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Optimum design of spherical 4-R function generating mechanisms
Mechanism and Machine Theory, 1974, Vol. 9, pp. 405-410. Pergamon Press. Printed in Great Britain
Optimum Design of Spherical 4-R Function Generating Mechanisms S. S. Rao*
and A. G. Ambekart
Received 25 May 1973 Abstract The problem of synthesizing spherical 4-R function generating mechanisms for minimum structural error is presented as a mathematical programming problem. The interior penalty function method is used to solve the constrained optimization problem. Three sample problems have been solved to demonstrate the method of synthesis. Introduction THE PROBLEM of optimal synthesis of function generators has received full attention in planar mechanisms. In [l], Fox and Wilimert considered the design of planar curve generating linkages as a nonlinear programming problem. In the case of 4-revolute spherical function generators, Z i m m e r m a n [ 2 ] presented an algorithm for 4-precision point synthesis. A more recent work on the synthesis of 4-R spherical m e c h a n i s m s is due to L a k s h m i n a r a y a n a [ 3 ] , wherein a five precision point algorithm has been presented. Sridhar and Torfason[4] dealt with the optimal synthesis of spherical 4-R m e c h a n i s m s using R o s e n b r o c k ' s method of rotating coordinates. The work, however, does not incorporate the transmission angle factor in the formulation. While the precision point methods have proven to be excellent for motion g e o m e t r y designs, they are usually not as effective if the required design has also to satisfy certain inequality constraints. Since the conventional methods seek to design the mechanism by a trial and error process to satisfy the prescribed constraints, it seems appropriate to resort to mathematical p r o g r a m m i n g techniques which can handle the constraints in a direct manner. In the present work, the structural error synthesis problem for a 4-R spherical function generator has been posed as a mathematical programming problem. The sum of the squares of structural error at a large n u m b e r of points, in the range of generation, f o r m s the objective function. Both inequality and equality constraints are imposed on *Assistant Professor, Department of MechanicalEngineering, Indian Institute of Technology, Kanpur-16, India. tLecturer, Department of Mechanical Engineering, S.G.S. Institute of Technology and Science, lndore, India.
405
406
the synthesis problem. T h u s the design problem can be stated as: Minimize F(Z) subject to the constraints g; ( Z ) _< 0, j -- 1 , 2 . . . . .
p
(1)
and t , ( z ) = o, j - 1, 2 . . . . .
q.
The interior penalty function m e t h o d has been used to solve this problem. Several e x a m p l e problems are given to illustrate the effectiveness of the synthesis method.
Problem Formulation Figure 1 shows a spherical 4-R m e c h a n i s m wherein the link lengths are m e a s u r e d in terms of the angles a~ s u b t e n d e d at the origin, b e t w e e n the axes zt and z~ + 1. T h e notation used is the same as that of H a r t e n b e r g and Denavit[5]. Thus the links z3
z4
12
F i g u r e 1. Illustration of t r a n s m i s s i o n a n g l e . n u m b e r e d 1, 2, 3, and 4 are respectively, the input link (crank), the f r a m e (fixed link), the output link and the coupler. If the position of the input link is m e a s u r e d as ~b~ and that of the o u t p u t link by ~bt, the d i s p l a c e m e n t equation is given by[2] At sin tpt + Bt cos tp; = C~
(2)
At = - sin ~bt
(3)
B~ = sin a, cot a , + cos ct, cos 4~
(4)
where
Ct = cos u, cot a z c o t
a 4 - sin O1~I cot a2COS ~,
COS
~3
sin a2 sin a4 "
(5)
T h u s the output angle is given by ~bt = 2 t a n , [ A , +
V(At2+Bt2-C' +C, 2)';2]B,
(6)
where V serves as a b r a n c h indicator having a value of 1 for the positive b r a n c h and - l
407
for negative branch. In the present formulation, the lengths of input link (a,), coupler (a3) and the output link (ct2), and the starting value of the input angle (6~) are taken as the design variables zl, z2, z3 and z4 respectively.
Objective function The objective or criterion function for minimization is taken as F(Z) = ~ (~b~,- ~r,) 2
(7)
where ~br, and ~, are the required and generated values of the output angle ~b at ith position of the input link, and n is the total number of points (positions) considered.
Constraints The necessity of using spherical trigonometrical relations in the calculation of transmission angle dictates the following constraints on link lengths
la, l_<
i
= 1,2,3,4.
