Hi Guys
While on the subject of Revtex I thought I'd point out a bug I found. It maybe
Lyx or Revtex. When I put a boldface character in an equation, when I print
it makes all characters that follow also bold upright which is incorrect even
though it is shown correctly in Lyx! Please see attached file for example.
Regards
Ben
--
_________________________________________
Ben Cazzolato
Fluid Dynamics and Acoustics Group
Institute of Sound and Vibration Research
University of Southampton,
Southampton, SO17 1BJ
UK
Email: [EMAIL PROTECTED], or
[EMAIL PROTECTED], or
[EMAIL PROTECTED], or
Work: +44 (0)1703 594 967
Fax: +44 (0)1703 593 190
Mobile: +44 (0)790 163 8826
Web Page : http://www.soton.ac.uk/~bscazz/
_________________________________________
#This file was created by <bscazz> Thu Aug 19 14:32:02 1999
#LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
\lyxformat 2.15
\textclass revtex
\begin_preamble
\pagestyle{myheadings} \markright{B.S. Cazzolato \& C.H. Hansen - J.A.S.A.}
% The following command gives single spaced preprints
% \tighten
% The folowing places the word Draft in the top RH corner of the first page
% \preprint{Draft}
\end_preamble
\options aps,manuscript
\language default
\inputencoding default
\fontscheme times
\graphics dvips
\paperfontsize default
\spacing single
\papersize a4paper
\paperpackage a4
\use_geometry 0
\use_amsmath 0
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation skip
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle headings
\layout Title
\added_space_top bigskip
Structural radiation mode sensing for active control of sound radiation
into enclosed spaces
\layout Author
\added_space_top bigskip
Ben S.
Cazzolato and Colin H.
Hansen
\layout Address
Department of Mechanical Engineering, University of Adelaide, SA, 5005,
AUSTRALIA.
\layout Standard
\added_space_top 5cm \added_space_bottom 0.3cm \pagebreak_bottom \align center \LyXTable
multicol5
4 2 0 0 -1 -1 -1 -1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
2 0 0 "" ""
8 0 0 "" ""
0 2 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
Total
\protected_separator
Number
\protected_separator
of
\protected_separator
Pages:
\newline
12
\newline
Number
\protected_separator
of
\protected_separator
Figures:
\newline
0
\newline
Number
\protected_separator
of
\protected_separator
Tables:
\newline
0
\newline
Number
\protected_separator
of
\protected_separator
Copies:
\newline
4
\layout Abstract
In the recent article by Cazzolato and Hansen [
\begin_inset Quotes eld
\end_inset
Active control of sound transmission using structural error sensing,
\begin_inset Quotes erd
\end_inset
J.
Acoust.
Soc.
Am 104, 2878-2889
\series medium
(
\series default
1998
\series medium
)]
\series default
it was shown that it is possible to derive for a structure some set of
surface velocity distributions, referred to as radiation modes, which are
orthogonal in terms of their contributions to the acoustic potential energy
of a coupled cavity.
The technique used an orthonormal decomposition to derive an expression
for the radiation modes which was based on prior work for free-field sound
radiation.
It will be shown in the following article that for the special case involving
the calculation of global internal potential energy it is possible to use
a simple approach which requires no orthonormal decomposition since the
expression for the global potential energy is already in a form that can
be easily diagonalised.
\layout Abstract
PACS numbers: 43.50.Ki
\layout Standard
\latex latex
\backslash
draft
\layout Standard
\latex latex
\backslash
pacs{989}
\layout Section
\pagebreak_top
Introduction
\layout Standard
In the recently published article by Cazzolato and Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
an expression for the structural radiation modes orthogonal to the acoustic
potential energy in an enclosure was derived.
