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202
STABILITY OF AN ELASTIC ROD WITH DYNAMIC ABSORBERS
UNDER HARMONIC TRANSVERSE VIBRATIONS
Anvar Kudratov, Shokhzod Khamitov
Uzbek-Finish pedagogical institute
Abstract.
This paper considers the stability problem of a simply supported
rod with two dynamic vibration absorbers and hysteretic energy dissipation.
Linearization of dependencies describing hysteretic energy dissipation due to
imperfect elasticity of the material is carried out using Pisarenko-Boginich
functional in a frequency-independent form
Keywords:
stability, rod, oscillations, kinematic excitations, linearization
INTRODUCTION
When studying the oscillatory movements of the mechanical systems under
consideration, it is often necessary to pay attention to the parameters of the system,
which make it possible, to one degree or another, to predict the desired movements.
Thus, there are many works [1-5] on the study of vibrations of mechanical systems
with distributed parameters under the influence of excitations and forces in different
directions, with different parameters of the mechanical system, as well as damping
harmful vibrations [6-8], methods for modeling mechanical systems with dynamic
absorber.
It is known that nonlinear oscillations of mechanical systems have special
mechanical effects. Thus, in [9-12], vibrations of elastic mechanical systems with
nonlinear elastically dissipative characteristics are considered, where it is shown that
such mechanical systems have highly absorbing properties reflected by a hysteresis
loop. But on the other hand, unstable oscillations are possible in such systems in the
form of an amplitude jump with the slightest changes in the oscillation frequency,
which proceeds from the fact that the frequencies of nonlinear oscillations depend on
the amplitudes. Methods of equivalent and harmonic linearization in frequency-
dependent and frequency-independent forms are presented. The main types of the
function characterizing the nonlinear dependence of stresses on deformations due to
imperfect elasticity of the material in the structural elements of the system, as well as
nonlinear dependencies between tensions and deformations, are given. The nonlinear
dependencies are based on the models proposed by N.N.Davydenkov, G.S.Pisarenko,
and O.E.Boginich. The stability of some oscillatory mechanical systems is
considered.
In [13-15], the problems of stability of stationary vibrations of mechanical systems
with elastic-dissipative characteristics of the hysteresis type under kinematic and
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Volume-40_Issue-2_June-2025
203
random excitations are considered. The conditions and areas of stability of the
considered systems are determined.
METHOD OF RESEARCH
In this paper, the stability of transverse vibrations of an elastic rod with two
dynamic vibration absorbers with elastic-dissipative characteristics of the hysteresis
type is researched. Based on the system of differential equations of motion, using the
method of slowly varying amplitudes, the system is brought to a normal form and the
amplitude-frequency characteristics of the system under consideration are found. The
characteristic equations are obtained, the stability conditions according to the Hurwitz
criterion are verified. In particular, graphs of expressions in these inequalities are
constructed. Stability conditions are analytically expressed depending on the system
parameters. The change in stability conditions is analyzed depending on changes in
the masses of dynamic absorbers, changes in the stiffness of the elastic elements of
dynamic absorbers and changes in the installation locations of dynamic absorbers.
The differential equations of a rod and two dynamic absorbers with hysteresis
energy dissipation under kinematic excitation are written as follows:
∂
2
М
∂х
2
+ ρF
∂
2
w
∂t
2
− c
1
R
1
δ
1
(x − x
1
)ζ
1
− c
2
R
2
δ
2
(x − x
2
)ζ
2
= −ρF
∂
2
w
0
∂t
2
;
m
1
∂
2
w(x
1
)
∂t
2
+ m
1
∂
2
ζ
1
∂t
2
+ c
1
R
1
ζ
1
= −m
1
∂
2
w
0
∂t
2
;
(1)
m
2
∂
2
w(x
2
)
∂t
2
+ m
2
∂
2
ζ
2
∂t
2
+ c
2
R
2
ζ
2
= −m
2
∂
2
w
0
∂t
2
,
where
М
–
bending moment
;
ρ
, F
–
the density of the material and the cross-sectional
area of the rod, respectively
;
w
–
rod deflection function
;
w
0
−
displacement of the
base
;
х
1
, х
2
;
w(x
1
), w(x
2
)
–
coordinates and displacements at the points of the rod in
which the dynamic absorbers are installed
;
c
1
, c
2
–
stiffness coefficients of the elastic
damping elements of the dynamic absorbers
;
m
1
, m
2
–
dynamic absorbers masses
;
ζ
1
, ζ
2
–
displacements of the dynamic absorbers relative to the rod
;
δ
1
(х − х
1
), δ
2
(х −
х
2
)
–
Dirac delta functions
;
R
1
= 1 + (−ν
1
+ jν
2
)[D
0
+ f(ζ
1ot
)];
R
2
= 1 + (−η
1
+ jη
2
)[E
0
+ g(ζ
2ot
)].