(8)
The lower bound on the link lengths is taken as 10° in the present work. Thus Iz,
7T
i -- 1,2,3.
(9)
Similarly the arc length AC in Fig. 1 should satisfy the inequality
IX'el-<
(10)
Inequalities (8) and (10) can be satisfied if the following inequalities are satisfied - 7r + IZ21 + IZ31 - 0
(11)
-- "~ "~ [ Z , ] -{- ] a t [ ~ 0 .
(12)
In order to avoid imaginary values for the tangent of output angle ~ in equation (6) the following constraint is imposed - A , 2 - B , : + C , ~ <- O,
i = 1, 2 . . . . .
n
(13)
where the quantities A, B,, and C, are as defined in equations (3)-(5). In order to keep the mechanical error within reasonable limits, the transmission angle y has been restricted to lie between 30 ° and 150°. This constraint can be expressed as ~ - - 3'~-
--~0, i = 1,2 . . . . . n.
(14)
As in the case of a planar four bar linkage, the transmission angle is taken as the smallest angle between the direction of the relative velocity vector VBA and the direction of the absolute velocity vector VB of the output link (Fig. 1). Accordingly 3, is
408 given by the smaller one out of the angles 3/ and its complement, where
~/=2tan
'tt
pin (s -1 3t). sin (s -1~4)]"2} sin(s).sin(s-lA-'CI)
J J
(15)
with s = (Io~=1 + Io.I + IA~I)/2,
(16)
A'-'C = c o s - ' (cos a , cos a , - sin a , sin a, cos ~b).
(17)
and
Finally, the equality constraint $~, = tkr, is imposed to ensure that the generated value of output angle $,, is equal to the required value ~,,. In this work the final design is not restricted to any particular type of m e c h a n i s m by not imposing the G r a s h o f ' s criterion.
Solution Procedure The design problem formulated in the previous section can be stated in the standard f o r m of equation (1). The interior penalty function method has been used for the solution of the constrained minimization problem. In this method a function P ( Z , r) is formulated as follows: q
P ( Z , r) = F ( Z ) - r
+
I~2(Z). j=l
(18)
It is shown that the solution to the original constrained problem, equation (1), can be obtained by carrying out the unconstrained minimization of P(Z, r) for a sequence of decreasing values of r. The D a v i d o n - F l e t c h e r - P o w e l l method has been used to solve the sequence of unconstrained minimization problems. Cubic interpolation has been used for the linear minimizations that arise in the unconstrained minimization. The required initial feasible point Zo is found by using the principles of 3-accuracy point method of function generation. For each value of r, the unconstrained minimization of P is terminated w h e n e v e r the predicted percentage difference between the current and the optimal P - v a l u e s falls less than a specified small quantity STVP i.e. - - - < 2P
e
(19)
where S is the search direction for minimization.
Numerical Results The generation of the functions y = sin x, y = log x and y = x 2 are considered as example problems. The fixed link length o/1 is taken as 90 ° in all the examples. The data and results of optimization are shown in Table I. F r o m this table, the reduction in the objective function can be seen to be about 68 per cent for the first two examples and 97 per cent for the third example. The values of the structural error at various points corresponding to the initial and optimal designs are shown in Table 2. It can be seen that the m a x i m u m structural error for the initial and final solutions are 3.29 and 1.67 per cent of the range of the output angle for the case of the function y = sinx. The corresponding figures are 10.20 and 5-62 per cent for the case y = log x and 4-53 and
409 Table 1. Results of optimization
Function
Range of generation
to
be generated y = sin x O°~X
Design vector Z Initial Optimum
AS
A$
90 °
90 °
_<90 °
76.32 °
73-38 °
66.54 °
88.10 °
58.14 °
-205.50 y = log X
60 °
90 °
- 21.37 °
l__
y=X
°
2
90 °
90 °
0-003319
0.098331
0-031148
0-033813
0"001132
188.70 ° -
11.53 ° 57.45 °
55.69 °
32-76 °
145-00 °
135.21 °
- 23"70 °
0.010607
56-83 ° -
29-17 °
l<__x__
Objectivefunction* (rad.2) Initial Final
-13"93
°
66.12 °
72.59 °
- 38.13 ° - 29.29 °
- 20.86 ° -
16.88 °
*For computing the objective function, 11 equispaced points are considered. 160 150 140 130 120 Iio IOO
~ 1o
9o
d
80
o
70
.~
60
E ~
5o
i---
40 30 20 i0 0
i
2
3
4
5
6
7
8
9
I0
II
Position of the input link
Figure 2. 1-18 per cent for the case y = x 2. The synthesis by optimization techniques thus reduced the maximum structural error by about 47 per cent in the case of first two examples and 78 per cent in the case of third example. The transmission angle at various positions of the input link is shown graphically in Fig. 2. The average computer time taken for the optimization of one example is about 15 min on an IBM 7044.
