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Analysis of the performance characteristics of a two-inertia power transmission system with a plate clutch
Vol. 24, No. 6, pp. 499-503, 1989 Printed in Great Britain
0094-114X/89 $3.00 + 0.00 Pergamon Press plc
Mech. Math. Theory
ANALYSIS OF THE PERFORMANCE OF A TWO-INERTIA POWER SYSTEM
WITH
A PLATE
CHARACTERISTICS TRANSMISSION CLUTCH
M. R. R A G H A V A N and R. J A Y A C H A N D R A N Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012, India
Abstract--The performance of a plate clutch in a two-inertia power transmission system is analysed assuming negligible compliance and using a piecewise linear function to represent the clutch torque characteristic. Expressions defining, for all linear segments of the clutch torque characteristic, dimensionless input and output velocities of the clutch and dimensionless slip period are presented. The use of these expressions in preparing design charts to aid analysis and design Of the plate clutch is outlined.
INTRODUCTION Using a piecewise linear function to represent the nonlinear clutch torque characteristic and assuming negligible compliance of shaft in a two-inertia power transmission system, expressions have been developed [1] for defining in engineering units the input and output velocities of the clutch and the slip period. The authors present in this paper expressions defining, for all linear segments of the clutch torque characteristic dimensionless input and output velocities of the clutch and dimensionless slip period. The use of these expressions in preparing design charts to aid analysis and design of the plate clutch in mechanical power transmission system is outlined. C O E F F I C I E N T OF F R I C T I O N The coefficient of friction, between the sliding surfaces in a clutch varies with the sliding velocity (Fig. la), r, the number of engagements of the clutch, pressure, and temperature. In the general case, the variation of # with v can be approximately represented by a piecwise linear function. Accordingly, between sliding velocities e,_ ~and v, the i ~hsegment of the # vs v curve is linear with slope mi; it is given by P = Po, - m,r
(1)
T O R Q U E CAPACITY OF A P L A T E C L U T C H With p defined as in equation (1), the clutch torque T at any instant t in the time interval ( t , - t , _ i) or in the ith linear segment of the clutch torque curve (Figs Ib and lc) is given by T = To,(l - ~t,co,(t))
(2)
where ~,(t) = to1 (t) - coat)
(2a)
co~(t) and co:(t) are input and output angular velocities of the clutch at the instant, t. TOi and :t, are defined by equations (3) and (4) respectively for uniform rate of wear theory and by equations (5) and (6) respectively for uniform pressure theory. To~ = ½PPoiRi (i + R 2 / R I ) :t i = ~ ( m i / # o , ) R I ( ( ( R 2 / R
(3) l )3 _ I ) i ( ( R : / R I
)2 _ l))
To, = ~Pkto, R, (((R2/R,)3 _ I)/((R,./R, )2 _ I)) :ti= ¼(m~/#o,)Rt (((R2/RI)4 _ I)/((R,./RI )3 _ I))
(4) (5) (6)
where P = normal load on the clutch plates; Rt and R: are the inner and outer radii of the clutch
plate. 499
500
m . R . RAGHAVANand R. JAYACHANDRAN
Tl
~
G ( i ÷ 1~
To ( i +
1)
t.t oi
II I \
I I I
/
,,ppro,,,,,otJo,~
I I I (i-1)
0
V
i-1
I
1
I.
2 1 0 fOr(t) Fig. lb. Actual and approximate clutch torque characteristics.
Fig. la. Variation of friction coefficient with sliding velocity.
i
For a multiple plate clutch, with the assumption that all the n pairs of sliding surfaces are subjected to the same normal load P, the clutch torque 7", is given by
7", = nT MATHEMATICAL
MODELS
FOR
(7) TWO-INERTIA
POWER
T R A N S M I S S I O N SYSTEMS The elements of a simplified two-inertia power transmission system without a gear box is shown in Fig. 2. Here Ii and/~ are the moment ofinertias on the prime mover and load sides of the clutch, Te and Tr are the prime mover and load torques respectively. The model shown in Fig. 2 represents the system with a gear box between the clutch and the load provided equivalent values of Tr and /2 are used in place of T~ and I; respectively. The equivalent of 7", and I:. are T, = T;/G; [;. =/2/G'-
(8)
where G is the reduction ratio of the gear box. T~ and 1'2 are the load torque and moment of inertia respectively on the output side of the gear box.
