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Uncertainty analysis in vibroacoustic panels with band gap
D. Beli1, A.T. Fabro2, M. Ruzzene3, J.R.F. Arruda1 1 University of Campinas, School of Mechanical Engineering, Department of Computational Mechanics,
Cidade Universitária Zeferino Vaz, Campinas-SP, 13083-080, Brazil
e-mail: [email protected] 2 University of Brasilia, Department of Mechanical Engineering,
Campus Universitário Darcy Ribeiro, Brasilia-DF, 70910-900, Brazil 3 Georgia Institute of Technology, School of Aerospace Engineering, School of Mechanical Engineering,
North Avenue NW, Atlanta, Georgia 30332, United States
Abstract Vibration and acoustic isolation by passive control methods have been extensively studied along the years.
In a recent past, phononic crystals and acoustic metamaterials have also been employed for this purpose.
These periodic structures present band gaps, which are used herein to reduce vibration and acoustic
radiation of panels. However, variability in the manufacturing process causes material and geometry
uncertainties that affect their band gap robustness and consequently their dynamic attenuation
performance. In this work, the robustness of the band gap phenomenon in vibroacoustic panels is
investigated by employing the WFE method together to WKB approximation and KL expansion. A two-
dimensional analysis where the panel is represented by either a phononic undulated beam or a
metamaterial straight beam with distributed resonators is performed. Spatial correlated variability of the
elastic modulus and resonator stiffness is considered. Band gap of both panels are shown to be robust with
respect to this variability. However, vibration and acoustic attenuation are reduced.
1 Introduction
Phononic crystals and acoustic metamaterials are periodic structures used to control and to manipulate
acoustic and elastic waves [1-3]. One of their properties is the occurrence of band gaps, which are the
range of frequencies where wave propagation is forbidden. Applications could involve vibration
attenuation [4, 5] and noise reduction [6, 7] as well as cloaking and tunneling of mechanical waves [8, 9].
While phononic crystals are produced by material or geometric periodic modulation, metamaterials have
inclusions that work like internal resonators [10]. Herein, these configurations are applied in vibroacoustic
panels to attenuate structural vibration and acoustic radiation by using undulated beams, which work as a
phononic crystal, and straight beams with attached resonators, which work as a metamaterial.
In undulated beams, the coupling between flexural and extensional waves opens the band gaps; moreover,
it was demonstrated that beam thickness and undulation are responsible, respectively, for the location and
width of the band gaps [11]. For a beam with attached resonators, the coupling between beam and
resonator produces a band gap in the tuned resonance [12]. This configuration leads an acoustic mode at
low-frequency, where beam and resonators move in phase, and an optical mode at higher frequencies,
where beam and resonators move out of phase [13]. Besides, a beam with internal resonance band gaps
presents a locally negative slope in the dispersion diagram that is related to vibration attenuation [14].
Herein, these structures interact with an external fluid at one side. This problem is solved using coupling
equations that provide better insight about the elastic and acoustic wave interaction. In addition, the
Sommerfeld radiation condition is satisfied. This general formulation can be used for light or heavy fluid
interaction and also for interior problems if the Sommerfeld radiation condition is not imposed.
1969
By using wave propagation methods, such as the Spectral Element (SE) method [15, 16] and the Wave
Finite Element (WFE) [17-21] method, dispersion diagrams and forced responses of periodic structures
can be obtained by modeling one unit cell. Dispersion diagrams are computed by considering free
traveling waves and by applying Block’s theorem in the homogeneous equations, while, for forced
response, the structure is finite with boundary conditions and applied external forces. SE and WFE save
computational time because they model a single unit cell of the whole structure, which leads to a suitable
framework for optimization and uncertainty analysis, where a large number of interactions is required. In
addition, by employing the Finite Element (FE) method to construct a unit cell [22], WFE provides
numerical solutions for problems where analytical solutions are not available [23].