Conclusion The present method of design can handle all the constraints imposed on the problem in an efficient manner. An additional advantage of the present synthesis method is that,
410
Table 2. Structural error at various points (in degrees)
Point i 1
2 3 4 5 6 7 8 9 10 11
For y = sin x Initial Optimum design design
For y = log x Initial O pt i mum design design
For y = x2 Initial O pt i mum design design
0.0 -2.8% -2.990 - 1.472 -0.143 - 0.015 - 0.884 - 1-989 - 2.568 - 2.042 - 0.022
0-0 -8.174 -9-271 - 7.327 -3-886 0.019 3.484 5.719 6.132 4.337 0.009
0.0 -2.533 -4-121 - 4-208 -2.638 0.037 2-780 4-592 4.835 3.277 0-047
0.0 - 1.521 -0.753 0.546 1.095 0-672 - 0-290 - 1-181 - 1.447 - 0-662 1.468
once the initial solution is guaranteed objective
function
of branches. errors
greatly reduces
Further
work
in the objective
0-0 -4.967 -5.114 - 3-438 - 1.145 1.038 2.652 3.226 2.474 0.195 - 3.719
against mixing of branches,
any possibility
can be done
0.0 -0-613 -0.740 - 0.432 0.113 0.603 0.755 0.400 - 0-390 - 1.068 0.455
the very nature
of getting a solution involving
by considering
both
structural
of the mixing
and mechanical
function.
References [1] FOX R. L. and W I L L M E R T K. D., Optimum design of c urve generating linkage with inequality constraints. J. Engng for lndustry 89, 144-152 (1967). [2] Z I M M E R M A N J. R., Four precision point synthesis of the spherical four bar function generators. J. Mech. 2, 133-139 (1967). [3] L A K S H M I N A R A Y A N A K., On the synthesis of the spherical four bar-five precision points. Mechanism and Machine Theory 7, 63-69 (1972). [4] S R I D H A R B. N. and T O R F A S O N L. E., Optimization of spherical four-bar path generators. A S M E P a pe r No. 70-Mech-46. [5] H A R T E N B E R G R. S. and D E N A V I T J., Kinematic Synthesis of Linkages. McGraw-Hill, N e w Y ork (~964).
OHTHMAJIbHOE HPOEIreuHe. Flpo6neMa CnHTe3a npHnnMaer alia : M u m f(z), r/le gj(z) < O; j -- 1, 2 . . . . . p n
ll(z) = 0,
j = 1, 2 . . . . .
q.
B CTarbe M~lmtM}13rlpyeMan qbyHKtlHfl a a n a ypaaHeurleM (7), a HepaBeHCTaa noKa3am,I a yp. (8-14). ~sm pemenHa ncno~b3OBaH aHyrpeaHH~ tulpa~HO~ MeTO~I C rlpHMeHeH~eM MeTOlla <(~][aaHB,OHOneqap-Hoaen, ann He npHnyx/leHHOlt MI,IHHMa3aHHH. ~aHb[ FlpHMepbl aocnpoH3ae/leHN~ qbynrun~ y -- sin x, y = logto x H y = x 2. Bo Bcex npuMepax yrnoBaa Hanna CTO~rn npHaaTa 90 r p a a . MHHHMH3HpyeMafl qbynKtma 6bma c o r p a m e a a na 60 ~ a rlepa~x aByx npnMepax H np~6~n3n~enb~o Ha 97 ~ - - B nocneaHeM. Ma~cnMan~noe oTrnonenne aennqartb~ aocnpon3BOglnMO~l rbyHruun OT 3aaaHHo~ co~pameno Ha 47 a nepab~x aByx cny~aax H na 7 8 ~ - - a nocneaneM. ~bIJIO 3aMeqeHo, qTo nepaaeHcrao yrna nepeaaqn nanfleTcfl arTHBHb~M ycnoflneM a ne~oTopblx cayqaax (pHc. 2). BpeMa ab~ncnermfl Ka~raoro npHMepa Ha DLIBM IBM 7044 cocTaanseT o~ono 15 Mauyr.
