Plane Thermoelastic Waves in Infinite Half-Space Caused
FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 12, N
o
3, 2014, pp. 325 - 337
1 ENERGY ANALYSIS OF FREE TRANSVERSE VIBRATIONS
OF THE VISCO-ELASTICALLY CONNECTED
DOUBLE-MEMBRANE SYSTEM
UDC 531+534+517. 93(045)=111
Julijana Simonović
1
, Danilo Karličić
2
, Milan Cajić
2
1
Faculty of Mechanical Engineering, University of Niš, Serbia
2
Mathematical Institute of the SASA, Serbian Academy of Science and Arts, Serbia
Abstract. The presented paper deals with the analysis of energy transfer in the visco-
elastically connected circular double-membrane system for free transverse vibration of
the membranes. The system motion is described by a set of two coupled non-homogeneous
partial differential equations. The solutions are obtained by using the method of
separation of variables. Once the problem is solved, natural frequencies and mode shape
functions are found, and then the form of solution for small transverse deflections of
membranes is derived. Using the obtained solutions, forms of reduced kinetic, potential
and total energies, as functions of dissipation of the whole system and subsystems, are
determined. The numerical examples are given as an illustration of the presented
theoretical analysis as well as the possibilities to investigate the influence of different
parameters and different initial conditions on the energies transfer in the system.
Key Words: Double-membrane system, Visco-elastic layer, Dissipation function,
Energy transfer, Multi-frequency vibration
1. INTRODUCTION
Membrane structures and compound systems are widely used in many industrial
applications. In addition, in the micro world the system of compound membranes may
represent a biological model of a cell membrane. The application involves several
disciplines and industrial contexts as, for example, microfiltration systems in biological,
medical, food, dairy and beverage products and application in aeronautics, cosmonautics,
civil and mechanical engineering (see Refs. [1] and [2]). Membrane, string, beam, plate
or cable [3] structures can perform linear and nonlinear vibrations. Despite the fact that
the membrane systems are, in general, strongly nonlinear systems, under certain assumptions
they can be considered as linear. By assuming that the amplitude of a membrane deflection is
Received June 1, 2014 / Accepted September 10, 2014
Corresponding author: Juliana Simonović
University of Niš, Faculty of Mechanical Engineering, A. Medvedeva 14, 18000 Niš, Serbia
E-mail: bjulijana@masfak.ni.ac.rs
Original scientific paper
326 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
smaller than its thickness and considering the initial tension on the membrane [4], the
dynamic behavior of such structures can be observed within the linear vibration theory.
Opposite to this, when no or small initial tension is applied and the membrane deflection
amplitude is close to the value of the membrane thickness, the system is performing
nonlinear vibration [4]. The linear as well as the nonlinear vibration analysis of membrane
systems is important from the theoretical and practical point of view. Several papers have
been published on the vibration analysis of elastically coupled double-membrane or
plate-membrane systems, continuously connected by an elastic layer. Oniszczuk [5, 6],
Noga [7, 8] and Karliĉić [9] present analytical expressions for the undamped free and
forced vibrations of double-membrane and plate-membrane systems connected with the
Winkler type of layers. Knowing the principles of phenomenological mapping and
mathematical analogy, [10], it is clear that the composite membrane systems can be
studied like coupled plate, beam or belt systems. Hedrih and Simonović [11-13] have
studied transverse vibrations of the rectangular and circular plates connected with elastic
or visco-elastic layers for both linear and nonlinear dynamic problems. Kelly [14]
examines vibration of the elastically connected multiple beams system. Energy transfer
problem is studied in [15-17] for the vibrations of double-membrane and double-plate
systems, connected with elastic or visco-elastic type of layers.
In spite of many vibration studies on various types of hybrid systems, where different
structures are coupled with different types of layers, according to the best of authors’
knowledge, an energy transfer analysis of free transverse vibration of the visco-elastically
connected circular double-membrane system has not been considered in the literature yet.
Under the assumptions of small transverse vibration of the membranes and constant
initial tension for both the membranes, we analyze our system within the linear vibration
theory. Governing equations of the system are derived and the solution is proposed by
using the method of separation of variables. Natural frequencies and mode shape functions
are determined by solving the boundary and initial value problem. The expressions for time
functions are used in order to find analytical forms of reduced kinetic, potential and total
energy of the system.
2. FORMULATION OF GOVERNING EQUATIONS
As a model problem, we consider two circular membranes connected through visco-
elastic layer, modeled by continuously distributed elements of the Kelvin-Voigt type. The
scheme of such a mechanical model is depicted in Fig. 1. Both the membranes are
assumed to be thin with mass densities ρi, for i=1,2 corresponding to the lower and upper
membrane, respectively, ideally elastic and with neglected thickness.
