viscoelastic damping: lecture notes 140202

Viscoelastic Damping

Mohammad Tawfik

Cairo University

Aerospace Engineering Department

2 February, 2014

Introduction

Viscoelastic Damping 2

Contents Introduction ............................................................................................................................................ 3

Classical Models ...................................................................................................................................... 3

Maxwell Model ................................................................................................................................... 3

Model Characteristics ..................................................................................................................... 4

Kalvin-Voigt Model .............................................................................................................................. 6

Zener Model ........................................................................................................................................ 8

The Area in the curve exist only when 0 ...................................................................................... 11

Golla-Hughes-McTavish (GHM) 1983 ................................................................................................ 11

Unconstrained Layer Damping .............................................................................................................. 17

Finite Element Model of Bars ........................................................................................................... 17

Composite Bar ................................................................................................................................... 18

Constrained Layer Damping .................................................................................................................. 18

Active Constrained layers damping ...................................................................................................... 27

Bibliography .......................................................................................................................................... 42

Introduction


Introduction

Objectives

• Recognize the nature of viscoelastic material

• Understand the damping models of viscoelastic material

• Dynamics of structures with viscoelastic material

What is Viscoelastic Material?

• Materials that Exhibit, both, viscous and elastic characteristics.

• The material may be modeled in many different ways. Classical models include:

– Mawxell Model

– Kalvin-Voight Model

Classical Models

Maxwell Model The Maxwell model describes the material as a viscous damper in series with an elastic stiffness

(Figure 1). When stress is applied, it is uniform through the element, in turn, we may write the total

strain of the viscoeleastic element as:

ds

Figure 1. Schematic for a viscoelastic element using the Maxwell model

According to this model, the stress is equal in both elements, which may be expressed by the

relation:

ddss CE

According to this relation, we may write:

dtCE d

d

s

s

According to this, the total strain may be expressed as:

Classical Models


Es

dtCd

Or

ds CE

Model Characteristics

When investigating the model characteristics in our context, we are interested in three aspects;

namely:

• Creep. When a material is loaded for a prolonged period of time, the strain tends to

increase, which leads, in turn, to failure. The phenomenon of the strain increase at constant

load is called creep.

• Relaxation. When materials are strained for a prolonged periods of time, the internal

stresses tend to decrease. The phenomenon of stress decrease at a constant strain value is

called relaxation.

• Storage and Loss Moduli. When the viscoelastic material is loaded harmonically, the stress-

strain relation may be presented by complex modulus of elasticity. The real part of the

complex modulus is called storage modulus while the imaginary part is called the loss

modulus.

To study the creep characteristics of the Maxwell model, we need to set the rate of change of stress

to zero in the stress-strain differential relation. Thus:

d

zero

s CE

Solving the differential equation, we get:

tCd

The resulting strain time function indicates that the strain will grow to an unbound value as time

increases!

To investigate the relaxation characteristics, the strain rate is set to be zero in the differential

relation, the resulting relation becomes:

ds CE

0

When solved, the above relation gives the stress time relation as:

Classical Models


ds CtEe

0

Where, 0 indicates the initial stress value. The above relation indicates that the stress will decrease

exponentially with time with an asymptotic value of zero.

When studying the response of the model under harmonic excitation, the excitation stress is

presented as:

tje 0

Thus, the strain response is presented as:

tje 0

Substituting in the differential equation, we get:

o

ds

dso

CjE

jCE

Giving:

o

ds

dssdo

CE

jCEEC

222

222

Separating the real and imaginary parts, we get:

o

ds

ds

ds

sdo

CE

CEj

CE

EC

222

2

222

22

Where, the storage modulus is:

222

22

'ds

sd

CE

ECE

And the loss modulus becomes:

222

2

"ds

ds

CE

CEE

The loss modulus, defines as the ratio between the storage and loss moduli, may be given as:

d

s

C

E

Now, the stress strain relation may be expressed as:

oo jE 1

Classical Models


Where the complex modulus is given by:

jEE 1*

Figure 2. The variation of the storage modulus and the loss factor with frequency according to Maxwell’s model

Figure 2 presents the variation of the storage modulus and the loss factor with frequency. Note that

according to Maxwell’s Model:

• Under static loading, the stiffness, storage modulus, is zero and the loss factor is infinity!

