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Formulation for an FE and BE coupled problem and its application to the earmuff-earcanal system
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Engineering Analysis with Boundary Elements 16 (1995) 305-315 Copyright © 1996. Publishedby ElsevierScienceLimited Printed in Great Britain. All rights reserved 0955-7997/95/$09.50
Formulation for an FE and BE coupled problem and its application to the earmuff-earcanal system Chang-Myung Lee University of Ulsan, Ulsan, Kyungnam, 680-749, South Korea
Larry H. Royster North Carolina State University, Raleigh, North Carolina 27695, USA
&
Robert D. Ciskowski IBM, Boston, Massachusetts 02254, USA
(Received 30 September 1995; revised version received 9 October 1995; accepted 13 October 1995) A direct coupling procedure of the finite element method (FEM) and the boundary element method (BEM) for structure-acoustic-cavity problems has been introduced. The Laplace transformed matrix equations for the structure and acoustic cavity are coupled directly satisfying the necessary equilibrium and compatibility conditions. The coupled FEM-BEM technique is verified using a box-type cavity for the steady-state and the transient predictions compared with analytical results. The verified coupled FEM-BEM code is utilized in the analysis of the earmuff-earcanal system. Key words: Boundary element method, finite element method, coupling, structure-
acoustic-cavity system, steady-state response, transient response, Laplace transform, earmuff-earcanal system.
1 INTRODUCTION
Wolf and Nefske. 1 They predicted the interior noise of the passenger compartment and verified the predicted response against experimental results. In this same time period, the application of FEM to the structureacoustic-cavity system was investigated by several authors: Shuka and Ishihara, 2 Petyt et al. 3 and Richards and Jha. 4 Nefske et al. 5 provide a good review of the application of the FEM to the structure-acousticcavity system. The boundary element method (BEM) has found widespread use in acoustics since the 1970s. Recently, BEM has been used for structure analysis as well as for the analysis of the structure-fluid interaction problem. Tanaka and Masuda 6 published the first complete BEM formulation for the analysis of the coupled structure-acoustic-cavity problem. They developed a BEM formulation for the cavity and plate acoustic structure separately and coupled the solutions. The complete three-dimensional BEM formulation of the coupled elastic structure-acoustic-cavity system for both the
With the recent advances in numerical techniques, various numerical discre,tization methods for obtaining approximate solutions are available to solve complex coupled structure-acoustic-cavity problems. One such technique is the finite element method (FEM). In the 1960s and 1970s, FEM was established as the preferred technique for structural analysis problems. Although it had been applied to acoustic problems, the technique does not le,nd itself to the unbounded (infinite) regions so often encountered there. In spite of this drawback, the application of FEM to the structureacoustic-cavity proble~t was implemented by several researchers. During the 1970s marly FEM codes became available utilizing computer graphic systems which permit convenient modeling of complex geometries. A general purpose FEM code, NASTRAN, was first adopted for automobile structure-a.coustic-analysis problems by 305
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C.-M. Lee, L. H. Royster, R. D. Ciskowski
steady-state and transient cases was investigated by Ciskowski and Royster. 7-9 They applied the developed algorithm to the plugged earcanal system to predict its steady-state and transient responses. Xie et al. 1° and Mourad et al) 1 extended the BEM analysis to the coupled viscoelastic structure-acoustic-cavity system. Recently, Sladek et al) 2 introduced BEM analysis to the cavity backed plate system using the multiple reciprocity method. Unlike FEM, BEM is ideally suited to deal with unbounded regions and can reduce human efforts during the modeling of an enclosure. Although BEM is being applied to structural problems with an ever increasing frequency, its unsuitability for nonlinear and highly nonhomogeneous problems keep FEM the preferred technique for structural analysis. In light of the strengths and weaknesses of each of these techniques, for coupled structure-acoustic-cavity problems it is only natural to use a combination to achieve a numerical solution. For this reason, coupled FEM-BEM solutions have been widely used in practice and have often provided more accurate and faster results than solutions emLaloying either technique alone. In 1989, Suzuki et a l ) ° coupled a plate structure and a complex shaped acoustic cavity with a complicated boundary condition in order to predict the steady-state response of the enclosed internal cavity. In their research, the structure and acoustic cavity were formulated by the FEM code, NASTRAN, and BEM, respectively. They only investigated the steady-state response. This research will present the direct coupling of an FEM model of a plate or shell-like structure with a BEM model of an acoustic cavity. Using this hybrid method, steady-state and transient responses will be predicted for a hearing protection device (HPD) such as an earmuff that attenuates the pressure wave prior to its reception by the eardrum. Since the earmuff-wearer system results in a coupled shell-type acoustic-cavity system, the ability to be able to model the shell-type acoustic-cavity system is important from the stand point of being able to predict the level of protection achievable for the wearer. Therefore, the coupled FEM-BEM technique is essential for the analysis of this kind of problem. The authors are not aware of any research that has tried to predict the attenuation level of the earmuff-wearer system using a coupled FEM-BEM technique. The coupled F E M - B E M code is verified using a rectangular box-type cavity with one pinned-pinned end plate for the steady-state and the transient acoustic pressure responses and the results are compared with a known analytical solution. Initially the Laplace transformation is applied to the governing equations to remove the time dependence. Then the transformed FEM equations for the structure are developed in matrix form and coupled to the transformed
BEM equations for the acoustic cavity. This coupling is accomplished directly by satisfying appropriate compatibility and equilibrium conditions at the interface. Among three commonly used types of BEM elements, namely constant, linear and quadratic, the advantages of the constant element in F E M BEM coupling will be demonstrated. Complete details of this hybrid formulation can be found in Ref. 14.
2 MATHEMATICAL FORMULATION AND NUMERICAL IMPLEMENTATION 2.1 FEM formulation for the structure
The Laplace transformed differential equation of motion for a general structure takes on the following form with zero initial conditions [s2[M] + s[C] + [K]]{~} = {0}
(1)
where s is the Laplace transform parameter, {~} is the Laplace transform of force, and {~} is the Laplace transform of displacement. By defining a distribution matrix IN], which is the interpolation functions used in the FEM model, it is possible to relate the force term {~} in eqn (1) to the actual values of forces, or {~} = IIflN]r{p} dA
(2)
where {if} is a column matrix of the actual values of the external pressure at the nodes. Now eqn (1), using eqn (2), can be represented symbolically as As~ = NIp
(3)
where As = [s2[M] + s[C] + [K]]
(4)
In order to combine the FEM and BEM models at the common interface, the compatibility and equilibrium conditions need to be satisfied. In addition, the finite element matrices must be transformed into equivalent boundary element matrices for the boundary element region. In this research, the structure is entirely in contact with the acoustic medium. As a result, the displacement fields i and ii represent the same vector. Therefore, eqn (3) can be reduced as follows:
where N[,~I are the shape functions and pressures, respectively, at the common interface.
