CONTROL OF VIBRATION AND WAVE PROPAGATION IN … · CONTROL OF VIBRATION AND WAVE PROPAGATION IN SANDWICH STRUCTURES WITH AUXETIC CELLULAR CORES FEMCI WORKSHOP 2001 NASA Goddard Space

CONTROL OF VIBRATION AND WAVE PROPAGATIONIN SANDWICH STRUCTURES

WITH AUXETIC CELLULAR CORES

FEMCI WORKSHOP 2001NASA Goddard Space Flight Center, Greenbelt MD, May 16-17, 2001.

Luca MazzarellaDipartimento di Meccanica

Politecnico di Torino10129 Torino, ITALY

Massimo RuzzeneMechanical Engineering Department

Catholic University of AmericaWashington, DC 20064

Panagiotis TsopelasCivil Engineering Department

Catholic University of AmericaWashington, DC 20064

Fabrizio ScarpaAerospace Division

Department of Mechanical EngineeringUniversity of Sheffield, UK

OVERVIEW and OBJECTIVES

Analysis of WAVE DYNAMICS in sandwich structures with core of twocellular materials alternating PERIODICALLY along the structure;

One-dimensional structures - beams

Two-dimensional structures - Plates

Material A Material B

x

y

Material B

Material A

• Elastic and inertial properties of the core vary with elementary geometryof honeycomb structure

EXAMPLES:

Material A Material B

Regularhexagon

Re-entrantgeometry


Alternating honeycombs of different geometry generates IMPEDANCE MISMATCH ZONES

along the structure

IMPEDANCE MISMATCH


Periodic impedance mismatch zones generate aPASS AND STOP BAND

pattern typical ofPERIODIC STRUCTURES

This concept can be used to designCOMPOSITE PANELS

that behave asDIRECTIONAL MECHANICAL FILTERS

without requiring additional devices for active and passive vibrationand wave propagation control.

SIZE AND WEIGHTOF PANELS ARE NOT COMPROMISED

CONTROL OF WAVE PROPAGATION IN

SANDWICH BEAMS

WITH AUXETIC CORE

Wave propagation in 1-Dimensional periodic structures

Periodic Structure

Unit cell Material and\orgeometrical discontinuity

where Yk = state vector and Tk = transfer matrix

U Lk-1

Cell k-1 Cell kFLk-1

FRk-1FLk

FRk

Interface k

U Rk-1U Lk

U Rk

[ ]1kL

Lk

kL

L

FU

TFU

−

⋅=

[ ] 1kkk YTY −⋅=or

Eigenvalue Problem:kkk YY]T[ ⋅λ=⋅

1k1kk YeYY −µ

− ⋅=⋅λ=

Where:

µ : PROPAGATION CONSTANT

Real(µ ): Amplitude decay

Imag(µ ): phase difference between adjacent cells

0 500 1000 1500 2000 2500 3000 3500-1

0

1

2

3

4

5

Rea

l(µ )

0 500 1000 1500 2000 2500 3000 3500-4

-3

-2

-1

0

1

Imag

(µ )

frequency [rad/s]

Stop Bands


Auxetic solids

Geometric Layout of an elementary cell of honeycomb material:

θθθθ l

h

t

θ< 0: θ< 0: θ< 0: θ< 0: Re-entrant geometry (AUXETIC SOLID)

Auxetic cellular solids feature NEGATIVE POISSON’S RATIO behavior

α=h/lβ=t/l

Auxetic solids

Poisson’s ratio vs. angle θθθθ

Density vs. angle θθθθ

Shear Modulus vs. angle θθθθ

Spectral Finite Element (SFE) modeling of sandwich beam

Geometric and kinematic relationships:

γ = − + ⋅1

21 3h

u u h wx[ ]

u u u h h wx2 1 31 31

2 2= + + − ⋅[ ( ) ]

Shear deformation and displacement of the core:

wx

u1

h1 Face sheet 1

Face sheet 2

Core

w

h2

h3

u3

γ


U D w dx E A u dx E A u dx G A dxt xx

L

xL

xL L

= ⋅ + + + ⋅ ⋅z z z z12

12

12

12

2

01 1 1

20 3 3 3

20 2 2

20

γ

T mw dx m u dx m u dxL L L

= ⋅ + ⋅ + ⋅z z z12

12

12

20 1 1

20 3 3

20

! ! !

