Vibration Damping: ���The Turbulence of Structural Mechanics?

George A. Lesieutre Department of Aerospace Engineering Center for Acoustics & Vibration

SDM Lecture AIAA Science and Technology Forum and Exposition (SciTech 2104)

National Harbor, MD

16 January 2014

Vibration Damping: ���The Turbulence of Structural Mechanics?

•  Well, no. – But there are ���

unresolved issues – Often neglected initially, ���

sometimes regretted – No agreement ���

on the key equations!

•  All structural vibration is damped – At multiple scales

Princeton!

We will survey some ���passive damping methods and models

•   Conventional damping treatments

•   Material behavior

•   Conventional damping models

NOT isolation, structural mods

Material damping models

•   Structural damping models

•   Unconventional damping treatments

C[ ] x{ }C

R

piezoVs

cG ′′w

Vibration damping matters

•  Damping is important in aerospace – Dynamic response, fatigue, noise – Aeroelastic response and stability – Margin for active structural control – Wave-based SHM and NDE – MEMS resonators, sensors

•  System benefits require considering damping during the design process

•  Effective damping methods are needed •  Accurate damping models are needed

Damping is an important aspect ���of spacecraft design and performance

•  Launch vehicles –  Payload isolation and vibroacoustics –  Solid propellant; Deployable nozzles

•  “Flexible” spacecraft –  Deployment –  High-BW precision pointing, disturbances –  Damped struts, equipment sandwich panels, ���

joints, isolators –  VEM (CLD via MSE), piezo, magnetic (eddy-current)

•  Ongoing concerns –  Meeting stringent system req’ts

•  Multi-stage isolation, damping, active pointing –  Thermal (cryo-) environments; outgassing –  Effects of cabling –  Ground-based validation; effects of gravity.

Modal damping in bare JWST composite structure ���at 35K is as low as 0.017% (!)

Abbruzzese, Parker, Lin, Innis (NGST), “Measurement of Damping ���

in Composite Structure Assembly at Cryogenic Temperatures,” ���AIAA 2008–2192, 49th AIAA SDM.

Damping is an important aspect ���of aircraft design and performance

Pressure loads decrease damping

Liguore, Montgomery, Foss (Boeing), “Prediction and Measurement of Structural Loss Factors in Damped Composite

Panels under Pressure Loads,” AIAA 2008–2236, 49th SDM.

•  Fatigue and strength –  Aerodynamic (buffet); Acoustic; Landing –  Damped composite bonded repairs –  Stress-coupled, co-cured damped composites

•  Payload isolation

•  Interior noise reduction –  Composite panel ���

acoustic attenuation •  Light, stiff, fewer joints

•  Aeroelastic response and stability –  Gusts; Flutter; ASE –  Design analysis and practical augmentation

•  Ongoing concerns –  Effectiveness over broad operating range –  Pressure and CF loads –  Joint modeling (LCO) –  Damped structural composite materials

•  Rotorcraft lag damping

Damping is an important aspect ���of engine design and performance

•  High-cycle fatigue –  Monolithic blisks (IBR)

•  Compressor, turbine; turbopumps –  Composite fan blades

•  Ongoing concerns –  Integral damping

•  Damped structural materials –  CF loads –  High temperature

Modal damping in Inconel 718 specimen���as low as 0.01% (!)

Min, Harris, Ting (NASA), “Advances in Ceramic Matrix Composite Blade

Damping for Aerospace Turbomachinery Applications,” ���AIAA 2011–1784, 52th AIAA SDM.

Let’s start from the beginning •  Nominal equations of motion

•  Sources of damping

•  Measures of damping

M[ ] !!q{ }+ C[ ] !q{ }+ K[ ] q{ } = f{ }

ζ ≈ η2

= ψ4π

notional energy dissipation

modal ���damping ratio

loss ���factor

specific ���damping capacity

Some amenable to design specification

… acoustic���attenuation

Intrinsic damping is often low

CSA Engineering!

�

˙ ̇ a m + 2ζmωm ˙ a m + ωm2 am = Am (t)

modal EOM

Layered damping treatments ���can be effective

VEM properties depend on f & T Variations: segmented; standoff; composites (bending, in-plane)

CSA Engineering!

