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Journal ArticleDOI

Active control of turbulent boundary layer-induced sound transmission through the cavity-backed double panels

26 May 2018-Journal of Sound and Vibration (Academic Press)-Vol. 422, pp 161-188

AbstractThis paper presents a theoretical study of active control of turbulent boundary layer TBL induced sound transmission through the cavity-backed double panels. The aerodynamic model used is based on the Corcos wall pressure distribution. The structural-acoustic model encompasses a source panel (skin panel), coupled through an acoustic cavity to the radiating panel (trim panel). The radiating panel is backed by a larger acoustic enclosure (the back cavity). A feedback control unit is located inside the acoustic cavity between the two panels. It consists of a control force actuator and a sensor mounted at the actuator footprint on the radiating panel. The control actuator can react off the source panel. It is driven by an amplified velocity signal measured by the sensor. A fully coupled analytical structural-acoustic model is developed to study the effects of the active control on the sound transmission into the back cavity. The stability and performance of the active control system are firstly studied on a reduced order model. In the reduced order model only two fundamental modes of the fully coupled system are assumed. Secondly, a full order model is considered with a number of modes large enough to yield accurate simulation results up to 1000 Hz. It is shown that convincing reductions of the TBL -induced vibrations of the radiating panel and the sound pressure inside the back cavity can be expected. The reductions are more pronounced for a certain class of systems, which is characterised by the fundamental natural frequency of the skin panel larger than the fundamental natural frequency of the trim panel.

Summary (4 min read)

1. Introduction

  • In high speed automotive, aerospace, and railway transportation, the turbulent boundary layer (TBL) is one of the most important sources of interior noise.
  • Consequently the trim panels radiate sound into the vehicle interior.
  • Between the different models developed over the years to describe the wall pressure fluctuations due to a TBL [3], the Corcos model is among the simplest [4, 5].
  • The passive sound transmission control is ineffective in the lowfrequency range since the passive sound absorptive materials can not attenuate the large-length waves.
  • Feedback control systems can be used instead, and some promising results have been reported in the last decade [14, 20, 21, 22].

2. The model problem

  • The analytical model outlined in this section is used to predict the noise transmission through an acoustically coupled double panel system into a rectangular cavity when an active vibration isolation unit is used.
  • The problem under analysis is physically given by the interaction of an aerodynamic model, that represents the TBL pressure fluctuations on the structure, and a structural-acoustic model, that gives the noise transmission and interior sound levels.

2.1. Aerodynamic model

  • The wall pressure field, generated by a fully developed TBL with zero mean pressure gradient, can be regarded as homogeneous in space and stationary in time [3, 28].
  • It is thus possible that the results presented in the paper qualitatively differ for different convective speeds and the dimensions of the panels.
  • In the following sections the properties of the physical system used are defined in detail.
  • The longitudinal and lateral decay rates of the coherences, αx and αy respectively, are normally chosen to yield good agreement with experiments.

2.2. Structural-acoustic model

  • The model encompasses a cavitybacked homogeneous double panel driven on one side by a stationary TBL.
  • Real sensor-actuator transducers would certainly exhibit more complex highfrequency behaviour and decrease stability margins of the active control systems considered in the study.
  • In the following section, the structural-acoustic model presented describes the interaction between the vibrating structure and the turbulent flow assuming a one-way interaction, i.e. the influence of the panel vibration on the boundary layer is neglected [33].
  • In Eq. (8)-(9), v represents the distribution of source volume velocity per unit volume (including the effect of the boundary surface vibration), c0, ρ0 and σac are the sound speed in air, mean density of air and acoustic damping ratio for both cavities, respectively.

3.1. Description of the simplified model

  • The stability and performance analysis of the active control system are carried out using a simplified model.
  • Note that the first mode of each air cavity is characterised by a uniform pressure variation throughout the cavity volume and by a zero natural frequency.
  • The resulting coupled system 9 then behaves exactly as a two degree of freedom (DOF) mechanical system which, as shown in Fig. 2, is represented through mass and stiffness parameters m1,m2, k1, k2, k3, and which is characterised by two coupled mode shapes and two natural frequencies.
  • The remaining geometrical and physical properties used throughout the paper are shown in Table 1.

3.2. Stability

  • A system is stable if all roots of its characteristic equation have negative real parts.
  • Additionally, all the principal diagonal minors ∆i of the Hurwitz matrix must be positive.
  • It can be stated that in the case with β < 1 the source body, having the larger uncoupled natural frequency than the radiating body, behaves more like a fixed reference base against which the actuator can react without pronounced feedthrough effect that could otherwise compromise stability properties.
  • The frequency response function between the reaction force component and the velocity sensor is thus responsible for the stability problems in case β >.
  • This is because frequency response functions between two different points of a flexible structure do not necessarily have their phases bound within a 180 degree range, whereas the driving point FRFs do have their phase limited within a 180 degrees range.