The approach used an orthonormal decomposition of the acoustic error weighting
matrix,
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
, to obtain the eigenvectors,
\begin_inset Formula \( \mathbf{U} \)
\end_inset
, and eigenvalues,
\begin_inset Formula \( \mathbf{S} \)
\end_inset
, of the matrix (ie,
\begin_inset Formula \( \boldsymbol \Pi =\mathbf{Z}^{\mathrm{H}}_{a}\boldsymbol \Lambda \mathbf{Z}_{a}=\mathbf{USU}^{\mathrm{T}} \)
\end_inset
, where
\begin_inset Formula \( \mathbf{Z}_{a} \)
\end_inset
is the structural-acoustic transfer function matrix and
\begin_inset Formula \( \boldsymbol \Lambda \)
\end_inset
is a diagonal weighting matrix).
The approach taken was primarily because of precedence, since the technique
had been used in the past for radiation into the free space.
However, on closer inspection of the governing equations it can be shown
that if global control of the acoustic space is the objective of an active
noise control system, then it is not necessary to decompose the radiation
matrix.
By collecting the appropriate terms that comprise the radiation matrix,
a pseudo eigenvector-eigenvalue decomposition is obtained, and while not
truly orthonormal, it does result in an adequate approximation of the exact
orthogonal data set.
\layout Standard
As already stated, the following approach is only applicable to radiation
modes which are orthogonal in terms of their contributions to the total
acoustic potential energy in the acoustic space, to cases for which the
modal density is low and the flexible structure forms a large part of the
bounding surface of the structure.
If the latter two conditions do not hold, then at high frequencies the
internal radiation mode shapes degenerate to approximately the free field
radiation mode shapes
\begin_inset LatexCommand \cite{Johnson:1996}
\end_inset
.
This is because the assumption (used in the current simplification) that
the acoustic mode shapes integrated over the radiating surface are orthogonal,
no longer holds.
As mentioned in passing by Cazzolato and Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
, it is possible to derive a set of radiation modes which are orthogonal
in terms of their contributions to the potential energy in some subspace
of the interior cavity, such as the space around a passenger's head.
The alternative approach which is presented here is not suitable for such
subspaces and the previous formulation
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
must be used.
\layout Section
Previous Formulation
\layout Standard
In the paper by Cazzolato and Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
, the theory of sound transmission through a structure into a contiguous
cavity was developed with the transmitted sound field derived in terms
of radiation modes.
Using a modal-interaction approach to the solution of coupled problems,
the response of the structure was modelled in terms of its
\shape italic
in vacuo
\shape default
mode shape functions and the response of the enclosed acoustic space was
described in terms of the rigid-wall mode shape functions
\begin_inset LatexCommand \cite{Fahy:1985}
\end_inset
.
The response of the coupled system was then determined by solving the
modal formulation of the Kirchhoff-Helmholtz integral equation.
The following two sections have been taken directly from Cazzolato and
Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
for the sake of completeness.
\layout Subsection
\begin_inset LatexCommand \label{General-Theory}
\end_inset
Global error criteria
\layout Standard
An appropriate global error criterion for controlling the sound transmission
into a coupled enclosure is the total time-averaged frequency dependent
acoustic potential energy,
\shape italic
\begin_inset Formula \( E_{p}(\omega ) \)
\end_inset
\shape default
, in the enclosure
\begin_inset LatexCommand \cite{Nelsonetal:1987c}
\end_inset
\begin_inset Formula
\begin{equation}
\label{eqn:A}
E_{p}(\omega )=\frac{1}{4\rho _{0}c_{0}^{2}}\int _{V}|p(\vec{\mathbf{r}},\omega )|^{2}d\vec{\mathbf{r}}
\end{equation}
\end_inset
where
\begin_inset Formula \( p(\vec{\mathbf{r}},\omega ) \)
\end_inset
is the acoustic pressure amplitude at some location
\begin_inset Formula \( \vec{\mathbf{r}} \)
\end_inset
in the enclosure,
\shape italic
\begin_inset Formula \( \rho _{0} \)
\end_inset
\shape default
is the density of the acoustic fluid (air),
\shape italic
\begin_inset Formula \( c_{0} \)
\end_inset
\shape default
is the speed of sound in the fluid and
\shape italic
\begin_inset Formula \( V \)
\end_inset
\shape default
is the volume over which the integral is evaluated.
The frequency dependence,
\begin_inset Formula \( \omega \)
\end_inset
, is assumed in the following analysis but this parameter will be omitted
in the following equations for the sake of brevity.
