Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer
In this paper the analytical method has been used for solving a problem of free vibration with clamping of the axially loaded sandwich beam. The sandwich beam consists of two external layers connected by viscoelastic two-directional Winkler’s interlayer. The upper external layer is described a...
Збережено в:
| Дата: | 2000 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2000
|
| Назва видання: | Проблемы прочности |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/46371 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 6. — С. 93-105. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-46371 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-463712013-06-29T19:51:03Z Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer Cabanska-Placzkiewicz, K. Научно-технический раздел In this paper the analytical method has been used for solving a problem of free vibration with clamping of the axially loaded sandwich beam. The sandwich beam consists of two external layers connected by viscoelastic two-directional Winkler’s interlayer. The upper external layer is described as the Bernoulli-Euler’s model, which is loaded by the constant axial force. The lower external layer is modeling as the Timoshenko’s model. The phenomenon of free vibration has been described using a homogenous system of conjugate partial differential equations. After separation of variables in the system of differential equations the boundary problem has been solved and three complex equations for definition of frequency and modes of free vibration have been obtained. The free vibration problem for arbitrarily assumed initial conditions and various axial forces has been considered. Предлагается аналитический метод решения задач о свободных колебаниях с затуханием слоистых балок, состоящих из двух внешних слоев, соединенных внутренним вязкоупругим слоем, который рассматривается как двунаправленное винклеровское основание. Верхний внешний слой, нагруженный осевой постоянной силой, описывается на основе модели Бернулли-Эйлера. Нижний внешний слой моделируется с помощью модели Тимошенко. Свободные колебания описываются однородной системой связанных дифференциальных уравнений в частных производных. После разделения переменных в исходной системе дифференциальных уравнений решается краевая задача. В результате получено три комплексных уравнения для определения частот и мод свободных колебаний. Задача о свободных колебаниях рассмотрена для произвольных начальных условий и различных осевых сил. Запропоновано аналітичний метод розв’язку задач про вільні коливання зі згасанням шаруватих балок, що складаються з двох зовнішніх шарів, з ’єднаних внутрішнім в ’язкопружним шаром. Останній розглядається в якості двонапрямленої вінклерівської основи. Верхній зовнішній шар, що навантажується осьовою постійною силою, описується на основі моделі Бернул- лі-Ейлера. Нижній зовнішній шар моделюється за моделлю Тимошенка. Вільні коливання описуються однорідною системою зв’язаних диференціальних рівнянь в частинних похідних. Після розділення змінних у вихідній системі диференціальних рівнянь розв’язується крайова задача. У результаті отримано три комплексних рівняння для визначення частот і мод вільних коливань. Задача про вільні коливання розглянута для довільних початкових умов і різних осьових сил. 2000 Article Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 6. — С. 93-105. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/46371 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. Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer Проблемы прочности |
| description |
In this paper the analytical method has been
used for solving a problem of free vibration
with clamping of the axially loaded sandwich
beam. The sandwich beam consists of two
external layers connected by viscoelastic
two-directional Winkler’s interlayer. The upper
external layer is described as the
Bernoulli-Euler’s model, which is loaded by
the constant axial force. The lower external
layer is modeling as the Timoshenko’s model.
