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The coefficient of friction for yarns
THE COEFFICIENT OF FRICTION FOR YARNS. IRVING J. SAXL, Ph.D., Consulting Physicist, Providence, R. I. SUMMARY.
In view of the importance of the frictional characteristics in the manufacture of yarns and in the final appearance of textile products, a m e t h o d has been devised with which it is possible to determine the coefficient of friction of individual yarns in absolute figures. It consists substantially in paralyzing the constant, rotary m o m e n t of an unbalanced wheel by the frictional resistance of a piece of yarn, loaded at each end with a definite weight. T h e yarn is in contact with one-half of the carefully standardized surface of the rim of the wheel. The mathematical analysis of the physical principle involved is given. The method described lends itself also to the investigation of the frictional characteristics of different weaving and knitting constructions, to the determination of the lubricating value of different throwing oil compounds and various oils, the influence of oxidation upon the lubricating value of oils, the change in the friction of yarns imparted to t h e m by sizes and other agents. The coefficients of friction for several differeat types of yarns and conditioned yarns have been tabulated. INTRODUCTION.
One of the factors t h a t influences the production of yarns, cloth and knitted goods is the friction of the individual yarn. It is the coefficient of friction, u, t h a t determines w h a t tension is appropriate during quilling or in the shuttle for giving the filling adequate resistance. Variations in yarn friction necessitate changes in the tension adjustments. If the surface of the yarn is rough, the tension weight must be reduced while for smooth, well oiled yarn, the tension has to be increased. Accordingly, the tension necessary and permissible during 789
790
IRVING
J. SAXL.
[J. F. I.
the drawing, warping and other operations varies with the coefficient of friction of the individual yarn. Furthermore, the frictional properties also influence the " h a n d l e and feel" of the finished cloth. Last but not least, the change in the coefficient of friction of yarn as imparted to the yarn by different throwing oils, sizes and other conditioning compounds is a factor, the quantitative knowledge of which is necessary in research as well as in the efficient production which is based on exact knowledge rather than rule of the t h u m b methods. The correct setting of the tension controls has to be in correspondence with the absolute, numerical values of yarn friction. A method has been developed, therefore, for determining objectively this i m p o r t a n t physical characteristic of textile materials: the coefficient of friction, u. While methods were known before for estimating qualitatively the frictional value of yarns 1 and whole skeins in relation to each other, the principle described hereafter permits the quantitative determination of absolute numerical figures of the coefficient of friction for single individual yarns as well as for yarn combinations. The qualitative methods referred to m a y be calibrated against this absolute method. The rigid theoretical proof given lends itself in an analogous manner for the determination of u as well for wires such as tungsten filaments, in weaving- and knitting-constructions as for any other type of combination of yarn with yarn, or yarn with other materials, such as lubricants. To consider extreme applications of the method: cloth for brake linings could be developed for m a x i m u m frictional characteristics with an apparatus based upon the same principle or the difference in the lubricating characteristics of various throwing oils can be determined for different yarns and also the o p t i m u m a m o u n t of total size to be deposited upon the yarn for best results. This lubricating efficiency is not constant as such b u t has been found to change with the age of 1W. B. Sellars, "Yarn Friction Tests," Textile World, Dec. (I934), pp. 24o8. John H. Skinkle and R. C. Morrison, "Lubricating Power of Oils," American Dyestuff Reporter, Vol. XXIV, No. 2, Jan. (I935). Mercier, American Wool and Cotton Reporter, Nov. i3, I93o. American Silk and Rayon Journal, Vol. LIV, No. to, Oct. (I935), p. 32.
June, I936.]
TIlE
COEFFICIENT
OF t ; R I E T I O N
FOR
YARNS.
791
the conditioned yarn, the degree of oxidation of the throwing oil compounds, etc. Furthermore, the influence of viscosity of oils and their total a m o u n t deposited upon the yarn in relation to ~, their proper combination with binders such as gelatines or starches can be studied with this method. DESION
ov
THE INSTRUMENT
Figure I is a photograph of the Friction Tester. The principle involved will become clear by studying Fig. 2, in Fro. I.
Tile friction tester, FIG. 2.
/2 i/4
16~ ~ ~17
Constructional diagram of the~lfriction tester.