Producing structures by 3D printing to control waves is one of the emerging topics in acoustic
metamaterials [24]. However, manufacturing processes produce material and geometrical uncertainties
[25] that modify the structural dynamic behavior affecting the stop band performance [26]. Herein,
uncertainty analyses are performed to predict the robustness of phononic and metamaterials panels to
variability by using the WKB method using the WFE approach [27].
The WKB approximation was proposed by Wentzel, Kramers, and Brillouin, who give their name to the
method, for solving the Schrodinger equation in quantum mechanics. It assumes slowly spatial variation of
the properties such that internal reflections due to local impedance changes are negligible [28]. Uniformly
valid solutions including the turning point can also be found (e.g. [29]).
In this paper a Gaussian spatial distribution for the elastic modulus is generated by the Karhunen-Loève
(KL) expansion [30]. Then, the forward and backward propagation matrices are computed via WKB
approximation, which allow obtaining the forced response with uncertainties via dynamic stiffness or
reflection matrix [31]. The KL expansion applied herein to decompose exponential decaying
autocorrelation functions provides analytical equations for the spectral uncertainty expansion. A numerical
approach is usually necessary [32].
Sections 2 and 3 present the theoretical background for the WFE and WKB approaches, respectively. In
the former, it is discussed how to obtain the wavenumbers and wave shapes by a generalized eigenvalue
problem, while in the latter the uncertainty expression obtained by the KL expansion is presented. A
spectral decomposition, as well as the dynamic stiffness matrix for a substructure with slowly spatial
variability are obtained. Then, in Section 4, results for homogeneous and non-homogenous cases are
discussed and a complete uncertainty analysis is performed. Finally, Section 5 draws some conclusion and
discusses possible next steps of this work.
2 Dynamic Behavior by the WFE Method
2.1 Unit Cell by the FE Method
In this work, a panel in contact with a fluid is represented, assuming in-plane behavior, by a beam with
two-dimensional fluid elements. The unit cell is composed by a beam element connected at one side to a
two-dimensional fluid domain (2D fluid elements). On the external fluid elements, at the top of the unit
cell, the Sommerfeld radiation condition is imposed through absorption elements, which truncate the
unbounded fluid domain, which necessary in exterior acoustic problems [33]. Infinite elements produce
the same effect [34]; however, cylindrical or spherical external geometry is needed, which makes their
application unfeasible in periodic structures in rectangular coordinates. The mass, damping and stiffness
finite element matrices for the kth unit cell, which are extracted from the FE software Ansys®, can be
written in symmetric form as [35]:
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0ff
uuu
K00
0K000K
uuu
C00
00C
0C0
uuu
M00
0M000M
f
s
a
f
s
af
ff
s
a
f
s
af
cf
cf
a
f
s
af
ff
S
, (1)
where the superscripts s, f and a correspond, respectively, to structure, fluid, and absorption degrees of
freedom. In the damping matrix are included the fluid-structure coupling matrices, cC , and the
Sommerfeld damping matrix, aC .
