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Experimental results in intermittent motion
EXPERIMENTAL RESULTS IN INTERMITTENT MOTION BY R. FRISCH-FAY 1 SUMMARY
A thin, elastic bar is forced to deflect between two tixed supports. Nonlinear deformations are utilized in finding the critical load at which the bar becomes unstable. The critical load is a function of the frictional resistance between the bar and the supports, hence by measuring the force the friction coefficient may be calculated. 1. INTRODUCTION
T h e non-continuous m o v e m e n t of sliding surfaces is called interm i t t e n t m o t i o n ; it is also referred to as "stick-slip." This u n e v e n m o v e m e n t occurs when one of the sliding surfaces has some elastic freedom. T h e static friction causes a 'stick'; when this resistance is broken between the surfaces a slip will occur during which the kinetic friction governs the m o v e m e n t . This #k will slow down the motion and at a certain velocity the surfaces will stick again; after t h a t the cycles are repeating. It has been shown (1) °- t h a t (F/W)mi;,~
:
2ldk - -
(1)
]'~s
where F is the frictional force and W is the load on the sliding surfaces. T h e kinetic and static friction coefficients are u~ and t~,rrespectively, (u, >
Fro. 1. Flexible bar oil fixed supports. The load ill the upper and in the lower positions is identical. The center position shows the shape of the bar when it is about to slip under P ...... 1 School of Civil Engineering, University of New South \\'ales, Sydney, Australia. The boldface numbers in parentheses refer to the references appended to this paper. 283
284
R. FRISCH-FAY
[J. F. I.
In order to find the frictional force corresponding to the weight let us assume that a thin, elastic bar is forced to deflect between two fixed supports (Fig. 1). The distance between the inside faces of the supports is 21 and the total load of the bar and of the applied weight is 2P. Previous investigation (2) has shown that the relationship between the load P and the end slope of the bar, % is expressed by 1
1 = ~ [-sin (7 - X)q~(p, n) + 2p cos (~, -- X) cos n~,
(2)
where k = [ E I cos(~, -
X) ]1[2
El' E1 = flexural rigidity - 1 - v2 p = elliptic modulus = sin (¼7r + ½X) n = amplitude of elliptic integrals [cos(3'-x) --sin(3'-x) = sin_ ~ 2 2 cos ½X + sin ½X X = tan -1 #, angle of friction between bar and supports ¢(p, n) = F(p, n) - K(p) + 2E(p) - 2E(p, n), in which F, K, and E are Legendre's complete and incomplete elliptic integrals, respectively. Work by Sonntag (3) and Gospodnetic (4) has shown that, generally, for every P there are two possible shapes of the bar capable of balancing the load. This is illustrated in Fig. 1 where the upper and P
l
2P FIG. 2.
Equilibrium at the contact point.
Apr., I962.]
EXPERIMENTS
IN
INTERMITTENT
MOTION
28 5
lower positions of the bar are due to the same load. Of the two, however, only the upper shape is stable. An increasing load in that range leads to increasing deflection until P reaches a maximum. Past this stage the bar will balance diminishing loads only. Figure 2 shows the equilibrium of the forces at the contact point. 2. RELATIONSHIP BETWEEN CRITICAL LOAD AND X
Expressing P from Eq. 2, we obtain EI EI P = - ~ £ 0(17, X ) = 1 2 cos (-r-X)[sin (~,-X)¢(p, n ) + 2 p cos ('y-X) cos n] ~.
The function P Y / E I = 0 (3', X) has been presented graphically for different values of X. The dotted line connecting the peak values of the separate curves gives the relationship between P,~o~12/EI and the angle of friction. The lowest curve (X = 0 °) refers to frictionless supports and is in agreement with Gospodnetic's finding (4). As seen in Fig. 3 the function 0(~', X) reaches a maximum for a
pza El 3"6 3.0 2"4
/ ±/
1'8
,,-7----1.2
0.6
Oo
15 ° FIG. 3.
:30 °
45 °
60 °
75 °
90 °
G r a p h s of Pl2/E[ = ®(% X) for X = 0 °, 5 °, 10 °, etc.
g
286
R.