The membranes are stretched and fixed along their entire boundaries in xy plane. The
tensions per unit length σi [N/m], caused by stretching, are the same at all points in all
directions and do not change during the motion. Small transverse deflections of
membranes wi(r,φ,t), i=1,2 are considered, where b[Ns/m] is denoting the damping
coefficient and c [N/m] is the constant stiffness coefficient per surface unit area of the
distributed visco-elastic layer between the membranes. Using the D`Alambert principle,
the governing system of non-homogenous coupled partial differential equations for free
vibration of the double-membrane system is expressed in the following form:
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 327
Fig. 1 The physical model of circular double-membrane compound system
2
2 21
12
( , , ) ( , , ) ( , , )
( , , ) 2 ( , , ) ( , , )i i i
i i i i i i
w r t w r t w r t
c w r t a w r t w r t
t tt
(1)
where i = 1,2; ci = (σi /ρi)
1/2
[m/s] are velocities of the transverse wave propagation for
both the membranes, is Laplacian operator, and ai
2
= c /ρi, 2δi=b/ρi are notations for
reduction coefficients.
Analytical solutions of the system of coupled partial differential equations are
obtained by using the method of separation of variables. For the system of two coupled
partial equations Eq. (1), for free vibration, we separate variables in wi(r,φ,t), i=1,2 and
considering the eigenamplitude functions as Wi(nm)(r,φ); n,m=1,2,...,∞ and the time
expansion with coefficients in the form of unknown time function as Ti(nm)(t) we describe
their time evolutions in the form:
( ) ( )
1 1
( , ) ( ), 1, 2
i i nm i nm
n m
w r W T t i
(2)
wherein m =1,2,...,∞ denotes an infinite number of possible vibration modes.
Here, eigenamplitude functions Wi(nm)(r,φ) are the same for both the membranes and
written in the following form:
W(i)nm(r,) = Jn(knmr)cos(n + (i)0n),
(3)
which are obtained for the decoupled system and for the same boundary conditions of
membranes. In Eq. (3) Jn(knmr) are Bessel`s functions of the first kind of n-th order, and
knm=xnm/a are characteristic numbers for every m-th root of n-th Bessel`s functions over
membrane radius a.
After introducing Eq. (2) into the governing system of Eq. (1), we obtain the
following system of the homogeneous second order ordinary differential equations with
respect to unknown time functions Ti(nm)(t) for nm-family mode, in the following form:
2 2
1( ) 1 1( ) 1( ) 1( ) 1 2( ) 1 2( )( ) 2 ( ) ( ) 2 ( ) ( ) 0nm nm nm nm nm nmT t T t T t T t a T t
2 2
2( ) 2 2( ) 2( ) 2( ) 2 1( ) 2 1( )( ) 2 ( ) ( ) 2 ( ) ( ) 0,nm nm nm nm nm nmT t T t T t T t a T t
(4)
328 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
where ωi
2
(nm)= ω0i
2
(nm)+ai
2
=ki
2
(nm)ci
2
+ai
2
; i=1,2; n,m =1,2,...,∞, are natural frequencies for
the first and second membrane for the nm-th mode of vibration. To solve the system of
two coupled ordinary differential Eqs. (4), it is necessary to form the frequent determinant
and determine the eigenvalues of the system. We introduce matrices corresponding to the
nm-th mode of the observed dynamical system: inertia Anm matrix, stiffness coefficients
matrix Cnm and damping coefficients matrix Bnm in the following forms:
2 2
1( ) 11 1
2 2
2 2 2 2( )
2 21 0
; ;
2 20 1
nm
nm nm nm
nm
a
a
A B C (5)
The coupled system of Eqs. (4) for time domain may be treated as a system of equations
corresponding to a two-degree-of-freedom discrete system with defined matrices in Eq. (5).
The characteristic equation of the linearized coupled system is given in the form:
2 2 2
1 1( ) 1 12
2 2 2
2 2 2 2( )
2 2
0
2 2
nm nm nm nm
nm nm nm nm nm
nm nm nm nm
λ a
λ
a λ
A B C . (6)
We obtain eigenvalues of the system in nm modes as two pairs of complex conjugate
roots in the form:
1,2( )( ) ( )( ) ( )( ) , 1, 2s nm s nm s nmλ i s (7)
where δ (s)(nm) and ω (s)(nm) are real and imaginary parts of the corresponding pair of roots of
the characteristic equation.