Optimum Design of Spherical 4-R Function Generating Mechanisms S. S. Rao*
and A. G. Ambekart
Received 25 May 1973 Abstract The problem of synthesizing spherical 4-R function generating mechanisms for minimum structural error is presented as a mathematical programming problem. The interior penalty function method is used to solve the constrained optimization problem. Three sample problems have been solved to demonstrate the method of synthesis. Introduction THE PROBLEM of optimal synthesis of function generators has received full attention in planar mechanisms. In [l], Fox and Wilimert considered the design of planar curve generating linkages as a nonlinear programming problem. In the case of 4-revolute spherical function generators, Z i m m e r m a n [ 2 ] presented an algorithm for 4-precision point synthesis. A more recent work on the synthesis of 4-R spherical m e c h a n i s m s is due to L a k s h m i n a r a y a n a [ 3 ] , wherein a five precision point algorithm has been presented. Sridhar and Torfason[4] dealt with the optimal synthesis of spherical 4-R m e c h a n i s m s using R o s e n b r o c k ' s method of rotating coordinates. The work, however, does not incorporate the transmission angle factor in the formulation. While the precision point methods have proven to be excellent for motion g e o m e t r y designs, they are usually not as effective if the required design has also to satisfy certain inequality constraints. Since the conventional methods seek to design the mechanism by a trial and error process to satisfy the prescribed constraints, it seems appropriate to resort to mathematical p r o g r a m m i n g techniques which can handle the constraints in a direct manner. In the present work, the structural error synthesis problem for a 4-R spherical function generator has been posed as a mathematical programming problem. The sum of the squares of structural error at a large n u m b e r of points, in the range of generation, f o r m s the objective function. Both inequality and equality constraints are imposed on *Assistant Professor, Department of MechanicalEngineering, Indian Institute of Technology, Kanpur-16, India. tLecturer, Department of Mechanical Engineering, S.G.S. Institute of Technology and Science, lndore, India.
405
406
the synthesis problem. T h u s the design problem can be stated as: Minimize F(Z) subject to the constraints g; ( Z ) _< 0, j -- 1 , 2 . . . . .
p
(1)
and t , ( z ) = o, j - 1, 2 . . . . .
q.
The interior penalty function m e t h o d has been used to solve this problem. Several e x a m p l e problems are given to illustrate the effectiveness of the synthesis method.
Problem Formulation Figure 1 shows a spherical 4-R m e c h a n i s m wherein the link lengths are m e a s u r e d in terms of the angles a~ s u b t e n d e d at the origin, b e t w e e n the axes zt and z~ + 1. T h e notation used is the same as that of H a r t e n b e r g and Denavit[5]. Thus the links z3
z4
12
F i g u r e 1. Illustration of t r a n s m i s s i o n a n g l e . n u m b e r e d 1, 2, 3, and 4 are respectively, the input link (crank), the f r a m e (fixed link), the output link and the coupler. If the position of the input link is m e a s u r e d as ~b~ and that of the o u t p u t link by ~bt, the d i s p l a c e m e n t equation is given by[2] At sin tpt + Bt cos tp; = C~
(2)
At = - sin ~bt
(3)
B~ = sin a, cot a , + cos ct, cos 4~
(4)
where
Ct = cos u, cot a z c o t
a 4 - sin O1~I cot a2COS ~,
COS
~3
sin a2 sin a4 "
(5)
T h u s the output angle is given by ~bt = 2 t a n , [ A , +
V(At2+Bt2-C' +C, 2)';2]B,
(6)
where V serves as a b r a n c h indicator having a value of 1 for the positive b r a n c h and - l
407
for negative branch. In the present formulation, the lengths of input link (a,), coupler (a3) and the output link (ct2), and the starting value of the input angle (6~) are taken as the design variables zl, z2, z3 and z4 respectively.
Objective function The objective or criterion function for minimization is taken as F(Z) = ~ (~b~,- ~r,) 2
(7)
where ~br, and ~, are the required and generated values of the output angle ~b at ith position of the input link, and n is the total number of points (positions) considered.
Constraints The necessity of using spherical trigonometrical relations in the calculation of transmission angle dictates the following constraints on link lengths
la, l_<
i
= 1,2,3,4.
