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