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Surface preparation of machine tool slideways and its influence on the contact stiffness
Surface preparation of machine tool slideways and its influence on the contact stiffness D. K. Pal* and S. K. Basut This paper describes an analytical approach to evaluate the contact stiffness of machine tool slideways under the influence of surface preparation, external pressure on the contact face and material characteristics. A simple test rig has been used to find the correlation between theoretical and experimental results.
A Aa AF E H L l W
Y x o
6t 8c
~n 2
5 ct C
working surface area of test pieces apparent area of contact real or factual area of contact Young's modulus maximum peak to valley height of surface irregularities thickness of intermediate C.I. ring combined thickness of the test pieces external load applied on the joint flow stress of contacting material any vertical distance from the highest peak in Abbot's bearing area curve mean applied pressure total deformation as observed in comparator (microns) contact deformation of each joint (microns) elastic compressions Of the combined test pieces (microns) elastic compression of the C.I. rings (microns) total contact deformation of all the four joints (microns) coefficient of contact compliance (micron X 10 2 mm2]kg)
A major part of the total displacement in a machine tool slideway is due to contact deformations which greatly affect the operating characteristics of a machine tool. * R.E. College, Durgapur-9, India. ~- MERADO,Poona-8, India.
172
TRIBOLOGYOctober 1973
Levina I reported that contact deformations in plane surfaces in sliding are directly proportional to normal pressure and are determined mainly by the compression of surface irregularities whereas in the case of large areas, deformation of macro waves also has a major role to play. Identical observations were also made by Ostrovskii 2 and Lapidus et al 3. They obtained empirical as well as theoretically calculated relationships for contact deformations which show that the method of machining and the machine properties of the mating surface considerably influence the contact deformation process. Theoretical analysis All surfaces are necessarily imperfect far beyond the scale of atomic dimensions. Hence, true contact, where atoms of one body are so intimately associated with atoms of the other body as with any other atom within the same body, can initially occur at three points. Being insufficiently strong to bear a large contact pressure, these points will deform in order to spread the load over an adequate area. If work-hardening is regulated and if the material is assumed to be isotropic, then the fraction of the overall surface area which will be brought into real contact is o[Y. On the other hand, Abbot's bearing area curve of any machined surface expresses the fractional area which comes in true contact with another mating surface as the crests are gradually deformed with the application of external load. Therefore, one of the co-ordinates of the bearing area curve, namely the variable AF]A can be replaced by o/Y whereas the other variable xlH can be replaced by 5/1-1. The theoretical relationships between mean joint pressure and contact
deformations are then obtained by taking the roughness of the mating surfaces into account. To evaluate the influence of surface preparation on joint stiffness, mating surfaces were finished by fine machining processes: polishing, grinding, finish shaping, conventional scraping and milling. Surface records of these surfaces were taken on a Talysurf model 4 and Abbot's bearing area curves were constructed as shown in Fig 1. Assuming normal distribution of the crests of the irregularities in height, equations of the following type were fitted to the bearing area curves: Y
=
- -
1
/(2,0o
IO It
12
14
13
9
6
7
(1)
e -(x-~)2/2°2
Where Y refers to A F / A , x refers to 6[H, and # and o are the mean and standard deviation of the distribution of the irregularities, respectively. For a more exact fit, certain corrections were necessary. After solving for the constants from the actual curves and taking into account the corresponding corrections the final equations shown in Table 1 were derived. The degree of fit of the final equations was estimated by a chi-square test. These equations now represented the theoretical relationships between mean joint pressure and contact deformation, taking into account the roughness of the mating surfaces.
2
S
4
+ ~",,4/I
/I
I - H¢ovy b o l l
/
/
/
I--.._+
I
/ / / / / / /
/
//
/
9-- Lever
2 - Cyllndrlc=l block (C +1)
I 0 - Ba II
3 - R e c o r d e r of the
II - H i n g e
electronic ¢ot~4r oto r
S-TT¢$t sp¢clmeh 6 - Intermcdlote CI. rin9
12-Position
I I - Claiming block (C+l.]
