Paper a Bon Illo

1

American Institute of Aeronautics and Astronautics

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:


[ ] . 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:


[ ] . 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).

16


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).

17


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).

18


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.

19


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).

20


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).


Solid bar: Primary structure. Dashed bar: Optimum mass configuration (4 kg).

21


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.

22


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.

23


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.


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.

24


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 %).

25


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 %).

26


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.

27


5Blandeau, V. P., and Joseph, P. F., Broadband noise due to rotor-wake/rotor interaction in contra-rotating open rotors, AIAA Journal, Vol. 48, No. 11, 2010, pp. 2674-2686.

6Bishop, D. E., Cruise flight noise levels in a turbojet transport airplane, Noise Control, Vol. 7, No. 2, 1961, pp. 37-42.

7Bhat, W. V., Use of correlation technique for estimating in-flight noise radiated by wing-mounted jet engines on a fuselage, Journal of Sound and Vibration, Vol. 17, No. 3, 1971, pp. 349-355.

8Metcalf, V. L., and Mayes, W. H., Structureborne contribution to interior noise of propeller aircraft, SAE Technical Paper 830735, 1983.

9Cole III, J. E., and Martini, K. F., Structureborne noise measurements on a small twin-engine aircraft, NASA CR-4137, 1988.

10Knutz, H. L., and Prydz, R. A., Interior noise in the untreated Gulfstream II propfan test assessment (PTA) aircraft, Journal of Aircraft, Vol. 27, 1990, pp. 647-652.

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.

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