Li Cheng
Acoustic Black Holes for Structural Wave Manipulation and Control
Department of Mechanical Engineering,Consortium for Sound and Vibration Research
The Hong Kong Polytechnic University, Hong Kong
OUTLINE
2
p Acoustic Black Holes (ABHs)
p Characterization of ABH effect by a wavelet-decomposed model
p Vibration control applications
p Conclusions
Ø Theoretical modelling
Ø Numerical results and experimental validation
Ø Phenomena exploration and parameters investigation
Ø Enhancement methods
Ø Compound ABH
Ø multiple simple ABHs
Ø Periodic compound ABHs
v ABH Applications- Energy harvesting- SHM- Biomedical- ……- Vibration
suppression- Noise control
Vib. & Noise control Energy Harvesting
Acoustic Black hole:• flexural waves incident at an arbitrary angle upon a power-law profile edge can be
trapped near the edge and therefore never reflect back
( ) ( 2)mh x x me= ³2 3
42
( ) ( ), ( )( ) 12(1 )
( ) 0 then 0
D x Eh xC D xh x
h x C
wr µ
= =-
® ®
01/4 1/2 1/2
( ) ( )d
( ) 12 ( )
2, ( )
x
mp
kS x k x x
k x k xwhen m k x
F
e
F
-
= =
=
³ ®¥ ®¥
ò
Introduction
Introduction
4
Motivation
4
Existingtheoreticalmodels
Geometricalacousticsapproach
Impedancemethod
Wavemodel
v FlexiblemodelwhichconsidersmorerealisticstructurestoguideABHstructuredesignforvariousapplications
dampinglayertruncation
Our objectives: v 1D and 2D model: finite size and boundary
v Full coupling between ABH part and damping layer
v High accuracy in characterizing ABH effect
v Optimization of damping layer deployment
v Embodiment of other control or energy harvesting elements
v Building blocks for periodic ABHlattice
1D example
5
Theoretical modelling
hd(x)o
z
x
damping layer
hb h(x)beam
translational spring k
rotational spring q
xb2 xb1
x0
xd2
f(t)
xF
xd1
Euler-Bernoulli beam theory, the displacement:
( , ) ( ) ( )j=å i ii
w x t a t x{ }, , ( , )¶ì ü= -í ý¶î þwu w z w x tx
k pd 0d ( ) ( )i i
L L L E E Wt a t a tæ ö¶ ¶
- = = - +ç ÷¶ ¶è ø&
2
2 222 b2
b22
F
1 d2
( , )1 1 1( ) d ( , )2 2 2( ) ( , )
k
p
wE Vt
w x twE EI x x kw x t qx x
W f t w x t
r ¶æ ö= ç ÷¶è ø
æ ö ¶¶ æ ö= + +ç ÷ ç ÷¶ ¶è øè ø= ×
ò
ò
[ ]{ ( )} [ ]{ ( )} { ( )}a t a t f t+ =M K&&
2[[ ] [ ]]{ } { }A Fw- =K M
Lagrange’s equations:
Theoretical Modelling
6
Mexican Hat Wavelet (MHW) expansion
MHW functions with j=0:
2124 22( ) [1 ]e
3j
- -= p -
x
x x221 ( )
24 2,
2( ) [1 ( ) ]e23
2j k
kj
xj jx kxj
-- -
= p - -MHW:scaling j
translation k
Characteristics of MHW:
p approximately localized in [-5, 5]
p Flexible scaling and translation
p Smoothness and its derivation
Particularly suitable for charactering the rapidly
varying characteristics of wavelength and
vibration amplitude
, ,=0 , ,
d( , ) ( ) ( ) 0d ( ) ( )
m
j k j kj k j k j k
L Lw x t a t xt a t a t
jæ ö¶ ¶
= - =ç ÷ç ÷¶ ¶è øåå &
b2
0, ( )d 0
x
j kxx xj ¹ò
[M]
[K]
{ }F
Numerical results
7
Tip thickness truncation x0=1
Frequency (Hz)
FEM Present approach
Error (%)
ω1 432.91 432.77 -0.033ω2 1669.5 1669.44 -0.004ω3 2972.8 2972.68 -0.004… … … …ω20 132390 132388.1 -0.001… … … …ω36 436520 436889 0.085ω37 461370 463679 0.501ω38 486900 493317 1.318
• Extremely high accuracy. • MHW suitable to
characterize wave-length fluctuation.
Geometricalparameters Materialparameters
Beamε=0.005 Eb=210 GPam=2 ρb=7800 kg/m3
hb=0.125 cm ηb=0.001 Damping layers
x0=1 cm Ed=5 GPaxb1=5 cm ρd=950 kg/m3
xb2=10 cm ηd=0.05
LL. Tang, L. Cheng, HL. Ji and JH. Qiu, J. Sound Vib., 374, 172-184, 2016.
8
Experimental validations
Loss of ABH effect!
