TRANSVERSE VIBRATION ANALYSIS OF A PRESTRESSED THIN
CIRCULAR PLATE IN CONTACT WITH AN ACOUSTIC CAVITY
Daniel G Gorman1 ,Chee K Lee and Ian A Craighead
Department of Mechanical Engineering, James Weir Building,
University of Strathclyde, Glasgow G1 1XJ
Jaromír Horáček
Institute of Thermomechanics, Academy of Sciences of the Czech Republic,
Dolejškova 5,182 00 Prague 8, Czech Republic
ABSTRACT
This paper describes the free transverse vibration analysis of a thin circular plate,
subjected to in plane stretching, whilst in interaction with a cylindrical acoustic
cavity. An analysis is performed which combines the equations describing the plate
and the acoustic cavity to form a matrix equation which, when solved, produces the
natural frequencies (latent roots) of the coupled system and associated latent vectors
which describe the mode shape coefficients of the plate. After assessing the numerical
convergence of the method, results are compared with those from a commercial finite
element code (ANSYS). The results analysis is then extended to investigate the effect
of stressing upon the free vibration of the coupled system.
Keywords
Vibrations, vibro-acoustic interaction, structural/acoustic.
1. INTRODUCTION
Owing to their wide application in mechanical systems ranging from musical
instruments to structural elements in industrial and space applications, the transverse
vibration of circular plates and membranes has been the subject of many
investigations from the end of nineteenth century. Of particular interest has been the
effect upon the natural frequencies and associated mode shapes of these structural
elements due to the inclusion of in-plane stressing as a result of thermal gradients and
more general forms of hydrostatic loading. An excellent and extensive overview of
much of this work is presented in reference [1]. In all of these studies it has been
demonstrated that the inclusion of in-plane stressing can have a significant effect upon
the natural frequencies of light thin plates where the restraining forces and moments
due to the in-plane stressing becomes comparable, if not in excess of, the retraining
forces and moments due to the inherent flexural rigidity of the plate. Much of the
same body of work has shown that although the associated mode shapes are altered by
the addition of the in-plane stressing, as compared to the in-plane stress free plate, the
change is not so pronounced as the changes in the natural frequencies. However, these
significant changes in natural frequencies, and less significant changes in mode
shapes will no doubt result in significant changes in vibratory response to general
dynamic loading of the plate as compared to the plate in a non pre-stressed state.
1
Corresponding author. Department of Mechanical Engineering, University of
Strathclyde, Glasgow G1 1XJ, Scotland, Daniel.gorman@strath.ac.uk
1
Furthermore, often plates are in contact with enclosed acoustic cavities, obvious
examples being musical percussion instruments and pressure vessel bursting discs.
Frequency-modal characteristics for incompressible fluid in a rigid cylindrical
container covered by a flexible circular membrane have been studied [2] and for
interaction of liquid in a rigid cylindrical tank with a circular flexible bottom plate [3].
Coupled plate–liquid natural vibrations for a simply supported rectangular plate
carrying liquid with reservoir conditions at its edges were studied in [4] and
interaction of a rectangular flexible panel with an acoustic cavity was experimentally
investigated in [5]. Recently, strong coupling between a clamped elastic rectangular
plate and a quadrilateral (parallelepipedic) water–filled rigid cavity was
experimentally studied in [6]. Vibroacoustic couplings and frequency modal
characteristics of a rigid rectangular fluid-filled cavity with a flexible plate on one of
its faces were theoretically studied [7,8]. Acoustic–structural couplings for an elastic
plate in interaction with a cylindrical fluid–filled cavity was investigated [9,10], and
similarly for a circular prestressed membrane [11,12].
In this paper we consider the free undamped vibration of a thin circular plate
subjected to in-plane pre-stressing and in contact with a cylindrical acoustic cavity .
Accordingly, an analytical/numerical treatise, based upon a combination of the EulerBernoulli and Helmholtz equations and the Ritz-Galerkin technique, of the system is
performed and focuses upon the free vibration of the structure and how this is affected
by gas coupling and stressing which can be due to pressure acting on, and/or
temperature of, the structure. The analysis is confined to the modal parameters of
natural frequencies and associated mode shapes.
2. THEORETICAL ANALYSIS
w(r, , t)
h
The equation of motion, describing the free small lateral vibration, w = w (r,  ,t), of
a circular disc subjected to constant in-plane load intensity, N, and in interaction with
the acoustic cavity, as shown in Figure 1, is
 2a
L
p(x, r, , t)
x r
2
Figure 1 – Schematic diagram
 d h a4  2w p a3
Na 2 2
,
 w
 w

