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The influence of the coulomb, viscous and acceleration-dependent terms of kinetic friction on the critical velocity of stick-slip motion
Wear,70 (1981)119 -123 Q ElsevierSequoia S.A., Lausanne- Printedin The Netherlands
119
Short Communication
The influence of the Coulomb, viscous and a~~eleration~ependent terms of kinetic friction on the critical velocity of stick-slip motion*
El.M. BELGAUMKAR Vainatheya, (India)
III Floor, No. 1 Flat, S.V. Road, Irk,
Vile-Parle
(West), Bombay
400056
(Received October 17,1980)
To explain the experimentally observed highly complex hehaviour of friction in stick-slip motion an acceleration-dependent term with a fictitious mass is added to the friction force function in the governing equation of motion.-A new concept of complex mass is introduced into the modified equation. The inertia effect and the energy-absorbing effect of the complex mass are discussed. The influences of the Coulomb and viscous terms in the friction force function are described and the nature of the influence of the acceleration-dependent term is explained. It is postulated that the influence of the acceleration-dependent term is due to a molecular dislocation phenomenon on the contacting surfaces. It is suggested that an approach based on dislocation dynamics is necessary to understand more fully the behaviour of stick-slip motion. 1. Introduction A satisfactory analytical solution explaining the complex behaviour of friction in all its aspects is not yet available. Friction behaviour in the stickslip motion of two sliding surfaces requires elucidation. When a thin film of lubricant is applied between two sliding surfaces an additional friction factor, other than the friction between the metalmetal contact, is introduced. If the lubricant film has a shear strength lower than that of the metal the coefficient of friction will be reduced. For a substantial reduction in the friction coefficient the lubricant must produce two conditions in the system: a low value of the real contacting zirea over which metal-metal contact exists and a low ratio of the shear strength of the lubricant film to the shear strength between the metal surfaces. As well as reducing friction and wear, a good lubricant has a third function: the elimination of stick-slip motion. *Paper presented at the 15th InternationalCongress on Theoretical and Applied Mechanics,Toronto, August 1980.
120
Considering an elastically restrained slider bearing, it can be shown that stick-slip behaviour occurs only when the rate of variation in the friction coefficient with respect to the relative velocity of the slider is negative. There is a critical velocity of drive below which stick-slip motion occurs and above which the body moves in a smooth and continuous manner. The expression for the critical velocity u, is given by LIF uc = A(~~)112 where AF is the difference between the static friction force and the kinetic friction force, A = {!&rc/(km)1~2}1~2is a dimensionless parameter and n, c and Fzare the well-known parameters of a vibrating system. The critical velocity at which stick-slip motion disappears and is replaced by smooth sliding has been explained by assuming a sudden drop in the static friction to a lower kinetic friction at the start of slip, followed by a constant or linearly dependent kinetic friction. Experimentally, it is shown that friction does not fall abruptly and that the behaviour of friction is highly complex, showing a memory effect evidenced by hysteresis. In the light of the above facts it has been suggested [l J that the problem should be approached from a new angle. A tentative hypothesis that friction depends on the acceleration as well as the velocity [2] has extended the work of Banerjee [3 - 51. An entirely kinetic friction-dependent analysis of the critical velocity for stick-slip motion was carried out by Banerjee [3] and the effects of force excitation on a stick-slip system in the direction of its motion have also been studied [4] . Some facts, which are not explained by the linearized theory and which can be explained effectively by introducing non-linearities in the system have been considered [ 51. The influence of the addition of an acceleration-dependent term to the friction function is considered in this paper. This term can be combined with the inertia term of the governing equation and treated mathematically. Although a classical mechanics approach does not present any difficulty in obtaining a solution of the equation the physical interpretation of the influence of the total mass, combined for mathematical convenience, needs careful attention. To bring this point out clearly, a new concept of “complex mass” is in~oduced in the mathematics analysis and the discussion that follows. 2. Analysis The kinetic friction Fk is represented in the analysis of the stick-slip motion of an elastically restrained slider bearing as Fk=Fo-&Z+pjC2+m&rl(R) where, except for the last term mf sgn(V), all symbols have the same meanings as those used by Banerjee [3].
