World Journal of Mechanics, 2012, 2, 361-368
doi:10.4236/wjm.2012.26041 Published Online December 2012 (http://www.SciRP.org/journal/wjm)
Copyright © 2012 SciRes. WJM
Effects of Residual Stress on the Hydro-Elastic Vibration of
Circular Diaphragm
Junfeng Zhao, Song Yu
Department of Engineering Mechanics, Shandong University, Jinan, China
Email: zhaojunfeng@sdu.edu.cn
Received October 12, 2012; revised November 11, 2012; accepted November 21, 2012
ABSTRACT
The effects of residual stress on the hydro-elastic vibration of circular diaphragm are theoretically investigated by using
the added mass approach. The Kirchhoff theory of plates is used to model the elastic thin circular diaphragm on an ap-
erture of an infinite rigid wall and in contact with a fluid on one side. The fluid is assumed to be incompressible and
inviscid and the velocity potential is used to describe its irrotational motion. A non-dimensional tension parameter is
defined, and the effects of the tension parameter on the frequency parameters and mode shapes of the diaphragm in the
air are presented. The Hankel transform is applied to solve the fluid-diaphragm coupled system; boundary conditions
are expressed by integral equations. Finally, the effects of residual stress on the non-dimensional added virtual mass
incremental (NAVMI) factors of the diaphragm contact with a fluid on one side are investigated. It is found that the
effects of the residual stress cannot be neglected when the edges of the circular diaphragm are clamped. The effects of
residual stress for NAVMI factors can be increases 11% when the non-dimensional tension parameter is 1000.
Keywords: Circular Diaphragm; Residual Stress; Hankel Transform; Frequency Parameter; Mode Shape; NAVMI
1. Introduction
Micro-machined diaphragm structures are extensively
used in micro-electromechanical systems (MEMS) as
biosensors in air and in liquid media [1-5]. These sensors
are excited by electro-static force so that they vibrate at
their resonance frequency. When a biological entity is
captured by an electrode of biosensor, the resonant fre-
quency will change. The capture mass can be measured
by detection of the resonant frequency shift. On the fab-
rications of these multilayer micro-diaphragms, a certain
amount of residual stress will be developed for a variety
of reasons [6-9]. The residual stress bend the micro-dia-
phragm downward or upward depending on whether the
residual stress is tensile or compressive. Moreover, the
residual stress is an important influencing factor for the
resonant frequency and sensitivity of the biosensors. Sev-
eral researchers investigated the effects of residual stress
on the diaphragm’s resonant frequency, either theoreti-
cally or by finite element analysis [10-12]. They con-
cluded that residual stress stiffens the diaphragm and
increases its resonant frequency. But, the vibration char-
acteristics of the diaphragm in fluid media don’t studied.
The fluid structure interaction problems of plate struc-
tures partially or totally immersed in fluid have also re-
ceived much attention due to their importance. Kwak [13]
and Amabili et al. [14] investigated the effect of fluid on
the natural frequencies of circular plates vibrating in
contact with an infinite liquid surface. Amabili and Kwak
[15] investigated the effect of free-surface waves on free
vibrations of circular plates resting on a free surface of
infinite liquid domain. Amabili et al. [16] and Liang et al.
[17] gave the natural frequencies of annular plates on an
aperture of an infinite rigid wall and in contact with a
fluid on one side. Bauer [18] presented the coupled hy-
dro-elastic frequencies of a liquid in a circular cylindrical
rigid container, of which the free liquid surface was fully
covered by a flexible membrane or an elastic circular
plate. Bauer and Chiba [19] extended the study of Bauer
[18] to the structure filled with incompressible viscous
liquid. Amabili [20] studied the free vibrations of circular
plates resting on a sloshing liquid free surface; the liquid
domain was limited by a rigid cylindrical surface and a
rigid flat bottom. When the biosensors are used in liquid
media, the effect of the fluid on the diaphragm must be
investigated [21,22]. However, the residual stress was
neglected in these studies.
The objective of the present paper is to investigate the
effects of residual stress on the hydro-elastic vibration of
circular diaphragm. The Kirchhoff theory of plates is
used to model the elastic thin circular diaphragm. The
governing equation of a circular diaphragm with the re-
sidual stress is obtained in Section 2. A non-dimensional
tension parameter is defined, and the effects of the ten-
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
362
sion parameter on the frequency parameters and mode
shapes of the diaphragm in the air are presented. The
fluid is assumed to be incompressible and inviscid and
the velocity potential is used to describe its irrotational
motion. The fluid formulation of the circular diaphragm
is obtained in Section 3. In Section 4, the effects of re-
sidual stress on the frequency parameters, the mode
shapes and the non-dimensional added virtual mass in-
cremental (NAVMI) factors of the diaphragm contact
with a fluid on one side are investigated. Finally, Section
5 gives concluding remarks.
2. Equation for Circular Diaphragm with
Residual Stress
Consider a thin circular diaphragm having thickness h,
mass density
D, radius a, initial tension of the dia-
phragm per unit of length T due to the residual stress,
which is vibrating in vacuum, as sketched in Figure 1.
The diaphragm is also assumed to be made of linearly
elastic, homogeneous and isotropic material. The effects
of shear deformation and rotary inertia are neglected. The
Kirchhoff theory of plates is used to model the elastic
thin circular diaphragm.
For a circular diaphragm with initial tension, the equa-
tion of motion for transverse displacement, w, of the
diaphragm is [23]
2
42
D20
w
DwTw h
t
 