Using the modal interaction approach to the problem
\begin_inset LatexCommand \cite{Fahy:1985}
\end_inset
, the acoustic pressure at any location within the cavity is expressed as
an infinite summation of the product of rigid-wall acoustic mode shape
functions,
\begin_inset Formula \( \phi _{i} \)
\end_inset
, and the modal pressure amplitudes,
\begin_inset Formula \( p_{i} \)
\end_inset
, of the cavity
\begin_inset Formula
\begin{equation}
\label{eqn:b}
p(\vec{\mathbf{r}})=\sum _{i=1}^{\infty }p_{i}\phi _{i}(\vec{\mathbf{r}})
\end{equation}
\end_inset
The modal expansion for the acoustic potential energy evaluated over
\shape italic
\begin_inset Formula \( n_{a} \)
\end_inset
\shape default
acoustic modes is then given by
\begin_inset Formula
\begin{equation}
\label{eqn:c}
E_{p}=\mathbf{p}^{\mathrm{H}}\boldsymbol \Lambda \mathbf{p}
\end{equation}
\end_inset
where
\series bold
\begin_inset Formula \( \mathbf{p} \)
\end_inset
\series default
is the (
\shape italic
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
\shape default
x
\protected_separator
1) vector of acoustic modal amplitudes and
\begin_inset Formula \( \boldsymbol \Lambda \)
\end_inset
is a (
\shape italic
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
\shape default
x
\protected_separator
\shape italic
\begin_inset Formula \( n_{a} \)
\end_inset
\shape default
) diagonal weighting matrix, the diagonal terms of which are
\begin_inset Formula
\begin{equation}
\label{eqn:d}
\Lambda _{ii}=\frac{\Lambda _{i}}{4\rho _{0}c_{0}^{2}}
\end{equation}
\end_inset
where
\begin_inset Formula \( \Lambda _{i} \)
\end_inset
is the modal volume of the
\begin_inset Formula \( i \)
\end_inset
th cavity mode, defined as the volume integration of the square of the mode
shape function,
\begin_inset Formula
\begin{equation}
\label{eqn:e}
\Lambda _{i}=\int _{V}\phi _{i}^{2}(\vec{\mathbf{r}})dV(\vec{\mathbf{r}})
\end{equation}
\end_inset
\layout Standard
The pressure modal amplitudes,
\series bold
\begin_inset Formula \( \mathbf{p} \)
\end_inset
\series default
, within the cavity, arising from the vibration of the structure are given
by the product of the (
\begin_inset Formula \( n_{s} \)
\end_inset
\protected_separator
x
\protected_separator
1) structural modal velocity vector,
\series bold
\begin_inset Formula \( \mathbf{v} \)
\end_inset
\series default
, and the (
\shape italic
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
\shape default
x
\protected_separator
\begin_inset Formula \( n_{s} \)
\end_inset
) modal structural-acoustic radiation transfer function matrix,
\series bold
\begin_inset Formula \( \mathbf{Z}_{a} \)
\end_inset
\series default
\begin_inset LatexCommand \cite{Snyderetal:1994a}
\end_inset
,
\begin_inset Formula
\begin{equation}
\label{eqn:f}
\mathbf{p}=\mathbf{Z}_{a}\mathbf{v}
\end{equation}
\end_inset
\layout Standard
The
\begin_inset Formula \( (l,i)^{th} \)
\end_inset
element of the radiation transfer function matrix
\series bold
\begin_inset Formula \( \mathbf{Z}_{a} \)
\end_inset
\series default
is the pressure amplitude of the acoustic mode
\shape italic
\begin_inset Formula \( l \)
\end_inset
\shape default
generated as a result of structural mode
\shape italic
\begin_inset Formula \( i \)
\end_inset
\shape default
vibrating with unit velocity amplitude.
Substituting Eq.
\begin_inset LatexCommand \ref{eqn:f}
\end_inset
into Eq.