D
D
D
t 2
4
(1)
where
 2 1 
1  
 , w  w / a , r = r/a and D  E h 3 / 12(1   2 ) ;
   2 
 2
2 
r r r  
 r
E is Young’s modulus,  is Poisson number and d is the plate density; a and h are the
radius and thickness of the plate, respectively; L is the length of the cylindrical cavity
and p is the pressure inside.
2
Now writing
w


w
m
,
(2)
m 0
where
wm

Wm ( r ) 
and
Wm (r ) cos m  e it


s 1
q ms ms (r ) ,
(3)
where  ms (r ) is the natural mode shape of the disc in vacuo and q ms is a constant for
that mode, generally referred to as the mode shape coefficient for the mode consisting
of m nodal diameters and s nodal circles. In this particular case, for a stressed disc
clamped at the periphery, the mode shapes,  ms (r ) , are according to [1]:
 ms (r ) 
 I m ( ms )
J m ( ms )
J m ( ms r )  I m ( ms r ) ,
(4)
where  ms and  ms are roots (values of s = 1, 2, 3 etc.) computed from the equation:
0   ms
J m1  ms 
I  
  ms m1 ms
J m  ms 
I m  ms 
(5)
N a2
2
,
(6)
  ms
D
Im and Jm are the Bessel functions.
For particular values of m and s, the natural frequency of free undamped vibration,
 ms , is then:
and
 ms
2
 ms

  ms  ms
D
.
d h a4
In the case where the plate is not pre-stressed, i.e., N = 0, then  ms   ms and
equations (4), (5) and (7) are altered accordingly. Now for a particular mode of
vibration for the disc in vacuo:
2
 ms
d h a 4
Na 2 2
 4  ms (r ) cos m  
  ms (r ) cos m  
 ms (r ) cos m .
D
D
Therefore combination of equations (1), (3) and (8) gives:
(7)
(8)
3
  




p
. ..
(9)
 d ha
m 0 s 1
We shall now establish the form of the acoustic pressure, p, acting on the disc by
2
ms
  2 q ms  ms (r ) cos m e it

reference to the acoustic cavity. Consider the acoustic cavity shown in Figure 1,
whose velocity potential,  = (x , r ,  , t) is described by
 2 1 
1  2  a   2


 
r 2 r r r 2  2  L  x 2
2
a  
  
,
2
 c  t
2
where 
2
(10)
  / ac, x  x / L and c is speed of sound. Now writing




m 0
m ,
 m  H m ( x ) . Qm (r ) cos m e it
where
and substituting equation (11) into (10) gives (for a set value of m)
2
 Q"m 1 Qm m 2

 a  H "m
 

 2  2    k 2 ,
 
 L  Hm
 Qm r Qm r

a
where  
c
(11)
(12)
and k is a constant. For the right hand side of equation (12) equal to
–k2 we have
~ Y ( r ) ,
Qm (r )  Bm J m ( r )  B
m m
where   2  k 2
~
or k  2   2 , B  0 since Qm (r ) must be finite
when r  0 . At r  1 for each value of m
 m
r
dQm
dr

 0. ,
(13)
Therefore for a set value of m, the condition (13) has roots mq (q = 1, 2, 3 etc.),
which satisfy the equation J m ( )  0 .
( )
Similarly H m  C cos ( mq
x)
( )
 mq

where
L
a
2
2   mq
since

dH m
0 ,
dx x 0
L
( )
k mq
.
a
Therefore equation (11), for a set value of m, becomes:
m



q 1
( )
Bmq cos( mq
x ) J m ( mq r ) cos m e it .
(14)
At x  1 , the axial component of the velocity of the gas and the lateral velocity of the
plate must be equal, i.e.,
4
c  m

L x x 1
wm
t
for 0  r  1 .
Therefore from equations (2), (3) and (14) for a set value of m we have

c 
( )
( )
  Bmq  mq
sin ( mq
) J m ( mq r )  i  q ms  ms (r ) .
L q 1
s 1


(15)
Multiplying both sides of equation (15) by r J m  mq r  and integrating between
0  r  1 according to [13] gives