Now, the solution of the system of ordinary differential Eqs. (4) can be expressed as:
1( )( ) 2( )( )
( )( )
( )( ) ( )
s nm s nm
st ts
i nmi nm is i nm
s
T C A e A e
. (8)
We obtain eigenamplitude numbers A
(s)
i(nm) and their conjugates
( )
( )
s
i nmA from:
( ) ( )
1( ) 2( )
( )( ) ( )
21( ) 22( )
s s
nm nm
s nms s
nm nm
A A
C
K K
or
( ) ( )
1( ) 2( )
2 2 2
1 1 ( )( ) ( )( ) 1 ( )( ) 1( )2 2
s s
nm nm
s
s nm s nm s nm nm
A A
C
a
(8a)
where K
(s)
2i(nm) are cofactors of the determinant in Eq. (6) and Cs(nm) are known constants
determined from the corresponding characteristic equation.
Considering the time functions in Eqs.(8) corresponding to frequencies ω (s)(nm) of the
damped coupling and taking into account conjugate complex roots (Eq. (7)) yields:
( )
2
( ) ( ) ( )( ) ( ) ( )( )
1
cos( ) sin( )s nm
t
i nm is nm s nm is nm s nm
s
T e u t v t
. (9)
The constants of integration uis(nm) and vis(nm) are defined as follows:
( ) ( )
( ) ( ) ( )2 2
Re( ) Im( )
s s
is nm s nm s nmi i
u A K B K ,
( ) ( )
( ) ( ) 2 2
Im( ) Re( )
s s
is nm s nm i is nm
v A K B K . (10)
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 329
As(nm) and Bs(nm) can be obtained through application of the initial conditions. The initial
conditions are assumed as known functions:
( , ,0) ( , ),i iw r g r (11)
0
( , , )
( , )
i
t i
w r t
g r
t
. (12)
Now, the particular solutions for a non-homogenous system of the coupled partial
differential equations, for free vibrations, as membrane deflections, are:
1( )
2
( ) 1( ) 1 1( ) 1
1 1
2( ) 2 2( ) 2
( , , ) ( , ) [ cos( ) sin( )]
[ cos( ) sin( )] .
nm
nm
t
i i nm i nm i nm
n m
t
i nm i nm
w r t W r e u t v t
e u t v t
(13)
The solutions from Eq. (13) are analytical results of our research on transversal vibrations
of the visco-elastically connected double circular membrane system.
On the basis of the orthogonality properties of the mode shape functions, unknown
constants As(nm) and Bs(nm) can be determined from the assumed initial conditions Eq. (11)
and Eq. (12). Introducing the known functions of membranes’ point displacements and
velocities Еq. (11) and Eq. (12) into the solutions Eq. (2) and applying the classical
orthonormality conditions of eigenamplitude functions Wi(nm)(r,φ) and Wi(sr)(r,φ), in the
form:
2
2
( ) ( ) ( ) 2
( ) ( )
0 0
0 0
0 ,
( , ) ( , )
[ ( , )] , 1, 2.
a
a
i nmsr i nm i sr
i nm i nm
sr nm
M W r W r rdrd
M W r rdrd i sr nm
, (14)
we obtain the values of the initial time functions:
2
( )
0 0
0( )
( )
( , ) ( , )
a
i nm
i nm
i nm
g r W r rdrd
T
M
and
2
( )
0 0
0( )
( )
( , ) ( , )
a
i nm
i nm
i nm
g r W r rdrd
T
M
. (15)
In order to find the final forms of the transverse vibrations, the initial-value problem has
to be solved:
0( ) ( ) ( ) ,nm nm nmT K α (16)
where T0(nm)=[T10(nm) T20(nm) T
10(nm) T
20(nm)] is the vector of the known functions Eq. (15),
α(nm)=[A1(nm) B1(nm) A2(nm) B2(nm)]
T
is the vector of unknown constants needed for solutions
and K(nm) is the functional matrix of the fourth order depending on the system’s properties,
given as:
2
222
2
222
2
222
2
222
1
221
1
221
1
221
1
221
2
212
2
212
2
212
2
212
1
211
1
211
1
211
1
211
2
22
2
22
1
22
1
22
2
21
2
21
1
21
1
21
Re
~
Im
~
Im
~
Re
~
Re
~
Im
~
Im
~
Re
~
Re
~
Im
~
Im
~
Re
~
Re
~
Im
~
Im
~
Re
~
ImReImRe
ImReImRe
KKKKKKKK
KKKKKKKK
KKKK
KKKK
nm
K
(16a)
330 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
From analytical solutions, Eq. (13), and corresponding solutions of constant system,
Eq. (16), we can conclude that in one mode of vibration, two circular frequencies of
coupling and a two-frequency time function Ti(nm)(t) correspond to one eigenamplitude
function. The first time mode is with lower damped frequency ω (1)(nm), and the second one
is with higher damped frequency ω (2)(nm). Hence, the visco-elastic layer introduces into
the system duplication of the number of circular frequencies which correspond to the one
eigenamplitude function of the nm-family mode n,m=1,2,...,∞.