(2)
Here
j
2
= −1; ν
1
, ν
2
, η
1
, η
2
– coefficients depending on the dissipative properties
of the materials of the elastic damping elements of the dynamic absorbers, determined
FIGURE 1.
Rod with two dynamic absorbers
x
1
x
2
elastic damping
element
solid
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204
from the corresponding dependencies of the contour of the hysteresis loop
;
f(ζ
1ot
), g(ζ
2ot
)
- the decrements of oscillations, represented in general terms, as
functions of the maximum (amplitude) values of relative deformation
ζ
1ot
, ζ
2ot
:
f(ζ
1ot
) = ∑ D
K
1
ζ
1ot
К
1
;
(3)
g(ζ
2ot
) = ∑ E
K
2
ζ
2ot
К
2
,
(4)
D
0
, D
1
, . . . , D
r
1
, E
0
, E
2
, . . . , E
r
2
– some numbers (parameters) of the hysteresis loop
depending on the damping properties of the materials of the elastic damping elements
of the dynamic absorbers and determined by the experimental selected curves
δ
1
=
f(ζ
1ot
)
and
δ
2
= g(ζ
2ot
)
points with coordinates
𝛿
1𝑖
,
(ζ
1ot
)
i
and
δ
2i
,
(ζ
2ot
)
i
accordingly.
EJ
∂
4
w
∂x
4
+ ρF
∂
2
w
∂t
2
− c
1
R
1
δ
1
(x − x
1
)ζ
1
− c
2
R
2
δ
2
(x − x
2
)ζ
2
= −ρF
∂
2
w
0
∂t
2
;
m
1
∂
2
w(x
1
)
∂t
2
+ m
1
∂
2
ζ
1
∂t
2
+ c
1
R
1
ζ
1
= −m
1
∂
2
w
0
∂t
2
;
(5)
m
2
∂
2
w(x
2
)
∂t
2
+ m
2
∂
2
ζ
2
∂t
2
+ c
2
R
2
ζ
2
= −m
2
∂
2
w
0
∂t
2
.
Where
J
- moment of inertia of the cross section of the rod.
To solve this system, the method of Fourier was used
w(x, t) = ∑ u
i
(x)q
i
(t)
∞
i=1
.
After some calculations, the system (1) is reduced to the form
q̈
i
+ p
i
2
q
i
− μ
1
μ
0i
n
1
2
u
i1
R
1
ζ
1
− μ
2
μ
0i
n
2
2
u
i2
R
2
ζ
2
= −d
i
W
0
;
u
i1
q̈
i
+ ζ̈
1
+ n
1
2
R
1
ζ
1
= −W
0
;
(6)
u
i2
q̈
i
+ ζ̈
2
+ n
2
2
R
2
ζ
2
= −W
0
.
where
p
i
– the own frequency of the rod;
μ
1
=
m
1
m
c
; μ
2
=
m
2
m
c
;
μ
0i
=
1
d
2i
; d
i
=
d
1i
d
2i
; d
1i
= ∫ u
i
dx; d
2i
= ∫ u
i
2
dx
l
0
l
0
;
m
c
= ρFl
– rod mass;
m
1
, m
2
– masses of dynamic absorbers;
u
i
(x)
–
the rod vibration's own shapes;
W
0
–
acceleration of the base,
u
i1
= u
i
(x
1
); u
i2
= u
i
(x
2
)
;
n
1
= √
c
1
m
1
,
n
2
= √
c
2
m
2
;
c
1
, c
2
;
ζ
1
, ζ
2
– oscillation frequencies; stiffness coefficients of elastic elements and relative
displacements of dynamic absorbers.