• For very high frequencies, the loss factor becomes zero!

Kalvin-Voigt Model The Kalvin-Voigt model describes the material as a viscous damper in parallel with an elastic stiffness

(Figure 3). When stress is applied, it is distributed through the element, while the strain in both

elements is equal.

Figure 3. Schematic for a viscoelastic element using the Kalvin-Voigt model

The stress strain relation may be written as:

ds

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Frequency

Mo

du

lus

E

u

Classical Models


ddss CE

No we come to the studying the Kalvin-Voigt Model characteristics. To study the creep we solve the

above equation for constant stress to get:

ds CtE

s

eE

1

Which indicates that the strain will grow to a constant value as time increases!

When studying the relaxation, we set the strain rate to zero, giving:

0 sE

Which means that the stress will stay constant as time grows for the same strain!

Now, we come to investigating the Storage modulus and Loss Factor. For harmonic stress and strain

we get:

tje 0

tje 0

Resulting in the relation:

ods CjE

Figure 4. The variation of the storage modulus and the loss factor with frequency according to the Kalvin-Voigt model

Figure 4 presents the variation of the storage modulus and the loss factor with frequency. Note that

according to the Kalvin-Voigt Model:

0

2

4

6

8

10

12

14

0 2 4 6 8 10

Frequency

Mo

du

lus

E

u

Classical Models


• Under all loading, storage modulus is equal to the stiffness of the spring, and the loss factor

is zero.

• For very high frequencies, the loss factor becomes unbound!

Zener Model The Zener model describes the material as a viscous damper in parallel with an elastic stiffness and

both are in series with stiffness (Figure 5). The strain may be written as:

1 s

Figure 5. Schematic for a viscoelastic element using the Zener model

Stress-Strain relation, according to the zener model, may be written as:

11 dpss CEE

From which we may write in Laplace domain:

dps

ssCEE

1,

Or:

dps

sdp

dps sCEE

EsCE

sCEE

Back to time domain, we get:

sdpdps EsCEsCEE

From which we get the differential equation:

dspdsps CEECEEE

Or:

EE

Classical Models


Studying Zener Model characteristics, we get for the creep:

0 EE

Giving:

s

t

E

e

E

0

And for the relaxation, we get:

E

Giving:

teE 100

While for the storage modulus and loss factor we get:

oooo jEjE

Rearranging, we get:

ooo

jE

j

jE

22

2

1

1

1

1

Or:

oo

jE

2222

2

11

1

Or simply:

oo jE 1

Classical Models


Figure 6. The variation of the storage modulus and the loss factor with frequency according to the Zener model

This is more realistic for the presentation of the material characteristics, however, is does not satisfy

the detailed studies needed for analysis of complex structures. Let’s recall the harmonic relations:

ti

o

ti

o

e

e

And the differential equation

EE

Which give:

ti

o

ti

o

ti

o

ti

o eiEeEeie

Expanding the complex exponentials, we get:

titiEtitE

tititit

oo

oo

sincossincos

sincossincos

Equating the real and imaginary parts:

tEtEtt oooo sincossincos

tEtEtt oooo cossincossin

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4

Frequency

Mo

du

lus

E

u

Classical Models


22'

2'2'

2'

'

''

''

'''

)(

)sin()(

sin1

cos

cossinsin

eo

oo

o

oedissipativ

edissipativ

d

elastic

eTotal

ooo

E

tEE

tEE

E

tE

tEtEt

oo

o

e

o

d

o

oed

eod

EEThis

EE

E

E

E

''

2

'

2

'

2'

2'2

2

22'

2

*2diameterminor & *2 diameter major with ellipserepresent equation

1

)(by Divide

)(

)(

Figure 7.