792
IRVING
J.
SAXL.
lJ, 1,. i.
which the construction details are depicted diagrammatically. Attached to the base plate (I) is a vertical support (2). This stand holds a carefully machined and uniformly chromium plated wheel (3), in a balance suspension. This wheel tends to turn in the direction of the arrow on account of the aluminum arm (4) which is securely fastened to it. This arm is designed in such a manner that a force of 3o grams applied tangentially at the periphery of the wheel is necessary to overcome the torque of the lever. To ascertain a standard surface of the wheel between which end the yarn the friction takes place, the wheel has been carefully machined and chromium plated thereafter. Chromium has been chosen because it permits a uniform polish and because its hardness and durability makes it possible to maintain standard conditions of its surface over a considerable period of time. It also can be cleaned well (for instance with a piece of cotton soaked with ether) to free it from adhering oil used in soaking experiments. The frictional resistance of the yarn which is loaded by the movable weights upon the arms, 5 and 8, acts against the moment of rotation imparted to the wheel, 3, by the arm, 4, which latter is securely attached to the wheel. In practice sometimes one single movable weight only is used, the weight of one lever proper being sufficient under some conditions. The yarn itself is fastened to the levers by the binding post, 7, and the turnable peg, 6. The latter permits the takeup of the stretch of the yarn which elongates under the load of the two levers. Again, the lever bearings must turn with a minimum of friction and are well oiled. The yarn is pulled down in proportion to the weight exerted by the arms, 5 and 8, on which the movable weights, lO and II, travel. Equilibrium is reached when all three pointers play on their setting marks, II, 12 and 13, which marks are attached to the vertical stands 14 and 15 . For avoiding transverse slippage of the yarn upon the drum, yarn guides I6 and 17 are inserted in the proper position. The lower arms are held in the sensitive bearings, I9 and 20. Sokolnikoff's method is used for calculating the coefficient of friction, 2 as expressed in the following differential equations. 2 I. S. and E. S. Sokolnikoff, "Higher Mathematics for Engineers and Physicists," p. 214-216; McGraw-Hill Book Company (I934).
June, I936. ]
THE
C O E F F I C I E N T OF FRICTION FOR YARNS.
793
THEORY.
Let To and T1 be the tensions of the yarn (Fig. 3), at the points A and B. Consider an element of length As of the yarn which subtends an angle zX0 at O, with end points P and Q. Let the tension at P be T, and at Q be Y + &T, and let the normal pressure per unit of length of the arc be p, so that the total normal force on the element of the arc As is pd~s. If the angle FIG. 3.
P
T*,aT
T B
0
A
AO is assumed to be small, the normal pressure may be thought of as acting in the direction of the line ON. From the definition of the coefficient of friction u, the frictional force is equal to the product of u by the normal pressure, so that the frictional force on PQ is upzis. Since AO is small, this frictional force m a y be assumed to act at right angles to ON. If it is assumed that the yarn is at the point of slipping, the components of force along ON must balance. Hence :
Tsin AO + (Y + &T) sin&--O= p&s 2
2
or
AO (2T + zXT) sin - - = pzXs. 2
(I)
IRVING J. SAXL.
794
[J. F. I.
S i m i l a r l y , b y e q u a t i n g t h e forces a c t i n g a t r i g h t angles t o ON, (T+
AO AO AT) c o s - - -- T c o s - - = #pAs 2
2
or
A0 A T cos - - = upAs.
(2)
2
E l i m i n a t i n g pAs b e t w e e n ( I ) a n d (2) leads t o 2 T -I- A T A0 tan .... AT 2
(3)
#
F o r small v a l u e s of AO, A0 A0 tan-- = -2
2
since tan--
2
= -- + 2
..., 3-
so t h a t (3) a p p r o x i m a t e s 2T + AT AT
A0 2
I
or
__ATAo= u ( T + A T T h e limit of this r e l a t i o n as A0 --~ o is
dT --=#T. dO U p o n s e p a r a t i o n of t h e v a r i a b l e s this b e c o m e s
dT - - = udO T a n d i n t e g r a t i n g gives log T = N0 + c or
T = cle ~°.
