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THE PROBLEM OF PASSIVE AND ACTIVE VIBRATION PROTECTION OF
DISSIPATIVE-INHOMOGENEOUS MECHANICAL SYSTEMS WITH A FINITE
NUMBER OF DEGREES OF FREEDOM
D.G.Rayimov
Asia International University
Kinematic excitation (Fig. 1 ) is used for vibration protection of technical devices, technological
apparatus, machines, i.e., electronic devices and equipment that are very sensitive to vibrations
of devices installed on moving objects.
Fig. 1. One-dimensional diagram of an active vibration protection system.
1 – plate (base); 2 – springs (deformable elements); 3 – intermediate plate; 4 – acceleration
sensor; 5 – viscous element; 6 – control unit; 7 – vibration protection object;
a
(
t
) – base
displacement
In active vibration protection systems (Fig. 1 ), the control object will be the vibration protection
object, and passive elements are included in the system as springs, dampers or their
combinations. If the system also includes a servo drive, this means that the system has a
regulator.
The input signals will be displacements, velocities and accelerations of points, angular
displacements, angular velocities, angular accelerations, forces and stresses , etc.
[ ]
[ ]
[ ]
[ ]
[ ]
),
(
;
;
)
(
)
(
)};
(
{
)}
(
{
}
{
}
{
}
{
}
{
}
{
1
3
1
0
1
2
3
1
1
2
2
1
1
2
2
x
x
K
u
x
K
K
u
K
u
u
Ri
x
x
Bl
dt
di
L
t
t
F
X
C
X
C
X
B
X
B
X
M
dp
dp
dp
bx
dp
bx
-
=
=
-
=
+
-
+
+
=
+
+
+
+
&
&
&
&
&
&
h
( 1 )
Here
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[ ]
[ ]
[ ]
[ ]
[ ]
1
1
1
2
1
2
2
2
1
2
3
3
3
1
1
2
2
2
2
1
2
1
2
2
3
3
2
0 0
0 0
0 0
0 0 0
0
0 ,
0
0 ,
0
0 ,
0
0
0 0
0 0
0 0 0
0 0
0 0 0
0
0
0 ,{ }
,{ }
,{ }
,
0 0 0
0
m
c
b
M
m
C
c
C
c
B
b
m
c
b
x
x x
B
b
X
x
X
x x
X
x
x
x x
=
=
=
=
-
=
=
=
-
=
-
&
&
&
&
&
&
Here
i
– current strength;
u
– voltage;
L , R
– inductance and resistance;
ℓ
– conductor length;
K
dp
– displacement sensor gain;
K
1
– tracking system gain,
B ℓ i
– electromechanical force ;
The proposed research method with servo links represents a new direction, which is based on the
application of the model as a system, which is built using multi-pole elements, which made it
possible to apply automatic control methods to vibration protection systems and
electromechanical systems.
Methods of vibration protection and vibration isolation of machines and devices are developing
to transform vibration protection systems into controlled systems.
Active vibration protection devices (AVD) have a lower frequency range limit of ≈ 2 Hz, as well
as a maximum vibration suppression coefficient of 35 to 40 dB, which is available at a frequency
of ≈ 10 Hz.
Methods for solving the problem of natural and forced vibrations ( passive and active
vibration protection) of dissipative-inhomogeneous mechanical systems
When considering natural oscillations, the right-hand side is identically equal to zero. We will
seek a solution in the form:
,
i t
j
j
q
A e
w
-
=
j =
1 , …,
N
,
Where
I
R
i
w
w
w
+
=
– complex natural frequency. The problem is reduced to a complex
algebraic eigenvalue problem of the form:
2
1
( (
( )
)) 0,
N
k
jk
R
jk
k
A C
а
w
w
=
-
=
1,2,...,
j
N
=
, ( 2 )
with a nonlinearly entering complex parameter
R
w
. The characteristic equation of problem ( 2 )
is solved numerically, using the Muller method. As an initial approximation, a solution close to
( 2 ) of the corresponding conservative problem is adopted. In this case, the determinant of
system ( 2 ) at each iteration of the Muller method is calculated using the Gauss method with the
selection of the main element by rows and columns. Thus, the solution of the problem of natural
oscillations using the Muller method does not require the disclosure of its determinant.
