Vibrations of a Complex System with a Viscoelastic Inertial Interlayer
The paper presents an analytical method for solving problems of free and forced vibrations with damping of complex systems whose loaded layers are made of homogeneous elastic inertial materials, and the middle one is made of viscoelastic inertial material. Small lateral vibrations of the compl...
Gespeichert in:
| Datum: | 2001 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2001
|
| Schriftenreihe: | Проблемы прочности |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/46722 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Vibrations of a Complex System with a Viscoelastic Inertial Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2001. — № 6. — С. 103-115. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-46722 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-467222013-07-16T18:56:53Z Vibrations of a Complex System with a Viscoelastic Inertial Interlayer Cabanska-Placzkiewicz, K. Научно-технический раздел The paper presents an analytical method for solving problems of free and forced vibrations with damping of complex systems whose loaded layers are made of homogeneous elastic inertial materials, and the middle one is made of viscoelastic inertial material. Small lateral vibrations of the complex systems are caused by distributed and movable loads. A dynamic analysis of laminated structures for a wide range of variation of the geometrical and mechanical characteristics of a layer from viscoelastic inertial material was performed. Предложен аналитический метод решения задач о затухании свободных и вынужденных колебаний сложных систем, несущие слои которых выполнены из однородного упругого, а средний - из вязкоупругого инерционного материала. Малые поперечные колебания сложных систем обусловлены распределенной и подвижной нагрузкой. Выполнен динамический анализ слоистых конструкций в широком диапазоне изменения геометрических и механических характеристик слоя из вязкоупругого инерционного материала. Запропоновано аналітичний метод розв’язку задач щодо згасання вільних та вимушених коливань складних систем, несучі шари яких виконано з однорідного пружного, а середній - з в’язкопружного інерційного матеріалу. Малі поперечні коливання складних систем зумовлені розподіленим і рухомим навантаженням. Виконано динамічний аналіз шаруватих конструкцій у широкому діапазоні зміни геометричних і механічних характеристик шару з в’язкопружного інерційного матеріалу. 2001 Article Vibrations of a Complex System with a Viscoelastic Inertial Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2001. — № 6. — С. 103-115. — Бібліогр.: 16 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46722 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Cabanska-Placzkiewicz, K. Vibrations of a Complex System with a Viscoelastic Inertial Interlayer Проблемы прочности |
| description |
The paper presents an analytical method for
solving problems of free and forced vibrations
with damping of complex systems whose
loaded layers are made of homogeneous elastic
inertial materials, and the middle one is made
of viscoelastic inertial material. Small lateral vibrations
of the complex systems are caused by
distributed and movable loads. A dynamic analysis
of laminated structures for a wide range of
variation of the geometrical and mechanical
characteristics of a layer from viscoelastic inertial
material was performed. |
| format |
Article |
| author |
Cabanska-Placzkiewicz, K. |
| author_facet |
Cabanska-Placzkiewicz, K. |
| author_sort |
Cabanska-Placzkiewicz, K. |
| title |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer |
| title_short |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer |
| title_full |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer |
| title_fullStr |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer |
| title_full_unstemmed |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer |
| title_sort |
vibrations of a complex system with a viscoelastic inertial interlayer |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2001 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46722 |
| citation_txt |
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2001. — № 6. — С. 103-115. — Бібліогр.: 16 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT cabanskaplaczkiewiczk vibrationsofacomplexsystemwithaviscoelasticinertialinterlayer |
| first_indexed |
2025-07-04T06:10:08Z |
| last_indexed |
2025-07-04T06:10:08Z |
| _version_ |
1836695588056334336 |
| fulltext |
UDC 539.4
Vibrations of a Complex System with a Viscoelastic Inertial Interlayer
K. Cabanska-Placzkiewicz
Bydgoszcz Academy, Department of Mathematics, Technology, and Natural Science
Institute of Technology, Bydgoszcz, Poland
УДК 539.4
Колебания сложных систем с вязкоупругим инерционным
заполнителем
К. Цабанска-Плашкевич
Академия, Отделение математических технологий и естественных наук Техничес
кого института, Быдгощ, Польша
Предложен аналитический метод решения задач о затухании свободных и вынужденных
колебаний сложных систем, несущие слои которых выполнены из однородного упругого, а
средний - из вязкоупругого инерционного материала. Малые поперечные колебания слож
ных систем обусловлены распределенной и подвижной нагрузкой. Выполнен динамический
анализ слоистых конструкций в широком диапазоне изменения геометрических и механи
ческих характеристик слоя из вязкоупругого инерционного материала.
