressure difference. If the detector gives an electrical output which is used to control the variable resistor, the bridge can be made self-balancing and will give an electrical readout of the roughness value. The design of a roughness meter of this type, its performance and potential for further development, and the dependence of the pneumatic roughness on the surface profile are discussed.
1. Introduction
The concept of using a pneumatic analogue of the Wheatstone bridge (Fig. 1) for roughness measurement has been implemented in three stages. In the first instruments [ 1, 21 the pressure source was a hand-operated bellows. The fixed resistors Ri and Rz were glass capillary tubes packed with stranded nylon thread to give a resistance of 2.6 X 10” Pa s mV3. The sensor (Fig. 2) was placed on the rough surface so that R4 was the resistance to air flow out from this sensor via the roughness elements. The variable resistor was formed by a lens rocked [1] or traversed [2] over a flat plate with a small hole. The gap between lens and hole is proportional to the square of the traverse distance from the datum position at which the lens is exactly *Paper presented at the Second International Conference on Metrology and Properties of Engineering Surfaces, Leicester Polytechnic, Leicester, Gt. Britain, April 14 - 16, 1982. 0043-1648/82/0000-0000/$02.75
0 Elsevier Sequoia/Printed
in The Netherlands
38
Fig. 1. Equivalent circuit of a pneumatic bridge: RI, Rz, Ri, fixed resistors; RJ, wirecontrolled resistor; W, S, water and solution sides of the detector; F, feedback circuit controlling R,; V, two-way valve. Fig. 2. Sensor and detail of rim shape.
over the hole. The conductance is roughly proportional to the 2.5 power of the gap and hence to the fifth power of the traverse distance. The traversing lens resistor has proved extremely stable and reliable and has been of great value throughout this research. The null detector was a small Perspex U-tube containing water with small particles in suspension. The channel joining the limbs was of small cross-sectional area, and a microscope was used to view particle motion in the channel. This gave an extremely sensitive way of detecting the pressure balance point when RJR3 = R2/R I. The micrometer reading was found [ 21 to vary linearly with RA1’3 where RA is the centre-line average roughness of the ground surfaces which were used for testing. The object of the second stage of development was to remove the need to observe particles through a microscope while manually balancing the bridge. If the detector can be made to give an electrical output and if this can then be used to control the fluid resistance R, then we have a feedback loop, the bridge becomes self-balancing and the electrical output can be used for a digital readout. The result [3] showed that this objective could be achieved using a fluid resistor controlled by the expansion of a heated wire. At this stage the system consisted of an array of separate elements (the power supply, the amplifier, the voltmeter, the wire-controlled resistor and the detector), and the RA value of the surface was roughly proportional to the square of the indicated voltage. The third stage has therefore been to incorporate all the elements in one unit and to add an electrical exponentiator and variable amplifier so as to give a readout indicating RA directly in microns. The design and performance of this instrument are described in the present paper, and the effect of surface profile on the relation between the pneumatic roughness RP and the RA value measured by profilometry is also discussed.
39
2. Elements
of the bridge
2.1. The pressure supply Air at about 5000 Pa gauge pressure obtained via a reducing valve from a compressed air supply is conveyed by a small-bore flexible tube to the point of entry to Ri and Rz. Here the pressure is reduced and controlled at 1400 Pa by a small weight on a hole 6 mm in diameter. This can be removed to give zero pressure when the instrument is not in use or for levelling the detector. 2.2. The fixed resistors R 1, R2 and Ri and valve V The fixed resistors are, as in the original instrument [ 11, glass capillary tubes packed with nylon thread. The size has been reduced to 0.5 mm bore and length 15 mm. The miniature two-way valve V is connected to the reference resistor R4) until the sensor is in position on the surface to be measured. 2.3. The electrically variable fluid resistor R3 This is the wire-controlled resistor described earlier [3] and shown in Fig. 3. Current through the wire causes it to expand, increasing the gap between nozzle and lever. The wire diameter is 75 pm and the Grashof number is lOA or less. In this range heat transfer is by conduction rather than by buoyant convection so the rise in wire temperature should be exactly proportional to the power dissipated. The wire resistance is expected to rise with temperature so the relation between the wire voltage V and the gap width h is expected to be of the form V2 = &(l + oh) where cy and fl are constants, and experiment confirms this [ 31. The moving magnet (Fig. 3) provides fine control for the wire tension and hence a zero setting for the gap width. Drift of zero can arise from changes in room temperature giving differential expansion, but this is readily corrected. Changes in sensitivity can arise from changes in air humidity which changes the air conductivity. To avoid this a very small flow of air from the supply is bled into the wire chamber to ensure that the air in the chamber has the same humidity as that flowing through R,.
NOZZLE
I
WIRE
1OOmm
I
Fig. 3. Wire-controlled resistor. The motion of the pallets for wire adjustment is shown by the arrows.
