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Combined two-line power transmission synthesis
Mechanism and Machine Theory, 1975, Vol. 10, pp. 3-9.
Pergamon Press. Printed in Great Britain
Combined Two-Line Power Transmission Synthesis I. I. A r t o b o l e v s k i i * K. A. I m e d a s h v i l i
and B. A. K o r d z a d z e
Received on 5 December 1972 Abstract The synthesis of a multistage electromechanical transmission consisting of two directly connected three-link gear differentials is proposed. The characteristic feature of the transmission is the fact that, for shifting control ranges, the effect of a structural change is used instead of a gear box. This effect is achieved by means of connecting electrical machines to the input shaft, output shaft or, finally, to the external support shaft. In this case, an expansion of the control range is achieved by means of a double change of electrical machine regimes. Continuous control and constant torque action are secured by means of shifting over at the moment when shifted links are in the zero-power regime. Introduction RECENT years have shown the ever increased use of so-called combined multiline transmissions in which differential gear mechanisms are used as torque dividers or as an element producing the misalignment error signal. As is known, variators of different physical nature--electrical, hydrodynamic, hydrostatic transmissions, pulse and frictional variators of different design are used. The term "variator" is used to denote any device to produce a speed ratio variation. In these systems a gear differential and an automatic speed variator mutually complement one another, actually realising an effective adjustable system with high efficiency, a system that may be successfully used both in power transmissions and in control systems. It is well known that the number of gear differential versions along with the different designs of flexible control branches and versions of mutual parallel branch coordination is enormous. However, only few of these designs are rational from the technical point of view [3]. In the process of directed search for a referred system's optimal version one has to deal with a large number of different, often contradictory, parameters. That is to say, in the synthesis of referred multiparameter or multicriteria systems, the task is complicated by the fact that the improvement of certain qualtities can only be achieved at the expense of a change for the worse of other qualities [1]. *Academician, Institute for the Study of Machines, Academy of Sciences of the U.S.S.R., Griboedova Street 4, Rm. 60, Moscow Center, U.S.S.R.
In this work the theory of continuous diagrams is used to solve the problem of optical system synthesis[2] and a new method of synthesis of multistage power electromechanical transmissions (EMT) is proposed. This method m a y be effectively used, for example, for automobiles working under difficult conditions of mountain roads. P o w e r transmissions for these machines must, first of all, meet the requirements of automatic gear shifting and of maintaining power torque when shifting. As is known, one stage E M T designs require the use of electric machines of relatively large size. Therefore, the one stage E M T is not likely to be used in this case. Multistage electromechanical transmissions are formed by means of introducing a gear box (GB) into the design. These gear boxes may be mounted on input or the output shafts, on electric machine shafts or in the design of the main differential gear mechanism. All of these alternative methods of gear box mounting allow considerable reduction in the sizes of electrical machines. But, in this case, the system as a whole acquires, as a rule, a new disadvantage consisting in the a p p e a r a n c e of speed and torque j u m p s when shifting gears. In our work we consider, therefore, the problem of synthesis and investigation of such multistage E M T designs which have no gear boxes and where the speed change is achieved by other possible means. We have in view in this case the gear shift in the zero-speed regime [2]. This kind of gear shift can be achieved, for example, in designs having gear boxes in their links. The shift of stages in zero-speed regimes is undesired, as the link of a zero-speed design is loaded b y a m o m e n t and its shift has to be executed under load. This, in turn, inevitably causes a jump-like change of torque at the output shaft and its waste when shifting gears. It is seen f r o m the a b o v e that there remains only one possibility--to carry out the shifting only of those links which, with a sufficient n u m b e r of revolutions per minute, remain in the idle-speed regime, i.e. to carry out the speed shift of the machines in their zero-power regime. The advantage of this shift in the unloaded link lies in the fact that it does not influence the main process of p o w e r transmission f r o m the engine to the load. In other words, this kind of speed shift does not cause traction effort losses at the output shaft. At the same time the speed and rotation m o m e n t s change continuously without jumps. As shown above, shifting in the zero-power regime, necessary for motor cars on mountain roads, may be carried out by means of multistage designs having gear b o x e s in their links. In this case each link has one gear box, consisting of a minimum of one differential. So it is necessary to introduce as many additional differentials into the transmission as there are links. The mechanical design of the transmission is in this case much more complex. Another principle of carrying out multistage designs is the principle of inserting electrical machines M I and M 2 of the transmission into some links of the main mechanism of Fig. 1. This insertion is accomplished by means of a gear reduction unit 1,2,3,4,5,6 and 7,8,11,12,9,10, but in this case this gear reduction unit is not a gear box (Fig. 1). The shifting of any machine f r o m one m e c h a n i s m link to another one is accomplished in the zero-power regime. Investigations have shown that kinetic designs without torque and speed jumps, when shifting the machine in the z e r o - p o w e r regime, can be achieved only with
/'-T
p
I
t x
/
not =0
bD/x
Figure 1 specially chosen p a r a m e t e r s of the gear reduction unit to which the machine is successively connected.