13-14 Position o1 ¢ o m p a r o t o r
4-Probe
o1 the ~ m p Q r o t o r
o f boll for off sct
I°adlo9 probe f o r measurement of tilt
Fig 2 General arrangement of the set-up for determining contact rigidity tO
213 0,8
0.6
"~1" o, O2
4 - C onvlntlom¢l
02
0-4
06
O.S
I0
xlH
Fig 1
Bearing area curves for different surfaces
12
E x p e r i m e n t a l procedure The experimental set-up to determine the contact deformations of a pair of guide materials finished by a particular machining process is shown in Fig 2. It consists of a massive cast iron (C.I.) base on which are mounted top and bottom test pieces respectively. In between the test-pieces, there is an intermediate ring made of cast iron. Over the top test piece, there is another cast iron block and the load was applied on the top of this such that the joints were uniformly loaded without tilting. Compression of the test pieces was measured with a Mercer Electronic Comparator with an 0.25 microtl accuracy. Materials selected for the test pieces were cast-iron, mild steel (M.S.) and perspex and they were made in the form of 40 mm diameter discs with 20 mm diameter bores cut from a sheet cast from a single melt. The intermediate cast iron ring was also of the same size. This arrangement gave rise to four loaded joints - one between the test piece and the clamping disc, two between the intermediate ring and the test pieces and the fourth between the test piece and the cast iron base. Contact deformation of each joint was then calculated from the observed value of the total deformation using the following formulae which take into account the elastic compression of the test pieces and the intermediate cast iron ring.
TR I E'OLOGY October 1973
173
Table 1 Theoretical relationships for calculating contact deformation for a given surface preparation. Method of surface preparation
Theoretical relationships for calculating contact deformation
Polishing on emery belt
-(alH- 1)2
(7 -- = 1.62 e
+ 0.0583
0.122
- 1.94
-
Y Grinding with wheel periphery
1.62 e
- ( a ] H - 1) 2
o
- 1.46 e
0.6
0.65 < a/H <-..1.0
[
(¢5//-D2 )
+ 0.295 6[H
0.15 -(6/H-
--=1.46e Y Finish shaping
o ~ = 1.46 e
o --
Y Conventional scraping
o.-i /
Milling
= 1.46 e
1)2
0.15
-0.767(a/H-0.44)
0.15
+ 0.28 a/1-1
0 . 4 4 < 6/H<~ 1.0
(6/H) 2
0.15
-
-
~
Y
O - - = 1.33 e Y
L .--X o E2
1 (~/~02 ~
o.- 1
0 < a/H <~0.43
-(,5/11-1)2 0.14
-
-
0 . 3 ( 6 ] H - 0.43)
0.43 < 8/H <~1.0
-(6[H- 1)2 0.18
0 < ~/H <~0.22
-(a[H- 1)2 0.18
-0.5(a/H-
0.22)
103
103
o W/A =
act
6c- 4
(2)
By increasing the dead load, the mean joint pressure was increased from 0 to approximately 12 X 10 -2 kg/mm 2 and for some stipulated pressures in between, contact deformation values of the C.I.-Perspex pair furnished by different machining methods, as already mentioned, were evaluated.
174
o < a l H < 0.44
0.44 < 8/H <~1.0
~Ct + (~n I + (~n)
=
!
0.8(~5/H - 0.44)
-(SIH- 1)2 o - - = 1.51 e 0.14 + 0 . 3 0 2 ~ / H Y
(7 --=1.33e
1 6na=o.E1X
0 < a/H<<,0.44
-(5/H-1)2
(7 - - = 1.51 e Y
6n2
0 < a/H ~ 0.65
Y (7
=
(61H) 2 '~
a/H
r 17
~t
(I
0.122
T R I B O L O G Y October 1973
0.22 < 6/H <~1.0
These values were then plotted against the mean joint pressure o, see Fig 3. The curves have been constructed from the arithmetical average values of mean gap reduction on five similar test pieces. Theoretical relationships to calculate contact deformation as tabulated in Table 1 are represented graphically in Fig 4. In order to compare theoretical results with experimental data, Fig 3 is superimposed on Fig 4. Comparison reveals that actual contact deformation values coincide fairly closely with the theoretical values obtained from the equations of the bearing area curves obtained on the assumption of normal distribution of the heights of the crests. To investigate contact elasticity of slideways of various material pairs, the same set-up was used. Three material pairs namely, C.I.-C.I., C.I.-Perspex and C.I.-M.S. were chosen; one of the materials for the pair always being C.I. Tables 2, 3, 4 show the calculations of contact deformation and contact elasticity coefficients for the three pairs. These tables have been prepared from an analysis of the test
20.0
Ig'O
f 160
16"0
t40 v
"i
• o1 2 0
d¢ ,="
!
