The effects of friction-induced vibration on friction and wear

The effects of friction-induced vibration on friction and wear

307 Wear, 80 (1982) 307 - 320 THE EFFECTS OF FRICTION-INDUCED AND WEAR VIBRATION ON FRICTION K. KATO, A. IWABUCHI and T. KAYABA Department of Mec...

943KB Sizes 2 Downloads 49 Views

307

Wear, 80 (1982) 307 - 320

THE EFFECTS OF FRICTION-INDUCED AND WEAR

VIBRATION

ON FRICTION

K. KATO, A. IWABUCHI and T. KAYABA Department of Mechanical Sendai 980 (Japan)

Engineering,

(Received November 12,198l;

Faculty

of Engineering,

Tohoku

University,

in revised form January 5, 1982)

Summary Friction tests were carried out with an elastic system where the specimen was supported by the elastic plate spring with strain gauges and with a rigid system where the vibration of the specimen was restricted. The coefficient of friction was obtained by two different methods: a deflection method with an assembly that consisted of a spring and strain gauges and an inertia method with a flywheel. The friction and wear properties of steelsteel, white metal-steel, phosphor bronze-steel and bronze-steel pairs were examined. When the frictional operation was conducted in the elastic system accompanied by frictional vibration, the coefficient of friction measured by the deflection method was generally larger than by the inertia method. The relationships between the coefficient of friction and the driving velocity for steel-steel and white metal-steel pairs obtained by the inertia method in the rigid system were significantly different from those obtained by the same method in the elastic system. Frictional vibration increased the wear of white metal-steel, phosphor bronze-steel and bronze-steel pairs but decreased that of a steel-steel pair. Wear measurements from the elastic system exhibited more scatter than from the rigid system.

1. Introduction In the study of friction, the frictional force is usually measured by deflection of the specimen support in the elastic system. The strain gauge assembly is well established and the principle is popular. The Bowden-Leven apparatus is the most typical and classical example of this type. The friction force-time (or sliding distance) relationship in the elastic system is generally recorded as a vibrational process. The representative kinetic friction force is considered to be given by the value at the middle line of the oscillatory friction curve where the inertia force of the system is zero when the friction force is constant in the vibrational process. The kinetic friction force is generally plotted against the 0043-1648/82/0000-0000/$02.75

@ Elsevier Sequoia/Printed in The Netherlands

308

driving velocity in the friction force-velocity relationship. However, with respect to such measurements, two important questions arise. (1) How does such a kinetic friction force correlate with the mean value of the friction force in the total friction process? (2) How does the friction force-velocity relationship change when the frictional vibration is sufficiently restricted? Simpson et al. [l] and Ko and Brockley [2] showed that the frictional resistance was not reversible in the processes of velocity increase and velocity decrease. Rabinowicz [ 31 also showed. the existence of many types of frictional force-velocity relationship. The value of the middle line of the oscillatory curve of the friction force, therefore, cannot always give the average value of the friction force in the unit cycle; this provides the first question. The real sliding velocity at the interface is different from the apparent driving velocity, especially in the lower velocity range where the stick-slip motion is prominent, as analysed by Derjaguins et al. [ 41. Thus the driving velocity must be modified to obtain the real sliding velocity at the interface, although this is very difficult in practice. More complicated effects of vibration on friction can be expected for the heat history, oxidation history, impact action and the presence of wear debris. Because of this, frictional tests with no vibration in a rigid system are needed to check the accuracy of the friction force-velocity relationship in the elastic system. For this purpose, an inertia method involving a flywheel was developed for friction measurements. The same principle also applies to wear. For example, Miyagawa et al. [ 51 showed that frictional vibration increased wear. With the above points in mind, friction and wear tests were conducted in two systems, the elastic system and the rigid system, and the effects of frictional vibration on friction and wear were examined.

2. Experimental

apparatus

and procedure

Figure 1 shows schematic diagrams of the experimental apparatus (1, upper stationary pin specimen and lower driving cylindrical specimen; 2, flywheel; 3, bearings; 4, velocity detector; 5, coupling for cutting off the driving source; 6, driving motor; 7, normal load; 8, specimen holder; 9, plate spring with strain gauges). Figure l(a) shows the set-up of the test in the rigid system where the pin specimen holder can slide in the long vertical bore and the stiffness of the support is sufficient to suppress frictional vibration. For the elastic system test, the system of Fig. l(b) was used where the pin specimen was located at the end of the plate spring which had a stiffness of 27800 N m-l. In both cases a load of 10.8 N was applied through the thin beam (stiffness, 49.0 N m-l) in order to arrest the impact-type loading that resulted from the vertical jumping of the pin specimen. Figures 2(a) and 2(b) show the pin and the cylindrical specimens respectively. The materials of the upper stationary pin specimen are 1.0% C

309

(b)

(4 Fig. 1. Friction apparatus.