\begin_inset LatexCommand \ref{eqn:c}
\end_inset
gives an expression for the acoustic potential energy with respect to the
normal structural vibration,
\begin_inset Formula
\begin{equation}
\label{eqn:g}
E_{p}=\mathbf{v}^{\mathrm{H}}\boldsymbol \Pi \mathbf{v}
\end{equation}
\end_inset
where the error weighting matrix
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
is given by
\begin_inset Formula
\begin{equation}
\label{eqn:h}
\boldsymbol \Pi =\mathbf{Z}_{a}^{\mathrm{H}}\boldsymbol \Lambda \mathbf{Z}_{a}
\end{equation}
\end_inset
\layout Standard
It should be noted that the error weighting matrix
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
is not necessarily diagonal which implies that the normal structural modes
are not orthogonal contributors to the interior acoustic pressure field.
It is for this reason that minimisation of the modal amplitudes of the
individual structural modes (or kinetic energy) will not necessarily reduce
the total sound power transmission.
\layout Subsection
Diagonalisation of the error criteria
\layout Standard
Since
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
is real symmetric it may be diagonalised (using a singular value decomposition)
to yield the orthonormal transformation;
\begin_inset Formula
\begin{equation}
\label{eqn:i}
\boldsymbol \Pi =\mathbf{USU}^{\mathrm{T}}
\end{equation}
\end_inset
where the unitary matrix
\series bold
\begin_inset Formula \( \mathbf{U} \)
\end_inset
\series default
is the (real) orthonormal transformation matrix representing the eigenvector
matrix of
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
and the (real) diagonal matrix
\series bold
\begin_inset Formula \( \mathbf{S} \)
\end_inset
\series default
contains the eigenvalues (singular values) of
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
.
The physical significance of the eigenvectors and eigenvalues is interesting.
The eigenvalue can be considered a radiation efficiency (or coupling strength
\begin_inset LatexCommand \cite{Bessacetal:1996}
\end_inset
) and the associated eigenvector gives the level of participation of each
normal structural mode to the radiation mode; thus it indicates the modal
transmission path
\begin_inset LatexCommand \cite{Bessacetal:1996}
\end_inset
.
\layout Standard
Substituting the orthonormal expansion of Eq.
\begin_inset LatexCommand \ref{eqn:i}
\end_inset
into Eq.
\begin_inset LatexCommand \ref{eqn:g}
\end_inset
results in an expression for the potential energy of the cavity as a function
of an orthogonal radiation mode set,
\begin_inset Formula
\begin{equation}
\label{eqn:j}
E_{p}=\mathbf{v}^{\mathrm{H}}\mathbf{USU}^{\mathrm{T}}\mathbf{v}=\mathbf{w}^{\mathrm{H}}\mathbf{Sw}
\end{equation}
\end_inset
where the elements of
\series bold
\begin_inset Formula \( \mathbf{w} \)
\end_inset
\series default
are the velocity amplitudes of the radiation modes defined by
\begin_inset Formula
\begin{equation}
\label{eqn:k}
\mathbf{w}=\mathbf{U}^{\mathrm{T}}\mathbf{v}
\end{equation}
\end_inset
Eq.
\begin_inset LatexCommand \ref{eqn:k}
\end_inset
demonstrates that each radiation mode is made up of a linear combination
of the normal structural modes, the ratio of which is defined by the eigenvecto
r matrix
\series bold
\begin_inset Formula \( \mathbf{U} \)
\end_inset
\series default
.
As the eigenvalue matrix,
\series bold
\begin_inset Formula \( \mathbf{S} \)
\end_inset
\series default
, is diagonal, Eq.
\begin_inset LatexCommand \ref{eqn:j}
\end_inset
may be written as follows,
\begin_inset Formula
\begin{equation}
\label{eqn:l}
E_{p}=\sum _{i=1}^{n}s_{i}|w_{i}|^{2}
\end{equation}
\end_inset
where
\begin_inset Formula \( s_{i} \)
\end_inset
are the diagonal elements of the eigenvalue matrix
\series bold
\begin_inset Formula \( \mathbf{S} \)
\end_inset
\series default
and
\begin_inset Formula \( w_{i} \)
\end_inset
are the modal amplitudes of the individual radiation modes given by Eq.