i 4
I .~
)
I0.0
o
o o
Pair :- C.I - Ptrspex t .Flhimhed by pOhhlng 2 - Flh~hed by grinding ) - Fhllsbnd by finish shaping 4 - F l , l S h l d by conventional scrclpang S - F i n l s h e d by miming
~.0
Po~ C.I. - P a r | p a x P - PolllhkHi G - Grktdl~l F S - F l f l i l h shaplnql ¢S - Conventional scraping M o Milling -Experimental - - - Theoretical
0 I'0
2.0
$0
4.0
SO
60
7"0
~10
9"0
~0
IPO
1.0
2.0
3.0
4.0
|.0
6.0
7.0
|0
a-O
I 0 " 0 I1.0
I~.0
Mean p r e n u r l , O ' , = I 0 " 2 ( k g l m = )
12"0
M e o n preslure,O', x I 0 - a ( k l l / m l 2 )
Fig 3 Showing relationship of 8 c with a for C.l.-Perspex pair with different surface preparation in the case of centralized loading
Fig 4 Showing comparison of theoretical and experimental results for C.l.-Perspex with different surface preparation
results. The experimental values of the reduction in the mean gap (St) are statistical values, determined from a small number of measurements (5) of a particular character. Comparability of obtained deformation values with the actual deformation values was tested by t-distribution and the calculations indicated that the deviations of population means do not exceed 9% of the mean value of deformations with a 0.95 probability. Joint contact deformation values for various pairs of mating surfaces are plotted against mean pressure in Fig 5.
The nature of the curve is the usual one for a centrally loaded joint and expresses the non-linear exponential relationship between 8 c and o. Table 5 shows the empirical formulae relating a with 8 c. These were derived by analysing the results on a double logarithmic grid using the leastsquares method. The table shows that the contact rigidity of a C.I.-Perspex pair is about three to six times less than that of a C.I.-C.I. and a C.I.-M.S. pair respectively. Variation of the coefficient of contact compliance with pressure has been graphically represented in Fig 6. This graph can be
Table 2
C.I.-C.I. pair
(kg/mm 2)
8t (micron)
/in 1 (rnlcron)
8n~ (rmcron)
4~c = [St - ( 8 . 1 + 8.2)] (micron)
8c (micron)
0.01 0.02 0.04 0.06 0.08 O.12 O.16
3.75 5.25 7.50 10.00 10.75 13.00 15.25
0.006 0.012 0.024 0.036 0.048 0.072 0.096
0.01 0.02 0.04 0.06 0.08 0.12 O. 16
3.734 5.218 7.436 9.904 10.622 12.808 14.994
0.933 1.304 1.859 2.476 2.655 3.202 3.720
o
C micron X mm 2 kg 93.3 65.2 46.5 41.3 33.2 26.7 23.4
TRIBOLOGY October 1973
175
Table 3
C.l.-Perspexpair
o (kg]mm 2)
6t (micron)
6n z (micron)
6n (micron)
4~c = [~t - (~n 1 + ~n2)] (micron)
~c (micron)
0.01 0.02 0.04 0.06 0.08 0.12 0.16
12.5 20.0 32.0 42.0 50.5 66.0 80.0
0.169 0.338 0.676 1.014 1.352 2.028 2.704
0.01 0.02 0.04 0.06 0.08 0.12 0.16
12.323 19.642 31.324 40.986 49.068 63.852 77.136
3.081 4.910 7.831 10.246 12.267 15.963 19.286
used for practical design calculations. Experiments to study the influence of lubrication on contact rigidity were performed and it was established that lubrication has negligible influence on contact rigidity.
Conclusions (i) With an increase in the accuracy of mating surfaces, contact elasticity coefficients decrease with a corresponding reduction in contact deformation. (ii) Theoretical relationships obtained on the basis of
C micron X mm 2 kg 308.1 245.5 195.8 170.8 153.3 133.0 120.5
Abbot's bearing area curves make it possible to reliably calculate joint stiffness of surfaces finished by different machining processes as the agreement of experimental data with theoretically calculated values is quite close. In the case of a polished surface the cumulative effect of elastic deformation of the crest of the irregularities, work-hardening of the material and the influence of flatness error results in a small variation. (iii) In the case of a centralized loading, the contact elasticity coefficient of the Perspex-C.I. pair is about three times higher than the C.I.-C.I. pair, whereas the contact
32
,
,
r
.
