Vibrations of a Complex Systems with Damping under Dynamic Loading
This paper introduces an analytical-numerical
 method for solving a problem related to free
 and forced damping vibrations of a sandwich
 system that consists of two plates coupled with
 a viscoelastic interlayer. On the basis of the designed
 method we perfor...
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| Veröffentlicht in: | Проблемы прочности |
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| Datum: | 2002 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2002
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| Zitieren: | Vibrations of a Complex Systems with Damping under Dynamic
 Loading / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2002. — № 2. — С. 82-101. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860083120238231552 |
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| author | Cabanska-Placzkiewicz, K. |
| author_facet | Cabanska-Placzkiewicz, K. |
| citation_txt | Vibrations of a Complex Systems with Damping under Dynamic
 Loading / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2002. — № 2. — С. 82-101. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | This paper introduces an analytical-numerical
method for solving a problem related to free
and forced damping vibrations of a sandwich
system that consists of two plates coupled with
a viscoelastic interlayer. On the basis of the designed
method we perform numerical analysis
by means of the Timoshenko and the
Kirchhoff-Love models and investigate new
mechanical effects attributable to the action of
various types of dynamic loading on the system
considered.
На основе разработанного аналитико-численного метода решения задачи о свободных и
вынужденных затухающих колебаниях на примере сложной системы, которая состоит из
двух пластин, соединенных вязкоупругим слоем, выполнен численный анализ и изучены новые
механические эффекты, обусловленные действием на рассматриваемую систему различного
типа динамического нагружения. Расчеты проводились на основе моделей Тимошенко и
Кирхгофа-Лява.
На основі розробленого аналітико-числового методу розв’язку задачі про
вільні і вимушені згасаючі коливання на прикладі системи, що складається з
двох пластин, з ’єднаних в ’язкопружним шаром, виконано числовий аналіз і
вивчено нові механічні ефекти, зумовлені дією на дану систему різного типу
динамічного навантаження. Розрахунки проводилися на основі моделей
Тимошенка і Кірхгофа-Лява.
|
| first_indexed | 2025-12-07T17:17:29Z |
| format | Article |
| fulltext |
UDC 539.4
Vibrations of a Complex Systems with Damping under Dynamic
Loading
K. Cabanska-Placzkiewicz
Bydgoszcz University, Institute o f Technology, Bydgoszcz, Poland
УДК 539.4
Колебания сложных систем с затуханием при динамическом
нагружении
К. Ц абанска-П лаш кевич
Университет, Институт технологии, Быдгощ, Польш а
На основе разработанного аналитико-численного метода решения задачи о свободных и
вынужденных затухающих колебаниях на примере сложной системы, которая состоит из
двух пластин, соединенных вязкоупругим слоем, выполнен численный анализ и изучены новые
механические эффекты, обусловленные действием на рассматриваемую систему различного
типа динамического нагружения. Расчеты проводились на основе моделей Тимошенко и
Кирхгофа-Лява.
Introduction. Compound systems coupled together by viscoelastic
constraints play an important role in various engineering and building structures.
Since 1923, Timoshenko’s model [1] for various compound constructions has
been applied. Vibration analysis of laminated plates was presented in [2, 3] and in
many other works.
The problem of nonaxisymmetric deformation of flexible rotational shells
was solved in [4] with the use of the classical Kirchhoff-Love model and
improved Timoshenko’s model. The dynamic problem of elastic homogeneous
bodies was presented in [5]. Vibration analysis of systems of solid and deformable
bodies for complex motion was considered in [6].
Vibrations of elastically connected rectangular double-plate compound
systems under moving loading are presented in [7]. Vibration analysis of
compound systems with vibration damping is a difficult problem. In the above
com plex cases, especially where viscosity and discrete elements occur, it is
recomm ended to adopt the m ethod o f solving a dynam ic problem for a system in
the domain changing of a complex function [8, 9]. The property of orthogonality
of free vibrations of complex types was first described in [8] for discrete systems
w ith damping, for discrete-continuous system s w ith damping in [9] and for
continuous systems with damping in [10].
The goal o f this paper is to present a method for solving the problem and
dynamic analysis o f free and forced vibrations for a com plex system w ith
damping, w hich consists o f tw o elastic plates coupled by a viscoelastic interlayer,
for various types of dynamic loading.
© K. CABANSKA-PLACZKIEWICZ, 2002
82 ISSN 0556-171X. Проблемы прочности, 2002, N 2
Vibrations o f a Complex System
Statem ent of the Problem . Let us consider a problem of free and forced
vibrations for a complex system with, damping. External layers of the complex
system are made from elastic materials as plates coupled by a viscoelastic
interlayer (Fig. 1). The elastic plates are simply supported at their ends and
described by Timoshenko’s model [1]. The viscoelastic interlayer has the
characteristics of a homogenous continuous unidirectional Winkler’s foundation
and is described by the Voigt-Kelvin model [11—13].
Fig. 1. Dynamic model of a complex system with damping.
In this paper, w e consider two cases. In the first case, small-frequency
transverse vibrations of a complex system w ith damping are excited by a
stationary dynamic load f j ( x , y , t) at the point x 0 , y 0 w ith the load varying in
time t. In the second case, small-frequency transverse vibrations of a complex
system w ith damping are exited b y a nonstationary dynam ic load f ̂ x , y , t)
varying in tim e t.
