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ELIMINATING TRANSVERSE VIBRATIONS ON
AN ELASTIC BEAM USING DAMPERS
A.E.Kudratov,
J.N.Kuljonov
Uzbek-Finish pedagogical instiute
Abstract.
This article examines the optimization of the parameters of an elastic
beam with dynamic vibration dampers during transverse vibrations. Methods for
damping transverse vibrations in the elastic beam under consideration were
analytically found. In particular, the change in the optimal parameters of the system
under consideration is analyzed depending on the mass ratio and changes in the
installation locations of dynamic vibration dampers. The article discusses the
optimization of the parameters of a system of elastic beams under stationary
vibrations with two dynamic vibration dampers. The solution to the problem of
transverse vibrations of a beam with two parallel installed dynamic vibration dampers
is considered, using the method of series expansion in vibration modes. This method
is more convenient for optimizing the parameters of dynamic vibration dampers for
various types of beam vibrations with boundary conditions, when it is necessary to
repeatedly calculate the amplitude-frequency characteristics of the system.
Keywords.
Elastic beam, Laplace operator, bending moment, dynamic vibration
damper, transverse vibrations, amplitude-frequency response
Introduction
Modern development of engineering and technology requires the development
of elastic beams with the most economical and less material-intensive design. In this
case, problems associated with transverse vibrations on elastic beams often arise.
Insufficient elaboration of issues to solve problems of transverse vibrations leads to
the fact that during the design, construction and commissioning of such elements
there is a need for additional changes in the design, which leads to an increase in
development time or to changes in the main characteristics of the product. These
disadvantages reduce the consumer properties of elastic beams [1, 5].
Many scientific articles are devoted to the problems of damping oscillations of
systems with distributed parameters using dynamic oscillation dampers. It is shown
in [2,6] that when a dynamic vibration damper is attached to a beam, a new natural
frequency of the system appears, close to the partial frequency of the damper,
which, depending on the parameters of the system, can take values less than, greater
than, or equal to the partial frequency of the damper.
Many scientific articles are devoted to the problems of damping oscillations of
systems with distributed parameters using dynamic oscillation dampers. It is shown
in [2,7] that when a dynamic vibration damper is attached to a beam, a new natural
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frequency of the system appears, close to the partial frequency of the damper, which,
depending on the parameters of the system, can take values less than, greater than, or
equal to the partial frequency of the damper.
In experimental studies [2,8], a comparative analysis of the vibrations of a beam
with two dynamic vibration dampers, symmetrically located relative to the ends of
the beam, is carried out. Differential equations of motion are nonlinear and require
the use of appropriate methods to solve.
In works [3, 4, 9], problems of nonlinear vibrations of a beam with a dynamic
vibration damper are considered, taking into account elastic-damping properties of
the hysteresis type under harmonic influences. A solution to the system was obtained
in the form of transfer functions.
The problems of dynamics [5, 6, 10] of nonlinear oscillations, as well as their
stability [7, 8, 11], were studied. Based on the above, it follows that the study of
vibrations and vibration damping of beams remains an urgent task of modern science.
The article discusses the optimization of system parameters during stationary
vibrations of a beam with two DVD.
A device is presented [5, 6, 12] consisting of compression and tension springs,
working together to withstand both vertical and horizontal loads resulting from
permanent, temporary and seismic influences. In addition to the main task of
absorbing vertical and horizontal loads, the device is capable of returning the span
structure to its original position after exposure to seismic influences. It also eliminates
resonance without increasing costs, both for the span and for supports and
foundations, and does not complicate the installation conditions of the structure.
Proposed Methodology, Experiments and Results
In the presented research, the task was set to dampen transverse vibrations on
an elastic beam using dynamic vibration dampers (DVD). The algorithm of the
sequence of work with the required properties is obtained by a sequence of a
number of operations. In Fig. 1 presents the algorithm of operations and stages of
the tactical process.
Let us consider the solution to the problem of transverse vibrations of a beam
with two parallel installed DVD using the method of series expansion according to
vibration modes. This method is more convenient for optimizing the parameters of
the DVD for various types of beam vibrations with boundary conditions, when it is
necessary to repeatedly calculate the amplitude-frequency characteristics (AFC) of
the system.
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Fig. 1. An algorithm of operations and stages of the tactical process.