We can rewrite the solutions for time functions in the form:
i
t
nminminmi
nmieDT
Y , (17)
where Yi(nm) are eigenvectors corresponding to following system matrix:
1 1
.nm
nm nm nm nm
0 I
Q
A C A B
(18)
The constants of integration Di(nm) can be calculated by solving a system of simultaneous
algebraic equations formulated in the matrix form as:
0( ) ( ) ( ) ,nm i nm i nm
i
DT Y (19)
where T0(nm) is the vector of inital condition constants given by Eq. (15). The system
matrix Eq. (18) has two pairs of complex conjugate eigenvalues the same as values Eq.
(7). Therefore, we conclude that the solutions given in Eq.(17), Eq. (8) and Eq. (9) are
actually the same.
3. ENERGY ANALYSIS
We can analyze energy transfer in the double-membrane system by using reduced
components of kinetic and potential energy of membranes, potential energy of the light
distributed visco-elastic layer and the reduced Rayleigh function of dissipation for
corresponding nm-family mode. Also, we can use the system of two ordinary differential
Eqs. (4) for corresponding nm-th mode, which is considered as a system with two degrees
of freedom. For the system of Eqs. (4) it is possible to write forms of kinetic and potential
energies by using the matrices Eqs. (5). Reduced forms of energies corresponding to the
nm-th mode are given as:
a) Reduced kinetic energy ( )k nmE ,n,m=1,2,...,∞:
( )1( ) 2 2
( ) 1( ) 2( ) 1 1( ) 2 2( )
2( ) ( )
1 1
( ) [ ( ) ( ) ]
2 2
k nmnm
k nm nm nm nm nm
nm i nm
ET
E T T T T
T M
A , (20)
where Ek(nm) and Mi(nm) are kinetic energy and orthogonality function Eq. (14), respectively.
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 331
b) Reduced potential energy ( )p nmE :
1( )
( ) 1( ) 2( )
2( )
( )2 2 2 2 2
2( ) 1( ) 1 01( ) 1( ) 2 02( ) 2( )
( )
1
( )
2
1
[ ] ( ) ( )
2
nm
p nm nm nm
nm
p nm
nm nm nm nm nm nm
i nm
T
E T T
T
E
c T T T T
M
C
, (21)
where Ep(nm) is potential energy.
c) Reduced Rayleigh function of dissipation ( )nm lay , which is related to the layer, is of
the form:
( )lay1( )1 1 1 1 2
( ) 1( ) 2( ) 1( ) 2( )
2( )2 2 2 2 ( )
2 21 1
( ) [ ] .
2 22 2
p nmnm
nm lay nm nm nm nm
nm i nm
T
T T b T T
T M
(22)
Also, we can separate terms for the kinetic and potential energies which correspond to
the first and the second membrane, [8, 14, 15]:
a.1) Reduced kinetic energy of the membranes:
( )( )2
( )( ) ( )
( )
1
( ) , 1,2
2
k nm i
k nm i i i nm
i nm
E
E T i
M
. (23)
b.1) Reduced potential energy of the membranes and reduced potential energy of a
visco-elastic layer for the corresponding membranes:
( )( )2 2
( )( ) ( ) ( )
( )
1
( )
2
p nm i
p nm i i i nm i nm
i nm
E
E T
M
(24)
b.2) Reduced potential energy of pure interaction between membranes induced by a
visco-elastic layer:
( )int2 2
( ) int 1 1 2 2 2( ) 1( )
( )
1
( )
2
p nm
p nm nm nm
i nm
E
E a a T T
M
. (25)
b.3) Reduced potential energy of the membranes without reduced part of the potential
energy of layer for the corresponding membranes:
( )( )2 2
( )( ) 0 ( ) ( )
( )
1
( )
2
p nm i b
p nm i b i i nm i nm
i nm
E
E T
M
. (26)
b.4) Full reduced potential energy of a visco-elastic layer interaction between the
membranes:
(1,2)( )2
(1,2)( , ) 2( , ) 1( )
( )
1
[ ( ) ( )] .