FRISCH-FAY
[J. F. I.
certain value of % For values of 0 ° < X < 40 ° the graph Pma,l~-/EI= = F(X) is shown in Fig. 4. F(X) = 6)(% X) ..... and hence Pma, = BEI,/l 2.
p£z El 1.28
1"20
/
1"12
/
I.Oz
1"91~
0"88
/
0"8%0 FIG. 4.
I
0 o
20 °
Relationship between/3 =
30 °
4 0 ° ,/N,,
P~=I~/EI a n d X for 0 ° < X < 40 °.
If P , , , , can be found experimentally, /3 becomes known and X m a y be calculated from X = 626.796 -- 4(497193.635 -- 125010.8~).
(4)
This equation gives X in degrees and is valid for 0.8345 < / ~ < 1.200. For/3 > 1.200, the values of X m a y be obtained from Table I. H a v i n g established, theoretically at least, the relationship t h a t exists between the critical load P~o= and the angle of friction X, the simple a p p a r a t u s shown in Fig. 1 will enable us to investigate the influence of the viscosity of oils on i n t e r m i t t e n t motion, also, the influence of t h e hardness of the sliding metals.
Apr., 1962.]
EXPERIMENTSIN INTERMITTENT MOTION
287
T~SI,E I. X = 0° fl = 0.8345
5° 0.8848
10 ° 0.9347
20 ° 1.046
30 ° 1.174
40 ° fl = 1.33
50 ° 1.532
60 ° 1.821
70 ° 2.377
3.349
X =
3. E X P E R I M E N T A L
80 °
RESULTS
In order to investigate the frictional resistance between different metals the brackets in Figs. 1 and 2 have interchangeable top parts. Cast iron, axle steel, mild steel, brass, alunlinunl, and copper were used ; the parts were given an M3 surface finish and then polished with enlery cloth. T h e c o n t a c t edge was rounded down to a radius of ~ in. T h e Brinell hardness n u m b e r s were 247, 182, 140, 112, 56 and 42 for cast iron, axle steel, mild steel, brass, a l u m i n u m , and copper, respectively. T h e bar used in the experiment was a high tensile spring steel, d r a w n and h e a t treated, with a cross section of 1.002 by 0.048 in. ; it showed a hardness of 37 on the Rockwell C scale which corresponds to Brinell 345. Based on the linear deflection t h e o r y the observed flexural rigidity was E 1 = 281 l b i n s. T h e limit of proportionality was in the vicinity of 90,000 lb/in 2, well above the actual stresses to which the bar was subjected during loading. Like the supports, the bar was also polished with fine e m e r y cloth. T h e load was applied to the axis of a wheel equipped with ball bearings to m a k e sure t h a t the load acted always on the lowest point of the bar; this assured the central position of the force. T h e load consisted of the bar itself, the wheel, and a container for shot. It was noticed during tests t h a t the load-deflection relationship was not a continuous function; for 3, > 30 ° the unlubricated bar sagged piecemeal for e v e r y 4 grams of additional load. Past a certain load, this "stick-slip" process ceased and the bar slipped t h r o u g h completely. W h e n P reached Per the slip t h a t followed the breaking down of the static friction did not result in a n o t h e r "stick" because the angle reached the critical value during the slip and the resulting vertical reaction was insufficient to balance the load; hence it slipped t h r o u g h between the supports. To increase the sensitivity of the bar m o v e m e n t a buzzer was fixed on the board to which the brackets were a t t a c h e d and this buzzer vibrated the board for a p p r o x i m a t e l y 0.1 sec. when P was nearing P .... This w a y the static friction was destroyed, slipping began, and, for P > Pot, the sliding could not be arrested and the bar fell through. Tests have also shown t h a t the bar, u n d e r increasing load, does not sag s y m m e t r i c a l l y with respect to the supports. On the other hand, the difference between the overhanging parts when the bar was about to slip has never been more t h a n 3 in. and frequently less t h a n
288
R. FR~SCl[-FAY
[J. F. I.
