- Email: [email protected]
On the friction of bodies rolling under gravity
Wear,
117
Technical
(1987)
109
109
- 112
Note
On the friction
of bodies rolling under gravity
S. LINGARD Department (Received
of Mechanical November
Engineering,
20, 1986;accepted
University
of Hong
December
Kong
(Hong
Kong)
22, 1986)
Experiments on the accelerations of cylinders and spheres rolling under gravity down inclined tracks have recently been described by Chaplin and Chilson [ 11, Shaw and Wunderlich [ 21 and Chaplin and Miller [ 31. The tracks were fitted with photodetectors and timing circuits so that the average speed of the body could be measured. Results were presented in the form of graphs of average acceleration against sin /3 where 0 is the angle of the track to the horizontal, or average speed against h, where h is the projected vertical distance through which the ball moved. Typical results from Chaplin and Chilson [l] and from Shaw and Wunderlich [2] are shown in Fig. 1 and Fig. 2 respectively. The results were interpreted as two intersecting lines representing, at the smaller angles of inclination, the motion of a rolling body and, at the higher angles of inclination, the motion of a rolling-sliding body. It was also suggested in both papers that the intersection point of the two theoretical lines indicates the onset of slipping and the two regimes provide evidence of a unique value of the coefficient of kinetic friction associated with the intersection point. The attempted correlation of the lines of average acceleration against slope with particular constant values of the coefficient of friction is neither 10
E
-
Fig. 7. Acceleration against sin 0 from cylinder on an aluminium plane.
0043-1648/871$3.50
Chaplin
and
@ Elsevier
Chilson
[ 11. Stainless
Sequoia/Printed
in The
steel
annular
Netherlands
110
,
ZM-
z
-
2
,’
” 210. I‘;
,I..’ / /’
.I ’ ,,A’
I ;&/’
170. ,I; ,I’
./
I” /’
130. ./I / 6
0 d-r ;:A
I2
Fig. 2. Average velocity against h’ ’ from Sha w and Wunderlich [ 21. Steel tracks, ball material not quoted: r = 19.02 mm; b = 1’7.88 mm. Distance travelled by the ball equals 1745 mm.
necessary nor realistic since from the equations of motion body the coefficient of friction can be directly deduced. The appropriate equations are
of the rolling
ri = g(sin p - p cos /3) for a circular cylinder
(
li =g sinP--
rolling down a plane track and
pr cos fl
b
j
for a sphere rolling down two inclined parallel rails, where p is the coefficient of friction defined as the ratio of tangential force to normal force at the rolling contact and g is the gravitational acceleration. h is the effective rolling radius of the ball, r is its overall radius and & is the linear acceleration in the plane of the track. The results of Figs. 1 and 2 are replotted in Figs. 3 and 4, showing the coefficient of friction as a function of track inclination. These are average values based on the average velocities measured in the experiments The data points correspond to the and assuming uniform acceleration. arithmetic mean of the range of average velocities in the results of Shaw and Wunderlich [2] and the bars indicate the extent of experimental variability. Figures 3 and 4 show that, far from having fixed values associated with two types of motion, the coefficient of friction varies in a continuous and systematic way with track inclination, increasing progressively to a maximum at a particular slope and, subsequently, diminishing again at greater angles of inclination. This behaviour invites a direct comparison with typical friction-sliding speed characteristics of a variety of rolling and wheel-rail systems in which friction forces arise, such as tyre-road
111
/u
OS0
0 32 0 30 0 26026024 022 020. 0 I8 0 I6 0 I4 0 12. 0 IO0 08 006. 0 04. 032I 06
1 0.4
I 02
I I.0
I 08
Sin n
Fig. 3. Coefficient
of friction
calculated
from the data of Chaplin and Chilson
in Fig. 1
P
0.2or 0.18 P
0.160.14 0.12 0.10. 0.08 0.06 -
11
I
I
0.1 a2 a3
i
I
I
/
I
I
I
0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.001
Om.2
am5
0.010 a020
QOM
Sin R
Fig. 4. Coefficient Fig. 2(u,
of friction
calculated
0.10 Sr
from the data of Shaw and Wunderlich
5. Coefficient of friction against slide-roll ratio sx from disc machine -- IQ)/(u~ + ~2) where u1 and u2 are the surface speeds of the two discs.
in Fig. 2. tests.
.sr =
contacts. Figure 5 shows an example obtained by measuring the frictional forces developed between an unlubricated crowned steel roller and a cylindrical roller on a two-disc machine. At or near conditions of pure rolling the resistance (arising largely from inelastic behaviour in the contact region) is very small (of the order p = 0.001). As the creep increases the frictional or traction force rises, approximately linearly at first, reaching a maximum at around 1% slip and then gradually declining as sliding speed is further increased.