2
p nm layer
p n m layer n m nm
i nm
E
E c T t T t
M
(27)
From the previous equations, we can see that the reduced potential energy is obtained
by considering membranes’ vibration on the elastic foundation of the Winkler type. Both
332 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
the membranes share potential energy of the visco-elastic layer, and only one part is
interaction between the membranes depending on layers rigidity and on both time
functions of the membranes. In the following, we will introduce:
c.1) Rayleigh function of dissipation-reduced part of a visco-elastic layer for the
corresponding membranes:
( )layer( )2
( )layer( ) ( )
( )
( )
p nm i
p nm i i i i nm
i nm
T
M
. (28)
c.2) Part of the Rayleigh dissipation function – pure interaction between the membranes
induced by visco-elastic layer
( )layer(int)
( )layer(int) 1 1 2 2 1( ) 2( )
( )
p nm
p nm nm nm
i nm
T T
M
. (29)
If we use the solutions from Eq. (9) of free vibrations and their derivatives with
respect to time, the reduced total membranes energy for the nm-th mode can be expressed
as follows.
Reduced total energy of the first, i=1, and second, i=2, membranes in the nm-mode
are:
( )( ) ( )( )2 2 2
( )( ) ( )( ) ( )( ) ( ) ( ) ( )
( )
1
( ) ( )
2
k nm i p nm i
nm mi k nm i p nm i i i nm i nm i nm
i nm
E E
E E E T T
M
. (30)
Therefore, the reduced total energy of both membranes is:
( )( 1, 2) ( )(1) ( )(1) ( )(2) ( )(2) ( )( 1) ( )( 2)nm m m k nm p nm k nm p nm nm m nm mE E E E E E E . (31)
Reduced total energy of the system in the nm-th mode is equal to the sum of the reduced
total energy of both the membranes and the reduced potential energy of pure interaction
between them (Eq. (25)):
( ) ( )( 1, 2) ( )intnm system nm m m p nmE E E . (32)
As concluded in [17] for transverse vibrations of the double membrane system with
elastic layer, the total energy of the system remains constant and equal to the energy in
the initial moment during the whole dynamic process which is indeed a property of
conservative systems. Here, for a non-conservative system, we have energy dissipation
which is equal to:
( )
( )system ( )2 2
nm system
nm nm lay
E
dt
, (33)
where Φ(nm)lay is the Rayleigh function.
This means that the dissipation function is a measure of the degradation of the
mechanical energy of system, notwithstanding the choice of system parameters and the
initial conditions that is also confirmed in the numerical results section.
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 333
4. NUMERICAL RESULTS FOR THE SYSTEM ENERGY
For the numerical calculation and representation of Eqs. (32) and (33), the following
parameter values are considered: tension per unit length σ1=600[N/m], density
ρ1=200[kg/m
3
] and radius a=1m for the upper membrane and for the lower membrane we
consider the same radius but different tension and density. For the discussion of the
results their values are presented in Table 1. We introduce the following coefficients
ξ=ρ2/ρ1 and η=σ2/σ1 with values presented in Table 1. All the simulations are performed
in the first shape mode nm=01 when k01=2.40483.
When we have ξ=η=1, it could be noticed that the damped higher frequency of the
second time mode is constant ω 2=4.16528[s
-1
]=k01(σ1 /ρ1)
1/2
=const and that mode is not
damped δ 2=0 for different values of layer parameters. The lower frequency aptly
corresponds to the changes of stiffness coefficient c and damping coefficient b. Lower
natural frequency ω 1 increases for an increase of the layers stiffness and it decreases for
an increase of the damping coefficient.
