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Optimization of structural countermeasures for noise attenuation
in aircraft cabins
Alejandro Bonillo Coll1
Universitat Politcnica de Catalunya (UPC-BarcelonaTech), 08034
Barcelona, Spain
A simulation methodology is proposed for the optimization of
structural
countermeasures to be integrated in the airframe of typical
turbopropeller aircraft
with the objective of providing significant cabin noise
attenuation levels in the low-
and mid-frequency range. A number of available countermeasures
is considered,
ranging from local structural modifications, i.e. local
stiffening and punctual mass
addition, to single- and multiple-degree-of-freedom passive
vibration control
devices such as dynamic vibration absorbers (DVAs). The
optimization
methodology benefits from separate modeling of the primary
structure, i.e. the
airframe, and structural countermeasures, thus allowing for the
implementation of
mathematical optimization algorithms which yield optimum
countermeasure
configurations at low computational cost.
Nomenclature
= speed of bending waves = bending stiffness = elastic modulus =
frequency of oscillation = area moment of inertia = length = mass =
viscous damping ratio = wavelength = surface density = angular
frequency of oscillation = natural angular frequency
I. Introduction
IRCRAFT cabin noise or interior noise has been an active
research field in structural dynamics and
acoustics for the past 60 years1. The aircraft interior noise
problem is generally known as the
transmission of noise from aircraft sources, i.e. propulsion
systems and turbulent boundary layers, into the
cabin through both airborne and structure-borne transmission
paths.
The characterization of airborne noise sources varies notably
with the type of aircraft which is subject
to study as well as with the operating regime of its power
plant. Thus, multi-engine propeller-driven
aircraft, with maximum cruise speeds ranging from less than 240
km/h to about 450 km/h, present typical
tonal excitations related to the blade passing frequency
(BPF)2,3
in the frequency range between 100 Hz
and 250 Hz. The advent of the advanced turboprop, e.g. Airbus
Military A400M, and open-rotor concepts
has added extra complexity to this problem due to the effects of
transonic and supersonic helical tip
speeds4 and rotor-wake/rotor interaction effects in
counter-rotating open rotor (CROR) systems
5. Turbojet
and turbofan aircraft, typically with cased and wing-mounted
engines, are found to be prone to jet noise,
which constitutes the dominant noise source under low-speed
conditions such as those encountered in the
climb stage of the flight profile. Interior noise for turbojet
and turbofan aircraft under cruise conditions is
generally dominated by boundary layer excitation of fuselage
panels, although jet noise might also have
an influence as the distance between engines and fuselage
decreases6. Fan noise is another typical
airborne noise source found in turbofan aircraft which presents
significant contribution to interior noise
only at low flight speeds, e.g. during takeoff and initial
climb7, and is related to a tonal excitation at the
BPF of the fan.
1 PhD student, Department of Mechanical Engineering, ETSEIAT
(UPC-BarcelonaTech), 08222
Terrassa, Barcelona, Spain. AIAA Young Professional Member.
A
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Structure-borne noise is principally associated to engine
vibration and propeller wake-wing
interaction. Engine-based structure-borne noise components are
related to the BPF of rotational
machinery elements, with fan contribution usually being
dominant. Therefore, in the case of propeller-
driven aircraft, structure-borne noise might be superposed to
airborne propeller noise sources at discrete
tonal frequencies. Results obtained for typical turboprop
aircraft have shown that structure-borne
components usually fall between 10 dB and 20 dB below airborne
components in terms of interior noise,
thus leaving structure-borne noise sources at a secondary
role8-12
.
Figure 1 provides a schematic representation of aircraft
interior noise sources which are typically
found in turboprop and turbojet/turbofan aircraft.
The present study focuses on the interior noise problem
identified in turbopropeller aircraft related to
BPF in the low-frequency range, i.e. up to 220 Hz. Cabin noise
prediction and attenuation in
turbopropeller aircraft has been addressed in previous research
and development work in collaboration
with EADS-CASA and Airbus Military14,15
, which has arisen the need of taking a step further in
simulation methodologies for higher attenuation of
predicted and validated cabin noise levels.
A simulation methodology is presented for the
optimization of structural countermeasures
implemented in aircraft fuselage sections with the
objective of providing outstanding attenuation of
interior noise levels predicted for turbopropeller
aircraft. The range of structural countermeasures
considered for application covers local modifications
in terms of stiffness and mass, as well as the
implementation of passive vibration control devices
such as tuned and detuned DVAs. Such proposed
countermeasures are selected as they are potentially
applicable to the development of a vibro-acoustic
solution kit which can be introduced in late stages of
aircraft design or even during its service life, thus
affecting at a much reduced scale the conventional
design process.
The optimization methodology is applied to a
generic fuselage section of a typical turbopropeller
aircraft which defines an acoustic cavity referred as
the generic aircraft cabin. The fuselage section
contains all primary structural elements, e.g. skin, frames,
stringers, etc., and is henceforth referred as the
primary structure. At a secondary level, all structural
countermeasures, e.g. stiffeners, local masses,
DVAs, etc., are referred as substructure or secondary structure.
The optimization methodology is based
on separate modeling of the primary structure and the
substructure. Firstly, the primary structure is
subject to conventional finite element modeling and
vibro-acoustic simulation using currently available
commercial software packages such as MSC NASTRAN and LMS
Virtual.Lab. The effects of
countermeasure elements are then applied to the primary
structure using a method which computes
equivalent dynamic forces16
combined with frequency response functions (FRF) of the primary
structure.
The use of structural coupling techniques allows for
vibro-acoustic simulation of structural
countermeasure configurations at reasonably low computational
cost, i.e. some milliseconds, using an
appropriate MATLAB routine, which does not require recalculation
of the primary structure. Therefore,
simulation of multiple configurations within a reasonable period
of time becomes possible, thus allowing
for efficient application of optimization algorithms. As a final
result of the proposed optimization
methodology, an optimum configuration of structural
countermeasures, which provides highest
achievable noise reduction levels, is obtained based on any
number of initial considerations and
constrains related to the integration of such countermeasures in
the airframe.
In section II the primary structure is defined in terms of
geometry and mechanical properties. The
finite element model and the conventional methodology used for
vibro-acoustic simulation are also
presented together with baseline results in terms of cabin
noise. Section III deals with the description and
characterization of all structural countermeasures which are
taken into consideration within the scope of
the present study. These are divided into two major groups:
local structural modifications and passive
vibration control devices. Section IV constitutes a detailed
mathematical approach to the structural
coupling technique used to integrate the structural
countermeasures into the primary structure. Section V
is devoted to the formulation of the optimization algorithm
which is implemented in combination with the
Figure 1. Sources and transmission paths
of aircraft interior noise (picture taken
from Ref. 13)
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mathematical model for structural coupling developed in section
IV. Finally, all simulation results are
presented and compared in section VI, and conclusions are
summarized in section VII.