.
.
.
.
,
,
,
,
,
,
t4.0
IS.O
30 2e 161S 1111~1412
"
'
.
.
.
.
.
.
.
.
.
.
.
2.6
.
24 22
C I.- plrspex CI "C.I.
: "
•
2"0
C.I - M ~ .
joint
t 4 ;
7
I0 Oa 06 3
-
i
04 -
" ~020 0
I
2
3
4
S
6 7 IS g I0 fl M¢On pressure, cr. • 10"2(kglmm2)
f2
13
IA
IS
Fig 5 Showing relationship of ~c with o in the case of centralized loading
176
TRIBOLOGY October 1973
I~
~0
40
s.O 6 . 0 7 0
s.O
-0
-
~.0
.
G
HO
I 0
Mean prcnuregO, • 10"2J kg/mm2)
Fig 6 Showing relationship of C with o in the case of centralized loading
Table 4
C.I.-M.S. pair
C 5nl (micron)
8n~ (rmcron)
48c = [/it - (8nl +/i.2)]
/ic
(kg/mm 2)
/it (micron)
(micron)
(micron)
0.01 0.02 0.04 0.06 0.08 0.12 0.16
2.00 3.00 4.75 6.00 7.00 8.75 10.50
0.003 0.006 0.012 0.018 0.024 0.036 0.048
0.01 0.02 0.04 0.06 0.08 0.12 0.16
1.987 2.974 4.698 5.922 6.896 8.594 10.292
0.497 0.743 1.174 1.480 1.724 2.148 2.573
o
Table 5
Empirical formulae for contact deformation
Pair
Empirical formulae
C.I.-M.S.
/ic = 0.497 o 0.58
C.I.-Perspex
/ic = 3.081 00.66
C.I.-C.I.
/ic = 0.99
o 0"5
micron X mm 2 kg 49.7 37.1 29.4 24.7 21.5 17.9 16.1
elasticity of the C.I.-M.S. pair is half the value of the C.I.C.I. pair. Contact rigidity of a pair remains practicaUy unaffected even with the introduction of a lubricant.
References I Levina, Z.M. 'Calculations of contact deformations in siideways', Machines and Tooling, No 1 (1966) pp 8-17. 2 Ostrovskii, V. I. 'The influence of machining methods on contact stiffness', Machines and Tooling, Vol 36, No 1 (1965). 3 Lapidus, A.S. and Maiorova, E. A. 'Nylon facing strips for machine tool slideway', Machines and Tooling, No 9 (1965).
TRIBOLOGY October 1973
177
A Aa AF E H L l W
Y x o
6t 8c
~n 2
5 ct C
working surface area of test pieces apparent area of contact real or factual area of contact Young's modulus maximum peak to valley height of surface irregularities thickness of intermediate C.I. ring combined thickness of the test pieces external load applied on the joint flow stress of contacting material any vertical distance from the highest peak in Abbot's bearing area curve mean applied pressure total deformation as observed in comparator (microns) contact deformation of each joint (microns) elastic compressions Of the combined test pieces (microns) elastic compression of the C.I. rings (microns) total contact deformation of all the four joints (microns) coefficient of contact compliance (micron X 10 2 mm2]kg)
A major part of the total displacement in a machine tool slideway is due to contact deformations which greatly affect the operating characteristics of a machine tool. * R.E. College, Durgapur-9, India. ~- MERADO,Poona-8, India.