Thus, the dynamic stiffness matrix for one unit cell is kkkki KCMD 2*
, with ω the angular
frequency and )1( iss KK , where η is the structural damping. *
kD is reorganized in submatrices that
are associated to left (L) and right (R) cross-section or interior (I) degrees of freedom (DOF). Considering
that no external forces are applied to the interior nodes, these can be condensed [17]
****
IBIIBIBBBB DDDDD-1
with the B subscript being to L or R. This operation reduces the dimension of
the matrix used to compute the eigenvalue problem and gives the dynamic reduced matrix as:
k
kR
L
kR
L
kRRRL
LRLLkk
FF
F
U
UDDDD
UD ˆˆ
ˆ
ˆ
ˆˆ
. (2)
2.2 Dispersion Relation
Assuming the fluid-structure problem free of external forces, the compatibility of internal displacements
as well as internal forces at the interface of two consecutive unit cells is given by )1()(
ˆˆ
kLkR
qq and
)1()(
ˆˆ
kLkR
ff , respectively. In addition, considering the relations between internal and external
displacements and forces, kRLkRL ]ˆˆ[]ˆˆ[ UUqq and
kRLkRL ]ˆˆ[]ˆˆ[ FFff - , the state vector for two
consecutive cross-sections are related by a transfer matrix which is written in terms of the condensed
dynamic stiffness matrix as:
)(11
11
)1(
)1( ˆ
ˆ
ˆ
ˆkLk
kL
L
kLRRRLLLRRRRL
LRLLLR
kL
L
kLψS
f
q
DDDDDD
DDD
f
qψ
. (3)
If the dynamic stiffness matrix is symmetric, it can be proved the S is symplectic; moreover, S is Δ-
periodic (Δ is the length of unit cell), which makes this matrix invariant under a translation of length Δ
[19]. For a periodic structure, Bloch’s theorem relates the state vectors of two consecutive cross-sections
by the wavenumbers Λ, T
k
T
L
T
L
iT
k
T
L
T
Lke ]ˆˆ[]ˆˆ[
)1(fqfq
λ. Replacing this equation in Eq. (3) yields the
eigenvalue problem, kkkk
ψμψS , where T
k
T
L
T
Lk ]ˆˆ[ fqψ and
ki
ke
λμ . This eigenproblem provides
2n eigenvalues and respective eigenvectors, where n corresponds to the number of DOF associated with
each cross section. While the eigenvalues (µ) are associated to phase change or attenuation along the beam
length, the eigenvectors (Φ), or wave mode shapes, indicate the spatial distribution of the displacements
and forces on the cross section [17-18].
Because of the symplectic nature of matrix S, wave modes appear in pairs
kkΦμ , and
kkΦμ , that
correspond to n waves that propagate to the right and left, respectively. Also, the eigenvalues related to
forward and backward going waves are linked by kkμμ 1 , with 1
kμ and 1
kμ . Moreover, the
eigenvectors can be partitioned in displacement and force components as T
fq][ ΦΦΦ .
METAMATERIALS 1971
A modified eigenvalue problem can be written as proposed by Zhong and Williams [36] to avoid ill-
conditioning on the inversion of matrix DLR. The eigenproblem in Eq. (4) is well conditioned because the
eigenvectors are described just by the displacement components, which avoid the large discrepancy that
occur when they are considered together with the force components [20, 21].
μLwNw , where
LRLL
n
DD
0IL ,
RRRL
n
DD
I0N and
q
q
μΦ
Φw (4)
The relation between eigenvectors and wave mode shapes is given by LwΦ . The modified eigenvalue
problem of Eq. (4) can yield wave modes that do not satisfy exactly the relation kkμμ 1 , which results
in an ill-conditioned matrix system to compute the forced response. A regularization procedure was
proposed to solve this issue for symmetric unit cells with respect to their mid-plane [20], μμ 1 ,
RΦΦ and ff
RΦΦ , where R is a symmetry transformation matrix.
3 Stochastic Behavior by WKB Approximation
The WKB approximation is generally used to solve differential equations with variable coefficients
[37,38]. Herein, this procedure is applied to calculate the forced response of a periodic structure with slow
material property variation, which leads to wavenumbers dependent of space, λ(x). The WKB
approximation provides two wave properties, which capture the spatial fluctuation - the phase change and
the amplitude change, used as inputs for wave propagation methods, such as the WFE. However, this
approach fails when the properties have abrupt changes or when the oscillating waves reach a cut-off
section, breaking the propagation [31]. In the next subsection, we introduce how to represent a random
field with correlated spatial properties by the KL Expansion. Then, the propagation matrix composed by
the phase change and amplitude change are obtained, and finally it is used to compute the forced response
in the presence of uncertainties.