(4)
T h e a r b i t r a r y c o n s t a n t cl w h i c h e n t e r s i n t o t h e s o l u t i o n of t h e d i f f e r e n t i a l e q u a t i o n c a n be d e t e r m i n e d f r o m t h e initial
June, I936.]
THE
COEFFICIENT OF FRICTION
FOR YARNS.
795
condition T = To when 0 = o. Substituting these values in (4) gives, T = Toe ~° so that the tension TI corresponding to the angle of the lap is
(5)
T1 = Toe "'~
T h e angle a is d e t e r m i n e d in radians. It is .in our case 7r degrees. T h e equation (5) becomes therefore T1 = Toe""
or
TI To
e."
(6)
and finally, In T1 -- In To u =
(7)
7F
With the aforementioned method the friction has been determined for several yarns. T h e following table contains the coefficients of friction for a number of yarns that are used in the textile industry. CoeB~cients of Yarn Friction. Each figure represents the average of 5 independent measurements. T y p e of X_rarn.
Rayon 200/80 (natural) . . . . . . . . . . . . . . . . . . . Rayon 200/80 (plus conditioning oil) . . . . . . . . Acetate 200/64 . . . . . . . . . . . . . . . . . . . . . . . . . . . Acetate 15o/5 ° . . . . . . . . . . . . . . . . . . . . . . . . . . . Spun Rayon 1 / 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . Sized Acetate lOO/4O. . . . . . . . . . . . . . . . . . . . . .
Tl.
7"0.
9° IO3 88 85 III 85
6T 76 56 56 88 57
In T ~ - - I n 7"0 ~r
O.126 O.IOI O.144 O.I35 o.0536
o.I27
CONCLUSIONS.
I. T h e coefficient of friction for synthetic yarn is comparatively low. A lz of o.I26 as is the case with viscose t y p e y a r n is only a fraction compared to the friction of leather on d r y metals with a ~ of a b o u t o.56. 2. T h e coefficient of friction in spun yarns is particularly lOW. 3
3. Conditioning reduces y a r n friction. 4. T h e friction of acetate t y p e yarns is generally higher t h a n the friction of viscose t y p e yarns. 3 This is one of the reasons why it is often possible to spin and weave rayon staple without slashing. VOL. 221, ~O. I32~-55
In view of the importance of the frictional characteristics in the manufacture of yarns and in the final appearance of textile products, a m e t h o d has been devised with which it is possible to determine the coefficient of friction of individual yarns in absolute figures. It consists substantially in paralyzing the constant, rotary m o m e n t of an unbalanced wheel by the frictional resistance of a piece of yarn, loaded at each end with a definite weight. T h e yarn is in contact with one-half of the carefully standardized surface of the rim of the wheel. The mathematical analysis of the physical principle involved is given. The method described lends itself also to the investigation of the frictional characteristics of different weaving and knitting constructions, to the determination of the lubricating value of different throwing oil compounds and various oils, the influence of oxidation upon the lubricating value of oils, the change in the friction of yarns imparted to t h e m by sizes and other agents. The coefficients of friction for several differeat types of yarns and conditioned yarns have been tabulated. INTRODUCTION.
One of the factors t h a t influences the production of yarns, cloth and knitted goods is the friction of the individual yarn. It is the coefficient of friction, u, t h a t determines w h a t tension is appropriate during quilling or in the shuttle for giving the filling adequate resistance. Variations in yarn friction necessitate changes in the tension adjustments. If the surface of the yarn is rough, the tension weight must be reduced while for smooth, well oiled yarn, the tension has to be increased. Accordingly, the tension necessary and permissible during 789
790
IRVING
J. SAXL.
[J. F. I.