К л ю ч е в ы е с л о в а : колебания, сложная система, вязкоупругий инерционный
заполнитель, аналитическое решение.
Introduction. Compound systems coupled together by elastic constraints
play an important role in various engineering and building structures. Vibration
problems in engineering were considered by Timoshenko [1]. Vibration analysis
for laminated plates is presented in [2-3] and in many other works. The
stress-strain state of laminated orthotropic nonhomogeneous plates and shells was
considered in [4-5]. Vibrations of an elastically connected rectangular
double-plate compound system with moving loads are given in [6 ].
Vibration analysis of layered systems with vibration damping is a difficult
problem. In the above complex cases, especially where viscosity and discrete
elements occur, it is recommended to adopt a method of solving a dynamic
problem for such a system in complex functions [7-8]. For the first time the
property of orthogonality of complex modes of free vibrations for discrete
systems with damping was presented in [7], for discrete-continuous systems with
damping in [8 ], and for continuous systems with damping in [1 5 , 16].
The goal of this paper is to present the solution and dynamic analysis of free
and forced vibrations for a complex system with damping, which consists of a
plate, viscoelastic inerlayer, and stiff foundation.
Statement of the Problem. Let us consider a problem of free and forced
vibrations for a complex system with a viscoelastic inertial interlayer. The
external layers of the complex system are made from an elastic plate and stiff
© K. CABANSKA-PLACZKIEWICZ, 2001
ISSN 0556-171X. Проблемы прочности, 2001, N 6 103
K. Cabanska-Placzkiewicz
Fig. 1. Dynamic model for a complex system of an elastic plate with a viscoelastic inertial interlayer
resting on stiff foundation.
foundation and are shown in Fig. 1. The elastic plate is described by the
Kirchhoff-Love model and is simply supported at its ends. The viscoelastic
inertial interlayer has the characteristics of a homogeneous continuous
unidirectional Winkler’s foundation and was described by the Voigt-Kelvin
model [9-11].
In this paper, we consider two cases. In the first case, small-frequency
transverse vibrations of a complex system are exited by a stationary dynamic load
f 1 (x ,y , t ). In the second case, small-frequency transverse vibrations of a complex_ _ >ls
system are exited by a non-inertial moving load f 1 (x , y , t ), y = v t with the
speed v *.
The phenomenon of small-frequency transverse vibrations for an elastic plate
with a viscoelastic inertial interlayer resting on a stiff foundation is described by
the following non-homogeneous system of conjugate partial differential
equations:
̂2d W1
n r
d ) ^dw
1 + c — \E —
d t ) dz
= f 1 (X ,y , t), ( 1 )
z= 0
1+ c — \Eh
d 2 w
d t ) dt
= 0 , (2 )
where
*1 4 n 4 n 4A 2 d Wi d Wi d Wi
A2 w1 = ---- -- + 2— + ■ 1
dx dx 2 dy 2 dy
(3)
E 1 h3
D 1 = ---------- — , X 1 = P 1 h 1 , x = p h .
1 2 (1 - v ) (4)
Here w1 = w1 (x , y , t ), w(x, y , z , t ) are the transverse deflections of the plate and
the viscoelatic inertial interlayer, respectively, E 1 and E are Young’s moduli of
the material of the plate and the interlayer, c is the damping coefficient of the
104 ISSN 0556-171X. npoôëeMbi nponnocmu, 2001, № 6
Vibrations o f a Complex System
interlayer (retardation time), p j and p are the mass densities of the material of the
plate and the interlayer, respectively, h j and h are the thickness of the plate and
the interlayer, a and b are the dimensions of the plate, v lp is Poisson’s ratio, x and
y are the coordinate axes, and f j(x , y , t ) is the dynamic load acting on the
complex system.