Acceleration of the base during harmonic oscillations is
W
0
= w
0
cos ω t,
where
w
0
– the amplitude value of the acceleration;
ω
– frequency.
RESEARCH RESULTS
The transfer functions of system (2) are found in the form
q(jω) = −
B
1
(ω) + jB
2
(ω)
B
3
(ω) + jB
4
(ω)
∙ w
0
;
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205
ζ
1
(jω) = −
B
5
(ω) + jB
6
(ω)
B
3
(ω) + jB
4
(ω)
∙ w
0
;
ζ
2
(jω) = −
B
7
(ω) + jB
8
(ω)
B
3
(ω) + jB
4
(ω)
∙ w
0
;
where
B
1
(ω) = d
i
ω
4
− [T
1
n
1
2
(1 − ν
1
N
1
) + T
2
n
2
2
(1 − θ
1
N
2
)]ω
2
+
+n
1
2
n
2
2
T
3
[(1 − ν
1
N
1
)(1 − θ
1
N
2
) − ν
2
θ
2
N
1
N
2
];
B
2
(ω) = −[T
1
n
1
2
ν
2
N
1
+ T
2
n
2
2
θ
2
N
2
]ω
2
+ n
1
2
n
2
2
T
3
[(1 − ν
1
N
1
)θ
2
N
2
+
+(1 − θ
1
N
2
)ν
2
N
1
];
B
3
(ω) = −ω
6
+ [p
i
2
+ n
1
2
T
6
(1 − ν
1
N
1
) + n
2
2
T
7
(1 − θ
1
N
2
)]ω
4
−
−{n
1
2
p
i
2
(1 − ν
1
N
1
− η
c
ν
2
N
1
) + n
2
2
p
i
2
(1 − θ
1
N
2
) +
+n
1
2
n
2
2
T
8
[(1 − ν
1
N
1
)(1 − θ
1
N
2
) − ν
2
θ
2
N
1
N
2
]}ω
2
+
+n
1
2
n
2
2
[(1 − ν
1
N
1
)(1 − θ
1
N
2
) − ν
2
θ
2
N
1
N
2
];
B
4
(ω) = [n
1
2
T
6
ν
2
N
1
+ n
2
2
T
7
θ
2
N
2
]ω
4
− {n
1
2
p
i
2
[η
c
(1 − ν
1
N
1
) + ν
2
N
1
] +
+n
2
2
p
i
2
θ
2
N
2
+ n
1
2
n
2
2
T
8
[(1 − ν
1
N
1
)θ
2
N
2
+
+(1 − θ
1
N
2
)ν
2
N
1
]}ω
2
+ n
1
2
n
2
2
[(1 − ν
1
N
1
)θ
2
N
2
+ (1 − θ
1
N
2
)ν
2
N
1
];
B
5
(ω) = (1 − d
i
u
i1
)ω
4
− [p
i
2
+ T
4
n
2
2
(1 − θ
1
N
2
)]ω
2
+n
2
2
p
i
2
[1 − θ
1
N
2
];
B
6
(ω) = −n
2
2
T
4
θ
2
N
2
ω
2
+ n
2
2
p
i
2
θ
2
N
2
;
B
7
(ω) = (1 − d
i
u
i2
)ω
4
− [p
i
2
+ T
5
n
1
2
(1 − ν
1
N
1
)]ω
2
+ n
1
2
p
i
2
[1 − ν
1
N
1
];
B
8
(ω) = −(η
c
p
i
2
+ n
1
2
T
5
ν
2
N
1
)ω
2
+ n
1
2
p
i
2
ν
2
N
1
;
where
T
1
= d
i
+ μ
0i
μ
1
u
i1
;
T
2
= d
i
+ μ
0i
μ
2
u
i2
;
T
3
= d
i
+ μ
0i
(μ
1
u
i1
+ μ
2
u
i2
);
T
4
= 1 + μ
0i
μ
2
u
i2
(u
i2
− u
i1
) − u
i1
d
i
;
T
5
= 1 + μ
0i
μ
1
u
i1
(u
i1
− u
i2
) − u
i2
d
i
;
T
6
= 1 + μ
0i
μ
1
u
i1
2
;
T
7
= 1 + μ
0i
μ
2
u
i2
2
;
T
8
= 1 + μ
0i
(μ
1
u
i1
2
+ μ
2
u
i2
2
).