Formulation for an FE and BE coupled problem 2.2 BEM formulation for the acoustic cavity
The Laplace transformed 3-D scalar wave equation is given by V2p(£, s) - ~/~(£, s) = 0
(6)
Using the second form of Green's theorem, eqn (6) can be written as
g(~,% s) ~ ( ~ , s)
(7)
where g(Y, jT,s ) = the fimdamental solution of the Laplace transformed 3-1) scalar wave equation and C = 1/2 for a smooth boundary. For a given system, the boundary is discretized into n elements. Discretization of the boundary produces the discretized scalar wave equation that can be used to predict the acoustic pressure response over an arbitrary boundary, or k= 1
JPk
,, apk , =
Imposing known values of ~ and ~a on the boundary, eqn (14) can now be rearranged in accordance with the unknowns under consideration. When all the unknowns are arranged on the left-hand side and the knowns are arranged on the fight-hand side, the following matrix equation is obtained:
[-H~ - ms2Gd o.~
p0S2Gk][ ~k ] (15)
y, cp(As) + Irp(Z,s ) ~.~(£, o .. s) dr(£)
=
307
~-b-~- Jr g(~,y., s) drk(£)
(8)
where Pk, ~ak are the known acoustic pressure and displacement on the boundary, Hk, Gk are matrices consisting of columns of H and G corresponding to the knowns, Pk and ~ak, on the boundary, 1~, Uau are the unknown pressure and acoustical displacement on the boundary and Hu, Gu are matrices consisting of columns of H and G corresponding to the boundary unknowns and 0auIntroducing the matrix notation, [G] = -pos2[G]
(16)
and considering the acoustic impedance and structure interface boundary conditions, substituting eqn (16) into eqn (15) yields [-- ["a]I [Gu]I -- ["u]B [(~u]B -- [Hu]Z [Gu]z] [ {Pu}I '
Introducing the following notation:
½+ Hkn =
Irk
ag(e, yn, s) dFk(£) Ot,I
fr °g(g' ~7"'s) dFk(£) k
On
n= k (9)
n#k
x
=[Hk _Gk]/Pk/F"1 {Oaa}B L~ J {~}B
(17)
and Gk,
[ g(£,y.,s)drk(g) ,11k
(10)
eqn (8) can be rewritten as
• Hkn~k
~
k=l
k=l
=
G
kn~~Ok
(ll)
Symbolically eqn (11) may be represented as H~ = Gpn
(12)
where the matrices H and G are obtained by evaluating eqns (9) and (10), respectively, and ~ and Pn correspond to the transformed pressure, p k , and normal derivative of the transformed pres.~,ure, o~k/On, respectively. The normal directional pres~mre distribution Pn inside of the acoustic cavity, using the acoustical displacement, fia, and the air density, Po, is given by Pn ~- --poS2Ua
(13)
Using eqn (13), rewriting eqn (12) gives -pos2Gfia = Hp
(14)
where [Hu]I, [(~u]I are matrices consisting of columns of [H] and [G] corresponding to unknown interface boundary values of {Pu}1 and {Qaa}I, respectively, [Ha!s, [(]u]n are matrices consisting of columns of [H] and [G] corresponding to unknown acoustic boundary values of {Pu}B and {aaa}B, respectively, and [Ha]z, [(]u]z are matrices consisting of columns of [I-I] and [G] corresponding to unknown impedance boundary values of {Pu}z and {0an}Z, respectively. 2.3 Coupling conditions
At the interface between the structure and acoustic cavity, the boundary conditions must satisfy the compatibility and equilibrium coupling conditions. Examining the boundary interface, it is observed that the displacement of the structure coincides with the displacement of the acoustic particle. The Laplace transformed compatibility condition at the interface is expressed using a shape function
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C.-M. Lee, L. H. Royster, R. D. Ciskowski
[N] as
the coupled FEM and BEM system for the structureacoustic-cavity problem as follows:
Off = _poS2[N]{a } .n
(18)
On
As
0
NT
The equilibrium condition requires that the normal force produced by the vibrating structure at the interface be equal in magnitude and opposite in direction to the total acoustic pressure over the interface area for the enclosed cavity in the Laplace transformed domain,
0
0]
-[rInh {i},
i.e.
:[T Hko
×
0] (~k
Pk
(26)
iia*
. 2.4 Coupling the matrix equations In the analysis of an acoustic cavity, the acoustic impedance boundary condition is considered among unknown boundary conditions. The impedance boundary equation may be written in vector form as poS _
Pn = ---~-P
(20)
Then using eqns (13) and (20), a new displacement and pressure relationship including the impedance boundary condition is obtained: {fiau}Z = [Za]{~,}z
(21)
where 1 [Za] - ~-~
(22)
Before the FEM and BEM models can be coupled, using eqn (21), the BEM matrices of eqn (17) must be rewritten as
{i}I -[I-i.],
I
[G.]B [fio] ]
iUau}B[ • {p.}z I
=[Hk-Gk](
]~k ~k
(23)
where [~q] = [Gu]I[N] and
[I~] z = -[nu] z + [t~u]z[Za] (24)
2.5 Numerical transform inversion To obtain the steady-state response for the coupled system, a numerical formulation is used with s = i~v. For transient response, a numerical Laplace inversion is needed to recover the response in the time domain. Several methods have been suggested for the numerical inversion scheme, For the numerical implementation, Banerjee and Ahmad t5 used a FFT-based inversion algorithm which is suggested by Durbin) 6 However, in order to further reduce the computing time required, Wilcox and Gibson 17 presented a discrete Laplace transform pair for the analysis of the transient behavior. This algorithm requires far fewer sample points in the transform domain than are required by Durbin's algorithm. Actually, Wilcox's method takes more time during the Laplace transformation than the standard method (such as Durbin's) since the standard method utilizes the fast Fourier transform techniques. However, since Wilcox's method achieves savings in computing time if fewer than about 128 Laplace transform domain points are sufficient to discretize the given time domain, this method is utilized in this research. For the estimation of the transient response, external force functions are conveniently described in the time domain. The FEM-BEM coupled formulation requires, however, that this information be available in the Laplace transform domain. In particular, external forcing conditions are required at specific values of the complex valued Laplace transform parameter, s, as specified by the particular numerical inversion scheme employed. The algorithm used in this research is that of Narayanan and Beskos. is They described the piecewise linear excitation function on the Laplace transform domain as follows
The FEM matrices of eqn (5) also can be rewritten
as
~11 F(s)
=
i=1
1 s2 t i + l _
ti
[S(ti+ 1 -- t i ) ( f ( t i ) e
-st'
- f ( t i + l ) e -st'+') + (f(ti+l)
Combining eqns (23) and (25) gives the final form of
-- f(ti))(e
-sti - - e -st'+'
)]
(27)
Formulation for an FE and BE coupled problem
309
To determine the force conditions in the Laplace transform domain, eqn (27) is evaluated repeatedly using the information provided in l%e time domain.
Two BEM Elements & Nodes
2.6 Selection of boundary element type In discretizing a three-dhnensional cavity, two possible geometric shapes for the boundary element are triangular or quadrilateral with straight or curved edges. Selecting an element with a curved surface will produce more accurate modefing of the boundary of the physical problem. However, more effort will be required to numerically implement this system. The authors also felt that, for the types of geometries found in the applications to which their work would be applied, namely HPDs, the straight-edge element would provide an acceptable approximation to this geometry. For this reason and ease of nu~aerical implementation, a triangular shaped boundary element with straight edges was selected. Once the BEM element is chosen according to geometric shapes and curvatures, the order of the variation for the variables of the scalar wave equation over the element must be selected. There are three possible types of variation: constant, linear or quadratic. The element with a constant variation is usually called a constant element or a piston element, and it has its node at the centroid of the element. In order to define a linear element, only the vertices are specified. For higher order geometric variations like a quadratic element, additional poinl:s are needed on the edges. Because of the simplification of the numerical implementation and reasonab]le accuracy offered by the constant element, it has been used frequently in the boundary discretizatiorL of acoustic cavities even though the level of the: accuracy is more dependent upon the size of the element than with higher order elements. Higher order elements provide improved accuracy with fewer elements than are required by the constant element. However, higher order elements require more complicated numerical analysis and are more difficult to implement. When coupling BEM with FEM formulations, the selection of a constant BEM element type helps to avoid other numerical difficulties in the coupling process. If linear or quadratic BEM elements are selected for coupling to FEM elements, boundary conditions along the edge would h~tve to be addressed leading to numerical complexities. If, however, the constant BEM element is utilized for the discretization of the acoustic cavity, the boundary condition of the structure does not affect the coupling at the edge since the node of a constant BEM element would not coincide with the node of an FEM element at the edge. For the constant triangular BEM element,, three FEM triangular elements will be needed to couple with one BEM element since the node of the constant BEM element is located at the centroid of the element. Figure 1 shows the arrangement of
/
r Boundary Nodes of Structure
Internal Node of Structure or Acoustic Node
/ Three FEM Elements per BEM Element
Fig. 1. Node correlations between FEM and BEM elements during coupfing. FEM and BEM elements using six FEM elements and two BEM elements at the interface. For all of the above reasons, the constant BEM element is used in this research.