The phenomenon of small-frequency transverse vibrations for a complex
system w ith damping is described by Timoshenko’s model w ith the following
non-homogeneous system of conjugate partial differential equations:
D,
D,
S2W 11 + SV u
Sx 2 Sy 2
S 2 W12 + S 2 W 12
S x 2 + Sy2
Sx
Sx
St'
2
, „ t Sw1 , , ) „ S ^ 1 2 „
+ H A ^ T + W 12 | — s 1 „ 2 “ 0St
S W1 ( o SW 11 SW 12 1 ( S
V 1 ^~ 2 T — H 1I V w 1 + ^ T — + l + (w1 — w 2 A k + c — 1 = f 1( x ,y ,0 ,
D
D
St
S 2 W 21 + S 2 W 21
Sx 2 Sy 2
S 2 W 22 + S 2 W 22
Sx 2
Sx Sy St
Sx
I TT I Sw2 , , I — S W21 n
+ H 2^ + W21 l — s 2 ~ - 0
( 1)
Sy Sx
St
2Sw 2 1 S y 22
+ H 2I ^ + y 22 I - S 2 — ^ - 0,
St
V 2
2
S w 2
St
2 , SW 21 , SW 22 1 ( - S
V w 2 +---------- +
2 Sx Sy — ( w1 — w 2 )\ k + c — I- 0
ISSN 0556-171X. ÏpoôneMbi npounocmu, 2002, № 2 83
K. Cabanska-Placzkiewicz
where
, H 1 — K [G ih i, H 2 — K 2G 2 h2 ’
dx 2 dy 2 ’
V li —Y li + $ l i , V 2i —Y 2i + $ 2i , i — 1, 2
Here w l — w ^ x ,y , t) and w 2 — w 2 ( x ,y , t) are the transverse deflections of
p la tes I and II, respec tive ly ; V 11 —V i i ( x ,y , t ), V l2 —V l2 ( x ,y , t), V 21 —
— V 2l( x , y , t), and V 22 — V 22( x , y , t) are the full angles of rectilinear elements of
plates I and II turn; E i and E 2 are Young’s moduli of the material of plates I
and II, respectively; E is Young’s modulus of the material of the interlayer; Gi
and G 2 are the shear moduli of the material of plates I and II, respectively; p i
and p 2 are the mass densities of the material of plates I and II; K [ and K 2 are
the shear coefficients; k is the coefficient of elasticity of the interlayer; c is the
coefficient of viscosity of the interlayer; hi and h 2 are the thickness of plates I
and II; h is the thickness of the interlayer; a and b are the dimensions of the
plates; y n and y 2i are the angles of the rectilinear element turn due to a cross
shear; $ u and $ 2i are the angles of the middle surface turn in plane for
a —const and fl — const; v ip and v 2p are Poisson’s ratios; t is time; x , y ,and
z are the coordinate axes; f i ( x ,y , t) is the dynamic load acting on the complex
system .
Separation of Variables. The analytical numerical method is based on the
separation of variables. Presenting the solution of the problem considered in the
form
and substituting (3) in the system of differential equations (l), by the assumption
that f i( x ,y , t ) — 0, w e obtain a homogenous system of ordinary differential
equations describing complex modes o f free vibrations of a com plex system w ith
damping:
W i(x , y , t) W i(x , y )
V i i (x ,y , t) ^ n (x ,y )
V n ( x , y , t) W „ ( x , y )
(3)
V 21(x ,y , t) ^ 2l (x ,y )
V 22(x ,y , t) ^ 22(x ,y )
(4a)
84 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, N2 2
Vibrations o f a Complex System
h 11 v 2W +
dW ii dW
D-
D-
dx
+ 12
dy
- (Wi - W2 )(k + icv ) + n iWiv 2 = 0,
d 2 W21
dx 2
d 2 W 22
+ 21
dy
2
dx 2
+
2
W22
+ H + W
dy
2\ dx
+ H , I— 2 + W
21
( dW 2
!2i dx 22
+ S 2 W 2iv = °
(4b)
dW 2l dW
dx
+ 22
dy
+ (W 1 - W2 )(k + icv ) + n 2W2v 2 = 0,
2
where i = - 1 . Here W1 = W1( x ,y ) and W2 = W2 ( x ,y ) are the complex
transverse vibration modes of plates I and II, v is the complex eigenfrequency of
free vibrations of the complex system with damping.
Solution of the B oundary Value Problem . The solution of the differential
equation (4) is sought in the form [1]
Wi ( x , y ) = X i ( x )Yi ( y X
Wi i ( x , y ) = @ i i ( x W ii ( y ),
_ W12 (x ,y ) = © i2 ( x ) r i2(y X
W2 ( x , y ) = X 2 ( x )Y 2 ( y X
W 2i (x>y ) = © 2i ( x ) r 2i (y X
W 22(x >y ) = © 22(x )T 22(y )■
Searching for a general solution of the system of differential equations (4) in the
form
Wi ( x , y ) 'A
W ii( x , y ) C ii
Wi2( x , y ) C i2
W2 ( x , y ) B
W2i( x , y ) D 2 i
W22( x , y )_ D 22 _
exp( rix )exp( T2 y ), (6)
w e obtain a homogeneous system of algebraic equations:
H i 2
A T T ri + C i i si = ° D i
H 1 o **
A D r2 + C i2s i = 0
(7a)
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 85
K. Cabanska-Placzkiewicz
where
A (r? + r% + p * * ) + Bp* - C n rx - C n r2 = 0,
B
B
H
D :
H
D '
2 2
r\ + D 2 1s 2 = 0
2 2
r2 + D 22 s 2 = 0,
A p 2 + B ( r\ + r2 + p 2 ) D 21 r1 D 22 r2 = 0
* 1 ** 1
p 1 =■— { k + icv ),
H 1
p 1 =
* 1 ** 1
p 2 = — {k + icv ),
H 2
p 2 =
* 2 2 H 1 —1 **
s 1 = r1 + r2
D 1
r + ----v ,
1 D 1 ’ s1
* 2 2 H 2 —2 **
s 2 = r1 + r2
D 2
r + ---- v ,
1 D 2 ’
s 2
2 . 2 H 1
1 + r2 - D
2 2 H 2
—L v 2 ,
D i
(7b)
-----r2 + —— V 2.D 2 r2 D 2
(8)
Constructing the determinant of the characteristic matrix of the system of
equations (7) and equating it to zero
H
D
H
D
1 2
~ ri
1 2
- r2
19 9 *:
r1 + r2 + p i
0
0
*
p 2
*
s 1 0 0 0 0
0
**
s 1 0 0 0
- r 1 - r 2
*
p 1 0 0
0 0
H 2 2-----r,
D 2 1
*
s 2 0
0 0 2
2r
2
2
0
**
s 2
0 0
9 9 **
r1 + r2 + p 2 - r 1 - r2
= 0, (9)
w e obtain the characteristic equation in the form of the following algebraic
equation:
r1 + a 17 r1 + a 16 r1 + a 15 r15 + a 14 r1 + a 13 r1 + a 12 r1 + a 11r1 + r2 + a 27 r2 +
--0 (10)
with the following roots: r1 j = ( - 1) J ~1 iX1v, r2 j = ( - 1) J ~1 iX2v , j = (2v - 1), 2v ,
and v = 1, 2, 3, 4. Here a 1 7 , a 16, a 1 5 , a 14 , a 13 , a 1 2 , a n , a 2 7 , a 26, a 2 5 , a 24 , a 23 ,
a 2 2 , a 2 1 , and a 0 are constant coefficients.