2ABH
2Unif
< >Γ =10log< >VV
LL. Tang, L. Cheng, Applied Physics Letter, 109(1),104102, 2016.
System Analysis and Design
9
p Effect of the basic ABH design parameters
p Enhancement of ABH effectØ Extended platform
Ø Modified thickness profile
Ø Compound ABH structures
Ø Multiple simple ABHs
Ø Periodic ABH lattice
Ø …….
Ø m, ε…..
Ø Tip truncations
Ø Damping layer deployment: location and shape…
Ø Enhancement methods
Optimization shape area of damping layer
Damping layer deployment
Tip truncations2
ABH2
Unif
< >Γ =10log< >VV
LL. Tang, L. Cheng, J. Sound Vib., 391, 116-126, 2017.
10
FEM Model ( COMSOL𝐓𝐌 )
• Double-leaf compound ABH structure
• 2D & solid mechanics interface
• Clamped-free boundary condition
• Dynamic input close to free end
• Damping neglected in static analysis
• 3 study cases with different ℎ$
Modal analysis of CABH
Compound ABH
(c)
• Two types of modes
Vibration Control Application
Comparison between Compound and Simple ABH
CABH
Reference
SABH
• SABH&CABHhavesame cross-sectionthickness• Sameamount ofdampinglayersattachedatcorrespondingABHportions
SIMULATION
11
“Dynamic” Analysis
NormalStraindistributionsof8th-modeCABH
UnitMoment
Stress Concentration Factor ( SCF )
Strength
12
SIMULATION “Static” Analysis
CABHSABH
Index Definition: for Evaluating Static PropertiesStiffness
UnitForce
Equivalent Compliance Factor ( ECF )
156 5.66ECF
182 8.63SCF
SABH CABH
Comparison between Compound and Simple ABH
FE model is similar to dynamic analysis & damping layers are neglected.
The lower ECF, the better stiffness The lower SCF, the better strength
yu
yABH
𝜎u
𝜎ABH
=𝑦()*𝑦+
=𝜎()*𝜎+
Ø Stiffness increased by 27 timesØ Stress reduced by 21 times
13
SIMULATION Additional Platform
Additional Platform Enhance Damping and Strength of CABH
• p is equal to x0, i.e. half length of added platform.• Other geometric parameters in above figures are same as previous study case,
meaning after x0, nothing changes.
BasicGeometryp
Trade-offORBalance
OnlyforCABH
Compound ABH
14
Experimental investigation
1. CABH beam: steel & made by EDM
2. Damping layer: 3M F9473PC (multi-layer free damping treatment)
Experiment setup
Experimentwasconductedtoverify2DFEMmodelandtheABHeffectofCABH
3. Excited by shaker & measured by laser
withoutdamping withdamping
Vibration control application Multiple simple ABHs
15
p Improvement the low frequency performance without increasing the ABH dimension. p Possible accumulated ABH effect and wave filter effect.p Possible broad band gaps at low frequency without attaching additional elements and creating
multiple interfaces
Geometrical parameters:
hb =0.32 cm h0 = 0.02 cm
lABH = 2 cm a =8 cm
without damping layers with damping layers
p Transmission significantly
reduced in ‘attenuation band’
p Appearance before f c.
p Reduction enhancement as
number of ABHs increases
p Damping shows little influence
on attenuation band but reduce
transmission at resonant f
Apply developed model!