What do we need ���to design a layered damping treatment?

•  Models (placement and sizing) – Proportional viscous damping – Viscoelastic damping (freq-domain)

•  Complex modulus; loss factor •  Modal Strain Energy method (FE)

•  VEM materials selection •  Reduced-frequency nomogram

Viscous damping models are ���simple and popular

•  Proportional Damping: linear and viscous

–  Convenient, not physics-based: dissipation in time-domain model, real vibration modes

M[ ] q{ } + C[ ] q{ } + K[ ] q{ } = Q{ }C[ ] = β K[ ] + γ M[ ]

•  Strain-based (stiffness-proportional) viscous damping yields modal damping that increases with frequency

•  Motion-based (mass-proportional) viscous damping yields modal damping that decreases with frequency

Normalized Frequency, ω!1.0! 10.!0.1!

Nor

mal

ized

D

ampi

ng!

Proportional damping model!

Typical material damping behavior!

�

γ ω2

�

β2ω

(Curves offset vertically for clarity.)!

Frequency Response models avoid some difficulties of viscous damping •  Modify viscous damping matrix

– Ω is a harmonic forcing frequency

•  Complex structural stiffness

– Complex modulus – Possible frequency-dependent properties

C[ ]hysteretic

1Ω

C[ ]viscous =γΩ

K[ ]

M[ ] !!q{ }+ γ

ΩK[ ] !q{ }+ K[ ] q{ } = f{ }eiΩt

−Ω2 M[ ]+ 1+ iγ( ) K[ ]⎡⎣ ⎤⎦ q(Ω){ } = f{ }σ (Ω){ } = (1+ iη) E[ ] ε(Ω){ }

= E '[ ] + i E"[ ]⎡⎣ ⎤⎦ ε(Ω){ }

“structural” ���damping

loss factor η = ′′E′E

Myklestad, N. (1952). The concept of complex damping. Journal of Applied Mechanics, 19(3):284–286.!

Modal damping can be estimated ���for non-specific material damping

•  Dynamic response ���via modal superposition

•  Canonical SDOF modal eqns of motion

•  How to find modal damping?

�

˙ ̇ a m + 2ζmωm ˙ a m + ωm2 am = Am (t)

ζ m > 0

Modal Strain Energy (MSE) method provides results and insight

•  Damping of a mode of a structure is a weighted sum of the damping of its parts – Weighting factors are the ���

fraction of strain energy stored in each part

•  Applicable to multi-mode deformation,

composite materials, built-up structures

ηm =Upart

UTOTAL

⎛⎝⎜

⎞⎠⎟ m

ηpartparts∑

Ungar, E. E. and Kerwin Jr., E. M. (1962). Loss factors of viscoelastic systems in terms of energy concepts. Journal of the Acoustical Society of America, 34(7):954–957.!

High-damping materials (VEM) exhibit ���frequency- and temperature-dependence

•   3M ISD 112, for instance

modulus

loss factor

Variations possible (multilayer CLD, copolymer blends)

T

f

Modal Strain Energy method ���is very suitable for use with FEA

CSA Engineering!Johnson, C. D. and Kienholz, D. A. (1981). Finite element prediction of damping in

beams with constrained viscoelastic layers. Shock and Vibration Bulletin, 51:71–81.!

•  Models – Time-domain viscoelasticity

•  Temperature dependence •  Wave propagation

– Nonlinear elastomers (helo lag damper) –  “Geometric” viscous damping

•  Shear beams (spacecraft cabling) – Membrane loads

•  Methods – Damped composites – Piezoelectric materials

•  Resistive shunts •  Energy harvesting

– Particle damping

Potpourri of models and methods are ���available for accuracy and design options

CR

piezoVs

Better damping models are available •   Experiments on built-up structures show relatively

weak frequency-dependence •   More accurate damping models are available

–   Complex modulus models can yield frequency-independent modal damping, but useful mainly in frequency domain

•   Fractional derivative –   Frequency-dependence of high-loss viscoelastic materials

•   Time-domain internal variable models

�

ε

�

εA

�

σ

�

Δ +1Δ

Eu

Ω

�

Eu

Δ

�

Eu

Fractional Derivative model ���captures weak frequency dependence

•  Compact frequency domain model – Captures relatively weak frequency

dependence of damping and stiffness

m is order ���of time derivative

E * (iω ) = E0 + E1(iω )m

1+ b(iω )m

Normalized Frequency, ω!1.0! 10.!0.1!