Method of Synthesis A method of kinematic synthesis of electromechanical transmissions without torque and speed j u m p s is given below. Let us introduce the following designations: nBx----input shaft rotation velocity; n = nBb,~--output shaft rotation velocity; no,, n0s--coordinates of M I machine zero regimes at zero velocities of machine M 1; no2, no~--the same for M 2 machine; C,, C2, Cs, C , - - p a r a m e t e r s depending on the main m e c h a n i s m links; Msx--input shaft rotation m o m e n t ; M = MBb,x--output shaft rotation m o m e n t ; M~,~,MMz--rotation m o m e n t s of machines M I and M2; =~ n~
]~t -
M MBx
relative values.
Let us suppose that during the control process the machine M 2 is not shifted while machine M l is. Let us determine what it is possible to get in this case. T o do this, let us write the velocity and torque equations of machines M l and M 2 before and after the process of shifting. Before the process of shifting (the first stage) we h a v e ,~M, = c , ( n
- no,);
~M: -- c : ( , ~
- no:);
] (1)
=
J
and after the process of shifting (the second stage): r~M, = C 3 ( n - ~o~);
nM2 = C 2 ( n - r~02);
__ n-r~o2 _. M, - C ~ ( n o 3 no:)'
M
]
-_ n-~_ ~. " 2 - c : n ( n o 2 - n0~)J
(2)
Comparing equations (l) and (2), we find that only the law of gear shift of the unshifted machine remains the same for both stages. When machine M 1 is shifted over,
its velocity characteristic (Fig. 2) is sharply c h a n g e d - - l i n e A B is replaced by line BC, this character of change taking place at --
h = no2,
--I
11
rIMi
= hM2
i.e. c,(no~
-
no,) = c.(a,,.
-
~o3),
f r o m which we find the slope of line B C : C~ = C, ~o: - ~o,
(3)
# ' / o 2 - no3"
It is seen f r o m equation (3) that the slope of line B C depends on the slope of line A B and the velocity coordinates of zero regimes: h0,, t~o2, t~03. F r o m expression (2) for the M I machine m o m e n t after shifting, it is seen that, at t~ = t~02, this machine passes a z e r o - p o w e r regime. According to Fig. 2, exactly at the m o m e n t of shifting the machine M1, the absolute value of its rotation velocity achieves its m a x i m u m (in point B) and the m o m e n t has a zero value. This is the case of zero-power regime shift. Equation (3) shows how to choose the slope C3 of line B C for a given slope C, of line A B . The choice of C3 according to C, means the choice of the gear ratios of gears 1,2,5,6-i3 according to the gear ratio of gears 1,2,3,4-il. N o w let us choose ~o3 according to ho,, iio,_ Fig. 2. Let us draw the straight line of the machine M 2 velocity up to the point of intersection E with the limiting straight line BE. Let n-03'be the coordinate of point E. For the points t~03 and rio3 to coincide, what is necessary for the normal shifting of the M 2 machine in its zero p o w e r regime at t1 = tio3 (see equation 2 for MM:), is that C2 = - C3 and C1 = C2, i.e. sections B C and D E must be diagonals of a right angle quadrangle D B E C . In other words sections A B and D E must be parallel. At P'l =
J'102."