\begin_inset LatexCommand \ref{eqn:k}
\end_inset
.
\layout Standard
The potential energy contribution from any radiation mode is equal to the
square of its amplitude multiplied by the corresponding eigenvalue.
The radiation modes are therefore independent (orthogonal) contributors
to the potential energy and the potential energy is directly reduced by
reducing the amplitude of any of the radiation modes.
As mentioned previously, the normal structural modes are not orthogonal
radiators since the potential energy arising from one structural mode depends
on the amplitudes of the other structural modes.
The orthogonality of the radiation modes is important for active control
purposes as it guarantees that the potential energy will be reduced if
the amplitude of any radiation mode is reduced
\begin_inset LatexCommand \cite{Johnsonetal:1995}
\end_inset
.
\layout Section
Alternative Formulation
\layout Standard
It will be shown here that the previous approach to diagonalise the error
weighting matrix
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
used in Section
\begin_inset LatexCommand \ref{General-Theory}
\end_inset
via the orthonormal transformation was unnecessary for the cost function
being global potential energy.
This is because in this case the expression used to define
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
was already written in terms of a diagonal matrix
\begin_inset Formula \( \boldsymbol \Lambda \)
\end_inset
and a fully populated matrix
\begin_inset Formula \( \mathbf{Z}_{a} \)
\end_inset
and its Hermitian transpose.
In situations where the control objective is not global but rather a subspace,
the following approach cannot be used because the error weighting matrix
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
does not have an inner matrix which is diagonal.
For example, the error weighting matrix
\begin_inset Formula \( \boldsymbol \Pi \)
\end_inset
for minimising the sum of the squared pressures over some subspace is given
by
\begin_inset LatexCommand \cite{Cazzolato:1999}
\end_inset
\begin_inset Formula
\begin{equation}
\boldsymbol \Pi =\mathbf{Z}_{a}^{\mathrm{H}}\mathbf{Z}_{w}\mathbf{Z}_{a}
\end{equation}
\end_inset
where
\begin_inset Formula \( \mathbf{Z}_{w}=\boldsymbol \Phi _{e}^{*}\boldsymbol \Phi _{e}^{\mathrm{T}} \)
\end_inset
and
\begin_inset Formula \( \boldsymbol \Phi _{\mathbf{e}} \)
\end_inset
is the mode shape matrix at the error sensor locations within the subspace.
Clearly
\begin_inset Formula \( \mathbf{Z}_{w} \)
\end_inset
is not diagonal (unlike
\begin_inset Formula \( \boldsymbol \Lambda \)
\end_inset
) but fully populated and therefore it is necessary to use the approach
in Section
\begin_inset LatexCommand \ref{General-Theory}
\end_inset
.
\layout Standard
The reformulation of the radiation modes orthogonal to the internal potential
energy will now be presented.
The interior acoustic potential energy is given by
\begin_inset Formula
\begin{equation}
E_{p}=\mathbf{v}^{\mathrm{H}}\mathbf{Z}_{a}^{\mathrm{H}}\boldsymbol \Lambda \mathbf{Z}_{a}\mathbf{v}
\end{equation}
\end_inset
where
\begin_inset LatexCommand \cite{Snyderetal:1994a}
\end_inset
\begin_inset Formula
\begin{equation}
Z_{a}(l,i)=\frac{j\rho _{0}S\omega }{\Lambda _{l}(\kappa _{l}^{2}+j\eta _{a_{l}}\kappa _{l}k-k^{2})}B_{l,i}
\end{equation}
\end_inset
where
\begin_inset Formula \( B_{l,i} \)
\end_inset
is the
\begin_inset Formula \( (l,i)^{th} \)
\end_inset
element of the (
\begin_inset Formula \( n_{a} \)
\end_inset
x
\begin_inset Formula \( n_{s} \)
\end_inset
) non-dimensional coupling coefficient matrix,
\begin_inset Formula \( \mathbf{B} \)
\end_inset
\begin_inset LatexCommand \cite{Snyderetal:1994a}
\end_inset
,
\begin_inset Formula \( \kappa _{l} \)
\end_inset
and
\begin_inset Formula \( \eta _{a_{l}} \)
\end_inset
are the wavenumber and modal loss factor of the
\begin_inset Formula \( l^{th} \)
\end_inset
acoustic mode respectively, and
\begin_inset Formula \( S \)
\end_inset
is the total surface area of the bounding structure.