3 NUMERICAL RESULTS FOR THE STRUCTURE-ACOUSTIC-CAVITY SYSTEM
3.1 Box-type cavity with pinned-pinned plate 3.1.1 Steady-state response The F E M - B E M program developed herein is for any general shell-like structure coupled to an acoustic cavity. One example where this coupled program could be utilized is for the analysis of a flat plate coupled to an acoustic cavity. In order to verify the F E M - B E M coupled code, it will be used to model an acoustically hard-walled cubic cavity where one end consists of a simply-supported flat plate. The size of the cavity is 0.2 x 0.2 x 0.2m. The simply-supported flat plate is assumed to be made of brass with a thickness of 0-91 x 10-4m. The acoustic cavity is discretized into 96 constant triangular BEM elements with 50 elements at the interface between the plate and cavity. The plate is discretized into 150 triangular FEM elements in order to obtain three structural elements for each acoustic element at the interface as shown in Fig. 2. To obtain visual images of the mode shapes of the plate, the 256 element discretization was utilized for better
310
C.-M. Lee, L. H. Royster, R. D. Ciskowski
/
/
/l 'y )cx
be f i
f
96 Boundary Element
150 Finite Element
Discretization
Discretization
Fig. 2. Boundary element discretization for the rectangular box cavity and finite element discretization for the pinned-pinned plate. resolution in drawing the mode shapes. The refined discretization yielded no significant improvement in the predicted plate natural frequencies. The FEM calculated mode shape for each natural frequency is presented in Fig. 3. One approach to verifying accurate coupling between the FEM and BEM models involves statically loading the pinned-pinned end plate and then comparing the predicted increase in the internal volume's pressure with the known analytical solution. This comparison is shown in Table 1. Good agreement was obtained. Next, the steady-state pressure response of the coupled plate-box cavity system is investigated by applying a harmonic pressure excitation on the structure elements at the outside of the box and varying the pressure's excitation frequency from 10Hz up to 1 kHz. The predicted pressure response is calculated at the center of the acoustic element at the inside end of the box (0.2, 0.1, 0.1 m). An analytical solution for this problem was obtained by Guy and Bhattacharya. 19 They also experimentally investigated this system for steady-state pressure excitation.-Figure 4 shows the steady-state response comparing the numerically and the analytically predicted 201og(Pi/P2) at the receiver point located on the center of the acoustic element Table I. Static pressure response of a cavity backed plate system under static uniform pressure excitation (external pressure P1 = 11~), (0"2 x 0"2 x 0-2 m box)
Analytical FEM-BEM
Static pressure response inside of the box
Pressure difference ratio 201oglo(P2/ P l )
0.179 psi 0-180 psi
14-9dB 14.8 dB
at the inside end of the box. The analysis of the steady-state response reveals that the predicted panelcavity resonant frequencies are 90Hz, 400Hz, 680 Hz and 860 Hz. The first three of these natural frequencies correspond to the (1,1), (1,3) and (3,3) symmetrical panel modes, respectively. The 860 Hz is due to the fundamental plane wave room resonant frequency. The predicted steady-state response indicates good agreement between the F E M ' B E M model and the analytical solution 19 with typically less than 2dB differences noted. Furthermore, the predictions indicate good agreement with the experimental data. 3.1.2 Transient response Using the same plate-cavity configuration (0.2 x 0.2 x 0.2m) as was introduced in section 3.1.1, the transient response of the coupled system is now predicted using the F E M - B E M code. A triangular pressure impulse of a simulated gun shot is applied to the cavity backed plate. The assumed gun shot time history is presented in Fig. 5. The determined parameters for the time history signature are rl = 1.4 x 10-3 s, 7"2 3.1 x 10-3s, 7"3 = 3.4 x 10-3s and Pmax = 0"29psi, Pmin = -0"0725 psi. The predicted pressure response at the receiver point (0"2, 0" 1, 0" 1 m), is plotted against the analytical prediction in Fig. 6 for the assumed gun shot time-history. The F E M - B E M predictions are in good agreement with the analytical solution which was obtained by applying the numerical inverse Laplace transform to the Laplace transformed pressure response of the cavity backed plate. The description of the Laplace transformed transient pressure response of the cavity backed plate on the Laplace transform domain can be found in Ref. 14. The approximate period of the =
Formulation for an FE and BE coupled problem
77 Hz
311
(2, i )
193 Hz
(l, 2)
(~. I)
302 Hz
488 Hz
488 !-~
(2,2)
(2.3)
(3.2)
(i ,3)
386 Hz (3, i)
Fig. 3. Mode shapes of the end plate for the rectangular box configuration. resulting pressure response is 0.012 s. Therefore the frequency associated with the pressure wave is about 84Hz and is close to the fundamental frequency for the pinned-pinned panel. The resulting noise reduction is 17 dB.
3.2 Application to the a~aalysis of the earmuff-earcanal system
3.2.1 Steady-state response The coupled F E M - B E M model of the structureacoustic-cavity system has been verified using the boxtype cavity with pirmed-pirmed end plate. The model will now be applied to predict the response of a practical, real-world example of a structure-acoustic-cavity system, namely the eartauff-earcanal system in Fig. 7. The earmuff normally consists of a plastic hemispherical-shaped earcup with a cushion for sealing the
earcup to the flesh around the ear. When the earmuff is worn, an air cavity is created within the earcup. This cavity includes the extension into the earcanal up to the eardrum. The eardrum represents an impedance for the cavity as shown in Fig. 8. Data for the impedance of a typical eardrum are provided in Ref. 20. In a normal situation where the earmuff is worn for hearing protection, an acoustic pressure wave impinges on the outside surface of the earcup, causing it to vibrate. This vibration excites the interior cavity which is then transmitted to the eardrum. An effective earmuff design attenuates or reduces the pressure levels experienced by the eardrum compared to what impinged on the external surface of the earmuff. For the earmuff-earcanal system modelled, the earmuff is selected to be an EAR Model 1000.21 Dimensions for the system are given in Fig. 8. The earcup
C.-M. Lee, L. H. Royster, R. D. Ciskowski
312
i
Ill =
=
x
FEM-BEM ANALYTICAL EXPERIMENTA (G,-,.,, et ~_!)
:
: FEM-.E.
Ill 6
1. ~o
g L I1 P""° wi.,, = 0.0034 s¢¢.
I
(294 Hz)
° ° , ,,
1 oo
10
0
looo
0.01
0.02 TIME (sec.)
FREQUENCY (Hz)
Fig. 4. Comparison of numerical, analytical, and experimental predictions of 201og(P2/P1) for the rectangular box (0.2 x 0.2 x 0-2 m) with one end a pinned-pinned plate. was discretized using 132 flat shell finite elements as shown in Fig. 9. The earcup sealing cushion was modeled using linear spring elements connected to the element nodes at the base of the earcup. The acoustic cavity, consisting of the earcup and earcanal volumes, was discretized using 100 boundary elements. The boundary of the cavity not coupled to the earcup is assumed to be acoustically hard except for one portion where the eardrum impedance is located. The pressure wave impinging on the earcup results in an applied pressure on the flat shell elements. The internal pressure level response is predicted at the
0.0..3
Fig. 6. Comparison of predicted transient response against analytical results for the rectangular box (0-2 x 0.2 x 0.2m) with pinned-pinned end plate.