Nor

mal

ized

D

ampi

ng!

Typical material damping behavior!

m −m

Some utility in time domain Bagley and Torvik, “Fractional calculus – a different approach to the analysis of

viscoelastically damped structures,” AIAA Journal, 1983, 21(5):741–748.

Internal Variable viscoelastic models can capture frequency-dependence

•   Total displacement field is the sum of elastic and anelastic parts (ADF model)

•   Constitutive eqns

•   Evolution eqns

σ = Eu ′u − ′u A( )σ A = Eu ′u − c ′u A( )

ρu − Eu ′′u − ′′u A( ) = p x,t( )

cEu

Ω′′u A − Eu ′′u − c ′′u A( ) = 0

u x,t( ) = uE x,t( ) + uA x,t( )

�

ε

�

εA

�

σ

�

Δ +1Δ

Eu

Ω

�

Eu

Δ

�

Eu

�

ε

�

εA

�

σ

cEu

Ω

c −1( )Eu

�

Eu

Δ relaxation strength

motion

relaxation

ADF Finite Element (Rod)

m 3 m 6 0 0m 6 m 3 0 00 0 0 00 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u1u2u1A

u2A

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

+

0 0 0 00 0 0 00 0 ck Ω − ck Ω0 0 − ck Ω ck Ω

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

u1u2u1A

u2A

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

+

k −k k −k−k k −k kk −k ck −ck−k k −ck ck

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

u1u2u1A

u2A

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

=

f1f200

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

A time-domain model Lesieutre, G.A., “Finite Elements for Dynamic Modeling of Uniaxial Rods with Frequency-Dependent

Material Properties,” International Journal of Solids and Structures, Vol. 29, No. 12, 1992, pp. 1567–1579.!

Single-ADF FE Results Modal damping vs. modal frequency (~15 elements)

Hysteresis loops at different frequencies

Mod

al d

ampi

ng ra

tio

Stre

ss (P

a)

Frequency (r/s)

Strain

100 (r/s)

10,000 (r/s) 1000 (r/s)

Multiple ADF model can capture weaker-frequency dependence

•  Complex modulus – Single-ADF building blocks

– Unrelaxed and relaxed modulus – Relaxation strength

E * ω( ) = Er 1+ Δn

ω Ωn( )2

1+ ω Ωn( )2n∑

⎛

⎝⎜

⎞

⎠⎟

storage modulus

+ i Er Δn

ω Ωn( )1+ ω Ωn( )2

n∑

⎛

⎝⎜

⎞

⎠⎟

loss modulus

Δ total = Δnn∑

Eu = Er 1+ Δ total( )M

odul

us

Multiple ADFs ���can capture material behavior well

•  ISD112 •  1 ADF (single)

Single-ADF building

blocks

Multiple ADFs ���can capture material behavior well

•  ISD112 •  3 ADFs

Lesieutre, G.A., and E. Bianchini, “Time-Domain Modeling of Linear Viscoelasticity using Anelastic Displacement Fields,” Journal of Vibration and Acoustics, Vol. 117, No. 4, October, 1995, pp. 424–430. !

Ω1

Ω2Ω3

ADF Finite Elements used to model ���Active Constrained Layer damping

Elastic model

ADF model Lesieutre, G.A., and Lee, U. “A Finite Element Model for Beams Having Segmented Active Constrained Layers with Frequency-Dependent Viscoelastic Material Properties,” Smart Materials and Structures, Vol. 5, 1996, pp. 615–627. !

piezo visco elastic

•  Predict wave propagation ���in viscoelastic rods, ���

shock absorption ���in mechanical filters

•  3-D, ADF-based FE for axisymmetric geometry

•  Prediction vs. experiment: ���velocity of mass on 3-mm filter –  Matches magnitude, ���

frequency content •  Captures geometric dispersion, ���

viscoelastic dispersion ���and attenuation

ADF model can capture behavior ���of shock filters on Hopkinson bar

Air plenum Projectile Force gauge

Rubber filter

Accelerometer and 30-g mass

Rusovici, R., Lesieutre, G. and Inman, D. J. “Modeling of Shock Propagation and Attenuation in Viscoelastic Components”, Shock and Vibration Digest, Vol. 8, No. 5, 2001, pp. 287–302. !