The Area in the curve exist only when 0

Golla-Hughes-McTavish (GHM) 1983

Simple mass+visco elastic material

Classical Models


0

0

-

- 2

0

0 0

0

0

0

0

-

-

2 0

0 0

1 0

0

02.&

01

0

02

02

21

.. )(

int

2

2

2

02

21

0

0

n

00

00

2

n

2

n

2

n

0

0

00

00

20200

00

2

2

00

2

2

2

22

2

2

22

000

22000

22

2

22

22

nnn

n

n

n

n

nn

n

n

n

nn

nnn

nn

n

nn

n

Z

Z

ZSSS

SS

SSS

systemofDEinZsubstitute

vaiableernalZ

ZZZ

ZZZ

SSZlet

SS

SSS

Complex stiffeners is given as

Classical Models


3 2S

2 2S

DOF internal Z

Z variableinternal the

1 02

21S

-:spring elastic viscoaon supported mass

prametersunknown are , ,

2

21

2

21

2

21

22

22

2

n

22

22

n

22

2*

22

2*

22

2*

nn

nn

nn

n

nn

n

nn

n

nn

n

SZZ

S

assume

SS

SS

SS

SS

SS

SSGG

SS

SS

From equation 1&2

Classical Models


0

-

- 12

0

0 0

0

0

0 -

- 1

2 0

0 0

1 0

0

02 3 .

01

01

0

02

02

21

0

0

n

00

00

2

n

2

n

2

n

0

0

n00

00

2

n

20

n

00

00

2

2

2

2

2

22

2

2

2

2

n

n

n

nn

n

n

n

eqn

domaintimein

S

S

ZSS

S

SS

SSS

Classical Models


Visco elastic material has its own internal DOF =Z

In general X&Z are vectors

22

2

2

21

nnn

nn

nnSS

SS

Summary

The original system has (X ) DOF

The system+vesco elastic material has (X+Z) DOF

Entire system order has increased

X=primary DOF

Z=secondary DOF

Use static condensation method (Guyan reduction method )

I.e condensation =eliminate the secondary DOF&only maintain the primary DOF.

Static Condensation

*consider only the stiffnes matrix

01 1

1

1

-

- 11 1

1

1

1

0 -

- 1

consider

matrixstiffnesredusedF

F

F

Classical Models


redusedredusedredused

total

totalreduced

C

F

F

F

termsenergycompare

0000

0

n

00

2

n

0 -

- 12

0

0 0

0

0

01 1

0

0 -

- 1

-:follow as also obtained becan redused ""

DOF.primary ofenergy strain system entire ofenergy strain

2

1

2

1

2

1

:

Visco Elastic Material Damping

*Golla-Hughes-McTavish (GHM) model

Stiffners complex modulas (longitudinal or sheer)

22

2

*

2

21

nn

n

SS

SS

Unconstrained Layer Damping


For structure& V E M-------------system dynamics

0 -

- 12

0

0 0

0

0

0

0

n

00

00

2

n

F

**GHM model when augmented with structural model can be written as:-

1-frequency domain

2-time domain

Other Models

• Some, more accurate, models were developed to represent the behavior of viscoelastic

material

• The greatest concern was paid for the modeling in the time domain.

• The most famous models are:

– Golla-Hughes-McTavish

– Augmented Temperature Field

Fractional Derivative

Unconstrained Layer Damping

• The most common way of using viscoelastic material in damping is by

bonding it to the surface of the structure!

• The viscoelastic material will be strained with the structure resulting in

energy losses in the surface layer

Finite Element Model of Bars

• Recall the stiffness and mass matrices of a bar:

• It is possible, in the above model, to superimpose more than one

element!