II. Primary Structure
A. Definition The primary structure used for the implementation
of the proposed optimization methodology consists
of a generic fuselage section of a typical turbopropeller
aircraft which integrates all primary constitutive
elements: skin, frames, stringers and floor panels. A schematic
view of the proposed structure is presented
in figure 2.
With respect to materials definition for the fuselage structure,
a generic 2024 aluminum-copper alloy
was used for the skin, frames and stringers, whereas floor
panels were defined as a composite structure
constituted by an internal honeycomb layer covered by two
external layers of the same aluminum-copper
alloy.
B. Finite element model The primary structure is subject to
conventional finite element (FE) modeling oriented to numerical
simulation using MSC NASTRAN. Only a representative range of
fuselage sections were considered for
shell modeling in order to keep the FE model to a reasonable
number of elements. The dynamic behavior
of the cockpit was represented by its global mass, whereas rear
fuselage, wings and tail planes are
replaced by a fully clamped boundary condition along the
perimeter of the fuselage. The overall mesh
size is set to a value which allows appropriate representation
of the smallest wavelength scales expected
in the dynamic behavior of the primary structure. Using the
standard criterion of six elements per
wavelength, the maximum frequency at which the FE model provides
representative results is related to
the global element size, for thin-walled structures,
through17
(1)
where denotes the minimum wavelength which can be reproduced by
the FE model, equal to six times the highest element size, and
denotes the speed of wave propagation in bending through the
structure, which is related to mechanical properties and back to
the angular frequency of bending waves
through
(2)
According to a BPF of 102 Hz and a maximum analysis frequency of
220 Hz, the overall mesh size is
set to 50 mm by application of equations (1) and (2). The FE
model generated for the primary structure is
presented in figure 3.
Figure 2. Geometrical definition of the primary structure. Left:
Complete model of a generic
turbopropeller aircraft. Right: Detail of a fuselage
section.
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C. Vibro-acoustic simulation The vibro-acoustic response of the
primary structure is obtained following a two-step uncoupled
simulation process which consists on a modal frequency response
analysis in MSC NASTRAN (SOL
111) which yields the displacement field in the structure, and a
subsequent acoustic response analysis in
LMS Virtual.Lab to obtain acoustic pressure levels inside the
cabin. The fluid-structural coupling inside
the aircraft cabin is considered to be sufficiently weak so that
both analysis can be performed uncoupled
and subsequently instead of running a fully coupled case which
would imply higher computational
requirements.
Source characterization 1.According to the simulation process
described above, generic turbopropeller vibro-acoustic
excitation
is defined by means of a typical pressure distribution applied
to the outer skin of the aircraft fuselage over
a bounded region of influence. Source characterization is kept
simple as such process is not directly
related to the objectives included in the scope of the study:
any typical source characterization is expected
to provide similar results regardless its degree of complexity.
Figure 4 presents an overview of the
application of the vibro-acoustic excitation to the FE model of
the aircraft fuselage.
The excitation frequency is related to BPF of turbopropeller
engines and is arbitrarily set to 102 Hz.
Figure 3. Finite element model generated for the primary
structure (70853 elements).
Figure 4. Vibro-acoustic excitation on the FE model of the
aircraft fuselage (pressure
distribution represented as a set of equivalent point
forces).
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Structural response analysis 2.The structural response of the
primary structure is computed by means of a modal frequency
response
analysis using MSC NASTRAN (SOL 111) thus allowing for easy
recalculation of multiple load cases, a
key feature later on in the optimization process. As most of the
fuselage structure is basically made of
aluminum, a modal damping ratio of 2% is considered to be a
sufficiently good representation of the
structural damping in the structure. Nevertheless, as long as
the structure is lightly damped, the actual
value for such parameter is not relevant to the results of the
present study. The resulting displacement
field constitutes the main output of this step and is taken as
an input for prediction of acoustic pressure
levels inside the aircraft cabin.
Acoustic response analysis 3.Once the structural response of the
primary structure is obtained, the acoustic pressure levels
radiated
inside the aircraft cabin can be computed in LMS Virtual.Lab by
application of an acoustic response
analysis to the boundary element (BE) model of the aircraft
cabin. The BE model defines all acoustic
cavities inside the fuselage and relates the internal acoustic
pressure field to the displacement field in the
fuselage structure. Therefore, the previously computed
displacement field is herein taken as the acoustic
excitation. The BE model should be constructed based on mesh
size criteria related to pressure wave
propagation through the air, which generally results in coarser
meshes in comparison with FE models. For
the present study, a BE model with an overall mesh size of 200
mm is considered to represent
appropriately the displacement field as well as to comply with
the aforementioned mesh size criteria. An
overview of the BE model in comparison with the FE model is
shown in figure 5.
It can be observed from figure 5 that the FE model and the BE
model do not cover the same extent in
length of the aircraft fuselage. This corresponds to the fact
that, whereas shell modeling in the FE model
can be limited to a certain number of fuselage sections
according to proximity to the source, the BE
model needs to cover the whole extent of the acoustic cabin.
Figure 5 also provides a detailed view for
easy comparison of both mesh sizes. It can be noted that all
nodes in the BE model have correspondence
to a node in the FE model.
A set of 20 microphones is distributed inside the aircraft cabin
in order to determine the interior
overall sound pressure level (OASPL), which constitutes the main
output of the vibro-acoustic simulation
process as well as the control variable for vibro-acoustic
optimization. Furthermore, two field point
planes, one horizontal and one vertical, are defined for better
visualization of the acoustic pressure field
inside the aircraft cabin. The vertical position of all discrete
microphones and the horizontal field point
plane is established at a typical height for seated passenger
ears, whereas the vertical field point plane is
located in the plane of turbopropeller excitation. The position
of all field point elements is represented in
relation to the FE model in figure 6.
Figure 5. Boundary element model generated for the aircraft
cabin (5988 elements). Detail of
the FEM/BEM fitting.