2 in. T h e e x t r e m e c a s e g i v e s a c a l c u l a t e d e r r o r of a p p r o x i m a t e l y ± 1.0 p e r c e n t in t h e v e r t i c a l c o m p o n e n t o f t h e r e a c t i o n . The minimum friction coefficient for unlubricated intermittent m o t i o n , /Zo = 2/~k - #~, is t a b u l a t e d in T a b l e I I f o r s l i d i n g of d i f f e r e n t "l'Am,E l l . - - F r i c t i o n of Metals on Steel (unlubricated). 1',..... lb. fl x u,, = 2uk - #, Mild steel Axle steel C a s t iron Brass Alunfinum Copper
2.628 2.578 2.587 2.672 2.705 2.698
0.9474 0.9294 0.9326 0.9633 0.9752 0.9727
11°22' 9032 ' 9052 ' 12°59 ' 14o12 ' 13 °56'
0.201 0.170 0.174 0.230 0.253 0.248
m e t a l s o n s p r i n g steel. /3 = P,,a~12/EI is b a s e d o n a d i s t a n c e 2l = 2 0 . 1 3 in. b e t w e e n t h e s u p p o r t s . The theoretical considerations are based on a knife-edge support which for practical purposes has been replaced
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
TABLE I II.--Friction between Steel and Steel. P ...... lb. ~ X 2.552 2.574 2.490 2.598 2.608 2.576 2.482
0.9200 0.9279 0.8977 0.9366 0.9402 0.9287 0.8948
8°35 ' 9o23 ' 6 °20' 10o16 ' 10o38 ' 9o28 ' 603 '
TABLE IV.--Friction between Steel and Brass. P ...... lb. ~ x 2.585 2.567 2.496 2.634 2.669 2.652 2.506
0.9319 0.9254 0.8998 0.9496 0.9622 0.9561 0.9031
9048 ' 908 ' 6033 ' 11 °35' 12%2' 12014 ' 6053 '
TABLE V.--Friction between Steel and Aluminum. P ...... lb. /~ x 2.679 2.584 2.566 2.689 2.622 2.636 2.560
0.9658 0.9315 0.9251 0.9694 0.9453 0.9503 0.9229
13014 ' 9045 ' 906 ' 13 °36' 11 °8' 11 °40' 8o53 '
uo 0.151 0.165 0.111 0.181 0.188 0.167 0.106
uo 0.173 0.161 0.115 0.205 0.228 0.217 0.121
~o
0.235 0.172 0.160 0.242 0.197 0.206 0.156
Apr., 1962.]
EXPERIMENTS IN INTERMITTENT MOTION
28 9
by a contact edge having a radius of ~ in. ttowever, the span 2! is calculated for a bar having an end slope of ~, = 40 ° so t h a t 20.13 in. is very close to the theoretical span when the bar is about to slip. To assess the effect of lubricants a few drops of oil were applied on the contact edges of the supports and also on the bar. The following oils have been tested at room t e m p e r a t u r e (68-70 F.) : (i) Blend of high viscosity index, solvent extracted, bright stock and m e d i u m viscosity index (paraffinic) oil having 1080 sec. Redwood 1 viscosity at 140 F. (ii) High viscosity index, solvent extracted, bright stock of 850 sec. Redwood 1 viscosity at 140 F. (iii) S.A.E. 140 viscosity hypoid gear lubricant containing a sulfurchlorine-phosphorus type of "extreme pressure" additive. (iv) Heavy d u t y petrol crankcase oil, S.A.E. 10 rating, solvent extracted, contains anti-oxidant additives. (v) As (iv) b u t S.A.E. 30 rating. (vi) As (iv) b u t S.A.E. 50 rating. (vii) C o m p o u n d e d gear lubricant based on mineral oil and neatsfoot oil together with anti-oxidant, anti-corrosion, oiliness, and antifoam additives; 191 centistoke viscosity at 140 F. W h e n changing to a new type of oil in the experiments the surfaces were cleaned with carbontetrachloride. T h e friction coefficient ~o for lubricated metals is calculated in Tables III, IV, and V for different metals. 4. DISCUSSION OP RESULTS
T h e underlying principles in this experiment are different from those t h a t apply to the Bowden-Leben a p p a r a t u s (1, p. 74). In the latter experiment the conditions after the breaking down of the "stick" and subsequent "slip" were the same as before, in other words, the stickslip process went on repeating itself at regular time intervals. This is not so in the present discussion. T h e buzzer breaks down the "stick" b u t thereafter, because of the nonlinear relationship between the frictional force F and the load P, no "stick" will follow the ensuing "slip." T h u s the tabulated values refer to (F/W)m~ = 2~k -- ~. This explains the unusually low results for unlubrlcated surfaces. T h e results show the expected correlation between friction and hardness. The Brinell hardness numbers and the corresponding/z, for unlubricated metals are plotted in Fig. 5. The points lie approximately on an S-shaped line and it is felt t h a t for hardness numbers within the 50-250 range the values of/~o will roughly follow the given curve provided/~o is measured under similar conditions. Since the applied load is 2P, it appears from the tables t h a t the critical loads were in the 5.00-5.400-1b. range. Each test has been
290
R . FRISCH-FAY
[J. V. I.