86 ISSN 0556-171X. npo6n.eubi npounocmu, 2002, № 2
Vibrations o f a Complex System
Applying Euler’s formulas, w e construct the solution of the system of
differential equations (4) in the form of the following system of solutions:
Wl (x, y) = 1 (A* sin X lvx + A** cos X lvx )(A*** sinX 2vy + A**** cosX 2v y),
V=l
4
V u ( x , y) = 1 (Cu* cosX lvx + Cu* sinXlvx)(C iiV cos X 2v y + C n V \ i n X 2* y),
V=l
4
^ 12 (x y ) = 1 ( C 12 v cos X lvx + Cl2*v sin X lv x)(C l2*v* cos X 2v y + C 12v* sin X 2v y l
4=l (n )
W2 (x, y) = 1 (B v* sinX lvx + B** cosXlvx)(B *** sinX 2vy + B**** cosX 2v y),
V=l
4
^ 2 1 (x> y) = 1 (Dllv cos X lvx + D 21v sin X lvx ')(D21v cos X 2v y + D 2lV* sin X 2v y ),
V=l
4
^ 22 (x y ) = 1 (D22v cos X lvx + D 22v sin X lvx W l * cos X 2v y + D 22* * sin X 2v y )■
V=1
Here Av , Av , Av , Av , B v , B v , B v , B v , C n v , C n v , C n v , C n ,
C 12v , C !2v , C !2v , C 12v , D 2\v , D 2\v , D 2\v , D 2\v , D 22v , D 22v , D 22v , and
D v are constants and X1v = a 1v + ip 1v and X 2v = a 2v + ip 2v are the
parameters describing the roots of the characteristic equation (10).
In accordance with (7), the following relations exist between the constants of
(9) : * « . . . . . . .
B v B v* B V ** D
B Va v =
A* ’
av ** , A v**
* C . v ** **
C llv
c llv = A * ’A v
c llv ** , A v**
* C llv ** C llv
c l v
= A * ’Av*
c llv ** , A v**
d*lv
D llv
' B * ’
d llv
D llv
= B ** B v
d*iv
D *iv
* ,
B v*
**
d 22v
D llv
**
B V
*** , Av***
***
a v
C llv
*** , Av***
_***
c llv
C llv
*** , Av***
***
c llv
D llv ***
d llv*** ,
B v
D llv
d llv*** ,
B V
Av
C llv
c llv = ' „■ . . . .
Av
= ^ = C llv
c \2v * * * , c \2v * * * * , ( l l )
Av
D 21v
, d 2lv „ .. . ß .
B v
D 22v
where
, d 22v B .
B v
Xlv X 2v + Pl + i(c llv Xlv + c l2v X 2v )
Pl
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 87
K. Cabanska-Placzkiewicz
~ ^ 2iv - i d 1 X 1v + D v 2* *** D \ D 1 ** ****
c 1 1v — c 1 1v — c llv — _ ’ c llv — - c 1 1 v - - c 1 1v ,
_ 1 i2
D 1 K
- X 2v - X 2v + D V 2* *** D i D i ** ****
c 12v — c 12v — c 12v — _ , c 12v — - c 12v — - c 12v ,
—1 X \vD 1 2v
i2 _ 2 * 2 2
1v i D 1v ^ D ^* *** D 2 D 2 j** j ****
d 21v — d 21v — d 21v — _ , d 21v — - d 21v — - d 21v ,
__2 o2
D 2 * 1v
- X \v - * _ ^ X 2v + D 2 v 2
* *** D 2 D 2 i** 7****
d 22v — d 22v — d 22v — _ , d 22v — - d 22v — - d 22v ■
_ 2 ,2
D 2 1 2v
(13)
On substitution of (12) in (11), the general solution of the system of
differential equations (4) takes the following form:
4
2 * ** *** ****
(A v sin X 1vx + A v cos X 1vx )(A v sin X 2v y + A v cos X 2v y),
V—1
4
^ 1 1 (x, y ) cUv (A v* cos X 1v x - Av** sin X 1v x ) (A *** cos X 2 v У - Av**** sin X 2 v У1
V—1
4
4 12 (x, У) c12v (A v* cos X 1v x - 4 * sin X 1v x ) (A *** cos X 2v У - 4 * * * sin X 2v У),
v —1
4
2 * ** *** ****
a v (A v sin X 1vx + A v cos X 1vx ) ( A v sin X 2v у + A v cos X 2v у),
v —1
4
4 21(x , У) a v d 21v (4 cos X 1v x - K * sin X 1v x ) (A v*** cos X 2 v У - 4 * * * sin X 2 v У1
v —1
4
4 22(x , У) av d 22v ( A cos X 1v x - A * sin X 1v x )(4 * * cos X 2 v У + 4 * * * sin X 2v У).
v —1
(14)
In order to solve the boundary value problem , the following boundary conditions are
used:
88 ISSN 0556-171X. Проблемы прочности, 2002, N 2
Vibrations o f a Complex System
W1 x—0
W1
y—0
dWn
— o,
= o,
dx
dWn
x—0
dy y—0
W — 0,
Wi b — 0,1 y—b
— 0,
dWii
— 0,
dx
dW i2
dy y—b
W21 x—0 = °>
W>\ — = 0
— 0,
dW 21
dx
dW 22
— 0,
x—0
dy y—0
W — 0,
W2 b — 0,2 y—b
dW 2i
— 0,
dx
dW 22
— 0, (15)
dy
— 0.
y—b
Substituting (14) in (15), w e obtain a homogenous system of algebraic
equations, which in the matrix notation has the following form:
Here
or
YX — 0.