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D. Baseline results
Modal analysis 1. Even though modal analysis is not directly
related to the main simulation process for the vibro-
acoustic response of the primary structure, it can provide
useful information with respect to modal density
distribution over the frequency range of interest as well as
typical mode shapes corresponding to natural
frequencies which are close to the main excitation frequency of
102 Hz. A modal analysis case is run over
the frequency range between 0 Hz and 220 Hz using the Lanczos
method for eigenvalue extraction with
MSC NASTRAN (SOL 103). Modal analysis results yield a number of
993 eigenmodes in this frequency
range, 56 of these being comprised in the 10 Hz frequency band
centered at the excitation frequency, i.e.
between 97 Hz and 107 Hz. Three representative normal modes are
represented in figure 7.
Structural response analysis 2. The displacement field of the
primary structure resulting from BPF excitation at 102 Hz is
presented
in figure 8.
Figure 6. Definition of microphone positions (purple) and field
point planes (yellow) for acoustic
pressure field prediction.
Figure 7. Three eigenvalues obtained in the frequency range
between 97 Hz and 107 Hz. Left:
Mode 168 at 97.9 Hz. Center: Mode 187 at 102.0 Hz. Right: Mode
203 at 105.2 Hz.
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Acoustic response analysis 3.Acoustic pressure fields radiated
from BPF excitation at 102 Hz are presented in figure 9.
Results in terms of OASPL are presented in figures 10 and 11.
Contributions to the OASPL from skin
panels and floor panels along the longitudinal axis are compared
for further understanding of vibro-
acoustic transmission paths.
Figure 8. Displacement field in mm obtained for the primary
structure from BPF excitation at
102 Hz.
Figure 9. Interior acoustic pressure fields from BPF excitation
at 102 Hz. Left: Horizontal field
point plane (upper view, back to front). Right: Vertical field
point plane corresponding to the
propeller plane (front view).
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III. Structural Countermeasures
The range of structural countermeasures included in the scope of
the present study is divided into two
different categories: local structural modifications and passive
vibration control devices. The main
difference between these two groups lies on whether any degrees
of freedom are added to the primary
structure or not. Whereas local structural modifications are
herein understood as local changes in stiffness
and mass properties of the primary structure thus affecting to
relations between existing degrees of
freedom, passive vibration control devices pursue the effective
transfer of kinetic energy from the primary
structure to the degrees of freedom of attached elements.
Rather than providing a detailed description on how these
countermeasures are to be found in real
applications related to aerospace industry, this section is
aimed at defining how all proposed
countermeasure elements are managed in the vibro-acoustic
simulation process with respect to their
implementation in the primary structure FE model.
Figure 10. Contribution from structural elements to OASPL in the
aircraft cabin from BPF
excitation at 102 Hz.
Figure 11. Contribution from fuselage sections (front to back)
to OASPL in the aircraft cabin
from BPF excitation at 102 Hz. Solid line: Total OASPL. Dashed
line: Skin panels contribution.
Dotted line: Floor panels contribution.
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A. Local structural modifications
Local stiffness modifications 1.Local stiffness modifications
are implemented in the FE model of the primary structure by
adding
discrete beam elements, also referred further in this paper as
stiffeners. Their application to the numerical
simulation methodology is based on four major assumptions:
All stiffeners are attached to internal nodes of the primary
structure, i.e. coincidence between stiffener attachments and
boundary conditions is not considered.
Stiffeners do not transfer torsional loads, i.e. torsional
stiffness is set to zero for all elements.
All stiffeners are considered to have the same cross section and
to be made of the same material, i.e. same values of moments of
inertia and elastic modulus apply to all elements.
The dynamic effect related to the mass of the stiffeners is
disregarded, i.e. density is set to zero for all elements.
According to these assumptions, the stiffness matrix, which is
defined as the relationship between
element forces and node displacements as
{ } [ ]{ } (3)
for element forces applied at the nodes { } { } and
nodal displacements { } { }, is written as18
[ ]
[ ]
(4)
where
according to the hypotheses above.
Local mass modifications 2.Local mass modifications are
implemented in the FE model of the primary structure by adding
discrete mass elements. Their application to the numerical
simulation methodology is based on three
major assumptions:
All mass elements are attached to internal nodes of the primary
structure, i.e. coincidence between mass elements and boundary
conditions is not considered.
Mass inertias are not considered.
All elements are considered to have the same mass value.
Based on equation (3), an equivalent dynamic stiffness matrix is
defined as
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[ ] [
] (5)
B. Passive vibration control devices
Dynamic vibration absorbers (DVA) 1.Dynamic vibration absorbers
(DVAs) are implemented in the FE model of the primary structure as
a
number of single degree-of-freedom systems composed by a mass, a
spring and a damper, which are
attached to the primary structure at predefined positions. It is
important to note that, even though each
DVA introduces an additional degree of freedom to the structure,
only their dynamic effect on the
primary structure is of interest. Their application to the
numerical simulation methodology is based on
three major assumptions:
All DVAs are attached to internal nodes of the primary
structure, i.e. coincidence between DVA attachments and boundary
conditions is not considered.
Same mass, stiffness and damping values are considered for all
elements.
Dynamic vibration absorbers (DVAs) are studied for two different
types of attachment conditions. On
the one hand, compressive DVAs oscillate perpendicularly to the
structural element to which they are
attach and aim at counteracting its out-of-plane bending motion.
On the other hand, shear DVAs might
provide some kinetic energy absorption in the two in-plane
directions.
Based on equation (3), an equivalent dynamic stiffness is
defined as a scalar magnitude, for both
compressive and shear DVAs, as
(6)
where denotes the DVA mass, denotes the natural frequency of the
DVA as a single-degree-of-freedom system (the tuning frequency),
and denotes the DVA damping ratio. For the particular case in which
the DVA is tuned to the excitation frequency, i.e. , equation (6)
is simplified to
( )
(7)
IV. Structural Coupling
Structural coupling between the primary structure and structural
countermeasures is performed by
applying a structural coupling method based on the frequency
response functions (FRF) of the primary
structure16
. Substructural elements are represented by the equivalent
dynamic forces that they exert on the
primary structure so they can be easily included or excluded
from the analysis by modifying the load
case. Once the FRFs of the primary structure are computed at a
preliminary stage, recalculation of the
modified structure is performed at very low extra computational
cost.
The following transfer functions need to be obtained for the
primary structure:
Transfer functions between external loads and pre-defined
vibro-acoustic output positions, which is equivalent to the
vibro-acoustic response of the primary structure at the output
positions.
Transfer functions between external loads and candidate
positions subject to either local structural modification or
attachment of passive control devices, which is equivalent to the
vibro-
acoustic response of the primary structure at candidate
positions.