(a)

(b)

Fig. 2. Specimen shapes and dimensions.

steel, white metal, phosphor bronze and bronze. The material of the driving cylindrical specimen is 1.0% C steel. The surface of each specimen was abraded with grade 0 emery paper and cleaned with acetone just before the test. The whirl of the rotational centre of the cylindrical specimen was less than 5 pm. In the friction tests, the flywheel and the cylindrical specimen were driven up to a certain speed and then the driving source was cut off suddenly at the coupling 5. The revolution number n(t) was recorded in the process of slowing down. In the elastic system tests the friction force was measured by two methods simultaneously: the deflection method (with a plate spring and strain gauges) and the inertia method (with the flywheel). In the rigid system tests, the friction force was measured only by the inertia method. The frictional force in the inertia method was derived as follows. In the simplified model of the rigid system shown in Fig. 3, the flywheel with a moment I of inertia rotates around the x axis. The specimen Tz of outer radius r is mounted on the same axis and the specimen T1 rides on Tz under a load W. If the driving force of the flywheel is suddenly cut off after a certain number of revolutions, the angular velocity of the flywheel decreases from the initial value o. to w after a period t and the equation of energy balance is given by t

+do2 - ila2 =JF(rw)rw 0

dt + _/G’(r~)rw 0

dt

(1)

310

Fig. 3. The rigid system of the fiywheei for friction force measurements.

where F(W) is the frictional force the sliding velocity, G(F~) is the bearings and the resistance caused moment of G(W). Differentiation F(W)

= -;

dt

at the contact interface as a function of sum of the frictional resistance at the by air and T;is the equivalent radius of the of eqn. (1) with respect to t gives

- ; G(W)

(2)

In eqn. (Z), F(W) is equal to zero when the load is not applied. If we introduce the function w’(t) at time t under no load for the angular velocity of the flywheel, eqn. (2) gives I do’(t) --F dt If w(t) = w’(t’) I do’(f) - -

r

dt

By substituting by

= -;G{m’(t)} at the different

(3) times t and t’, eqn. (3) gives

= - fG{rw(t)}

(4)

eqn. (4) into eqn. (2), the frictional

force at time t is given

In eqn. (5), m(t) gives the sliding velocity u(t) at the interface between specimens T1 and T,. The numbers pt(t) and n’( t’) of revolution under load and under no load respectively are related to o(t) and w’(f) by w(t)

= 2nn(t)

By using those relations,

w’(f)

= 2nn’(t’)

eqn. (5) becomes

Therefore the frictional force at a certain velocity is given by eqn. (6) by obtaining dn(t)/dt and dn’(t’)/dt from the experimental results. The coefficient of friction is calculated from F(u (t)}/ W. For more acurate measurement, the experimental condition n(0) = n’(0) is desirable.

311

Wear tests were also conducted at a certain driving velocity for a certain period in addition to the friction tests. The wear loss was measured with an electric balance with an accuracy to 0.1 mgf (9.81 X 10e6 N). All tests were conducted at room temperature (about 18 - 20 “C) and a relative humidity of about 55% - 62%.

3. Experimental results 3.1. The effect of frictional vibration on the coefficient of friction Figure 4 shows a typical record of the plate spring deflection x-time relationship obtained by the deflection method in the elastic system. The average of the peak and valley values of the record was calculated as the representative value of the oscillatory frictional force given by x for a certain period.

1 0

X

Fig. 4. The record

of the plate spring deflection

x us. the time 6

Figure 5 shows examples of n(t) and n’(t) in the elastic and rigid systems where the driving force was cut off in each case at the same number of revolutions. n(t) and n’(t) gradually go to zero because of the various types of frictional resistances. The forms of the curves are not simple and they cannot be represented by any simple functions within moderate accuracy. Therefore the derivatives of n(t) and n’(t) were determined by approximating them as dn(t) -% dt

dn’( t)

An(t)

-z dt

At

Fig. 5. The deceleration

process

of the flywheel.

