REVISTA ROMÂNĂ DE AUTOMATICĂ - IPA SA

REVISTA ROMÂNĂ DE AUTOMATICĂ
25
ω4
− ω1
z ω3
− ω
= − ;
ω − ω z ω − ω
2
1
2
'
4
4
1
1
"
4
z
= −
z
3
.
(3)
from which, with the notations (1) he or she
deduced the equalities:
ω 3 = ω1
( 1−
i) + ω2i
; ω 4 = ω1( 1+
i1
) + ω2i1
. (4)
The kinetic energy of the system:
1 2
2 2 2 2 2
E C = ( J1ω1
+ m4R
ω1
+ J2ω2
+ J3ω3
+ J4ω4)
. (5)
2
with the notations (4) and (2) has the
equation:
2
2
( Aθ&
− Bθ&
θ&
+ Cθ
)
1
E = 1 2 &
C 1 2 2 . (6)
2
The mechanic power at a certain time is given
by:
P = M
M , (7)
1ω1
+ M 2ω2
+ 3ω3
or on the basis of the other relation (4)
[ M1 + M 3( 1−
i)
] ω1
+ ( M 2 + M3
) ω2
P = i
(8)
and from here are deduced the generalized
equations:
Q1 = M1
+ M3 (1 − i)
; Q2 = M 2 + M3i
. (9)
Lagrange equations. Knowing the fact that
ω 1 = θ & 1 ; ω 2 = θ& 2 , (10)
3. THE MOVEMENTS STUDY
Permanent movement. The permanent
movement is deduced from the conditions
ω 1 = 0;
2 = 0
ω , (13)
that leads to these equations
[ 1 + M3( 1−
i)
] C + ( M 2 + M3i)
B = 0
[ + M 1−
i)
] B + ( M + M i)
A 0
M ;
M (14)
1 3( 2 3 =
and since AC − B 2 > 0 , we obtained the
equations
M + M − i)
0 ; M + M i 0 . (15)
1 3 (1 =
2 3 =
The sources of power are present by the
characteristics movement-angularly speed
through the relations Mi = M i ( ωi
) and goes to
* * *
the values: ω 1 , ω 2 , ω 3 of the angularly
velocities that are obtained the permanent
condition that are deduced from the equation
system:
*
*
*
*
M ( ω ) + (1 − i)
M ( ω ) 0 ; M ( ω ) + i M ( ω ) 0 ;
ω
1 1
3 3 =
*
* *
3 = ( 1−
) ω1
+ iω2
2 2 3 3 =
i . (16)
If these values M ( * 2 ω 2)
, M ( * 3 ω 3 ) are related to
the value of the moment M ( * 1 ω 1)
of the heat
engine, one obtained the relations (17) with
the graphic representations captured in figure
2 and figure 3.
M 2 ( ω ) i
= −
*
M ( ω ) i −1
1
*
2
1
;
M 3 ( ω ) 1
= −
*
M ( ω ) i −1
1
*
3
1
, (17)
from Lagrange equations are obtained the
differential equations
ω &
1 − Bω&
2 = M1
+ M 3( 1−
i);
− Bω&
1 + Cω&
2 = M 2 + M i ,
A 3
or
ω&
1
ω&
2
=
=
[ M + M ( 1−
i)
] C + ( M + M i)
1
[ M − M ( 1−
i)
] B + ( M + M i)
1
3
3
AC − B
2
AC − B
2
2
2
3
3
B
;
A
.
(11)
(12)