40
2.4. The detector This is as described earlier [ 3] but is smaller and is improved in detail. The chambers of the Perspex U-tube are 20 mm in diameter and 20 mm deep; one contains distilled water and the other contains dilute KC1 solution. Electrodes are placed at each end of the connecting channel, the small-bore part of which is 0.43 mm in diameter and 5 mm long. One of the electrodes is a dressmaker’s pin 0.6 mm in diameter which acts as a needle valve to permit adjustment of the fluid resistance of the channel. Excess pressure on one side or the other of the U-tube increases or decreases the concentration of solution in the channel, thus changing the electrical conductivity. A voltage of 1.0 V at 4 kHz is applied across the channel in series with a 56 ka resistor. The fraction developed across this resistor goes to a rectifying detector and amplifier giving a d.c. voltage output. For input amplitudes II less than u1 the d.c. output V is zero. As u rises from u1 to ua, V rises linearly from zero to 1.0 V, at which it remains if u rises further, thus preventing over-heating of the wire. To prevent oscillation and to reduce response time a differentiator and adder are included in the circuit to give proportional plus derivative feedback. A simplified analysis [3] has been given to show the effect of this on the system. Oscillation can still occur, particularly in the sensitive condition when the bridge arms are of equal resistance. Optimum conditions are obtained by setting the system into oscillation at this condition and then increasing the channel resistance by the needle valve until the oscillation is lightly damped. 2.5. The sensor The sensor (Fig. 2) is as described earlier [3]. The annulus is formed by lapping against two mating spherical surfaces, one convex and one concave, with a radius of curvature of 110 mm, to produce two approximately conical surfaces of slope l/25. A final brief lap on a flat surface gives the flat portion a width of 75 pm, ensuring perfect flatness of the rim and making it more durable. 2.6. Overall performance and the modified display The simplest analysis of the system would suggest that the resistances R, and R4 would be equal, and balance would be obtained, if the gap above the nozzle in R, were propo~ion~ to the gap between the sensor and the surface in R4 and hence to the surface roughness. Since the gap in Rs is roughly proportional to V2, we would expect V2 to be proportional to the surface roughness to be measured. The real situation is more complex in two respects. One is the nonlinearity arising from the change in wire resistance with temperature (Section 2.3). The other is that the different flow configurations of R, and R4 are likely to give differences in the resistance uersus gap relations and hence to give deviations from direct proportionality of one gap to the other for the condition of balance.
41
The matter was investigated experimentally [3] and it appeared that the two complicating factors roughly compensated one another and that V2 was at least approximately proportional to the surface roughness RA. At that time the voltage V was displayed, but it was suggested that display of V2 would be desirable. It is, however, almost as easy (by substituting a variable resistor for a fixed resistor) to display an adjustable power V” and, by adding a variable amplifier, to display FZV” where both 12 and n are adjustable. The extra degree of freedom makes it possible to adjust the calibration to ensure equality of pneumatic roughness R, and the profilometric RA at the maximum RA and at an intermediate value as well as at zero. This has now been done, and the display shows directly the surface roughness in microns. Adjustment of the index n effectively bends the calibration curve at the lower end. If any adjustment at the upper end is needed, advantage can be taken of the fact that at this end the resistances of the tubing connecting the detector to R, and R4 become comparable with Rs and R+ Ideally these series resistances should be equal, but deliberate inequalities can be used to bend the curves if required.
3. Effects of surface profile on the relation of pneumatic to profilometric roughness 3.1. Variation in resistance with sensor gap Consider a sensor held so that the minimum gap between it and a smooth flat surface is h. The way in which the fluid conductance varies with h depends on the form of the sensor. For a flat annulus (Fig. 4(a)) with negligible edge effects the conductance is proportional to h3. For the doublecone form (Fig. 4(d)) with a sharp edge it is proportional to h2. These can be regarded as extremes between which lie more practical forms, including the toroid (Fig. 4(b)) which has conductance proportional to h2.5 and the flattened double cone (Fig. 4(c)). This has conductance proportional to h3(1 + h/cd)-’ which approximates to the h2.5 law if h/d is not too far from unity. It therefore seems appropriate to assume an h2e5law in order to examine the effects of surface profile on pneumatic roughness. a
c
b
I
I
d
I
Fig. 4. Illustration of possible sensor designs.