An’(t) At

(7)

312

where An(t) and An’(t) are the amounts of reduction in the number of revolutions within the period At. In this investigation At was taken as 10 s and F{v(t)} was calculated from eqn. (6). Figures 6 - 9 show the relationships between the coefficient of friction and the driving velocity obtained for steel-steel, white metal-steel, bronzesteel and phosphor bronze-steel pairs respectively in the elastic system. Open circles and broken lines show the relationships obtained by the deflection method and full lines show the relationships obtained at the same time by the inertia method. 0.6 -

a 02 -

0

02 -

2

4

6

6

10

Xl0

0

2

4

6

v (cmlscc) Fig. ---,

6. The coefficient inertia method;

-

of friction-driving - -, 0, deflection

velocity method.

relationship

Fig. 7. The coefficient of friction-driving velocity relationship pair: -, inertia method; -- - -, 0, deflection method.

0

I

1

2

4

6

8

v ( CrnlSec Fig. 8. The coefficient -, inertia method;

6

10

X10

V ( cm/set)

IO

I

Xl0

0

I

4

6

I

I

6

IO

V (cmlsec)

)

velocity relationship 0, deflection method.

pair:

for a white metal-steel

I

2

of friction-driving velocity relationship - - -, 0, deflection method.

Fig. 9. The coefficient of friction-driving steel pair: -, inertia method; - - -,

for a steel-steel

for a bronze-steel

for a phosphor

xl0

I

pair:

bronze-

The coefficient of friction obtained by the deflection method is larger than that by the inertia method for each frictional pair. The difference between them is greatest for the bronzesteel pair. The most probable explanation for such differences between the two measurements for each pair may be that frictional vibration involves a vertical jumping motion and this makes the actual frictional work smaller than the apparent work according to the record of the frictional force.

The coefficient of friction obtained by the deflection method varies complicatedly in each frictional pair at the lower velocity range. Coefficient of friction obtained by the inertia method, by contrast, are nearly constant over a wide range of velocity, especially for steel-steel, white metal-steel and bronze-steel pairs. It is difficult to know why such differences in the relationship exist between the results from two methods. It is of interest to know how the dependence of the coefficient of friction obtained by the inertia method would change with a change in the support system from elastic to rigid. In Figs. 10 and 11, full lines indicate the results observed in the rigid system and the broken lines in the elastic system for steel-steel and white metal-steel pairs respectively. In Fig. 10, the coefficient of friction measured in the rigid system increases from 0.4 to 0.7 as the driving velocity increases but the coefficient of friction measured with the elastic system remains at a fairly constant value of 0.35 as the driving velocity increases from 20 to 110 cm s-l. In Fig. 11, by contrast, the coefficient of friction measured in the rigid system decreases from 0.78 to 0.38 as the driving velocity increases but the coefficient of friction measured in the elastic system remains constant at 0.45 over a wide range of velocities.

0

2

4

6

8

v (cm/set)

10

12 x10

0

2

L

6 V

8

(0

12X

(cmlsec)

Fig. 10. The coefficient of friction-driving velocity relationship for a steel-steel pair by the inertia method: -, rigid system; - - -, elastic system. Fig. 11. The coefficient of friction-driving velocity relationship for a white metal-steel pair by the inertia method: -, rigid system; -- - -, elastic system.

It is evident from these results that the stiffness of the specimen support system greatly affects dynamic friction. The rigid system makes the difference in frictional properties between different friction pairs clearer. The frictional vibration in the elastic system appears to reduce the difference in frictional properties between different friction pairs. It is generally believed that friction pairs of the same material give the large frictional resistance which leads to extensive seizure because of their mutual weldability. Dissimilar material pairs give a small frictional resistance because of the lack of weldability. From this point of view, a white metal-steel pair is expected to have better frictional properties than a steel-steel pair. Observations with the rigid system in Figs. 10 and 11 appear to confirm this. Thus the coefficient of friction measured in the rigid system gives more reliable values for practical use than that measured in the elastic system.