Separation of Variables. Presenting the solution of the problem under
consideration in the form
щ ( x , y , t ) W1 (x ,y )
_ 4 x , y , z , t )_ W ( x , y , z )_
exp( iv t ) (5)
and substituting (5) in the system of differential equations (1), (2), by an
assumption that f 1(x ,y , t ) = 0 , we obtain a homogeneous system of conjugate
ordinary differential equations describing complex modes of free vibration of the
plate and the viscoelastic inertial interlayer:
2 2 d ~W
D 1A 2W 1 - u 1v 2 W1 - (1 + ic v ) E -----
dz
= 0 ,
z=0
(6 )
where
d 2W
d z 2
+ X2W = 0 ,
л 2 d 4 W, d 4 W, d 4 W,
A2 W1 = -----Л + 2 „ \ +■ 1
dx dx 2dy 2 dy
(7)
(8 )
A = , i2 = - 1 (9)
E h (1 + icv ) w
Here W 1( x , y ) and W (x , y , z ) are the complex modes of free vibration of the
plate and the interlayer, and v is the complex eigenfrequency of the complex
system with damping.
Solution of a Boundary Value Problem. The solution for the inertial
viscoelastic interlayer (7) is presented in the following form:
W ( x , y , z ) = C j(x , y )sin Xz + C 2( x , y )cos X z , (10)
where C i(x ,y ) and C 2 (x ,y ) are constant coefficients.
We have assumed the following geometric conditions:
W|z= 0 = Wi, W|z=A = 0. (11)
On substitution of (10) and (11) in (7), Eq. (10) can be rewritten in the following
form:
ISSN 0556-171X. Проблемы прочности, 2001, N2 6 105
K. Cabanska-Placzkiewicz
W ( x ,y , z ) = W 1( x ,y ) (cos Xz — ctg Xz — ctg Xh sin X z). (1 2 )
Substituting (12) in Eq. (6 ), we obtain
D 1A 2W 1 — [/m1v 2 + (1+ icv)E X ctg X h ]W1 = 0. (13)
The solution for W i is similar to that in [1]:
W 1 (x , y ) = X 1 (x ) ^ ( y ). (14)
Substituting (9) and (14) in (13), we can rewrite Eq. (13) in the following form:
X ( V Y 1 + 2 X 1 Y \ + X 1Y ( V — Z 4 X 1Y 1 = 0, (15)
where
Z 4 = - ^ [ м ^ 2 + (1 + ic v )EX ctg X h ]. (16)
^ 1
The quantities X 1(x ) and Y 1( y ) can be separated as follows [1]:
Y II y i vY1 2 Y1 4
— = — d , — = g . ( 17)
Representing the solution of the differential equation (15) in the forms of
X 1 = A exp( r1x ) and Y1 = B exp( r2 y ), (18)
we obtain a characteristic equation in the form of an algebraic equation
[r 4 — 2 d 2 r12 + (d 4 — Z 4)]( r22 + d 2) = 0 , d 4 = g 4 (19)
with the following roots:
r = ± i a v , a v = y j—0 2 ± Z 2 , r2 = ± i 3 , 3 = d . (20)
The solution of the differential equation (6 ) consists of a system of solutions:
2
W1 (x ,y ) = ^ [A * sina vx + A** cos a vx ][B* sin j3y + B ** cos j3y ], (2 1 )
U= 1
* * * *
where A v , A v , B , and B are constant coefficients.