(2)
121
The term m, sgn(R) is added also to account for the dependence of friction on acceleration. Evidently a signum function is required so that the force has the correct sign for all accelerations and positions. It may be observed that the linearity of the term m, sgn(R) ensures the parabolic friction-velocity relation assumed earlier [ 31. The general equation of motion of the stick-slip system becomes m*if+c(i-v)+k(x-ut-kO)+FO-&--fl2=0
(3)
where m* = m + mf. Since m* in eqn. (3) is mathematically of the same form as m in eqn. (1) of ref. 3, what follows is that the entire mathematical analysis of Banerjee will be valid here, except that wherever m occurs it should be replaced by m* which cannot be disposed of as an equivalent mass. It needs a different treatment. The influence of m on the system is an inertia effect while that of mf, which is a fictitious mass, is energy dissipation. The two effects are not in phase. Because there is a phase difference of n/2 between the two effects, m + mf should be used vectorially. The concept of a complex modulus E* is widely used [6]. A similar notion is required here to explain the function of m* , which is introduced as the complex mass and is defined as m* = mf + im
(4)
where mf and m are the real and imaginary parts respectively of the complex mass. Although mf is fictitious, because it produces a damping effect in the system it is represented on the real axis in the above definition of the complex mass. The magnitude of mf is given by (5)
mf =mcot6 where 6 is the angle that mf makes with m* in the phase plane. Equation (4) may be written as m* = m,(l
+ id)
(4a)
where, as for a complex modulus, d is the loss factor and is defined by d=tan6
=m/m,
(W
Thus it is possible to determine m* by measuring the real part m, and the loss factor d. It will be shown later that this loss factor depends on parameters such as the range of stress, the crystal structure, the density and the type of dislocation. To estimate the loss factor in any stick-slip system, measuring devices and a measuring method need to be developed, e.g. by using the so-called Young’s fringe technique (laser-generated speckles) for displacement and strain analysis.
122
3. Discussion of the friction function Consideration of the influence of lm*l on each term in eqn. (2) affecting the behaviour of stick-slip motion requires careful study. The term F, represents the Coulomb kinetic friction force at zero sliding velocity and the well-known relation F, = pN is a simplified model. The response of the system where F,, remains constant is well known and needs no further attention. F, does not remain generally constant but varies both in magnitude and in direction with time, giving rise to a non-linear equation of motion. This case has been analysed [ 7, 81 and it has been shown that oscillations persist indefinitely with time, contrary to the result obtained when F, is constant. The resulting damping force has a viscous character, despite the fact that it arises from the Coulomb friction. The second term, &, represents the influence of newtonian viscous drag in the system, while the third term, f13i2,accounts for the non-newtonian viscous drag that introduces non-linearity into the system. The influences of (Yand fl are discussed by Banerjee [3] who uses some known facts concerning stick-slip motion. It is argued that in the term lc - 2fiiI which is the absolute value of the slope of the assumed friction function, as cxincreases, the critical velocity u, increases while, as /3 increases, u, decreases. Damping of the acceleration-dependent forces has been considered by Wilms [9] without aiding the understanding of the influence of the mf sgn(jf) term on stick-slip motion behaviour. In order to explain the memory effect evidenced by hysteresis it is necessary to depend on much more fundamental studies of friction forces. Some of the results of Simkins [lo] and of Savenko et al. [ 111 may offer a means of approach to the solution of the complex problem. It was shown [lo] by carefully designed and conducted experiments that an extensive sliding contact on a microscopic scale was observed before gross (visible) sliding. A close examination of ref. 10, Fig. 6, which exhibits a plot of the friction force against the displacement, shows that, as with the load-extension diagram of any ductile steel, there is an initial straight steep inclined portion of the curve indicating an apparent elastic resistance to displacement. This is followed by a sharp bend and a straight horizontal line corresponding to yield followed by plastic flow in the load-extension diagram of steel. The curve then rises with a reverse bend followed by a straight inclined portion not so steep as that at the initial stage, thus indicating an apparent strain-hardening effect. The remaining portion of the curve repeats several stages similar to the initial stage. It appears that the observed phenomenon is due to some type of dislocation mechanism between the molecules of the sliding surfaces. Savenko et al.