Rubber filter

Accel / mass force ���gauge projectile

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

msec

m/s

predictedexperiment

msec

m/s

Internal variable models can also capture temperature-dependence

•  Add thermal evolution equations – Conduction and heat generation

– Relaxation of anelastic displacement is faster at high temperature: shift function

cv T − k ′′T = r(x,t) r(x,t) = cGu

Ω′u A( )2

αT

cGu

Ω!′′u A −Gu ′′u − c ′′u A( ) = 0 αT T( ) = ea

1T( )− 1

Tref( )⎡⎣⎢

⎤⎦⎥

ADF-T shear FE ���includes thermal dynamics

m 0 0 00 1 0 00 0 αT c Ω( )k 00 0 0 C

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

!!u!u!uA!T

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

+

0 k −k 0−1 0 0 00 −k ck 00 0 0 K

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

!uuuA

T

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

=

F00R

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

Three coupled fields

ADF-T model can capture ���“thermal runaway”

Tip displacement

Tip temperature Force-displacement

hysteresis

Lesieutre, G.A., and Govindswamy, K.M., “Finite Element Modeling of Frequency–Dependent and Temperature–Dependent Dynamic Behavior of Viscoelastic Materials in Simple Shear,” International Journal of Solids and Structures, Vol. 33, No. 3, 1995, pp. 419–432. !

displacement

forc

e

time

time

disp

lace

men

t te

mpe

ratu

re

-1 -0.5 0 0.5 1-10

-5

0

5

10

-20 -10 0 10 20-60

-40

-20

0

20

40

60

Stre

ss (p

si)!

Strain (%)!

Variation of material hysteresis with strain amplitude

Stre

ss (p

si)!

Strain (%)!

Nearly elliptical at low amplitude!

E 100 W, Room Temperature, 1 Hz!

Significant slope and area changes!

Elastomeric materials can exhibit nonlinear behavior with increasing strain

Helo lag damper app

A nonlinear elastomeric model:���Nonlinear Multi-ADF with Friction (NMAF)

∑∑==

+−=M

j

fj

N

i

Aiuu GG

11σεεσ

),,1( Ni

FcGGGciNL

Aiiuu

Ai

i

ui

…

!

=

=+−Ω

εεε

Friction Stress Contribution!

(Rate Independent Nonlinearity)!

Nonlinear Viscous Force (Rate Dependent

Nonlinearity)!

Current Model:!!3 internal fields (N=3)!!3 friction elements (M=3)!

Model Schematic:!

Simple Shear!

Brackbill, C.R., E.C. Smith, G.A. Lesieutre, “Application of a Refined Time Domain Elastomeric Damper Model to Helicopter Rotor Aeroelastic Response and Stability,” AHS Journal, Vol. 47, No. 3, July 2002, pp. 186–197. !

Shear Stress (psi)

-20

-10

0

10

20

-10 -5 0 5 10

Stre

ss (p

si)

Shear Strain (%)

Experimental DataPrediction

10% Strain1% Strain

-0.5

0

0.5

-0.1 0 0.1

0.1% Strain

E100W, 75°F, 1Hz!

Nonlinear model validated ���with experimental data

Data used in fit: storage modulus and loss factor ��� vs. amplitude and frequency

Viscous damping models are ���simple and popular

•  Proportional Damping: linear and viscous

–  Convenient, not physics-based: dissipation in time-domain model, real vibration modes

M[ ] q{ } + C[ ] q{ } + K[ ] q{ } = Q{ }C[ ] = β K[ ] + γ M[ ]

•  Strain-based (stiffness-proportional) viscous damping yields modal damping that increases with frequency

•  Motion-based (mass-proportional) viscous damping yields modal damping that decreases with frequency

Normalized Frequency, ω!1.0! 10.!0.1!