Dimensionless angular velocities and slip period The two-inertia power transmission system is analysed assuming that (i) the compliance of the shaft is negligible and (ii) during any time interval, (ti - ti_ ~), the prime mover output torque, T~,, the load torque, T,~, the slope parameter of the clutch torque characteristic, ~ (Fig. ic), are constant.
Slope
.
EI
i
CZ i
.
(i-t)
~rCt) Fig. lc. A linear segment of clutch torque characteristic.
Two-inertia power transmission system
I
Prime mover
501
i
l,, I
Te
Ldt(t)
~2(t)
Iz
Fig. 2. Two-inertia power transmission system.
During the time interval ( t i - t,_ i), with t,_ ~ as the reference time, i.e., t~_ t = 0, the equations of motion for this system are Ii ( do91/dt ) - ~, To,(cot ( t ) - oJ,( t ) ) = T,i - To,
(9)
1 2 ( d w l / d t ) + :~,To,(Col(t) - o 9 , . 0 ) ) = T o , - T,,
(10)
Expressions for cot (t) and co2(t) are obtained by simultaneously solving the differential equations (9) and (10). The dimensionless angular velocities ~cot(/) and a:o2(t) at any time. t are given by ~,co t (t) = P~,(exp(STo,/T~,AL) - 1) - Q , S + ~ , ~ (t,_t)
(11)
aio~,_(t) = P,.i(exp(STo~/T, iAi ) - 1) - Q i S + ~io~,.(ti_ t)
(12)
P,, = A, ((( T,,/To,) - l) + Q,( T,,/To,) + ~,oJ,(ti_ ,))
(1 la)
Q, = ((Tr,/T,,) - 1)/(1 + (6111))
(llb)
A, = (1:/11)1(1 + (I, II, )
(llc)
where
(lid)
S = ~ T , it/ll
(lle)
~or(i -- 1) = COt (/i_ t) -- ~.,(t,- t)
P: = A, (11/I2 ) (( 1 -- Tei/To,) + Oi ( T~i/To,)) - (I./12 )~, (-.Or(i -- 1 ))
(12a)
The dimensionless slip velocity at any time, t in the time interval t,_~ to t, is derived from equations (11) and (12) as :~,eg,(t) = A,((I + (t:/I,.))~,og,(i - 1) + ((7",,/To,) - 1) - (I.//2)(1 - ( T , , / T o , ) ) e x p ( S T o , / L ,
AI) - A.((L,/To,)
- 1) - I./I,.(1 - Tr,/To,)))
(13)
The dimensionless slip period. S, for the dimensionless slip velocity to change from :~ge~,(i- 1) to ~ico,(t) is given by S - tTe'~'-
I,
1 In (C,~,ogr(t)) + B, (To,/T,,)A~ (C,~,oJ,(i - 1)) + &
(14)
where B, = (1211 ~)(1 - (To, IT,,)) - ((To, IT,,) - (T,,/T,i))
(14a)
C, = (! + (I:II~))(To, IT,,)
(14b)
Equation (14) defines uniquely for all the linear segments of a piecewise linearised clutch torque characteristic the dimensionless slip period S or tT¢,o~,/I t in terms of the dimensionless parameters To,/T,i, T,,/Te~,I,_/ll. ~ , o ~ , ( i - 1) and ~o~,(t). Equations (11) and (12) define the dimensionless
502
M.R.
RAGHAVAN a n d R. JAYACHANDRAN
ecitOr(i-1)= 0.025
0.6
~.i(.Or(i ) = 0
°t
0.5
Tri = 0 . 8 T ¢ i
0.4 Si
0.3
= 10
0.2 0.1 0.0
1
0
2
3
4
5
ToilT¢i
0.7 ~i tOr ( i - I ) =0.1 ctiCOr(i) =0 Tri = O. 8T¢i
0.6 0.50.4Si
12111=10
0a-
2 1
0.20.10.0
I
0
2
0.7
////
0.6
5
~iLOr(i)=O
Tri = O. 8 T¢i
/\\ \
0.4 Si
3
o,
~/1/
0.5
Toi I Tei
= lo
0.3 0.2 0.10.0 .l ,
I
I
I
I
1
2
3
4
5
Toil Tel
Fig. 3. Effect of T~,/T,, on Si at various /2/Ti.
