A Compact Matrix Formulation Using the Impedance and Mobility Approach for the Analysis of Structural-Acoustic Systems
0022Â±460X/99/210097 Â‡ 17 $30.00/0 # 1999 Academic Press
A COMPACT MATRIX FORMULATION USING
THE IMPEDANCE AND MOBILITY APPROACH
FOR THE ANALYSIS OF STRUCTURAL^
ACOUSTIC SYSTEMS
S. M. KIM AND M. J. BRENNAN
Institute of Sound and Vi
ation Research, University of Southampton,
Southampton, Hampshire, SO17 1BJ, U.K.
(Received 20 March 1997, and in Â®nal form 16 December 1998)
This paper describes a compact matrix formulation for the steady-state
analysis of structuralÂ±acoustic systems. A new approach to the problem is
adopted that uses impedance and mobility methods commonly found in the
analysis of purely structural or purely acoustic systems. The advantages of the
approach are that an investigation into the coupling between the structural and
acoustic systems is made easier, and it facilitates improved physical insight into
the behaviour of structuralÂ±acoustic systems. In addition, because the equations
describing the complete system are in matrix form, they can be solved easily
using a computer. Due to the mismatch of dimensions between structural
mobility and acoustic impedance, new terms are introduced for the coupled
system analysis; the coupled acoustic impedance and the coupled structural
mobility. FÂ±u (forceÂ±velocity) and pÂ±Q (pressureÂ±source strength) diagrams are
also introduced for impedance and mobility representations of a complete
coupled system. Experimental work is presented, in which a simple rectangula
acoustic enclosure with Â®ve rigid and one Â¯exible side was used, to validate the
analytical model and to investigate structuralÂ±acoustic coupling.
# 1999 Academic Press
1. INTRODUCTION
The interaction between an acoustic space and its Â¯exible boundaries is an
important problem in the Â®eld of acoustics. Analysis of this interaction has been
of interest to many researchers during the last half a century, as reviewed by Pan
et al. [1, 2] and Hong and Kim [3]. A comprehensive theoretical model fo
coupled responses in a structuralÂ±acoustic coupled system has been presented by
Dowell et al. [4]. They provided solutions for coupled responses in terms of the
modal characteristics of the uncoupled structural and acoustic systems. This
paper considers the analysis of a similar coupled system, but uses the impedance-
mobility approach, which results in a compact matrix formulation. The
impedance-mobility approach is well known to electrical engineers and physicists
and is particularly applicable to the analysis of coupled systems, which are
Journal of Sound and Vi
ation XXXXXXXXXX), 97Â±113
Article No. jsvi XXXXXXXXXX, available online at http:
www.ideali
ary.com on
98 S. M. KIM AND M. J. BRENNAN
composed of several individual linear systems. Each system at the connection
can be characterised by impedance or mobility, and the dynamics of a complete
coupled system can be described at some or all of the points of interest. They are
particularly useful concepts to judge the degree of coupling when two or more
systems are connected, and are often used for the analysis of electrical systems
[5]. In the 1950s, the method was adapted by mechanical engineers, who applied
it to mechanical vi
ation problems [6]. A general theory of the approach and
application examples to mechanical systems can be found in reference [7].
Furthermore, the approach has been successfully applied to various sound and
vi
ation problems, such as the coupling between actuators and substructures,
sound radiation from a plate to the acoustic free Â®eld, and wave propagation
through media with different physical properties, as can be found in many
textbooks, for example references [8Â±10].
In this paper, the classical theory by Dowell et al. [4] is re-examined from the
impedance-mobility point of view, and a general method of structuralÂ±acoustic
coupling analysis is presented. The basic theory of the impedanceÂ±mobility
approach is considered in Section 2 with a simple conceptual structuralÂ±acoustic
coupled system. In fact this represents the coupling between a single structural
mode and a single acoustic mode. The approach is extended in Section 3 to
u
ZCA
ZS
FS
F
FA
YCS
QS
YA
p
(b)
(a)
Q
Figure 1. Impedance and mobility representation of a conceptual structuralÂ±acoustic system
excited by (a) structural excitation, force F, and (b) acoustic excitation, source strength Q. (a) FÂ±u
epresentation for structural excitation; (b) pÂ±Q representation for acoustic excitation.
STRUCTURALÂ±ACOUSTIC COUPLED SYSTEM 99
analyse general structuralÂ±acoustic coupled systems in modal co-ordinates. The
methodology can also be applied to structuralÂ±acoustic systems described by
their physical co-ordinates, and this has been described in detail by Kim [11]. A
criterion to establish whether or not a structuralÂ±acoustic system is strongly o
weakly coupled is proposed in Section 3, and this criterion is presented in terms
of acoustic impedance and structural mobility. In Section 4, some experimental
esults are presented that validate the analytical model developed, and illustrate
the effects of structuralÂ±acoustic coupling. Finally, some conclusions are drawn
in Section 5. There is also an Appendix to this paper that gives the relevant
equations for the model problem used in the simulations and the experimental
work.