Bmq

 2i L

c
s 1

( )
mq
sin( 
( )
mq
q ms K msq

m 2  2

) 1  2 J m  mq 
  
mq 

,
(16)
where K msq   r  ms ( r ) J m  mq r  dr
1
(17)
0
the value of which can be obtained through standard numerical integration.
Now the pressure, p, at the surface of the plate is given by:
p    f ac

t
,
x 1
where  f is the fluid density.
Therefore combining equations (14) and (16) we have:
 
q ms K msq J m  mq r 
p  2 2 aL  f 
cos m e it .
( )
( )
2
2
2



tan

1

m
/

J

s 1 q 1
mq
mq
mq
m
mq



(18)
Substituting equation (18) into equation (9) gives:
 

s 1
2
ms

  2 q ms  ms (r )
 2 2
f L
d h

q ms K msq J m  mq r 


s 1 q 1

( )
mq
( )
tan  mq

2 

1  m  J m2  mq 
 2 
mq 

.
Multiplying both sides by r J m  mq r  and integrating between 0  r  1 we have:


s 1
where 



 2
q ms K msq  ms
  2 1  ( )
( )
 mq tan  mq







  0 ,
 


(19)
f L
mass of gas

.
d h
mass of plate
5
Now, since
2
 ms
2
2
  ms
 ms
D
 d ha 4
we can introduce a quantity  instead of  by the relation:
D
2   4
.
 d ha 4
Hence equation (19) can be re-written as




 2 2
4
q
K

ms
msq  ms  ms   1 
( )
( )
 mq tan  mq
s 1




( )

where  mq
(20)


 = 0 ,
 


(21)
L
D
2
4
  mq
.
2 2
a
 d ha c
Equation (21) can be represented in matrix form as
 a 11   a 12    a 1n     q m1   0 
 a   a    a    q   0 
22
2n
 21
  m2   

   




   ,
a qs  
   q ms   0 
 

   




   
 a n1   a n2    a nn    q mn  00
(22)
where



2
2
 ms
  4 1  ( )
aqs() = K msq  ms
( )
 mq tan  mq



 
 .
 

(23)
Hence values of  can be obtained (iterated upon) which renders the determinant of
matrix (22) equal to zero. Consequently for each of these values (roots) of  we can
then obtain the corresponding values of mode shape coefficients q m1, q m2, ………
q mn., normalised to q m1.
3. RESULTS AND DISCUSSION
In this study, since in all cases we are dealing with some degree of structural/fluid
vibration interaction, it would be erroneous to describe any mode of vibration as
either purely a structural mode or an acoustic (fluid) mode. Rather we will refer to the
modes as either structural/acoustic (st/ac) to denote modes which are predominantly
structural with acoustic interference and likewise acoustic/structural (ac/st) to denote
modes which are predominantly acoustic but with structural interference.
Also we shall define the parameter  , as reported in reference [1], as