angular velocities ~ioot(t) and ~iCO2(t) during this period. The total time of engagement or slip period (t,),, for the angular velocity to decrease from co,(o) to co,(m) is given by
(t,)~ = ~ tei i=1
where t,~ is the slip period (equation 14) for the dimensionless relative angular velocity to decrease from ~tta;,(i - 1) to ~to,(i). a~,(i) is the relative angular velocity at which the slope parameter ~i, changes from ~ti to cti÷.. Design charts or tables giving the dimensionless slip period & and the dimensionless angular velocities ct~coz(t) and ~o92(t) during this slip period can be developed for
Two-inertia power transmission system
503
all practical values of the parameters Toi/T,,, I2/lt, ~ico,(i I) and ~co,(i). These charts/tables are useful for estimating quickly the total slip period (t,),, and the corresponding variations in cot (t) and co2(t) in a two-inertia power transmission system with a non-linear clutch torque characteristic. -
Design charts A computer programme is used to study through equations (11), (12) and (14) the effect of the parameters To~/T~, T,~/T,i, I2/I1, aico,(i - 1) and c~ico,(i) on Si and the corresponding changes in the dimensionless angular velocities ~col(t) and ~ico2(t). Design charts are prepared to adequately cover the ranges over which the above mentioned five parameters vary in practice. Each design chart present for constant values of parameters: ~co,(i - 1), ~,co,(i) and T,,/Te~, the effect of To~/T,i on S~ at three or four values of 12/11 varied over the range 1-10; a supplementary figure shows the variations in ~co~(t) and ~co2(t) during the dimensionless slip period, St, For purposes of illustration, three of these design charts are presented in Figs 3-5 and the salient features highlighted. In all these three charts ~co,(i) and T,/Te, are kept constant at zero and 0.8 respectively. The values of ~cor(i - 1) corresponding to Figs 3, 4 and 5 are 0.025, 0.1 and 0.25 respectively. Curves presented in each figure show the variation of 5,. with To~/Te~ for 3 or 4 values of I2/I~ in the range 1-10. A study of these figures reveal the following features: (i) At ~cor(i - !) values of 0.025 and 0.I, the variations of S~ with To~/T,~ is insignificant for To~/Te~ values above 1.5 and 2.5 respectively, However it is found that at Gcico~(i - l) above 0.5, the variation of S; with To~/T¢~ is insignificant only at To~/T,~ above 4. (ii) for ~,co,(i - 1) ~<0.1 and (To~/T,i) >12, S~ is rather insensitive to variation of 12/11. (iii) The total slip period, (t,), can be reduced by decreasing the slope parameter ~ of the linear segments in the clutch torque characteristic. This can be achieved by choosing a lining material with high #o~, low m~, high permissible lining pressure and low rate of wear. CONCLUSIONS For a linear segment of a clutch torque characteristic (approximated by a piecewise linear function), expressions are given for dimensionless slip period, input and output dimensionless angular velocities of the plate clutch in a two-inertia power transmission system. Design charts based on these expressions aid significantly the analysis and design of the clutch in a power transmission system. REFERENCE 1. Z. J. Jania and D. Sinclair, Friction clutchesand brakes, In Mechanical Design and Systems Handbook (Edited by H. A. Rothbart), pp. 28-1 to 28-35. McGraw-Hill,New York (1964). N A C O M M 87 ANALYSE DER CHARAKTERISTIKA EINES ZWEI-TR.~GHEIT-KRAFT~BERTRAGUNGS SYSTEMS MIT EINER SCHEIBENKUPPELUNG Zusammenfassung--Die Aufflahrungeiner Scheibenkuppelung in einem Zwei-Tr§.gheits-Krafttabertragungs systems wird analysiert unter Annahme der vernachl~issigbarer Nachigiebigkeit. Die Kuppelungsmomentenkurve wurde dutch eine st~ckweiserlinearen Funktion ersetet. Ausdriicke werden pr~sentiert, die f~r alle linearen Stficke der Momentenkurve, die normierten Ein- und Ausgangs geschwindigkeitender Kuppelung und die normierte Schluptdauer definieren. Die Verwendungdieser AusdrOcke f13rdie Vorbereitungvon kurventafelnffir die konstructionwird gezeigt, um die Analyseund den Entwurf von Selaeibenkupplungenzu unterst~tzen.