The phenomenon of free vibration has been
described using a homogenous system of
conjugate partial differential equations. After
separation of variables in the system of
differential equations the boundary problem has
been solved and three complex equations for
definition of frequency and modes of free
vibration have been obtained. The free
vibration problem for arbitrarily assumed initial
conditions and various axial forces has been
considered. |
| format |
Article |
| author |
Cabanska-Placzkiewicz, K. |
| author_facet |
Cabanska-Placzkiewicz, K. |
| author_sort |
Cabanska-Placzkiewicz, K. |
| title |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer |
| title_short |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer |
| title_full |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer |
| title_fullStr |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer |
| title_full_unstemmed |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer |
| title_sort |
free vibration of the axially loaded system of two beams connected by two-directional viscoelastic interlayer |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2000 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/46371 |
| citation_txt |
Free Vibration of the Axially Loaded System of Two Beams Connected by Two-Directional Viscoelastic Interlayer / K. Cabanska-Placzkiewicz // Проблемы прочности. — 2000. — № 6. — С. 93-105. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT cabanskaplaczkiewiczk freevibrationoftheaxiallyloadedsystemoftwobeamsconnectedbytwodirectionalviscoelasticinterlayer |
| first_indexed |
2025-07-04T05:36:32Z |
| last_indexed |
2025-07-04T05:36:32Z |
| _version_ |
1836693473595490304 |
| fulltext |
UDC 539.4
Free Vibration of the Axially Loaded System of Two Beams Connected
by a Two-Directional Viscoelastic Interlayer
K. Cabanska-Placzkiewicz
Pedagogical University, Bydgoszcz, Poland
УДК 539.4
Свободные колебания системы из двух слоистых балок,
соединенных внутренним вязкоупругим слоем и нагруженных
осевой силой
К. Ц абанска-П лаш кевич
Педагогический университет, Быдгощ, Польша
Предлагается аналитический метод решения задач о свободных колебаниях с затуханием
слоистых балок, состоящих из двух внешних слоев, соединенных внутренним вязкоупругим
слоем, который рассматривается как двунаправленное винклеровское основание. Верхний
внешний слой, нагруженный осевой постоянной силой, описывается на основе модели
Бернулли-Эйлера. Нижний внешний слой моделируется с помощью модели Тимошенко. Сво
бодные колебания описываются однородной системой связанных дифференциальных урав
нений в частных производных. После разделения переменных в исходной системе диф
ференциальных уравнений решается краевая задача. В результате получено три комплекс
ных уравнения для определения частот и мод свободных колебаний. Задача о свободных
колебаниях рассмотрена для произвольных начальных условий и различных осевых сил.
Introduction. In recent years, the Bernoulli-Euler and Timoshenko models
have been applied to the solution of various mechanical and building vibration
problems. The Bernoulli-Euler model has been used for the solution of the
problem of vibration of sandwich beams [1-3]. The problem of free vibration of
two axially loaded Bernoulli-Euler beams transversally coupled with discrete
springs without damping was studied in [1]. The problem of a complex
continuous dynamical system was considered in [2 ] with the use of the classical
method and the complete theory of non-damped vibrations. Vibrations of two
Bernoulli-Euler elastic beams connected by an elastic interlayer with moving
loads was solved in [3].
For the first time, the influence of transverse forces and rotational inertia
with the shearing coefficient in a beam was considered in [4]. Natural frequencies
for continuous Timoshenko models were studied in [5], and, for discrete-
continuous Timoshenko models, this problem was solved in [6].
The property of orthogonality of the complex modes of free vibration for
continuous systems with damping was demonstrated in [7-11]; for discrete and
discrete-continuous systems with damping, it was demonstrated in [12, 13].
The general method for the solution of problems of free vibration for
complex continuous one- and two-dimensional systems with damping for various
© K. CABANSKA-PLACZKIEWICZ, 2000
ISSN 0556-171X. Проблемы прочности, 2000, N 6 93
K. Cabanska-Placzkiewicz
boundary conditions and different initial conditions was presented in [1]. The
application of the Bernoulli-Euler and Timoshenko models to the solution of
free-vibration problems for different sandwich beams with damping was
considered in [2-4].
The purpose of this paper is the solution of the problem of free vibration of
an axially loaded sandwich beam with damping for various axial forces. The
calculations of the dynamic displacements for a two-directional interlayer are
compared with similar results for a one-directional interlayer.
S tatem ent of the Problem. The physical model of the structural system is
an axially loaded sandwich beam with damping, which consists of two
homogenous elastic parallel beams of equal length coupled together by a soft
viscoelastic interlayer (Fig. 1). The upper external layer is simulated by the
Bernoulli-Euler model and is loaded by a constant axial force P . The lower
external layer is simulated by the Timoshenko model. The beams are supported at
their ends. The viscoelastic interlayer has the characteristics of a homogenous
continuous two-directional Winkler base [5] and is described by the Voigt-Kelvin
model [14-18].
Fig. 1. Dynamical model of an axially loaded system of two beams connected by a two-parameter
viscoelastic interlayer.