Table 1 The eigenvalues of the system in Eq. (7) for proper 01-mode, for different
system parameters
a=1[m] 50. 1 2
c
[N/m]
b
[Ns/m]
50. 1 50. 1 50. 1
100
100 λ1,2(1)
λ1,2(2)
-0.75 ± 4.28i
0 ± 4.17i
-0.53 ± 5.87i
-0.22 ± 4.27i
-0.27 ± 4.16i
-0.23 ± 3.06i
-0.5 ± 4.25i
0 ± 4.16i
-0.26 ± 4.2i
-0.11 ± 2.14i
-0.26 ± 4.19i
-0.11 ± 3.0i
200 λ1,2(1)
λ1,2(2)
-1.50 ± 4.07i
0 ± 4.16i
-1.10 ± 5.52i
-0.39 ± 4.46i
-0.5 ± 3.89i
-0.5 ± 3.22i
-1.0 ± 4.16i
0 ± 4.16i
-0.51 ± 4.11i
-0.24 ± 2.17i
-0.53 ± 4.07i
-0.21 ± 3.06i
300 λ1,2(1)
λ1,2(2)
-2.25 ± 3.71i
0 ± 4.16i
-1.9 ± 5.02i
-0.35 ± 4.69i
-0.3 ± 3.61i
-1.2 ± 3.31i
-1.5 ± 4.01i
0 ± 4.16i
-0.76 ± 3.97i
-0.36 ± 2.21i
-0.84 ± 3.86i
-0.28 ± 3.18i
250
100 λ1,2(1)
λ1,2(2)
-0.75 ± 4.53i
0± 4.16i
-0.57 ± 6.02i
-0.18 ± 4.34i
-0.32 ± 4.26i
-0.18 ± 3.16i
-0.5 ± 4.43i
0 ± 4.16i
-0.27 ± 4.29i
-0.1 ± 2.22i
-0.28 ± 4.29i
-0.09 ± 3.05i
200 λ1,2(1)
λ1,2(2)
-1.50 ± 4.34i
0 ± 4.16i
-1.18 ± 5.72i
-0.32 ± 4.48i
-0.64 ± 4.01i
-0.35 ± 3.31i
-1.0 ± 4.34i
0 ± 4.16i
-0.54 ± 4.21i
-0.21 ± 2.24i
-0.57 ± 4.18i
-0.17 ± 3.1i
300 λ1,2(1)
λ1,2(2)
-2.25 ± 4.00i
0 ± 4.16i
-1.94 ± 5.28i
-0.31 ± 4.66i
-1.21 ± 3.6i
-0.29± 3.55i
-1.5 ± 4.19i
0 ± 4.16i
-0.81 ± 4.07i
-0.32 ± 2.28i
-0.89 ± 4i
-0.23 ± 3.18i
500
100 λ1,2(1)
λ1,2(2)
-0.75 ± 4.93i
0 ± 4.16i
-0.62 ± 6.27i
-0.13 ± 4.42i
-0.38 ± 4.47i
-0.12 ± 3.27i
-0.5 ± 4.7i
0 ± 4.16i
-0.29 ± 4.45i
-0.08 ± 2.32i
-0.30 ± 4.46i
-0.07 ± 3.11i
200 λ1,2(1)
λ1,2(2)
-1.50 ± 4.75i
0 ± 4.16i
-1.27 ± 6.03i
-0.23 ± 4.51i
-0.78 ± 4.29i
-0.22 ± 3.36i
-1.0 ± 4.62i
0 ± 4.16i
-0.58 ± 4.38i
-0.17 ± 2.34i
-0.62 ± 4.37i
-0.12 ± 3.15i
300 λ1,2(1)
λ1,2(2)
-2.25 ± 4.45i
0 ± 4.16i
-2 ± 5.66i
-0.25 ± 4.63i
-1.27 ± 4.0i
-0.22 ± 3.48i
-1.5 ± 4.48i
0 ± 4.16i
-0.87 ± 4.25i
-0.25 ± 2.37i
-0.96± 4.23i
-0.17 ± 3.2i
It is interesting to note that the same is true for ξ=η=const, which is marked with gray
cells in Table 1. This mechanism of changes is also the same for other arbitrary values of
σ1 and ρ1 when ξ=η=const, where we have obtained ω 2=k01(σ1 /ρ1 )
1/2
=const and δ 2=0 for
all values of layers parameters.
Comparing to the other values in Table 1, it is obvious that the damped natural
frequencies increase for an increase of the layers stiffness, whereas damped lower
frequency ω 1 decreases and higher frequency ω 2 increases for an increase of damping
coefficient. The final forms of time functions, Eqs.(9) rely on the values of vector α(nm).
The relations for time functions and their time derivatives determine the values of
reduced energies, Eqs. (30-32), and dissipative function, Eq. (33), of subsystems and the
334 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
system itself. It is possible to obtain the diagrams for the system and subsystems energies
for every particular value of parameters. For the qualitative analyses several different
parameters for membranes and layers are selected for the diagrams of energies and
presented in the following Figures.
Fig. 2a shows the reduced total energy of the system, Eq. (32), in appropriate 01-
mode for different values of visco-elastic layer stiffness and the same initial conditions.
The parameters used in simulation for this Figure are σ1=2σ2=600 [N/m], ρ1=5ρ2=
200 [kg/m
3
], b=200 [Ns/m] for three different values of c ={100, 250 and 500}[N/m], and
initial conditions w10=0.0025W01, w20=0.001W01, 10w = 20w =0. As can be seen from Fig.
2a, the total energy of the system is larger and more slowly dissipated for higher values
of the visco-elastic layer stiffness coefficient than for the lower values. At initial time,
total energy depends on the energy given to the system by appropriate initial conditions,
as they are the same for those three lines starting from the same appropriate point.