3.3. Performance

  • The performance of the active control system is analysed by using two different load conditions.
  • Firstly, a white noise forcing is assumed on m1, Fig. 2, where the mass m1 represents the source panel and is thus referred to as the source body.
  • Then, the mean squared velocity of the mass m2 is found by calculating the integral over all frequencies of the squared magnitude of the transfer mobility, Eq. (28), of the system in Fig.
  • Note that the mean squared displacement of the mass m2 is proportional to the mean squared pressure in the cavity c2 according to the two mode assumption, see Eq. (29).
  • The frequency- and space-averaged velocity PSD of the radiating panel is used.

3.3.1. Point force excitation

  • The mean squared velocity of the radiating body is plotted as function of the passive and active damping ratio in Fig.
  • This is however only due to a) the limitations of the reduced order model and its inability to capture the high-frequency behaviour of the double panel structure, and b) the fact that idealised sensor and actuator transducers are assumed.
  • The passive damping ratio is again set to the optimal value that lies on the white dash-dotted line in Fig.
  • In addition, the roll-off at higher dimensionless frequencies, above approximately 2.1, is compromised as shown in Fig. 5-a.
  • In conclusion, the inconsistent impact of the passive damping ratio at various frequency ranges is the reason why the frequency averaged kinetic energy of the radiating panel plotted in Fig. 6-a has a minimum which corresponds to the optimal passive damping coefficient ηopt.

3.3.2. Turbulent boundary layer excitation

  • The performance of the control system in reducing the TBL−induced sound transmission is studied by using the simplified, reduced order model.
  • As with the point force excitation, the radiating panel kinetic energy decreases monotonically with ξ when β < 1 Fig. 7-(b) and Fig. 8-(b).
  • By comparing the situation with the TBL excitation to the scenario with the white noise force excitation discussed in the previous subsection, it can be stated that the stochastic TBL pressure distribution results in a steeper high frequency roll-off of the velocity PSD, shown in Fig.
  • This requires careful tuning of the passive and active damping ratios to the optimum combination.

4. Results with the full order model

  • The stability and performance of the active control system are again discussed in terms of the mean squared vibration velocity of the radiating panel and the mean squared acoustic pressure in the cavity c2.
  • The number of modes used enables accurate calculation of the mean squared pressure and velocity up to 1000 Hz.
  • Two load conditions are analysed: 1) white noise point force excitation; and 2) TBL excitation.
  • The large number of modes precludes the use of the Routh-Hurwitz criterion to assess the stability of the active system such that the Nyquist criterion is used instead.

4.1. Stability

  • The sensor-actuator OL − FRF can be used to analyse the stability of the closed loop control system by using the Nyquist criterion.
  • In Fig. 11 - 12, the corresponding Bode and Nyquist plots are shown.
  • Thus the system with β < 1 is unconditionally stable under the assumption of ideal sensor-actuator frequency response.
  • Regarding the high-frequency behaviour, it is still interesting that the phase of the OL − FRF is bound between ±90 degrees, given the fact that the sensor and the actuator are not dual and collocated.
  • Therefore the relative importance of the non-collocated FRF decreases as the frequency increases.

4.2. Performance

  • In the following subsection, the performance of the control system assuming a full order model is analysed for both load conditions: point force and TBL.
  • The frequency-averaged vibration velocity of the radiating panel and the sound pressure in the acoustic enclosure c2 are used as the metrics for the quality of the sound transmission control.

4.2.1. Point force excitation

  • In the case of a point force excitation on the radiating panel, the mean squared velocity at the centre of the radiating panel and the mean squared acoustic pressure near the corner of 24 the cavity c2 are calculated using the procedure described in Section 2.2.
  • The squared magnitude of either frequency response function is integrated numerically over the frequency range 0 − 1000 Hz.
  • The amplitude of the velocity at the centre of the radiating panel per unit excitation force on the source panel is shown in Fig. 14-b, and the amplitude of the sound pressure per unit excitation force at the pressure monitoring point in the cavity c2 is shown in Fig. 15-b.
  • Again, the results with the active approach are contrasted to the results using a fully passive approach.