314

3.2. The effect

of vibration on wear

Wear tests were conducted in both the elastic and the rigid systems to determine the effect of frictional vibration on wear. The experimental conditions were the same as those for the friction tests except that a constant velocity of 0.91 m s-l was used. Figures 12 - 15 show the relationships between the wear loss A W of the pin specimen and the sliding distance L with steel-steel, white metal-steel, phosphor bronze-steel and bronze-steel pairs respectively. Full circles show the results obtained for the elastic system and open circles show the results measured in the rigid system. Two lines for each system show the upper and lower limits of the scatter in the experimental observations. In Fig. 12, for a steel-steel pair, the extent of the scatter is much larger for the elastic system than for the rigid system. The lower limits with the

4-

0

2

4

6

6

IO

12

14

0

2

4

6

XIOZ

L Cm)

6

10

12

L(m)

Fig. 12. The wear loss-sliding 0, elastic system.

distance

relationship

for a steel-steel

Fig. 13. The wear loss-sliding system; 0, elastic system.

distance

relationship

for a white metal-steel

14 x102

pair: 0, rigid system;

pair: 0, rigid

6-

0

2

4

6

Fig. 14. The wear loss-sliding distance 0, rigid system; 0, elastic system.

relationship

for a phosphor

Fig. 15. The wear loss-sliding system; 0, elastic system.

relationship

for a bronze-steel

distance

6 L (m)

10

bronze-steel

pair:

12

pair:

0, rigid

14 Xl02

315

elastic system are ~pproxima~ly close to zero. In comparison, the extent of scatter for a white metal-steel pair is relatively small and similar in the elastic and the rigid systems. For the phosphor bronze-steel pair, the extent of scatter is very small for the rigid system but for the elastic system it is about 4 - 5 times as large as that at the later stage of sliding. For the bronzesteel pair, the extent of scatter with the elastic system is about 2 - 3 times as large as that with the rigid system. From the results it is evident that frictional vibration increases the extent of scatter in me~ummen~ of wear. However, the extent and the manner of the effect of vibration on wear differ greatly according to the frictional material. The wear of steel against steel seems to be the most sensitive to frictional vibration. Table 1 shows the wear loss of the cylindrical steel specimen in each friction pair after a sliding distance of 1.4 km, where positive values indicate loss from the specimen by wear and negative values indicate the increase in the weight of the specimen after sliding. The wear loss of the cylindrical steel specimen paired with the same material is positive and large in both the elastic and the rigid systems. The wear loss obtained in the elastic system is a little smaller than that obtained in the rigid system, which corresponds to the fact that the wear loss of the pin specimen obtained in the elastic system is smaller than that obtained in the rigid system, as shown in Fig. 12. For the steel-steel pair, therefore, frictional vibration reduces the wear of both pin and cylindrical specimens. For the white metal-steel pair, the wear loss of the steel specimen is negative with both the elastic and the rigid systems, which means that the white metal transfers to the steel surface by adhesion and the wear of the pin specimen occurs at the interface between the surface of the white metal specimen and the transferred white metal layer on the steel. The fact that the mass of adhesive transfer is larger with the elastic system than with the rigid system means that frictional vibration aids the adhesive transfer of white metal to steel. For the phosphor bronze-steel pair, the wear loss of the steel specimen is also negative with both the elastic and the rigid systems and their values are small and equal. Adhesive transfer of phosphor bronze to steel, therefore, seems so poor that frictional vibration does not increase effectively the TABLE 1 Wear loss data for the cylindrical steel specimens Frictional