Crossed transfer functions between candidate positions and
pre-defined vibro-acoustic output positions in order to construct
the response of the modified structure using equivalent forces.
A. Mathematical model The displacement at each node affected by
the implementation of countermeasures is written as a
superposition of terms from external loads and equivalent forces
for all countermeasure elements. Node
displacements are then written as
{ } { } [ ]{ } (8)
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where { } denotes the vector of node displacements for the
modified structure, { } denotes the vector of node displacements
for the primary structure, [ ] denotes the matrix of transfer
functions between affected nodes, and { } denotes the vector of
equivalent forces resulting from the implementation of
countermeasure element configurations. All displacements and forces
are referred to the global coordinate
system of the primary structure.
The vector of equivalent forces at the affected nodes needs to
be constructed from individual element
forces by appropriate coordinate transformation and matrix
composition, i.e. forces related to a given
individual element need to be applied to corresponding positions
of the global force vector { }. Furthermore, element forces in
equation (3) produce reactions on the primary structure which are
of same
magnitude but opposite direction, thus requiring sign inversion
of individual force vectors { } in the construction of the global
force vector { }. The vector of node equivalent forces is therefore
written, in terms of individual vectors of element forces, as
{ } [ ][ ] [ ]{ } (9)
where { } is a vector constructed from individual vector forces
{ } for all elements as
{ }
{
{ } { }
{ }}
(10)
[ ] is a diagonal matrix assembled from a number of identity
submatrices which select the elements used in the countermeasure
configuration, [ ] is a square banded matrix such that
[ ]
[ [ ] [ ] [ ]]
(11)
in which [ ] is the coordinate transformation matrix for element
, and [ ] is referred as the distribution matrix, assembled from a
number of identity submatrices, which relates forces at the nodes
of individual
elements to forces at the affected nodes of the primary
structure.
The extended individual force vector { } is written in terms of
the extended individual displacement vector { } in local
coordinates, as
{ } [ ]{ } (12)
where the extended dynamic stiffness matrix is defined as
[ ]
[ [ ] [ ] [ ]]
(13)
The extended individual displacement vector is written in terms
of global node displacements as
{ } [ ][ ] { } (14)
Substitution of equations (9), (12) and (14) into equation (8)
allows for the reformulation of the
mathematical problem as a classical system of linear equations
in terms of global node displacements as
[ ]{ } { } (15)
with
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[ ] [ ] [ ][ ][ ] [ ][ ][ ][ ] (16)
and
{ } { } (17)
Once equation (15) is solved for global node displacements,
global forces are obtained as
[ ] [ ][ ] [ ][ ][ ][ ] { } (18)
The vibro-acoustic response at output positions is finally
obtained using the principle of superposition
as
{ } { } [ ]{ } (19)
where { } denotes the baseline response of the primary
structure, and [ ] denotes the matrix of transfer functions between
affected nodes and output positions. It should be noted that
transfer functions in [ ] may relate equivalent forces to either
structural displacements, velocities and accelerations, or to
acoustic
pressure values at given pre-defined output positions. Validity
of such operations is exclusively stated on
the initial assumption of linear vibro-acoustic behavior of all
elements in the problem of interest.
B. Particularities
Local stiffness modifications 1.The implementation of a
stiffener into the primary structure establishes a relationship
between one
element and two nodes in all six degrees of freedom. Therefore,
for any problem consisting on the
attachment of stiffeners to attachment nodes, the following
particularities need to be taken into consideration with respect to
the mathematical model above:
Vectors and matrices related to nodes of the primary structure
satisfy { } { } { } and
[ ] . Extended force and displacement vectors, { } { } , are
constructed from individual
force vectors { } and individual displacement vectors { }
respectively, with twelve components each.
The selection matrix, [ ] , is assembled from identity
submatrices and allows for straightforward elimination of any
stiffener from a given initial configuration.
The extended coordinate transformation matrix, [ ] , is
constructed from individual coordinate transformation matrices, [
]
, which transform local coordinates for each stiffener to global
coordinates of the primary structure.
The distribution matrix, [ ] , is assembled from identity
submatrices and relates forces at the nodes of individual
stiffeners to forces at the affected nodes of the primary
structure.
Local mass modifications 2.The implementation of a mass into the
primary structure establishes a relationship between one
element and one node in all three translational degrees of
freedom. Therefore, for any problem consisting
on the attachment of masses to attachment nodes, with , the
following particularities apply:
Vectors and matrices related to nodes of the primary structure
satisfy { } { } { } and
[ ] . Extended force and displacement vectors { } { } , are
constructed from individual force
vectors { } and individual displacement vectors { }
respectively, with three components each. The selection matrix, [ ]
, is assembled from identity submatrices and
allows for straightforward elimination of any mass element from
a given initial configuration.
The extended coordinate transformation matrix [ ] becomes the
identity matrix in due to the absence of local coordinates for mass
elements.
The distribution matrix [ ] becomes the identity matrix in due
to the one-to-one correspondence between mass elements and affected
nodes of the primary structure.
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Dynamic vibration absorbers 3.The implementation of a dynamic
vibration absorber (DVA) into the primary structure establishes
a
relationship between one set of elements and one node in one
translational degree of freedom. Therefore,
for any problem consisting on the attachment of DVAs to
attachment nodes, with , the following particularities apply:
Vectors and matrices related to nodes of the primary structure
satisfy { } { } { } and
[ ] . Extended force and displacement vectors { } { } , are
constructed from individual forces
and individual displacements respectively. The selection matrix,
[ ] , allows for straightforward elimination of any DVA from a
given initial configuration.
The extended coordinate transformation matrix [ ] becomes the
identity matrix in due to the absence of local coordinates for
DVAs.
The distribution matrix [ ] becomes the identity matrix in due
to the one-to-one correspondence between DVAs and affected nodes of
the primary structure.
V. Optimization Tool
The optimization tool applied to structural countermeasures for
the primary structure is based on a
brute force algorithm which fundamentally consists on sequential
simulation of a number of candidate
configurations and direct comparison of results. Besides the
structural definition of the primary structure
and the countermeasure elements which are considered in the
optimization problem, a set of candidate
elements attached to candidate positions needs to be pre-defined
at an early stage of the process. The
objective of the optimization tool is then to determine which
configuration constructed as a combination
of any number of candidate elements provides best results in
terms of vibro-acoustic response.
Although the mathematical model presented above allows for
vibro-acoustic recalculation of
countermeasure configurations at very low computational cost,
the number of configurations which need
to be processed escalates with the number of candidate elements.