Nor

mal

ized

D

ampi

ng!

Proportional damping model!

Typical material damping behavior!

�

γ ω2

�

β2ω

(Curves offset vertically for clarity.)!

Strain-based viscous damping yields damping that increases w/ modal frequency

•  Modal damping

strain-based viscous damping (stiffness-proportional)

�

ρA ˙ ̇ w + cK ′ ′ ′ ′ ˙ w + EI ′ ′ ′ ′ w = 0

�

˙ ̇ a m +cK

ρAmπL

⎛ ⎝

⎞ ⎠

4

˙ a m +1ρA

EI mπL

⎛ ⎝

⎞ ⎠

4⎡

⎣ ⎢

⎤

⎦ ⎥ am = 0

�

ζK m =cK

mπL

⎛ ⎝

⎞ ⎠

2

2 ρA EI( )1 2=cK ωm

2EI

Motion-based viscous damping yields damping that decreases w/ modal frequency

•  Modal damping

motion-based viscous damping (mass proportional)

�

ρA ˙ ̇ w + cM ˙ w + EI ′ ′ ′ ′ w = 0

�

˙ ̇ a m +cM

ρA˙ a m +

1ρA

EI mπL

⎛ ⎝

⎞ ⎠

4⎡

⎣ ⎢

⎤

⎦ ⎥ am = 0

�

ζM m =cM

2 ρA EI( )1 2 mπL

⎛ ⎝

⎞ ⎠

2 =cM

2EI ωm

•  Modal damping

“Geometric” viscous damping yields relatively constant modal damping

�

ρA ˙ ̇ w − cG ′ ′ ˙ w + EI ′ ′ ′ ′ w = 0

�

˙ ̇ a m +cG

ρAmπL

⎛ ⎝

⎞ ⎠

2

˙ a m +1ρA

EI mπL

⎛ ⎝

⎞ ⎠

4⎡

⎣ ⎢

⎤

⎦ ⎥ am = 0

�

ζG m =cG

2 ρAωm

mπL

⎛ ⎝

⎞ ⎠

2

=cG

2 ρA EI( )1 2

“geometric” viscous damping (flexural structures)!

Geometric Damping, SS BC, FEA – ���Modal damping is very nearly constant •  50 finite elements, 1st 49 modes

Norm

aliz

ed m

odal

dam

ping

Mode number

1.00

0.97

0.5%

2.5%

Non-proportional damping, but REAL modes!

Lesieutre, G.A., “Frequency-Independent Modal Damping for Flexural Structures via a Viscous ‘Geometric’ Damping Model,” Journal of Guidance, Control, and Dynamics, 2010, v 33, n 6, p 1931–1935. !

Damping model for shear beams ���captures dynamics of spacecraft cabling

•  Behavior separated into bending- ���and shear-dominated regimes –  Corresponding physical insight

•  Freq-independent modal damping ���achievable in bending region –  Can control damping ���

in shear regime

ζ =αϕ

2 ρAEI1+ αβ

αϕεm2

1+ εm2

−ρA !!w +κAG − ′ϕ + ′′w( ) = −q −αβ! ′β −αϕ ! ′ϕ

Kauffman, J.L., Lesieutre, G.A. and Babuska, V., “Damping Models for Shear Beams with Applications to Spacecraft Wiring Harnesses,” Journal of Spacecraft and Rockets, 2013; doi:10.2514/1.A32440!

Mode number, m M

odal

dam

ping

Mode number, m Norm

mod

al d

ampi

ng

bending shear

Membrane loads affect the ���modal damping of flexural structures

•  Rotor blades •  Airplane fuselages •  Actuators; Acoustic transducers •  MEMS resonators

HSCT

Pressurization can decrease damping ���of some fuselage modes by 80%

•   Damping of thin sandwich shell

Of concern for interior noise Lesieutre, Wodtke, Zapfe; Damped Composite Honeycomb Sandwich Panels ���

for High-Speed Aircraft Interior Noise Reduction, BCA, 1995.

n!

m!