~ , = C,(ho. - ~io,) = h .
(4)
....
At
(-%
~MIMM
B
I//
.%,
Figure 2
E
//N .~/
/-~,~/
n~l" ~ [ n- n°~)
I
\
_ I
/
I
.o~-',,,,
A:D/ / I \
/ \i
/I I
o. o
MM~ __
/
o
Comparing (4) and (5), we find Cl(n02
- - ~01) = C 2 ( a 0 3 - - ~'~02)
or (6)
ho2- rio1 - ho3 - rio2
as C, = C2. Equation (6) shows that, when reversing the regime of machine M 1, it is necesary, in order to reverse the regime of machine M2, to take the value rio3 according to a uniform distribution. In the same way it is possible to find the regime boa, and so on. We have proved a rather important condition concerning the necessity of a uniform distribution of zero-regime coordinates rio,, ao2, ~o3, ~o4 over the fi range in order to carry out the machine shifting in zero-power regimes without velocity and torque jumps. The character of the uniform distribution of no,, no2, ri03, a04 is shown in Fig. 3. i NM
f
2
I
0-~,-o
'),~/eo~
±
%~"',~ ,.
-%,,
N
-2
Figure 3 As a result of some investigation, it was determined that diagrams with uniform distribution of zero regimes over the range have the same value of the total rated moment irrespective of the uniform distribution spacing, namely: E[MNI = 1.333. This condition may be generalized by proving the following:
Theorem. The total rated moment of machines with multistage electromechanical transmissions does not depend upon the distribution spacing Amp, but depends only upon ~,~ max (the maximum boundary rotation velocity of the machine) and upon the common control range % Theoretical evidence. Let ~A be the given output shaft maximum velocity, and 7 the given c o m m o n range of transmission velocity control. In this case the minimum rotation velocity of the output shaft, when the electrical drive has been working for a long time, will be: ~a = - - . Y
(7)
The rated regimes of electrical machines are achieved in points that are within the limits of the first stage. Let us find the rated moments of electrical machines M1 and M2: 1
t~ - - t l o z
glM, =c,(ao,_~02). -
1
MM~_-- C~(ao2 - no,)" rio, = O.
a fi -
--
no2
c,~a02' rio2
a
n
I
= C2no~
(8)
L e t us s u b s t i t u t e fi = tia = ( f i - / 7 ) into the e x p r e s s i o n 1VIM, 1~1.,
=
1
(h./T)-
- c,a,>2
ho..
h. - ~/h,,m~_
(9)
- C,ao:n."
(a~,13.)