pair

Steel-steel White metal-steel Phosphor bronze-steel Bronze-steel

Wear loss (mgf) Elastic system

Rigid system

+106.2 -0.7 -0.1 -0.2

+111.1 -0.5 -0.1 +0.5

316

adhesive transfer of phosphor bronze. For the bronze-steel pair, the wear loss is positive with the rigid system but negative with the elastic system, which means that the friction must have changed from sliding between bronze and steel in the rigid system to that between bronze and transferred bronze in the elastic system. By comparing Figs, 12 - 15 and Table 1, it may be suggested that the generation of the adhesive transfer layer of the soft metal to the driving steel surface is promoted by frictional vibration. When such a transferred layer of soft metal is formed, the wear rate becomes nearly constant as for the white metal-steel pair and the phosphor bronzesteel pair as shown in Figs. 13 and 14. In Fig. 15 the transferred layer is formed by friction with the elastic system and the wear rate tends to increase with sliding distance. If the transferred layer is not formed, as for the steel-steel pair with both systems as shown in Fig. 12 and for the bronze-steel pair with the rigid system as shown in Fig. 15, the wear rate tends to decrease with sliding distance. The wear properties of steel itself may be important in this case. 4. Discussion In order to understand friction and wear properties in a vibrational process, the exact relationship between the frictional force and the real sliding velocity must be deduced from measurements. The effect of heating on frictional resistance must be known. Vertical motion is also an important factor to be considered in the analysis of friction and wear in a vibrational process. The general solutions to these problems appear to be difficult to achieve. Every friction pair has different friction and wear properties. With steel, for example, the formation of an oxide layer reduces frictional resistance and wear and wear is much more sensitive to the formation of an oxide layer than frictional resistance. The large scatter in the wear data for a steelsteel pair with the elastic system shown in Fig. 12 indicates the complex effects. Wear at the upper limits of measurement indicates severe metallic wear and that at the lower limits indicates mild wear controlled by the oxide layer on the surface. The surfaces of steel specimens at the lower limits were covered with layers of red or black oxides. Wear observations with the rigid system were close to the upper limits of wear observations in the elastic system and the specimen surfaces exhibited severe wear. Frictional vibration, therefore, should be assumed to induce sometimes the transition from severe to mild wear with the elastic system. Then the real sliding velocity in the vibrational process is needed for analysis of the problem. However, in practice it is difficult to carry out the exact dynamic analysis on friction measurements of induced vibration that is necessary to obtain the real sliding velocity and the frictional force. In Appendix A a simplified technique to determine the maximum sliding velocity in the vibrational process is developed. As an example, the maximum sliding velocity with the elastic system shown in Fig. 12 was calculated by eqn. (A6) in Appendix A. This velocity

317

was 1.24 m s-’ compared with the nominal driving velocity of 0.91 m s-l, i.e. 1.36 times the driving velocity. The velocity of 1.24 m s-l is very close to the critical velocity for steel at the transition from severe to mild wear shown for similar experimental conditions in the report of the Japan Society of Lubrication Engineers committee [ 61. Variation in the wear mode with the elastic system of Fig. 12, therefore, might be assumed to be due to the transition from severe to mild wear as a result of extensive frictional vibration which enabled the formation of an oxide layer.

5. Conclusions To elucidate the effect of friction-induced vibration on friction and wear, frictional resistance and wear were observed in elastic and rigid systems by a deflection method and an inertia method. By comparing the results for steel-steel, white metal-steel, phosphor bronze-steel and bronze-steel pairs with the two systems, the following conclusions were obtained. (1) In the elastic system, the coefficient of friction determined by the deflection method is greater than that determined by the inertia method. (2) The relationship between the coefficient of friction and the driving velocity in the presence of frictional vibration differs greatly from that with no vibration. The original frictional property of the frictional pair is obtained with the rigid system where frictional vibration is suppressed. (3) Frictional vibration changes the wear state differently with each friction material. The wear of white metal, phosphor bronze and bronze against steel tends to increase with frictional vibration but the wear of steel against steel tends to decrease with frictional vibration.

References 1 2 3 4

J. B. Simpson, F. Morgan, D. W. Reed and M. J. Muscut, J. Appl. Phys., 14 (1943) 689. P. L. Ko and C. A. Brockley, J. Lubr. Technol., 92 (October 1970) 543. E. Rabinowicz, Friction and Wear ofMaterials, Wiley, New York, 1965, p. 101. B. R. Derjaguins, V. E. Push and D. M. Toistoi, hoc. Int. Conf. on Lubrication and Wear, London, 1957, Institution of Mechanical Engineers, London, 1958, p. 257. 5 Y. Miyagawa, K. Seki and T. Nishimura, J. Jpn. Sot. Lubr. Eng., 18 (1973) 323 (in Japanese). 6 T. Sata et al., J. Jpn. Sot. Lubr. Eng., 17 (1972) 113 (in Japanese).

Appendix A In Fig. Al, the mass of the rider is denoted by m and it is attached to a spring with spring constant h. The base surface moves at a velocity V to give a resultant frictional force F at the interface between the rider and the base surface. In such a frictional system, the displacement x of the rider can be

318

I-x

Fig. Al.