Fig. 2b shows the double reduced Rayleigh function of the dissipation, Eq. (29), in the
upper part of diagrams, and time exchange of reduced total energy of the system (nm)system/dt
in the lower part of diagrams. Those diagrams are obtained for the same values of
parameters and initial conditions as in Fig. 2a. Also, as it stems from Eq.(33), time
exchange of total energy is equal to the negative double value of dissipation function and
diagrams from Fig. 2b are completely symmetric.
Figs. 3a and 3b show similar diagrams as Figs. 2a and 2b, respectively, with the same
initial conditions but for different system parameters: σ1=5σ2=600 [N/m], ρ1=ρ2/2=200 [kg/m
3
],
c =250[N/m] and for varied values of damping coefficient b={100, 200 and 300}[Ns/m].
In this case, the total system energy dissipates faster as the damping constant increases.
In order to investigate the effect of change of the initial conditions, numerical
simulations are performed for the same stiffness and damping coefficients. Diagrams in
Figs. 4a and 4b are similar to those in Figs. 2a and 2b, respectively. However, in this case
calculations are performed for different system parameters: σ1=5, σ2=600 [N/m],
ρ1=ρ2/2=200 [kg/m
3
], c =250[N/m], b=300 [Ns/m]. In addition, three different cases of
initial displacements and velocities on the first and the second membranes are employed:
I) w10=0.002W01, w20=0, 10w = 20w =0, the lower membrane is at rest in the initial moment;
II) w10=0.002W01, w20=0.002W01, 10w = 20w =0, the membrane points are having the
opposite initial positions corresponding to the first amplitude shape function, so-called
initially anti-phased membranes, and III)w10=w20=0.002W01, 10w = 20w =0, the membrane
points are having the same initial positions corresponding to the first amplitude shape
function, so-called initially phased membranes. The shapes of membranes at the initial
moment for the three cases are presented in Figs. 5a, 5b and 5c, respectively.
The forms of membrane deflection and velocity at the initial moment determine the
mechanical energy given to the double membrane system. In Fig. 4a one can see different
starting points of the total energy function for different initial conditions. The largest
values obtained for the case of initially anti-phase membranes, since the system has
received the greatest potential energy in the initial moment when the membrane points
are having the opposite positions corresponding to the first amplitude shape function.
Since the system is damped, after some period of time the total energy of the system is
dissipated to the zero value. As can be seen from Fig. 4 for all the three simulated conditions
the period of time needed for dissipation of the total energy does not change and we can say
that the system’s damped properties depend only on the system parameters.
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 335
a) b)
Fig. 2 a) The reduced total energy of the system and b) the double reduced Rayleigh
function of dissipation and time exchange of the system’s reduced total energy in
appropriate 01-mode for system parameters and different values of stiffness
coefficient c
a) b)
Fig. 3 a) The system’s reduced total energy and b) the double reduced Rayleigh function
of dissipation and time exchange of the system’s reduced total energy in
appropriate 01-mode for system parameters and different values of damping
coefficient b
a) b)
Fig. 4 a) The system’s reduced total energy and b) the double reduced Rayleigh function
of dissipation and time exchange of the system’s reduced total energy in
appropriate 01-mode for system parameters and different three values of initial
displacement and velocity on the first and the second membranes (I, II, III)
336 J. SIMONOVIĆ, D. KARLIĈIĆ, M. CAJIĆ
Fig. 5 The shape of membranes at different initial moment: a) the lower membrane is at
rest at the initial moment; b) so-called initially anti-phased membranes, and c) so-
called initially phased membranes
5. CONCLUSIONS
In this paper, the free vibration and energy analysis of a double-membrane system
joined by the Kelvin-Voigt type layer is performed analytically and numerically. The
obtained forms of solutions for time functions Eq.(9) in proper mode shape are rather
down-to-earth solutions since they are applicable in numerical experiment. Furthermore,
those forms of solutions could be used in the analysis of forced and nonlinear system
oscillations. The analytical analysis showed that the Kelvin-Voigt type layer is responsible
for the appearance of two-frequency regimes, which corresponds to one eigenamplitude
function. Time functions for different modes of vibrations are uncoupled and energy
transfer appears in a single mode. The system is non-conservative and damped by a visco-
elastic layer, so the dissipation function is a measure of the degradation of system’s
mechanical energy, notwithstanding the choice of system parameters and the change of
the initial conditions. The mechanical energy given to the system at the initial time decreases
until the whole energy of the system is dissipated. It can be concluded that influence of
the system parameters is more significant for the mechanical energy analysis than the
change of the initial conditions, which is, in general, a property of the linear systems.
Acknowledgements: This research is sponsored by the research grant of the Serbian Ministry of
Education, Science and Technological Development under the number ON 174001.