3.1 KL Expansion
A spatial random distribution can be represented by an expansion composed by a complete set of
deterministic functions with corresponding random coefficients [32]. By using the KL expansion the total
mean square error is minimized. Thus, it is an optimal procedure to express the information in a random
field by a series. Otherwise, representing a random process by point discretization methods would need a
large number of samples for a good approximation. In the KL expansion, the eigenvalues and eigenvectors
of the covariance function provide the magnitudes and functions of the orthogonal deterministic basis,
respectively.
A random field, H(x,τ), defined in a finite domain D (x ϵ D) and in a probability space Ω (τ ϵ Ω) with mean
Ho(x) and finite variance E[H(x,τ)-Ho(x)]2, can be represented by a spectral decomposition (like a
generalized Fourier series) as in Eq. (12). Herein, Ho(x) describes the deterministic solution part of a
physical behavior [32].
1
0,
j
jjjxfxHxH (12)
where χ and f (x) are the eigenvalues and eigenvectors of the covariance function C(x1, x2) that satisfy the
Fredholm integral equation of the second kind with orthogonal eigenvectors, Eq. (13) [30]. The covariance
function is assumed bounded, symmetric and positive definite. In addition, ξ corresponds to uncorrelated
random coefficients.
)()(,ojj
Diijoi
xfdxxfxxC and ijD
jidxff (13)
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Considering that the properties vary in a one-dimensional space, a Gaussian autocorrelation function can
be expressed by an exponential decay as bxx
oiexxC 21,
, with b the correlation length in the interval
L/2 < x < L/2, L is the domain length, and xi and xo are any two points within this interval, represented,
respectively, by the kth and kth+Nuc cross-sections, where Nuc is the number of unit cells between these
points. For a zero mean and NKL terms of the series, truncated because of numerical implementation
purposes, the analytical decomposition in descending order of eigenvalues is [30, 32]:
KLN
j
jjjjjjxwxxwxxH
1
2211cossin, (14)
where ξj are Gaussian random coefficients with mean and covariance function given by E[ξi(τ)] = 0 and
E[ξi(θ)ξj(τ)] = δij [31]. Moreover, αj, βj, w1j, w2j, are given by Eq. (15) and Eq. (16) obtained by the
analytical spectral decomposition of Eq. (13) [30].
j
j
jjw
LwL
1
1
12
sin
2/ , where
22
1
1
2
cw
c
j
j
and 02
tan11
jj
wL
wc (15)
and
j
j
jjw
LwL
2
2
22
sin
2/ , where
22
2
2
2
cw
c
j
j
and 02
tan22
c
Lww
jj (16)
with c = 1/b.
The random variation of physical properties such as elastic modulus, density and geometry in space, can
be represented by a KL expansion, ,1,0
xHx , where φ0 is the nominal value and σ the
standard deviation. The assumption of Gaussian probability distribution allows negative values for the
material and geometric properties that contradict the physical meaning; however, the random uncorrelated
coefficients obtained by Monte Carlo sampling can be chosen within an interval to avoid this issue. In
addition, the correlation length can be appropriately chosen to produce a smooth distribution, as the larger
the correlation length is, the smoother is the spatial distribution [30].
3.2 Propagation Matrix
The spatial part of the spectral solution can be written as a function of the local wavenumber. Then, the
eikonal function S(x) is employed to seek a wave solution as ˆ S x i xU x e U x e
, with
xixUxS ~
ln [27, 28]. The propagation matrices for a nonhomogeneous periodic structure with
slow variation of the properties are composed by the phase change and amplitude change. The phase
change for a substructure composed by Nuc unit cells is computed integrating the local wavenumbers on
their domain, Eq. (17). The Gauss-Legendre numerical integration is used; therefore, the wavenumbers
need to be evaluated only just at the integration points, which must to be kept a minimum to avoid excess
computation. This procedure considers a homogeneous cross-section with discretization in the mid-point.