106 ISSN 0556-171X. Проблемы прочности, 2001, N 6
Vibrations o f a Complex System
In order to solve the boundary value problem, the following boundary
conditions are used:
W 1 x= 0 = 0 Wll x=a = 0 W \ y= 0 = 0 W \ y=b = 0
d 2W 1
= 0 ,
d 2W 1
x= 0
= 0 ,
d 2W 1
dy*
= 0 ,
d 2W 1
y= 0
d y
= 0 . (2 2 )
y=b
Substituting the sequences a 1n = a 2n = a and f i n in Eqs. (12) and (21),
we obtain the following two complex sequences of free vibration modes for a
plate and viscoelastic inertial interlayer:
W 1 nln2 (x, y ) = sin a n x sin ß n2 y ,
W ^ ( x , y , z ) = sin a ni x sin ß „2 y[cos k h sin k z ],
(23)
(24)
where
a
n i n „ n 2 n
n1 = J ß n2 = ~r~> ni = 1,2, 3, ■■■, n 2 = 1, 2, 3, .... (25)
For r1n = ± i a n and r2n = ± i 8 , on substituting Eqs. (16) and (25) in
1 n 1 2 n 2
(2 0 ) and carrying out the transformations, we obtain the following equation of
frequency:
where
and
2 2 2
n 1 n , n 2 n = z 2
2 + , 2 = Z n1n 2 a b
k =k n 1 n 2
n 1 n 2
E h (1 + icV n1 n2)
(26)
z «1 n2 (27)
(28)
from which a sequence of complex eigenfrequencies is determined:
(29)
Solution of the Initial Value Problem. Free vibration of a complex system
with a viscoelastic inertial interlayer is represented in the form of a Fourier series
based on the complex eigenfunctions, i.e.,
2
ISSN 0556-171X. npoôneMbi npouuocmu, 2001, № 6 107
K. Cabanska-Placzkiewicz
Wi(X,y , t )
w( X, y , z , t )
2 2 W!«1«2 (X, y )
«1 = 1 «2 = 1
00 00
2 2 w «1«2 ( x ’ y ’z )
n = 1 « 2 = 1
$ n1 n2 exp( iv „ 1„ 2 t), (30)
where $ is the Fourier coefficient.n1n 2
From the system of equations (6 ) and (7) performing some algebraic
transformations, adding the equations together and then integrating them on both
sides within the limits from 0 to l, we obtain the property of orthogonality of
eigenfunctions for a complex system with an inertial viscoelastic interlayer:
a b
f f
0 0
i(V n + v m ) V1W1nW1m + f V W nW m d z d x d y +
L a b d W d W
+ c f f f —Wn---- Wm d x d y d z = N n ô ,
0 0 0
d z d z (31)
where
N nn2 = f f
0 0
d x d y +
+ c
a b h / d W \m l tt r f „
I
0 0 0
dz
d xd yd z . (32)
Here ô nm is Kronecker’s delta, and n = (n 1, n 2), m = (m 1,m 2 ).
The following initial conditions are the basis for solving the problem of free
vibrations:
W1 (x , y ,0 ) = W0 1 , W1 (x , y , z , o ) = w 0 1 , W0 (x , y ,0 ) = W0 . (33)
By applying conditions (33) in series (30) and taking into account the property of
orthogonality (31), the formula for the complex Fourier coefficient is obtained:
$ = 1
n 1 n 2 N f f
n1n 2 0 0
o
V1 ( i v n1n2 W1n1n2 w 0 1 + W1n1n2 w0 1 ) + f MWn1n2 w 0 d z
0
a bc h dWn1n2 d w {
+ c f f f
0 0 0
_______ v_0_
d z d z
d x d y d z .
d x d y +
(34)
0
0
2
108 ISSN 0556-171X. npoôneMbi npoHHoemu, 2001, № 6
Vibrations o f a Complex System
Free vibration of the complex system with a viscoelastic inertial interlayer is
obtained in the following form:
00 00
w 1 = 2 2 e"
« ! = 1 «2 = 1
Ini «2
n 1 n 2 C0 s ( ° n , t + p n , + x 1«1 «2 )’ (35)
0 0
w = 2 2 '
ln1 n2 W,n1n 2
$ n1n 2 C0s(° m n2 t + P n1n2 + X n n2 ) ’ (36)
where
= modWi, Wnn1n2 = mod W n
1n1n 2 ’
X 1n2 n2 = arg W1n1n2 ’ X n1n2 = arg $ n ^ .
(37)
Solution of the Forced Vibration Problem. The first case concerns
stationary forced vibrations. Small-frequency transverse vibrations of a complex
system with damping are excited by dynamic loading f 1(x , y , t ) at the points x 0
and y 0 varying in time t (Fig. 1):
f 1 (x ,y , t ) = p 1 ^ ( x - x o)d(y - y o )sinoo t , (38)
where P 1 is the force, d(x — x 0) and d( y — y 0) are Dirac delta functions, and
rn 0 is the real frequency of stationary forced vibrations.
The solution for complex modes of stationary forced vibrations for a
complex system with damping are written in the form
W1 ( x , y ) = W1 (x , y ) + W1 (x , y ). (39)
The general solution of the differential equation (1) consists of the system of
solutions
W1*(x ,y ) = 2 K * s i n + A * * cosa„x][B * sinßy + B ** cosß y ]. (4 0 )
U= 1
In the case v = 0 0 in Eq. (20)
a ^ a 2 = V - ß 2 + Z 0 , ß = ’ (41)
where
Z 0 = ± J D [ ̂ 1° 2 + (1+ ic 0 o )EX o Ctg A 0 h ]. (42)
w1 = 1 2̂ = 1
ISSN 0556-171X. npoôneMbi npounocmu, 2001, № 6 109
K. Cabanska-Placzkiewicz
Here
À o —
/Wa 0
Eh( 1 + ica) o )
(43)
from which the frequencies of real stationary forced vibrations are determined:
m 0 = {0 ...1 0 0 0 0 0 }.