A s s e e n f r o m e q u a t i o n (8), the m o m e n t 1VIM2d o e s not d e p e n d on ~. T h e r e f o r e , w e c a n i m m e d i a t e l y find t h e total r a t e d m o m e n t o f t h e m a c h i n e s . W e h a v e : (10) It is s e e n f r o m e q u a t i o n s (8) a n d (9) that within t h e first stage b o t h m o m e n t s a r e p o s i t i v e (Ct = C,. > 0, fib - ~/fi0: < 0). T h e r e f o r e e q u a t i o n (10) m a y b e w r i t t e n in t h e f o l l o w i n g f o r m :
:CllqI~I=M~,,+MM_Vno:-r~. "
/'~B
1
1
1
C I ~02 -[- C2]v~02 -- C z n 0 2
(vno~-n.+l)_ \
nB
~,
C2~B"
A s C2tio2 = tiM ~x w h i c h w e find f r o m t h e f o r m u l a f o r tiM2 a n d fir = /ion = 3fio2 then: XlMNI = nM3'.... • 3fio.; rio2
and, finally, w e h a v e : ~ll~lt,r I -
T 3fi~ ma~"
(11)
A s s e e n f r o m e x p r e s s i o n (11) EI1VIN[ d o e s n o t d e p e n d on AtOp.T h i s f o r m u l a is t r u e f o r a n y d i s t r i b u t i o n of Afip. F o r e x a m p l e , at 3' = 8 a n d flu .... = 2 w e find t h a t E[MN[ = 1.333. H e r e w e h a v e o b t a i n e d t h e s a m e r e s u l t as a b o v e . F o r m u l a ( 1 l ) m a y b e o f use f o r c a l c u l a t i n g t h r e e s t a g e e l e c t r o m e c h a n i c a l t r a n s m i s s i o n s at d i f f e r e n t g i v e n v a l u e s o f 3' a n d h~ ..... So, w i t h t h e g i v e n d i f f e r e n t i a l g e a r m e c h a n i s m d e s c r i b e d h e r e , t h e a b o v e g i v e n m e t h o d a l l o w s t h e c h o i c e o f s y s t e m p a r a m e t e r s (Fig. 1) in such a w a y as to b e s t m e e t t h e g i v e n c o n d i t i o n s of c o n t i n u o u s t o r q u e a c t i o n a n d a u t o m a t i c c o n t r o l .
References [l] A R T O B O L E V S K I I I. I. and I M E D A S H V I L I K. A., Dynamics of Differential Gear-Train Mechanisms. J. Mechanisms 4, 79-92 (1%9). [2] I M E D A S H V I L I K. A., Dynamics of a Differential Gear Mechanism in Combined Systems. MachinoStroienie Academy of Sciences of U.S.S.R, Moscow (1%5). [3] I V A N C H E N K O P. N. et al., Electromechanical Transmissions. Moshgiz, M.-L. (1962). CHHTE3
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Pergamon Press. Printed in Great Britain
Combined Two-Line Power Transmission Synthesis I. I. A r t o b o l e v s k i i * K. A. I m e d a s h v i l i
and B. A. K o r d z a d z e
Received on 5 December 1972 Abstract The synthesis of a multistage electromechanical transmission consisting of two directly connected three-link gear differentials is proposed. The characteristic feature of the transmission is the fact that, for shifting control ranges, the effect of a structural change is used instead of a gear box. This effect is achieved by means of connecting electrical machines to the input shaft, output shaft or, finally, to the external support shaft. In this case, an expansion of the control range is achieved by means of a double change of electrical machine regimes. Continuous control and constant torque action are secured by means of shifting over at the moment when shifted links are in the zero-power regime. Introduction RECENT years have shown the ever increased use of so-called combined multiline transmissions in which differential gear mechanisms are used as torque dividers or as an element producing the misalignment error signal. As is known, variators of different physical nature--electrical, hydrodynamic, hydrostatic transmissions, pulse and frictional variators of different design are used. The term "variator" is used to denote any device to produce a speed ratio variation. In these systems a gear differential and an automatic speed variator mutually complement one another, actually realising an effective adjustable system with high efficiency, a system that may be successfully used both in power transmissions and in control systems. It is well known that the number of gear differential versions along with the different designs of flexible control branches and versions of mutual parallel branch coordination is enormous. However, only few of these designs are rational from the technical point of view [3]. In the process of directed search for a referred system's optimal version one has to deal with a large number of different, often contradictory, parameters. That is to say, in the synthesis of referred multiparameter or multicriteria systems, the task is complicated by the fact that the improvement of certain qualtities can only be achieved at the expense of a change for the worse of other qualities [1]. *Academician, Institute for the Study of Machines, Academy of Sciences of the U.S.S.R., Griboedova Street 4, Rm. 60, Moscow Center, U.S.S.R.