/
E*rmuf~ Cushion
~ _ _
Acoustic Pressure Wave
~
~
Acoustic Cavity
Fig. 7. Earmuff-earcanal system. eardrum receiver location (6"2, 0-1, 0.1 cm) over the
frequency range of 100Hz, to 5kHz. The predicted acoustic pressure response of the earmuff-earcanal system is validated by several criteria. b-
10.2 Cm
i.-- -I LtJ t~ be) 03 ~ hJ tie" O..
4.5 c m - - e ~
-..I i I 1.glcm
eareup
cavity
I
ca
al ......
2.8,Cm L
TIME (see.)
Fig. 5. Time history of assumed gun shot transient input pressure time response.
Fig. 8. Dimensions of acoustic cavity of earmuff-earcanal system.
Formulation for an FE and BE coupled problem
132
affected by the rigid body motion of the earcup on the acoustic cavity and the sealing cushion. The mass of the earmuff and the effective spring constants of the acoustic cavity and the sealing cushion together act like a single degree-of-freedom system. The response of a single degree-of-freedom system generally consists of the resonant frequency peak with a stiffnesscontrolled region to the left (lower frequencies) and a mass-controlled region to the right. Given the mass and stiffness of the single degree-offreedom system, straight forward calculations ~ can be used to arrive at the resonant frequency, static deflection in the stiffness-controlled region, and slope (or fall off) in the mass-controlled region. With the mass of the earmuff at 1.48 x 10-4 kg and the effective spring constants of the acoustic cavity at 103 Nm and the sealing cushion at 6Nm, the resonant frequency peak should be at 137Hz, the static pressure should be 0.719psi, and the slope should be 12dB/octave. The F E M - B E M model predicted 140Hz, 0.741psi and 12.4dB/octave. Model values are within 3% of calculated values.
Irmi~ E l ~ n t Diseretizafionof Cup
tO0B ~ a = 7 E I ~
~
o
o
of Cavit7
Fig. 9. Discretization of earmuif-earcanal system.
The response is characte]~_ed by two primary types of behavior. Above 1 kHz, tbe response is the result of the coupling of the vibration of the earcup with that of the acoustic cavity. The two major structure resonances, 2200 Hz and 4150 Hz, indicated in Fig. 10 are associated with the first and second t;ymmetrical vibrational modes of the earmuff's shell. Below 1 kHz, the response is 0 0
e FEM-BE M
o-
0 L~ m
/\
K
T
n v
t71 0 0 O4 0 I
0 0
,
110
,
0
.,,.
J
100
.
.
0
.=.
1 000
313
04
FREQUENCY (Hz)
Fig. 10. Steady-state response of earmuff-earcanal system for uniform external pressure.
3.2.2 Transient response A triangular pressure impulse of a simulated gun shot is applied uniformly over the shell to get the transient response at the eardrum with eardrum impedance. The assumed pulse shape was presented in Fig. 5. The assumed shape parameters are 7-I = 1.4 x 10-3 s, r 2 = 3.1 x 10-°s, r3 = 3.4 x 10-3 s and Pmax = 9"16psi, Pn~n = - 1"67psi. For the duration of the impulse, if we assume that rigid body motion is dominant, the earmuff-earcanal system may be modeled as a single degree-of-freedom system for analytical comparison purposes. The mass of the earmuff and the effective supporting spring stiffnesses such as the effective spring constant for the acoustic cavity and the spring constant for the cushion are assumed to act as a single degree-of-freedom. The acoustic impedance at the eardrum is assumed as the effective damping constant. The displacement response of the single degree-of-freedom system on the frequency domain is converted to the acoustic pressure response assuming the earmuff internal cavity as an equivalent rectangular box cavity. The calculated pressure response of the single degree-of-freedom system on the time domain for analytical purposes is obtained from the frequency domain using the numerical inversion scheme. The resulting predicted transient pressure time history responses from the F E M - B E M coupled code and the analytical result at the eardrum is shown in Fig. 11. The period of the predicted pressure response is 0.0075s, which translates into a basic frequency of 133Hz. A good agreement is indicated when the F E M - B E M result is compared with the analytical prediction.
314
C.-M. Lee, L. H. Royster, R. D. Ciskowski tO
1 - O ANALYTICAL Excitafioa Pulse
=
= F E M - BEM
NRp¢~= 3.3
AA "t'?' q")
03
~tO W 2~ ~ W
dB
0
'
'
,
'
°
I
I
,
0.015
0.02
,
Pulse Width = 0.0034 sec. o
I 0
I
(,2.94
5 x 1 0 -3
0.01
TIME (see.)
Fig. 11. Transient response of earmuff-earcanal system for the duration of the impulse (0.0034 s). 4 CONCLUSIONS A coupled F E M and BEM model has been developed to predict the steady-state and transient responses for structure-acoustic-cavity systems. The model was validated using a box-type cavity with one pinned-pinned end plate. G o o d agreement between the model and available analytical or experimental data for the configuration studied has been achieved for steady-state and transient responses. For the first time, such an F E M - B E M model has been applied to the analysis of the practical problem involving hearing protection, namely an earmuff-earcanal system. Preliminary results appear good. Future work will compare with experimental results.
ACKNOWLEDGEMENTS This work was supported by Kia Motors Corp. of South Korea. The first author also received support from the Mechanical Engineering Department o f North Carolina State University and a scholarship from the N o r t h Carolina Regional Chapter of the Acoustical Society of American and the Cabot Foundation.
REFERENCES 1. Wolf, J. A. & Nefske, D. J. NASTRAN, 1975. 2. Shuku, T. & Ishihara, K. The analysis of the acoustic field in irregularly shaped room by the finite element method. J. Sound Vib., 1973, 29, 67-76. 3. Petyt, M., Lea, J. & Koopmarm, G. H. A finite element
method for determining the acoustic modes of irregularly shaped cavities. J. Sound Vib., 1976, 45, 495-502. 4. Richard, T. L. & Jha, S. K. A simplified finite element method for determining the acoustic modes of irregularly shaped cavities. J. Sound Vib., 1979, 63, 61-72. 5. Nefske, D. J., Wolf, J. A., Jr. & Howell. L. J. Structural acoustic finite element analysis of the automotive passenger compartment: a review of current practice. J. Sound Fib., 1982, 80, 247-66. 6. Tanaka, M. & Masuda, Y. Boundary integral equation approach to structural-acoustic coupling problems. In Boundary Element IX, Proc. 9th Int. Conf., ed. C. A. Brebbia, W. L. Wendland & G. Kulm. Computational Mechanics Publications, Springer-Vedag, Berlin, 1987. 7. Ciskowski, R. D. & Royster, L. H. Boundary element solution for a coupled elastodynamic and wave equation system to predict forced responses of a plugged acoustic cavity. J. Acoust. Soc. Amer., Suppl. I, 1987, 112, $64 8. Ciskowski, R. D. & Royster, L. H. Boundary element solution to predict steady-state response of a 3-D fluidstructure system. Proc. NOISE-CON 88, ed. R. J. Bernhard & J. S. BoRon. Noise Control Foundation, New York, 1988, pp. 579-84. 9. Ciskowski, R. D. & Royster, L. H. Boundary element solution to predict transient response of a 3-D coupled fluid cavity-elastic structure system. In Boundary Element Methods in Applied Mechanics, ed. M. Tanaka & T. A. Cruse. Pergamon Press, New York, 1988, pp. 54554. 10. Xie, K-J., Ciskowski, R. D. & Royster, L. H. An investigation of wave propagation in viscoelastic media modeled by fractional derivative using the boundary element method. In Boundary Element Methods in Applied Mechanics, ed. M. Tanaka & T. A. Cruse. Pergamon Press, New York, 1988, pp. 523-31. 11. Mourad, K. M., Royster, L. H. & Ciskowski, R. D. Using the boundary element method to investigate internal earmuff-earcanal resonances. In Advances in Boundary Elements, Vol. 2, ed. C. A. Brebbia & J. J. Connor. Computational Mechanics Publication, Southampton, 1989, pp. 297-308. 12. Sladek, V., Sladek, J. & Tanaka, M. Boundary element solution of some structural-acoustic coupling problems using the multiple reciprocity method. Commun. Numer. Meth. Engng, 1994, 10, 237-48. 13. Suzuki, S., Maruyama, S. & Ido, H. Boundary element analysis of cavity noise problems with complicated boundary conditions. J. Sound Vib., 1989, 130(1), 79-91. 14. Lee, Chang-Myung, Application of finite element and boundary element methods to a coupled shell-type acoustic-cavity system. PhD thesis, North Carolina State University, Raleigh, 1992. 15. Banerjee, P. K. & Ahmad, S. Advanced 3-D dynamic analysis by boundary element methods. In Advanced Topics in Boundary Element Analysis, Iiol. 72, ed. T. A. Cruse et al. AMD, 1985, 65-81. 16. Durbin, F. Numerical inversion of Laplace transform: an efficient improvement to Dubner and Abates' method. Comput. J., 1974, 17, 371-6. 17. Wilcox, D. J. & Gibson, I. S. Numerical Laplace transformation and inversion in the analysis of physical systems, lnt. J. Numer. Meth. Engng, 1984, 20, 1507-19. 18. Narayanan, G. V. & Beskos, D. E. Numerical operational methods for time-dependent linear programs. Int. J. Numer. Meth. Engng, 1982, 18, 1829-54. 19. Guy, R. W. & Bhattacharya, M. C. The transmission of sound through a cavity-backed tLrtiteplate. J. Sound Vib., 1973, 27(2), 207-23.