The amplitude-frequency characteristics are obtained by calculating the absolute
values of the transfer functions.
Solutions of the system (2) are sought in the form of
q
i
= a
i
cos( ωt + α
i
);
ζ
1
= b
1
cos( ωt + β
1
);
(7)
ζ
2
= b
2
cos( ωt + β
2
),
where the amplitudes and phases
a
i
, α
i
, b
1
, β
1
, b
2
, β
2
of the oscillations are
considered as slowly varying
.
Substituting (7) into equations (6), equating the second derivatives to zero, we find
the equations of motion in normal form
ȧ
i
= −
1
2ω
[−d
i
w
0
sin α
i
− μ
1
μ
0i
n
1
2
u
i1
b
1
H
1
− μ
2
μ
0i
n
2
2
u
i2
b
2
sinφ
2
];
α̇
i
= −
1
2ωa
i
[−d
i
w
0
cos α
i
+ (ω
2
− p
i
2
)a
i
+ μ
1
μ
0i
n
1
2
u
i1
b
1
H
2
+ μ
2
μ
0i
n
2
2
u
i2
b
2
cosφ
2
];
World scientific research journal
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206
b
1
̇ = −
1
2ω
[−(1 − d
i
u
i1
)w
0
sin β
1
+ n
1
2
(1 + μ
1
μ
0i
u
i1
2
)ν
2
N
1
b
1
+ u
i1
a
i
p
i
2
sin φ
1
+ μ
2
μ
0i
n
2
2
u
i1
u
i2
b
2
sinφ
3
];
(8)
β
1
̇ = −
1
2ωb
1
[−(1 − d
i
u
i1
)w
0
cos β
1
+ (ω
2
− n
1
2
(1 + μ
1
μ
0i
u
i1
2
)(1 − ν
1
N
1
)) b
1
+ u
i1
a
i
p
i
2
cos φ
1
− μ
2
μ
0i
n
2
2
u
i1
u
i2
b
2
cosφ
3
] ;
b
2
̇ = −
1
2ω
[−(1 − d
i
u
i2
)w
0
sin β
2
+ u
i2
a
i
p
i
2
sin φ
2
− μ
1
μ
0i
n
1
2
u
i1
u
i2
b
1
sinφ
3
];
β
2
̇ = −
1
2ωb
2
[−(1 − d
i
u
i2
)w
0
cos β
2
+ (ω
2
− n
2
2
(1 + μ
2
μ
0i
u
i2
2
)) b
2
+ u
i1
a
i
p
i
2
cos φ
2
− μ
1
μ
0i
n
1
2
u
i1
u
i2
b
1
cosφ
3
],
where
H
1
= (1 − ν
1
N
1
) sin φ
1
+ ν
2
N
1
cos φ
1
; H
2
= (1 − ν
1
N
1
) cos φ
1
− ν
2
N
1
sin φ
1
;
φ
1
= β
1
− α
i
, φ
2
= β
2
− α
i
, φ
3
= β
2
− β
1
.