The mathematical model of the problem is represented by the following
system of conjugate partial differential equations describing small transverse
vibrations of the physical system:
Ei, I i , u i
x
(1)
94 ISSN 0556-171X. npodneMbi npoHHOcmu, 2000, № 6
Free Vibration o f the Axially Loaded System
where
fl1 = p 1F 1, 2 = p 2F 2 , “2 = p 2 12 , kp = 2(1
W = Wj(x, t ) and w2 = w2(x , t ) are transverse deflections of beams I and II,
= ^ i ( x , t ) and ^ 2 = ^ 2(x , t ) are the angles of rotation of the cross sections
of beams I and II, E j and E 2 are the elastic moduli of the material for beams I
and II, 11 and 12 are the moments of inertia of the cross sections of beams I
and II, P is the axial force, F j and F 2 are the areas of the cross sections of
beams I and II, G 2 is the Kirchhoff modulus of the material of beam II, p j and
p 2 are the mass densities of the material of beams I and II, k ' is the shearing
coefficient, k is the transverse coefficient of elasticity of the interlayer, k p is the
longitudinal coefficient of elasticity of the interlayer, c is the coefficient of
viscosity of the interlayer, h and ^2 are the heights of beams I and II, ^0 is
the height of the interlayer, and l is the length of beams I and II. The bending
moment and transverse force in beam II were determined by Timoshenko [9] in
the form
for w 1, w 2 , and \p 2 in the system of differential equations (1), we represent the
homogenous system of conjugate ordinary differential equations describing the
complex modes of vibration of the beams in the following form:
3W 2
, M 2 = — R 2 ~ 5 Q 2 = k G 2 F 2 Y 2 5 (2)
d x
dw 1 dw 2
where ----- = ^ 1, ------ = ^ 2 + Y2 , and y 2 = y 2(x , t) is the angle of shearing in
d x d x
beam II.
Solution of the Boundary-Value Problem. Substituting
w 11 T W1( x )
w2 = W 2(x ) exp( iv t)
V 2 J L ^ 2(x).
(3)
\ d x 2 d x )
(4)
d W 2
d x
- W 2 + S 2 W2 v 2 - k
ISSN 0556-171X. npoôneMbi npouuocmu, 2000, № 6 95
K. Cabanska-Placzkiewicz
where
k ' = —1- k k '' = k k '' = k ' = —1 —2 k Г31'k pi = 4 k p ’ k p - = 4 k p ’ k pi = k p - = 4 k p [31’
W x = W1( x ) and W2 = W 2 (x ) are the complex transverse vibration modes of
beams I and II, ^ = ^ ( x ) is the complex rotation mode of vibration of beam
II, v is the complex frequency of vibration of beams I and II, and t is time.
Seeking a particular solution of the system of differential equations (4) in the
form
W 1 ' A '
W 2 = B
У 2 - 0
exp( r x ), (5)
we obtain the following homogeneous system of linear algebraic equations:
\ A [ R lr 4 — (P + k'p l ) r 2 — n xv 2 + k + icv ]— B ( k + ic v ) — 0 k'p2 r = 0,
A ( k + ic v ) + B( N r 2 + jU 2 v 2 — k — ic v ) — 0 N r = 0, (6)
A k ,p l r — B N r — 0 ( R2 r 2 — N + S 2 v 2 — kp2 ) = 0.
Expanding the determinant of the characteristic matrix of the system of
equations (6) and equating it to zero, namely,
R r ■( P + k ”p \) r r
( k + ic v )
kp ir
+ n 1 —( k + ic v )
( N r 2 — n - )
N r
—k 'p 2 r
—N r
R 2 r 2 = N + S 2 v 2 k k p2
= 0, (7)
we obtain the characteristic equation in the form of an algebraic equation, namely,
r 8 + a n r 6 + a — r 4 + a 33 r 2 + a 44 = 0, (8)
with the roots r j = (— 1) J—1 ik v , j = (2u — 1), 2u, v = 1, 2, 3, 4, where n1 = k +
2 2 + icv — ju1v , n 2 = k + icv — ц 2v , and a 11, a 22, a 33 , and a 44 are constant
coefficients.