out
in
20 log wTw
=
16
Modelling infinite periodic structuresMultiple simple ABHs
nknn
pn
nL EL E+¥ +¥
=-¥ =-¥
= = -å åThe Lagrangian of the system:
2
22
2
( )1 ( ) d2
( )1 d2
n
nk
np
n
w xE EI x xx
w xE Vt
r
æ ö¶= ç ÷¶è
ìïïíïï
¶æ ö= ç ÷¶è ø
øîò
ò
1
1
( ) ( )
( ) ( )n n
n n
jka
jka
w x a e w x
w x a e w x+
+
ü+ = ïý
¢¢ ¢¢+ = ïþ
, ,
d 0d ( ) ( )
n n
i s i s
L Lt a t a tæ ö¶ ¶
- =ç ÷ç ÷¶ ¶è ø&
For the (n+1)th unit cell:
2qjkan n
n qL L L e
+¥ +¥
=-¥ =-¥
= =å å
21
jkan nL e L+ = 2qjka
n q nL e L+ =Similarly
17
Modelling infinite periodic structuresMultiple simple ABHs
( ) (0) :jkan nw a e w=
( ) (0) :jkan nw a e w¢ ¢=
:( ) ..( .0)jkan nw a e w¢¢ ¢¢= ( ) (0) : ...jka
n nw a e w¢¢¢ ¢¢¢= -
p One unit cell should satisfy the Lagrange’s equation and the following periodic boundary conditions
, ,=0 1
( , ) ( ) ( ) ( )m n
i s i s i ii s i
w x t a t x a xj j=
= =åå å ( )1
( ) (0) 0n
jkai i i
ia e aj j
=
- =å1
1i
n
n ii
a al-
=
=å [ ]1
1( , ) ( ) ( )
n
i i n ii
w x t x x aj lj-
=
= -å
2
11 1 1
ni i
n ii n n
a al ll l
-
-= - -
æ ö¢-= ç ÷¢- +è øå
2
1 11 1 1 1 1
( , ) ( ) ( ) ( )n
i i i ii n n i n i
i n n n n
w x t x x x al l l lj j l l jl l l l
-
- -= - - - -
ì üé ù¢ ¢- -ï ï= + + -í ýê ú¢ ¢- + -ï ïë ûî þå
i
i
mm
mn
AA
l = ( ) (0)i i i
m m jka mA a ej j= - m is derivative order
We can get:
where:
2[[ ] [ ]]{ } 0Aw- =K M
p Submitting displacement expression into , dispersion curve
can be obtained by, ,
d 0d ( ) ( )
n n
i s i s
L Lt a t a tæ ö¶ ¶
- =ç ÷ç ÷¶ ¶è ø&
Band structures
18
Numerical validation
138%
121%
Bandwidthp w and w’ sufficient to
describe the band structures’
p Band structures obtained
based on single element.
p Ultra-wide local resonance
bandgaps due to ABH effect
p Bandgaps coincide with
attenuation gaps
19
Periodic Structures with simple ABHs Parametric analyses
ü Better ABH effect, increasing m or decreasing h0, boundary of bandgaps decrease
and bandwidths enlarge overall
p For m=2 and h0=0.005Very few elements needed to achieved considerable attenuation
LL. Tang, L. Cheng, Journal of Applied Physics, in press
20
Phononic beams with compound ABHs
p fs is the whole frequency range, equaling to 7.2 in present case.
p Enlarged bandwidth at mid-high frequencies for Bragg scattering
p Further increasing tune bandwidth and reduce the passband as a whole
Bandwidth percentage up to 92%!
0 0.00025h =
21
2DplatewithABH
0 10 20 30 40 50 60 70 80 90 1000
500100015002000250030003500
Eige
nfre
quen
cy
FEM L12m7 L16m7
Number of eigenfrequency
100 1000 3000-85-80-70-60-50-40-45
Mea
n Sq
uare
Ve
loci
ty (d
B)
Frequency (Hz)
FEM Numerical
98th mode
100 1000 3000-85-80-70-60-50-45
Mea
n Sq
uare
Ve
loci
ty (d
B)
Frequency (Hz)
Whole flat plate+0.5h0 damping layers Quarter ABH +2h0 damping layers
Lighter structure, better dynamic properties!
22
2DLatticewithcut-outs
0.05 m
0.05 m
0.05 m
0.05 m
1 m
1 m
WaveletParameterL=16,m=7
124th mode(894.56Hz)35th mode(224.82Hz)
W. Huang, HL. Ji, JH. Qiu, L. Cheng Journal of Vibration and Acoustics, ASME, 138, 061004-1, 2016.
AnalysesofflexuralrayTrajectoriesinimperfectABHindentation
p Power flow through different cross-section
{ }122 ImxDm'I Gd'
=Powerflow:
sp ds= ò I
l Thespotsizeis 6.7% of the width of plate.l 58.25% of the energy propagates through the section.
Conclusions
24
q Fullcouplingbetweenthedampinglayersandstructures
q HighaccuracyincharacterizingwavefluctuationofABHeffectusingMHW
q SystemOptimizationspacetoachievemaximumABHeffect
q Flexibility ofembeddingotherelementsforvariouspotentialABHapplications
Ø Flexible models considering more realistic structure developed: 1D and 2D
Ø Avoidance of Loss of ABH effect by predicting local resonant frequencies
Ø Enhancement of ABH effect by modified thickness profile and extend platform
Ø Compound ABH structures to ensure ABH effect while improving the static properties
Ø Model for periodic ABHs developed and broadband local resonance bandgaps obtained
Ø Compound ABH lattice to achieve ultra-wide band gaps in a broad frequency ranges
Ø Laser-ultrasonic experimental facilities for time-domain wave visualization
Ms. Liling TangMs. Li MaMr. Tong ZhouDr. Su ZhangMs. Wei HuangMs. Jing Luo………..Prof. Jihao Qiu (NUAA, China)Dr. Hongli Ji……….
Acknowledgement