Loss

fact

or

•  High Q (>106) attributed to tensile stress The precise mechanism by which tension increases Q even in the presence of increased material damping and boundary losses “remains unknown.”

Tension decreases modal damping ���in MEMS resonators

Verbridge, Shapiro, Craighead, Parpia, “Macroscopic Tuning of Nanomechanics: Substrate Bending For Reversible Control of Frequency And Quality Factor of Nanostring Resonators,” Nano Letters, Vol. 7, No. 6, 2007, pp. 1728–1735.

Model: Tension decreases ���modal damping of a SS beam

•  Modal damping

T

ρA w + cs ′′′′w − T ′′w + EI ′′′′w = 0

strain-based viscous damping

T Pcr

ζEI m =cEI

mπL

⎛⎝⎜

⎞⎠⎟2

2 ρA EI( )1 2ζEI m 0

1

1+ Tm2Pcr

⎛⎝⎜

⎞⎠⎟

1 2 =ζEI m0

1+ T Pcrm2

⎛⎝⎜

⎞⎠⎟1 2

Tension increases modal frequencies ���and decreases modal damping

0.001

0.01

0.1

1

Rela

tive

Mod

al L

oss

Fact

or

5 61

2 3 4 5 610

2 3 4 5 6100

2

Normalized Modal Frequency

m = 1

m = 5

0 -0.5

+0.5

5

50

500

T/Pcr

Lesieutre, G.A., “How Membrane Loads Influence the Modal Damping of Flexural Structures,” AIAA Journal, Vol. 47, No. 7, July 2009, pp. 1642–1646. !

Modal Strain Energy method ���provides insight

•  Consider the fraction of strain (potential) energy associated with each contributor to the stiffness

ηm =UEI

UTOTAL

⎛⎝⎜

⎞⎠⎟ m

ηEI +VT

UTOTAL

⎛⎝⎜

⎞⎠⎟ m

ηT

mat’l loss factor loss factor assoc. w/ tension = 0

=UEI

UEI +VT

⎛⎝⎜

⎞⎠⎟ m

ηEI =1

1+ T Pcrm2

⎛⎝⎜

⎞⎠⎟ηEI

Non-strain-based damping needed ���—highly distributed vibration absorbers

•  Metamaterials? Zapfe, J.A. and G.A. Lesieutre, “Broadband Vibration Damping Using Highly Distributed Tuned

Mass Absorbers,” 1996, AIAA Journal, Vol. 35, No. 4, April 1997, pp. 753–756. !

F s( ) = ki s2mi

s2mi + kii=1

N

∑ u s( )

Hébert, C.A. and G.A. Lesieutre, “Rotorcraft Blade Lag Damping using Highly Distributed Tuned Vibration Absorbers,” AIAA 98–2001, 39th SDM, 1998, 2452–2457. !

Mod

al lo

ss fa

ctor

Modal frequency (Hz)

200-1000 Hz!

10-1000 Hz!

CLD T

Sandwich panel damping ���increased via internal VEM layers

Zapfe, J.A., and G.A. Lesieutre, “A Discrete Layer Finite Element for the Dynamic Analysis of Composite Sandwich Beams with Integral Damping Layers,” Computers and Structures, Vol. 70, 1999, pp. 647–666. !

VEM just inside facesheet Discrete layer finite element model linear u, quadratic w; lock-free (shear stress ~ constant)

Mode number

Mod

al lo

ss fa

ctor

0.0

0.2

baseline

Frequency (Hz)

Acce

lera

nce

(mag

, kg-

1 )

High-damping fiber ���would enhance composite damping

•  Most of the strain energy of deformation is in the fiber

•  Modal strain energy ���shows leverage of fiber damping

ηEL=

vf E f

EL

⎛⎝⎜

⎞⎠⎟η f +

1− vf( )Em

EL

⎛

⎝⎜

⎞

⎠⎟ ηm

EL = vf E f + 1− vf( )Em

Yarlagadda, S., and G.A. Lesieutre, “Fiber Contribution to Modal Damping of Polymer Matrix Composite Panels,” Journal of Spacecraft and Rockets, Vol. 32, No. 5, September-October, 1995, pp. 825–831.