(1)
where

32
12 1DEh
 is the flexural rigidity of the
diaphragm,
is the Poisson ratio of the material, E is the
Young’s modulus of the material.

01Th
is the
tension force of the diaphragm per unit of length due to
the residual stress,
0 is the residual stress. In addition
22
2
222
11
rr
rr

 

(2)
is the Laplace operator in the polar co-ordinates r and
.
For vibration analysis of the circular diaphragm in
vacuum, the transverse displacement w is assumed as
 
i
00
,,, emn t
mn
nm
wr tW r



 (3)
in which m and n are the numbers of nodal diameters and
circles. Substituting Equation (3) into Equation (1), we
Figure 1. Schematic of a circ ular diaphragm with the initial
tension.
obtain

42 2
D
22 220
mnmnmnmn
mnmn mn
DWTWh W
W



 (4)
in which
222
D
222
D
14
2
14
2
mn mn
mn mn
TDT
D
TDT
D




(5)
where
mn and
mn are the frequency parameters which
are determined by the boundary conditions, which satisfy
the following equation

2
22
mn mn
Ta
aa
D
 (6)
where
is a non-dimensional tension parameter which is
to determined whether the diaphragm is tension domi-
nated for the pure membrane behavior or flexural rigidity
dominated for the pure plate behavior. After substituting
Equation (6) into Equation (5), we obtain


42
22
D
42
22
D
4
1
2
4
1
2
mn
mn
mn
mn
a
aD
a
aD





(7)
The Equation (4) is satisfied when the solutions of the
following equations are satisfied

22 22
0, 0
mn mnmn mn
WW

 (8)
It is possible to separate variables by substituting

,cos
mn mn
WrRr m
(9)
Substituting Equation (9) into Equation (8), then we
obtain
22
2
22
dd
10
d
d
mn mn
mn mn
RR mR
rr
rr

 


(10)

22
2
22
dd
1i0
d
d
mn mn
mn
mn
RR mR
rr
rr

 


(11)
Equations (10) and (11) are Bessel’s equations of frac-
tional order. The solutions can be expressed as a series
form. The solution of Equation (10) is in terms of Bessel
functions of the first and second kind, Jm (
mnr) and Ym
(
mnr). The solution of Equation (11) is in terms of
modified Bessel functions of the first and second kind, Im
(
mnr) and Km (
mnr).
For the general solution of circular diaphragm, the so-
lution Rmn (r) is obtained
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
363

 
 
mnmnm mnmnm mn
mnm mnmnm mn
Rr AJrBYr
CIr DKr



 (12)
in which Amn, Bmn, Cmn and Dmn are the mode shape con-
stants that are determined by the boundary conditions.
Both Ym (
mnr) and Km (
mnr) are singular at r = 0. Thus,
for a diaphragm with no central hole, we set Bmn = Dmn =
0. The Equation (12) becomes

 
mnmnm mnmnm mn
Rr AJrCIr

 (13)
Substituting Equation (13) into Equation (9), we ob-
tain the mode shape as follows

 
,cos
mnmnm mnmnm mn
WrAJ rCIrm



(14)
To simplify the computation, the mode shape con-
stants Amn and Cmn are normalized so that