(8)
The lower bound on the link lengths is taken as 10° in the present work. Thus Iz,
7T
i -- 1,2,3.
(9)
Similarly the arc length AC in Fig. 1 should satisfy the inequality
IX'el-<
(10)
Inequalities (8) and (10) can be satisfied if the following inequalities are satisfied - 7r + IZ21 + IZ31 - 0
(11)
-- "~ "~ [ Z , ] -{- ] a t [ ~ 0 .
(12)
In order to avoid imaginary values for the tangent of output angle ~ in equation (6) the following constraint is imposed - A , 2 - B , : + C , ~ <- O,
i = 1, 2 . . . . .
n
(13)
where the quantities A, B,, and C, are as defined in equations (3)-(5). In order to keep the mechanical error within reasonable limits, the transmission angle y has been restricted to lie between 30 ° and 150°. This constraint can be expressed as ~ - - 3'~-
--~0, i = 1,2 . . . . . n.
(14)
As in the case of a planar four bar linkage, the transmission angle is taken as the smallest angle between the direction of the relative velocity vector VBA and the direction of the absolute velocity vector VB of the output link (Fig. 1). Accordingly 3, is
408 given by the smaller one out of the angles 3/ and its complement, where
~/=2tan
'tt
pin (s -1 3t). sin (s -1~4)]"2} sin(s).sin(s-lA-'CI)
J J
(15)
with s = (Io~=1 + Io.I + IA~I)/2,
(16)
A'-'C = c o s - ' (cos a , cos a , - sin a , sin a, cos ~b).
(17)
and
Finally, the equality constraint $~, = tkr, is imposed to ensure that the generated value of output angle $,, is equal to the required value ~,,. In this work the final design is not restricted to any particular type of m e c h a n i s m by not imposing the G r a s h o f ' s criterion.
Solution Procedure The design problem formulated in the previous section can be stated in the standard f o r m of equation (1). The interior penalty function method has been used for the solution of the constrained minimization problem. In this method a function P ( Z , r) is formulated as follows: q
P ( Z , r) = F ( Z ) - r
+
I~2(Z). j=l
(18)
It is shown that the solution to the original constrained problem, equation (1), can be obtained by carrying out the unconstrained minimization of P(Z, r) for a sequence of decreasing values of r. The D a v i d o n - F l e t c h e r - P o w e l l method has been used to solve the sequence of unconstrained minimization problems. Cubic interpolation has been used for the linear minimizations that arise in the unconstrained minimization. The required initial feasible point Zo is found by using the principles of 3-accuracy point method of function generation. For each value of r, the unconstrained minimization of P is terminated w h e n e v e r the predicted percentage difference between the current and the optimal P - v a l u e s falls less than a specified small quantity STVP i.e. - - - < 2P
e
(19)
where S is the search direction for minimization.
Numerical Results The generation of the functions y = sin x, y = log x and y = x 2 are considered as example problems. The fixed link length o/1 is taken as 90 ° in all the examples. The data and results of optimization are shown in Table I. F r o m this table, the reduction in the objective function can be seen to be about 68 per cent for the first two examples and 97 per cent for the third example. The values of the structural error at various points corresponding to the initial and optimal designs are shown in Table 2. It can be seen that the m a x i m u m structural error for the initial and final solutions are 3.29 and 1.67 per cent of the range of the output angle for the case of the function y = sinx. The corresponding figures are 10.20 and 5-62 per cent for the case y = log x and 4-53 and
409 Table 1. Results of optimization
Function
Range of generation
to
be generated y = sin x O°~X
Design vector Z Initial Optimum
AS
A$
90 °
90 °
_<90 °
76.32 °
73-38 °
66.54 °
88.10 °
58.14 °
-205.50 y = log X
60 °
90 °
- 21.37 °
l__
y=X
°
2
90 °
90 °
0-003319
0.098331
0-031148
0-033813
0"001132
188.70 ° -
11.53 ° 57.45 °
55.69 °
32-76 °
145-00 °
135.21 °
- 23"70 °
0.010607
56-83 ° -
29-17 °
l<__x__
Objectivefunction* (rad.2) Initial Final
-13"93
°
66.12 °
72.59 °
- 38.13 ° - 29.29 °
- 20.86 ° -
16.88 °
*For computing the objective function, 11 equispaced points are considered. 160 150 140 130 120 Iio IOO
~ 1o
9o
d
80
o
70
.~
60
E ~
5o
i---
40 30 20 i0 0
i
2
3
4
5
6
7
8
9
I0
II
Position of the input link
Figure 2. 1-18 per cent for the case y = x 2. The synthesis by optimization techniques thus reduced the maximum structural error by about 47 per cent in the case of first two examples and 78 per cent in the case of third example. The transmission angle at various positions of the input link is shown graphically in Fig. 2. The average computer time taken for the optimization of one example is about 15 min on an IBM 7044.