After application of the Euler formulas, the solution of the system of
differential equations (4) consists of the fundamental system of solutions
4
W1( x ) = ^ АЦ, sinX vx + A Jj* cosX vx ,
U=1
96 ISSN 0556-171X. Проблемы прочности, 2000, N2 6
Free Vibration o f the Axially Loaded System
4
W 2( x ) = ^ B U sinA^,x + B U* cosX t
U=1
4
W2(x ) = ^ © U cosÀ vx + ©** sinAUx , (9)
U=1
dW,
w here W-(x) = ------ , A'U, A'U, B U, B U, ©V, and ©V are constants, and
dx
Av = a v + i/3v is a parameter that describes the roots of the characteristic
equation (8).
In agreement with (6), the constants in (9) satisfy the following relations:
* _ B U ** _ B U 7 * _ ©U 7 ** _ ©U
a U = ,* 5 a U = 5 bU = ,* 5 bU = _
A.U A U A U A
(10)
U
where
bU = bU=
kp2 r
= a = k Pi N r 2 - (k + ic v )R R 2
( N r ) 2 + NN 2 R R 2 ’
kP, N r 2 — ( k + icv )RR 2
RR, — ( k + icv ) ^ --------------- 2
( N r ) 2 + N N 2 R R 2
(11)
, b U =
R R , = R , r 4 — (P + k'p1) r 2 + n ,, RR2 = R 2 r 2 — N + v 2S 2 = k"
NN2 = N r 2 - n 2.
Substituting (10) in (9), we obtain the general solution of the system of
differential equations (4) in the following form:
W ,(x) = ^ AU sinAUx + A** cos AUx ,
U=1
4
W,(x ) = ^ A UAU cosAUx — AUAU* sin AUx ,
U=1
4
W 2(x ) = ^ a U(AU sinAUx + AU* cos AUx ),
U=1
4
^ 2(x ) = ^ b U(AU cos AUx — AU* sinAUx ).
(12)
U=1
ISSN 0556-171X. npoôneMbi npouuocmu, 2000, № 6 97
K. Cabanska-Placzkiewicz
In order to solve the boundary-value problem, we use the following
boundary conditions:
W,(0) = 0 , W ,( l) = 0 ,
w 2(0) = 0 , W2 (l) = 0 ,
d—, d—,
— ^(0) = 0, — k l) = 0 ,
dx dx
(13)
d—2
dx
-(0) = 0 ,
d—2
dx -(l ) = 0.
Substituting (12) in (13), we obtain the homogenous system of linear
algebraic equations. The matrix of the system obtained has the following form:
YX = 0, (14)
where X = [A , , A2,A3 , A 4 , A ™, A 2 , A 3 , A 4* ]T is the vector of unknowns of the
system of equations and
Y = [Y,*j ]8*8 (15)
is the characteristic matrix of the system of equations (14).
The first four equations in (14) are represented in the form
1 1 1 1
a, a 2 a 3 a 4
—a2
22A— — A —a24
—A1b 1 A 2 b 2 A 3 b 3 4b4A—
r A p l
A **
A2
A **
A3 |
L A4J
|
|= 0. (16)
It follows from the system of equations (16) that A“ = A 2* = A3* = A4* = 0.
The other four equations in (14) give the following system of equations:
sin A, l
a , sin A, l
-A 2 sin A, l
-A ,b , sin A, l
sin A 2 l
a 2 sin A 2 l
-A 2 sin A2 l
sin A 3 1
a 3 sin A 3 l
—A3 sin3 l
—A 2 b2 sin A 2 l —A 3 b3 sin A 3 1 —A 4 b4 sin A 4 1
sin A 4 1
a 4 sin A 4 1
—A4 sin A41
14 b4 .
r A* l
A 2
A3
|= 0.
(17)
L a 4 J
A condition for the solvability of the system of equations (17) is vanishing of
the characteristic determinant, i.e.,
linsi sin A 2 1 l3Ainsi l4Ainsi
linsi a 2 sin A 2 1 l3Ainsi3a l4Ainsi4a
—A", sin A, l l2Ainsi22A— l3Ainsi23A— l4Ainsi2
4A—
—A,b, sin A, l —A 2b2 sin A 2 1 —A 3 b3 sin A 3 1 —A 4 b 4 sin A 4 l
= °- (18)
98 ISSN 0556-171X. npodneMbi npouuocmu, 2000, № 6
Free Vibration o f the Axially Loaded System
Expanding the determinant (18), we obtain the following characteristic
equation:
sin X l l sin X 2l sin X 31 sin X 41 = 0 , (19)
where Xi = X2 = X3 = X4 = X.