REFERENCES
1. Postlethwaite, J., Lamping, S.R., Leach, G.C, Hurwitz, M.F., Lye, G.J., 2004, Flux and transmission
characteristics of a vibrating microfiltration system operated at high biomass loading, Journal of Membrane
Science 228, pp. 89–101.
2. Jaffrin, M.Y., 2008, Dynamic shear-enhanced membrane filtration: A review of rotating disks, rotating
membranes and vibrating systems, Journal of Membrane Science 321 , pp. 7–25.
3. Vassilopoulou, I., Gantes, C. J., 2010, Vibration modes and natural frequencies of saddle form cable nets,
Computers & structures, 88(1), pp. 105-119.
4. Bao, Z., Mukherjee, S., Roman, M., Aubry, N., 2004, Nonlinear vibrations of beams, strings, plates, and
membranes without initial tension, Journal of Applied Mechanics, 71(4), pp. 551-559.
5. Oniszczuk, Z., 1998, Vibrations of elastically connected circular double-membrane compound system,
Scientiific Works of Rzeszow University of Technology, Civil Engineering 132, pp. 61-81.
An Energy Analysis of Visco-Elastically Connected Double-Membrane System 337
6. Oniszczuk, Z., 1999, Transverse vibrations of elastically connected rectangular double- membrane compound
system, Journal of Sound and Vibration 221(2), pp. 235-250.
7. Noga, S., 2010, Free transverse vibration analysis of an elastically connected annular and circular double-
membrane compound system, Journal of Sound and Vibration 329, pp. 1507–1522.
8. Noga, S., 2008, Numerical analysis of a free transverse vibration of an elastically connected annular double-
membrane compound system, Acta Mechanica Slovaca 3-A , pp. 307–312.
9. Karliĉić, D., 2012, Free transversal vibrations of a double-membrane system, Scientific Technical Review 62,
pp. 77–83.
10. Simonović, J., 2008, Dynamics of Mechanical Systems of Complex Structure, Magister thesis, Faculty of
Mechanical Engineering, University of Niš , in Serbian, pp. 249.
11. Hedrih, K. S., 2006, Transversal vibrations of double plate systems, Acta Mechanica Sinica, 22, pp. 487-501.
12. Hedrih, K. S., Simonović, J., 2007, Transversal vibrations of a non-conservative double circular plate system,
Facta Universitatis, Series Mechanics, Automatic Control and Robotics 6(1), pp. 19-64.
13. Hedrih, K. S., Simonović, J. D., 2010, Non-linear dynamics of the sandwich double circular plate system,
International Journal of Non-Linear Mechanics 45(9), pp. 902-918.
14. Kelly, S. Graham, 2007, Advanced Vibration Analysis, Boca Raton, FL: CRC/Taylor & Francis, p. 1-637.
15. Hedrih, K. S., Simonović, J. D., 2012, Energies of the Dynamics in a Double Plate Nonlinear System,
International Journal of Bifurcation and Chaos 21(10), pp. 2993-3011.
16. Hedrih, K. S., 2009, Energy transfer in the hybrid system dynamics (energy transfer in the axially moving
double belt system), Archive of Applied Mechanics 79, pp. 529-540.
17. Karliĉić, D., Cajić, M., 2012, Energy Transfer Analysis of an Elastically Connected Circular Double-Membrane
Compound System, Book of abstracts, 8th European Solid Mechanics Conference, Graz (2012).
ENERGIJSKA ANALIZA SLOBODNIH TRANSVERZALNIH
VIBRACIJA SISTEMA VISKO-ELASTIČNO SPREGNUTIH
MEMBRANA
Predstavljeni rad je posvećen analizi prenosa energije kod slobodnih transverzalnih oscilacija
sistema visko-elastično spregnute dve membrane. Kretanje sistema je opisano sistemom dve
spregnute nehomogene parcijalne diferencijalne jednačine. Rešenja su dobijena primenom metode
razdvajanja promenljivih. Rešavajući problem dobijaju se sopstvene kružne frekvencije i osnovni
amplitudni oblici oscilovanja sistema, a potom i oblici rešenja za male transverzalne pomeraje
membrana. Koristeći dobijena rešenja određene su redukovane vrednosti kinetičke, potencijalne
energije i funkcije rasipanja kako celog sistema tako i podsistema. Primeri numeričkog proračuna
su dati kao ilustracija prikazane teorijske analize i kao mogućnost da se prouče uticaji različitih
parametara sistema i različitih početnih uslova na prenos energije u sistemu.
Kljuĉne reĉi: sistem dve membrane, visko-elastični sloj, funkcija rasipanja, prenos energije, više-
frekventne oscilacije.