In this work the theory of continuous diagrams is used to solve the problem of optical system synthesis[2] and a new method of synthesis of multistage power electromechanical transmissions (EMT) is proposed. This method m a y be effectively used, for example, for automobiles working under difficult conditions of mountain roads. P o w e r transmissions for these machines must, first of all, meet the requirements of automatic gear shifting and of maintaining power torque when shifting. As is known, one stage E M T designs require the use of electric machines of relatively large size. Therefore, the one stage E M T is not likely to be used in this case. Multistage electromechanical transmissions are formed by means of introducing a gear box (GB) into the design. These gear boxes may be mounted on input or the output shafts, on electric machine shafts or in the design of the main differential gear mechanism. All of these alternative methods of gear box mounting allow considerable reduction in the sizes of electrical machines. But, in this case, the system as a whole acquires, as a rule, a new disadvantage consisting in the a p p e a r a n c e of speed and torque j u m p s when shifting gears. In our work we consider, therefore, the problem of synthesis and investigation of such multistage E M T designs which have no gear boxes and where the speed change is achieved by other possible means. We have in view in this case the gear shift in the zero-speed regime [2]. This kind of gear shift can be achieved, for example, in designs having gear boxes in their links. The shift of stages in zero-speed regimes is undesired, as the link of a zero-speed design is loaded b y a m o m e n t and its shift has to be executed under load. This, in turn, inevitably causes a jump-like change of torque at the output shaft and its waste when shifting gears. It is seen f r o m the a b o v e that there remains only one possibility--to carry out the shifting only of those links which, with a sufficient n u m b e r of revolutions per minute, remain in the idle-speed regime, i.e. to carry out the speed shift of the machines in their zero-power regime. The advantage of this shift in the unloaded link lies in the fact that it does not influence the main process of p o w e r transmission f r o m the engine to the load. In other words, this kind of speed shift does not cause traction effort losses at the output shaft. At the same time the speed and rotation m o m e n t s change continuously without jumps. As shown above, shifting in the zero-power regime, necessary for motor cars on mountain roads, may be carried out by means of multistage designs having gear b o x e s in their links. In this case each link has one gear box, consisting of a minimum of one differential. So it is necessary to introduce as many additional differentials into the transmission as there are links. The mechanical design of the transmission is in this case much more complex. Another principle of carrying out multistage designs is the principle of inserting electrical machines M I and M 2 of the transmission into some links of the main mechanism of Fig. 1. This insertion is accomplished by means of a gear reduction unit 1,2,3,4,5,6 and 7,8,11,12,9,10, but in this case this gear reduction unit is not a gear box (Fig. 1). The shifting of any machine f r o m one m e c h a n i s m link to another one is accomplished in the zero-power regime. Investigations have shown that kinetic designs without torque and speed jumps, when shifting the machine in the z e r o - p o w e r regime, can be achieved only with
/'-T
p
I
t x
/
not =0
bD/x
Figure 1 specially chosen p a r a m e t e r s of the gear reduction unit to which the machine is successively connected.
Method of Synthesis A method of kinematic synthesis of electromechanical transmissions without torque and speed j u m p s is given below. Let us introduce the following designations: nBx----input shaft rotation velocity; n = nBb,~--output shaft rotation velocity; no,, n0s--coordinates of M I machine zero regimes at zero velocities of machine M 1; no2, no~--the same for M 2 machine; C,, C2, Cs, C , - - p a r a m e t e r s depending on the main m e c h a n i s m links; Msx--input shaft rotation m o m e n t ; M = MBb,x--output shaft rotation m o m e n t ; M~,~,MMz--rotation m o m e n t s of machines M I and M2; =~ n~
]~t -
M MBx
relative values.