172
TRIBOLOGYOctober 1973
Levina I reported that contact deformations in plane surfaces in sliding are directly proportional to normal pressure and are determined mainly by the compression of surface irregularities whereas in the case of large areas, deformation of macro waves also has a major role to play. Identical observations were also made by Ostrovskii 2 and Lapidus et al 3. They obtained empirical as well as theoretically calculated relationships for contact deformations which show that the method of machining and the machine properties of the mating surface considerably influence the contact deformation process. Theoretical analysis All surfaces are necessarily imperfect far beyond the scale of atomic dimensions. Hence, true contact, where atoms of one body are so intimately associated with atoms of the other body as with any other atom within the same body, can initially occur at three points. Being insufficiently strong to bear a large contact pressure, these points will deform in order to spread the load over an adequate area. If work-hardening is regulated and if the material is assumed to be isotropic, then the fraction of the overall surface area which will be brought into real contact is o[Y. On the other hand, Abbot's bearing area curve of any machined surface expresses the fractional area which comes in true contact with another mating surface as the crests are gradually deformed with the application of external load. Therefore, one of the co-ordinates of the bearing area curve, namely the variable AF]A can be replaced by o/Y whereas the other variable xlH can be replaced by 5/1-1. The theoretical relationships between mean joint pressure and contact
deformations are then obtained by taking the roughness of the mating surfaces into account. To evaluate the influence of surface preparation on joint stiffness, mating surfaces were finished by fine machining processes: polishing, grinding, finish shaping, conventional scraping and milling. Surface records of these surfaces were taken on a Talysurf model 4 and Abbot's bearing area curves were constructed as shown in Fig 1. Assuming normal distribution of the crests of the irregularities in height, equations of the following type were fitted to the bearing area curves: Y
=
- -
1
/(2,0o
IO It
12
14
13
9
6
7
(1)
e -(x-~)2/2°2
Where Y refers to A F / A , x refers to 6[H, and # and o are the mean and standard deviation of the distribution of the irregularities, respectively. For a more exact fit, certain corrections were necessary. After solving for the constants from the actual curves and taking into account the corresponding corrections the final equations shown in Table 1 were derived. The degree of fit of the final equations was estimated by a chi-square test. These equations now represented the theoretical relationships between mean joint pressure and contact deformation, taking into account the roughness of the mating surfaces.
2
S
4
+ ~",,4/I
/I
I - H¢ovy b o l l
/
/
/
I--.._+
I
/ / / / / / /
/
//
/
9-- Lever
2 - Cyllndrlc=l block (C +1)
I 0 - Ba II
3 - R e c o r d e r of the
II - H i n g e
electronic ¢ot~4r oto r
S-TT¢$t sp¢clmeh 6 - Intermcdlote CI. rin9
12-Position
I I - Claiming block (C+l.]
13-14 Position o1 ¢ o m p a r o t o r
4-Probe
o1 the ~ m p Q r o t o r
o f boll for off sct
I°adlo9 probe f o r measurement of tilt
Fig 2 General arrangement of the set-up for determining contact rigidity tO
213 0,8
0.6
"~1" o, O2
4 - C onvlntlom¢l
02
0-4
06
O.S
I0
xlH
Fig 1
Bearing area curves for different surfaces
12
E x p e r i m e n t a l procedure The experimental set-up to determine the contact deformations of a pair of guide materials finished by a particular machining process is shown in Fig 2. It consists of a massive cast iron (C.I.) base on which are mounted top and bottom test pieces respectively. In between the test-pieces, there is an intermediate ring made of cast iron. Over the top test piece, there is another cast iron block and the load was applied on the top of this such that the joints were uniformly loaded without tilting. Compression of the test pieces was measured with a Mercer Electronic Comparator with an 0.25 microtl accuracy. Materials selected for the test pieces were cast-iron, mild steel (M.S.) and perspex and they were made in the form of 40 mm diameter discs with 20 mm diameter bores cut from a sheet cast from a single melt. The intermediate cast iron ring was also of the same size. This arrangement gave rise to four loaded joints - one between the test piece and the clamping disc, two between the intermediate ring and the test pieces and the fourth between the test piece and the cast iron base. Contact deformation of each joint was then calculated from the observed value of the total deformation using the following formulae which take into account the elastic compression of the test pieces and the intermediate cast iron ring.