M M T 24 6--D
0094-114X/89 $3.00 + 0.00 Pergamon Press plc
Mech. Math. Theory
ANALYSIS OF THE PERFORMANCE OF A TWO-INERTIA POWER SYSTEM
WITH
A PLATE
CHARACTERISTICS TRANSMISSION CLUTCH
M. R. R A G H A V A N and R. J A Y A C H A N D R A N Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560 012, India
Abstract--The performance of a plate clutch in a two-inertia power transmission system is analysed assuming negligible compliance and using a piecewise linear function to represent the clutch torque characteristic. Expressions defining, for all linear segments of the clutch torque characteristic, dimensionless input and output velocities of the clutch and dimensionless slip period are presented. The use of these expressions in preparing design charts to aid analysis and design Of the plate clutch is outlined.
INTRODUCTION Using a piecewise linear function to represent the nonlinear clutch torque characteristic and assuming negligible compliance of shaft in a two-inertia power transmission system, expressions have been developed [1] for defining in engineering units the input and output velocities of the clutch and the slip period. The authors present in this paper expressions defining, for all linear segments of the clutch torque characteristic dimensionless input and output velocities of the clutch and dimensionless slip period. The use of these expressions in preparing design charts to aid analysis and design of the plate clutch in mechanical power transmission system is outlined. C O E F F I C I E N T OF F R I C T I O N The coefficient of friction, between the sliding surfaces in a clutch varies with the sliding velocity (Fig. la), r, the number of engagements of the clutch, pressure, and temperature. In the general case, the variation of # with v can be approximately represented by a piecwise linear function. Accordingly, between sliding velocities e,_ ~and v, the i ~hsegment of the # vs v curve is linear with slope mi; it is given by P = Po, - m,r
(1)
T O R Q U E CAPACITY OF A P L A T E C L U T C H With p defined as in equation (1), the clutch torque T at any instant t in the time interval ( t , - t , _ i) or in the ith linear segment of the clutch torque curve (Figs Ib and lc) is given by T = To,(l - ~t,co,(t))
(2)
where ~,(t) = to1 (t) - coat)
(2a)
co~(t) and co:(t) are input and output angular velocities of the clutch at the instant, t. TOi and :t, are defined by equations (3) and (4) respectively for uniform rate of wear theory and by equations (5) and (6) respectively for uniform pressure theory. To~ = ½PPoiRi (i + R 2 / R I ) :t i = ~ ( m i / # o , ) R I ( ( ( R 2 / R
(3) l )3 _ I ) i ( ( R : / R I
)2 _ l))
To, = ~Pkto, R, (((R2/R,)3 _ I)/((R,./R, )2 _ I)) :ti= ¼(m~/#o,)Rt (((R2/RI)4 _ I)/((R,./RI )3 _ I))
(4) (5) (6)
where P = normal load on the clutch plates; Rt and R: are the inner and outer radii of the clutch
plate. 499
500
m . R . RAGHAVANand R. JAYACHANDRAN
Tl
~
G ( i ÷ 1~
To ( i +
1)
t.t oi
II I \
I I I
/
,,ppro,,,,,otJo,~
I I I (i-1)
0
V
i-1
I
1
I.
2 1 0 fOr(t) Fig. lb. Actual and approximate clutch torque characteristics.
Fig. la. Variation of friction coefficient with sliding velocity.
i
For a multiple plate clutch, with the assumption that all the n pairs of sliding surfaces are subjected to the same normal load P, the clutch torque 7", is given by
7", = nT MATHEMATICAL
MODELS
FOR
(7) TWO-INERTIA
POWER
T R A N S M I S S I O N SYSTEMS The elements of a simplified two-inertia power transmission system without a gear box is shown in Fig. 2. Here Ii and/~ are the moment ofinertias on the prime mover and load sides of the clutch, Te and Tr are the prime mover and load torques respectively. The model shown in Fig. 2 represents the system with a gear box between the clutch and the load provided equivalent values of Tr and /2 are used in place of T~ and I; respectively. The equivalent of 7", and I:. are T, = T;/G; [;. =/2/G'-
(8)
where G is the reduction ratio of the gear box. T~ and 1'2 are the load torque and moment of inertia respectively on the output side of the gear box.