Hamilton’s Principle:

Strain energy:

Kinetic energy:

δ( )t

t

T U dt1

20z − =

BENDING EXTENSION SHEAR


Equations of motion:

uK

G Ah

m mK

G Ah

m uK

G Ah

h h h m wxx x11

2 2

22

21 2 1

1

2 2

22

22 3

1

2 2

22

1 3 22

1 14

1 14

18

= − + − + ⋅ + + −[ ( )]u ( ) ( )ω ω ω

uK

G Ah

m uK

G Ah

m mK

G Ah

h h h m wxx x33

2 2

22

22 1

3

2 2

22

23 2 3

3

2 2

22

1 3 22

1 14

1 14

18

= − + ⋅ + − + + − −( ) [ ( )]u ( )ω ω ω

wD

G A hh

m uD

G A hh

m u mD

wD

G A hh

h h m wxxxxt

xt

xt t

xx= − ⋅ − − ⋅ + + − −1 18

1 18

116

2 2

22

22 1

2 2

22

22 3

2 2 22

22

1 32

22( ) ( ) ( ( ) )ω ω ω ω

Z A Zx ( ) ( )x x= ⋅

State Space Formulation & Solution:

where:

Z ZA( ) ( )x e x= ⋅⋅ 0

Z = [ ]u u w w u u w wx x x xxx xxT

1 3 1 3

Transfer Matrix and propagation constants

Y G G YA( ( )L) e L= ⋅ ⋅ ⋅⋅ −1 0

Y = [ ]u u w w N N S MxT

1 3 1 3State Vector:

Generalized Displacements

Generalized Forces

(TRANSFER MATRIX)T G GA= ⋅ ⋅⋅ −e L 1

where:G = − − −diag K K D Dt t([ ])1 1 1 1 1 3

x=0 x=L

Y(0) Y(L)

MODEL VALIDATION: Beam with Uniform Core

Beam Geometry:

Material Length [m] Thickness [m] Width [m]

Face sheet 1 Aluminum 1 1e-3 0.04

Core Regular Nomex 1 2e-3 0.04

Face sheet 2 Aluminum 1 1e-3 0.04

ANSYS Mesh:• Core: 480 PLANE42 elements;• Face Sheets: 240 BEAM54 elements.

MATLAB Mesh:• 20 Sandwich-beam elements.

MODEL VALIDATION: Beam with Uniform Core

Tip response for harmonic base motion

Performance of sandwich beams with periodic core

Periodic Sandwich Beam:

Material A:Regular Honeycomb

Material B:Auxetic solid

Elementary cell:

LA LB

Performance is evaluated for:• MATERIAL A: given regular honeycomb;• MATERIAL B: Auxetic solids of different geometries (angle θθθθ);• Varying length ratios (LB/L).

Cell length:L= LA +LB

Properties of sandwich core

(A) - Regular Honeycomb (Hexagonal NOMEX):θ=30o

α=1; β=0.05(B) - Auxetic Honeycomb:

θ= - 60o : -10o

α=2; β=0.05

-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -100.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -100.5

1

1.5

2

2.5

3

3.5

θθθθ θθθθ

Relative Shear Modulus Relative Density

GB/GA ρρρρB/ρρρρA

Propagation Constants

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

00.10.20.30.40.50.60.70.80.9

LB /L=0.5

Uniform beamLB /L=0.25

Real(µµµµ)

Frequency [Hz]

Im(µµµµ)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5

3

3.5

θ=-60o

Uniform beamθ=-30o

Frequency [Hz]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.10

0.10.20.30.40.50.60.70.80.9

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.5

1

1.5

2

2.5

3

3.5

LB /L=1θ=-60o

Propagation Constants

Frequency [Hz] Frequency [Hz]

θθθθ

θθθθ

θθθθ

θθθθ

Real(µµµµ)

Real(µµµµ)

Real(µµµµ)


Real(µµµµ)

LB/L=0.1LB/L=0.25

LB/L=0.5 LB/L=0.75

Response of periodic sandwich beam: harmonic motion of the base

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-60

-40

-20

0

20

40

60

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-50

-40

-30

-20

-10

0

10

20

30

40

50


LB /L=0.5

Uniform beamLB /L=0.25

θ=-60o

Uniform beamθ=-30o

Tip

dis

plac

emen

t [dB

]

Tip

dis

plac

emen

t [dB

]

CONTROL OF WAVE PROPAGATION IN

SANDWICH PLATES

WITH AUXETIC CORE

(Work in progress)

Wave Propagation in 2-Dimensional Periodic Structures

x

y

θ

Direction of wave propagation

θ

2-Dimensional periodic structure Unit Cell

BOTTOM

TOP

LEFT

RIG

HT

{ } { }q FLB LB, { } { }q FB B, { } { }q FRB RB,

{ } { }q FLT LT, { } { }q FT T, { } { }q FRT RT,

{ } { }q FR R,{ } { }q FL L,

{ }qij

{ }Fij

Vector of generalized displacements at interface ij

Vector of generalized forces at interface ij

( ){ } { }− + =ω 2[ ] [ ]M K q F

Cell’s equation of motion (Finite Element Formulation):

Relation between interface displacements (Bloch’s theorem):

{ } { }q e qR Lx= µ ;

Relation between interface forces (Bloch’s theorem and equilibrium conditions):