X — [Ai , A 2 , A 3 , A 4 , Ai , A 2 , A 3 , A 4 ]
X = [Ai , A 2 , A 3 , A 4 , A i , A 2 , A 3 , A 4 ]
are the vectors of the unknowns in the system of equations and
Y = \Yi* j ]8*g
is the characteristic matrix of the system of equations (16).
The first four equations (16) are presented in the form
1 1 1 1
**-
A 1
- X n c U 1 - X 12 c 112 - X 13 c 113 - X 14 c 114
**
A 2
Ü1 a 2 a 3 a 4 A 3
_- X 11a 1d 211 - X 12 a 2 d 212 - X 13 a 3 d 213 X 14 a 4 d 214 .A 4
(16)
(17)
— 0 (18)
where A 1 = A 2 = A 3 = A 4 = 0.
The remaining four equations (16) give the following system of equations:
SS11 ss12 ss13 ss14 A* "A{
- X 11c111SS11 - X 12 c112 ss12 - X 13 c113 ss13 - X 14 c114 ss14
*
2
'K
a^ssn a 2 ss12 a 3 ss13 a 4 ss14 A *
A 3
_- X 11a 1d211ss11 - X 12 a 2 d212 ss12 - X 13 a 3 d213 ss13 X 14 a 4 d214 ss14 A *
_ A4 _
— 0,
(19)
where ssn = sin X 11a , ss12 = s in X 12a , ss13 = sinX 13 a, and ss14 = sinX 14a.
The condition of solving the system of equations (19) is vanishing of the
characteristic determinant, i.e.,
ISSN 0556-171X. npoôëeMbi npounocmu, 2002, N 2 89
K. Cabanska-Placzkiewicz
SSu ssi2 ssi^ ss 4̂
- X i i ci n ssn - X 12cn 2ss12 - X 13 c113ss13 - X 14c114ss14
a 1ssn a 2 ssi2 a 3 ssi3 a 4 ssi4
~X11a 1d2i i ssii - X 12a2d212ss12 - X 13 a 3d213ss13 - X 14a 4d214ss14
= 0. (20)
Expanding determinant (10), w e obtained the following characteristic
equation:
sinXl l a s inX l2a sin X l3 a sin Xl4a — 0, (2 1 )
where X ll — X l2 — X l3 — X l4 — X]_.
The characteristic equation (21) m ay be rewritten in the form
sin Xla — 0, (2 2 )
where, in the general case,
X l — a l + ifll (23)
are complex numbers.
Substituting (23) in (22), w e obtain the following equation:
sin a l a cosh f l l a + icos a l a sinh f l l a — 0, (24)
w hich has the following roots:
n ft
a
n Ji
= - 1- , P - = 0, ni = 1 ,2 ,3 , . . . . (25)
Taking into account (25) in (23), w e obtain the following identity:
X n = a = ~ . (26)a
By analogy with equations (16)-(24), w e obtain
n 2 J
a —2 b , —2 0, П2 1, 2, 3, """,
= = n 2 J
X n, = a n, = L "
(2 1 )
Substituting r1 = iX and r2 = iX n in equation (10) and carrying out all
transformations, w e obtain the following equation of frequency:
8 . # 1 . # 6 . # 5 . # 4 . # 3 . # 2 . # . # nv + а ц v + a i6 v + a i5 v + а ы v + a ^ v + a u v + a n v + a 0 = 0 (28)
90 ISSN 0556-171X. Проблемы прочности, 2002, N 2
Vibrations o f a Complex System
from w hich a sequence o f com plex eigenfrequencies is determined:
v HiH2 = iV HiH2 ~ ® HiH2 , (29)
# # # # # # # 1 # •where a 11, a 16, a 15, a 14, a 13, a 12, a 11,a n d a 0 are constant coefficients.
By substituting (29) into (13) and (14), the following formulas for the
coefficients o f amplitudes are obtained:
- ^ 1ri1 H2 — ^ 2»1»2 1 P 1 1 i(c 11»1 n2 ̂ 1U1H2 1 C12H1H2 ̂ 2»1»2 )
a fan, = * 5
1 2 P 1
— ;2 — ; i 2
1n1 n2 1 D 1 1n2 + D 1 V n1n2
where
c 11П1П2 H , -__L ;2
D Л\пхпг
I 2 — H L 2 + —L 2
2п1 n2 1 d l 2nin2 + D l VnLn2
Cl2nl n2 H
— }2D л 2nln2
— }2 — H 2 2 2
Lnl n2 l D nLn2 + D V nLn2
d
2 lnLn2 H 2 2
D ̂ nL n2
H— — ~~~ 2 2 + —2 2
2 nl n2 l D 2nl n2 + d V nLn2
d 22n n = ------------------- --------- ' (30)__2 у 2
D 2 2 nLn2
p* = H1 ( k + iCV nL n2 \ P \* = H f ^ * LV — k — icV nxn1 ),
P 2 = H f ( k + iCV nL n2 \ P 2* = H j~ ( * 2V \ * 2 — k — iCV nln2 )-
(31)
Substituting the sequences к щ , X^ and а ЩПг, С ц ^ , c UniП2, d 2^ ,
d 22nitli in (14), w e obtain the following six sequences of complex modes of free
vibrations for a complex system with damping:
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K. Cabanska-Placzkiewicz
Wjnjn2 (x ,y ) = sin X ni X sin X ni y ,
^ j j n j n2(x , y ) = c jjnjn2 cos X nj x C0s X n2 y >
^12nJn2(x >y ) = c J2nJn2 cos XnJ x cos Xn2 y ,
W2nJ n2(x , y ) = a nJn2 sin X nJ x sin X n2 y ,
W 2JnJn2( x ,y ) = a nJn2 d 2JnJn2 cos Xnj x cos Xn2 y ,
^ 2 2 nJn2( x , y ) = a nJn2 d 22nxn2 cos X nJ x cos X n2 y-
(32)
Solution of the Initial Value Problem . In the case of v = v .
complex equation of motion
T = 0 exp( iv t)
can be written in the following form:
Tnj n2 = 0 Hj n2 exp ( iv njn2 t);
the
(33)
(34)
where O is the Fourier coefficient.