4.2.2. Turbulent boundary layer excitation

  • For the full order model assuming the TBL excitation on the source panel, the performance of the control system is analysed in terms of: a) the velocity PSD at the centre of the radiating panel, pr, and b) the pressure PSD near the corner of the cavity c2, see Fig.
  • The pressure PSD near the corner of c2 is plotted versus frequency in Fig. 17-b with β < 1 and Fig. 19-b with β > 1 for three cases: without control, with passive control using cp only, and finally with active control using both cp and g.
  • By comparing the right-hand side plot of Fig. 16 to the right hand side plot of Fig. 17, it can be concluded that large contributions to the interior sound pressure are due to the two lowest double panel modes.
  • Additional mode shapes obtained using the full order model are shown and discussed.

5. Conclusions

  • The active control of TBL noise transmission through a cavity-backed double panel is investigated.
  • Stability and performance of the velocity feedback active control system are carried out for a reduced order model and an increased order model.
  • The theoretical analysis indicates that the first fundamental mode and the mass-air-mass mode of the coupled system are the strongest radiators of sound into the back cavity.
  • Closed form expressions for these stability limits are given in terms of the minimal/maximal active damping ratio.
  • The fundamental resonance frequency of the source panel is larger than the fundamental resonance frequency of the radiating panel, which results in unconditionally stable control systems such that very large feedback gains can be used.

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Citation
A. Caiazzo, N. Alujevic, B. Pluymers and W. Desmet, (2018)
Active control of turbulent boundary layer-induced sound transmission
through the cavity-backed double panels
Journal of Sound and Vibration, 422, 161-188.
Archived version
Author manuscript: the content is identical to the content of the published
paper, but without the final typesetting by the publisher.
Published version
https://www.sciencedirect.com/science/article/pii/S0022460X18301172
Journal homepage
http://www.sciencedirect.com/science/journal/0022460X
Author contact
Anna.Caiazzo@kuleuven.be
+ 32 (0)16 37 20 48
(article begins on next page)

Active control of turbulent boundary layer-induced sound
transmission through the cavity-backed double panels
A. Caiazzo
a,b,
, N. Alujevi
´
c
a
, B. Pluymers
a,b
, W. Desmet
a,b
a
KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B, 3001 Heverlee, Belgium.
b
Member of Flanders Make.
Abstract
This paper presents a theoretical study of active control of turbulent boundary layer
(
TBL
)
in-
duced sound transmission through the cavity-backed double panels. The aerodynamic model
used is based on the Corcos wall pressure distribution. The structural-acoustic model encom-
passes a source panel (skin panel), coupled through an acoustic cavity to the radiating panel
(trim panel). The radiating panel is backed by a larger acoustic enclosure (the back cavity). A
feedback control unit is located inside the acoustic cavity between the two panels. It consists of a
control force actuator and a sensor mounted at the actuator footprint on the radiating panel. The
control actuator can react o the source panel. It is driven by an amplified velocity signal mea-
sured by the sensor. A fully coupled analytical structural-acoustic model is developed to study
the eects of the active control on the sound transmission into the back cavity. The stability and
performance of the active control system are firstly studied on a reduced order model. In the
reduced order model only two fundamental modes of the fully coupled system are assumed. Sec-
ondly, a full order model is considered with a number of modes large enough to yield accurate
simulation results up to 1000 Hz. It is shown that convincing reductions of the TBL-induced
vibrations of the radiating panel and the sound pressure inside the back cavity can be expected.
The reductions are more pronounced for a certain class of systems, which is characterised by the
fundamental natural frequency of the skin panel larger than the fundamental natural frequency
of the trim panel.
Keywords: Turbulent Boundary Layer, Corcos, Active vibroacoustic control, Direct velocity
feedback, Active control of sound transmission, Stability of active systems
1. Introduction
In high speed automotive, aerospace, and railway transportation, the turbulent boundary layer
(TBL) is one of the most important sources of interior noise. For example, in large passenger
aircraft, the principal source of interior noise is the TBL during typical cruise conditions [1, 2].
The stochastic pressure distribution associated with the turbulence is able to significantly excite
structural vibrations of vehicle exterior panels. These vibrations are transmitted to the interior
Corresponding author
Email address: Anna.Caiazzo@kuleuven.be (A. Caiazzo)
Preprint submitted to Journal of Sound and Vibration February 23, 2018