Indeed, for pre-defined candidate elements, the number of
configurations is obtained as
(18)
thus limiting severely the applicability of the optimization
tool to global solutions distributed all over the
primary structure. In order to overcome such drawback, an
enhancement of the original brute force
optimization algorithm is proposed for higher efficiency and
hence lower computational requirements.
The proposed algorithm is based on the decomposition of the
structural optimization problem into a finite
number of subdomains, which are individually optimized following
an iterative process until results
convergence is reached.
For a given structural domain to be optimized, denoted as , the
proposed optimization algorithms is described as follows:
The structural domain is partitioned into subdomains . 1.
Brute force optimization is performed for subdomain 2.
Brute force optimization is performed for subdomain keeping
optimum configurations for all 3.subdomains with .
Once subdomain is processed, results are compared to those
obtained prior to processing 4.subdomain .
The loop 2-4 is repeated until differences between both results
fall below a given reference value. 5.
For all loops after the first one, optimum configurations for
all subdomains with are maintained from the previous loop.
It should be noted that the optimization algorithm described
above might lead to different optimum
configurations and optimum values depending on initial numbering
of subdomains. The optimization
algorithm allows finding local optimum values but cannot ensure
overall minimum results.
Furthermore, if the primary structure presents low modal density
in the frequency range of interest, an
additional problem might arise. If the excitation frequency
falls close to a natural frequency of the
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primary structure and the vibro-acoustic response is highly
dominated by the corresponding structural
mode, local application of countermeasures in a single subdomain
might not be sufficiently intrusive to change such a modal
composition of the response. Therefore, the optimization tool might
not be able to
find any optimum configuration for any individual subdomain thus
failing to initiate the optimization
process. This can be exemplified for the particular case of
stiffener optimization. If the excitation
frequency falls slightly above a natural frequency of the
primary structure, local stiffening of a single
subdomain might not be sufficient to shift that natural
frequency well above the excitation frequency thus failing to
provide any improvement in the vibro-acoustic response of the
primary structure. If this
happens for all subdomains, the optimization process never
starts. However, this does not mean that there
does not exist any optimum stiffener configuration, as a higher
number of stiffeners distributed over the
whole domain might yield the expected response. To overcome this
limitation, an extra step is added to the optimization process
described above. Prior to performing individual optimization of
subdomain , both baseline primary structure and primary structure
with all candidate elements are compared in terms
of vibro-acoustic results. Optimization of subdomain is then
performed parting from either configuration, which provides better
vibro-acoustic results.
It can be stated from the description of the optimization
algorithm above that the selection of
candidate countermeasure elements and candidate positions in the
primary structure becomes a key factor
in the process of finding an effective optimum configuration.
This step cannot be automatized but is left
to engineering considerations based on the characteristics of
the primary structure and the assessment of
the baseline vibro-acoustic response, e.g. geometry constraints
morphology of both the displacement field
and the most contributing structural modes, presence of large
radiating surfaces, etc. More than one
candidate set of elements and positions might eventually be
required for a most effective approach to the
structural optimization problem.
VI. Results
A. Local stiffness modifications A candidate set of stiffeners
and attachment positions is defined on the frames of the aircraft
fuselage
under the restriction of not affecting the integrity of the
skin. The spatial distribution of candidate
stiffening elements is designed, at a first stage, to be
homogeneous over the four frames forward to the
excitation. Although other alternative candidate sets might be
also of particular interest, the proposed set
is expected to provide relevant results with respect to the
applicability of the optimization methodology.
Candidate set definition for stiffeners on frames is
schematically presented in figure 12.
It should be emphasized that the objective of the optimization
process for stiffener configurations
consists on finding which combination of any number of
stiffeners from those shown in figure 12
provides highest noise reduction levels at predefined output
positions. The optimization process is
performed for stiffeners with inertia values within a predefined
range. Results in terms of OASPL
reduction levels are presented in figure 13.
Figure 12. Candidate set for stiffener configurations on
fuselage frames (64 elements).
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From figure 13 it can be observed that optimum configurations
only provide relevant OASPL reductions
with respect to full configurations for stiffness ratios
(between bending stiffness of the stiffeners and
bending stiffness of the original frame) below 10. In this
range, full configurations prove to be
excessively stiff thus causing noise amplification. Optimum
configurations, which are composed of about
one third of the candidate elements, allow for slight reductions
below 1 dB. As the stiffness ratio is
increased beyond 10, optimum configurations almost coincide with
full configurations, and both curves in
figure 13 merge to an asymptotic behavior of OASPL reduction
with the stiffness ratio.
According to manufacturability considerations, a stiffness ratio
of 10 with a corresponding OASPL
reduction of 3.9 dB is set as a limit value, and therefore is
taken for further analysis through conventional
vibro-acoustic simulation. Vibro-acoustic results for this
configuration are compared to those presented
for the baseline primary structure in section II. Such optimum
stiffener configuration is found to contain
63 out of the 64 candidate elements chosen for optimization. The
displacement field of the optimum
configuration is compared to that of the primary structure
resulting from BPF excitation at 102 Hz in
figure 14.
Acoustic pressure fields radiated from BPF excitation at 102 Hz
are compared in figure 15.
Figure 13. OASPL reduction levels from optimization of stiffener
configurations. Solid line:
OASPL reduction levels provided by optimum configurations found
for discrete values of
stiffener-to-frame stiffness ratio. Dashed line: OASPL reduction
levels provided by
corresponding full configurations.
Figure 14. Displacement field in mm from BPF excitation at 102
Hz. Left: Primary structure.
Right: Optimum stiffener configuration (stiffness ratio 10).
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Finally, global comparison of OASPL contributions is presented
in figure 16, and comparison of
OASPL contributions along the longitudinal axis of the airframe
is presented in figure 17.
Figure 15. Interior acoustic pressure fields from BPF excitation
at 102 Hz. Up: Primary
structure. Down: Optimum stiffener configuration (stiffness
ratio 10).
Figure 16. Contribution from structural elements to OASPL from
BPF excitation at 102 Hz.
Solid bar: Primary structure. Dashed bar: Optimum stiffener
configuration (stiffness ratio 10).
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From figure 14 and figure 15 it can be observed that frame
stiffening provides an homogeneous
reduction of both the displacement field and the interior
acoustic pressure field, especially for the vertical
propeller plane. Figure 16 also shows an homogeneous reduction
of about 4 dB for both floor panels and
skin components, and figure 17 reveals very little changes with
respect to panel contributions along the
longitudinal axis. Therefore, it can be stated that frame
stiffening produces slight homogeneous
attenuation of the vibro-acoustic behavior of the primary
structure.
