Effects of Residual Stress on the Hydro-Elastic Vibration of Circular Diaphragm

doi:10.4236/wjm.2012.26041

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World Journal of Mechanics, 2012, 2, 361-368

doi:10.4236/wjm.2012.26041 Published Online December 2012 (http://www.SciRP.org/journal/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: zhaojunf[email protected]

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

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]

D20

DwTw h





 

 (1)

where



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

222





 





(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

 

,,, emn t

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

22 220

mnmnmnmn

mnmn mn

DWTWh W







 (4)

in which

222

mn mn

TDT

































(5)

where



mn and



mn are the frequency parameters which

are determined by the boundary conditions, which satisfy

the following equation



mn mn









 (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











































(7)

The Equation (4) is satisfied when the solutions of the

following equations are satisfied





22 22

0, 0

mn mnmn mn





 (8)

It is possible to separate variables by substituting





,cos

mn mn

WrRr m



 (9)

Substituting Equation (9) into Equation (8), then we

obtain

mn mn

RR mR







 





(10)



1i0

mn mn

RR mR







 





(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

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



0d1





 (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

 



,,,Re,, iemnt

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

2222

11 0in

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]

 

cos on

zmn

vRrmS





 

 (18)

The condition of an impermeable rigid wall on S2 is

0onS





 (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 S

 













(20)

Equation (17) can be simplified by separation of vari-

ables





,,, cosrzrzm





 (21)

where  is the solution of

222

10in

rzr

 





 (22)

The boundary conditions for  are directly obtained

from Equations (18)-(20) as





,on

rz Rr S

z

 

 (23)

0on

z

 

 (24)







,, ,0

for ,on

rz rz

rz S



 









(25)

Using the Hankel transformation [24], we obtain the

J. F. ZHAO, S. YU

364

relation

 

,,d

zrrzJrr







 (26)

By integrating by parts, we can derive the relation



 

,d ,

rJrr

rrzJ rrz









 









 



(27)

At the same time, we have for the third term of Equa-

tion (22)

 

rJrrz







 (28)

Using Equations (27) and (28) reduces the partial dif-

ferential Equation (22) to the ordinary differential equation

 

d,,0

dzz





(29)

Boundary condition Equation (25) requires that the

solution of Equation (29) consists of the attenuating part

only

 

,e0

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

 

 

,,d

rzz Jr

BaJ r



















(31)

Inserting Equation (31) into the boundary condition

Equations (23) and (24) give

 

0d,0

mmn

BaJrRrr a



 

 (32)



0d0,

BaJrr a





 (33)

Introducing the non-dimensional variables as

 

raA B



 

  (34)

The integral Equations (32) and (33) can be written in

the following standard form

 

0d,01

mmn

AJ aR

 

 

 (35)

 

0d0, 1

 



 (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

 

mn mmn

AaRJ aH

 

 

 (38)

in which [25]









mnmnAmnmn Cmn

HAH CH





 (39)







22 2

Amn

mn mmnmmmnm

aJaJJa J



















(40)







22 2

Cmn

mnmmnmmmnm

aIaJIa J















(41)

Inserting Equation (38) into Equation (31) and using

Equation (34), we obtain the function  at the fluid-plate

interface

 

,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

vacuum





 (43)

fluid

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



 (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

365



 (46)

The AVMI factor can be made dimensionless

mn mn







 (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

,0 ,0cosd d

Trmrr







 



 (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 d

Fθmn

θmn

TrRrrr









 



(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



Fθmn

TaH









 (50)

The kinetic energy of the circular diaphragm is

 

2π22

00 cosdd

Dmn

ThRrmrr





 (51)

Integration with respect to



, and using the normaliza-

tion criterion Equation (15), then we obtain



22 2

Dθmn

θmn

ThRrrr

ha Rha







 







(52)

By using Equations (46), (47), (50) and (52), the

NAVMI factors are found to be given by



mn mn

Th H









 

 (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

Wra







 (54)

Using Equation (14) and Equation (54), we obtain











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





 

mmn mmn

mnm mnmnm mn

Ja Ia

JaIa









 (56)

The characteristic equation of the circular diaphragm

with clamped edge can be obtained



 



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





 (58)

This gives the mode shapes expression









cos

mmn

mnmn m mnm mn

mmn

WAJ rIrm

















(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

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

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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