2. BASIC THEORY OF THE IMPEDANCE-MOBILITY APPROACH
In this section a simple model of a conceptual structuralÂ±acoustic system is
described, which forms the basis of the comprehensive model of a general
structuralÂ±acoustic system discussed in Section 3. The conceptual model could,
in fact, be used to describe the behaviour of a single structural mode coupled
with a single acoustic mode.
In a single input structural system, the frequency domain quantities of
mobility YS and impedance ZS are deÂ®ned as [7]:
YS Âˆ u
F
, ZS Âˆ F
u
, Â…1a, bÂ†
where the subscript S denotes the structural system and F and u are applied
force and resulting velocity, respectively. In a single input acoustic system, the
impedance and mobility are deÂ®ned as [9]:
ZA Âˆ p
Q
, YA Âˆ Q
p
, Â…2a, bÂ†
where the subscript A denotes the acoustic system and Q and p are the source
strength and acoustic pressure, respectively. It is important to note that the
dimensions of impedance and mobility in structural and acoustic systems are
different; the dimension of structural impedance being [Ns/m] and the dimension
of acoustic impedance being [Ns/m5]. This dimension difference makes the
theoretical description for structuralÂ±acoustic coupled systems different from
that for general mechanical systems considered in textbooks, for example
eference [7].
Consider the conceptual structuralÂ±acoustic coupled system consisting of
impedances ZS and ZCA excited by a single known structural force F as shown in
Figure 1(a). The impedance ZS is deÂ®ned as the uncoupled structural impedance
and is the ratio of the effective force applied to the structure FS to the velocity u.
The impedance ZCA represents the acoustic reaction force FA to the structural
input velocity u and may be deÂ®ned as the coupled acoustic impedance. Thus,
ZS Âˆ FS
u
, ZCA Âˆ FA
u
: Â…3a, bÂ†
100 S. M. KIM AND M. J. BRENNAN
Using the force equili
ium condition, F=FS+FA, one gets an expression fo
the velocity of the structure u in terms of the structural mobility YS and the
coupled acoustic impedance ZCA.
u Âˆ YS
1Â‡ YSZCA F, Â…4Â†
where YS=1/ZS. When a single acoustic source of strength Q excites the
conceptual structuralÂ±acoustic coupled system, it can be represented by the series
combination of mobilities YA and YCS as shown in Figure 1(b). Hereafter it is
called the pÂ±Q representation since the physical parameters are pressure and
source strength, while the diagram in Figure 1(a) is called the FÂ±u representation,
i.e., the forceÂ±velocity representation. The mobility YA is deÂ®ned as the
uncoupled acoustic mobility and is the ratio of the effective source strength QA
acting on the acoustic system to the acoustic pressure p. The mobility YCS
epresents the induced structural source strength QS to the acoustic pressure p
and is deÂ®ned as the coupled structural mobility. thus,
YA Âˆ QA
p
, YCS Âˆ Ã¿QS
p
: Â…5a, bÂ†
Note the minus sign of YCS because the direction of QS is deÂ®ned opposite to the
u
ZCAYS
SZAQF
QS
ZAYCS
SYSF
p
Q
(b)
(a)
Figure 2. FÂ±u and pÂ±Q representation for the conceptual structuralÂ±acoustic system with both
structural and acoustic excitation. (a) FÂ±u representation; (b) pÂ±Q representation.
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STRUCTURALÂ±ACOUSTIC COUPLED SYSTEM 101
acoustic pressure. Since both source strengths are acting toward the acoustic
system, the effective source strength acting on this system is QA=Q+QS. Thus,
p Âˆ ZA
1Â‡ ZAYCS Q, Â…6Â†
where ZA=1/YA. When there is both force and acoustic excitation, a coupling
factor, which connects the FÂ±u and the pÂ±Q representations is required. For the
conceptual system studied in this section an area S may be simply used to match
the dimensions. Thus, the relationship between the coupled and uncoupled
acoustic impedances and the coupled and uncoupled structural mobilities are
given by:
ZCA Âˆ S2ZA, YCS Âˆ S2YS: Â…7a, bÂ†
Conversions between the FÂ±u representation and the pÂ±Q representation can be
achieved by using Thevenin and Norton's theorems [7]. The FÂ±u and the pÂ±Q
epresentations for the conceptual structuralÂ±acoustic system subject to both
structural and acoustic excitation are given in Figure 2(a) and (b), respectively.
The equations relating the structural velocity and the acoustic pressure to the
applied force and acoustic strength are given by:
u Âˆ 1
1Â‡ YSZCA YSÂ…FÃ¿ SZAQÂ†, p Âˆ
1
1Â‡ ZAYCS ZAÂ…QÂ‡ SYSF Â†: Â…8a, bÂ†
These are the key equations for the analysis of general coupled systems, and can
e extended to vector and matrix forms to deal with multi-degree-of-freedom
systems with several excitation points. In Section 3 these equations are expanded
to model a general structuralÂ±acoustic system.