trim panels through the structural and acoustical paths between the interior and exterior panels.
Consequently the trim panels radiate sound into the vehicle interior.
The TBL excitation is random and broadband. In order to model this excitation, semi-empirical
models for the wall pressure distribution are usually used. Between the dierent models de-
veloped over the years to describe the wall pressure fluctuations due to a TBL [3], the Corcos
model is among the simplest [4, 5]. This is because the space variables are separated and the
phase variation is only accounted for along the streamwise direction. It should be noted that
modelling the low-wavenumber region of the TBL pressure spectrum is still an active area of
research [6, 7]. In particular, it is known that Corcos model overpredicts experimental results
in terms of the spatially integrated correlation functions. It nevertheless enables rather realistic
representations of the wall pressure distribution at and near the convective wavenumber, where
most of the energy in the boundary layer pressure fluctuations is concentrated. Therefore, the
Corcos model is often used as a reference for other models [8, 9, 10], such as, for example, the
recently developed Generalized Corcos model [7].
The disturbance produced by TBL typically excites the vehicle exterior surface in a broad band
of frequencies including the low-frequency range. The resulting transmission of sound into the
interior is usually controlled by passive means. However, the eectiveness of passive noise con-
trol treatments, such as, for example, sound absorbing materials, is limited to frequencies where
the acoustic wavelengths are short, in the medium and high frequency range. As a rule of thumb,
the sound absorbing layers should be at least as thick as 1/4 of the acoustic wavelength in order
to absorb the sound eectively. The passive sound transmission control is ineective in the low-
frequency range since the passive sound absorptive materials can not attenuate the large-length
waves. On the other hand, double leaf partitions may be used whose transmission loss increases
more rapidly with frequency than that of single leaf partitions. Therefore, it is often the case
that the vehicles are equipped with additional interior trim panels. In such arrangements, the
sound absorbing layers are packed in the space available between the two panels. Even with
such integrated approach, the transmission loss of double leaf partitions is still rather poor at the
frequencies below the mass-air-mass resonance [11, 12].
For these reasons, interest has grown in investigating active control of low-frequency sound
transmission through double panels [13, 14]. Active control of sound transmission can be used
both with deterministic and stochastic disturbances. Feed-forward and feedback active control
systems have been considered in this sense. Although feed-forward control methods are more
easily applied in situations with deterministic, tonal disturbances [15, 16], it is also possible to
use them to control noise of stochastic origin [17]. This, however, requires that creative arrange-
ments are used to obtain the reference signals well correlated to the disturbance [18, 19]. Here a
good balance between the causality margin and the coherence with respect to the reference and
disturbance signals is essential [19].
Feedback control systems can be used instead, and some promising results have been reported
in the last decade [14, 20, 21, 22]. The use of feedback control is appealing since the reference
signals are not necessary. In this context, some very eective feedback control methods have
been proposed for vibration isolation [23, 24, 25]. These include, for example, the so-called sky-
hook damping and integral force feedback [26, 27]. Such methods can overcome the diculties
with traditional passive vibration isolation systems. The diculties include the intrinsic trade-o
in the vibration isolation performance between the low-frequency resonance-controlled transmis-
sion and the high-frequency mass-controlled transmission. With active vibration isolation the low
frequency vibration transmission may be attenuated without compromising the high-frequency
roll-o of the vibration transmissibility with frequency. Such active vibration isolation methods
2

have been typically considered in the situations where the mechanical systems at hand can be
accurately represented through their lumped parameter approximation [23, 24, 25].
In this paper, it is shown that feedback vibration isolation techniques can be very eective to ac-
tively control the sound transmission in distributed parameter systems with complex structural-
acoustic coupling. In particular, the TBL-induced sound transmission from a flexible panel (the
source panel) through an acoustic medium to another flexible panel (the radiating panel) and fur-
ther into additional acoustic enclosure, is treated as an active vibration isolation problem. Thus
the problem studied roughly approximates the problem of the transmission of aerodynamic noise
into high-speed vehicles. The stability and performance of the control system based on direct
velocity feedback, are addressed in detail. It is shown that they are substantially aected by the
passive properties of the double panel system before control. Particularly acoustic reductions
of the transmitted sound are feasible with a certain class of double panels that are characterised
by the fundamental natural frequency of the source panel larger than that of the radiating panel.
Then the frequency spectra of sound pressure in the acoustic enclosure approximating the vehicle
interior can be successfully attenuated in the low frequency range without an overshot in the high
frequency range. As a result, significant broadband sound transmission control can be realised,
even though only a single control unit is used.
The paper is structured as follows. In Section 2, an outline of the mathematical model used
to couple the aerodynamic, control and vibroacoustic aspects of the problem is given. In Sec-
tion 3, the stability and performance of the active control system are discussed on a reduced
order model, whereas in Section 4 a full-order model is considered. First a point force excitation
with white noise spectral distribution is used as a reference case, and then, the stochastic TBL
pressure distribution is considered - both for the reduced and the full order models.
2. The model problem
The analytical model outlined in this section is used to predict the noise transmission through
an acoustically coupled double panel system into a rectangular cavity when an active vibra-
tion isolation unit is used. The problem under analysis is physically given by the interaction
of an aerodynamic model, that represents the TBL pressure fluctuations on the structure, and a
structural-acoustic model, that gives the noise transmission and interior sound levels.
2.1. Aerodynamic model
The wall pressure field, generated by a fully developed TBL with zero mean pressure gra-
dient, can be regarded as homogeneous in space and stationary in time [3, 28]. Under such
conditions, for a flow in the x-direction over the (x, y) plane, the generated pressure field can
be expressed by a cross power spectral density
(
CPSD
)
function decaying with spatial and time
separations and convected with the flow as follows,
Ψ
pp
(ζ
x
, ζ
y
, ω) = φ(ω)Ψ (ζ
x
, ζ
y
, ω), (1)
in which ζ (ζ
x
, ζ
y
, 0) is the spatial separation vector, φ(ω) is the single-point wall-pressure
spectrum (i.e., auto-spectrum) and Ψ(ζ
x
, ζ
y
, ω) is a spatial correlation function, that, according to
Corcos, is given by
Ψ (ζ
x
, ζ
y
, ω) = e
ik
ω
ζ
x
e
−|ζ
x
|k
ω
α
x
−|ζ
y
|k
ω
α
y
. (2)
The convective wavenumber is k
ω
= ω/U
c
, given by the angular frequency ω and the convective
velocity U
c
= 0.7U
, with U
= 240m s
1
. In this study the dimensions of the structure and the
3