We will seek the solution to the problem of forced oscillations of the system ( 2 ) in the
form:
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,
1,...,
i t
j
j
q
A e
j
N
l
-
=
=
, ( 3 )
Where
j
A
– the sought complex amplitudes. The problem of steady-state forced oscillations is
reduced to a system of non-homogeneous algebraic equations:
2
1
(
( )
)
( ),
N
jm
jm
m
j
j
э
m
C
a
A
f
q
l
l
h
=
-
=
+
where the solution is carried out by the Gauss method. The result of solving the problem of
forced oscillations is obtaining the amplitude-frequency characteristics (AFC) of the mechanical
system. If the mechanical system has one degree of freedom, then the equation of motion of the
system is written as:
2
1
1
1
1
''
(
) ( )
sin
( )sin
t
э
y
y
R t S y S dS C
pt
q
pt
w
h
-
+
-
-
-
= -
,
( 4 )
where
p
is the frequency of the external influence. The particular solution of equation ( 4 ) has
the form:
)
(
sin
cos
1
2
1
1
t
y
t
p
d
t
p
d
=
+
,
Where
2
1
1
1
2
2
2
4 2
1
1
1
(
( ))
( )
;
(1
( ))
( )
э
S
C
S
c
q
p
d
p
p
p
h
w
w
w
- +
G
=
- G
-
+ G
2
2
1
1
1
2
2
2
2
4 2
1
1
1
(
( ))
(1
( ))
;
(1
( ))
( )
э
C
C
S
c
q
p
p
d
p
p
p
h
w
w
w
-
- G
-
=
- G
-
+ G
;
cos
)
(
)
(
0
1
1
=
t
t
t
d
p
R
p
F
C
1
1
0
( )
( )sin
.
S
F p
R
p d
t
t t
=
When the viscous properties of deformable elements are taken into account through viscous
friction, then in matrix form relative to the matrix – column of
� = ����� �
1
, …, �
�
integro-
differential equations (IDE) ( 4 ) take the form
{ }
{ }
[
]
{
}
{ } { } {
}
[ ]
[ ]
(
)
( )
[ ]
( )
t
э
M X
C X
R t
X
d
K X
f
q
t
t
t
h
-
+
-
-
+
=
+
&&
&
,
( 5 )
where [
M
] is a positive definite matrix, the elements of which denote concentrated masses, [
C
]
is the matrix of damping coefficients of deformable bodies and [
K
] is the matrix of elements of
rigidity characteristics of deformable bodies (square symmetric matrix),
[
]
(
)
R t
t
-
-
viscosity
matrix without mass elements. The disturbing force is denoted by the column
{ }
f
vector .
The above matrices have a physical meaning, i.e.
jk
M
,
jk
C
and
jk
K
– respectively, the
elements of the mass, damping and stiffness matrix. All matrices are quadratic,
j
– number of
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lines,
k
– the number of columns. The elements of this equation ( 2 ) are obtained from the
system of differential equations ( 1 ).
Equation of motion of vibration protection systems with a finite number of degrees of
freedom.
The equations of vibrations of machine elements can be written in the form:
[ ]
{ }
[ ]
{ }
[ ]
{ } {
}
)
(
t
F
B
C
M
=
+
+
j
j
j
&
&
&
, ( 6 )
where
{ }
j
is the matrix of generalized coordinates, [
M
], [
B
], [
C
] are the matrices of inertial,
dissipative and elastic loads, respectively;
F ( t )
is the matrix of external loads.
For active vibration protection systems, its system of equations can be represented by differential
equations:
{ } {
}
[ ]
{ }
x
A
t
F
x
+
=
)
(
&
, ( 7 )
where
x
is the state matrix;
[ ]
A
is a square matrix.
If a complex system is considered, then [
B
] and [
C
] may be "dense" matrices.
The reactions of servolinks can be determined by structural diagrams that are equivalent to
automatic control systems. This approach can also be used for systems for which mass is taken
as an inertial element of a rigid div.
It is known that it is impossible to be distracted from the implementation of servolinks . They
can be implemented not passively, not by simply touching bodies, but by mechanisms that
generate various generalized forces, for example, mechanical, electromechanical, pneumatic,
hydraulic, electrohydraulic, etc.
Conclusions.
methods of solving the problem and the algorithm of combined vibration protection of a
mechanical system with a finite number of degrees of freedom based on the constraint of the
mechanical system by servo links have been developed . Based on the analysis of the obtained
numerical results, it has been established that the efficiency of including active vibration
protection in the support, in parallel with the passive one, is determined by the gain in the
feedback circuit. It has been substantiated that an appropriate choice of the gain can achieve
unlimitedly high efficiency, however, with large values of the gain in the feedback circuit, the
system may lose stability.
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2.
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3.
Ismoil Safarov, Muhsin Teshaev, Sharifboy Axmedov, Doniyor Rayimov, Farhod Homidov.
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Properties.
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of
Conferences
264,
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(2021)
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