Let us suppose that during the control process the machine M 2 is not shifted while machine M l is. Let us determine what it is possible to get in this case. T o do this, let us write the velocity and torque equations of machines M l and M 2 before and after the process of shifting. Before the process of shifting (the first stage) we h a v e ,~M, = c , ( n
- no,);
~M: -- c : ( , ~
- no:);
] (1)
=
J
and after the process of shifting (the second stage): r~M, = C 3 ( n - ~o~);
nM2 = C 2 ( n - r~02);
__ n-r~o2 _. M, - C ~ ( n o 3 no:)'
M
]
-_ n-~_ ~. " 2 - c : n ( n o 2 - n0~)J
(2)
Comparing equations (l) and (2), we find that only the law of gear shift of the unshifted machine remains the same for both stages. When machine M 1 is shifted over,
its velocity characteristic (Fig. 2) is sharply c h a n g e d - - l i n e A B is replaced by line BC, this character of change taking place at --
h = no2,
--I
11
rIMi
= hM2
i.e. c,(no~
-
no,) = c.(a,,.
-
~o3),
f r o m which we find the slope of line B C : C~ = C, ~o: - ~o,
(3)
# ' / o 2 - no3"
It is seen f r o m equation (3) that the slope of line B C depends on the slope of line A B and the velocity coordinates of zero regimes: h0,, t~o2, t~03. F r o m expression (2) for the M I machine m o m e n t after shifting, it is seen that, at t~ = t~02, this machine passes a z e r o - p o w e r regime. According to Fig. 2, exactly at the m o m e n t of shifting the machine M1, the absolute value of its rotation velocity achieves its m a x i m u m (in point B) and the m o m e n t has a zero value. This is the case of zero-power regime shift. Equation (3) shows how to choose the slope C3 of line B C for a given slope C, of line A B . The choice of C3 according to C, means the choice of the gear ratios of gears 1,2,5,6-i3 according to the gear ratio of gears 1,2,3,4-il. N o w let us choose ~o3 according to ho,, iio,_ Fig. 2. Let us draw the straight line of the machine M 2 velocity up to the point of intersection E with the limiting straight line BE. Let n-03'be the coordinate of point E. For the points t~03 and rio3 to coincide, what is necessary for the normal shifting of the M 2 machine in its zero p o w e r regime at t1 = tio3 (see equation 2 for MM:), is that C2 = - C3 and C1 = C2, i.e. sections B C and D E must be diagonals of a right angle quadrangle D B E C . In other words sections A B and D E must be parallel. At P'l =
J'102."
~ , = C,(ho. - ~io,) = h .
(4)
....
At
(-%
~MIMM
B
I//
.%,
Figure 2
E
//N .~/
/-~,~/
n~l" ~ [ n- n°~)
I
\
_ I
/
I
.o~-',,,,
A:D/ / I \
/ \i
/I I
o. o
MM~ __
/
o
Comparing (4) and (5), we find Cl(n02
- - ~01) = C 2 ( a 0 3 - - ~'~02)
or (6)
ho2- rio1 - ho3 - rio2
as C, = C2. Equation (6) shows that, when reversing the regime of machine M 1, it is necesary, in order to reverse the regime of machine M2, to take the value rio3 according to a uniform distribution. In the same way it is possible to find the regime boa, and so on. We have proved a rather important condition concerning the necessity of a uniform distribution of zero-regime coordinates rio,, ao2, ~o3, ~o4 over the fi range in order to carry out the machine shifting in zero-power regimes without velocity and torque jumps. The character of the uniform distribution of no,, no2, ri03, a04 is shown in Fig. 3. i NM
f
2
I
0-~,-o
'),~/eo~
±
%~"',~ ,.