To study the stability of stationary oscillations of the system, we will use the
Lyapunov method as a first approximation. By varying equations (8), it is possible to
obtain a system of equations in variations, from which we obtain the characteristic
equation
λ
6
+ A
1
λ
5
+ A
2
λ
4
+ A
3
λ
3
+ A
4
λ
2
+ A
5
λ + A
6
= 0,
(9)
in this case, the Hurwitz criterion will look like
A
1
> 0,
A
1
A
2
− A
3
> 0, −A
1
2
A
4
+ A
1
A
2
A
3
+ A
1
A
5
− A
3
2
> 0,
A
1
2
A
2
A
6
− A
1
2
A
4
2
− A
1
A
2
2
A
5
+ A
1
A
2
A
3
A
4
− A
1
A
3
A
6
+ 2A
1
A
4
A
5
+ A
2
A
3
A
5
− A
3
2
A
4
− A
5
2
> 0,
−A
1
3
A
6
2
+ 2A
1
2
A
2
A
5
A
6
+ A
1
2
A
3
A
4
A
6
− A
1
2
A
4
2
A
5
− A
1
A
2
2
A
5
2
−
−A
1
A
2
A
3
2
A
6
+ A
1
A
2
A
3
A
4
A
5
− 3A
1
A
3
A
5
A
6
+ 2A
1
A
4
A
5
2
− A
2
A
3
A
5
2
+ A
3
3
A
6
−
A
3
2
A
4
A
5
− A
5
3
> 0,
(10)
(−A
1
3
A
6
2
+ 2A
1
2
A
2
A
5
A
6
+ A
1
2
A
3
A
4
A
6
− A
1
2
A
4
2
A
5
− A
1
A
2
2
A
5
2
− A
1
A
2
A
3
2
A
6
+
A
1
A
2
A
3
A
4
A
5
− 3A
1
A
3
A
5
A
6
+
+2A
1
A
4
A
5
2
+ A
2
A
3
A
5
2
+ A
3
3
A
6
− A
3
2
A
4
A
5
− A
5
3
)A
6
> 0.
For numerical analysis, let's take a rod with dimensions
l = 25 ∙ 10
−2
m; b =
10
−2
m; h = 2 ∙ 10
−3
m
, made of steel grade 40X, masses and installation
coordinates of the dynamic absorbers are
m
1
= m
2
= 0.0027335 kg; x
1
=
l
3
,
x
2
=
2l
3
. Coefficients
ν
1
=
3
8
; ν
2
=
1
2π
; ν
2
N
1
= 817684.6 ∙ a
1∗
2
− 1.920207 ∙ 10
12
∙ a
1∗
4
+
1.556859 ∙ 10
18
∙ a
1∗
6
.
The first four inequalities of the six inequalities of the Hurwitz criterion are
satisfied. The graphs of the coefficients of the last two inequalities depending on the
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change in the stiffness of the elastic elements of the dynamic absorbers are shown in
Figure 2. It can be seen from the graphs that the functions of these inequalities have
negative values, which indicates the presence of unstable stationary amplitudes.
FIGURE 2
. Graphs of the last two terms of Hurwitz inequalities
FIGURE 3
. Amplitude-Frequency Characteristics of the vibrations of a rod with
two dynamic absorbers is the ratio of the masses of the dynamic absorbers and the
rod, respectively, 0.03 (gray), 0.033 (blue) and 0.035 (red)
In Figure 3, graphs of amplitude-frequency characteristics are plotted depending
on the ratio of the masses of the dynamic absorbers and the rod for the first oscillation
form. From Fig.3 it can be concluded that with an increase in the ratio of the masses
of the dynamic absorbers and the rod from 0.03 to 0.035, the maximum values of the
oscillation amplitudes decrease. At the same time, areas of unstable amplitudes
remain. The frequencies of unstable amplitudes depart from the antiresonance
frequency.
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FIGURE 4
. Amplitude-Frequency Characteristics of vibrations of a rod with two
dynamic absorbers with a change in installation locations: red curve for
𝑥
1,2
=0,5l
±
0,11
l
, blue curve for
𝑥
1,2
=0,5l
±
0,09
l
and gray curve two dynamic
absorbers combined into one
The graphs of the Amplitude-Frequency Characteristics of the considered system
with different installation locations of dynamic absorbers are constructed. Thus, in
Fig. 4, the Amplitude-Frequency Characteristics of a rod with two dynamic absorbers
is constructed to dampen the first form of vibration near the center of the rod and in
the center of the rod. It can be seen from the figure that if two dynamic dampers are
combined into one, the oscillation amplitudes will increase much more than in the
case when the dynamic dampers are located separately.
CONCLUSIONS
Differential equations of a kinematically excited rod with two dynamic vibration
dampers and hysteresis energy dissipation in normal form are constructed. The
amplitude-frequency characteristics of this system are found. It is shown that some
terms of the Hurwitz inequalities have negative values for the stability of the system
under consideration, which indicate the presence of unstable stationary amplitudes
under certain conditions in the form of an abrupt change in oscillation amplitudes.
These results ensure accuracy in the selection of these rod parameters in practical
projects, when checking the dynamics and stability of the system in mathematical
modeling.
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