12
0d1
mn
R

(15)
where ra
is the ratio between the radius r and the
maximum radius a.
3. Fluid Formulation
We consider the free vibrations of the circular diaphragm
placed in an aperture of an infinite rigid wall and in con-
tact with a fluid on one side. The mode shapes of the
diaphragm vibrating in constant with a fluid are assumed
to be equal to those of the diaphragm vibrating in a vac-
uum. This hypothesis was also used to study vibrations
of circular plates in contact with fluids [13-17]. In fact,
although natural frequencies are considerably reduced by
the presence of a fluid, mode shapes undergo only small
changes. We study the velocity potential of an incom-
pressible and inviscid fluid in contact with a circular
diaphragm on one side. The fluid movement, considered
only due to the diaphragm’s vibration, is assumed to be
irrotational. This movement can be described by the ve-
locity potential
 

i
,,,Re,, iemnt
mn
rzt rz

 (16)
where
is the spatial distribution of the velocity poten-
tial, mn
is the circular frequency of the diaphragm in
contact with fluid. As a consequence of the hypotheses,
(r,
, z) satisfies the Laplace equation
222
2
2222
11 0in
F
rr
rrz
 
 
 

(17)
where F represents the unbounded fluid domain. The
fluid is in contact with the diaphragm through the surface
denoted by S1 and in contact with an infinite rigid wall
through the surface denoted by S2, as sketches Figure 2.
Due to the fluid domain is unbounded, the surface S
exists at infinity.
Figure 2. The circular diaphragm and the fluid domain.
As assumed in the hypotheses, the diaphragm mode
shapes in a vacuum and in fluids are unchanged, there-
fore, the displacement w obtained from Equation (3) is
imposed at the fluid-plate interface. The contact between
the fluid and diaphragm is assured by the equation [13,14]
 
1
cos on
zmn
vRrmS
z
 
(18)
The condition of an impermeable rigid wall on S2 is
2
0onS
z
(19)
Equations (18) and (19) give a Neumann problem.
Moreover, there is the radiation condition that the spatial
velocity potential
and the velocity of the fluid go to
zero when the distance from the diaphragm becomes
very large [13,14]


,, ,,
,, ,,0
for ,on
rz rz
rz rz
rz S
 




(20)
Equation (17) can be simplified by separation of vari-
ables

,,, cosrzrzm

 (21)
where is the solution of
222
222
10in
m
F
rr
rzr
 

 (22)
The boundary conditions for are directly obtained
from Equations (18)-(20) as

1
0
,on
mn
z
rz Rr S
z

(23)
2
0
0on
z
S
z

(24)


,,
,, ,0
for ,on
rz rz
rz rz
rz S
 


(25)
Using the Hankel transformation [24], we obtain the
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
364
relation
 
0
,,d
m
zrrzJrr


(26)
By integrating by parts, we can derive the relation

 
22
22
0
22
0
1d
,d ,
m
m
m
rJrr
rr
rr
rrzJ rrz


 



 
(27)
At the same time, we have for the third term of Equa-
tion (22)
 
22
22
0
d
d,
d
m
rJrrz
zz


(28)
Using Equations (27) and (28) reduces the partial dif-
ferential Equation (22) to the ordinary differential equation
 
2
2
2
d,,0
dzz
z


(29)
Boundary condition Equation (25) requires that the
solution of Equation (29) consists of the attenuating part
only
 
,e0
z
zBa z

  (30)
where B (a
) is a function that must be determined. From
the inversion formula for the Hankel transformation
along with Equation (30), we obtain
 
 
0
0
,,d
ed
m
z
m
rzz Jr
BaJ r



(31)
Inserting Equation (31) into the boundary condition
Equations (23) and (24) give
 
2
0d,0
mmn
BaJrRrr a

 
(32)

2
0d0,
m
BaJrr a


(33)
Introducing the non-dimensional variables as
 
,,
raA B
a
 
  (34)
The integral Equations (32) and (33) can be written in
the following standard form
 
3
0d,01
mmn
AJ aR
 
 
(35)
 
0d0, 1
m
AJ
 

(36)
where

 
mnmnm mnmnm mn
RAJaCIa
 
 (37)
The solution of the integral Equations (35) and (36),
can be obtained by using the properties of the Hankel
transform [24]. In fact, since the Equation (36) is equal to
zero, we obtain
 
1
33
0d
mn mmn
AaRJ aH
 
 
(38)
in which [25]

mnmnAmnmn Cmn
HAH CH

 (39)