Schematic

diagram

of the frictional

Fig. A2. The displacement-time

relationship

system. of the slider.

described schematically as shown in Fig. A2, where the type of vibration in the unit cycle ABC may sometimes be stick-slip vibration and at other times may be self-excited sinusoidal vibration. The equilibrium equation of force in the vibrational process, except for the stick process, is given by mX+kx=F

By supposing

(Al)

that F is constant,

x = a cos{w(t

the solution

of eqn. (Al) is

- t,,)} + F/k

(-42)

where a and t,, are constants determined by the initial boundary conditions and w = (k/m)1’2. The formula of eqn. (A2) is the same as the experimental formula of frictional vibration obtained by Simpson et al. [Al]. The velocity of frictional vibration is given by the first derivative of x in eqn. (A2): i =--a0

sin{w(t - tb)}

(A3)

The real sliding velocity u of the rider against the base surface is given by the relative velocity between them : u=V-3?=V+awsin{w(t-tt,)}

(-44)

At a time t = (l/o)(nn + 7r/2) + tb, where jt = 0, the maximum u is given by the equation U max =

V + aw

This can be written, U max =

value u,,,

of (A5)

with the introduction

V + 27raf

of w = 27rf, as (W

where f is the frequency of vibration. The displacement x, at the time when urnax is attained is equal to the mean value of x in the unit cycle and is related to the frictional force by the following formula: F=kx,

(-47)

319

Fig. A3. The displacement-time

relationship of the slider in stick-slip motion.

When the frictional vibration is continuous sinusoidal vibration, the maximum sliding velocity u,,, and the corresponding frictional force F can be determined from eqns. (A6) and (A7) by measuring a, f and x, on the friction record. In the vibration of stick-slip motion (Fig. A3), two different processes are included in the unit cycle and then eqn. (A6) does not hold for the determination of urnax. If we suppose that stick starts when the sliding velocity is equa LOzero, the time tv of stick is given by equating u to zero in eqn. (A4) as follows: V = ---a~ sin{o(tv

- tb)}

(A81

By introducing the function r defined by the equation t,=t,+L+r

(A9)

0

and substituting eqn. (A9) into eqn. (A8), V is given by V = aw sin(or)

(AlO)

where 0 < r < n/Zw. Stick will not occur when V > aw. If it is assumed that the initiation of slip is not affected by the period of stick, slip starts when x = x, + a. In this case, the time t, of slip is given bv ”

tr=tV+Xm+a-~ V

where xv is the value of x at tv. Substitution of eqns. (A2) and (A9) into eqn. (All) gives t,=t,+II+r+--

w

a(1 + cos(wr)} V

(A=)

The frequency f in the vibration of stick-slip motion, however, is given by 1 - = t, - tb

f

Substituting eqn. (A12) into eqn. (A13), we obtain

(Al3)

(A14)

Elimination 1

of r from eqns. (AlO) and (A14) gives

a[1 + {1-(V/aw)2}1’2]

-=

V

f The following

approximations

{ 1 - (V/uC#}1’2 arcsin( V/au) Substitution

71

1

V

w

w

i aw i

+ - + - arcsin -

6415)

are possible when 0 < V/a < 1:

= 1 - (V/aw)2/2

(A=)

= V/au

of eqn. (A16) into eqn. (A15) gives

V

+?+-=o 2aw2 w

The solution

2a

(Al7)

Vf

of eqn. (A17) gives the following

equation:

‘=;[jl+$(&-l)~“2-1] w Substitution of eqn. (AH) into eqn. (A5) gives the maximum ity urnax for the vibration of stick-slip motion as follows

6418) sliding veloc-

The critical value of the velocity V for the transition from stick-slip motion to self-excited sinusoidal vibration is given by V = ao where stick initiates. By substituting V/au = 1 into eqn. (A15), the critical velocity V,, for stick initiation is given by

v,, =

1 + ; af i ) In conclusion, urnax is given by eqn. (A19) for V < V,, and is given by eqn. (A6) for V> V,,. The frictional force corresponding to umaXis always given by eqn. (A7). It is evident that the measurements of the amplitude a, frequency f and the mean value X, on the record of the oscillating frictional force are readily obtained. Although three important assumptions were introduced concerning the frictional resistance and the initiation of stick and slip in this theory, eqns. (A6), (A19) and (A20) provide a simple technique for the analysis of frictional data. Reference Al

for Appendix

J. B. Simpson, 689.

A

F. Morgan, D. W. Reed and M. J. Muscut, J. Appl. Phys.,

14 (1943)