B. Local mass modifications In order to approach SPL
contributions from skin panels and floor panels separately, the
optimization
methodology is applied independently to a given set of candidate
masses and positions for each structural
component. However, whereas floor panels can be approached
directly, masses aimed at reducing the
SPL contribution from skin panels are attached to the frames
under the restriction of not affecting the
integrity of the skin. In both cases, candidate sets are defined
on the basis of homogeneous distribution of
a reasonable number of elements into the airframe. In order to
portrait candidate set definition, mass
configurations which contain all candidate elements for both
cases are presented in figure 18.
It should be emphasized that the objective of the optimization
process for mass configurations consists
on finding which combination of any number of masses from those
shown in figure 18 provides highest
noise reduction levels at predefined output positions.
Optimization of frame configurations 1.The optimization process
is performed for mass elements with discrete values between 1 kg
and 15
kg. It should be noted that, for a total number of 64 candidate
elements, this can result in a maximum total
attached mass of 960 kg, corresponding to the 16% of the
airframe mass. Results in terms of OASPL
reduction levels are presented in figure 19.
Figure 17. Contribution from fuselage sections (front to back)
to OASPL from BPF excitation
at 102 Hz. Left: Primary structure. Right: Optimum stiffener
configuration (stiffness ratio 10).
Solid line: Total OASPL. Dashed line: Skin panels contribution.
Dotted line: Floor panels
contribution.
Figure 18. Candidate sets for mass configurations. Left: Frame
configuration (64 elements).
Right: Floor configuration (48 elements).
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OASPL reduction levels presented in figure 19 (solid line) can
be separated into two components
according to two different source effects. On the one side,
OASPL reduction can be achieved by
appropriate reconfiguration of the natural frequencies of the
primary structure in relation to the excitation
frequency, which is exclusively a mass effect and is not related
to the spatial configuration of the attached
elements. On the other hand, an optimum spatial distribution of
attached elements might also yield to
further reduction of OASPL due to spatial effects such as slight
morphing of mode shapes or phase
cancellation effects. The dashed line in figure 19 represents
the mass effect component associated to the
total amount of mass added by each optimum configuration.
Therefore, the difference between both lines
in figure 19 reflects the impact of spatial effects in the
overall OASPL reduction levels. It is interesting to
find that OASPL reduction levels present a monotonous increase
with the element mass which is not
translated to a parallel increase of mass effect reductions. In
fact, for the range between 6 kg and 11 kg,
the increase in element mass yields optimum configurations with
less elements thus keeping the total
mass and, consequently, the mass effect at a roughly constant
level.
It is important to note that, as optimum configurations
associated to discrete values of element mass
might have uncorrelated number of elements, the dashed line in
figure 19 does not constitute an
appropriate representation in the mass domain. Alternatively,
such representation is given in the domain
of total mass in figure 20.
Figure 19. OASPL reduction levels from optimization of frame
configurations of mass elements.
Solid line: OASPL reduction levels provided by optimum
configurations found for discrete
values of element mass. Dashed line: OASPL reduction levels
provided by full configurations
with equivalent total mass.
Figure 20. OASPL reduction levels due to the mass effect in the
domain of total mass.
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From comparison between figure 19 and figure 20 it can be stated
that there is a certain similitude in
the shape of both solid curves, the mass domains being
equivalent. In fact, OASPL reduction levels from
optimum configurations in figure 19 present two regions of
maximum steep, up to 6 kg and 13 kg
respectively, which are corresponding with the local maxima in
the mass effect curve in figure 14.
However, reduction levels provided by optimum configurations (13
dB and 20 dB respectively) are
significantly higher than those associated to the mass effect (8
dB and 15 dB respectively).
In order to provide further insight on the nature of OASPL
reduction levels produced by optimum
mass configurations, the optimum configuration for an element
mass of 4 kg is subject to conventional
vibro-acoustic simulation, and the obtained results are compared
to those presented for the baseline
primary structure in section II. An element mass of 4 kg is
chosen because it provides significant OASPL
reduction (10.7 dB) at a relatively low total mass increase of
168 kg (about 3% of the airframe mass). The
optimum configuration for an element mass of 4 kg consists of 42
elements, out of the 64 candidate
elements, which are distributed as shown in figure 21.
The displacement field of the optimum configuration is compared
to that of the primary structure
resulting from BPF excitation at 102 Hz in figure 22.
Acoustic pressure fields radiated from BPF excitation at 102 Hz
are compared in figure 23.
Figure 21. Optimum mass configuration for mass elements of 4
kg.
Figure 22. Displacement field in mm from BPF excitation at 102
Hz. Left: Primary structure.
Right: Optimum mass configuration (4 kg).
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Finally, global comparison of OASPL contributions is presented
in figure 24, and comparison of
OASPL contributions along the longitudinal axis of the airframe
is presented in figure 25.
Figure 23. Interior acoustic pressure fields from BPF excitation
at 102 Hz. Up: Primary
structure. Down: Optimum mass configuration (4 kg).
Figure 24. Contribution from structural elements to OASPL from
BPF excitation at 102 Hz.
Solid bar: Primary structure. Dashed bar: Optimum mass
configuration (4 kg).
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From comparison between figure 22 for mass configurations and
figure 14 for stiffener configurations
it can be observed that the implementation of optimum mass
configurations does not provide further
substantial attenuation of the displacement field in the
airframe. Nevertheless, figure 23 does show much
more significant attenuation of the interior acoustic pressure
field, both for the horizontal and the vertical
plane. A reasonable explanation for such unexpected phenomena
can be found in figure 24, in which a
reduction of about 7 dB of the skin component combines with an
outstanding reduction of 14 dB for the
floor panel component for an overall OASPL reduction of about 10
dB. This very same effect can be also
observed in figure 25, where the curve corresponding to floor
panel contribution presents a remarkable
shift downwards from the skin contribution curve, which
virtually coincides with the OASPL curve after
the implementation of the optimum mass configuration. In
conclusion, it can be stated that, whereas the
impact on the skin contribution to the OASPL presents little
improvement, the implementation of mass
configurations produces an outstanding attenuation effect on the
floor panel contribution which proves to
affect sensitively to the global reduction levels.