0'26 0'24" 0'22 0"20 0'18 0'16 0"14 o12 O
l
I
25
50
I
75
I
IOO
I
125
I
150
Hardness
I
175
I
I
I
200
225
250
(Brin¢ll)
FIG. 5. #o plotted against hardness.
carried out five times for unlubricated metals and three times when lubricants were used ; the load was weighed to the nearest gram. Pm°x was t a k e n as half of the a r i t h m e t i c m e a n of the m e a s u r e m e n t s and it is n o t e w o r t h y t h a t for every set of readings the deviations from the average were less t h a n 4-10 grams for the lubricated sliding and less t h a n + 12 grams for unlubricated metals. Based on an average load of 5.2 lb. this represents an error of 4-0.4 per cent, t h a t is, less t h a n half the error due to uneven sagging. It m a y be assumed, therefore, t h a t part of the error in -Pm,~ is due to the lack of s y m m e t r y of the slipping bar. T h e u n e v e n sagging is, of course, the direct result of the irregularities of the surface of the spring steel. Allowing for an additional 4-0.2 per cent error in the flexural rigidity and the square of the span, Eq. 4 is acc u r a t e within AX = ~ 0.54 ° . This is based on A/3 = 4- 0.6 per cent and/3 = 0.93. T h e c o n t a c t stresses between the bar (E = 28.9 X l0 Glb./in2.) and the various supports can be obtained by applying the formulae of H e r t z (5). T a k i n g an average end slope ")'av = 42 ° (Fig. 3), the maxim u m stresses in bearing are between 12,000 lb./in 2. and 16,900 lb./in 2. Hence the test can be carried out for c o m p a r a t i v e l y high bearing stresses y e t requiring m o d e r a t e loads only.
Acknowledgment T h e a u t h o r is indebted to Prof. C. H. Munro, H e a d of School of Civil Engineering, T h e University of N e w South Wales, for his assistance during the preparation of this paper. Also, the cooperation of the Shell C o m p a n y of Australia Limited is gratefully acknowledged.
Apr., I962.]
EXPERIMENTS IN INTERMITTENT 1VIOTION
291
REPERENCES
(1) F. P. BOWm~.NAnD D. TABOR, "The Friction and Lubrication of Solids," Oxford, Clarendon Press, 1950, p. 106. (2) R. FRISCH-FAv, "Particular Cases of Large Deflections," Aust. J. Appl. Sci., Vol. 11, pp. 443-450 (1960). (3) R. SONNTAG, "Der beiderseits gestuezte, symmetrisch belastete gerade Stab," Ingen.Arch., Vol. 12, pp. 283-306 (1942). (4) D. GOSPODNETIC, "Deflection Curve of a Simply Supported Beam," J. Appl. l~[ech., Vol. 26; Trans. ASME, Vol. 81, Series E, pp. 675-676 (1959). (5) S. TIMOSHENKO, "Strength of Materials," Vol. 2, New York, D. Van Nostrand Co., 1956, p. 342.