Formulation for an FE and BE coupled problem 20. Schroeter, J. & Poesselt, C. The use of acoustical test fixtures for the measurement of hearing protector attenuation. Part II: Modeling the external ear, simulating bond conduction, and comparing text fixture and real-ear data. J. Acoust. Soc. Amer., 15686,80(2), 505-27.
315
21. Berger, E. H., Ward, W. D., Morrill, J. C. & Royster, L. H. Noise and hearing conservation manual, 4th edn. American Industrial Hygiene Association, 1986. 22. Harris, C. M. Shock Vibration Handbook. Cyril M. Harris, 1988.
ELSEVIER
Engineering Analysis with Boundary Elements 16 (1995) 305-315 Copyright © 1996. Publishedby ElsevierScienceLimited Printed in Great Britain. All rights reserved 0955-7997/95/$09.50
Formulation for an FE and BE coupled problem and its application to the earmuff-earcanal system Chang-Myung Lee University of Ulsan, Ulsan, Kyungnam, 680-749, South Korea
Larry H. Royster North Carolina State University, Raleigh, North Carolina 27695, USA
&
Robert D. Ciskowski IBM, Boston, Massachusetts 02254, USA
(Received 30 September 1995; revised version received 9 October 1995; accepted 13 October 1995) A direct coupling procedure of the finite element method (FEM) and the boundary element method (BEM) for structure-acoustic-cavity problems has been introduced. The Laplace transformed matrix equations for the structure and acoustic cavity are coupled directly satisfying the necessary equilibrium and compatibility conditions. The coupled FEM-BEM technique is verified using a box-type cavity for the steady-state and the transient predictions compared with analytical results. The verified coupled FEM-BEM code is utilized in the analysis of the earmuff-earcanal system. Key words: Boundary element method, finite element method, coupling, structure-
acoustic-cavity system, steady-state response, transient response, Laplace transform, earmuff-earcanal system.
1 INTRODUCTION
Wolf and Nefske. 1 They predicted the interior noise of the passenger compartment and verified the predicted response against experimental results. In this same time period, the application of FEM to the structureacoustic-cavity system was investigated by several authors: Shuka and Ishihara, 2 Petyt et al. 3 and Richards and Jha. 4 Nefske et al. 5 provide a good review of the application of the FEM to the structure-acousticcavity system. The boundary element method (BEM) has found widespread use in acoustics since the 1970s. Recently, BEM has been used for structure analysis as well as for the analysis of the structure-fluid interaction problem. Tanaka and Masuda 6 published the first complete BEM formulation for the analysis of the coupled structure-acoustic-cavity problem. They developed a BEM formulation for the cavity and plate acoustic structure separately and coupled the solutions. The complete three-dimensional BEM formulation of the coupled elastic structure-acoustic-cavity system for both the
With the recent advances in numerical techniques, various numerical discre,tization methods for obtaining approximate solutions are available to solve complex coupled structure-acoustic-cavity problems. One such technique is the finite element method (FEM). In the 1960s and 1970s, FEM was established as the preferred technique for structural analysis problems. Although it had been applied to acoustic problems, the technique does not le,nd itself to the unbounded (infinite) regions so often encountered there. In spite of this drawback, the application of FEM to the structureacoustic-cavity proble~t was implemented by several researchers. During the 1970s marly FEM codes became available utilizing computer graphic systems which permit convenient modeling of complex geometries. A general purpose FEM code, NASTRAN, was first adopted for automobile structure-a.coustic-analysis problems by 305
306
C.-M. Lee, L. H. Royster, R. D. Ciskowski
steady-state and transient cases was investigated by Ciskowski and Royster. 7-9 They applied the developed algorithm to the plugged earcanal system to predict its steady-state and transient responses. Xie et al. 1° and Mourad et al) 1 extended the BEM analysis to the coupled viscoelastic structure-acoustic-cavity system. Recently, Sladek et al) 2 introduced BEM analysis to the cavity backed plate system using the multiple reciprocity method. Unlike FEM, BEM is ideally suited to deal with unbounded regions and can reduce human efforts during the modeling of an enclosure. Although BEM is being applied to structural problems with an ever increasing frequency, its unsuitability for nonlinear and highly nonhomogeneous problems keep FEM the preferred technique for structural analysis. In light of the strengths and weaknesses of each of these techniques, for coupled structure-acoustic-cavity problems it is only natural to use a combination to achieve a numerical solution. For this reason, coupled FEM-BEM solutions have been widely used in practice and have often provided more accurate and faster results than solutions emLaloying either technique alone. In 1989, Suzuki et a l ) ° coupled a plate structure and a complex shaped acoustic cavity with a complicated boundary condition in order to predict the steady-state response of the enclosed internal cavity. In their research, the structure and acoustic cavity were formulated by the FEM code, NASTRAN, and BEM, respectively. They only investigated the steady-state response. This research will present the direct coupling of an FEM model of a plate or shell-like structure with a BEM model of an acoustic cavity. Using this hybrid method, steady-state and transient responses will be predicted for a hearing protection device (HPD) such as an earmuff that attenuates the pressure wave prior to its reception by the eardrum. Since the earmuff-wearer system results in a coupled shell-type acoustic-cavity system, the ability to be able to model the shell-type acoustic-cavity system is important from the stand point of being able to predict the level of protection achievable for the wearer. Therefore, the coupled FEM-BEM technique is essential for the analysis of this kind of problem. The authors are not aware of any research that has tried to predict the attenuation level of the earmuff-wearer system using a coupled FEM-BEM technique. The coupled F E M - B E M code is verified using a rectangular box-type cavity with one pinned-pinned end plate for the steady-state and the transient acoustic pressure responses and the results are compared with a known analytical solution. Initially the Laplace transformation is applied to the governing equations to remove the time dependence. Then the transformed FEM equations for the structure are developed in matrix form and coupled to the transformed
BEM equations for the acoustic cavity. This coupling is accomplished directly by satisfying appropriate compatibility and equilibrium conditions at the interface. Among three commonly used types of BEM elements, namely constant, linear and quadratic, the advantages of the constant element in F E M BEM coupling will be demonstrated. Complete details of this hybrid formulation can be found in Ref. 14.