convective speed are fixed. It is thus possible that the results presented in the paper qualitatively
dier for dierent convective speeds and the dimensions of the panels.
In the following sections the properties of the physical system used are defined in detail. In this
study, an aerospace application is considered in which the model approximates a rectangularly
shaped aircraft multi panel wall. The longitudinal and lateral decay rates of the coherences,
α
x
and α
y
respectively, are normally chosen to yield good agreement with experiments. In the
following analysis, the empirical parameters are chosen as α
x
= 0.10 and α
y
= 0.77, [29, 30].
In Eq. (1), the single-point wall-pressure spectrum, φ(ω), is given by a semi-empirical formula
defined by Goody [31],
φ(ω) =
3
(
δ/U
)
3
(
ωτ
ω
)
2
h
(
ωδ/U
)
0.75
+ 0.5
i
3.7
+
h
1.1R
0.57
T
(
ωδ/U
)
i
7
, (3)
in which δ is the boundary layer thickness, τ
ω
= U
2
t
ρ is the wall shear stress, with air density ρ,
and R
T
is the Reynolds number dependent factor given by
R
T
=
U
2
τ
δ
U
ν
, (4)
where U
τ
is the friction velocity and ν is the kinematic viscosity. Such a model compares well
with experimental data over a large range of Reynolds numbers [32] and it is able to describe
the essential properties of the single point wall pressure spectrum with a limited numbers of
variables.
2.2. Structural-acoustic model
In Fig. 1, the structural-acoustic model problem is shown. The model encompasses a cavity-
backed homogeneous double panel driven on one side by a stationary TBL. Thus, the acoustic
enclosure, c
2
, filled with air, has ve rigid walls and one flexible double wall. The two flexible
panels are acoustically coupled with the air in cavity c
1
between them. As shown in Fig. 1, the
source panel, p
s
, is excited by the grazing flow. Its vibrations generate sound waves which excite
the radiating panel, p
r
. Finally, the vibrations of the radiating panel generate sound waves which
are radiated to the acoustic enclosure c
2
.
The active control approach in this study is based on isolating the radiating panel from vibrations
coming from the source panel through an active vibration isolation unit. A sensor is placed near
the centre of the radiating panel, which is collocated to a force actuator. The actuator generates
the control force while reacting against the source panel. In parallel to the actuator, a passive
damper is mounted with a damping coecient c
p
, as seen in Fig. 1. In this paper, idealised
sensor-actuator transducers are considered, that is, the sensor-actuator internal dynamics are not
taken into account. Real sensor-actuator transducers would certainly exhibit more complex high-
frequency behaviour and decrease stability margins of the active control systems considered in
the study. However, in this paper the stability and performance characteristics of the active
control system, which are independent of the sensor-actuator dynamics, are investigated. Also,
by neglecting the dynamics of the transducers, benchmark results can be obtained. Additionally,
sensors and actuators would certainly add some mass to the system. However, a miniature voice-
coil actuator that can develop an equivalent of 140 grams of peak force (approx. 1.4 N) has a total
mass (coil and magnet) of only about 6 grams with a stroke of about 3 mm. Furthermore such
actuators do not inflict any stiness between their two terminals like, for example, piezoelectric
4

Figures (20)
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