The particular solution of the differential equation (1) consists of the system
of solutions
** .
W 1 (x,y ) =
P ̂ 1 a b
= 7 T 2 ----a / / sin a u (x — T 1 )^( T 1 — x 0 )sin 3( y — T 2 >̂ ( T 2 — y 0 )dT 1 dT 2 .
D1 u= 1 a ^ 3 0 0
(44)
On substituting (40) and (44) into (39), the complex modes of the stationary
forced vibrations for a complex system with a viscoelastic inertial interlayer can
be written in the following form:
W1( x , y ) = (A* sin a 1x + A2* sin a 2 x )B* sin 3 y +
+ — sin a 1( x — x 0)H ( x — x o) + — sin a 2( x — x 0)
a . a -
x
x ~fisin fi( y — y o)H ( y — y o). (45)
In order to solve the problem of forced vibrations, the boundary conditions
(22) have been applied. The constants occurring in Eq. (45) are described in the
following form:
A 1 — A 2 — 0 A 1
1 sin a 1( a — x o)
A 2 —
1 sin a 2(a — x o)
B — 0, B —
a 1 sin a 1a a 2 sin a 2 a
p 1 1 sin fi( b — y o) (46)
D 1 fi sin fib
Forced vibrations of a complex system with damping are stationary and have
the following form:
W1 ( x , y , t ) = W1 ( x , y )exp( im 0 1). (47)
Substituting (39) into (47) and performing trigonometric and algebraic
transformations, we obtain forced vibrations of a complex system with a
viscoelastic inertial interlayer:
110 ISSN 0556-171X. npoôneMbi npouuocmu, 2001, № 6
Vibrations o f a Complex System
w 1 = |W1 |sin( rn o t + # j), (48)
where \W ^ is the amplitude of an elastic plate with a viscoelastic inertial
interlayer and # 1 is argW ^x,y ).
The second case concerns non-stationary forced vibrations. Small-frequency
transverse vibrations of a complex system with damping are excited by a*
non-inertial moving load f 1( x ,y , t) [12, 13] with the speed v (Fig. 1).
f i (x ,y ,t) = P jd (y — y ). (49)
_ _ *
Here P 1 is the force, o ( y — y ) is the Dirac delta function, y = v t , x = 0.5a, and
*
v = const.
In order to solve the differential equations (1), (2). the function of the load
(49) is expanded by the operational method [14]:
00 00
f 1 (x , y, t) = (^1W1«1«2 + W ni « 2 ) f «1 « 2 , (50)
n1 =1 2̂ = 1
where W \nini and have been described by Eqs. (23) and (24).
The function of the displacement of a complex system with damping is
presented in the form of a Fourier series as
Wi
w = 2 2
Wiini fto
W nn 1 n 2
T -L (51)
Substituting (50) and (51) into the differential equations (1), (2), we obtain
the following equation of motion:
T n1 n2 ni n^ ni n2 j ni n2 ^ (52)
where T n n is the coefficient of the distribution of the dynamic loading function
in the Fourier series.
Applying the property of orthogonality of eigenfunction (31), we derive
formulas for the coefficients of load distribution, namely:
f = ----- i—
i n 2 D i N n n
-IV,nin 2
2 0 0
u W nin2 ( x , y ) + / Wnin2 (x , y , z ) d z Piô( y — y )d xd y .