TR I E'OLOGY October 1973
173
Table 1 Theoretical relationships for calculating contact deformation for a given surface preparation. Method of surface preparation
Theoretical relationships for calculating contact deformation
Polishing on emery belt
-(alH- 1)2
(7 -- = 1.62 e
+ 0.0583
0.122
- 1.94
-
Y Grinding with wheel periphery
1.62 e
- ( a ] H - 1) 2
o
- 1.46 e
0.6
0.65 < a/H <-..1.0
[
(¢5//-D2 )
+ 0.295 6[H
0.15 -(6/H-
--=1.46e Y Finish shaping
o ~ = 1.46 e
o --
Y Conventional scraping
o.-i /
Milling
= 1.46 e
1)2
0.15
-0.767(a/H-0.44)
0.15
+ 0.28 a/1-1
0 . 4 4 < 6/H<~ 1.0
(6/H) 2
0.15
-
-
~
Y
O - - = 1.33 e Y
L .--X o E2
1 (~/~02 ~
o.- 1
0 < a/H <~0.43
-(,5/11-1)2 0.14
-
-
0 . 3 ( 6 ] H - 0.43)
0.43 < 8/H <~1.0
-(6[H- 1)2 0.18
0 < ~/H <~0.22
-(a[H- 1)2 0.18
-0.5(a/H-
0.22)
103
103
o W/A =
act
6c- 4
(2)
By increasing the dead load, the mean joint pressure was increased from 0 to approximately 12 X 10 -2 kg/mm 2 and for some stipulated pressures in between, contact deformation values of the C.I.-Perspex pair furnished by different machining methods, as already mentioned, were evaluated.
174
o < a l H < 0.44
0.44 < 8/H <~1.0
~Ct + (~n I + (~n)
=
!
0.8(~5/H - 0.44)
-(SIH- 1)2 o - - = 1.51 e 0.14 + 0 . 3 0 2 ~ / H Y
(7 --=1.33e
1 6na=o.E1X
0 < a/H<<,0.44
-(5/H-1)2
(7 - - = 1.51 e Y
6n2
0 < a/H ~ 0.65
Y (7
=
(61H) 2 '~
a/H
r 17
~t
(I
0.122
T R I B O L O G Y October 1973
0.22 < 6/H <~1.0
These values were then plotted against the mean joint pressure o, see Fig 3. The curves have been constructed from the arithmetical average values of mean gap reduction on five similar test pieces. Theoretical relationships to calculate contact deformation as tabulated in Table 1 are represented graphically in Fig 4. In order to compare theoretical results with experimental data, Fig 3 is superimposed on Fig 4. Comparison reveals that actual contact deformation values coincide fairly closely with the theoretical values obtained from the equations of the bearing area curves obtained on the assumption of normal distribution of the heights of the crests. To investigate contact elasticity of slideways of various material pairs, the same set-up was used. Three material pairs namely, C.I.-C.I., C.I.-Perspex and C.I.-M.S. were chosen; one of the materials for the pair always being C.I. Tables 2, 3, 4 show the calculations of contact deformation and contact elasticity coefficients for the three pairs. These tables have been prepared from an analysis of the test
20.0
Ig'O
f 160
16"0
t40 v
"i
• o1 2 0
d¢ ,="
!
i 4
I .~
)
I0.0
o
o o
Pair :- C.I - Ptrspex t .Flhimhed by pOhhlng 2 - Flh~hed by grinding ) - Fhllsbnd by finish shaping 4 - F l , l S h l d by conventional scrclpang S - F i n l s h e d by miming
~.0
Po~ C.I. - P a r | p a x P - PolllhkHi G - Grktdl~l F S - F l f l i l h shaplnql ¢S - Conventional scraping M o Milling -Experimental - - - Theoretical
0 I'0
2.0
$0
4.0
SO
60
7"0
~10
9"0
~0
IPO
1.0
2.0
3.0
4.0
|.0
6.0
7.0
|0
a-O
I 0 " 0 I1.0
I~.0
Mean p r e n u r l , O ' , = I 0 " 2 ( k g l m = )
12"0
M e o n preslure,O', x I 0 - a ( k l l / m l 2 )
Fig 3 Showing relationship of 8 c with a for C.l.-Perspex pair with different surface preparation in the case of centralized loading
Fig 4 Showing comparison of theoretical and experimental results for C.l.-Perspex with different surface preparation
results. The experimental values of the reduction in the mean gap (St) are statistical values, determined from a small number of measurements (5) of a particular character. Comparability of obtained deformation values with the actual deformation values was tested by t-distribution and the calculations indicated that the deviations of population means do not exceed 9% of the mean value of deformations with a 0.95 probability. Joint contact deformation values for various pairs of mating surfaces are plotted against mean pressure in Fig 5.