A thin, elastic bar is forced to deflect between two tixed supports. Nonlinear deformations are utilized in finding the critical load at which the bar becomes unstable. The critical load is a function of the frictional resistance between the bar and the supports, hence by measuring the force the friction coefficient may be calculated. 1. INTRODUCTION
T h e non-continuous m o v e m e n t of sliding surfaces is called interm i t t e n t m o t i o n ; it is also referred to as "stick-slip." This u n e v e n m o v e m e n t occurs when one of the sliding surfaces has some elastic freedom. T h e static friction causes a 'stick'; when this resistance is broken between the surfaces a slip will occur during which the kinetic friction governs the m o v e m e n t . This #k will slow down the motion and at a certain velocity the surfaces will stick again; after t h a t the cycles are repeating. It has been shown (1) °- t h a t (F/W)mi;,~
:
2ldk - -
(1)
]'~s
where F is the frictional force and W is the load on the sliding surfaces. T h e kinetic and static friction coefficients are u~ and t~,rrespectively, (u, >
Fro. 1. Flexible bar oil fixed supports. The load ill the upper and in the lower positions is identical. The center position shows the shape of the bar when it is about to slip under P ...... 1 School of Civil Engineering, University of New South \\'ales, Sydney, Australia. The boldface numbers in parentheses refer to the references appended to this paper. 283
284
R. FRISCH-FAY
[J. F. I.
In order to find the frictional force corresponding to the weight let us assume that a thin, elastic bar is forced to deflect between two fixed supports (Fig. 1). The distance between the inside faces of the supports is 21 and the total load of the bar and of the applied weight is 2P. Previous investigation (2) has shown that the relationship between the load P and the end slope of the bar, % is expressed by 1
1 = ~ [-sin (7 - X)q~(p, n) + 2p cos (~, -- X) cos n~,
(2)
where k = [ E I cos(~, -
X) ]1[2
El' E1 = flexural rigidity - 1 - v2 p = elliptic modulus = sin (¼7r + ½X) n = amplitude of elliptic integrals [cos(3'-x) --sin(3'-x) = sin_ ~ 2 2 cos ½X + sin ½X X = tan -1 #, angle of friction between bar and supports ¢(p, n) = F(p, n) - K(p) + 2E(p) - 2E(p, n), in which F, K, and E are Legendre's complete and incomplete elliptic integrals, respectively. Work by Sonntag (3) and Gospodnetic (4) has shown that, generally, for every P there are two possible shapes of the bar capable of balancing the load. This is illustrated in Fig. 1 where the upper and P
l
2P FIG. 2.
Equilibrium at the contact point.
Apr., I962.]
EXPERIMENTS
IN
INTERMITTENT
MOTION
28 5
lower positions of the bar are due to the same load. Of the two, however, only the upper shape is stable. An increasing load in that range leads to increasing deflection until P reaches a maximum. Past this stage the bar will balance diminishing loads only. Figure 2 shows the equilibrium of the forces at the contact point. 2. RELATIONSHIP BETWEEN CRITICAL LOAD AND X
Expressing P from Eq. 2, we obtain EI EI P = - ~ £ 0(17, X ) = 1 2 cos (-r-X)[sin (~,-X)¢(p, n ) + 2 p cos ('y-X) cos n] ~.
The function P Y / E I = 0 (3', X) has been presented graphically for different values of X. The dotted line connecting the peak values of the separate curves gives the relationship between P,~o~12/EI and the angle of friction. The lowest curve (X = 0 °) refers to frictionless supports and is in agreement with Gospodnetic's finding (4). As seen in Fig. 3 the function 0(~', X) reaches a maximum for a
pza El 3"6 3.0 2"4
/ ±/
1'8
,,-7----1.2
0.6
Oo
15 ° FIG. 3.
:30 °
45 °
60 °
75 °
90 °
G r a p h s of Pl2/E[ = ®(% X) for X = 0 °, 5 °, 10 °, etc.
g
286
R.