-%,,
N
-2
Figure 3 As a result of some investigation, it was determined that diagrams with uniform distribution of zero regimes over the range have the same value of the total rated moment irrespective of the uniform distribution spacing, namely: E[MNI = 1.333. This condition may be generalized by proving the following:
Theorem. The total rated moment of machines with multistage electromechanical transmissions does not depend upon the distribution spacing Amp, but depends only upon ~,~ max (the maximum boundary rotation velocity of the machine) and upon the common control range % Theoretical evidence. Let ~A be the given output shaft maximum velocity, and 7 the given c o m m o n range of transmission velocity control. In this case the minimum rotation velocity of the output shaft, when the electrical drive has been working for a long time, will be: ~a = - - . Y
(7)
The rated regimes of electrical machines are achieved in points that are within the limits of the first stage. Let us find the rated moments of electrical machines M1 and M2: 1
t~ - - t l o z
glM, =c,(ao,_~02). -
1
MM~_-- C~(ao2 - no,)" rio, = O.
a fi -
--
no2
c,~a02' rio2
a
n
I
= C2no~
(8)
L e t us s u b s t i t u t e fi = tia = ( f i - / 7 ) into the e x p r e s s i o n 1VIM, 1~1.,
=
1
(h./T)-
- c,a,>2
ho..
h. - ~/h,,m~_
(9)
- C,ao:n."
(a~,13.)
A s s e e n f r o m e q u a t i o n (8), the m o m e n t 1VIM2d o e s not d e p e n d on ~. T h e r e f o r e , w e c a n i m m e d i a t e l y find t h e total r a t e d m o m e n t o f t h e m a c h i n e s . W e h a v e : (10) It is s e e n f r o m e q u a t i o n s (8) a n d (9) that within t h e first stage b o t h m o m e n t s a r e p o s i t i v e (Ct = C,. > 0, fib - ~/fi0: < 0). T h e r e f o r e e q u a t i o n (10) m a y b e w r i t t e n in t h e f o l l o w i n g f o r m :
:CllqI~I=M~,,+MM_Vno:-r~. "
/'~B
1
1
1
C I ~02 -[- C2]v~02 -- C z n 0 2
(vno~-n.+l)_ \
nB
~,
C2~B"
A s C2tio2 = tiM ~x w h i c h w e find f r o m t h e f o r m u l a f o r tiM2 a n d fir = /ion = 3fio2 then: XlMNI = nM3'.... • 3fio.; rio2
and, finally, w e h a v e : ~ll~lt,r I -
T 3fi~ ma~"
(11)
A s s e e n f r o m e x p r e s s i o n (11) EI1VIN[ d o e s n o t d e p e n d on AtOp.T h i s f o r m u l a is t r u e f o r a n y d i s t r i b u t i o n of Afip. F o r e x a m p l e , at 3' = 8 a n d flu .... = 2 w e find t h a t E[MN[ = 1.333. H e r e w e h a v e o b t a i n e d t h e s a m e r e s u l t as a b o v e . F o r m u l a ( 1 l ) m a y b e o f use f o r c a l c u l a t i n g t h r e e s t a g e e l e c t r o m e c h a n i c a l t r a n s m i s s i o n s at d i f f e r e n t g i v e n v a l u e s o f 3' a n d h~ ..... So, w i t h t h e g i v e n d i f f e r e n t i a l g e a r m e c h a n i s m d e s c r i b e d h e r e , t h e a b o v e g i v e n m e t h o d a l l o w s t h e c h o i c e o f s y s t e m p a r a m e t e r s (Fig. 1) in such a w a y as to b e s t m e e t t h e g i v e n c o n d i t i o n s of c o n t i n u o u s t o r q u e a c t i o n a n d a u t o m a t i c c o n t r o l .
References [l] A R T O B O L E V S K I I I. I. and I M E D A S H V I L I K. A., Dynamics of Differential Gear-Train Mechanisms. J. Mechanisms 4, 79-92 (1%9). [2] I M E D A S H V I L I K. A., Dynamics of a Differential Gear Mechanism in Combined Systems. MachinoStroienie Academy of Sciences of U.S.S.R, Moscow (1%5). [3] I V A N C H E N K O P. N. et al., Electromechanical Transmissions. Moshgiz, M.-L. (1962). CHHTE3
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