% [ 111 results appear to confirm this. Savenko et al. [ll] studied the generation of dislocations in ionic crystals caused by boundary friction both in the presence and in the absence of a lubricant and they showed .that the formation of dislocations is mainly dependent on the tangential stresses due to the adhesion of contacting, bodies and the stresses arising from their displacement. A similar effect may be expected in the influence of the complex mass. The increase in the
123
absolute value of the complex mass is expected to reduce the critical velocity and to explain the memory effect. It may be concluded that a dislocation dynamics approach may be necessary to elucidate further the behaviour of stick-slip motion. 1 B. M. Belgaumkar, Ten years of tribology, Presidential address, 15th Congr. of the Indian Society of Theoretical and Applied Mechanics, Sindri, 1970. 2 B. M. Belgaumkar, Some topics in tribology, 1st World Conf. on Industrial Tribology, New Delhi, 19 72. 3 A. K. Banerjee, Influence of kinetic friction on the critical velocity of stick-slip motion, Wear, 12 (1968) 107 - 116. 4 A. K. Banerjee, Forced oscillations in a stick-slip system, Rev. Roum. Sci. Tech., Ser. Met. Appl., 16 (1969) 219 - 228. 5 A. K. Banerjee, On the mechanics of stick-slip motion and its elimination, Ph.D. Thesis, Indian Institute of Technology, Kharagpur, 1969. 6 B. M. Belgaumkar and A. S. R. Murty, Internal friction of materials, 1st World Conf. on Zndustrial Tribology, New Delhi, 19 72. 7 E. V. Wilms, Forced vibration of a single-degree-of-freedom system with Coulomb bearing friction, J. Appl. Mech., 36 (1969) 571 - 573. 8 E. V. Wilms and H. Cohen, A two-degree-of-freedom system with Coulomb bearing friction, J. Appl. Mech., 46 (1979) 217 - 218. 9 E. V. Wilms, Damping of mechanical systems by acceleration dependent moments, 2. Angew. Math. Mech., 20 (1969) 988 - 990. 10 T. E. Simkims, The mutuality of static and kinetic friction, Lubr. Eng., 23 (1967) 26 - 31. 11 V. I. Savenko, L. A. Kochanova and E. D. Shchukin, An investigation of boundary friction by ultramicrosclerometry, Wear, 56 (1979) 297 - 307.
119
Short Communication
The influence of the Coulomb, viscous and a~~eleration~ependent terms of kinetic friction on the critical velocity of stick-slip motion*
El.M. BELGAUMKAR Vainatheya, (India)
III Floor, No. 1 Flat, S.V. Road, Irk,
Vile-Parle
(West), Bombay
400056
(Received October 17,1980)
To explain the experimentally observed highly complex hehaviour of friction in stick-slip motion an acceleration-dependent term with a fictitious mass is added to the friction force function in the governing equation of motion.-A new concept of complex mass is introduced into the modified equation. The inertia effect and the energy-absorbing effect of the complex mass are discussed. The influences of the Coulomb and viscous terms in the friction force function are described and the nature of the influence of the acceleration-dependent term is explained. It is postulated that the influence of the acceleration-dependent term is due to a molecular dislocation phenomenon on the contacting surfaces. It is suggested that an approach based on dislocation dynamics is necessary to understand more fully the behaviour of stick-slip motion. 1. Introduction A satisfactory analytical solution explaining the complex behaviour of friction in all its aspects is not yet available. Friction behaviour in the stickslip motion of two sliding surfaces requires elucidation. When a thin film of lubricant is applied between two sliding surfaces an additional friction factor, other than the friction between the metalmetal contact, is introduced. If the lubricant film has a shear strength lower than that of the metal the coefficient of friction will be reduced. For a substantial reduction in the friction coefficient the lubricant must produce two conditions in the system: a low value of the real contacting zirea over which metal-metal contact exists and a low ratio of the shear strength of the lubricant film to the shear strength between the metal surfaces. As well as reducing friction and wear, a good lubricant has a third function: the elimination of stick-slip motion. *Paper presented at the 15th InternationalCongress on Theoretical and Applied Mechanics,Toronto, August 1980.