2 MATHEMATICAL FORMULATION AND NUMERICAL IMPLEMENTATION 2.1 FEM formulation for the structure
The Laplace transformed differential equation of motion for a general structure takes on the following form with zero initial conditions [s2[M] + s[C] + [K]]{~} = {0}
(1)
where s is the Laplace transform parameter, {~} is the Laplace transform of force, and {~} is the Laplace transform of displacement. By defining a distribution matrix IN], which is the interpolation functions used in the FEM model, it is possible to relate the force term {~} in eqn (1) to the actual values of forces, or {~} = IIflN]r{p} dA
(2)
where {if} is a column matrix of the actual values of the external pressure at the nodes. Now eqn (1), using eqn (2), can be represented symbolically as As~ = NIp
(3)
where As = [s2[M] + s[C] + [K]]
(4)
In order to combine the FEM and BEM models at the common interface, the compatibility and equilibrium conditions need to be satisfied. In addition, the finite element matrices must be transformed into equivalent boundary element matrices for the boundary element region. In this research, the structure is entirely in contact with the acoustic medium. As a result, the displacement fields i and ii represent the same vector. Therefore, eqn (3) can be reduced as follows:
where N[,~I are the shape functions and pressures, respectively, at the common interface.
Formulation for an FE and BE coupled problem 2.2 BEM formulation for the acoustic cavity
The Laplace transformed 3-D scalar wave equation is given by V2p(£, s) - ~/~(£, s) = 0
(6)
Using the second form of Green's theorem, eqn (6) can be written as
g(~,% s) ~ ( ~ , s)
(7)
where g(Y, jT,s ) = the fimdamental solution of the Laplace transformed 3-1) scalar wave equation and C = 1/2 for a smooth boundary. For a given system, the boundary is discretized into n elements. Discretization of the boundary produces the discretized scalar wave equation that can be used to predict the acoustic pressure response over an arbitrary boundary, or k= 1
JPk
,, apk , =
Imposing known values of ~ and ~a on the boundary, eqn (14) can now be rearranged in accordance with the unknowns under consideration. When all the unknowns are arranged on the left-hand side and the knowns are arranged on the fight-hand side, the following matrix equation is obtained:
[-H~ - ms2Gd o.~
p0S2Gk][ ~k ] (15)
y, cp(As) + Irp(Z,s ) ~.~(£, o .. s) dr(£)
=
307
~-b-~- Jr g(~,y., s) drk(£)
(8)
where Pk, ~ak are the known acoustic pressure and displacement on the boundary, Hk, Gk are matrices consisting of columns of H and G corresponding to the knowns, Pk and ~ak, on the boundary, 1~, Uau are the unknown pressure and acoustical displacement on the boundary and Hu, Gu are matrices consisting of columns of H and G corresponding to the boundary unknowns and 0auIntroducing the matrix notation, [G] = -pos2[G]
(16)
and considering the acoustic impedance and structure interface boundary conditions, substituting eqn (16) into eqn (15) yields [-- ["a]I [Gu]I -- ["u]B [(~u]B -- [Hu]Z [Gu]z] [ {Pu}I '
Introducing the following notation:
½+ Hkn =
Irk
ag(e, yn, s) dFk(£) Ot,I
fr °g(g' ~7"'s) dFk(£) k
On
n= k (9)
n#k
x
=[Hk _Gk]/Pk/F"1 {Oaa}B L~ J {~}B
(17)
and Gk,
[ g(£,y.,s)drk(g) ,11k
(10)
eqn (8) can be rewritten as
• Hkn~k
~
k=l
k=l
=
G
kn~~Ok
(ll)
Symbolically eqn (11) may be represented as H~ = Gpn
(12)
where the matrices H and G are obtained by evaluating eqns (9) and (10), respectively, and ~ and Pn correspond to the transformed pressure, p k , and normal derivative of the transformed pres.~,ure, o~k/On, respectively. The normal directional pres~mre distribution Pn inside of the acoustic cavity, using the acoustical displacement, fia, and the air density, Po, is given by Pn ~- --poS2Ua
(13)
Using eqn (13), rewriting eqn (12) gives -pos2Gfia = Hp
(14)
where [Hu]I, [(~u]I are matrices consisting of columns of [H] and [G] corresponding to unknown interface boundary values of {Pu}1 and {Qaa}I, respectively, [Ha!s, [(]u]n are matrices consisting of columns of [H] and [G] corresponding to unknown acoustic boundary values of {Pu}B and {aaa}B, respectively, and [Ha]z, [(]u]z are matrices consisting of columns of [I-I] and [G] corresponding to unknown impedance boundary values of {Pu}z and {0an}Z, respectively. 2.3 Coupling conditions
At the interface between the structure and acoustic cavity, the boundary conditions must satisfy the compatibility and equilibrium coupling conditions. Examining the boundary interface, it is observed that the displacement of the structure coincides with the displacement of the acoustic particle. The Laplace transformed compatibility condition at the interface is expressed using a shape function
308
C.-M. Lee, L. H. Royster, R. D. Ciskowski
[N] as
the coupled FEM and BEM system for the structureacoustic-cavity problem as follows:
Off = _poS2[N]{a } .n
(18)
On
As
0
NT
The equilibrium condition requires that the normal force produced by the vibrating structure at the interface be equal in magnitude and opposite in direction to the total acoustic pressure over the interface area for the enclosed cavity in the Laplace transformed domain,
0
0]
-[rInh {i},
i.e.
:[T Hko
×
0] (~k
Pk
(26)
iia*
. 2.4 Coupling the matrix equations In the analysis of an acoustic cavity, the acoustic impedance boundary condition is considered among unknown boundary conditions. The impedance boundary equation may be written in vector form as poS _
Pn = ---~-P
(20)
Then using eqns (13) and (20), a new displacement and pressure relationship including the impedance boundary condition is obtained: {fiau}Z = [Za]{~,}z
(21)
where 1 [Za] - ~-~
(22)
Before the FEM and BEM models can be coupled, using eqn (21), the BEM matrices of eqn (17) must be rewritten as
{i}I -[I-i.],
I
[G.]B [fio] ]
iUau}B[ • {p.}z I
=[Hk-Gk](
]~k ~k
(23)
where [~q] = [Gu]I[N] and
[I~] z = -[nu] z + [t~u]z[Za] (24)
2.5 Numerical transform inversion To obtain the steady-state response for the coupled system, a numerical formulation is used with s = i~v. For transient response, a numerical Laplace inversion is needed to recover the response in the time domain. Several methods have been suggested for the numerical inversion scheme, For the numerical implementation, Banerjee and Ahmad t5 used a FFT-based inversion algorithm which is suggested by Durbin) 6 However, in order to further reduce the computing time required, Wilcox and Gibson 17 presented a discrete Laplace transform pair for the analysis of the transient behavior. This algorithm requires far fewer sample points in the transform domain than are required by Durbin's algorithm. Actually, Wilcox's method takes more time during the Laplace transformation than the standard method (such as Durbin's) since the standard method utilizes the fast Fourier transform techniques. However, since Wilcox's method achieves savings in computing time if fewer than about 128 Laplace transform domain points are sufficient to discretize the given time domain, this method is utilized in this research. For the estimation of the transient response, external force functions are conveniently described in the time domain. The FEM-BEM coupled formulation requires, however, that this information be available in the Laplace transform domain. In particular, external forcing conditions are required at specific values of the complex valued Laplace transform parameter, s, as specified by the particular numerical inversion scheme employed. The algorithm used in this research is that of Narayanan and Beskos. is They described the piecewise linear excitation function on the Laplace transform domain as follows
The FEM matrices of eqn (5) also can be rewritten
as
~11 F(s)
=
i=1
1 s2 t i + l _
ti
[S(ti+ 1 -- t i ) ( f ( t i ) e
-st'
- f ( t i + l ) e -st'+') + (f(ti+l)
Combining eqns (23) and (25) gives the final form of
-- f(ti))(e
-sti - - e -st'+'
)]
(27)
Formulation for an FE and BE coupled problem
309
To determine the force conditions in the Laplace transform domain, eqn (27) is evaluated repeatedly using the information provided in l%e time domain.