The characteristic equation (19) can be rewritten in the form
sin Xl = 0 , (20)
where
X = a + ifi (21 )
are complex numbers in the general case.
Substituting (21) in (20), we get the equation
sin a l ch f i l + i cos a l sh f i l = 0 , (22)
which has the following roots:
s n
a s = — , s = 0, s = 1 ,2 ,3 ,.... (23)
In view of (23), relation (21) yields the following identity:
X s = a s = y ■ (24)
Substituting r = iX s in equation (8) and carrying out the corresponding
transformations, we obtain the equation for frequency
V 6 + b n v 5 + b22v 4 + b33 V 4 + b 55v + b 66 = ^ (25)
from which we determine the sequence of complex natural frequencies
v n = in n ± m n , (26)
where n = (3s — 2), (3s — 1), 3s, and b 11, b 22, b 33 , b 44, b 55 , and b 66 are constant
coef^cients.
Substituting equation (26) in equations (11), we obtain the following
formulas for the coefficients of amplitudes:
k'p ! NX2s + ( к + icv n ) RR 2
(NXs ) 2 - N N 2RR 2
n ^ - Л ! NX2s + ( к + icv n )RR 2
RR! - (k + iCV n ) - ybn = 1
n ik p2X s
(27)
( NX s У - N N 2 RR 2
ISSN 0556-171X. Проблемы прочности, 2000, N 6 99
K. Cabanska-Placzkiewicz
RRl = R A4, + (P + h'pX) A2S + nx, RR 2 = - R 2 A2 - N + v 2nE 2 - k'p2 ,
N N 2 = - N A2 - n 2.
Substituting the sequences A and a n , bn in (12), we get the following four
sequences of modes for the free vibration of two beams:
W1n (x ) = sin A sx ,
^ ln (x) = As COS Asx ,
W2n (x ) = a n sin Asx ,
^2n (x ) = bn COS AsX.
(28)
Solution of the Initial-Value Problem. The complex equation of motion
T = $exp( iv t), (29)
in the case v = v n can be rewritten in the form
Tn = $ n exP( iv ntX (30)
where $ n is the Fourier coefficient.
Free vibration of beams is represented in the form of a Fourier series based
on the complex eigenfunctions [17], i.e.,
> n '
w2n
9 2n
r ” 1
2 W1n (X )
n=1
00
2 W2n ( x )
n=1
0
2 W2n (x )
L n=i
$ n exP( iv nt). (31)
J
From the system of equations (4), after performing algebraic transformations,
adding the equations together, and then integrating them on the sides from 0 to l ,
we establish the property of orthogonality of the eigenfunctions for two beams
coupled together by a two-directional viscoelastic interlayer:
l
f [i(v n + v m ) (M 1W1nW1m + M 2W2nW2m + “ 2^ 2n^ 2m ) +
+ C(Wm - W2n )(W1m - W2m )]dx = N nd t (32)
where d is the Kronecker delta and
100 ISSN 0556-171X. npodneMbi npouuocmu, 2000, № 6
0
Free Vibration o f the Axially Loaded System
N n = f [2iv n ( M W n +Ц 2W2n + z 2 ̂ n + С (Wln - W2 n )2] dx . (33)
The following initial conditions form the basis for the solution of the
problem of free vibrations:
Wi(x ,0) = w 01, w 2 ( x ,0) = w 02, p 2 ( x ,0) = p 02 ,
w°(x ,0) = w01’ w2 (x ,0) = w 02 , p 2 (x ,0) = p 02 ■ ( )
Applying conditions (34) to series (31) and taking into account the property of
orthogonality (32), we obtain the following formula for the Fourier coefficients:
ф = —-
n NN n
where
Un
(35)
U n = f {M 1( iv nW1nw01 + W1nw01) + M 2 ( iv nW2nw 02 + W2nw02) +
+ S 2( iv n XV2nV 02 + XV2nV 02) + c [(W1n - W2n ) ( w 01 - w0 2 )]}dx. (36)
Substituting (28), (30), and (35) in (31) and performing trigonometric and