GL
o
i
N
m
mm
x
xoi xWdxxxx )(,θ (17)
where xm and Wm are the sampling points and weights used in the Gauss-Legendre integration with NGL
integration points. Considering the energy conserving property as a consequence of WKB approximation,
the amplitude change is computed by Eq. (18).
io
oi
o
i
oixxi
xxi
Q
Qxx
q
H
f
q
H
f
ΦΦ
ΦΦγ
Re
Relog
2
1log, (18)
METAMATERIALS 1973
The local wavenumbers, λ (xm), as well as the wave shapes on Eq. (18) are obtained using the WFE
method, as described in subsection 2.2. Thus, considering the eikonal solution, forward and backward
propagation matrices with oscillating and evanescent waves are obtained by Eq. (19) [31].
oioioi
oioioi
xxxxidiagxx
xxxxidiagxx
,γ,θexp,Γ
,γ,θexp,Γ
(19)
3.3 Dynamic Stiffness Matrix by WFE Approach
In the WFE method, the dynamic stiffness matrix is obtained by relating the displacements and forces at
the ends of the element. Moreover, this matrix can be developed for an undefined number of unit cells [18-
21], and only at the boundary conditions or locations with external forces the geometry needs to be
meshed. By using Block’s theorem, the state vector can be expanded in terms of the complete wave basis,
j
k
jjjkQ )1(
Φψ . Considering a periodic system composed by Nuc unit cells between the points xo and
xi, the state vector can be evaluated at the ends of the system to write displacements and forces as a
function of the wavenumbers, wave modes and wave amplitudes [18-21]:
Q
Q
ΦΓΦ
ΓΦΦ
R
L
U
Uˆ
ˆ and
Q
Q
ΦΓΦ
ΓΦΦ
ff
ff
R
L
F
Fˆ
ˆ (20)
where L and R are the coordinates correspondent to kth and kth+Nuc cross-sections, respectively.
Eliminating the vectors of wave amplitudes, forces are related to displacements in the frequency domain
by a dynamic stiffness matrix, UDF ˆˆ .
4 Numerical Results
The unit cell geometries, with length Δ = 0.0252m and thickness h = 0.00252m, for the undulated beam
and for the straight beam with resonators are shown in Fig. 1. These structures are made of polyamide -
density ρ = 1000kg/m3, elastic modulus E = 1GPa and structural damping η = 0.01 - and interact with air -
density ρo = 1.2kg/m3 and sound speed c0 = 341m/s. The polyamide was chosen because it is a commonly
used material in 3D printing. For the undulated beam, the orthogonal undulation of the neutral beam axis
is given by l(x) = l0 cos(2πx/Δ) [11], where l0 is the undulation amplitude; l0/Δ = 0.015 is used in this work.
Moreover, a resonator with mass mr = 5mg (7.9% of the total weight), stiffness kr = 0.5MN/m and
damping ηr = 0.01 is placed in the middle of the unit cell. Timoshenko beam elements (BEAM188) and
acoustic fluid elements (FLUID29) are used. The fluid is truncated at y = 0.126m, which was a sufficient
distance to guarantee the Sommerfeld radiation condition. The mesh respects the criterion of 8 elements
per wavelength to reach numerical convergence, which gives 6 elements in the x-direction and 30
elements in the y-direction.
(a)
(b)
Figure 1: Unit cells for an undulated beam (a) and a straight beam with internal resonator (b) interacting
with acoustic elements and imposing the Sommerfeld radiation condition.
1974 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
In the dispersion diagram for a straight beam with air interaction, Fig.2 (aP) and (aE), the two flexural
wavenumbers, green and red thick lines, the extensional wavenumber, orange thick line, as well as the
acoustic wavenumbers can be observed. These waves do not interact and this configuration does not
produce a band gap. The number of acoustic wavenumbers correspond to the number of fluid DOF
because a numerical WFE approach is employed. The speed of the acoustic waves approximate the sound
speed at higher frequencies, while their evanescent part decrease. They are always confined between the
sound speed, thick blue line, and the x-axis. In the undulated beam, Fig. 2(bP) and (bE), the flexural and
extensional wavenumbers present a coupling by the curvature that produces internal wave interference,
opening the band gap. This behavior appears as a straight band in both flexural and extensional elastic
waves; in addition, the acoustic waves are not directly modified by this interaction. The pure evanescent
flexural wave is not influenced, while the pure travelling elastic waves in the stop band zone have a small
attenuation part. Coupling between resonator and beam open the band gap at the tuned resonator
frequencyrrr
mk , Fig. 2(cP) and (cE). The coupling just modifies the two flexural wavenumbers,
the pure evanescent and the pure oscillating flexural waves become complex, with negative-positive
derivatives in the real and imaginary components. These effects on the dispersion diagram are associated
with the vibration attenuation zone on the forced response.