Now
\begin_inset Formula \( \mathbf{Z}_{a} \)
\end_inset
can be written in matrix form,
\begin_inset Formula
\begin{equation}
\mathbf{Z}_{a}=\boldsymbol \Upsilon \mathbf{B}
\end{equation}
\end_inset
where
\begin_inset Formula \( \boldsymbol \Upsilon \)
\end_inset
is the (
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
x
\protected_separator
\begin_inset Formula \( n_{a} \)
\end_inset
) diagonal acoustic resonance matrix whose elements are given by
\begin_inset Formula
\begin{equation}
\boldsymbol \Upsilon _{l,l}=\frac{j\rho _{0}S\omega }{\Lambda _{l}(\kappa _{l}^{2}+j\eta _{a_{l}}\kappa _{l}k-k^{2})}
\end{equation}
\end_inset
\layout Standard
Therefore, the potential energy may be expressed as
\begin_inset Formula
\begin{equation}
E_{p}=\mathbf{v}^{\mathrm{H}}\mathbf{B}^{\mathrm{H}}\boldsymbol \Upsilon ^{*}\boldsymbol \Lambda \boldsymbol \Upsilon \mathbf{Bv}
\end{equation}
\end_inset
or
\begin_inset Formula
\begin{equation}
\label{eqn-radiation:alternative-formulation-Ep}
E_{p}=\mathbf{y}^{\mathrm{H}}\boldsymbol \Omega \mathbf{y}
\end{equation}
\end_inset
where
\begin_inset Formula \( \mathbf{y} \)
\end_inset
is the (
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
x
\protected_separator
1) modal amplitude column vector of the radiation modes given by
\begin_inset Formula
\begin{equation}
\mathbf{y}=\mathbf{Bv}
\end{equation}
\end_inset
and the (
\begin_inset Formula \( n_{a} \)
\end_inset
x
\begin_inset Formula \( n_{a} \)
\end_inset
) diagonal frequency-dependent weighting matrix,
\begin_inset Formula \( \boldsymbol \Omega \)
\end_inset
, is given by
\begin_inset Formula
\begin{equation}
\boldsymbol \Omega =\boldsymbol \Upsilon ^{*}\boldsymbol \Lambda \boldsymbol \Upsilon
\end{equation}
\end_inset
\layout Standard
Evaluating the diagonal weighting matrix, the elements are given by
\begin_inset Formula
\begin{equation}
\label{eqn-rm:diag-weightin-matrix-alternative}
\Omega _{ll}=\frac{\rho _{0}c(Sk)^{2}}{4\Lambda _{l}\left( (\kappa _{l}^{2}-k^{2})^{2}+(\eta _{a_{l}}\kappa _{l}k)^{2}\right) }
\end{equation}
\end_inset
\layout Standard
It is clear that Eq.
(
\begin_inset LatexCommand \ref{eqn-radiation:alternative-formulation-Ep}
\end_inset
) is the same format as Eq.
(
\begin_inset LatexCommand \ref{eqn:j}
\end_inset
) with a fully-populated participation matrix and a diagonal weighting matrix.
The radiation efficiency filters used by Cazzolato and Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite[FIG 1]{Cazzolatoetal:1998a}
\end_inset
to weight the modal amplitudes to provide the inputs to the active noise
control system are therefore given by the square root of the diagonal weighting
matrix,
\begin_inset Formula \( \boldsymbol \Omega \)
\end_inset
, which is equal to the magnitude of the product of the (
\begin_inset Formula \( n_{a} \)
\end_inset
\protected_separator
x
\protected_separator
\begin_inset Formula \( n_{a} \)
\end_inset
) diagonal frequency-dependent acoustic resonance matrix,
\begin_inset Formula \( \boldsymbol \Upsilon \)
\end_inset
, and the square root of the modal volume matrix,
\begin_inset Formula \( \boldsymbol \Lambda \)
\end_inset
.