Constrained Layer Damping


21

12

6&

11

11 ALM

L

EAK

Composite Bar

• The effect of each part of the bar may be added to the other part

linearly incorporating the effect of both materials

21

12

6

11

11

LAA

M

L

AEAEK

VVBBC

VVBBC

Homework #9

• Use the datasheet of the DYAD606 viscoelastic material to calculate the

bar response with modulus of elasticity varying with frequency


• When the viscoelastic layer is covered, constrained, from the top side,

sheer stresses are generated between the different surfaces.

• Viscoelastic materials are characterized by having much higher losses in

the case of sheer than in the case of axial strain.




Sheer Stresses



2hx

x

uE

2

22

2

2

hEx

u

1

0

22

*

2

2

h

uu

hE

G

x

u

Axial Displacement

• The axial displacement relation becomes:

1

0

22

*

2

2

h

uu

hE

G

x

u

02

2

*

122 uux

u

G

hhE

02

2* uu

x

uB

• The axial displacement relation becomes:

• Solving:

xuuB xx 0

*

xB

xCha

B

xShau 0*1*1

*

*

*

0

2B

lCh

B

xShB

xu

Sheer Strain



1

0

1

0

h

xu

h

uu

*

*

*

0

2B

lCh

B

xShB

Lost Energy

2/

2/

*2

*2

1

2*22/

2/

2

12

l

l

o

l

l

dxBxShBlChh

BGdxGhW

Note that

tan )1cos()tan1cos()sin(cos

)2/sin(l

)2/cos(

)cos()(

)2/cos()sin()2/sin()(4

)2/1(

*

221

0

00

0

21

2

02

v

ass

iGGiGG

G

EhhlA

Ach

Ash

llhh

W

Example

''28.3*28.3

''110

10*01.*01.

0

3

7

0

optimumL

22

**2/

2/

*2 lBlSh

BdxBxSh

l

l



For unconstrained layer damping

1

)(

2

0

''

11

2

0

''

11

Lhh

dissipatedenergytotalW

hhWd

In the constrained case

2''

11

2/

2/

2

1

2/

2/

2''

1

2/

2/

''

)(

o

l

l

nedunconstrai

dconstraine

l

l

l

l

dconstraine

Lhh

dxGhh

W

W

dxGhhWdW

''' '''

1''''

1''''

*

*

GG

ii

iGiGGG



3/1''

''

3

''

3

'G''G'

3

'G'

0.5, VEMfor

ratio spoisson' 12

''

G

G

1000124.0*10*1*3

2

10G

1

45 1

124.0coscosh

2/cossin2/sin

sin

coscosh

2/cossin2/sin

sin3

2

3

1

4

42

1

2

o

2

1

2

2/

2/

2

2

R

h

h

Ash

l

Ash

lGh

h

dxlW

WRatio

v

o

o

l

louncon

con

Summary

*constraining the VEM makes it deforms in sheer & results in significantly high

energy dissipation characteristics

Notes

The plunkett & Lee analysis assumes:-

1-quasi-static analysis (satisfied by the force that the constraining layer

thickness is small (its inertia can be neglected)



2-A general base structure

3-longitudinal vibration

beam

0max

2

2

2

2

.max

tan

tcons

wM

wdofVEM

For the beam:-

Energy dissipated



2/

2/

2

2222

1

*1

*

*

0

2/

2/

2''

1

2/

2/

*)/(

*)/(*''

2

-

)(

l

l

xx

l

l

l

l

dconstraine

dxlch

shWdGhhW

lchh

sh

dxGhhWdW

Exercise

Show that the above composite has

1

2

e

1

2

1

2

1

*

2*

e

*

*

23*

11

r

)1()1(r :