If the system is excited by a structural source and the structure responds
predominantly as though it was in vacuo then the coupled acoustic impedance
has a negligible effect on the structure. In this case the system is said to be
weakly coupled. Moreover, if the system is excited acoustically and the cavity
esponds predominantly as though the structure were inÂ®nitely rigid it is also
said to be weakly coupled. These conditions can be examined mathematically
using equations (8a) and (8b). If one sets Q=0 in equation (8a) then u=YSF
provided that |YSZCA|5 1 so that one can set YSZCA=0. If one sets F=0 in
equation (8b) then p=ZAQ provided that |ZAYCS |5 1 so that one can set
ZAYCS=0. Noting the relationship between the coupled and the uncoupled
acoustic impedance and structural mobility in equations (7a) and (7b), one can
see that YSZCA=ZAYCS, and thus the condition for weak coupling is
independent of the type of excitation. If there is both structural and acoustic
excitation in a weakly coupled conceptual system then the equations for the
structural velocity and acoustic pressure are given by
u Âˆ YSÂ…FÃ¿ SZAQÂ†, p Âˆ ZAÂ…QÂ‡ SYSFÂ†: Â…8c, dÂ†
The FÂ±u and the pÂ±Q representations for a weakly coupled conceptual
structuralÂ±acoustic system are shown in Figure 3(a) and (b), respectively.
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102 S. M. KIM AND M. J. BRENNAN
3. STRUCTURALÂ±ACOUSTIC COUPLING THEORY IN MODAL
COORDINATES
In this section the impedance and mobility approach described in Section 2 is
used to analyse the dynamic behaviour of an a
itrary shaped enclosure
su
ounded by a Â¯exible structure and an acoustically rigid wall such as that
shown in Figure 4. The acoustic source strength density function s(x, o) and the
force distribution function f(y, o) excite the cavity and the Â¯exible structure,
espectively. Co-ordinate x is used for the acoustic Â®eld in the cavity, and co-
ordinate y is used for vi
ation on the structure.
It is assumed that coupled responses can be described by Â®nite sets of
uncoupled acoustic and structural modes. The uncoupled modes are the rigid-
walled acoustic modes of the cavity and the in vacuo structural modes of the
structure. Full coupling is considered between the Â¯exible structure and the
acoustic cavity system. However, weak coupling is assumed between the Â¯exible
structure and the acoustic Â®eld outside the cavity. This is because it is assumed
that the vi
ation of the structure is not inÂ¯uenced by the radiated acoustic Â®eld
outside the cavity.
The acoustic pressure and the structural vi
ation are described by the
summation of N and M modes, respectively. Hence, both the acoustic pressure p
at x inside the enclosure and the structural vi
ation velocity u at y are given by
[4]:
u
YS
SZAQ
F
QS
ZA
SYSF
p
Q
(b)
(a)
Figure 3. FÂ±u and pÂ±Q representations for a weakly coupled conceptual structuralÂ±acoustic
system with both structural and acoustic excitation. (a) FÂ±u representation; (b) pÂ±Q represen-
tation.
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STRUCTURALÂ±ACOUSTIC COUPLED SYSTEM 103
pÂ…x, oÂ† Âˆ
XN
nÂˆ1
cnÂ…xÂ†anÂ…oÂ† Âˆ CCCTa, Â…9aÂ†
uÂ…y, oÂ† Âˆ
XM
mÂˆ1
fmÂ…yÂ†bmÂ…oÂ† Âˆ FFFTb, Â…9bÂ†
where, the N length column vectors CCC and a consist of the a
ay of uncoupled
acoustic mode shape functions cn(x) and the complex amplitude of the acoustic
pressure modes an(o), respectively. Likewise the M length column vectors FFF and
consist of the a
ay of uncoupled vi
ation mode shape functions fm(y) and
the complex amplitude of the vi
ation velocity modes bm(o), respectively. The
superscript T denotes the transpose. The mode shape functions cn(x) and fm(y)
satisfy the orthogonal property in each uncoupled system, and are normalised as
follows:
V Âˆ
Â…
V
c2nÂ…xÂ† dV, Sf Âˆ
Â…
Sf
f2mÂ…yÂ† dS, Â…10a, bÂ†
where V and Sf are the volume of the cavity and the surface area of the Â¯exible
structure, respectively. The complex amplitude of the nth acoustic mode unde
structural and acoustic excitation is given by [4, 12]:
anÂ…oÂ† Âˆ roc
2
o
V
AnÂ…oÂ†
Â…
V
cnÂ…xÂ†sÂ…x, oÂ† dVÂ‡
Â…
Sf
cnÂ…yÂ†uÂ…y, oÂ† dS
!
Â…11Â†
where r0 and co denote the density and the speed of sound in air, respectively.
Volume V
Acoustically rigid