The nature of the curve is the usual one for a centrally loaded joint and expresses the non-linear exponential relationship between 8 c and o. Table 5 shows the empirical formulae relating a with 8 c. These were derived by analysing the results on a double logarithmic grid using the leastsquares method. The table shows that the contact rigidity of a C.I.-Perspex pair is about three to six times less than that of a C.I.-C.I. and a C.I.-M.S. pair respectively. Variation of the coefficient of contact compliance with pressure has been graphically represented in Fig 6. This graph can be
Table 2
C.I.-C.I. pair
(kg/mm 2)
8t (micron)
/in 1 (rnlcron)
8n~ (rmcron)
4~c = [St - ( 8 . 1 + 8.2)] (micron)
8c (micron)
0.01 0.02 0.04 0.06 0.08 O.12 O.16
3.75 5.25 7.50 10.00 10.75 13.00 15.25
0.006 0.012 0.024 0.036 0.048 0.072 0.096
0.01 0.02 0.04 0.06 0.08 0.12 O. 16
3.734 5.218 7.436 9.904 10.622 12.808 14.994
0.933 1.304 1.859 2.476 2.655 3.202 3.720
o
C micron X mm 2 kg 93.3 65.2 46.5 41.3 33.2 26.7 23.4
TRIBOLOGY October 1973
175
Table 3
C.l.-Perspexpair
o (kg]mm 2)
6t (micron)
6n z (micron)
6n (micron)
4~c = [~t - (~n 1 + ~n2)] (micron)
~c (micron)
0.01 0.02 0.04 0.06 0.08 0.12 0.16
12.5 20.0 32.0 42.0 50.5 66.0 80.0
0.169 0.338 0.676 1.014 1.352 2.028 2.704
0.01 0.02 0.04 0.06 0.08 0.12 0.16
12.323 19.642 31.324 40.986 49.068 63.852 77.136
3.081 4.910 7.831 10.246 12.267 15.963 19.286
used for practical design calculations. Experiments to study the influence of lubrication on contact rigidity were performed and it was established that lubrication has negligible influence on contact rigidity.
Conclusions (i) With an increase in the accuracy of mating surfaces, contact elasticity coefficients decrease with a corresponding reduction in contact deformation. (ii) Theoretical relationships obtained on the basis of
C micron X mm 2 kg 308.1 245.5 195.8 170.8 153.3 133.0 120.5
Abbot's bearing area curves make it possible to reliably calculate joint stiffness of surfaces finished by different machining processes as the agreement of experimental data with theoretically calculated values is quite close. In the case of a polished surface the cumulative effect of elastic deformation of the crest of the irregularities, work-hardening of the material and the influence of flatness error results in a small variation. (iii) In the case of a centralized loading, the contact elasticity coefficient of the Perspex-C.I. pair is about three times higher than the C.I.-C.I. pair, whereas the contact
32
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Fig 5 Showing relationship of ~c with o in the case of centralized loading
176
TRIBOLOGY October 1973
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Fig 6 Showing relationship of C with o in the case of centralized loading
Table 4
C.I.-M.S. pair
C 5nl (micron)
8n~ (rmcron)
48c = [/it - (8nl +/i.2)]
/ic
(kg/mm 2)
/it (micron)
(micron)
(micron)
0.01 0.02 0.04 0.06 0.08 0.12 0.16
2.00 3.00 4.75 6.00 7.00 8.75 10.50
0.003 0.006 0.012 0.018 0.024 0.036 0.048
0.01 0.02 0.04 0.06 0.08 0.12 0.16
1.987 2.974 4.698 5.922 6.896 8.594 10.292
0.497 0.743 1.174 1.480 1.724 2.148 2.573
o
Table 5
Empirical formulae for contact deformation
Pair
Empirical formulae
C.I.-M.S.
/ic = 0.497 o 0.58
C.I.-Perspex
/ic = 3.081 00.66
C.I.-C.I.
/ic = 0.99
o 0"5
micron X mm 2 kg 49.7 37.1 29.4 24.7 21.5 17.9 16.1
elasticity of the C.I.-M.S. pair is half the value of the C.I.C.I. pair. Contact rigidity of a pair remains practicaUy unaffected even with the introduction of a lubricant.
References I Levina, Z.M. 'Calculations of contact deformations in siideways', Machines and Tooling, No 1 (1966) pp 8-17. 2 Ostrovskii, V. I. 'The influence of machining methods on contact stiffness', Machines and Tooling, Vol 36, No 1 (1965). 3 Lapidus, A.S. and Maiorova, E. A. 'Nylon facing strips for machine tool slideway', Machines and Tooling, No 9 (1965).
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