Dimensionless angular velocities and slip period The two-inertia power transmission system is analysed assuming that (i) the compliance of the shaft is negligible and (ii) during any time interval, (ti - ti_ ~), the prime mover output torque, T~,, the load torque, T,~, the slope parameter of the clutch torque characteristic, ~ (Fig. ic), are constant.
Slope
.
EI
i
CZ i
.
(i-t)
~rCt) Fig. lc. A linear segment of clutch torque characteristic.
Two-inertia power transmission system
I
Prime mover
501
i
l,, I
Te
Ldt(t)
~2(t)
Iz
Fig. 2. Two-inertia power transmission system.
During the time interval ( t i - t,_ i), with t,_ ~ as the reference time, i.e., t~_ t = 0, the equations of motion for this system are Ii ( do91/dt ) - ~, To,(cot ( t ) - oJ,( t ) ) = T,i - To,
(9)
1 2 ( d w l / d t ) + :~,To,(Col(t) - o 9 , . 0 ) ) = T o , - T,,
(10)
Expressions for cot (t) and co2(t) are obtained by simultaneously solving the differential equations (9) and (10). The dimensionless angular velocities ~cot(/) and a:o2(t) at any time. t are given by ~,co t (t) = P~,(exp(STo,/T~,AL) - 1) - Q , S + ~ , ~ (t,_t)
(11)
aio~,_(t) = P,.i(exp(STo~/T, iAi ) - 1) - Q i S + ~io~,.(ti_ t)
(12)
P,, = A, ((( T,,/To,) - l) + Q,( T,,/To,) + ~,oJ,(ti_ ,))
(1 la)
Q, = ((Tr,/T,,) - 1)/(1 + (6111))
(llb)
A, = (1:/11)1(1 + (I, II, )
(llc)
where
(lid)
S = ~ T , it/ll
(lle)
~or(i -- 1) = COt (/i_ t) -- ~.,(t,- t)
P: = A, (11/I2 ) (( 1 -- Tei/To,) + Oi ( T~i/To,)) - (I./12 )~, (-.Or(i -- 1 ))
(12a)
The dimensionless slip velocity at any time, t in the time interval t,_~ to t, is derived from equations (11) and (12) as :~,eg,(t) = A,((I + (t:/I,.))~,og,(i - 1) + ((7",,/To,) - 1) - (I.//2)(1 - ( T , , / T o , ) ) e x p ( S T o , / L ,
AI) - A.((L,/To,)
- 1) - I./I,.(1 - Tr,/To,)))
(13)
The dimensionless slip period. S, for the dimensionless slip velocity to change from :~ge~,(i- 1) to ~ico,(t) is given by S - tTe'~'-
I,
1 In (C,~,ogr(t)) + B, (To,/T,,)A~ (C,~,oJ,(i - 1)) + &
(14)
where B, = (1211 ~)(1 - (To, IT,,)) - ((To, IT,,) - (T,,/T,i))
(14a)
C, = (! + (I:II~))(To, IT,,)
(14b)
Equation (14) defines uniquely for all the linear segments of a piecewise linearised clutch torque characteristic the dimensionless slip period S or tT¢,o~,/I t in terms of the dimensionless parameters To,/T,i, T,,/Te~,I,_/ll. ~ , o ~ , ( i - 1) and ~o~,(t). Equations (11) and (12) define the dimensionless
502
M.R.
RAGHAVAN a n d R. JAYACHANDRAN
ecitOr(i-1)= 0.025
0.6
~.i(.Or(i ) = 0
°t
0.5
Tri = 0 . 8 T ¢ i
0.4 Si
0.3
= 10
0.2 0.1 0.0
1
0
2
3
4
5
ToilT¢i
0.7 ~i tOr ( i - I ) =0.1 ctiCOr(i) =0 Tri = O. 8T¢i
0.6 0.50.4Si
12111=10
0a-
2 1
0.20.10.0
I
0
2
0.7
////
0.6
5
~iLOr(i)=O
Tri = O. 8 T¢i
/\\ \
0.4 Si
3
o,
~/1/
0.5
Toi I Tei
= lo
0.3 0.2 0.10.0 .l ,
I
I
I
I
1
2
3
4
5
Toil Tel
Fig. 3. Effect of T~,/T,, on Si at various /2/Ti.