11
22 2
Amn
mn mmnmmmnm
mn
H
aJaJJa J
a



(40)




11
22 2
Cmn
mnmmnmmmnm
mn
H
aIaJIa J
a



(41)
Inserting Equation (38) into Equation (31) and using
Equation (34), we obtain the function at the fluid-plate
interface
 
0
,0 d
mn m
aH J


(42)
Using the Rayleigh quotient for the coupled vibrations,
the squares of the natural frequencies of the diaphragm in
a vacuum are proportional to the ratio between the
maximum potential energy of the diaphragm VD and its
kinetic energy TD [26]. Due to the mode shapes of the
diaphragm vibration in constant with a fluid are assumed
to be equal to those of the diaphragm vibration in a vac-
uum, the squares of the natural frequencies in fluid are
proportional to the ratio between the maximum potential
energy of the diaphragm VD and the sum of the kinetic
energies of both the diaphragm TD and the fluid TF.
Therefore, we can obtain
2
vacuum
D
V
D
V
fT


 (43)
2
fluid
D
F
DF
V
fTT



(44)
Since the mode shapes in a vacuum and in fluids are
assumed to be the same, the maximum potential energies
VD and the kinetic energy TD are not changed when
evaluated in vacuum or in fluids. By using Equations (43)
and (44), the following relation between natural frequen-
cies in a vacuum fV and natural frequencies in fluids fF is
obtained
1
V
F
mn
f
f
(45)
where
mn is the added virtual mass incremental (AVMI)
factor [13,14]. This factor is given by the ratio between
the kinetic energy of the fluids and the kinetic energy of
the diaphragm
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
365
F
mn
D
T
T
(46)
The AVMI factor can be made dimensionless
F
mn mn
D
a
h
 (47)
where
F and
D are the fluid’s and diaphragm’s mass
density, respectively, and mn is the non-dimensionalized
added virtual mass incremental (NAVMI) factor [15-17].
Therefore, natural frequencies of a circular diaphragm
vibrating in contact with a fluid are related to the modal
properties of vibration in a vacuum.
By using the hypothesis of irrotational movement of
the fluid and simple connection of the fluid domain, the
kinetic energy of the fluid can be evaluated from its
boundary motion [13-17]. In fact, as a consequence of
Green’s Theorem applied to the harmonic function
, we
obtain

 
2π2
00
,0 ,0cosd d
2
F
F
r
Trmrr
z

 
 (48)
The function z
 is always zero on the boundary
of the fluid domain, except for the surface in contact with
the plate, therefore Equation (48) reduces to
 
 
0
1
2
0
,0 d
2
,0 d
2
a
F
Fθmn
F
θmn
TrRrrr
aR

 
 
(49)
where
= 2 for m = 0,
= for m > 0. Upon substi-
tuting Equation (42) into Equation (49) and using Equa-
tion (38), the result is found to be

32
0d
2
F
Fθmn
TaH

(50)
The kinetic energy of the circular diaphragm is
 
2π22
00 cosdd
2
a
D
Dmn
ThRrmrr
 (51)
Integration with respect to
, and using the normaliza-
tion criterion Equation (15), then we obtain


2
0
1
22 2
θ
0
d
2
d
22
a
D
Dθmn
DD
θmn
ThRrrr
ha Rha

 

(52)
By using Equations (46), (47), (50) and (52), the
NAVMI factors are found to be given by

2
0d
FD
mn mn
DF
Th H
Ta
 
(53)
This quantity must be evaluated numerically, because
the integral cannot be expressed in terms of elementary
functions. However, upon inserting the modal parameters
mn,
mn, Amn and Cmn, the NAVMI factors can be quickly
computed.
4. Numerical Results and Discussion
4.1. Numerical Results
To illustrate the effects of the residual stress, the fre-
quency parameters,
mna, the mode shapes and the non-
dimensional added virtual mass incremental (NAVMI)
factors are been studied for the circular diaphragm with
the clamped edge. For the circular diaphragm with clamped
edge, the boundary conditions are
0,0 on
mn
mn
W
Wra
r

(54)
Using Equation (14) and Equation (54), we obtain

11
0
0
mnmmnmn mmn
mnmnmmnmnmnmmn
AJaCIa
AJ aCI a

 



(55)
Due to Equation (55) has non-zero solution, the coef-
ficient determinant of Equation (55) must be zero, which
gives