algebraic transformations, we determine the final free vibration of beams [2-4]:
00
w 1 = ^ e-tlnt\W1n\ф n| [cos( m nt + X in + P n X
n=1
0
w 2 = ^ e~ Vnt\W2n\|Ф n | [c°S( m nt + x 2n + P n X (37)
n=1
0
V 2 = ^ e~Vnt | ^ 2n| \Ф n\[cos( m nt + ° 2n + Pn X
n=1
where
№ы\ Ч X 2 + Y12n , \w 2n\ = V X 2n + Y'ln , 1^2n\ = ^ A 22n + Q L ,
x 1n = arg w1n , x 2n = arg W2n , ° 2n = arg ̂ 2 n , (38)
Ф n| = J c I + d I , P n = arg Ф n ,
and
X 1n = ReW1n , Y1n = Im W2 n , X 1n = ReW2n, Y2 n = ImW2 m ,
Л 2n = Re ̂ 2n , Q 2n = Im ̂ 2n , C n = ReФ n , D n = Im Ф n . (39)
ISSN 0556-171X. Проблемы прочности, 2000, N 6 101
0
0
K. Cabanska-Placzkiewicz
W1 , m
W2 , m
Fig. 2. Distribution of the dynamic displacements w(x, t) (a) and w2(x, t) (b) of beams I and II for
the axial force P = —4 -105 N and x =0.5/.
Numerical Results. On the basis of the method developed, the
investigation of sandwich system was carried out. Calculations were performed
for the following data: E 1 = E 2 = E = 2.1-1011 N -m - 2 , E 0 = 108 N -m - 2 , k =
= ( E 0 b 0) / h0 , k p = k / [2(1+ v 0)], l = 6 m, p 1 = p 2 = 7.8-103 N -s2 /m 4 ,
k ' = 0.84, P = { - 4-105, 4-105} N, G 2 = E 2 /[2 (1+ v 0)], c = 2-102 N -s-m - 2 ,
b1 = b2 = b0 = 0.07 m, h1 = 0.1 m, h2 = 0.2 m, h0 = 0.2m, F1 = b1h1, F 2 = b2h2 ,
11 = (b1h3 ) / 12, 12 = (b2h23 ) / 12, v 0 = 0.2.
In order to find the Fourier coefficient ^ n (35), the following initial
conditions were assumed:
102 ISSN 0556-171X. npoôneMbi npouHocmu, 2000, № 6
Free Vibration o f the Axially Loaded System
Wi, m
W01 = 0.01s i n ^ j , w°1 = 0,
w°2 = —0.001sin j , w02 = 0 ,
( n x \
\p02 = —0.001cos | , \p02 = 0.
(40)
Fig. 3. Distribution of the dynamic displacements w1(x, t) (a) and w2(x, t) (b) of beams I and II for
the axial force P = 4-105 N and x =0.5l.
The effect of various axial forces in the sandwich beams with damping is
shown in Figs. 2-3. The distributions of displacements for external layers
described according to the Bernoulli-Euler model are displayed in Figs. 2a and
ISSN 0556-171X. npoôëeMbi npounocmu, 2000, N2 6 103
K. Cabanska-Placzkiewicz
3 a. The distributions of displacements for lower external layers represented using
the Timoshenko model are displayed in Figs. 2b and 3b. Figures 2-3 show the
motion of the sandwich beams with time t for x = 0.5l. Calculations of the
dynamic displacements for a two-directional interlayer k Ф 0 and k p Ф 0 are
compared with a one-directional interlayer k Ф 0 and k p = 0. It follows from the
comparison of the results for one- and two-directional interlayers that, as time t
increases, the dynamic displacements rapidly decay in the sandwich beams for
which the interlayer corresponds to a two-directional Winkler base k p Ф 0. This
difference becomes more substantial in the case o f loading o f sandwich
constructions by compressing forces.
The dynamic displacements w1 (x , t) and w 2 (x , t ) of beams I and II for
various loadings by axial forces rapidly decay with time t in the sandwich beams
for which the interlayer corresponds to a two-directional Winkler base k p Ф 0.