Free vibration of a complex system with damping is presented in the form of
a Fourier series based on the complex eigenfunctions, i.e.:
2 2 w jnJn2( x ,y )
nj =J n2 =J
00 0C
^ JJnjn2 ( x ,y )
nj =1 n2 =J
0 0
2 2 ^ J2njn2 ( x ,y )
nj =J n2 =J
0 0
2 2 W2nJn2 (x ,y )
nj =J n2 j
0 0
2 2 ^ 2jnjn2 (x ,y )
Wj( x , y , t)
V jj( x , y , t)
V j 2 ( x , y , t)
w 2 ( x ,y , t)
V 2j ( x ,y , t)
V 22( x ,y , t)
nj =j n =j
0 0
2 2 ^ 22nj n2 ( x , y )
0 njn2 exp (iv njn2 t ) - (35)
From the system of equations (4), performing some algebraic
transformations, adding the equations together and then integrating them on both
sides within the limits from 0 to a, and from 0 to b, w e obtain the property of
orthogonality o f eigenfunctions for a com plex system w ith damping using
Timoshenko’s model:
Hj =j n2 =j
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Vibrations o f a Complex System
a b
î î [и ! {Wlm Vln + WlnVlm ) + /Л 2 (W 2m V2n + W2nV2m ) +
0 0
+ ^ l (^U m Q n n + ^ l l n Q llm + ^ 12mQ\2n + ^ 12nQ l2m ) +
+ S 2 (^ 2lmQ 2ln + ^ 2lnQ 2lm + ^ 22mQ 22n + ^ 22nQ 22m ) +
+ c(Wln - W2n )(Wlm - W2m )]dxdy = NnÔ nm , (36)
where
a b
N n = î î ^ Л lWlnVln + Л 2W2nV2n + “ l (^ llnQ lln + 4*l2nQ l2n ) +
0 0
+ “ 2 (^ 2lnQ 2ln + ^ 22nQ 22n ) + c(Wln - W2n Ÿ \dxd y , (37)
Vln = iv nWln (x >УX V2n = iv nW2n (x >УX
Vlm = iv m Wlm ( x , У X V2m = iv mW2m ( x , У X
Q lln = iv n ^ l ln ( x ,У X Q 2ln = iv n ^ 2ln( x ,УX
Q llm = iv m ̂ l lm ( x , У ), Q 2lm = iv m ̂ 2lm ( x , УX
Q l2n = iv n^ l2n ( x ,У X Q 22n = iv n^ 22n ( x ,У X
Q l2m = iv m ̂ l2m (x ’У X Q 22m = iv m ̂ 22m (x ’У X
(38)
Here d nm is Kronecker’s delta, n = (n 1, n 2 ),a n d m = (m1, m2 ).
The following initial conditions are the basis for solving the problem of free
vibrations:
w l ( x , y ,0) = w ou W2 ( x , y ,0) = w 02 ,
V 11(x , y ,0) = V 011, V 2l (x , y ,0) = V 02U (39)
V 12( x ,y ,0) = XP 012 , V 22( x ,y ,0) = XP 022-
By applying conditions (39) in series (35) and taking into account the
property of orthogonality (36), the formula for a complex Fourier coefficient is
obtained:
1 a b o
^ ni n2 = N S S ( V l (V1ni n2 W01 W1nin2 w 01)
N n1n2 0 0
+ Л 2 (V 2nln2 w 02 + W2nln2 w 02 ) +
О О
+ “ l (Q llnln2У 0ll + '^ lln ln2 У 0l l + Q l2nln2У 0l2 + ''^l2nln2 У 0l2) +
+ “ 2 (Q 21n1n2V 021 + ^ 21n1n2 V 021 + Q 22n1n2V 022 + ^ 22пхп2 У 022) +
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K. Cabanska-Placzkiewicz
+ c[(W inin2 - W2n1n2)(Woi - w 02 )]}dxdy. (40)
By substituting (32), (34), and (40) into (35) and performing the
trigonometric and algebraic transformations, the final form of free vibration of a
complex system with damping is obtained:
00 X
Wi = 2 2 e
n =1 n2 =1
0 0
V ii = 2 2 '
V 1 2 = 2 2 e
n =i n =i
o ,
ini n2l o ,ni n2 w iinin2
o ,ni n2 w i2n n
cos( (° n̂ 2 t + $ n̂ 2 + 0 iinin2 )’
cos(O ni n2 t + $ ni n2 + 0 i2ni 2̂
ni =i n2 =
0 0 (4 i)
w2 = 2 2 e
n =i n =i
Ini n2' o W cos( O nin2 t + $ nin2 + X 2nin2 l
ni =i n2 =
0 0
y 2i = 2 2 i
ni =i n2 =i
0 0
V 22 = 2 2 '
lni n2
o ni n2 w 2in n
o w 22 ni n2
cos(O nin2 t + $ nin2 + 0 2inin2 X
cos(O nin2 t + $ nin2 + 0 22 ni n2 X
n =i n =i
where
W, = v W
w iinin2 A i in n + ^ i i n ^2 , w 2inin2 A 2inin2 + Q 2in^ 2 ,
w i2nin2 w 22nin2 22nin2 + Q 22nin2 ,
X inin2 arg Winin2 , X 2nin2 argW 2n̂ 2 ,
0 iinin2 = arg W iinin2 , 0 2in^2 = arg W 2inin2
= arg Wi2nv = arg W 22n„
o ni n2 A/C ni n2 + D n̂ 2 , $ ni n2 arg 0 nin2 ,
X in^2 R eW inin2 , Yinin2 lm W inin2 , X 2nin2 R eW 2nin2 , Y2n̂ 2 lm W 2n,n.i n2
A iinin2 Re W iinin2 ’ Q iinin2 Im W iinin2 ’
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ni =i n2 =i
Vibrations o f a Complex System
^ 12n̂ 2 W j2njn2 ’ ^ 12ntn2 W j2njn2 ’
^ 22njn2 = W 22njn2 ’ ^ 22njn2 = W 22n j^ ’
C njn2 ^ njn2 ’ D njn2 ^ njn2 ' (42)
Solution of the Forced V ibration Problem. In order to solve the differential
equations (l), the function of loading is expanded from the operator method [15]
00 X
f j (x ’ y ’ t) = jWjnjn 2 ^ ß 2W2njn 2 ^
n, =j n 1 =1
+ û j (Wjjnjn2 W j2njn2 ) “ 2 (W2 jnjn2 W 22njn2 ) ] fnjn2 ’ (43)
w h e r e Wlnln2 , W llnln2 , V 2 lnln2 , W2nln2 , V 12nln2 , an d V 22nxn2 h a v e b e e n
described by equations (32).