Na 2
14.68D
6
for a circular plate clamped around its boundaries. The significance of  is that it is a
ratio of the lateral restraining force of the plate due to the in-plane stressing to that
due to the flexural rigidity and for values of    1 the plate will be in a buckled
state. Therefore we shall consider the case where  is greater than -1.
3.1 Convergence
As explained earlier, the roots  (from which the natural frequencies of the coupled
system can be obtained) and the mode shape coefficients q ms are obtained by iterating
values of  which renders the determinant of the matrix equation (22) equal to zero.
The determinant of this matrix equation is obtained by performing the LU
decomposition [14], whereupon the value of the determinant is the product of the
diagonal terms. Subsequently these root values of  which render the determinat zero
are substituted back into equation (22) to obtain the corresponding values of the mode
shape coefficients, q ms, (normalised to q m1, ) which describe which structural modes
are present and dominate. Of immediate interest therefore is the convergence of the
solution with respect to size of the square dimensions of the [A] matrix selected, i.e.,
the solutions obtained from the first n rows and columns of the matrix. For this
convergence analysis, the following parameters were used:
radius of cylinder (a) = 38 mm
plate thickness (h) = 0.38 mm
length of cylinder (L) = 255 mm
density of air (f )= 1.2 kg/m3
density of plate (d) = 7800 kg/m3
Poisson ratio () = 0.3
Young’s modulus (E) = 2.1x 1011 Pa
speed of sound in air (c ) = 343 m/s.
These parameters were selected upon the basis that in the absence of any inplane
stressing, the natural frequency of the first axisymetric mode (m = 0, s =1) of
vibration of the plate in vacuo corresponds closely to the natural frequency of the first
axisymetric mode (m =0, q = 1) of vibration of the acoustic cavity if the disc was
( )
assumed rigid and  mq
contained in equation (14) is π. In other words we will have
strong structural/acoustic coupling between these two modes around this common
frequency. Accordingly, prior to listing the table which demonstates the convergence
of the technique, Table 1 lists (a) the natural frequencies of axisymetric modes of the
disc if it were in vacuo, and, (b) natural frequencies of the axisymetric modes of the
acoustic cavity if the disc was to be rigid.
7
m
0
0
0
ω [Hz]
671.83
2615.5
5859.8
s
1
2
3
ξ2 [eq. (20)]
10.216
39.771
89.104
(a)
m q
ω [Hz]
( )
[eq. (14)]
 mq
0
0
0
0
0
0
0
0
0
672.55
1345.1
2017.6
2891.4
3117.2
3461.0
3864.1
4035.8
4306.9
π
2π
3π
π
2π
3π
π
2π
3π
1
1
1
2
2
2
3
3
3
(b)
Table 1 Calculated natural frequencies: a) for the circular plate in vacuo, b) for the
acoustic cavity if the disc was rigid.
Table 2 lists natural frequencies and corresponding modal coefficients, qms , for values
of n = 2, 4, 6 and 8 with m = 0 (axisymetric mode) and in-plane load intensity N = 0.
Table 2 demonstrates the convergence of the technique (with respect to n x n) for
computing the natural frequencies and corresponding mode shape coefficients, q m1,
q m2, … q mn., normalised to q m1 for the axisymetric modes (m = 0) of the coupled
system .
8
n=2
636.33
n=4
636.99
(1, 6.44x10-4)
(1, 2.58x10-4,
2.17x10-6)
708.19
707.59
(1,-7.619x10-4)
(1, -3.074x10-4,
-2.713x10-6)
1347.1
1347.3
(1, -1.032x10-2,
-7.078x10-5)
2018.4
(1, -4.99x10-2,
-2.038x10-4)
2607.6
(1, -12.57,
-5.189x10-4)
(1,-2.526x10-2)
2017.5
(1,-0.122)
2601.1
(1,-13.312)
n=6
637
(1, 2.49x10-4,
1.79x10-6,--- 4.7x10-11)
707.55
(1, -2.97x10-4,
--- -3.53x10-11)
n=8
637
(1, 2.49x10-4,----1.4x10-14)
1347.3
(1, 1x10-2, ---- -1.09x10-9)
2018.4
(1, -4.84x10-2,
--- -3.02x10-9)
2607.8
(1, -12.58,----- -6.8x10-9)
1347.3
(1, 9.9x10-3,
--- -1x10-12)
2018.4
(1, -4.8x10-2,
--- -2x10-12)
2607.8
(1, -12.58,
--- -4.4x10-12)
707.55
(1, -2.94x10-4,
--- -4x10-14)
Comments
1st st/ac
(s=1, q=1),
strong coupling
with 1st ac/st
1st ac/st
(q=1, s=1),
strong coupling
with 1st st/ac
2nd ac/st
(q=1, s=1),
weak coupling
3rd ac/st
(q=1, s=1)
2nd st/ac
(s=2, q=1)
Table 2 Convergence of the natural frequencies  [Hz] and the mode shapes
coefficients (q01 , q02 , q03 , ... q0n )
3.2 Comparison with results obtained from ANSYS2
In the construction of the finite element model, the same physical parameters of the
disc were selected as that for the convergence test in 3.1 above. However in this case
the length of acoustic cavity, L, was set as 350mm.
The three-dimensional model uses 6000 elements (type FLUID30) for the fluid in the
cylinder and 300 elements (type SHELL63) for the plate. The cylinder walls were
assumed rigid and the plate was fully fixed at the edges. The plate and fluid elements
that are in contact are coupled for fluid-structure interaction. The plate is first prestressed by heating (or cooling) followed by a modal analysis of the combined system.
The Lanczos unsymmetric eigensolver method is used for the mode extraction during
the solution process. It is worth noting that, for prestressing to work correctly in
ANSYS, it is necessary to select all the elements of the model (not just the plate itself)
