1 Introduction
In this paper the problem of computing the mode shapes and resonant frequencies
of a linear elastic structure in contact with an acoustic medium is considered.
When the structure is made up wholly of material that is of much greater
density than the acoustic medium, such as a thick steel structure in air, then
acoustic loading can be safely neglected in the mathematical model. However, in the
cases when the structure is of similar density to the medium, such as steel
in water, or when the structure is flimsy, such as a noise shield in air, then
acoustic loading must be included in the model.
Acoustic loading is similar to viscous damping, that is the influence
of a fluid filled medium in contact with the structure, generally
having the effect of reducing the amplitude of the vibration when compared
to the structure's in vacuo response. Techniques
already exist for accounting for viscous damping in the vibratory analysis
of structures [1].
However, these techniques are not based on a mathematical model
of the acoustic field and are useful only in giving a crude
representation of the effect of the fluid on the vibratory properties.
Moreover, in applying these techniques,
it is assumed that the mode shapes of the structure in contact with the fluid
are the same as those of the isolated structure.
In this paper we consider only the development of a deliberate mathematical
model of the acoustic medium. Thus a consistent numerical method may be derived
for determining the coupling effect of the acoustic loading on the in vacuo vibratory
properties of the structure. The model may then be extended further to
give a method for computing the ensuing properties of the acoustic
field.
The finite element method is a well-established computational method
for modelling the vibratory properties of linear elastic structures
[1].
In this paper it is assumed
that the vibratory properties of an isolated linear elastic
structure can be readily computed.
The acoustic field is governed by the wave equation. This can then be reduced
to the Helmholtz equation when one particular frequency is considered in isolation.
The Helmholtz equation is amenable to solution by the finite element, finite
difference or integral equation methods.
Should the finite element or finite difference methods be used
then the physical problem reduces to a generalised linear
eigenvalue problem, which can be solved using a standard numerical algorithm.
The solution of this eigenvalue problem yields the mode shapes and
resonant frequencies of the acoustically-loaded structure. However, the
full domain of the acoustic fluid must be discretised so that when the
domain is large a costly method will result.
In the case when the acoustic domain is large or is a domain simply exterior
to the structure, an integral equation method for modelling the acoustic
field is attractive. Reformulation of the Helmholtz equation governing
the solution in the domain reduces it to an integral equation involving
points on the boundary only, thus the dimension of the problem, as
far as the acoustic modelling is concerned, is
effectively reduced by one. The standard solution of coupled
acoustic-structure problems, using the finite element method to model the
structural vibration
and integral equation methods to model the acoustic field, has received
some attention from researchers in recent decades, see references
[2-4].
For modal analysis, the application of an integral
equation method reduces the acoustic-structure problem to that of
solving a non-linear eigenvalue problem of the form
where the components of the matrix A(k) are complex-valued
continuously differentiable
functions of the wavenumber k. Unfortunately, standard algorithms for
solving non-linear
eigenvalue problems are not generally available.
Hence the application of integral equation
methods to the modal analysis of coupled acoustic-structure
problems is not straightforward.
In this paper a method for solving the non-linear eigenvalue
problem of the form (1) is used as a means to obtaining
a modal analysis of the acoustic-structure problem.
The wavenumber or frequency
range is divided into subranges and the matrix A(k) that arises
is approximated by a quadratic polynomial in k in each subrange.
This allows us to reduce the non-linear eigenvalue problem
of the form (1) to a standard generalised eigenvalue problem.
Previous work on the application of interpolation (with respect to the
wavenumber k) to acoustic problems is given
in references [5,6].
The application of interpolation to coupled acoustic-structure problems
is considered in references [7-8].
However, the main background work to the method
considered in this paper is the authors' previous work [9-10],
where a
similar method was used to find the eigenfrequencies
of the interior Helmholtz equation.
Two-dimensional problems only are considered in this paper, although the
extension of the central ideas to three-dimensional problems is straightforward.
The structure we consider is that of a steel sheet with each side fixed at
a hinged joint. We consider the situation of the sheet being
fixed to the side of a square with the acoustic fluid being
exterior to the square. The two most common acoustic media are
considered, that is air and water. The effect of the acoustic loading on
the in vacuo mode shapes and resonant frequencies of the
structure is explored.