112
The initial (creep) region can be explained, at least partly, by elastic compliance and microslip in the contact zone [4] but it is necessary to differentiate it from a condition of pure rolling where there is no macroscopic creep. An important feature of the friction-sliding speed curve is the region beyond the maximum where increased sliding produces diminishing frictional forces. Here the motion tends to become unstable as the force opposing slip falls in response to any small disturbance which increases the sliding. There is a prima facie case for arguing that the friction characteristics in the inclined track tests (Figs. 3 and 4) occur as a result of gradually increasing slip as the track inclination increases and are comparable with Fig. 5 as far as general trends are concerned. If this is accepted it follows that at high angles of inclination, where cc decreases with increased sliding, the acceleration is unlikely to be constant and any single value quoted for the coefficient of friction must be simply a notional figure representing an average for the particular track length employed. The difference between the two regimes is reflected in the much higher degree of experimental variability at high track angles. Also noteworthy is the insensitivity of the accelerations or average velocity measurements to changes in the coefficient of friction, so that the range of average velocities in the tests of Shaw and Wunderlich at the highest range of the coeffiinclination is only 270 - 278 cm s ’ . The corresponding cient of friction is 0.143 - 0.023, a sixfold reduction. The insensitivity is due to the relatively small magnitude of the forces resisting motion to the gravitational accelerating force acting down the track. For these reasons it must be concluded that the inclined track experiment as reported is not suitable for the accurate determination of the coefficient of friction, except perhaps in the stable low slip region of the characteristic. However, it can be usefully employed in illustrating the form of the coefficient of friction against sliding speed relationship and to determine roughly the maximum value of /J (usually lower in rolling-sliding than in purely sliding systems 141) which corresponds approximately to the point of intersection of the two average acceleration lines. The experiment might be improved, at the expense of considerable complication, by including facilities for measuring the rotational velocity of the ball, and hence the slip.
1 H.
L. Chaplin
I’. B. Chilson. The coefficient of klnetlc friction I’or alumlnrum. 213. 2 D. E. Shaw and F. J. Wunderlich, Study of the slipping of a rolling sphere. Am. J. Phys., 52 (1984) 997. 3 R. L. Chaplin and M. G. Miller, The coefficient of friction for a sphere. Am. J. Phys.. 52 (1984) 1108. 4 J. Hailing. The rolling of a ball subjected to normal and tangential loads. Wear, 7 (1964) 516.
Wear, 107 (1986)
and
117
Technical
(1987)
109
109
- 112
Note
On the friction
of bodies rolling under gravity
S. LINGARD Department (Received
of Mechanical November
Engineering,
20, 1986;accepted
University
of Hong
December
Kong
(Hong
Kong)
22, 1986)
Experiments on the accelerations of cylinders and spheres rolling under gravity down inclined tracks have recently been described by Chaplin and Chilson [ 11, Shaw and Wunderlich [ 21 and Chaplin and Miller [ 31. The tracks were fitted with photodetectors and timing circuits so that the average speed of the body could be measured. Results were presented in the form of graphs of average acceleration against sin /3 where 0 is the angle of the track to the horizontal, or average speed against h, where h is the projected vertical distance through which the ball moved. Typical results from Chaplin and Chilson [l] and from Shaw and Wunderlich [2] are shown in Fig. 1 and Fig. 2 respectively. The results were interpreted as two intersecting lines representing, at the smaller angles of inclination, the motion of a rolling body and, at the higher angles of inclination, the motion of a rolling-sliding body. It was also suggested in both papers that the intersection point of the two theoretical lines indicates the onset of slipping and the two regimes provide evidence of a unique value of the coefficient of kinetic friction associated with the intersection point. The attempted correlation of the lines of average acceleration against slope with particular constant values of the coefficient of friction is neither 10
E
-
Fig. 7. Acceleration against sin 0 from cylinder on an aluminium plane.
0043-1648/871$3.50
Chaplin
and
@ Elsevier
Chilson
[ 11. Stainless
Sequoia/Printed
in The
steel
annular
Netherlands
110
,
ZM-
z
-
2
,’
” 210. I‘;
,I..’ / /’
.I ’ ,,A’
I ;&/’
170. ,I; ,I’
./
I” /’
130. ./I / 6
0 d-r ;:A
I2
Fig. 2. Average velocity against h’ ’ from Sha w and Wunderlich [ 21. Steel tracks, ball material not quoted: r = 19.02 mm; b = 1’7.88 mm. Distance travelled by the ball equals 1745 mm.