Ef ≫ Em

Damping of carbon fibers is low, ���but can be improved

•  Nominal loss factor of P-100 fiber: 10-4

•  Bromine intercalation increases 10–40X

400 Hz

Lesieutre, Eckel, DiCarlo, “Damping of Bromine-Intercalated P-100 Graphite Fibers,” Carbon, Vol. 29, No. 7, 1991, pp. 1025–32

Temperature (K)

Dam

ping

cap

acity

High-damping piezoelectric fiber ���might be possible

•  Piezo (3–3) –  Stiffness ~ aluminum –  Loss factor ~ 30%

•  Whisker •  Resistance

•  4% loss factor, 10% vf –  Small capacitance (~stray)

•  Challenges –  Poling, resistance, capacitance, freq-dependence

Yarlagadda, S., G.A. Lesieutre, S. Yoshikawa, and J. Witham, “Resistively-Shunted Piezocomposites for Passive Vibration Damping,” 37th AIAA/ASME Adaptive Structures Forum, Salt Lake City, UT, April 18–19, 1996, pp. 217–227.

Shape Memory Alloys ���can provide high damping — at high strain

Thomson, Balas and Leo, “The use of shape memory alloys for passive ���structural damping,” Smart Materials and Structures, 1995, v4, 36–42.

Strain

Stre

ss (N

/mm

2 )

Piezoelectric shunting can ���provide damping, vibration reduction

•  Mechanical structure interacts with electrical circuit via (capacitive) piezo transducer –  Capacitive shunting yields ���

frequency-dependent stiffness –  Resistive shunting yields ���

frequency-dependent damping –  Inductive shunting yields ���

electrical resonant absorber –  Switch-shunting could provide ���

broadband damping, stored energy

CR

piezoVs

C

piezoVs

Zsh

CL

piezoVs

R

CC

piezoVs sh

Lesieutre, G.A., “Vibration Damping and Control using Shunted Piezoelectric Materials,” Shock and Vibration Digest, Vol. 30, 1998, pp. 187–195. !

Resistive shunts provide damping (dissipation)

10-1 100 1010

0.1

0.2

0.3

0.4

Loss

fact

or

10-1 100 1011

1.5

2

Non-dimensional frequency

Non-

dim

ensio

nal m

odul

us

!τ = τ S

1− k2= R CS

1− k2

Loss factor

CR

piezoVs

coupling coefficient k = 0.70

0.35

k = 0.35 η = 0.070 k = 0.70 η= 0.343

η(ω ) =ηmax

2(ω !τ )1+ (ω !τ )2

ηmax =k2

2 1− k2

Davis, C.L., and G.A. Lesieutre, “A Modal Strain Energy Approach to the Prediction of Resistively–Shunted Piezoceramic Damping,” Journal of Sound and Vibration, Vol. 184, No. 1, 1995, pp. 129–139.

Frequency

Loss

fact

or

Semi-passive switch shunting ���can also be effective

•  Can remove electrical energy from piezo / structure

– Semi-passive damping

•  May be able to store it – Energy harvesting (and damping)

C

piezo

sV DCV+

+

-

-

Lefeuvre, E., Badel, A., Petit, L., Richard, C., and Guyomar, D., “Semi-Passive Piezoelectric Structural Damping by Synchronized Switching on Voltage Sources,” Journal of Intelligent Material Systems and Structures, Vol. 17, Nos. 8–9, 2006, pp. 653–660.

A self-powered energy harvesting ���circuit also provides damping

Measured damping due to energy harvesting agrees well with theory

�

foc = 53.50 Hzηoc = 0.173

fsc = 51.50 Hzηsc = 0.174

k = 0.2709

ηharvest = 2k 2

π 2 − k 2( ) = 0.024

ηmeasured = 0.197Δηmeasured = 0.024

Excellent agreement with theory Lesieutre, G.A., Hofmann, H., and Ottman, G., “Damping as a Result of Piezoelectric Energy

Harvesting,” Journal of Sound and Vibration, v 269, 22 January 2004, pp. 991–1001. !

Particle Impact Damper can be effective, but nonlinearity complicates design

•  Enclosure / void partially filled with particles (sand, BBs, etc.)