The function o f the displacement o f a com plex system w ith damping is
presented in the form of a Fourier series:
Wj
V jj
V j2
Wj
V 2j
V 22
0 0
= 2 2
n =j n =j
W jnj n2
W jjnjn2
W j2njn2
WoW 2njn2
W 2jnjn2
W 22njn2
T (44)
Substituting (43) and (44) into the differential equations (l), w e obtain the
following equation o f motion:
T njn2 ̂V njn2 Tnj n2 f nj n2 ’ (45)
where T are the coefficients of the distribution of the loading function in a
Fourier series.
Applying the property of the eigenfunction orthogonality (36), w e derive the
coefficients for the load distribution, namely:
f
j
n jn 2 N NN j N n jn 2 0 0
ijn 2I f Wjn + W-, + W jjn jn 2 + W j2n jn 2 +
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K. Cabanska-Placzkiewicz
+ ^ 21n1n2 + 'V 22n1n2 V 1(x , y , t )dxdy.
The solution of the differential equation (46) has the form [15]
(46)
T
nin2 0
i V n 1n 2 )( t - r ) - W n 1n 2( r )dx. (47)
On substituting (46) and (47) into (44), equation (44) can be rewritten in the
following form:
T
n1 n2 COS($ 1n̂ 2 n1^2 ^̂
n1 —1 n2 —
00 X
V 11 ^ 11n1 H2 Tn H2 sin( ̂ 11 1̂ H2 ^ U1^2 )
n —1 n2 —1
0 0
<pn — 2 2 r 12 n1 n2 T
n1 n2 sin(0 12n̂ 2 £ n1n2 X
0 0
W2 |W2n̂ 2
n1 —1 n2 —1
0 0
V 21 — 2 2 | ^ 21n1 n
T.n1n2
T
cos($ 2n1 n2 n1n2 ^’
sin(0 21n1n2 ^ n1n2
n1 —1 n2 —1
0 0
v 22 — 2 2K22n̂ 2 T
n1 n2
(48)
where
$ 1n1 n2 — arg W1n1n2 ’ $ 2 n n — argW2 n n ’
0 11n̂ 2 — arg ^ 11n̂ 2 ’ 0 21n̂ 2 — arg ̂ 21n̂ 2 ’
0 12n1n2 — arg ̂ 1 2 ’ 0 22n̂ 2 — arg ^ 22n n ’ £ n n — arg Tn n ' (49^
Solution of the Problem for Various Types of Dynamic Loading. Let us
consider various types of dynamic loading for a complex system with damping.
External layers of the complex system are made as plates of elastic material
coupled by a viscoelastic interlayer and simply supported at their ends. The elastic
plates are described by the Timoshenko and Kirchhoff-Love models. The
viscoelastic interlayer has the characteristics o f a homogenous continuous
unidirectional Winkler’s foundation and is described by the Voigt-Kelvin model
[11-13].
n1 —1 n2 —1
n —1 n2 — 1
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Vibrations o f a Complex System
We consider two cases. In the first case, small-frequency transverse
vibrations of a complex system with damping are excited at the point x 0 = 0.6a ,
y 0 = 0.6b by a stationary dynamic load varying in time t:
f j ( x ,y , t ) = P jd(x - x 0 )d (y - y 0)sin(rn01). (50)
In the second case, small-frequency transverse vibrations o f the com plex
system with damping are excited by the dynamic nonstationary load f j(x , y , t)
varying in time t [j0]. W e consider various types of nonstationary loading.
W ith the first type, small-frequency transverse vibrations o f a com plex
system w ith damping are excited by a non-inertial m oving concentrated load
f ̂ x , y , t) varying in time t for y = const:
f ̂ x * ,y , t ) = Pjd(x - x *) + M x - x *). (5j)
With the second type, small-frequency transverse vibrations of a complex
system with damping are excited by an inertial moving concentrated load
f x , y , t ) varying in time t for y = const:
, , * „ d w (x , y , t) * d
f i (x ,y , t) = - mi —2 ° (x - x ) - r ^ —y
d t2 d t2
dw(x , y , t)
dx
ô '(x - x ).
(52)
W ith the third type, small-frequency transverse vibrations o f a com plex
system with damping are excited by a nonuniform load f j(x , y , t ) varying in time
t:
f !( x u y ! , t) = b(t) - mi
d w (x u y u t)
d t2
ô(x - x i )ô(y - y 1). (53)
Here Pj is the force; M j is the moment; mj is the rubble mass; Tj = m jr ; r is
the radius of the gyration of the mass; d(...) and d '(...) are the Dirac delta;
H (...) is the Heaviside function; x = v t (v is a constant speed); y = 0.5b;
w (x , y , t ) is the first iteration of the dynamic displacement of the complex system
with damping from the force; f ̂ x j , y j , t ) is the displacement of the complex
system w ith damping under rubble; v n are com plex eigenfrequencies o f forced
vibrations; x 0 and y 0 are coordinate rubble for time t = 0; and b (t) is the
displacement of the rubble in the direction of the axis z.