Optimization of floor configurations 2.The optimization process
is performed for mass elements with discrete values between 100
grams and
5 kg. It should be noted that, for a total number of 48
candidate elements, this can result in a maximum
total attached mass of 240 kg, corresponding to the 4% of the
airframe mass. Results in terms of OASPL
reduction levels are presented in figure 26. Additionally, OASPL
reduction levels due to the mass effect,
i.e. obtained by implementation of full mass configurations, are
presented in figure 27.
Figure 25. Contribution from fuselage sections (front to back)
to OASPL from BPF excitation at
102 Hz. Left: Primary structure. Right: Optimum mass
configuration (4 kg). Solid line: Total
OASPL. Dashed line: Skin panels contribution. Dotted line: Floor
panels contribution.
Figure 26. OASPL reduction levels from optimization of floor
panel configurations of mass
elements. Solid line: OASPL reduction levels provided by optimum
configurations found for
discrete values of element mass. Dashed line: OASPL reduction
levels provided by full
configurations with equivalent total mass.
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First of all, from both figure 26 and figure 27, it can be
stated that adding mass to the floor panels is
not generally advisable, as one of their first natural
frequencies is shifted down to the excitation frequency
thus producing floor panels resonance for a total mass of 66 kg.
Besides this particular effect, the
homogeneous addition of mass to the floor panels proves to be
limited to a maximum OASPL reduction
of about 3 dB. In this case, the optimization of mass elements
might be providing two positive effects
with respect to the limitations of homogeneous mass addition. On
the one hand, for each value of element
mass, the number of elements is adjusted so that undesired
resonance is avoided. On the other hand,
optimum configurations might benefit from phase cancellation
effects between different panels thus
allowing for significant noise reduction. From figure 26 it can
be observed that optimum configurations
yield OASPL reduction levels up to about 6 dB.
When it comes to noise attenuation from elementary structural
elements, e.g. beams, plates,
membranes, etc., which possess a low number of structural modes
in the frequency range of interest, it is
generally advisable to study such modal composition before
launching any optimization tool. For this
study case, the sharp OASPL amplification peak found in figure
27 for a total mass of 66 kg might be an
indicator that one of the first modes of the floor panels,
probably with high noise radiation efficiency,
appears at a natural frequency slightly above the excitation BPF
frequency of 102 Hz. Therefore, as floor
panels mass is increased, such natural frequency is shifted
towards the BPF frequency eventually
producing floor panels resonance. Nevertheless, this observation
does not necessarily result in discarding
the optimization of mass configurations attached to floor panels
as optimum positioning of mass elements
might benefit from phase cancellation (or subtraction) of
individual panel noise contributions, even if they
are high due to structural response close to resonance.
From conventional vibro-acoustic simulation of the optimum
configuration for an element mass value
of 3 kg, global OASPL contributions are compared in figure 28,
and OASPL contributions along the
longitudinal axis of the airframe are compared in figure 29.
Figure 27. OASPL reduction levels due to the mass effect in the
domain of total mass.
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From figure 28 it can be observed that the optimum configuration
has little effect on the OASPL
contribution from skin panels, and the total OASPL reduction
level of about 4 dB is related to an
attenuation of the contribution from floor panels of 6.5 dB.
This effect is also observed in figure 29,
where the dashed line for skin panels contribution presents very
little variation when compared to the
baseline results.
C. Dynamic vibration absorbers The optimization problem for
dynamic vibration absorbers (DVAs) is identical to optimization
of
frame configurations of mass elements with respect to the
definition of candidate nodes and candidate
elements. However, in this case, DVA masses are only allowed to
oscillate in the perpendicular direction
with respect to the frames to which they are attached. The
optimization process is performed, both for
tuned ( ) and detuned ( ) DVAs in the mass range up to 6 kg.
Compressive dynamic vibration absorbers 1.At a first stage,
compressive DVAs are forced to a single-degree-of-freedom natural
frequency equal
to the excitation frequency, i.e. 102 Hz, which hence
establishes a direct relationship between the DVA
mass and the stiffness of the DVA spring. Therefore, the
optimization process needs to account for spatial
distribution of DVAs as well as two parameters: mass and damping
ratio. Results in terms of OASPL
reduction levels, for the mass range up to 6 kg and for discrete
values of damping ratio, are compared to
results obtained from the optimization of mass configurations in
figure 30.
Figure 28. Contribution from structural elements to OASPL from
BPF excitation at 102 Hz.
Solid bar: Primary structure. Dashed bar: Optimum floor mass
configuration (3 kg).
Figure 29. Contribution from fuselage sections (front to back)
to OASPL from BPF excitation at
102 Hz. Left: Primary structure. Right: Optimum floor mass
configuration (3 kg). Solid line:
Total OASPL. Dashed line: Skin panels contribution. Dotted line:
Floor panels contribution.
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It can be observed from figure 30 that optimum DVA
configurations only provide higher reductions
than optimum mass configurations for the range of element mass
up to 2 kg, and the corresponding curves
do not present an increasing trend as the element mass is
increased. Instead, some optimum values of
DVA mass are found for each value of viscous damping ratio.
At a second stage, the analysis focuses on the effect of tuning
the DVAs at other frequencies rather
than the excitation frequency. For such extended analysis only
DVAs with mass value of 1.2 kg and
damping ratio of 0.1 % are taken, which corresponds to the
optimum configuration of tuned compressive
DVAs with highest OASPL reduction level (6.7 dB). Results in
terms of OASPL reduction levels are
presented in figure 31.
Shear dynamic vibration absorbers 2.Following the same process
than for compressive DVAs, results in terms of OASPL reduction
levels
for shear DVAs tuned at the excitation frequency, i.e. 102 Hz,
are presented in figure 32.
Figure 30. OASPL reduction levels from optimization of
compressive DVA configurations for
discrete values of viscous damping ratio.
Figure 31. OASPL reduction levels from optimization of detuned
compressive DVA
configurations for tuning frequencies in the range between 92 Hz
and 112 Hz (mass 1.2 kg,
damping ratio 0.1 %).
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At second analysis stage, a shear DVA with mass value of 1.2 kg
and damping ratio of 5 %,
corresponding to the optimum configuration of tuned shear DVAs
with highest OASPL reduction level
(9.9 dB) is chosen. Results in terms of OASPL reduction levels
are presented in figure 33.
It is interesting to note that both compressive and shear DVAs
present a similar behavior with respect
to tuning frequency, a peak OASPL reduction level is reached at
a tuning frequency slightly below the
excitation frequency, 98 Hz for both cases, and a rapid decay
rate for higher tuning frequencies.