–  Kinetic energy dissipated via impacts!•  Adds mass and damping

Master curve collapses performance on frequency, gap size, particle size, and N layers

ÄÄ0

forfor

CeC

DPEC

C

<ΓΓ<<

⎩⎨⎧

Δ= −

Δ

4

2

3

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.2 0.4 0.6 0.8 1.0Displacement / (#Layers * Gap)

Dis

sipa

ted

Pow

er E

ffic

ienc

y

1/16" 10.5 grams3/32" 10.5 grams1/8" 10.5 grams5/32" 10.5 grams3/16" 10.5 grams1/16" 14 grams3/32" 14 grams1/8" 14 gramsCurve FitCurve Fit

Purely empirical

�

DPE ≡PdissipatedPtotal

= Real(P)ω mbed V

2Dissipated Power Efficiency

Γ

Yang, M.Y., G.A. Lesieutre, S.A. Hambric, and G.H. Koopmann, “Development of a Design Curve For Particle Impact Dampers,” Noise Control Engineering Journal, v 53, n 1, 2005, p 5–13. !

An event-based model ���was developed to predict performance

e=0.1, d=3.0!

•  Parameters –  Non-dimensional gap, d = Δ / A!–  Non-dimensional base acceleration, β = Aω2/g!–   Effective coefficient of restitution, e!

Time simulation for β=10.0!

time di

spla

cem

ent

High loss factors correspond to ���two-impact-per-cycle solutions

Continuation plot!

e=0.1, d=3.0!

Design: optimum gap to yield high / constant loss factors •  Given M and operating range of β, ���

select e and d for two-impact-per-cycle behavior •   Adjust μ to change loss factor

Ramachandran, S., and G. Lesieutre, “Dynamics and Performance of a Vertical Impact Damper,” Journal of Vibration and Acoustics, v 130, n 2, April, 2008, #021008. !

acceleration lo

ss fa

ctor

acceleration

phas

e at

impa

ct

A broad variety of damping methods and models are available

Layered treatments can be effective

Viscoelastic polymers have high damping

–   f, T

Proportional viscous damping is inadequate

MSE+FE can predict modal damping

Complex modulus can be useful for FRF

ADF viscoelastic model useful in time domain

–   f, T, heating –   Nonlinear versions

“Geometric” viscous damping is ~constant

Tensile loads reduce modal damping

Piezo shunting and energy harvesting can provide damping

Particle damping is nonlinear, can be effective

K[ ] !x{ }C

R

piezoVs

cG ′′w

helo lag damper

spacecraft cable

fuselage, rotor blade

Challenging req’ts and environments demand advances in damping

•  Effectiveness over broad range ���of operating conditions

•  System benefits require ���damping-by-design

•  Close integration ���with load-bearing structure

•  Joints, cabling, multi-D models •  Materials

–  Active fiber composites; others •  Micro-nano tailoring

– Materials genome – Metamaterials

•  Additive manufacturing C.T. Chan, HKUST

Continuum Control Corp.

Acknowledgments •  Colleagues, students and

post-docs: –  Ted Nishimoto –  Jim DiCarlo –  Ed Smith –  Hans-Walter Wodtke –  Shoko Yoshikawa –  Farhan Gandhi –  Mikael Enelund –  Y.C. Yiu –  Vikram Kinra –  Dan Inman –  Gary Koopmann –  Steve Hambric –  Heath Hofmann –  John Kosmatka –  Norm Wereley –  Vit Babuska

–  Jeff Zapfe

–  Emanuele Bianchini –  Shridhar Yarlagadda –  Kiran Govindswamy –  Chris Brackbill –  Eric Ruhl –  Lynn Byers –  Chris Davis –  Chad Hébert –  Julien Bernard –  Razvan Rusovici –  Deepak Ramrakhyani –  Geff Ottman –  Mike Yang –  Sanjiv Ramachandran –  Jeff Kauffman

•  Sponsors –  CIRA, NSF, UTRC, Lord,

Boeing (BCA, Helo), Bell, Army (RCOE), ONR, Sandia, AFRL, AFOSR, NASA, PCB

And remember … Don’t dampen a

vibrating structure Damp it!