(aP)
(aE)
(bP)
(bE)
(cP)
(cE)
Figure 2: Propagative (P) and evanescent (E) parts of the dispersion diagram for a straight beam (a),
undulated beam (b) and straight beam with internally resonators (c) with weak interaction.
METAMATERIALS 1975
The whole structure is composed by 30 unit cells, L = 0.756m, and with baffle boundary conditions at the
ends; the structural excitation is imposed at x = 0.2268m, the structural and acoustic forced responses are
measure at x = 0.504 m, and for the acoustic response y = 0.0504m. For the non-homogenous analysis
using the WKB approach, this structure is portioned in three. Two integration points are employed for
each substructure, and also the waves need be computed at the ends, which gives nine WFE evaluation
points. In all the results, the correlation length b = L and a standard deviation σ = 0.05 were used.
Furthermore, variability in the elastic modulus is considered for straight and undulated beams, while just
variability in the resonator stiffness is assumed for straight beams with distributed resonators.
The forced response obtained by WKB approach was compared with the FE method for non-homogenous
cases using one sample. For the straight beam, Fig. 3, structural and acoustic responses are in agreement;
however, small errors at some frequency zones appear which can be explained by the approximation of the
WKB method or by ill-conditioning of the dynamic stiffness matrix obtained using the WFE approach.
The use of the WFE with reflection matrix formulation, which should provide well-conditioned dynamic
responses, is left for future work. Moreover, non-homogeneity in the elastic modulus produces more
vibration modes at higher frequencies, as well as a shift in the resonance frequencies. The acoustic
response also captures these modes, which propagate by the fluid generating more noise at higher
frequencies.
(a)
(b)
Figure 3: Structural (a) and acoustical (b) responses for homogeneous (green) and non-homogeneous, by
using WKB-WFE (red) and FE (black), straight beam using one sample with σ = 0.05 and b = L.
(a)
(b)
Figure 4: Structural (a) and acoustical (b) responses for homogeneous (green) and non-homogeneous, by
using WKB-WFE (red) and FE (black), undulated beam using one sample with σ = 0.05 and b = L.
1976 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
The structural and acoustic forced responses for the undulated beam, Fig. 4, and straight beam with
periodically distributed resonators, Fig. 5, exhibit attenuations around 5000Hz and 1500Hz, respectively.
Even for non-homogenous spatial variation of these properties, the band gaps are opened. More evident
numerical issues could be observed mainly in the transition of stop to pass bands for both cases, which is
worse for the undulate beam. These transitions could produce internal reflections where the WKB
assumption is broken; however, for frequencies below or above as well as within the band gap, WKB and
FE results are in a good agreement. These results show that the WKB method gives a fairly accurate
response prediction in all frequency ranges for structures that present band gaps, which justifies its
application in uncertainty analysis, since this method provides an approximate response saving
computational time (around 30% of computational cost compared to the FE approach) for the examples
studied in this work.
(a)
(b)
Figure 5: Structural (a) and acoustical (b) responses for homogeneous (green) and non-homogeneous, by
using WKB-WFE (red) and FE (black), straight beam with internally attached resonators using one sample
with σ = 0.05 and b = L.
The complete uncertainty results performed by using 200 Monte Carlo samples are presented by
employing the mean and the 95% confidence bounds, Fig.6. Only the structural dynamic behavior is
presented because similar conclusions can be transferred to the fluid response. For the straight beam, Fig.