120
Considering an elastically restrained slider bearing, it can be shown that stick-slip behaviour occurs only when the rate of variation in the friction coefficient with respect to the relative velocity of the slider is negative. There is a critical velocity of drive below which stick-slip motion occurs and above which the body moves in a smooth and continuous manner. The expression for the critical velocity u, is given by LIF uc = A(~~)112 where AF is the difference between the static friction force and the kinetic friction force, A = {!&rc/(km)1~2}1~2is a dimensionless parameter and n, c and Fzare the well-known parameters of a vibrating system. The critical velocity at which stick-slip motion disappears and is replaced by smooth sliding has been explained by assuming a sudden drop in the static friction to a lower kinetic friction at the start of slip, followed by a constant or linearly dependent kinetic friction. Experimentally, it is shown that friction does not fall abruptly and that the behaviour of friction is highly complex, showing a memory effect evidenced by hysteresis. In the light of the above facts it has been suggested [l J that the problem should be approached from a new angle. A tentative hypothesis that friction depends on the acceleration as well as the velocity [2] has extended the work of Banerjee [3 - 51. An entirely kinetic friction-dependent analysis of the critical velocity for stick-slip motion was carried out by Banerjee [3] and the effects of force excitation on a stick-slip system in the direction of its motion have also been studied [4] . Some facts, which are not explained by the linearized theory and which can be explained effectively by introducing non-linearities in the system have been considered [ 51. The influence of the addition of an acceleration-dependent term to the friction function is considered in this paper. This term can be combined with the inertia term of the governing equation and treated mathematically. Although a classical mechanics approach does not present any difficulty in obtaining a solution of the equation the physical interpretation of the influence of the total mass, combined for mathematical convenience, needs careful attention. To bring this point out clearly, a new concept of “complex mass” is in~oduced in the mathematics analysis and the discussion that follows. 2. Analysis The kinetic friction Fk is represented in the analysis of the stick-slip motion of an elastically restrained slider bearing as Fk=Fo-&Z+pjC2+m&rl(R) where, except for the last term mf sgn(V), all symbols have the same meanings as those used by Banerjee [3].
(2)
121
The term m, sgn(R) is added also to account for the dependence of friction on acceleration. Evidently a signum function is required so that the force has the correct sign for all accelerations and positions. It may be observed that the linearity of the term m, sgn(R) ensures the parabolic friction-velocity relation assumed earlier [ 31. The general equation of motion of the stick-slip system becomes m*if+c(i-v)+k(x-ut-kO)+FO-&--fl2=0
(3)
where m* = m + mf. Since m* in eqn. (3) is mathematically of the same form as m in eqn. (1) of ref. 3, what follows is that the entire mathematical analysis of Banerjee will be valid here, except that wherever m occurs it should be replaced by m* which cannot be disposed of as an equivalent mass. It needs a different treatment. The influence of m on the system is an inertia effect while that of mf, which is a fictitious mass, is energy dissipation. The two effects are not in phase. Because there is a phase difference of n/2 between the two effects, m + mf should be used vectorially. The concept of a complex modulus E* is widely used [6]. A similar notion is required here to explain the function of m* , which is introduced as the complex mass and is defined as m* = mf + im
(4)
where mf and m are the real and imaginary parts respectively of the complex mass. Although mf is fictitious, because it produces a damping effect in the system it is represented on the real axis in the above definition of the complex mass. The magnitude of mf is given by (5)
mf =mcot6 where 6 is the angle that mf makes with m* in the phase plane. Equation (4) may be written as m* = m,(l
+ id)
(4a)
where, as for a complex modulus, d is the loss factor and is defined by d=tan6
=m/m,
(W
Thus it is possible to determine m* by measuring the real part m, and the loss factor d. It will be shown later that this loss factor depends on parameters such as the range of stress, the crystal structure, the density and the type of dislocation. To estimate the loss factor in any stick-slip system, measuring devices and a measuring method need to be developed, e.g. by using the so-called Young’s fringe technique (laser-generated speckles) for displacement and strain analysis.