The results of the above works confirm that with a sufficiently large decrement
of vibrations of the material of the elastic-damping element of the DVD, the
nonlinearity of the internal resistance characteristics of the beam material has little
effect on the vibrations of the beam and the determination of the optimal parameters
of the DVD. A beam of length
l
, width
b
, height
h
, is fixed on a vibrating base; its
movement is specified along the
Oz
axis. At points of the beam with coordinates X1,
X2, DVDs are installed (Fig. 2.).
Fig. 2. Design diagram of an elastic beam (EB) with two dynamic vibration
dampers (DVD).
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Analysis of Experiments and Research Results
We write the differential equations of an elastic beam with two DVD under
kinematic excitation in the following form:
;
)
(
)
(
2
0
2
2
2
2
2
1
1
1
1
2
2
2
2
t
w
F
x
x
c
x
x
c
t
w
F
x
M
;
)
(
2
0
2
1
1
1
2
1
2
1
2
1
2
1
t
w
m
c
t
m
t
x
w
m
:
)
(
2
0
2
2
2
2
2
2
2
2
2
2
2
2
t
w
m
c
t
m
t
x
w
m
where
M
is the bending moment;
-material density;
F
is the cross-sectional
area of the beam;
w
-beam deflection function;
0
w
-moving the base;
)
(
),
(
2
1
x
w
x
w
-
movement of the point of the beam in which the DVDs are installed;
2
1
,
c
c
-stiffness
coefficients of the elastic damping elements of the DVD;
2
1
,
m
m
-
mass of DVD;
2
1
,
-displacements of the DVD relative to the beam;
)
(
),
(
2
2
1
1
x
x
x
x
- Dirac
delta functions;
2
1
,
x
x
-are the coordinates of the DVD installation;
To solve this system of equations, we used the method of separation of
variables:
1
).
(
)
(
)
,
(
i
i
i
t
q
x
u
t
x
w
After some calculations, the system of equations is reduced to the form (1):
;
0
2
2
2
2
0
2
1
1
2
1
0
1
2
W
d
u
n
u
n
q
p
q
i
i
i
i
i
i
i
i
;
0
1
2
1
1
1
W
n
q
u
i
i
(1)
:
0
2
2
2
2
2
W
n
q
u
i
i
where
pi
-is the natural frequency of the beam;
;
2
1
1
m
m
;
2
1
c
m
m
;
2
0
i
i
d
l
;
2
1
i
i
i
d
d
d
;
0
1
l
i
i
dx
u
d
;
0
2
2
l
i
i
dx
u
d
Fl
m
c
–beam mass;
,
1
m
,
2
m
– masses of
dynamic vibration dampers;
)
(
x
u
i
– natural vibration modes of the beam;
0
W
– base
acceleration,
),
(
1
1
x
u
u
i
i
);
(
2
2
x
u
u
i
i
2
1
,
x
x
-
coordinates of the DVD installation;
,
1
1
1
m
c
n
;
2
2
2
m
c
n
2
1
2
1
,
;
,
c
c
-oscillation frequencies; stiffness coefficients of
elastic elements and relative movements of the DVD.
Acceleration of the base during harmonic vibrations
,
cos
0
0
t
w
W
where
w0
- is the amplitude value of acceleration;
ω
– frequency.
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214
We look for solutions to the system in the form:
);
cos(
i
i
i
t
a
q
);
cos(
1
1
1
t
b
(2)
);
cos(
2
2
2
t
b
Substituting these expressions into the differential equations of motion and
assuming that the coefficients vary slowly, we obtain the following normal
equations for the system under consideration:
];
sin
sin
sin
[
)
2
(
2
2
2
2
2
1
1
2
1
1
0
1
b
n
l
b
n
l
w
d
a
i
i
i
];
cos
cos
cos
[
)
2
(
2
2
2
2
2
1
1
2
1
1
2
0
1
b
n
l
b
n
l
a
w
d
a
i
i
i
i
i
];
sin
sin
sin
)
1
[(
)
2
(
1
2
1
3
2
1
2
1
2
1
0
1
1
1
i
i
i
i
i
i
a
p
u
b
u
n
l
w
u
d
b
(3)
];
cos
cos
cos
)
1
[(
)
2
(
1
2
1
3
2
1
2
2
2
2
1
6
2
1
1
1
0
1
1
1
1
i
i
i
i
i
i
a
p
u
b
u
n
l
b
T
n
b
w
u
d
b
];
sin
sin
sin
)
1
[(
)
2
(
2
2
2
3
1
2
2
1
1
2
0
2
1
2
i
i
i
i
i
i
a
p
u
b
u
n
l
w
u
d
b
(4)
];
cos
cos
cos
)
1
[(
)
2
(
2
2
2
3
1
2
2
1
1
2
2
7
2
2
2
2
0
2
1
2
2
i
i
i
i
i
i
a
p
u
b
u
n
l
b
T
n
b
w
u
d
b
where
;
1
1
i
;
2
2
i
;
1
2
3
;
1
0
1
1
i
i
u
l
.