(53)
ni=i n2=i
0
ISSN 0556-171X. npoôneMbi npounocmu, 2001, № 6 111
K. Cabanska-Placzkiewicz
The solution of the differential equation (52) has the form [14]
1 t
Tnin2 = ------- f [exp( iv n in i)( t _ r ) - 1 l/n in2 (T)dx.
n1n 2 0
On substituting (53) and (54) into (51), Eq. (51) can be rewritten in the
following form:
00 00
^ 1 Klnin2 Tnin2 c° s(^ ini n2 nin2 ),
n1= 1 n2 = 1
(55)
where
^ ^ Tnin2 COs(̂ ni n2 nin2 ), (56)
^ 1nin2 ar§ W1nin2 ’ ^ nin2 ar§ Wnin2 ’ ^ nin2 arg Tnin2 ■ (57)
Calculations. The calculations for a complex system with a viscoelastic
inertial interlayer are presented. The external layers of the complex system are
made of an elastic plate and stiff foundation. The thickness and mechanical
characteristics of the plate, foundation, and the interlayer do not change.in _2
Numerical results are presented for the same parameters: E i = 10 N- m ,
p 1 = 2-103 N -s 2 -m_4, v 1p = 0.3, h 1 = 0.2 m, h = 0.5 m, P = 2-104 N,
* o
a = 10 m, b = 1000 m, c = 2.5 s, and v = 120 m-s .
The amplitude-frequency diagrams for a complex system with damping for
real stationary frequencies in the range 0 < r n 0 < 1 0 0 0 0 0 are presented in
Figs. 2-3.
In the first case, small-frequency transverse vibrations of the complex system
are excited by the force f 1(x , y , t ) = P 1̂ ( x _ x 0 )d( y _ y 0 )sin( rn 0 1) acting at the
point x 0 = 0.5a, y 0 = 0.5b and varying in time t. The amplitude-frequency
diagrams of the complex system are presented in Fig. 2. Changes in the amplitude
W1 I for the complex system at the point x = 0.55a, y = 0.55b (variant “a”) and at
the point x = 0.7a, y = 0.8b (variant “b”) are also given there. In the case of
variant “b” considered, the amplitude \W2 of the complex system is 50% smaller
than the amplitude \W^ of the complex system for variant “a”.
The changes in the amplitude of the viscoelastic inertial interlayer at the
point x = 0.7a, y = 0.8b are shown in Fig. 3 for two thicknesses of the interlayer.
In variant “a” for the thickness z = 0.2h, the amplitude \W^ of the interlayer is
36% smaller than the amplitude of the plate for z = 0. In variant “b” for the
thickness z = 0.5h, the amplitude \W^ of the interlayer is 65% smaller than the
amplitude of the plate for z = 0. After analyzing the results presented in Figs. 2-3,
we state that a viscoelastic inertial interlayer can be a vibration damper for a plate
loaded by the force f 1(x , y , t ) acting at the point x 0 , y 0 and varying in time t.
ni= 1 n2 = 1
112 ISSN 0556-171X. npodneMbi npouuocmu, 2001, № 6
Vibrations o f a Complex System
a
b
Fig. 2. The amplitude-frequency diagrams for an elastic plate with a viscoelastic inertial interlayer at
the points: (a) x — 0.55a, y — 0.55b; (b) x — 0.7a, y — 0.8b.
a
Fig. 3. The amplitude-frequency diagrams for an elastic plate with a viscoelastic inertial interlayer at
the points: (a) x — 0.7a, y — 0.8b, z — 0.2k; (b) x — 0.7a, y — 0.8b, z — 0.5h.
ISSN 0556-171X. npoôëeMbi nponnocmu, 2001, № 6 113
K. Cabanska-Placzkiewicz
0 . 0 0 1
0 . 0 0 0 5
- 0 . 0 0 0 5
- 0 . 001
- 0 . 0 0 1 5
- 0 . 0 0 2
Fig. 4. The trajectory of dynamic displacement of a moving point of an elastic plate with a
viscoelastic inertial interlayer for the concentrated non-inertial moving force for x = 0.5a for time
t = 0.03 s.
In the second case, small-frequency transverse vibrations of a complex
system are exited by the moving concentrated force f 1 (x , y , t ) = P1d(y — y ) with
*the speed v . The effect of the non-inertial moving force in the complex system
with an inertial viscoelastic interlayer is presented in diagram 4. The diagram
shows the real part of the trajectory of dynamic displacement of a moving point of*
the complex system w 1( x ,y , t ) for x = 0.5a, y = v t for time t = 0.03 s.
A model for a complex system consisting of an elastic plate with a
viscoelastic inertial interlayer resting on a stiff foundation can play the role of a
runway in an airport. In the first case, we might have been considering free and
forced vibrations of an aircraft, which touches the runway of the airport. In the
second case, we might have been considering the trajectory of dynamic
displacement of an aircraft moving with the speed v * after touching the runway
of the airport. A complex system with a viscoelastic inertial interlayer of big
thickness can be used for damping vibrations in practical problems.