122
3. Discussion of the friction function Consideration of the influence of lm*l on each term in eqn. (2) affecting the behaviour of stick-slip motion requires careful study. The term F, represents the Coulomb kinetic friction force at zero sliding velocity and the well-known relation F, = pN is a simplified model. The response of the system where F,, remains constant is well known and needs no further attention. F, does not remain generally constant but varies both in magnitude and in direction with time, giving rise to a non-linear equation of motion. This case has been analysed [ 7, 81 and it has been shown that oscillations persist indefinitely with time, contrary to the result obtained when F, is constant. The resulting damping force has a viscous character, despite the fact that it arises from the Coulomb friction. The second term, &, represents the influence of newtonian viscous drag in the system, while the third term, f13i2,accounts for the non-newtonian viscous drag that introduces non-linearity into the system. The influences of (Yand fl are discussed by Banerjee [3] who uses some known facts concerning stick-slip motion. It is argued that in the term lc - 2fiiI which is the absolute value of the slope of the assumed friction function, as cxincreases, the critical velocity u, increases while, as /3 increases, u, decreases. Damping of the acceleration-dependent forces has been considered by Wilms [9] without aiding the understanding of the influence of the mf sgn(jf) term on stick-slip motion behaviour. In order to explain the memory effect evidenced by hysteresis it is necessary to depend on much more fundamental studies of friction forces. Some of the results of Simkins [lo] and of Savenko et al. [ 111 may offer a means of approach to the solution of the complex problem. It was shown [lo] by carefully designed and conducted experiments that an extensive sliding contact on a microscopic scale was observed before gross (visible) sliding. A close examination of ref. 10, Fig. 6, which exhibits a plot of the friction force against the displacement, shows that, as with the load-extension diagram of any ductile steel, there is an initial straight steep inclined portion of the curve indicating an apparent elastic resistance to displacement. This is followed by a sharp bend and a straight horizontal line corresponding to yield followed by plastic flow in the load-extension diagram of steel. The curve then rises with a reverse bend followed by a straight inclined portion not so steep as that at the initial stage, thus indicating an apparent strain-hardening effect. The remaining portion of the curve repeats several stages similar to the initial stage. It appears that the observed phenomenon is due to some type of dislocation mechanism between the molecules of the sliding surfaces. Savenko et al.% [ 111 results appear to confirm this. Savenko et al. [ll] studied the generation of dislocations in ionic crystals caused by boundary friction both in the presence and in the absence of a lubricant and they showed .that the formation of dislocations is mainly dependent on the tangential stresses due to the adhesion of contacting, bodies and the stresses arising from their displacement. A similar effect may be expected in the influence of the complex mass. The increase in the
123
absolute value of the complex mass is expected to reduce the critical velocity and to explain the memory effect. It may be concluded that a dislocation dynamics approach may be necessary to elucidate further the behaviour of stick-slip motion. 1 B. M. Belgaumkar, Ten years of tribology, Presidential address, 15th Congr. of the Indian Society of Theoretical and Applied Mechanics, Sindri, 1970. 2 B. M. Belgaumkar, Some topics in tribology, 1st World Conf. on Industrial Tribology, New Delhi, 19 72. 3 A. K. Banerjee, Influence of kinetic friction on the critical velocity of stick-slip motion, Wear, 12 (1968) 107 - 116. 4 A. K. Banerjee, Forced oscillations in a stick-slip system, Rev. Roum. Sci. Tech., Ser. Met. Appl., 16 (1969) 219 - 228. 5 A. K. Banerjee, On the mechanics of stick-slip motion and its elimination, Ph.D. Thesis, Indian Institute of Technology, Kharagpur, 1969. 6 B. M. Belgaumkar and A. S. R. Murty, Internal friction of materials, 1st World Conf. on Zndustrial Tribology, New Delhi, 19 72. 7 E. V. Wilms, Forced vibration of a single-degree-of-freedom system with Coulomb bearing friction, J. Appl. Mech., 36 (1969) 571 - 573. 8 E. V. Wilms and H. Cohen, A two-degree-of-freedom system with Coulomb bearing friction, J. Appl. Mech., 46 (1979) 217 - 218. 9 E. V. Wilms, Damping of mechanical systems by acceleration dependent moments, 2. Angew. Math. Mech., 20 (1969) 988 - 990. 10 T. E. Simkims, The mutuality of static and kinetic friction, Lubr. Eng., 23 (1967) 26 - 31. 11 V. I. Savenko, L. A. Kochanova and E. D. Shchukin, An investigation of boundary friction by ultramicrosclerometry, Wear, 56 (1979) 297 - 307.