the drawing, warping and other operations varies with the coefficient of friction of the individual yarn. Furthermore, the frictional properties also influence the " h a n d l e and feel" of the finished cloth. Last but not least, the change in the coefficient of friction of yarn as imparted to the yarn by different throwing oils, sizes and other conditioning compounds is a factor, the quantitative knowledge of which is necessary in research as well as in the efficient production which is based on exact knowledge rather than rule of the t h u m b methods. The correct setting of the tension controls has to be in correspondence with the absolute, numerical values of yarn friction. A method has been developed, therefore, for determining objectively this i m p o r t a n t physical characteristic of textile materials: the coefficient of friction, u. While methods were known before for estimating qualitatively the frictional value of yarns 1 and whole skeins in relation to each other, the principle described hereafter permits the quantitative determination of absolute numerical figures of the coefficient of friction for single individual yarns as well as for yarn combinations. The qualitative methods referred to m a y be calibrated against this absolute method. The rigid theoretical proof given lends itself in an analogous manner for the determination of u as well for wires such as tungsten filaments, in weaving- and knitting-constructions as for any other type of combination of yarn with yarn, or yarn with other materials, such as lubricants. To consider extreme applications of the method: cloth for brake linings could be developed for m a x i m u m frictional characteristics with an apparatus based upon the same principle or the difference in the lubricating characteristics of various throwing oils can be determined for different yarns and also the o p t i m u m a m o u n t of total size to be deposited upon the yarn for best results. This lubricating efficiency is not constant as such b u t has been found to change with the age of 1W. B. Sellars, "Yarn Friction Tests," Textile World, Dec. (I934), pp. 24o8. John H. Skinkle and R. C. Morrison, "Lubricating Power of Oils," American Dyestuff Reporter, Vol. XXIV, No. 2, Jan. (I935). Mercier, American Wool and Cotton Reporter, Nov. i3, I93o. American Silk and Rayon Journal, Vol. LIV, No. to, Oct. (I935), p. 32.
June, I936.]
TIlE
COEFFICIENT
OF t ; R I E T I O N
FOR
YARNS.
791
the conditioned yarn, the degree of oxidation of the throwing oil compounds, etc. Furthermore, the influence of viscosity of oils and their total a m o u n t deposited upon the yarn in relation to ~, their proper combination with binders such as gelatines or starches can be studied with this method. DESION
ov
THE INSTRUMENT
Figure I is a photograph of the Friction Tester. The principle involved will become clear by studying Fig. 2, in Fro. I.
Tile friction tester, FIG. 2.
/2 i/4
16~ ~ ~17
Constructional diagram of the~lfriction tester.
792
IRVING
J.
SAXL.
lJ, 1,. i.
which the construction details are depicted diagrammatically. Attached to the base plate (I) is a vertical support (2). This stand holds a carefully machined and uniformly chromium plated wheel (3), in a balance suspension. This wheel tends to turn in the direction of the arrow on account of the aluminum arm (4) which is securely fastened to it. This arm is designed in such a manner that a force of 3o grams applied tangentially at the periphery of the wheel is necessary to overcome the torque of the lever. To ascertain a standard surface of the wheel between which end the yarn the friction takes place, the wheel has been carefully machined and chromium plated thereafter. Chromium has been chosen because it permits a uniform polish and because its hardness and durability makes it possible to maintain standard conditions of its surface over a considerable period of time. It also can be cleaned well (for instance with a piece of cotton soaked with ether) to free it from adhering oil used in soaking experiments. The frictional resistance of the yarn which is loaded by the movable weights upon the arms, 5 and 8, acts against the moment of rotation imparted to the wheel, 3, by the arm, 4, which latter is securely attached to the wheel. In practice sometimes one single movable weight only is used, the weight of one lever proper being sufficient under some conditions. The yarn itself is fastened to the levers by the binding post, 7, and the turnable peg, 6. The latter permits the takeup of the stretch of the yarn which elongates under the load of the two levers. Again, the lever bearings must turn with a minimum of friction and are well oiled. The yarn is pulled down in proportion to the weight exerted by the arms, 5 and 8, on which the movable weights, lO and II, travel. Equilibrium is reached when all three pointers play on their setting marks, II, 12 and 13, which marks are attached to the vertical stands 14 and 15 . For avoiding transverse slippage of the yarn upon the drum, yarn guides I6 and 17 are inserted in the proper position. The lower arms are held in the sensitive bearings, I9 and 20. Sokolnikoff's method is used for calculating the coefficient of friction, 2 as expressed in the following differential equations. 2 I. S. and E. S. Sokolnikoff, "Higher Mathematics for Engineers and Physicists," p. 214-216; McGraw-Hill Book Company (I934).
June, I936. ]
THE
C O E F F I C I E N T OF FRICTION FOR YARNS.
793
THEORY.
