INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 192
DETERMINATION OF KINEMATIC PARAMETERS OF THE BISATELLITE
POINT IN A BIPLANETARY MECHANISM
Shamanov G'ulom Zohidovich
Tashkent Institute of Chemical Technology,Republic of Uzbekistan, Tashkent
Xaydarova Sharifa Kamilovna
Tashkent Institute of Chemical Technology,Republic of Uzbekistan, Tashkent
Shokirov Asliddin Pazlitdin o’g’li
Tashkent Institute of Chemical Technology,Republic of Uzbekistan, Tashkent
Jobborova Maftuna Akrom qizi
Tashkent Institute of Chemical Technology,Republic of Uzbekistan, Tashkent
Radjabova Sunbula Rajabovna
Tashkent Institute of Chemical Technology,Republic of Uzbekistan, Tashkent
Abstract:
This study investigates the kinematic parameters of the bisatellite point in a
biplanetary mechanism used in periodic dough mixing machines. Using the motion inversion
method, we derive equations for angular velocities, trajectory, speed, and acceleration of the
bisatellite point. Computational analysis in MathCAD 15 evaluates the influence of carrier
rotation speed on extreme kinematic values, providing insights for optimizing biplanetary
drive performance.
Keywords:
biplanetary mechanism, kinematic parameters, bisatellite, dough mixing machine,
motion inversion, angular velocity, trajectory optimization
One of the ways to improve the efficiency of periodic dough mixing machines is the use of a
biplanetary mechanism as the drive for the working div. The biplanetary drive of the
working div enables high-quality mixing of ingredients to produce the final dough. The
work presents various designs of biplanetary drives for working bodies of periodic dough
mixing machines. It has been established that the trajectory and kinematic parameters of the
bisatellite points play a crucial role in achieving high-quality mixing.
To analyze the kinematics of the biplanetary mechanism, we use the motion inversion
method.
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 193
Fig.
1.
Schematic
for
deriving
kinematic equations
of
the
biplanetary
mechanism
For this mechanism (Fig. 1), the following relation can be written
u
4H
= (1 − u
41
(H)
)
By imparting all links of the mechanism (Fig. 1) with an angular velocity –
ω
H
,
equal in magnitude and opposite in direction to the velocity of the main carrier
H
, the carrier
H
becomes stationary, while gear 1 acquires an angular velocity –
ω
H
.
This motion inversion
results in a conventional planetary mechanism with two simple gear pairs (1 and 2). Its
transmission ratio is:
u
41
(H)
= u
21
(1 − u
43
(h)
)
or
u
41
(H)
=−
z
1
z
2
(1 − u
43
(h)
)
where
(1 − u
43
h
)
is the transmission ratio of the satellite planetary mechanism, which can be
determined using a second motion inversion applied exclusively to this mechanism.
then
u
43
(h)
=−
z
3
z
4
that
u
41
(H)
=−
z
1
z
2
(1 +
z
3
z
4
) =−
z
1
z
2
−
z
3
z
1
z
4
z
2
Considering this, we can write
u
4H
= (1 +
z
1
z
2
+
z
1
z
3
z
2
z
4
)
Thus, the absolute rotation angle of the bisatellite relative to
Ο
Η
is:
ϕ
4
= ϕ
Η
1 +
z
1
z
2
+
z
3
z
4
z
1
z
2
The absolute angular velocity of the bisatellite is
ω
4
= ω
Η
1 +
z
1
z
2
+
z
1
z
2
z
3
z
4
the angular
velocity of the bisatellite 4 relative to the bicarrier
H
is
ω
4h
= ω
Η
z
1
z
2
z
3
z
4
the angular velocity
of the bisatellite 4 relative to the carrier
H
is
ω
4H
= ω
Η
z
1
z
2
+
z
1
z
2
z
3
z
4
, and the absolute angular
velocity of satellite 2 is
ω
2
= ω
Η
1 +
z
1
z
2
.
Proceeding to the kinematics of the biplanetary mechanism, based on, the kinematic
equations for point "C" of the bisatellite in the biplanetary mechanism can be written as:
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 194
x
c
t = Α cos ω
Η
t + Β cos k
1
ω
Η
t + D cos k
2
ω
Η
t ;
y
c
t = Α sin ω
Η
t + Β sin k
1
ω
Η
t + D sin k
2
ω
Η
t ,
where
k
1
= u
2Η
= 1 − u
21
Η
or
k
1
= 1 +
z
1
z
2
k
2
= u
4Η
= 1 − u
21
Η
− u
43
h
= 1 +
z
1
z
2
+
z
1
z
2
z
3
z
4
4
3
2
1
,
,
,
r
r
r
r
- are the pitch circle radii of gears 1, 2, 3, 4.
The velocities
C
v
of point “С” of bisatellite 4 are determined by:
(
)
(
)
(
)
(
)
(
)
(
)
.
cos
cos
cos
;
sin
sin
sin
2
2
1
1
2
2
1
1
t
k
k
D
t
k
k
t
v
t
k
Dk
t
k
k
t
v
cy
cx
H
H
H
H
H
H
H
H
H
H
H
H
+
B
+
A
=
-
B
-
A
-
=
w
w
w
w
w
w
w
w
w
w
w
w
The absolute velocity of point "C" of bisatellite 4 is:
2
2
2
2
cy
cx
c
c
c
v
v
dt
dx
dt
dy
v
+
=
+
=
Similarly, the acceleration of point "C" of bisatellite 4 is determined by:
(
)
(
)
(
)
(
)
(
)
(
)
,
sin
sin
sin
;
cos
cos
cos
2
2
2
2
1
2
2
1
2
2
2
2
2
1
2
2
1
2
t
k
k
D
t
k
k
t
a
t
k
Dk
t
k
k
t
a
cy
cx
H
H
H
H
H
H
H
H
H
H
H
H
-
B
-
A
-
=
-
B
-
A
-
=
w
w
w
w
w
w
w
w
w
w
w
w
and the absolute acceleration of point "C" of bisatellite 4 is:
2
2
2
2
2
2
2
cy
cx
c
c
c
a
a
dt
y
d
dt
x
d
a
+
=
+
=
The equations describing the kinematic characteristics of the biplanetary mechanism for the
drive of a periodic dough mixing machine were implemented on a computer using MathCAD
15.
To study the influence of carrier rotation speed on the kinematic parameters of the bisatellite
point,
we
determined
the
extreme
values
of
velocities
and
accelerations
min
max
min
max
,
,
,
a
a
v
v
, and the oscillation ranges
min
max
v
v
H
v
-
=
,
min
max
a
a
H
a
-
=
as shown in Table 1.
Table 1.
Kinematic characteristics of the bisatellite point "C"
№/n
Variation of the carrier rotation speed for point “C” of the bisatellite.
n
Η
, (rpm)
max
v
(m
/s)
min
v
(m/s)
v
H
(m/s)
max
a
(m/s²)
min
a
(m
/s²)
a
H
(m/s
²)
1
20
3,11
0,34
2,77
49,15
18,49
30,66
2
30
4,67
0,51
4,16
110,6
41,6
69
3
40
6,22
0,68
5,54
196,6
73,6
123
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 195
4
50
7,78
0,85
6,93
307,2
115,5
191,7
5
60
9,33
1,02
8,31
442,4
166,4
276
he obtained results are currently used to determine rational parameters for the biplanetary
drive of working bodies in periodic dough mixing machines.
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