 
11
0
mmn mmn
mnm mnmnm mn
Ja Ia
JaIa



(56)
The characteristic equation of the circular diaphragm
with clamped edge can be obtained
 

1
10
mn mmnmmn
mn mmnmmn
aJaIa
aIaJa


(57)
Using Equations (6) and Equation (57), the accurate
frequency parameters,
mna, of the circular diaphragm
with clamped edge can be determined.
To find the mode shapes, we formulate from Equation
(55)


mmn
mn mn
mmn
J
a
CA
I
a
 (58)
This gives the mode shapes expression




cos
mmn
mnmn m mnm mn
mmn
Ja
WAJ rIrm
Ia






(59)
Setting Equation (59) equals to zero defines the node
lines. It turns out that there will be concentric circles and
diametrical lines. The number of concentric circles will
be n and the number of diametrical lines will be m.
For illustration purpose, the circular diaphragm con-
sidered here is taken to be made of silicon with the fol-
lowing material properties: E = 170 GPa,
= 0.3. The
frequency parameters, (
mna), are available for circular
plates which were given by Vogel and Skinner [27] and
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
366
for circular membranes which were given by E. C. Wente
[28]. We have computed the frequency parameters and
mode shapes in order to provide more accurate results.
The frequency parameters can be obtained by the Equa-
tion (57) and the mode shapes can be obtained by Equa-
tion (59). The roots of these equations were found by
using the software Fortran95.
4.2. Discussion of Results
In Table 1, the frequency parameters, (
mna), for the
circular diaphragm with the clamped edge are reported
for the different non-dimensional tension parameter (
).
When the non-dimensional tension parameter is zero, the
diaphragm model is reduced to the plate [27]. It is found
the results are the same with the circular plates when the
tension parameter is zero. When the non-dimensional
tension parameter is infinite, the diaphragm model is
reduced to the membrane [28]. The frequency parameters
are the same with the circular membranes. It indicates
that the present model is valid for the diaphragm with the
clamped edge.
In Figure 3, the frequency parameters of the four dif-
ferent models for the circular diaphragms with clamped
edges are investigated. It is clearly seen that the frequen-
cies parameters decreases rapidly with the increasing
value of the non-dimensional tension parameter. When
the tension parameter is very small, the frequency pa-
rameters close to the results of the plates.
Table 1. Comparison of frequency parameters, (
mna), for
circular diaphragms with clamped edge,
= 0.3.
Present work
plates diaphragm
m n
Plates
Vogel et al.
[27]
= 0
= 102
= 104
Membrane
E. C.Wente [28]
0 0 3.19622 3.19622 2.65624 2.42910 2.40483
0 1 6.30644 6.30644 5.99518 5.57569 5.52008
1 0 4.61090 4.61090 4.20075 3.87036 3.83171
1 1 7.79927 7.79927 7.55770 7.08615 7.01559
Figure 3. Frequency parameters of the clamped circular
diaphragm as f unction of the t ension par ameter . m, n values:
, 0, 0; , 1, 0; , 0, 1; , 1, 1.
When the tension parameter is very larger, the fre-
quency parameters close to the results of the membranes.
Therefore, the vibration of a diaphragm structure depends
on whether the structure behaves as a tension dominated
membrane or a flexural rigidity dominated plate.
In Figure 4, the first mode shapes of the circular dia-
phragm with clamped edges for the different non-di-
mensional tension parameters are investigated. Mode
shapes with zero nodal diameters (m = 0), zero nodal
circles (n = 0) and
= 0.3 are assumed. It is clearly ob-
served from Figure 4 that the normalized maximum am-
plitude of the first mode shape by the present model is
about 3.31 when the non-dimensional tension parameter
is equal to 0. However, the normalized maximum ampli-
tude of the first mode shape by the present model is
about 2.75 when the non-dimensional tension parameter
is equal to 10,000. The normalized maximum amplitude
by the present model when the non-dimensional tension
parameter is equal to 10,000 decreases approximately 17
percent than that predicted by the plate model.
In Figure 5, the second mode shapes of the circular
diaphragms with clamped edges for the different non-
dimensional tension parameters are investigated. It is
clearly observed from Figure 5 that the normalized
Figure 4. Mode shapes comparison of circular diaphragm
for different tension parameter.
Figure 5. A comparison of mode shapes of circular dia-
phragm for different tens ion parameters.
J. F. ZHAO, S. YU
Copyright © 2012 SciRes. WJM
367