Accelerations occurring in equations (52) and (53) are expanded using the
Renaudot formula [j4]
2 2 * 2 * 2 * d r _ d w(x , y, t ) d w(x , y, t ) * d w(x , y, t ) * 2
— r[w (x , y, t)] = -
dt2
■+ 2 -
dxdt
-v +-
d t2
dw(x , y, t )
dx
d 3 w(x *, y, t) + 2 d 3 w(x *, y, t ) v * + d 3 w(x *, y, t ) *)2 (54)
dxdt dx 2 dt dx3
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d
K. Cabanska-Placzkiewicz
O _
Calculations are made for the following data: E l = E 2 = 10 N- m , E =
= 105 N -m ~2 , p 1 = p 2 = 2-103 N - s 2 - m _ 4 , h 1 = h 2 = 0.04 m, h = {0.08,
0.12} m , v 1p = v 2p = 0.3, a = 4 m , b = 2 m , c = 2-104 N - s - m _ 2 , P1 = 1 0 4
N, v * = 20 m - s ~ 2.
In order to solve the boundary value problem, the following boundary
conditions are used:
for Timoshenko’s model
W1 x=0 = 0
dW u
dx
= 0,
x=0
W21 x=0 = 0
d^ 2 1
dx x=0
W1 = 0,1 x=a
dW 11
dx
W2 \2
= 0,
dW 21
dx
= 0,
x=a
= 0,
= 0,
W1 l y=0 = 0,
dW,12
d
w 2 y=0
dW 22
= 0,
y=0
= 0,
dy
= 0,
y=0
W1 b = 0,1 y=b
dW u
dy
= 0,
y=b
W2 \ = 0,2 y= b
dW 22
dy
= 0,
y=b
(55)
for the Kirchhoff-Love model
W1 l x=0
d 2W1
= 0,
dx
= 0,
x=0
W2I „ . = 0,
d 2W■
x=0
2
dx x=0
WA1 x=l
d 2W1
= 0,
dx 2
W2\21 x=
d 2W0
dx 2
= 0,
= 0,
x=a
. = 0,
= 0,
W1 Ï y=0 = 0,
d 2W1
dy
W2I
d 2W■
= 0,
y=0
2
y=0
= 0,
dy y=0
W1 b = 0,1 y=b
d 2W1
dy
= 0,
y=b
W2I , = = 0,
= 0,
d 2W■
y=b
2 = 0.
y=b
(56)
In the first case, the effect o f the stationary dynamic force (5 0) in a com plex
system , w hich consists o f elastic plates coupled b y a viscoelastic inertial
interlayer resting on a stiff foundation W2 = 0, is presented in Figs. 2 and 3.
Figures 2 and 3 present the amplitude-frequency diagrams for complex systems
with damping (c = 10—6 s) with viscoelastic interlayers of small h = 0.08 m (Fig.
2) and large h = 0.12 m (Fig. 3) thickness for real stationary frequencies in the
range 1700 < m 0 < 2000 at the point x = 0.7 a , y = 0.7b.
The amplitudes o f forced vibrations o f the plates o f a com plex system w ith
damping loaded b y a concentrated m oving force calculated on the basis o f
Timoshenko’s model are approximately 9-12% higher than those of the plates
obtained with the use of the Kirchhoff-Love model (Figs. 2 and 3).
In the latter case, small-frequency transverse vibrations o f a com plex system
with damping are excited by a nonstationary dynamic load f 1(x , y , t ) varying in
tim e t.
98 ISSN 0556-171X. npoôëeubi npounocmu, 2002, N2 2
Vibrations o f a Complex System
Amplitude [m]
Fig. 2. The amplitude-frequency diagram for a complex system with damping with a stationary
dynamic force f[(x, y, t) acting at the point x0, y0 for h = 0.08 m: (a) for the Kirchhoff-Love
model W*l; (b) Timoshenko’s model \wt \.
Amplitude [m]
Fig. 3. The amplitude-frequency diagram for a complex system with damping with a stationary
dynamic force f 1(x, y, t) acting at the point x0, y0 for h = 0.12 m: (a) for the Kirchhoff-Love
model |w*|; (b) Timoshenko’s model jwj.
Small-frequency transverse vibrations of a complex system w ith damping are
sjs
excited by a non-inertial moving force f j (x , y , t ) = P1o(x — v t ) with the speed
v for the parameters: y = 0.5b and h = 0.08 m.
The effect o f non-inertial moving loading in a complex system with a
viscoelastic interlayer is shown in Figs. 4 and 5 (the Kirchhoff-Love model (Fig. 4)
and Timoshenko’s model (Fig. 5)).
Calculations on the basis of Timoshenko’s model for a complex system with
damping loaded b y a concentrated m oving force gave the amplitudes o f forced
vibrations of the plates that are approximately 60% larger than those obtained
w ith the use o f the K irchhoff-Love model.
Similar results were obtained for the problems o f vibration o f a com plex
system w ith damping excited b y an inertial m oving concentrated mass and
nonuniform loading varying in tim e t.
ISSN 0556-171X. Проблемы прочности, 2002, № 2 99
K. Cabanska-Placzkiewicz
w,
0.00004
0.00003
0.00002
0.00001
- 0.00001
m
t , s
Fig. 4. Forced vibrations of a complex system with damping induced by a nonstationary force
f (x, y, t) moving with the speed v = 20 m • s_ , h = 0.08 m.
w, m
t , s
Fig. 5. Forced vibrations of a complex system with damping induced by a nonstationary force
f (x, y, t) moving with the speed v = 20 m • s_ , h = 0.08 m.
Conclusions. The analytical-numerical method presented in this paper can
be applied to solutions of free and forced vibrations o f various engineering
structures consisting of plates and shells coupled by viscoelastic constraints for
various types o f dynam ic loading.
Numerical investigations revealed that the problem of free and forced
vibrations for various engineering structures consisting o f plates and shells
subjected to the action of a stationary dynamic force can be solved using the
Kirchhoff-Love model. When w e consider a problem of free and forced
vibrations for various engineering structures under m oving concentrated loading,
it is necessary to use Timoshenko’s model.