By induction, it is possible to define a corresponding mode shape matrix
\begin_inset Formula
\begin{equation}
\label{eqn-radiation:corresponding-mode-shape}
\boldsymbol \Xi =\boldsymbol \Psi \mathbf{B}^{\mathrm{T}}
\end{equation}
\end_inset
where
\begin_inset Formula \( \boldsymbol \Psi \)
\end_inset
is the structural mode shape matrix.
\layout Standard
Pre-multiplying Eq.
(
\begin_inset LatexCommand \ref{eqn-radiation:corresponding-mode-shape}
\end_inset
) by
\begin_inset Formula \( \boldsymbol \Psi ^{\mathrm{T}} \)
\end_inset
and integrating over the surface of the structure gives
\begin_inset Formula
\begin{equation}
\frac{1}{S}\int _{s}\boldsymbol \Psi ^{\mathrm{T}}(\vec{\mathbf{x}})\boldsymbol \Xi (\vec{\mathbf{x}})dS(\vec{\mathbf{x}})=\frac{1}{S}\int _{s}\boldsymbol \Psi ^{\mathrm{T}}(\vec{\mathbf{x}})\boldsymbol \Psi (\vec{\mathbf{x}})\mathbf{B}^{\mathrm{T}}dS(\vec{\mathbf{x}})
\end{equation}
\end_inset
and using the principle of modal orthogonality, the following expression
is obtained
\begin_inset Formula
\begin{equation}
\label{eqn-rm:proof-1}
\frac{1}{S}\int _{s}\boldsymbol \Psi ^{\mathrm{T}}(\vec{\mathbf{x}})\boldsymbol \Xi (\vec{\mathbf{x}})dS(\vec{\mathbf{x}})=\mathbf{MB}^{\mathrm{T}}
\end{equation}
\end_inset
where
\begin_inset Formula \( \mathbf{M} \)
\end_inset
is the (
\begin_inset Formula \( n_{s} \)
\end_inset
x
\begin_inset Formula \( n_{s} \)
\end_inset
) diagonal matrix with diagonal elements given by
\begin_inset Formula
\begin{equation}
M_{i}=\frac{1}{S}\int _{s}\Psi _{i}^{2}(\vec{\mathbf{x}})dS(\vec{\mathbf{x}})
\end{equation}
\end_inset
\layout Standard
The left hand term of Eq.
(
\begin_inset LatexCommand \ref{eqn-rm:proof-1}
\end_inset
) is the same as the expression for the non-dimensional coupling coefficient
matrix
\begin_inset Formula \( (\mathbf{B}^{\mathrm{T}}=\frac{1}{S}\int _{s}\boldsymbol \Psi ^{\mathrm{T}}(\vec{\mathbf{x}})\boldsymbol \Phi (\vec{\mathbf{x}})dS(\vec{\mathbf{x}})) \)
\end_inset
with the exception that the mode shape matrix of the radiation mode has
been used in place of the acoustic mode shape matrix corresponding to the
acoustic mode shape at the enclosure boundary.
Therefore it follows that the radiation mode shape matrix is identical
to the acoustic mode shape matrix in which column
\begin_inset Formula \( i \)
\end_inset
scaled by some scalar term
\begin_inset Formula \( M_{i} \)
\end_inset
, ie
\begin_inset Formula
\begin{equation}
\label{eqn-radiation:alternative-mode-shape}
\boldsymbol \Xi (\vec{\mathbf{x}})=\boldsymbol \Phi (\vec{\mathbf{x}})\mathbf{M}
\end{equation}
\end_inset
\emph on
cf
\emph default
the same expression in terms of the structural mode shapes
\begin_inset Formula \( \boldsymbol \Xi (\vec{\mathbf{x}})=\boldsymbol \Psi (\vec{\mathbf{x}})\mathbf{U} \)
\end_inset
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
.