Two BEM Elements & Nodes
2.6 Selection of boundary element type In discretizing a three-dhnensional cavity, two possible geometric shapes for the boundary element are triangular or quadrilateral with straight or curved edges. Selecting an element with a curved surface will produce more accurate modefing of the boundary of the physical problem. However, more effort will be required to numerically implement this system. The authors also felt that, for the types of geometries found in the applications to which their work would be applied, namely HPDs, the straight-edge element would provide an acceptable approximation to this geometry. For this reason and ease of nu~aerical implementation, a triangular shaped boundary element with straight edges was selected. Once the BEM element is chosen according to geometric shapes and curvatures, the order of the variation for the variables of the scalar wave equation over the element must be selected. There are three possible types of variation: constant, linear or quadratic. The element with a constant variation is usually called a constant element or a piston element, and it has its node at the centroid of the element. In order to define a linear element, only the vertices are specified. For higher order geometric variations like a quadratic element, additional poinl:s are needed on the edges. Because of the simplification of the numerical implementation and reasonab]le accuracy offered by the constant element, it has been used frequently in the boundary discretizatiorL of acoustic cavities even though the level of the: accuracy is more dependent upon the size of the element than with higher order elements. Higher order elements provide improved accuracy with fewer elements than are required by the constant element. However, higher order elements require more complicated numerical analysis and are more difficult to implement. When coupling BEM with FEM formulations, the selection of a constant BEM element type helps to avoid other numerical difficulties in the coupling process. If linear or quadratic BEM elements are selected for coupling to FEM elements, boundary conditions along the edge would h~tve to be addressed leading to numerical complexities. If, however, the constant BEM element is utilized for the discretization of the acoustic cavity, the boundary condition of the structure does not affect the coupling at the edge since the node of a constant BEM element would not coincide with the node of an FEM element at the edge. For the constant triangular BEM element,, three FEM triangular elements will be needed to couple with one BEM element since the node of the constant BEM element is located at the centroid of the element. Figure 1 shows the arrangement of
/
r Boundary Nodes of Structure
Internal Node of Structure or Acoustic Node
/ Three FEM Elements per BEM Element
Fig. 1. Node correlations between FEM and BEM elements during coupfing. FEM and BEM elements using six FEM elements and two BEM elements at the interface. For all of the above reasons, the constant BEM element is used in this research.
3 NUMERICAL RESULTS FOR THE STRUCTURE-ACOUSTIC-CAVITY SYSTEM
3.1 Box-type cavity with pinned-pinned plate 3.1.1 Steady-state response The F E M - B E M program developed herein is for any general shell-like structure coupled to an acoustic cavity. One example where this coupled program could be utilized is for the analysis of a flat plate coupled to an acoustic cavity. In order to verify the F E M - B E M coupled code, it will be used to model an acoustically hard-walled cubic cavity where one end consists of a simply-supported flat plate. The size of the cavity is 0.2 x 0.2 x 0.2m. The simply-supported flat plate is assumed to be made of brass with a thickness of 0-91 x 10-4m. The acoustic cavity is discretized into 96 constant triangular BEM elements with 50 elements at the interface between the plate and cavity. The plate is discretized into 150 triangular FEM elements in order to obtain three structural elements for each acoustic element at the interface as shown in Fig. 2. To obtain visual images of the mode shapes of the plate, the 256 element discretization was utilized for better
310
C.-M. Lee, L. H. Royster, R. D. Ciskowski
/
/
/l 'y )cx
be f i
f
96 Boundary Element
150 Finite Element
Discretization
Discretization
Fig. 2. Boundary element discretization for the rectangular box cavity and finite element discretization for the pinned-pinned plate. resolution in drawing the mode shapes. The refined discretization yielded no significant improvement in the predicted plate natural frequencies. The FEM calculated mode shape for each natural frequency is presented in Fig. 3. One approach to verifying accurate coupling between the FEM and BEM models involves statically loading the pinned-pinned end plate and then comparing the predicted increase in the internal volume's pressure with the known analytical solution. This comparison is shown in Table 1. Good agreement was obtained. Next, the steady-state pressure response of the coupled plate-box cavity system is investigated by applying a harmonic pressure excitation on the structure elements at the outside of the box and varying the pressure's excitation frequency from 10Hz up to 1 kHz. The predicted pressure response is calculated at the center of the acoustic element at the inside end of the box (0.2, 0.1, 0.1 m). An analytical solution for this problem was obtained by Guy and Bhattacharya. 19 They also experimentally investigated this system for steady-state pressure excitation.-Figure 4 shows the steady-state response comparing the numerically and the analytically predicted 201og(Pi/P2) at the receiver point located on the center of the acoustic element Table I. Static pressure response of a cavity backed plate system under static uniform pressure excitation (external pressure P1 = 11~), (0"2 x 0"2 x 0-2 m box)
Analytical FEM-BEM
Static pressure response inside of the box
Pressure difference ratio 201oglo(P2/ P l )
0.179 psi 0-180 psi
14-9dB 14.8 dB
at the inside end of the box. The analysis of the steady-state response reveals that the predicted panelcavity resonant frequencies are 90Hz, 400Hz, 680 Hz and 860 Hz. The first three of these natural frequencies correspond to the (1,1), (1,3) and (3,3) symmetrical panel modes, respectively. The 860 Hz is due to the fundamental plane wave room resonant frequency. The predicted steady-state response indicates good agreement between the F E M ' B E M model and the analytical solution 19 with typically less than 2dB differences noted. Furthermore, the predictions indicate good agreement with the experimental data. 3.1.2 Transient response Using the same plate-cavity configuration (0.2 x 0.2 x 0.2m) as was introduced in section 3.1.1, the transient response of the coupled system is now predicted using the F E M - B E M code. A triangular pressure impulse of a simulated gun shot is applied to the cavity backed plate. The assumed gun shot time history is presented in Fig. 5. The determined parameters for the time history signature are rl = 1.4 x 10-3 s, 7"2 3.1 x 10-3s, 7"3 = 3.4 x 10-3s and Pmax = 0"29psi, Pmin = -0"0725 psi. The predicted pressure response at the receiver point (0"2, 0" 1, 0" 1 m), is plotted against the analytical prediction in Fig. 6 for the assumed gun shot time-history. The F E M - B E M predictions are in good agreement with the analytical solution which was obtained by applying the numerical inverse Laplace transform to the Laplace transformed pressure response of the cavity backed plate. The description of the Laplace transformed transient pressure response of the cavity backed plate on the Laplace transform domain can be found in Ref. 14. The approximate period of the =
Formulation for an FE and BE coupled problem
77 Hz
311
(2, i )
193 Hz
(l, 2)
(~. I)
302 Hz
488 Hz
488 !-~
(2,2)
(2.3)
(3.2)
(i ,3)
386 Hz (3, i)
Fig. 3. Mode shapes of the end plate for the rectangular box configuration. resulting pressure response is 0.012 s. Therefore the frequency associated with the pressure wave is about 84Hz and is close to the fundamental frequency for the pinned-pinned panel. The resulting noise reduction is 17 dB.