Maximum OASPL reduction provided by shear DVAs with a DVA mass
of 1.2 kg proves slightly higher
than maximum OASPL reduction provided by compressive DVAs also
with a DVA mass of 1.2 kg,
reaching in both cases the OASPL reduction levels provided by
mass configurations for an element mass
of 6 kg.
VII. Conclusions
From the results presented above and their interpretation, it
can be stated that the proposed
optimization methodology constitutes an effective tool not only
for the attenuation of cabin noise in
turbopropeller aircraft but also for the enhanced understanding
of its vibro-acoustic behavior. Results in
terms of displacement fields and acoustic pressure fields, among
other magnitudes, can be obtained at
Figure 32. OASPL reduction levels from optimization of shear DVA
configurations for discrete
values of viscous damping ratio.
Figure 33. OASPL reduction levels from optimization of detuned
shear DVA configurations for
tuning frequencies in the range between 92 Hz and 112 Hz (mass
1.2 kg, damping ratio 0.5 %).
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relatively low computational cost, which therefore enables the
analyst to tailor the approach to any
particular aspect of the vibro-acoustic performance of the
primary structure, e.g. OASPL reduction at
given output positions, contribution from structural elements
and sections, attenuation of the vibration
amplitude for a particular structural element, modal composition
and contribution to the vibro-acoustic
response, etc. Furthermore, as the proposed optimization
methodology is stated on generic terms, it can be
easily applied to any particular structure with any particular
set of requirements only by applying the
corresponding restrictions to the matrix equation above
accordingly.
The contents of the present document constitute an initial stage
or level zero in the development
process of a mature optimization tool which is ready for
application to real vibro-acoustic problems in
aerospace industry. It is also aimed at defining a roadmap for
such development which consists on
leveling up each of the particular topics which are involved,
i.e. structural coupling methods, optimization
algorithms, exploration of countermeasures, etc., by increasing
their level of complexity, eventually
resulting in lower computational requirements, more detailed
insight in the vibro-acoustic problem,
higher noise reduction levels, etc. This is summarized in table
1.
Acknowledgments
The present study is developed in the framework of a PhD program
in aerospace engineering at UPC-
BarcelonaTech, co-funded by SENER Ingeniera y Sistemas, S.A. The
author is very grateful to the Noise
and Vibration Group in SENER Ingeniera y Sistemas, and
especially to fellows Pierre Huguenet, Ben
Park and Emiliano Tolosa for their active support and valuable
advice throughout all the research
activities.
References 1Wilby, J. F., Aircraft interior noise, Journal of
Sound and Vibration, Vol. 190, No. 3, 1996, pp. 545-564. 2Wilby, J.
F., and Wilby, E. G., Measurements of propeller noise in a light
turboprop airplane, Journal of
Aircraft, Vol. 26, No. 1, 1989, pp. 40-47. 3Ewing, M. S., Kirk,
M. A., and Swearingen, J. D., Beech 1900D flight test to
characterize propeller noise on the
fuselage exterior, 7th AIAA/CEAS Aeroacoustics Conference,
AIAA-2001-2110, Maastricht, NL, 2001. 4Farassat, F., Dunn, M. H.,
Tinetti, A. F., and Nark, D. M., Open-rotor noise prediction at
NASA Langley A
technology review, 15th AIAA/CEAS Aeroacoustics Conference,
AIAA-2009-3133, Miami, FL, 2009.
Level Zero Level Up Target
Selection of
Countermeasures
Stiffeners, masses
and DVAs
Multiple-DoF DVAs,
absorption panels,
mechanical properties
Elaboration of a library
of vibro-acoustic
countermeasures
Structural
Coupling
FRF coupling at
discrete nodes
FRF and modal coupling
at continuous boundaries
Selection of structural
coupling techniques
according to the
selection of
countermeasures
Optimization
Algorithm
Enhanced brute force
optimization
Study and
implementation of
sophisticated algorithms
for mathematical
optimization
Selection of the
optimization algorithm
according to problem
requirements
Coding Simple MATLAB
routines
Code debugging and
improvement
User-oriented
simulation tool with
well-defined inputs and
outputs
Real
Implementation -
Test-based Monte Carlo
Simulation for optimum
countermeasure
configurations
Implementation of
Monte Carlo
Simulation in the
optimization process
Table 1. Level zero and roadmap for further development of the
proposed methodology for the
optimization of structural countermeasures for noise attenuation
in aircraft cabins.
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5Blandeau, V. P., and Joseph, P. F., Broadband noise due to
rotor-wake/rotor interaction in contra-rotating open rotors, AIAA
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7Bhat, W. V., Use of correlation technique for estimating
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8Metcalf, V. L., and Mayes, W. H., Structureborne contribution
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9Cole III, J. E., and Martini, K. F., Structureborne noise
measurements on a small twin-engine aircraft, NASA CR-4137,
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10Knutz, H. L., and Prydz, R. A., Interior noise in the
untreated Gulfstream II propfan test assessment (PTA) aircraft,
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11Cole III, J. E., Stokes, A. W., Garrelick, J. M., and Martini,
K. F., Analytical modeling of the structureborne noise path on a
small twin-engine aircraft, NASA CR-4136, 1988.
12Unruh, J. F., Aircraft propeller induced structure-borne
noise, NASA CR-4255, 1989. 13Wilby, J.F., Interior noise of general
aviation aircraft, SAE Technical Paper 820961, 1982. 14Rodriguez
Ahlquist, J., Huguenet, P., and Palacios Higueras, J.I., Coupled
FEM/BEM vibroacoustic modeling
of turbopropeller cabin noise, 16th AIAA/CEAS Aeroacoustics
Conference, AIAA-2010-3948, Stockholm, SE, 2010. 15Huguenet, P.,
Rodriguez Ahlquist, J., and Bonillo Coll, A., Coupled FEM-BEM
modeling of turbopropeller
cabin noise, 2011 LMS European Aerospace Conference, Toulouse,
FR, 2011. 16Liu, C.Q., and Liu, X., A new method for analysis of
complex structures based on FRFs of substructures,
Shock and Vibration, Vol. 11, No. 1, 2004, pp. 1-7. 17Palacios
Higueras, J.I., Desarrollo y validacin de una metodologa de
prediccin de los campos acsticos
resultantes de la aplicacin de sistemas de control activo de
ruido, Ph.D. Thesis, Departament dEnginyeria Mecnica, Universitat
Politcnica de Catalunya, Barcelona, ES, 2009.
18Peery, D.J., and Azar, J.J., Aircraft Structures, McGraw-Hill,
New York, 1983, Chap. 7.