Let To and T1 be the tensions of the yarn (Fig. 3), at the points A and B. Consider an element of length As of the yarn which subtends an angle zX0 at O, with end points P and Q. Let the tension at P be T, and at Q be Y + &T, and let the normal pressure per unit of length of the arc be p, so that the total normal force on the element of the arc As is pd~s. If the angle FIG. 3.
P
T*,aT
T B
0
A
AO is assumed to be small, the normal pressure may be thought of as acting in the direction of the line ON. From the definition of the coefficient of friction u, the frictional force is equal to the product of u by the normal pressure, so that the frictional force on PQ is upzis. Since AO is small, this frictional force m a y be assumed to act at right angles to ON. If it is assumed that the yarn is at the point of slipping, the components of force along ON must balance. Hence :
Tsin AO + (Y + &T) sin&--O= p&s 2
2
or
AO (2T + zXT) sin - - = pzXs. 2
(I)
IRVING J. SAXL.
794
[J. F. I.
S i m i l a r l y , b y e q u a t i n g t h e forces a c t i n g a t r i g h t angles t o ON, (T+
AO AO AT) c o s - - -- T c o s - - = #pAs 2
2
or
A0 A T cos - - = upAs.
(2)
2
E l i m i n a t i n g pAs b e t w e e n ( I ) a n d (2) leads t o 2 T -I- A T A0 tan .... AT 2
(3)
#
F o r small v a l u e s of AO, A0 A0 tan-- = -2
2
since tan--
2
= -- + 2
..., 3-
so t h a t (3) a p p r o x i m a t e s 2T + AT AT
A0 2
I
or
__ATAo= u ( T + A T T h e limit of this r e l a t i o n as A0 --~ o is
dT --=#T. dO U p o n s e p a r a t i o n of t h e v a r i a b l e s this b e c o m e s
dT - - = udO T a n d i n t e g r a t i n g gives log T = N0 + c or
T = cle ~°.
(4)
T h e a r b i t r a r y c o n s t a n t cl w h i c h e n t e r s i n t o t h e s o l u t i o n of t h e d i f f e r e n t i a l e q u a t i o n c a n be d e t e r m i n e d f r o m t h e initial
June, I936.]
THE
COEFFICIENT OF FRICTION
FOR YARNS.
795
condition T = To when 0 = o. Substituting these values in (4) gives, T = Toe ~° so that the tension TI corresponding to the angle of the lap is
(5)
T1 = Toe "'~
T h e angle a is d e t e r m i n e d in radians. It is .in our case 7r degrees. T h e equation (5) becomes therefore T1 = Toe""
or
TI To
e."
(6)
and finally, In T1 -- In To u =
(7)
7F
With the aforementioned method the friction has been determined for several yarns. T h e following table contains the coefficients of friction for a number of yarns that are used in the textile industry. CoeB~cients of Yarn Friction. Each figure represents the average of 5 independent measurements. T y p e of X_rarn.
Rayon 200/80 (natural) . . . . . . . . . . . . . . . . . . . Rayon 200/80 (plus conditioning oil) . . . . . . . . Acetate 200/64 . . . . . . . . . . . . . . . . . . . . . . . . . . . Acetate 15o/5 ° . . . . . . . . . . . . . . . . . . . . . . . . . . . Spun Rayon 1 / 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . Sized Acetate lOO/4O. . . . . . . . . . . . . . . . . . . . . .
Tl.
7"0.
9° IO3 88 85 III 85
6T 76 56 56 88 57
In T ~ - - I n 7"0 ~r
O.126 O.IOI O.144 O.I35 o.0536
o.I27
CONCLUSIONS.
I. T h e coefficient of friction for synthetic yarn is comparatively low. A lz of o.I26 as is the case with viscose t y p e y a r n is only a fraction compared to the friction of leather on d r y metals with a ~ of a b o u t o.56. 2. T h e coefficient of friction in spun yarns is particularly lOW. 3
3. Conditioning reduces y a r n friction. 4. T h e friction of acetate t y p e yarns is generally higher t h a n the friction of viscose t y p e yarns. 3 This is one of the reasons why it is often possible to spin and weave rayon staple without slashing. VOL. 221, ~O. I32~-55