3.2 Application to the a~aalysis of the earmuff-earcanal system
3.2.1 Steady-state response The coupled F E M - B E M model of the structureacoustic-cavity system has been verified using the boxtype cavity with pirmed-pirmed end plate. The model will now be applied to predict the response of a practical, real-world example of a structure-acoustic-cavity system, namely the eartauff-earcanal system in Fig. 7. The earmuff normally consists of a plastic hemispherical-shaped earcup with a cushion for sealing the
earcup to the flesh around the ear. When the earmuff is worn, an air cavity is created within the earcup. This cavity includes the extension into the earcanal up to the eardrum. The eardrum represents an impedance for the cavity as shown in Fig. 8. Data for the impedance of a typical eardrum are provided in Ref. 20. In a normal situation where the earmuff is worn for hearing protection, an acoustic pressure wave impinges on the outside surface of the earcup, causing it to vibrate. This vibration excites the interior cavity which is then transmitted to the eardrum. An effective earmuff design attenuates or reduces the pressure levels experienced by the eardrum compared to what impinged on the external surface of the earmuff. For the earmuff-earcanal system modelled, the earmuff is selected to be an EAR Model 1000.21 Dimensions for the system are given in Fig. 8. The earcup
C.-M. Lee, L. H. Royster, R. D. Ciskowski
312
i
Ill =
=
x
FEM-BEM ANALYTICAL EXPERIMENTA (G,-,.,, et ~_!)
:
: FEM-.E.
Ill 6
1. ~o
g L I1 P""° wi.,, = 0.0034 s¢¢.
I
(294 Hz)
° ° , ,,
1 oo
10
0
looo
0.01
0.02 TIME (sec.)
FREQUENCY (Hz)
Fig. 4. Comparison of numerical, analytical, and experimental predictions of 201og(P2/P1) for the rectangular box (0.2 x 0.2 x 0-2 m) with one end a pinned-pinned plate. was discretized using 132 flat shell finite elements as shown in Fig. 9. The earcup sealing cushion was modeled using linear spring elements connected to the element nodes at the base of the earcup. The acoustic cavity, consisting of the earcup and earcanal volumes, was discretized using 100 boundary elements. The boundary of the cavity not coupled to the earcup is assumed to be acoustically hard except for one portion where the eardrum impedance is located. The pressure wave impinging on the earcup results in an applied pressure on the flat shell elements. The internal pressure level response is predicted at the
0.0..3
Fig. 6. Comparison of predicted transient response against analytical results for the rectangular box (0-2 x 0.2 x 0.2m) with pinned-pinned end plate.
/
E*rmuf~ Cushion
~ _ _
Acoustic Pressure Wave
~
~
Acoustic Cavity
Fig. 7. Earmuff-earcanal system. eardrum receiver location (6"2, 0-1, 0.1 cm) over the
frequency range of 100Hz, to 5kHz. The predicted acoustic pressure response of the earmuff-earcanal system is validated by several criteria. b-
10.2 Cm
i.-- -I LtJ t~ be) 03 ~ hJ tie" O..
4.5 c m - - e ~
-..I i I 1.glcm
eareup
cavity
I
ca
al ......
2.8,Cm L
TIME (see.)
Fig. 5. Time history of assumed gun shot transient input pressure time response.
Fig. 8. Dimensions of acoustic cavity of earmuff-earcanal system.
Formulation for an FE and BE coupled problem
132
affected by the rigid body motion of the earcup on the acoustic cavity and the sealing cushion. The mass of the earmuff and the effective spring constants of the acoustic cavity and the sealing cushion together act like a single degree-of-freedom system. The response of a single degree-of-freedom system generally consists of the resonant frequency peak with a stiffnesscontrolled region to the left (lower frequencies) and a mass-controlled region to the right. Given the mass and stiffness of the single degree-offreedom system, straight forward calculations ~ can be used to arrive at the resonant frequency, static deflection in the stiffness-controlled region, and slope (or fall off) in the mass-controlled region. With the mass of the earmuff at 1.48 x 10-4 kg and the effective spring constants of the acoustic cavity at 103 Nm and the sealing cushion at 6Nm, the resonant frequency peak should be at 137Hz, the static pressure should be 0.719psi, and the slope should be 12dB/octave. The F E M - B E M model predicted 140Hz, 0.741psi and 12.4dB/octave. Model values are within 3% of calculated values.
Irmi~ E l ~ n t Diseretizafionof Cup
tO0B ~ a = 7 E I ~
~
o
o
of Cavit7
Fig. 9. Discretization of earmuif-earcanal system.
The response is characte]~_ed by two primary types of behavior. Above 1 kHz, tbe response is the result of the coupling of the vibration of the earcup with that of the acoustic cavity. The two major structure resonances, 2200 Hz and 4150 Hz, indicated in Fig. 10 are associated with the first and second t;ymmetrical vibrational modes of the earmuff's shell. Below 1 kHz, the response is 0 0
e FEM-BE M
o-
0 L~ m
/\
K
T
n v
t71 0 0 O4 0 I
0 0
,
110
,
0
.,,.
J
100
.
.
0
.=.
1 000
313
04
FREQUENCY (Hz)
Fig. 10. Steady-state response of earmuff-earcanal system for uniform external pressure.
3.2.2 Transient response A triangular pressure impulse of a simulated gun shot is applied uniformly over the shell to get the transient response at the eardrum with eardrum impedance. The assumed pulse shape was presented in Fig. 5. The assumed shape parameters are 7-I = 1.4 x 10-3 s, r 2 = 3.1 x 10-°s, r3 = 3.4 x 10-3 s and Pmax = 9"16psi, Pn~n = - 1"67psi. For the duration of the impulse, if we assume that rigid body motion is dominant, the earmuff-earcanal system may be modeled as a single degree-of-freedom system for analytical comparison purposes. The mass of the earmuff and the effective supporting spring stiffnesses such as the effective spring constant for the acoustic cavity and the spring constant for the cushion are assumed to act as a single degree-of-freedom. The acoustic impedance at the eardrum is assumed as the effective damping constant. The displacement response of the single degree-of-freedom system on the frequency domain is converted to the acoustic pressure response assuming the earmuff internal cavity as an equivalent rectangular box cavity. The calculated pressure response of the single degree-of-freedom system on the time domain for analytical purposes is obtained from the frequency domain using the numerical inversion scheme. The resulting predicted transient pressure time history responses from the F E M - B E M coupled code and the analytical result at the eardrum is shown in Fig. 11. The period of the predicted pressure response is 0.0075s, which translates into a basic frequency of 133Hz. A good agreement is indicated when the F E M - B E M result is compared with the analytical prediction.
314
C.-M. Lee, L. H. Royster, R. D. Ciskowski tO
1 - O ANALYTICAL Excitafioa Pulse
=
= F E M - BEM
NRp¢~= 3.3
AA "t'?' q")
03
~tO W 2~ ~ W
dB
0
'
'
,
'
°
I
I
,
0.015
0.02
,
Pulse Width = 0.0034 sec. o
I 0
I
(,2.94
5 x 1 0 -3
0.01
TIME (see.)
Fig. 11. Transient response of earmuff-earcanal system for the duration of the impulse (0.0034 s). 4 CONCLUSIONS A coupled F E M and BEM model has been developed to predict the steady-state and transient responses for structure-acoustic-cavity systems. The model was validated using a box-type cavity with one pinned-pinned end plate. G o o d agreement between the model and available analytical or experimental data for the configuration studied has been achieved for steady-state and transient responses. For the first time, such an F E M - B E M model has been applied to the analysis of the practical problem involving hearing protection, namely an earmuff-earcanal system. Preliminary results appear good. Future work will compare with experimental results.
ACKNOWLEDGEMENTS This work was supported by Kia Motors Corp. of South Korea. The first author also received support from the Mechanical Engineering Department o f North Carolina State University and a scholarship from the N o r t h Carolina Regional Chapter of the Acoustical Society of American and the Cabot Foundation.
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Formulation for an FE and BE coupled problem 20. Schroeter, J. & Poesselt, C. The use of acoustical test fixtures for the measurement of hearing protector attenuation. Part II: Modeling the external ear, simulating bond conduction, and comparing text fixture and real-ear data. J. Acoust. Soc. Amer., 15686,80(2), 505-27.
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21. Berger, E. H., Ward, W. D., Morrill, J. C. & Royster, L. H. Noise and hearing conservation manual, 4th edn. American Industrial Hygiene Association, 1986. 22. Harris, C. M. Shock Vibration Handbook. Cyril M. Harris, 1988.