Р е з ю м е
Запропоновано аналітичний метод розв’язку задач про вільні коливання зі
згасанням шаруватих балок, що складаються з двох зовнішніх шарів, з ’єд
наних внутрішнім в ’язкопружним шаром. Останній розглядається в якості
двонапрямленої вінклерівської основи. Верхній зовнішній шар, що наван
тажується осьовою постійною силою, описується на основі моделі Бернул-
лі-Ейлера. Нижній зовнішній шар моделюється за моделлю Тимошенка.
Вільні коливання описуються однорідною системою зв’язаних диференці
альних рівнянь в частинних похідних. Після розділення змінних у вихідній
системі диференціальних рівнянь розв’язується крайова задача. У результаті
отримано три комплексних рівняння для визначення частот і мод вільних
коливань. Задача про вільні коливання розглянута для довільних початкових
умов і різних осьових сил.
1. Kukla S. Free vibration of the system of two beams connected by many
translational springs // J. Sound Vibration. - 1994. - 172, N 1. - P. 130 -
135.
2. Oniszczuk Z. Vibration analysis of the compound continuous systems with
elastic constraints. -P u b l. of the RzeszowUniv. of Tech., Rzeszow, 1997.
3. Szczewniak W. Vibration of elastic sandwich and elastically connected
double-beam system under moving loads // Building Engineering. - 1998. -
132. - P. 111 - 151 (Publ. of the Warsaw Univ. of Tech.).
4. Timoshenko S. P., Young D. H., Weaver G. Vibration Problems in
Engineering. - New York: Wiley, 1974.
5. Wang T. M. Natural frequencies of continuous Timoshenko beams // J.
Sound Vibration. - 1970. - 13. - P. 406 - 414.
6. Pielorz A. Discrete-continuous models in the analysis of low structures
subject to kinematic excitations caused by transversal waves // J. Theor.
Appl. Mech. - 1996. - 34, N 3. - P. 547 - 566.
104 ISSN 0556-171X. Проблеми прочности, 2000, № 6
Free Vibration o f the Axially Loaded System
7. Winkler E. Die Lehre von der Elasticitat und Festigkeit. - Praga:
Dominicus, 1867.
8 . Cabanska-Placzkiewicz K. Free vibration of the system of two strings
coupled by a viscoelastic interlayer // J. Engng. Trans. - 1998. - 46, N 2. -
P. 217 - 227.
9. Cabanska-Placzkiewicz K. Free vibration of the system of two viscoelastic
beams coupled by a viscoelastic interlayer // J. Acoustic Bulletin. - 1999. -
1, N 2. - P. 3 -10.
10. Cabanska-Placzkiewicz K., Pankratova N. The dynamic analysis of the
system of two beams coupled by an elastic interlayer // XXXVIIIth Symp.
of Model. in Mech., 9, Silesian Univ. of Tech., 23-28, Gliwice, 1999.
11. Cabanska-Placzkiewicz K. Free vibration of the system of two Timoshenko
beams coupled by a viscoelastic interlayer // J. Engng. Trans. - 1999. - 47,
N 1. - P. 21 - 37.
12. Tse F., Morse I., Hinkle R. Mechanical Vibrations Theory and Applications.
- Boston: Allyn & Bacon, 1978.
13. Niziol J., Snamina J. Free vibration of the discrete-continuous system with
damping // J. Theor. Appl. Mech. - 1990. - 28, N 1-2. - P. 149 - 160.
14. Cremer L., Heckel M., Ungar E. Structure-Borne Sound: Structural
Vibrations and Sound Radiation at Audio Frequencies. - Berlin: Springer
Verlag, 1988.
15. N ashif D., Johnes D., Henderson J. Vibration damping // J. Vibration
Acoustic. - Moscow: Mir, 1988.
16. Nowacki W. The Building Dynamics. - Warsaw: Arkady, 1972.
17. Osinski Z. Damping of the Mechanical Vibration. - Warsaw: Polish Scient.
Publ., 1979.
18. Kasprzyk S. Dynamics of the Continuous System. - Krakow: Publ. of AHG,
1971.
Received 02. 10. 99
ISSN 0556-171X. npoôneMbi npouuocmu, 2000, № 6 105
|