angular velocities ~ioot(t) and ~iCO2(t) during this period. The total time of engagement or slip period (t,),, for the angular velocity to decrease from co,(o) to co,(m) is given by
(t,)~ = ~ tei i=1
where t,~ is the slip period (equation 14) for the dimensionless relative angular velocity to decrease from ~tta;,(i - 1) to ~to,(i). a~,(i) is the relative angular velocity at which the slope parameter ~i, changes from ~ti to cti÷.. Design charts or tables giving the dimensionless slip period & and the dimensionless angular velocities ct~coz(t) and ~o92(t) during this slip period can be developed for
Two-inertia power transmission system
503
all practical values of the parameters Toi/T,,, I2/lt, ~ico,(i I) and ~co,(i). These charts/tables are useful for estimating quickly the total slip period (t,),, and the corresponding variations in cot (t) and co2(t) in a two-inertia power transmission system with a non-linear clutch torque characteristic. -
Design charts A computer programme is used to study through equations (11), (12) and (14) the effect of the parameters To~/T~, T,~/T,i, I2/I1, aico,(i - 1) and c~ico,(i) on Si and the corresponding changes in the dimensionless angular velocities ~col(t) and ~ico2(t). Design charts are prepared to adequately cover the ranges over which the above mentioned five parameters vary in practice. Each design chart present for constant values of parameters: ~co,(i - 1), ~,co,(i) and T,,/Te~, the effect of To~/T,i on S~ at three or four values of 12/11 varied over the range 1-10; a supplementary figure shows the variations in ~co~(t) and ~co2(t) during the dimensionless slip period, St, For purposes of illustration, three of these design charts are presented in Figs 3-5 and the salient features highlighted. In all these three charts ~co,(i) and T,/Te, are kept constant at zero and 0.8 respectively. The values of ~cor(i - 1) corresponding to Figs 3, 4 and 5 are 0.025, 0.1 and 0.25 respectively. Curves presented in each figure show the variation of 5,. with To~/Te~ for 3 or 4 values of I2/I~ in the range 1-10. A study of these figures reveal the following features: (i) At ~cor(i - !) values of 0.025 and 0.I, the variations of S~ with To~/T,~ is insignificant for To~/Te~ values above 1.5 and 2.5 respectively, However it is found that at Gcico~(i - l) above 0.5, the variation of S; with To~/T¢~ is insignificant only at To~/T,~ above 4. (ii) for ~,co,(i - 1) ~<0.1 and (To~/T,i) >12, S~ is rather insensitive to variation of 12/11. (iii) The total slip period, (t,), can be reduced by decreasing the slope parameter ~ of the linear segments in the clutch torque characteristic. This can be achieved by choosing a lining material with high #o~, low m~, high permissible lining pressure and low rate of wear. CONCLUSIONS For a linear segment of a clutch torque characteristic (approximated by a piecewise linear function), expressions are given for dimensionless slip period, input and output dimensionless angular velocities of the plate clutch in a two-inertia power transmission system. Design charts based on these expressions aid significantly the analysis and design of the clutch in a power transmission system. REFERENCE 1. Z. J. Jania and D. Sinclair, Friction clutchesand brakes, In Mechanical Design and Systems Handbook (Edited by H. A. Rothbart), pp. 28-1 to 28-35. McGraw-Hill,New York (1964). N A C O M M 87 ANALYSE DER CHARAKTERISTIKA EINES ZWEI-TR.~GHEIT-KRAFT~BERTRAGUNGS SYSTEMS MIT EINER SCHEIBENKUPPELUNG Zusammenfassung--Die Aufflahrungeiner Scheibenkuppelung in einem Zwei-Tr§.gheits-Krafttabertragungs systems wird analysiert unter Annahme der vernachl~issigbarer Nachigiebigkeit. Die Kuppelungsmomentenkurve wurde dutch eine st~ckweiserlinearen Funktion ersetet. Ausdriicke werden pr~sentiert, die f~r alle linearen Stficke der Momentenkurve, die normierten Ein- und Ausgangs geschwindigkeitender Kuppelung und die normierte Schluptdauer definieren. Die Verwendungdieser AusdrOcke f13rdie Vorbereitungvon kurventafelnffir die konstructionwird gezeigt, um die Analyseund den Entwurf von Selaeibenkupplungenzu unterst~tzen.
M M T 24 6--D