necessary nor realistic since from the equations of motion body the coefficient of friction can be directly deduced. The appropriate equations are
of the rolling
ri = g(sin p - p cos /3) for a circular cylinder
(
li =g sinP--
rolling down a plane track and
pr cos fl
b
j
for a sphere rolling down two inclined parallel rails, where p is the coefficient of friction defined as the ratio of tangential force to normal force at the rolling contact and g is the gravitational acceleration. h is the effective rolling radius of the ball, r is its overall radius and & is the linear acceleration in the plane of the track. The results of Figs. 1 and 2 are replotted in Figs. 3 and 4, showing the coefficient of friction as a function of track inclination. These are average values based on the average velocities measured in the experiments The data points correspond to the and assuming uniform acceleration. arithmetic mean of the range of average velocities in the results of Shaw and Wunderlich [2] and the bars indicate the extent of experimental variability. Figures 3 and 4 show that, far from having fixed values associated with two types of motion, the coefficient of friction varies in a continuous and systematic way with track inclination, increasing progressively to a maximum at a particular slope and, subsequently, diminishing again at greater angles of inclination. This behaviour invites a direct comparison with typical friction-sliding speed characteristics of a variety of rolling and wheel-rail systems in which friction forces arise, such as tyre-road
111
/u
OS0
0 32 0 30 0 26026024 022 020. 0 I8 0 I6 0 I4 0 12. 0 IO0 08 006. 0 04. 032I 06
1 0.4
I 02
I I.0
I 08
Sin n
Fig. 3. Coefficient
of friction
calculated
from the data of Chaplin and Chilson
in Fig. 1
P
0.2or 0.18 P
0.160.14 0.12 0.10. 0.08 0.06 -
11
I
I
0.1 a2 a3
i
I
I
/
I
I
I
0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.001
Om.2
am5
0.010 a020
QOM
Sin R
Fig. 4. Coefficient Fig. 2(u,
of friction
calculated
0.10 Sr
from the data of Shaw and Wunderlich
5. Coefficient of friction against slide-roll ratio sx from disc machine -- IQ)/(u~ + ~2) where u1 and u2 are the surface speeds of the two discs.
in Fig. 2. tests.
.sr =
contacts. Figure 5 shows an example obtained by measuring the frictional forces developed between an unlubricated crowned steel roller and a cylindrical roller on a two-disc machine. At or near conditions of pure rolling the resistance (arising largely from inelastic behaviour in the contact region) is very small (of the order p = 0.001). As the creep increases the frictional or traction force rises, approximately linearly at first, reaching a maximum at around 1% slip and then gradually declining as sliding speed is further increased.
112
The initial (creep) region can be explained, at least partly, by elastic compliance and microslip in the contact zone [4] but it is necessary to differentiate it from a condition of pure rolling where there is no macroscopic creep. An important feature of the friction-sliding speed curve is the region beyond the maximum where increased sliding produces diminishing frictional forces. Here the motion tends to become unstable as the force opposing slip falls in response to any small disturbance which increases the sliding. There is a prima facie case for arguing that the friction characteristics in the inclined track tests (Figs. 3 and 4) occur as a result of gradually increasing slip as the track inclination increases and are comparable with Fig. 5 as far as general trends are concerned. If this is accepted it follows that at high angles of inclination, where cc decreases with increased sliding, the acceleration is unlikely to be constant and any single value quoted for the coefficient of friction must be simply a notional figure representing an average for the particular track length employed. The difference between the two regimes is reflected in the much higher degree of experimental variability at high track angles. Also noteworthy is the insensitivity of the accelerations or average velocity measurements to changes in the coefficient of friction, so that the range of average velocities in the tests of Shaw and Wunderlich at the highest range of the coeffiinclination is only 270 - 278 cm s ’ . The corresponding cient of friction is 0.143 - 0.023, a sixfold reduction. The insensitivity is due to the relatively small magnitude of the forces resisting motion to the gravitational accelerating force acting down the track. For these reasons it must be concluded that the inclined track experiment as reported is not suitable for the accurate determination of the coefficient of friction, except perhaps in the stable low slip region of the characteristic. However, it can be usefully employed in illustrating the form of the coefficient of friction against sliding speed relationship and to determine roughly the maximum value of /J (usually lower in rolling-sliding than in purely sliding systems 141) which corresponds approximately to the point of intersection of the two average acceleration lines. The experiment might be improved, at the expense of considerable complication, by including facilities for measuring the rotational velocity of the ball, and hence the slip.
1 H.
L. Chaplin
I’. B. Chilson. The coefficient of klnetlc friction I’or alumlnrum. 213. 2 D. E. Shaw and F. J. Wunderlich, Study of the slipping of a rolling sphere. Am. J. Phys., 52 (1984) 997. 3 R. L. Chaplin and M. G. Miller, The coefficient of friction for a sphere. Am. J. Phys.. 52 (1984) 1108. 4 J. Hailing. The rolling of a ball subjected to normal and tangential loads. Wear, 7 (1964) 516.
Wear, 107 (1986)
and