{ } { }q e qT By= µ ;

{ } { }q e qRB LBy= µ ; { } { }q e qLT LB

y= µ ; { } { }q e qLB RTx y= +µ µ ;

{ } { }F e FR Lx= − µ ; { } { }F e FT B

y= − µ ;

{ } { }F e FRB LBy= − µ ; { } { }F e FLT LB

y= − µ ; { } { };FeF RTyx

LBµµ +=

Wave Propagation in 2-Dimensional Periodic Structures


( ) { }− +

=ω µ µ µ µ2 0[ ( , )] [ ( , )]* *M Kqqq

x y x y

LB

L

B

Cell’s equation of motion is reduced as

Eigenvalue problem whose solution provides for an assigned pair the corresponding value of frequency ω

µ µx y,

µ µx y, : Propagation Constants along x & y

µµ

θx

y= tan : Direction of wave propagation

Example: spring mass system

Unit Cell:

m

ky

kx

Periodic Structure:

x

y

θ

Equation of motion of mass:

wi,j

( ) ( ) ( ) 02 11112 =+−+−++− +−+− j,ij,iyj,ij,ixj,iyxj,i wwkwwkwkkmwω

wi,j : off-plane displacement of mass i,j


STOP BANDS

ky /kx=1 ky /kx=0.57

PHASE CONSTANT ( ) SURFACESµ µx y,

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

0.5

1

1.5

2

30

210

60

240

90

270

120

300

150

330

180 0

DIRECTIVITY PLOTSLocation of stop bands vs. direction of wave propagation


ky /kx=1 ky /kx=0.57ω [rad/s]

θSTOP BANDS


ky /kx=0.57,Load applied in (21,21)

Response to harmonic point load for a 40*40 grid

ω=1.033 rad/s ω=1.181 rad/s

Example: periodic stiffened plate

x

y

Unit cell

Stiffened Periodic Plate

h1

h2

Example: stiffened plate

PHASE CONSTANT ( ) SURFACESµ µx y,

h2 /h1=2h2 /h1=1

Example: stiffened plate

h2 /h1=2

h2 /h1=1 h2 /h1=4

DIRECTIVITY PLOTSLocation of stop bands vs. direction of wave propagation

REFERENCES

Ruzzene, M., Scarpa, F. “Control of wave propagation in sandwich beams with auxetic core”, Submitted toJournal of Intelligent Material Systems and Structures, 2001.

Mead, D. J. "A New Method of Analyzing Wave Propagation in Periodic Structures; applications to PeriodicTimoshenko Beams and Stiffened Plates" Journal of Sound and Vibration (1986) 104(1), 9-27.

Mead D.J., “Wave propagation in continuous periodic structures: research contributions from Southampton1964-1995” Journal of Sound and Vibration 190(3), 495-524, 1996.

Mester, S. and H. Benaroya, “Periodic and Near Periodic Structures: Review”, Shock and Vibration, 2 (1), 69-95, 1995

Doyle, J. F., Wave propagation in Structures, 2nd Ed., Springer, New York, 1997.

Gopalakrishnan, S., Doyle, J. F., 1994 “Wave propagation in connected wave guides of varying cross section”,Journal of Sound and Vibration, Vol. 175, pp. 347-363.

Baz, A. “Spectral Finite Element modeling of longitudinal wave propagation in rods with Active ConstrainedLayer Damping”, J. of Smart Materials and Structures 2000.

REFERENCES

Langley, D. J. ”The response of 2-dimensional periodic structures to point harmonic forcing" Journalof Sound and Vibration (1996) 197(4), 447-469.

Scarpa F. and Tomlinson G., 2000. Theoretical characteristics of the vibration of sandwich plateswith in-plane negative Poisson’s ratio values. J. Sound Vib. 230(1), pp. 45-67.

Smith F. C., Scarpa F. and Chambers B., 2000. The electromagnetic properties of re-entrantdielectric honeycombs. IEEE Microwave and Guided Wave Letters. In press.

Gibson L. J. and Ashby M. F. Cellular solids: Structure and Properties, 2nd Edition, 1997, CanbridgeUniversity Press, Cambridge, UK.

Mead D. J. and Markus S., 1969. The forced vibration of a three layer, damped sandwich beam witharbitrary boundary conditions. J. Sound Vib., 10, pp. 163-175

Scarpa F. and Tomlin P. J., 2000. On the transverse shear modulus of negative Poisson’s ratiohoneycomb structures. Fatigue Fract. Engng. Mat. Struct. 23, pp. 717-720.

Documents

CONTROL OF VIBRATION AND WAVE PROPAGATION IN … · CONTROL OF VIBRATION AND WAVE PROPAGATION IN SANDWICH STRUCTURES WITH AUXETIC CELLULAR CORES FEMCI WORKSHOP 2001 NASA Goddard Space