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Elasto-optical investigations of rolling of a rigid cylinder on a visco-elastic base
30
\Vli,\
ELASTO-OPTICAL
INVESTIGATIONS
ON A VISCO-ELASTIC
1:
OF KOLLING OF A RIGID CYLINDER
BASE
J. B. HALAUNBRENNER Department of Physics, (Received
Technical University, Krakow (Poland)
March 2, 1964)
The rolling friction of metals is at least one order of magnitude smaller than their sliding friction, whereas that of rubber-like materials is approximately equal to their sliding friction on well-lubricated surfaces. The principal cause of rolling resistance on a visco-elastic base is undoubtedly hysteresis losses in the base materialr,2. Measurements of the rolling resistance of a ball on two kinds of rubber of different internal damping, but similar elastic moduli, showed the rolling resistance to be proportional to the damping coefficients. Other factors, such as adhesion of the body to the base, roughness of the surface, and rolling velocity influence the rolling resistance to a lesser degree. The theoretical work of calculating the rolling resistance of a ball or of a cylinder on a visco-elastic base as a function of the velocity and dynamic coefficients of elasticity and viscosity is well advanced, but experimental papers are few. The results of an investigation of the phenomenon of a rolling cylinder, using the elasto-optical method, are given in this report.
Fig. I. Apparatus rolling cylinder.
for observation
of e&to-optical
patterns
in a visco-elastic
base under a rigid
The visco-elastic base was made of epoxy resin, P52. A circular base, diameter 17 cm, was cut from a I cm-thick plate of resin. The base was set centrally on the axis of a motor whose speed was continually variable. A worm gear was used to obtain very low velocities. The base was supported from two sides by two rigid, smaller-diameter metallic supports Bi.2, in order to prevent buckling. The base is put between crossed Polaroids P~,z. In this way peripheral speeds from 0.1 cmjsec to 20 m/set were obtained. Wear, 8 (w65)
30-33
ROLLING OF A RIGID CYLINDER
Fig. z. Elasto-optical rigid 2.0 cm-diameter moment of loading.
pictures in a visco-elastic base (epaxide resin P 52) loaded suddenly by a cylinder of a normal load .I kg for the times 2, IO, 20, 30 and 40 see from the
Fig. 3. The isochromes and ieoclines of parameter o” in the visco-elastic base (a) with the cylinder resting 40 set on the base: (b)-(‘)1 with the cylinder rolling at the velocities z, = 0.16, 0.32, 0.62, ~252.50, 5, IO and 20 cm/see.
J. 13.HALAI’iTBRESNlil~
32 Figure
shows the elasto-optical base subjected suddenly to a normal load of 2.0 cm diameter for the times t = 2, IO, 20,30,40 set after the moment of loading. The stresses clearly penetrate into the base. The elasto-optical image of isochromes and isoclines o” is symmetrical to the plane, perpendicular to the base, and passing through the axis of the cylinder. The photograph in Fig. 3(a) shows the isochromes and isoclines of 0’ after the cylinder has rested for 40 set on the base. Figures 3(b)-(i) show the effect of rolling at the velocities ‘o = 0.16, 0.32, 0.62, 1.25, 2.50, 5.10 and 20 cm/set. The photographs must be made during the first revolution of the loading target, so that the cylinder rolls on a surface undeformed by earlier subjection to stress. As the velocity of rolling increases the number of the isochromes quickly decreases and the elasto-optical picture shrinks more and more to the subsurface regions. At relatively small velocities only the isoclines remain visible. The isocline o’, formerly vertical, is now curved and deviates from the vertical position in a direction opposite to that of the velocity. For the rolling velocities of 2.5 cmjsec and 2
I kg by a rigid cylinder of
5 cmjsec the isoclines were photographed between
o0 and 90’ at intervals
for values of the isocline parameter varying
of 15~. The principal stress trajectories
were then
drawn. Figure 4 presents successively the trajectories
(4
of the principal stresses when
(4
Fig. 4. The trajectories of the principal stresses in the visco-elastic base with a cylinder (diam. B = 2.0 cm, N = 0.5 kg). (a) Resting: (b) rolling at a velocity of 2.5 cm/set; (c) rolling with a velocity of 5 cm/set. Wear, 8 (19%) 30-33
ROLLING OF A RIGID CYLINDER
33
a 20 mm-diameter cylinder loaded at 0.5 kg (a) is resting; (b) is rolling with a velocity of 2.5 cm/set; and (c) is rolling with a velocity of 5 cm/set. From these photographs it may be concluded that: (I) with increasing rolling velocity, the stresses penetrate less and less into the material and are limited at high speeds to a thin subsurface layer ; (2) the stress distribution becomes asymmetric to the plane passing through the cylinder axis and perpendicular to the base. A force results which determines the friction moment opposite to the angular velocity, and the component of this force parallel to the surface is equal to the friction force. It should be noticed that when a rigid body rolls on the visco-elastic base there is no equilibrium between the normal load exerted by the rolling cylinder and the resultant of normal stresses in the base; the latter is always smaller than the former. There are, therefore, two different ways of defining the friction coefficient. The first is the ratio: friction force = ,ui; the second is the ratio: normal load friction force = ,UZ.It is clear that ,14 >,ui and both the coefficients resultant of normal stresses are different functions of rolling velocity. REFERENCES I D. TABOR, &it. J. Appl. Phys., 6 (I) (1955). 2 F. P. BOWDEN AND D. TABOR, The Friction and Lubvication ofSolids. Oxford University Press, London, 1964.