FRISCH-FAY
[J. F. I.
certain value of % For values of 0 ° < X < 40 ° the graph Pma,l~-/EI= = F(X) is shown in Fig. 4. F(X) = 6)(% X) ..... and hence Pma, = BEI,/l 2.
p£z El 1.28
1"20
/
1"12
/
I.Oz
1"91~
0"88
/
0"8%0 FIG. 4.
I
0 o
20 °
Relationship between/3 =
30 °
4 0 ° ,/N,,
P~=I~/EI a n d X for 0 ° < X < 40 °.
If P , , , , can be found experimentally, /3 becomes known and X m a y be calculated from X = 626.796 -- 4(497193.635 -- 125010.8~).
(4)
This equation gives X in degrees and is valid for 0.8345 < / ~ < 1.200. For/3 > 1.200, the values of X m a y be obtained from Table I. H a v i n g established, theoretically at least, the relationship t h a t exists between the critical load P~o= and the angle of friction X, the simple a p p a r a t u s shown in Fig. 1 will enable us to investigate the influence of the viscosity of oils on i n t e r m i t t e n t motion, also, the influence of t h e hardness of the sliding metals.
Apr., 1962.]
EXPERIMENTSIN INTERMITTENT MOTION
287
T~SI,E I. X = 0° fl = 0.8345
5° 0.8848
10 ° 0.9347
20 ° 1.046
30 ° 1.174
40 ° fl = 1.33
50 ° 1.532
60 ° 1.821
70 ° 2.377
3.349
X =
3. E X P E R I M E N T A L
80 °
RESULTS
In order to investigate the frictional resistance between different metals the brackets in Figs. 1 and 2 have interchangeable top parts. Cast iron, axle steel, mild steel, brass, alunlinunl, and copper were used ; the parts were given an M3 surface finish and then polished with enlery cloth. T h e c o n t a c t edge was rounded down to a radius of ~ in. T h e Brinell hardness n u m b e r s were 247, 182, 140, 112, 56 and 42 for cast iron, axle steel, mild steel, brass, a l u m i n u m , and copper, respectively. T h e bar used in the experiment was a high tensile spring steel, d r a w n and h e a t treated, with a cross section of 1.002 by 0.048 in. ; it showed a hardness of 37 on the Rockwell C scale which corresponds to Brinell 345. Based on the linear deflection t h e o r y the observed flexural rigidity was E 1 = 281 l b i n s. T h e limit of proportionality was in the vicinity of 90,000 lb/in 2, well above the actual stresses to which the bar was subjected during loading. Like the supports, the bar was also polished with fine e m e r y cloth. T h e load was applied to the axis of a wheel equipped with ball bearings to m a k e sure t h a t the load acted always on the lowest point of the bar; this assured the central position of the force. T h e load consisted of the bar itself, the wheel, and a container for shot. It was noticed during tests t h a t the load-deflection relationship was not a continuous function; for 3, > 30 ° the unlubricated bar sagged piecemeal for e v e r y 4 grams of additional load. Past a certain load, this "stick-slip" process ceased and the bar slipped t h r o u g h completely. W h e n P reached Per the slip t h a t followed the breaking down of the static friction did not result in a n o t h e r "stick" because the angle reached the critical value during the slip and the resulting vertical reaction was insufficient to balance the load; hence it slipped t h r o u g h between the supports. To increase the sensitivity of the bar m o v e m e n t a buzzer was fixed on the board to which the brackets were a t t a c h e d and this buzzer vibrated the board for a p p r o x i m a t e l y 0.1 sec. when P was nearing P .... This w a y the static friction was destroyed, slipping began, and, for P > Pot, the sliding could not be arrested and the bar fell through. Tests have also shown t h a t the bar, u n d e r increasing load, does not sag s y m m e t r i c a l l y with respect to the supports. On the other hand, the difference between the overhanging parts when the bar was about to slip has never been more t h a n 3 in. and frequently less t h a n
288
R. FR~SCl[-FAY
[J. F. I.