42
3.2. Equivalent gap for a rough surface If the sensor is placed on a rough surface and the fluid resistance is the same as that for gap hE with a smooth surface, hE can be called the equivalent gap. It depends on the surface roughness RA, the probability distribution of the surface and the load on the sensor. Consider for example a surface with two-dimensional roughness with a gaussian distribution P(Z) =
1 u( 2n)“2
z* exp - 22 ( 1
where cr is the standard deviation and p is the probability corresponding to height z. Suppose that a sensor of the type of Fig. 4(c) has weight W and nominal contact area A = ndl and that the microhardness of the surface is M. Then the ratio of the actual to the nominal contact area is W _ kMA
a _= A
where k is a constant which Uppal et al. [4] have shown is about very light loads. Then the sensor will rest at height zi such that a -= A
l Jexpi(J(2?7)“2 *,
-$I
0.4 for
dz
This procedure enables zi to be estimated for any probability distribution if the load and hardness are known. For the sensor used on mild steel surfaces a/A was about 1.35 X lop3 corresponding to zi = 30. If each element of surface is regarded as an elementary flat plate with gap z 1 - z, then by using the 2.5 power law we should find that the surface conductance is proportional to 1 o(2n)“2
21 (2, --z)*.~ exp - gz dz J _m i 1
and that the equivalent gap is this quantity raised to the power 0.4. Similarly, for any other distribution the equivalent gap is *I (21 - z)*.“p(z) dz o.4 J 1 -cc I where z 1 is the point giving the same bearing area. Clearly this treatment is an approximation only, which is likely to be reasonably good so long as the surface slopes are not too large. For a gaussian surface with z1 = 30, the equivalent gap hE is 3.2350 which is equal to 4.054RA. If we use these surfaces as standards for calibrating the roughness meter by adjusting it to show RA for these surfaces,
43
then for any other surface the indicated pneumatic roughness RP will be h,/4.054. Hence we can calculate the ratio of RP to RA for the surface as
RP - = 0.247 2 HA RA 3.3. Effects of skewness and of truncation on RP and on Rp/RA While the gaussian distribution is the most commonly quoted, in practice few surfaces are exactly gaussian. Surfaces produced by a single process such as grinding often have distributions similar to gaussian but with more or less negative skewness. Those produced by double processes such as grinding followed by lapping or running in or, conversely, polishing followed by wear producing scratches often resemble truncated gaussian or skewed gaussian distributions. It is therefore desirable to examine the effects both of gradual change of skewness and of truncation. For gradual change of skewness, the logarithmic gaussian
which becomes gaussian as a0 tends to zero and has a skewness which increases with a0 is convenient. The distribution with u. = 0.3 and skewness -0.95 is compared with a gaussian having the same mean and RA in Fig. 5. Figure 6 shows the Rp/RA ratio uersus skewness which has a steady decrease, reaching 0.736 at a skewness of -1. The effect of truncation of the distributions is shown in Figs. 5, 7 and 8. This changes the value of zi in the definition of hx. Figure 5 shows Rp/RA against the truncation position for the gaussian and for the logarithmic gaussian with o. = 0.3. In both cases RP/RA falls as the truncation increases (z 1
Fig. 5. Gaussian and logarithmic gaussian probability distributions and the effect of truncation on R~/RA.
r
t
L_--_I.
0
-0.5
SKEWNESS
-1.0
Fig. 6. Rp/RA vs.skewness
for the logarithmic
Fig. 7. Rp vs. the logarithm mic gaussian distributions.
of the bearing
Fig. 8. Variation in Rp with RA arising rithmic gaussian distributions.
from
gaussian
area ratio
distributions.
for truncated
progressive
truncation
gaussian
and logarith-
of gaussian
and loga-
falling), but reaches a minimum when the t~ncation is around the modal point of the distribution and then rises if zi falls further. Figure 7 shows, for the same distributions, RP uersus the logarithm to the base 10 of the bearing area ratio, i.e. the proportion of the total area that has been rendered flat by the t~ncation. A notable feature is the large difference between the slopes of the two curves when the bearing area ratio is small. This is relevant to the effect of loading of the sensor. For the gaussian, a change in load by a factor of 10 at low loads changes RP by about 22%, while for the skewed d~~ibu~on the corresponding change would be only 10%. This suggests that measurement of RP under varying sensor loads could be valuable for study of surface compliance. Figure 8 shows how RP varies with RA as the truncation point changes. For the gaussian surface with zl = 30, RP = RA. If z1 is reduced, for example by lapping against a flat surface, RP is at first rapidly reduced with very little change in RA. With greater truncation RA does fall. As already shown in Fig. 5, R~/RA falls to a minimum of 0.44 when z1 = 0 but rises back through unity when the roughness has been reduced to 0.067 of its original value.