6a, WKB and FE results are in perfect agreement and the uncertainties influence mainly the higher
frequencies. After 3000Hz the location of the resonance peaks cannot be determined with accuracy.
However, the band gap created by the undulated beam, Fig. 6b, exhibits robustness to material variability
thought its width was reduced. The band gap robustness can be explained by the fact that it is opened by
geometry modulation that depends only on the beam thickness and undulation; thus, an elastic modulus
variation has little effect on the band gap characteristics, which present a low standard deviation in this
zone. For the variability only in the internal resonator spring, Fig. 6c, WKB and FE results are also in
good agreement, and the band gap is also robust to variability, but it becomes wider and shallower.
Moreover, the uncertainty causes variability in the dynamic response before and mainly after the band gap
zone.
The formulation presented here is straightforward to apply in cases with strong fluid-structure interaction.
To exemplify, the undulated structure was surrounded by water – density ρ0 = 1030kg/m3 and sound speed
c0 = 1460m/s. The WKB and FE results do not agree in the band gap zone; main divergences appear in the
upper and lower confidence bounds. This issue needs more investigation. The band gap location is shift to
lower frequencies because the fluid adds mass to the system; in addition, its width becomes narrower.
However, vibration and acoustic attenuation are also robust in the undulated case with uncertainties in the
elastic modulus for cases with strong fluid interaction.
METAMATERIALS 1977
(a)
(b)
(c)
Figure 6: Mean and 95% confidence bounds for structural response of the straight beam (a), undulated
beam (b) and straight beam with periodically attached resonators (c) by using 200 Monte Carlo samples, σ
= 0.05 and b = L. Legend: WFE-WKB (red), FE (black) and homogeneous properties (green).
(a)
(b)
Figure 7: Mean and 95% confidence bounds for structural (a) and acoustic (b) response of undulated with
water interaction by using 200 Monte Carlo samples, σ = 0.05 and b = L. Legend: WFE-WKB (red), FE
(black) and homogeneous properties (green).
5 Conclusions
The WFE approach was applied to compute the dispersion relations of an undulated beam (phononic
crystal) and a straight beam with internally distributed resonators (metamaterial) that interact with external
fluid. Band gaps are observed in the structural and acoustic responses, which make these configurations
applicable in acoustic panel to improve vibration and sound attenuation. This work also uses the WKB
1978 PROCEEDINGS OF ISMA2016 INCLUDING USD2016
approximation to analyze the dynamic response with correlated spatial variability, which reduces the
computational time when compared to the FE method. Results are in good agreement, in all the analyzed
range of frequencies, with the FE method. However, issues occur in the transition stop band -pass band
that are more evident for the undulated beam. An approach to better treat these transitions is a point to be
improved in future work. The WKB approach presents more issues for the strong fluid-structure case,
which is more sensitive in these transitions.
The uncertainties affect the band gaps that become narrower for the undulated beam and wider for the
straight beam with periodic resonators; moreover, vibration and acoustic attenuations are reduced in both
cases. However, the band gaps can be considered robust to variability in the elastic modulus and in the
resonator stiffness. The undulated beam showed superior robustness due to the nature of its band gap,
which is open by geometry modulation. The band gap robustness in both cases needs to be confirmed by
investigating variability in other physical properties, such as thickness and undulation, which may have a
more significant influence in the case of the undulated beam. Moreover, the manufacturing process could
present a higher standard deviation or a shorter correlation length compared to those used in this work. An
experimental study is necessary to confirm these parameters.
Acknowledgements
The authors are grateful to the São Paulo Research Foundation (FAPESP) Project numbers 2014/19054-6
and 2015/15718-0, to the Federal District Research Foundation (FAPDF) Project number
0193001040/2015, and to the Brazilian National Council of Research (CNPq) Project number
445773/2014-6 for the financial support.
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