2 in. T h e e x t r e m e c a s e g i v e s a c a l c u l a t e d e r r o r of a p p r o x i m a t e l y ± 1.0 p e r c e n t in t h e v e r t i c a l c o m p o n e n t o f t h e r e a c t i o n . The minimum friction coefficient for unlubricated intermittent m o t i o n , /Zo = 2/~k - #~, is t a b u l a t e d in T a b l e I I f o r s l i d i n g of d i f f e r e n t "l'Am,E l l . - - F r i c t i o n of Metals on Steel (unlubricated). 1',..... lb. fl x u,, = 2uk - #, Mild steel Axle steel C a s t iron Brass Alunfinum Copper
2.628 2.578 2.587 2.672 2.705 2.698
0.9474 0.9294 0.9326 0.9633 0.9752 0.9727
11°22' 9032 ' 9052 ' 12°59 ' 14o12 ' 13 °56'
0.201 0.170 0.174 0.230 0.253 0.248
m e t a l s o n s p r i n g steel. /3 = P,,a~12/EI is b a s e d o n a d i s t a n c e 2l = 2 0 . 1 3 in. b e t w e e n t h e s u p p o r t s . The theoretical considerations are based on a knife-edge support which for practical purposes has been replaced
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
Oil (i) (ii) (iii) (iv) (v) (vi) (vii)
TABLE I II.--Friction between Steel and Steel. P ...... lb. ~ X 2.552 2.574 2.490 2.598 2.608 2.576 2.482
0.9200 0.9279 0.8977 0.9366 0.9402 0.9287 0.8948
8°35 ' 9o23 ' 6 °20' 10o16 ' 10o38 ' 9o28 ' 603 '
TABLE IV.--Friction between Steel and Brass. P ...... lb. ~ x 2.585 2.567 2.496 2.634 2.669 2.652 2.506
0.9319 0.9254 0.8998 0.9496 0.9622 0.9561 0.9031
9048 ' 908 ' 6033 ' 11 °35' 12%2' 12014 ' 6053 '
TABLE V.--Friction between Steel and Aluminum. P ...... lb. /~ x 2.679 2.584 2.566 2.689 2.622 2.636 2.560
0.9658 0.9315 0.9251 0.9694 0.9453 0.9503 0.9229
13014 ' 9045 ' 906 ' 13 °36' 11 °8' 11 °40' 8o53 '
uo 0.151 0.165 0.111 0.181 0.188 0.167 0.106
uo 0.173 0.161 0.115 0.205 0.228 0.217 0.121
~o
0.235 0.172 0.160 0.242 0.197 0.206 0.156
Apr., 1962.]
EXPERIMENTS IN INTERMITTENT MOTION
28 9
by a contact edge having a radius of ~ in. ttowever, the span 2! is calculated for a bar having an end slope of ~, = 40 ° so t h a t 20.13 in. is very close to the theoretical span when the bar is about to slip. To assess the effect of lubricants a few drops of oil were applied on the contact edges of the supports and also on the bar. The following oils have been tested at room t e m p e r a t u r e (68-70 F.) : (i) Blend of high viscosity index, solvent extracted, bright stock and m e d i u m viscosity index (paraffinic) oil having 1080 sec. Redwood 1 viscosity at 140 F. (ii) High viscosity index, solvent extracted, bright stock of 850 sec. Redwood 1 viscosity at 140 F. (iii) S.A.E. 140 viscosity hypoid gear lubricant containing a sulfurchlorine-phosphorus type of "extreme pressure" additive. (iv) Heavy d u t y petrol crankcase oil, S.A.E. 10 rating, solvent extracted, contains anti-oxidant additives. (v) As (iv) b u t S.A.E. 30 rating. (vi) As (iv) b u t S.A.E. 50 rating. (vii) C o m p o u n d e d gear lubricant based on mineral oil and neatsfoot oil together with anti-oxidant, anti-corrosion, oiliness, and antifoam additives; 191 centistoke viscosity at 140 F. W h e n changing to a new type of oil in the experiments the surfaces were cleaned with carbontetrachloride. T h e friction coefficient ~o for lubricated metals is calculated in Tables III, IV, and V for different metals. 4. DISCUSSION OP RESULTS
T h e underlying principles in this experiment are different from those t h a t apply to the Bowden-Leben a p p a r a t u s (1, p. 74). In the latter experiment the conditions after the breaking down of the "stick" and subsequent "slip" were the same as before, in other words, the stickslip process went on repeating itself at regular time intervals. This is not so in the present discussion. T h e buzzer breaks down the "stick" b u t thereafter, because of the nonlinear relationship between the frictional force F and the load P, no "stick" will follow the ensuing "slip." T h u s the tabulated values refer to (F/W)m~ = 2~k -- ~. This explains the unusually low results for unlubrlcated surfaces. T h e results show the expected correlation between friction and hardness. The Brinell hardness numbers and the corresponding/z, for unlubricated metals are plotted in Fig. 5. The points lie approximately on an S-shaped line and it is felt t h a t for hardness numbers within the 50-250 range the values of/~o will roughly follow the given curve provided/~o is measured under similar conditions. Since the applied load is 2P, it appears from the tables t h a t the critical loads were in the 5.00-5.400-1b. range. Each test has been
290
R . FRISCH-FAY
[J. V. I.