Р е з ю м е
На основі розробленого аналітико-числового методу розв’язку задачі про
вільні і вимушені згасаючі коливання на прикладі системи, що складається з
двох пластин, з ’єднаних в ’язкопружним шаром, виконано числовий аналіз і
вивчено нові механічні ефекти, зумовлені дією на дану систему різного типу
динамічного навантаження. Розрахунки проводилися на основі моделей
Тимошенка і Кірхгофа-Лява.
100 ISSN 0556-171X. Проблеми прочности, 2002, № 2
Vibrations o f a Complex System
1. S. P. Timoshenko and S. Wojnowsky-Krygier, Theory o f Plates and Shell,
Arkady, New York, Toronto, London (j959).
2. R. A . Di Taranto and J. R. McGraw, “Vibratory bending of damped
laminated plates,” Trans. ASM E, J. Eng. Industry, 91, j 0 8 j - j0 9 0 (j969).
3. W . Kurnik and A. Tylikowski, M echanics o f Lam inated Elements, Publ.
Warsaw Univ. of Techn., Warsaw (j997).
4. N. D. Pankratova, B. Nikolaev, and E. Switonski, “Nonaxisymmetrical
deform ation o f flexible rotational shells in classical and improved
statements,” J. Eng. M e c h 3, No. 2, 89-96 (j996).
5. V. T. Grinchenko, Equilibrium and Steady-State Vibration o f Elastic Bodies
o f Finite Dimensions [in Russian], Naukova Dumka, Kiev (j978).
6. V . I. Gulyaev and P. P. Lizunov, Vibration o f Systems o f Solid and
D eformable Bodies under Complex M otion [in Russian], Vyshcha Shkola,
K iev, (j989).
7. W . Szczecniak, “Vibration of elastic sandwich and elastically connected
double-plate systems under moving loads,” in: Building Engineering, Publ.
Warsaw Univ. of Techn., No. j3 2 , j5 3 - j7 2 (j998).
8. F. Tse, I. Morse, and R. Hinkle, M echanical Vibrations: Theory and
Applications, Allyn & Bacon, Boston (j978).
9. J. Niziol and J. Snamina, “Free vibration of the discrete-continuous system
with damping,” J. Theor. Appl. M e ch , 28, No. j-2 , j4 9 - j6 0 (j990).
10. K. Cabanska-Placzkiewicz, “Problems of vibration control in ecologically-
dangerous engineering systems,” in: The Role o f Universities in the Future
Information Society (RUFIS 2000), Kiev (2000), pp. 26-27.
11. W . N owacki, Building Dynamics, Arkady, Warsaw (j972).
12. Z. Osinski, Dam ping o f M echanical Vibration, PWN, Warsaw (j979).
13. D. Nashif, D. Jones, and J. Henderson, Vibration D am ping [Russian
translation], Mir, M oskw a (j988).
14. M. Renaudot, “Etude de l ’influence des charges en mouvement sur la
resistance, des ponts metallique droites,” Annales des Ponts et Chausses,
No. j , j4 5 -2 0 4 ( j 86j).
15. J. Cabanski, “Generalized exact m ethod o f free and forced oscillations in the
non-conservative physical system,” J. Techn. P h ys , 4, No. 4 j, 4 7 j-4 8 j
(2000).
Received 15. 05. 2001
ISSN 0556-171X. npoôneMbi npounocmu, 2002, N 2 101
|
| id | nasplib_isofts_kiev_ua-123456789-46749 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T17:17:29Z |
| publishDate | 2002 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Cabanska-Placzkiewicz, K. 2013-07-06T15:04:39Z 2013-07-06T15:04:39Z 2002 Vibrations of a Complex Systems with Damping under Dynamic
 Loading / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2002. — № 2. — С. 82-101. — Бібліогр.: 14 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/46749 539.4 This paper introduces an analytical-numerical
 method for solving a problem related to free
 and forced damping vibrations of a sandwich
 system that consists of two plates coupled with
 a viscoelastic interlayer. On the basis of the designed
 method we perform numerical analysis
 by means of the Timoshenko and the
 Kirchhoff-Love models and investigate new
 mechanical effects attributable to the action of
 various types of dynamic loading on the system
 considered. На основе разработанного аналитико-численного метода решения задачи о свободных и
 вынужденных затухающих колебаниях на примере сложной системы, которая состоит из
 двух пластин, соединенных вязкоупругим слоем, выполнен численный анализ и изучены новые
 механические эффекты, обусловленные действием на рассматриваемую систему различного
 типа динамического нагружения. Расчеты проводились на основе моделей Тимошенко и
 Кирхгофа-Лява. На основі розробленого аналітико-числового методу розв’язку задачі про
 вільні і вимушені згасаючі коливання на прикладі системи, що складається з
 двох пластин, з ’єднаних в ’язкопружним шаром, виконано числовий аналіз і
 вивчено нові механічні ефекти, зумовлені дією на дану систему різного типу
 динамічного навантаження. Розрахунки проводилися на основі моделей
 Тимошенка і Кірхгофа-Лява. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Vibrations of a Complex Systems with Damping under Dynamic Loading Колебания сложных систем с затуханием при динамическом нагружении Article published earlier |
| spellingShingle | Vibrations of a Complex Systems with Damping under Dynamic Loading Cabanska-Placzkiewicz, K. Научно-технический раздел |
| title | Vibrations of a Complex Systems with Damping under Dynamic Loading |
| title_alt | Колебания сложных систем с затуханием при динамическом нагружении |
| title_full | Vibrations of a Complex Systems with Damping under Dynamic Loading |
| title_fullStr | Vibrations of a Complex Systems with Damping under Dynamic Loading |
| title_full_unstemmed | Vibrations of a Complex Systems with Damping under Dynamic Loading |
| title_short | Vibrations of a Complex Systems with Damping under Dynamic Loading |
| title_sort | vibrations of a complex systems with damping under dynamic loading |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/46749 |
| work_keys_str_mv | AT cabanskaplaczkiewiczk vibrationsofacomplexsystemswithdampingunderdynamicloading AT cabanskaplaczkiewiczk kolebaniâsložnyhsistemszatuhaniempridinamičeskomnagruženii |