Complex modes of free vibrations and the property of orthogonality of those
modes presented in this paper are the basis for solving the problems of free and
forced vibrations of a complex system with a viscoelastic inertial interlayer.
Р е з ю м е
Запропоновано аналітичний метод розв’язку задач щодо згасання вільних та
вимушених коливань складних систем, несучі шари яких виконано з одно
рідного пружного, а середній - з в’язкопружного інерційного матеріалу.
Малі поперечні коливання складних систем зумовлені розподіленим і рухо
мим навантаженням. Виконано динамічний аналіз шаруватих конструкцій у
широкому діапазоні зміни геометричних і механічних характеристик шару з
в’язкопружного інерційного матеріалу.
1. S. P. Timoshenko, V ibra tion P ro b le m s in E n g in e e r in g , D. Van Nostrand
Company, New York, Toronto, London (1955).
2. R. A. Di Taranto and J. R. McGraw, “Vibratory bending of damped
laminated plates,” Trans. A S M E , J. E ng . In d u s try , 91, 1081-1090 (1969).
114 ISSN 0556-171X. П роблеми прочности, 2001, № 6
Vibrations o f a Complex System
3. W. Kurnik and A. Tylikowski, M e c h a n ic s o f L a m in a te d E le m e n ts , Publ.
Warsaw Univ. of Techn., Warsaw (1997).
4. V. Z. Vlasov and N. N. Leont’ev, B ea m s, P la te s , a n d S h e lls on an E la s tic
F o u n d a tio n [in Russian], Glav. Izdat. Fiz.-Mat. Literatury (1960).
5. N. D. Pankratova, A. O. Rasskazov, A. Bondar’, and A. G. Bondarskii, “On
the calculation of the stress state of shear-compliant multi-layer shells and
plates,” P rik l. M e k h ., 23, No. 7, 55-61 (1987).
6 . W. Szczeæniak, “Vibration of elastic sandwich and elastically connected
double-plate systems under moving loads,” B u ild in g E n g in e e r in g (Publ. of
the Warsaw Univ. of Techn.), No. 132, 153-172 (1998).
7. F. Tse, I. Morse, and R. Hinkle, M e c h a n ic a l V ibra tions: T h eo ry a n d
A p p lic a tio n s , Allyn & Bacon, Boston (1978).
8 . J. Niziol and J. Snamina, “Free vibration of the discrete-continuous system
with damping,” J. T heor. A pp l. M e c h ., 28, No. 1-2, 149-160 (1990).
9. W. Nowacki, T h e B u ild in g D y n a m ic s , Arkady, Warsaw (1972).
10. Z. Osinski, D a m p in g o f th e M e c h a n ic a l V ib ra tion , PWN, Warsaw (1979).
11. D. Nashif, D. Jones, and J. Henderson, V ib ra tio n D a m p in g [Russian
translation], Mir, Moscow (1988).
12. M. Renaudot, “Etude de l’influence des charges en mouvement sur la
resistance, des ponts metallique droites,” A n n a le s d es P o n ts e t C h a u sses ,
No. 1, 145-204 (1861).
13. N. Z. Yakushev, “Some problems of beam dynamics under the action of
moving loads,” in: In v e s tig a tio n in to th e T h eo ry o f P la te s a n d S h e lls [in
Russian], Izd. Kazan. Univ., Kazan (1974), pp. 199-220.
14. J. Cabanski, “Vibration of Timoshenko-Kelvin beam,” T h e R e se a rc h
C o pybook. M e c h a n ic s (Publ. of the Univ. of Techn. and Agriculture,
Bydgoszcz), 221, No. 44, 27-37 (1999).
15. K. Cabanska-Placzkiewicz, “Free vibration of the system of two Timoshenko
beams coupled by a viscoelastic interlayer,” J. E ng . T rans., 47, No. 1, 21-37
(1999).
16. K. Cabanska-Placzkiewicz, “Free vibration of three-layer constructional
system with damping,” J. M ech . E n g ., 1, No. 31, 36-39 (2000).
Received 30. 01. 2001
ISSN 0556-171X. npoôneMbi npouuocmu, 2001, № 6 115
|