It should be noted that since the mode shapes for the current formulation
obviously do not vary with frequency it is only appropriate to compare
this current formulation with that of the
\begin_inset Quotes eld
\end_inset
fixed-shape
\begin_inset Quotes erd
\end_inset
radiation modes presented in Section III of the previous paper.
\layout Standard
The approach just described is only suited to low frequencies where the
modal density of the acoustic system is low since this ensures that the
rows of the
\begin_inset Formula \( \mathbf{B} \)
\end_inset
matrix are unique (column-orthogonal).
As the number of the acoustic modes is increased, the likelihood of the
acoustic mode shapes across the vibrating surface being orthogonal decreases.
When non-orthogonality occurs, the advantage of this current approach begins
to break down.
To ensure uniqueness, it is possible to collect all the acoustic modes
which have the same surface pressure pattern into a
\begin_inset Quotes eld
\end_inset
single
\begin_inset Quotes erd
\end_inset
radiation mode.
This results in removal of the redundant line in the
\begin_inset Formula \( \mathbf{B} \)
\end_inset
matrix and adds the corresponding terms in the diagonal weighting matrix
\begin_inset Formula \( \boldsymbol \Omega \)
\end_inset
.
The SVD approach has the advantage that this occurs automatically.
It has been shown numerically and experimentally
\begin_inset LatexCommand \cite{Cazzolato:1999}
\end_inset
that, for low frequencies, the two approaches for calculating the radiation
mode shapes lead to identical levels of control.
This has been shown not to be the case at high frequencies
\begin_inset LatexCommand \cite{Johnson:1996}
\end_inset
, especially when the radiating structure is small compared to the bounding
surface of the cavity, which is when the internal radiation modes shapes
degenerate to approximately the free field radiation mode shapes.
This is because the acoustic response in the cavity becomes diffuse and
can no longer be considered modal.
In this situation
\series bold
\begin_inset Formula \( \mathbf{B} \)
\end_inset
\series default
is no longer column-orthogonal and therefore a SVD is necessary to orthogonalis
e the expression for the radiation matrix.
\layout Standard
The current formulation is not only applicable to active noise control but
has important implications for passive control of sound transmission into
cavities.
This shows that when attempting to minimise the sound transmission into
cavities it is just as important to have an understanding of the dynamics
of the receiving space as an understanding of the dynamics of the exciting
structure.
Dynamic absorbers and co-located sensor/actuator pairs act to increase
the impedance the structure
\begin_inset Quotes eld
\end_inset
sees
\begin_inset Quotes erd
\end_inset
at the mount point.
Therefore, using the acoustic mode shapes to guide placement of such devices
would likely achieve good results very quickly without having to analyse
the dynamics of the structure.
Obviously further refinement and optimisation would have to take into considera
tion the dynamics of both the structure and the cavity.
\layout Section
Conclusion
\layout Standard
The results presented by Cazzolato and Hansen
\latex latex
\latex default
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
still hold since no new assumptions have been presented.
The advantage of the current approach is that there is no need for the
SVD to derive the mode shape matrices of the radiation modes contributing
orthogonally to the global potential energy of the enclosure which not
only simplifies the analysis but also decreases computation times.
Note however that the approach outlined here is not suitable for cases
where the cost function is the potential energy in a subspace of the enclosure.
In this case the analysis presented previously in
\begin_inset LatexCommand \cite{Cazzolatoetal:1998a}
\end_inset
must be used.
\layout Standard
The approach presented here has important implications for the design of
active control systems using radiation modal control.
Only the dynamics of the cavity are required to design the control system.
The radiation mode shapes are identical in shape to the acoustic mode shapes
of the cavity, and the radiation efficiencies of the radiation modes can
be easily derived from the cavity resonance terms.
Therefore, the modal sensor shapes need to be identical to the acoustic
shapes at the enclosure boundary and the frequency weighting (radiation
efficiency) filters need to emulate the modal interface coupling that occurs
between the structure and the cavity to enable a successful active noise
control system to be implemented.
\layout Standard
\begin_inset LatexCommand \BibTeX[ieeetr]{/home/bscazz/latex/bibtex/references}
\end_inset
\the_end