during the prestressing phase of the solution.
For the particular plate/acoustic cavity configuration the natural frequencies were
computed by iteration of the determinant of the first three rows and columns of matrix
ANSYS User’s Manual for Revision 5.4, Swanson Analysis Systems, Inc,
Houston, 1997
2
9
equation (22) for a value of m = 0 (axisymmetric modes only). Corresponding values
of natural frequencies were obtained from the ANSYS analysis described above.
Tables 3a and 3b shows these values of natural frequencies. Also included in Tables
3a and 3b are the corresponding values of natural frequency associated with the plate
in vacuo, i.e., in the absence of any acoustic coupling effects, and, the acoustic cavity
alone if the plate was treated as a rigid boundary. From Table 3a (  =0) one can see
that the modes of natural frequencies for the coupled system resemble those for the
plate in vacuo and the acoustic cavity alone, in other words the system is fairly
uncoupled. However for the case where  = - 0.12752 (Table 3b) the fundamental
natural frequency of the plate in vacuo approaches a value close to the acoustic
natural frequency giving rise to strong interaction in the values for the coupled
system.
1
2
486 (1st ac/st)
674 (1st st/ac)
983 (2nd ac/st)
1471 (3rd ac/st)
1961 (4th ac/st)
2449 (5th ac/st)
486
674
984
1474
1967
(a)
3
4
490
671.8
980
1470
1960
2450
2616
 =0
1
2
3
4
st
429
505
983
1474
1967
442.3
490
2371
980
1470
1960
2450
428 (1 ac/st)
505 (1st st/ac)
982 (2nd ac/st)
1471 (3rd ac/st)
1960 (4th ac/st)
2365 (2nd st/ac)
(b)  = -0.12752
Table 3 Calculated natural frequencies: 1 is solution from eq. (22); 2 is ANSYS
solution; 3 is for plate in vacuo; 4 is for rigid closed acoustic cavity. All
frequencies are in Hertz.
3.3 General Results
As above we shall only consider results for the axisymmetric modes of vibration, i.e.,
m = 0. At this stage we introduce a coupling factor, F, defined as
F   do /  go ,
where  do is the fundamental natural frequency of the unstressed plate in vacuo (for
a particular value of m, zero in this case) and  go is the first and fundamental natural
frequency of the gas chamber (for the same value of m) when the plate is assumed
rigid.
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Figure 2 shows plots of non-dimensional natural frequencies  2 , calculated from
equation (22), against  for values of F equal to 0.5, 1 and 1.5. In all of the plots we
have highlighted (encircled) the cases where there exists strong acoustic-structural
coupling, illustrated as “veering”, or, avoidance. In these cases the coupled fluidmechanical system has two different natural frequencies for the same mode which are
neither purely acoustic nor purely structural, and the dynamic characteristics of this
system can not be studied separately for the structural and/or acoustic subsystems. Out
with these veering regions the modes can safely be described as either predominantly
structural or acoustic.
It is seen from the three plots that the introduction of in-plane stressing can result in
the plate moving into or away from zones of strong and weak structural/fluid vibration
interaction. For example, when F = 0.5 (Figure 2a) meaning that in the unstressed
state the plate fundamental natural frequency is half that of the fundamental natural
frequency of the gas column, we observe the following;
(a) as the stressing increased positively the natural frequency of the first
structural/acoustic mode moves gradually to a point where there would be
strong coupling with the first acoustic/structural mode, and
(b) by setting the first structural unstressed natural frequency to half of the first
acoustic natural frequency (F = 0.5) it is seen that in the absence of stressing,
the second structural/acoustic mode natural frequency is close to the second
acoustic/structural mode frequency, Furthermore strong structural/acoustic
interaction will manifest itself at moderately low levels of in-plane stressing.
4.
CONCLUSIONS
A theoretical - analytical method, based on the Galerkin method, has been developed
for the frequency-modal analysis of a coupled vibroacoustic system. The
convergence of the solution is fast and the numerical results are in good agreement
with the finite element analysis performed by the ANSYS code. It was shown that
strong structural-acoustic couplings can appear in the system, for example, by
changing the static in-plane prestress of the disc backed on the top of an acoustic
cylindrical cavity. In the region of parameters, where a strong acoustic -structural
coupling exists, the differences in the spectral characteristics between the associated
coupled and uncoupled systems can be substantial.
5.
REFERENCES
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This research was supported by the Royal Society (London) and the Grant Agency of
the Academy of Sciences of the Czech Republic by the project A2076101/01 Natural
vibration and stability of shells in interaction with fluid.
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