0'26 0'24" 0'22 0"20 0'18 0'16 0"14 o12 O
l
I
25
50
I
75
I
IOO
I
125
I
150
Hardness
I
175
I
I
I
200
225
250
(Brin¢ll)
FIG. 5. #o plotted against hardness.
carried out five times for unlubricated metals and three times when lubricants were used ; the load was weighed to the nearest gram. Pm°x was t a k e n as half of the a r i t h m e t i c m e a n of the m e a s u r e m e n t s and it is n o t e w o r t h y t h a t for every set of readings the deviations from the average were less t h a n 4-10 grams for the lubricated sliding and less t h a n + 12 grams for unlubricated metals. Based on an average load of 5.2 lb. this represents an error of 4-0.4 per cent, t h a t is, less t h a n half the error due to uneven sagging. It m a y be assumed, therefore, t h a t part of the error in -Pm,~ is due to the lack of s y m m e t r y of the slipping bar. T h e u n e v e n sagging is, of course, the direct result of the irregularities of the surface of the spring steel. Allowing for an additional 4-0.2 per cent error in the flexural rigidity and the square of the span, Eq. 4 is acc u r a t e within AX = ~ 0.54 ° . This is based on A/3 = 4- 0.6 per cent and/3 = 0.93. T h e c o n t a c t stresses between the bar (E = 28.9 X l0 Glb./in2.) and the various supports can be obtained by applying the formulae of H e r t z (5). T a k i n g an average end slope ")'av = 42 ° (Fig. 3), the maxim u m stresses in bearing are between 12,000 lb./in 2. and 16,900 lb./in 2. Hence the test can be carried out for c o m p a r a t i v e l y high bearing stresses y e t requiring m o d e r a t e loads only.
Acknowledgment T h e a u t h o r is indebted to Prof. C. H. Munro, H e a d of School of Civil Engineering, T h e University of N e w South Wales, for his assistance during the preparation of this paper. Also, the cooperation of the Shell C o m p a n y of Australia Limited is gratefully acknowledged.
Apr., I962.]
EXPERIMENTS IN INTERMITTENT 1VIOTION
291
REPERENCES
(1) F. P. BOWm~.NAnD D. TABOR, "The Friction and Lubrication of Solids," Oxford, Clarendon Press, 1950, p. 106. (2) R. FRISCH-FAv, "Particular Cases of Large Deflections," Aust. J. Appl. Sci., Vol. 11, pp. 443-450 (1960). (3) R. SONNTAG, "Der beiderseits gestuezte, symmetrisch belastete gerade Stab," Ingen.Arch., Vol. 12, pp. 283-306 (1942). (4) D. GOSPODNETIC, "Deflection Curve of a Simply Supported Beam," J. Appl. l~[ech., Vol. 26; Trans. ASME, Vol. 81, Series E, pp. 675-676 (1959). (5) S. TIMOSHENKO, "Strength of Materials," Vol. 2, New York, D. Van Nostrand Co., 1956, p. 342.