2
0
2
2
i
i
u
l
From the system of equations (4), putting zeros instead of derivatives on the
left side, we obtain the required stationary solutions in the following form:
;
5
2
4
4
3
6
2
2
1
4
A
A
A
A
A
d
a
q
i
i
ik
;
)
1
(
5
2
4
4
3
6
7
2
6
4
1
1
1
A
A
A
A
A
u
d
b
i
i
(5)
;
)
1
(
5
2
4
4
3
6
9
2
8
4
2
2
2
A
A
A
A
A
u
d
b
i
i
where
,
1
1
1
m
c
n
;
2
2
2
m
c
n
natural mode of vibration
,
sin
)
(
x
l
i
x
u
i
in this
case, in a particular case for we find;
,
8660254037
.
0
1
i
u
,
8660254035
.
0
2
i
u
;
2
2
0
i
d
l
as well as coefficients:
);
(
2
2
2
1
2
1
1
T
n
T
n
A
;
3
2
2
2
1
2
T
n
n
A
;
2
7
2
2
6
2
1
3
i
p
T
n
T
n
A
;
8
2
2
2
1
2
2
2
2
2
1
4
T
n
n
p
n
p
n
A
i
i
;
2
2
2
2
1
5
i
p
n
n
A
;
4
2
2
2
6
T
n
p
A
i
;
2
2
2
7
n
p
A
i
;
5
2
1
2
8
T
n
p
A
i
;
2
1
2
9
n
p
A
i
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Volume-40_Issue-2_June-2025
215
;
1
0
1
1
i
i
i
u
d
T
;
2
0
2
2
i
i
i
u
d
T
);
(
2
2
1
1
0
3
i
i
i
i
u
u
d
T
;
)
(
1
1
1
2
2
2
0
4
i
i
i
i
i
i
d
u
u
u
u
T
;
)
(
1
2
2
1
1
1
0
5
i
i
i
i
i
i
d
u
u
u
u
T
;
1
2
1
1
0
6
i
i
u
T
;
1
2
2
2
0
7
i
i
u
T
:
)
(
1
2
2
2
2
1
1
0
8
i
i
i
u
u
T
4. Results of numerical studies
Numerical analysis is carried out to determine the first eigenshape from two
separate cases: 1) first we carry out a numerical analysis with changes in the mass
ratios
μ1
and
μ2
, the ratios of the masses of the DVDs to the mass of the beam; 2)
from the obtained relationships, we construct graphs of the amplitude-frequency
characteristics (AFC) of the system and find approximate locations for installing a
DVD.
During the initial experiment, AFCs were constructed for the ratio of the
absorber masses to the beam mass from 0.04 to 0.1. In this case, the approximate mass
ratio for the system under consideration can be taken as 0.06 (Fig. 3).
Based on the results of the secondary experiment, the amplitude-frequency
characteristics were constructed for a ratio of the absorber masses to the beam mass
from 0.04 to 0.1. In this case, the approximate mass ratio for the system under
consideration can be taken as 0.06 (but in this case, the DVDs are not installed
symmetrically - l/3, 4l/5) (Fig. 4).
Fig. 3. AFCs when changing mass ratios 0.04; 0.06; 0.08; 0.1 (blue line
corresponds to 0.06).
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Fig. 4. AFCs when changing the mass ratio 0.04; 0.06; 0.08; 0.1 (red line
corresponds to 0.06).
5. Conclusion
The problem of optimizing the transverse vibrations of an elastic beam with
two parallel installed dynamic vibration dampers with elastic elements during
harmonic vibrations of the base is considered.
A principle has been developed for calculating the installation of dynamic
vibration dampers for an elastic beam, taking into account the amplitude-frequency
characteristics corresponding to transverse vibrations.
Analyzes of system vibrations were carried out with changes in the locations of
the DHA installations and changes in the ratio of the masses of the DHA to the mass
of the beam.
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