maximum amplitude of the second mode shape by the
present model is about 4.43 when the non-dimensional
tension parameter is equal to 0. However, the normalized
maximum amplitude of the second mode shape by the
present model is about 4.20 when the non-dimensional
tension parameter is equal to 10000. The normalized
maximum amplitude by the present model when the non-
dimensional tension parameter is equal to 10000 de-
creases approximately 5.2 percent than that predicted by
the plate model. Therefore, it is insufficient that the fre-
quency parameters and the mode shapes of the circular
diaphragm with the clamped edges are predicted when
the tension due to the residual stress is neglected.
The NAVMI factors, mn, are obtained by numerical
integration of Equation (53). These factors, for clamped
circular diaphragms having the non-dimensional tension
parameters
equal to 0, 10, 100, 1000 and 10000, are
given in Table 2 with six frequency parameters for m
and n. When the non-dimensional tension parameter goes
to zero, the diaphragm with the clamped edge becomes a
clamped plate. Table 2 is shown to give a comparison
between numerical results obtained by the present model
and the results of the Amabili [16] for circular plates.
Differences are always less than 1%; the error can be
attributed to the accuracy of the calculation. It indicates
that the present model is valid. Therefore, for the differ-
ent non-dimensional tension parameters, the NAVMI
factors, mn, are different. It is clearly observed from
Table 2 that the NAVMI factors are increase for the fre-
quency parameter n = 0, however, the NAVNI factors are
decrease for the frequency parameter n > 0 when the
non-dimensional tension parameters is smaller than 1000.
The NAVMI factors of Table 2 are presented in Figure
6 in a graph as a function of the non-dimensional tension
parameter.
It is clearly observed from Figure 6 that not all modes
have a monotonic behavior with the non-dimensional
tension parameter; moreover, the fundamental mode, (m
= 0, n = 0), shows the largest change with the non-di-
mensional tension parameter. The NAVMI factor for the
fundamental mode increases approximately 11% when
the non-dimensional tension parameter is equal to 1000.
Table 2. Comparison of NAVMI factors, mn, for circular
diaphragms with clampe d e dge,
= 0.3.
Present work
plates diaphragms
m n
Amabili [18]
plates
= 0
= 100
= 102
= 104
0 0 0.65381 0.65394 0.65493 0.69337 0.73936
0 1 0.27613 0.27617 0.27553 0.25751 0.25934
0 2 0.16513 0.16515 0.16496 0.15578 0.15097
1 0 0.29883 0.29883 0.29903 0.30914 0.32696
1 1 0.16914 0.16913 0.16903 0.16566 0.16872
1 2 0.11591 0.11591 0.11586 0.11337 0.11271
Figure 6. NAVMI factors of the clamped circular dia-
phragm as function of the tension parameter. m, n values:
, 0, 0; , 1, 0; , 0, 1; , 1, 1.
However, the NAVMI factor for the mode with m = 0
and n = 1 decrease approximately 7.3% when the non-
dimensional tension parameter is equal to 1000. There-
fore, the residual stress must be considered when we
calculate the NAVMI factors of the clamped circular
diaphragms.
5. Conclusion
The effects of residual stress on the hydro-elastic vibra-
tion of circular diaphragm are theoretically investigated
by using the added mass approach. The Kirchhoff theory
of plates is used to model the elastic thin circular dia-
phragm on an aperture of an infinite rigid wall and in
contact with a fluid on one side. The fluid is assumed to
be incompressible and inviscid and the velocity potential
is be used to describe its irrotational motion. It is as-
sumed that the mode shapes are not changed by the fluid.
The Hankel transform is applied to solve the fluid-dia-
phragm coupled system; boundary conditions are ex-
pressed by integral equations. Finally, the effects of re-
sidual stress on the non-dimensional added virtual mass
incremental (NAVMI) factors of the diaphragm contact
with a fluid on one side are investigated. It is found that
the effects of the residual stress cannot be neglected
when the boundary conditions of the circular diaphragm
is clamped. The effects of residual stress for NAVMI
factors can be increased 11% when the non-dimensional
tension parameter is 1000. The importance of these fac-
tors is that they are dimensionless, so that they can be
applied to circular diaphragm with different dimensions
and material, therefore the numerical data given in pre-
sent paper can be quickly used for the design of mi-
cro-diaphragm in MEMS.
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