Part II, Clarendon Press:
Wear. 8 (1965) 3~33
\Vli,\
ELASTO-OPTICAL
INVESTIGATIONS
ON A VISCO-ELASTIC
1:
OF KOLLING OF A RIGID CYLINDER
BASE
J. B. HALAUNBRENNER Department of Physics, (Received
Technical University, Krakow (Poland)
March 2, 1964)
The rolling friction of metals is at least one order of magnitude smaller than their sliding friction, whereas that of rubber-like materials is approximately equal to their sliding friction on well-lubricated surfaces. The principal cause of rolling resistance on a visco-elastic base is undoubtedly hysteresis losses in the base materialr,2. Measurements of the rolling resistance of a ball on two kinds of rubber of different internal damping, but similar elastic moduli, showed the rolling resistance to be proportional to the damping coefficients. Other factors, such as adhesion of the body to the base, roughness of the surface, and rolling velocity influence the rolling resistance to a lesser degree. The theoretical work of calculating the rolling resistance of a ball or of a cylinder on a visco-elastic base as a function of the velocity and dynamic coefficients of elasticity and viscosity is well advanced, but experimental papers are few. The results of an investigation of the phenomenon of a rolling cylinder, using the elasto-optical method, are given in this report.
Fig. I. Apparatus rolling cylinder.
for observation
of e&to-optical
patterns
in a visco-elastic
base under a rigid
The visco-elastic base was made of epoxy resin, P52. A circular base, diameter 17 cm, was cut from a I cm-thick plate of resin. The base was set centrally on the axis of a motor whose speed was continually variable. A worm gear was used to obtain very low velocities. The base was supported from two sides by two rigid, smaller-diameter metallic supports Bi.2, in order to prevent buckling. The base is put between crossed Polaroids P~,z. In this way peripheral speeds from 0.1 cmjsec to 20 m/set were obtained. Wear, 8 (w65)
30-33
ROLLING OF A RIGID CYLINDER
Fig. z. Elasto-optical rigid 2.0 cm-diameter moment of loading.
pictures in a visco-elastic base (epaxide resin P 52) loaded suddenly by a cylinder of a normal load .I kg for the times 2, IO, 20, 30 and 40 see from the
Fig. 3. The isochromes and ieoclines of parameter o” in the visco-elastic base (a) with the cylinder resting 40 set on the base: (b)-(‘)1 with the cylinder rolling at the velocities z, = 0.16, 0.32, 0.62, ~252.50, 5, IO and 20 cm/see.
J. 13.HALAI’iTBRESNlil~
32 Figure
shows the elasto-optical base subjected suddenly to a normal load of 2.0 cm diameter for the times t = 2, IO, 20,30,40 set after the moment of loading. The stresses clearly penetrate into the base. The elasto-optical image of isochromes and isoclines o” is symmetrical to the plane, perpendicular to the base, and passing through the axis of the cylinder. The photograph in Fig. 3(a) shows the isochromes and isoclines of 0’ after the cylinder has rested for 40 set on the base. Figures 3(b)-(i) show the effect of rolling at the velocities ‘o = 0.16, 0.32, 0.62, 1.25, 2.50, 5.10 and 20 cm/set. The photographs must be made during the first revolution of the loading target, so that the cylinder rolls on a surface undeformed by earlier subjection to stress. As the velocity of rolling increases the number of the isochromes quickly decreases and the elasto-optical picture shrinks more and more to the subsurface regions. At relatively small velocities only the isoclines remain visible. The isocline o’, formerly vertical, is now curved and deviates from the vertical position in a direction opposite to that of the velocity. For the rolling velocities of 2.5 cmjsec and 2
I kg by a rigid cylinder of
5 cmjsec the isoclines were photographed between
o0 and 90’ at intervals
for values of the isocline parameter varying
of 15~. The principal stress trajectories
were then
drawn. Figure 4 presents successively the trajectories
(4
of the principal stresses when
(4
Fig. 4. The trajectories of the principal stresses in the visco-elastic base with a cylinder (diam. B = 2.0 cm, N = 0.5 kg). (a) Resting: (b) rolling at a velocity of 2.5 cm/set; (c) rolling with a velocity of 5 cm/set. Wear, 8 (19%) 30-33
ROLLING OF A RIGID CYLINDER
33
a 20 mm-diameter cylinder loaded at 0.5 kg (a) is resting; (b) is rolling with a velocity of 2.5 cm/set; and (c) is rolling with a velocity of 5 cm/set. From these photographs it may be concluded that: (I) with increasing rolling velocity, the stresses penetrate less and less into the material and are limited at high speeds to a thin subsurface layer ; (2) the stress distribution becomes asymmetric to the plane passing through the cylinder axis and perpendicular to the base. A force results which determines the friction moment opposite to the angular velocity, and the component of this force parallel to the surface is equal to the friction force. It should be noticed that when a rigid body rolls on the visco-elastic base there is no equilibrium between the normal load exerted by the rolling cylinder and the resultant of normal stresses in the base; the latter is always smaller than the former. There are, therefore, two different ways of defining the friction coefficient. The first is the ratio: friction force = ,ui; the second is the ratio: normal load friction force = ,UZ.It is clear that ,14 >,ui and both the coefficients resultant of normal stresses are different functions of rolling velocity. REFERENCES I D. TABOR, &it. J. Appl. Phys., 6 (I) (1955). 2 F. P. BOWDEN AND D. TABOR, The Friction and Lubvication ofSolids. Oxford University Press, London, 1964.
Part II, Clarendon Press:
Wear. 8 (1965) 3~33