45
The corresponding curve for the skewed distribution starts at RP = 0.743 (the RP/RA ratio corresponds to a0 = 0.3 in Fig. 6) and behaves similarly except that the fall in RP arising from truncation is smaller. Despite the lower initial RP value, the minimum RP/RA is slightly higher at 0.47 and the original RP/RA value is regained when the roughness has fallen to 0.17 of the original value. It has been assumed that the roughness meter is calibrated using gaussian surfaces. If ordinary ground surfaces are used, they may have slightly skewed distributions lying between those shown in Fig. 5. The effects of lapping would then be less drastic than those for the gaussian in Fig. 8 and the minimum Rp/RA value reached by truncation would be higher. If a truly gaussian profile were then tested it would appear to have RP/RA > 1. Figure 8 suggests that pneumatic roughness is very sensitive to processes of wear or running in which flatten the tops of the distribution, and the rapid decrease in RP may give a better indication of the resulting improvement of the tribological properties of surfaces than would be obtained by profilometry . An interesting example of a profile with a low RP/RA value is a gramophone record. This has right-angled V-shaped grooves alternating with flats of about the same width; the calculated RP/RA value is about 0.4, and measurements give RP = 5.4 pm and RA = 13.8 pm. 4. Prospects Further
for further
development
developments
seem possible along the following
lines.
4.1. Use of smaller sensors The present sensor diameter of 8.55 mm makes it very sensitive to surface waviness and some reduction is desirable. Fortunately the conductance is more sensitive to h (being proportional to h’.‘), and hence a reduction in the diameter by a factor of 5 would involve a gap reduction in Rs of only a factor of 1.9. With a very small detector, more sophisticated means of handling might be required. 4.2. Use of slit sensors A slit, rather than an annular sensor, would give a more exact comparison with profilometry. The more important advantage, however, would be the possibility of using a slit sensor on cylindrical surfaces, and preliminary work has suggested that this is worthy of further study. 4.3. Study of surface profile effects Some of the effects to be expected for two-dimensional roughness have been discussed in Section 3, and while some experimental confirmation has been obtained further work is needed. A study of the effects of anisotropy is a further obvious need.
46
4.4. Study of effects of sensor load It was suggested in Section 3, and experiment confirms, that large changes of sensor load produce relatively small changes of conductance and hence of the apparent RP value. A study of the optimum load which would give the most consistent results without damaging the sensor or the surface is required, however. As already suggested, a sensor with a variable load might give a useful indication of the compliance of the surface.
5. Postcript:
design improvements
Following the acceptance in 1981 of this paper for the Conference, a number of design improvements have been made which increase by an order of magnitude the speed of response of the instrument. First, analysis showed that a better combination of response speed and rigidity would be obtained by using a thinner wire that pulled directly rather than at the 1:2.5 inclination shown in Fig. 3. The wire diameter has been reduced from 75 to 15 pm. The wire now controls a needle whose point passes through the nozzle of resistor R, and rocks a “seesaw” type of lever, the other end of which forms the gap over another nozzle which now forms the resistor Ri. Thus as R, falls, RI rises. This enables the ratio R3/R1 to vary over a wider range without involving very large values for either resistor. Thus the range and response speed are both improved. The range improvement is such that removing the sensor from the surface (thus reducing R4 to a very low value) no longer gives any ill effects. It is now unnecessary to switch between R4 and R/ when placing the sensor on, or removing it from, the surface. With the detector described in Section 2.4, the use of water, which has high and variable surface tension, made it necessary to use fairly large chambers for the U-tube. This has now been replaced by a smaller U-tube with chambers 6.3 mm in diameter which uses silicone oil. A rolling sphere 0.6 mm in diameter whose position variation alters the transmission of IR light from an emitter on one side to a detector on the other is now located in the connecting channel. The viscosity (100 mm* s-l) of the silicone oil was chosen to provide the correct damping. The reduction by a factor of about 30 in the chamber volume which this new detector has provided is an essential complement to the reduction of the wire diameter in improving the response speed of the instrument. The system now oscillates at 50 Hz compared with 4 Hz for the previous design. A stable reading is now obtained after a delay of 0.25 - 1.0 s, rising to 1.5 s only for the very smoothest surfaces. A situation where the improvement in response speed is particularly valuable arises in the use of a slit sensor with cylindrical surfaces. It is necessary to traverse the slit to obtain the minimum wire voltage corresponding to the minimum gap position, and it now seems likely that it will be possible to do this automatically and reasonably quickly.
47
References 1 L. H. Tanner, A pneumatic Wheat&one bridge for surface roughness measurement, J. Phys. E, 12 (1979) 957 - 960. 2 L. H. Tanner, An improved pneumatic Wheatstone bridge for roughness measurement, J. Phys. E, 13 (1980) 593 - 594. 3 L. H. Tanner, A self-balancing pneumatic potentiometer and Wheatstone bridge with electrical readout. Applications to surface roughness measurement, pneumatic gauging and measurement of pressure difference ratios, Precis. Eng., 3 (4) (1981) 201 - 207. 4 A. H. Uppal, S. D. Prober-t and T. R. Thomas, The real area of contact between a rough and a flat surface, Wear, 22 (1972) 163 - 183.