Conclusion The present method of design can handle all the constraints imposed on the problem in an efficient manner. An additional advantage of the present synthesis method is that,
410
Table 2. Structural error at various points (in degrees)
Point i 1
2 3 4 5 6 7 8 9 10 11
For y = sin x Initial Optimum design design
For y = log x Initial O pt i mum design design
For y = x2 Initial O pt i mum design design
0.0 -2.8% -2.990 - 1.472 -0.143 - 0.015 - 0.884 - 1-989 - 2.568 - 2.042 - 0.022
0-0 -8.174 -9-271 - 7.327 -3-886 0.019 3.484 5.719 6.132 4.337 0.009
0.0 -2.533 -4-121 - 4-208 -2.638 0.037 2-780 4-592 4.835 3.277 0-047
0.0 - 1.521 -0.753 0.546 1.095 0-672 - 0-290 - 1-181 - 1.447 - 0-662 1.468
once the initial solution is guaranteed objective
function
of branches. errors
greatly reduces
Further
work
in the objective
0-0 -4.967 -5.114 - 3-438 - 1.145 1.038 2.652 3.226 2.474 0.195 - 3.719
against mixing of branches,
any possibility
can be done
0.0 -0-613 -0.740 - 0.432 0.113 0.603 0.755 0.400 - 0-390 - 1.068 0.455
the very nature
of getting a solution involving
by considering
both
structural
of the mixing
and mechanical
function.
References [1] FOX R. L. and W I L L M E R T K. D., Optimum design of c urve generating linkage with inequality constraints. J. Engng for lndustry 89, 144-152 (1967). [2] Z I M M E R M A N J. R., Four precision point synthesis of the spherical four bar function generators. J. Mech. 2, 133-139 (1967). [3] L A K S H M I N A R A Y A N A K., On the synthesis of the spherical four bar-five precision points. Mechanism and Machine Theory 7, 63-69 (1972). [4] S R I D H A R B. N. and T O R F A S O N L. E., Optimization of spherical four-bar path generators. A S M E P a pe r No. 70-Mech-46. [5] H A R T E N B E R G R. S. and D E N A V I T J., Kinematic Synthesis of Linkages. McGraw-Hill, N e w Y ork (~964).
OHTHMAJIbHOE HPOEI
ll(z) = 0,
j = 1, 2 . . . . .
q.
B CTarbe M~lmtM}13rlpyeMan qbyHKtlHfl a a n a ypaaHeurleM (7), a HepaBeHCTaa noKa3am,I a yp. (8-14). ~sm pemenHa ncno~b3OBaH aHyrpeaHH~ tulpa~HO~ MeTO~I C rlpHMeHeH~eM MeTOlla <(~][aaHB,OHOneqap-Hoaen, ann He npHnyx/leHHOlt MI,IHHMa3aHHH. ~aHb[ FlpHMepbl aocnpoH3ae/leHN~ qbynrun~ y -- sin x, y = logto x H y = x 2. Bo Bcex npuMepax yrnoBaa Hanna CTO~rn npHaaTa 90 r p a a . MHHHMH3HpyeMafl qbynKtma 6bma c o r p a m e a a na 60 ~ a rlepa~x aByx npnMepax H np~6~n3n~enb~o Ha 97 ~ - - B nocneaHeM. Ma~cnMan~noe oTrnonenne aennqartb~ aocnpon3BOglnMO~l rbyHruun OT 3aaaHHo~ co~pameno Ha 47 a nepab~x aByx cny~aax H na 7 8 ~ - - a nocneaneM. ~bIJIO 3aMeqeHo, qTo nepaaeHcrao yrna nepeaaqn nanfleTcfl arTHBHb~M ycnoflneM a ne~oTopblx cayqaax (pHc. 2). BpeMa ab~ncnermfl Ka~raoro npHMepa Ha DLIBM IBM 7044 cocTaanseT o~ono 15 Mauyr.