1)1(31

h

hr

iriwhere

rr

rrrrr

h

e

he

he

rhe

tt



And show that:-

24232

4232

4641)1(

2463

hehehehehe

hehehhe

rrrrrrrrrr

rrrrrrrr

Take;-

2

2

4

1

2

00519.0

00502.0

10*585.3

1

e

h

r

h

hr

Kinematics of CLD

2

3113

33

21

)2

()(

2

2

h

UUW

Whh

UUUU

Wh

UU

Wh

UU

Ax

xA

x

x

Active Constrained layers damping


Wh

h

h

UU

h

Whhh

UU

h

WhUU

22

31

2

31

31

2

2

)22

()(




0X

U 2/

*

2212*

0

2*

L

G

hh

UU

Solution procedure

*solve for U

*determine γ

2

2'' hGW

Compute *

WdxW Compute *

*put in dimensionless form η



For ACLD

activepassive

If controller fails---------------system still “fail-safe” because of passive damping

00 )( t

dpp

Notes (viscoelastic)

dampingelastic

iwt

o

o

iwt

ii

i

FFw

F

iweiw

eif

F

if

0'

'

0

'''

'

)1(

)1(

2'2'

/2

0

22

0

2'

0

/2

0

2'/2

0

2

2)(cos

sin

oo

o

d

dttWEnergy

tfor

dtdtdt

dxFcycleperdissipatedEnergy

capacitydampingspecificW

W

W

W

WEnergypotential

e

e

e

W2

W 2

2

1

e

2

0

'



222

/2

0

222

/2

0

cosC

oo

o

oo

o

d

CC

dtt

dtCenergydissipatedW

CF

DampingViscous

Equivalent viscous damping to viscoelastic material

2 resonanceat

2

2

2

1

C

'

'

'

''

oC

Cratiodamping

C

2-Transeverse Vibration

Kinematics equation:

231

22

31

2h

hhh

Wh

h

h

UUx

U=longitudinal deflection of base structure



=shear angle

Wx=slope of deflection line

1 21

21

12

hWhU

hWhU

hWUh

=F xUh 111 Force on top layer per unit width =

*

211

*

111

)(

2

GhWhh

sshearstresGUhd

dF

Wh

hG

allongitudinhlet

Wh

h

hh

G

21

*

111

2211

*

Rigidity

NOTE

Bending in beam:

0

0

0

;

)i 1()(

4*

*

2

2*

**

WW

WD

mW

WmWD

motionofEquation

DD

B

t

t

tt



4/1*B

*

)(

0

B

)/(let

solution n propagatio one

*

mD

W

eWW

numberwavebendingwhere

t

wti B

)4/(

0

)4/()(

0

4/14/1

2/1* )

41(

) 1()/(

B

BB

eWW

eeWW

iimD

W

wti

B

t

B

22

0

)2/(2

0

2

)2

(dx

B

B

eWCenergyd

eWCCWEnergy

B

assemblylayersdconstraineEnergy

d

Energy

d

B

C

B

Energy/dx 2

2

Energy/dx

2

2v

221

*

)(

0

G' energy/dx d calculate *

*

*

; *

h

for

Wh

h

h

Gsolve

eWWput

forsolution

Bwti

C

Using the solution given in ''Damping of flexural waves by constrained layers ''

Journal of acoustic society of America, Vol 31, 7 pp952-962, 1959



iWh

h

h

G

WiW

eeWiW

B

B

iWtiB

B

3

221

*

3*

0

3* *

equationsatisfiesitthatcheck

WG

h

ih

B

B

)(

*1

21

2*2

Summary

Loss factor for constrained layer damping during transverse vibration

W

Wfactorloss

EnergyElastic

Energydissipated

W

Wdampingspecifi

D

D

2

2-for beam in bending

Equation of motion



0

0

0

;

)i 1()(

4*

*

2

2*

**

WW

WD

mW

WmWD

motionofEquation

DD

B

t

t

tt

4/12

B*

*

tD

mWnumberwavebendingwhere

)4/(

0

)4/()(

0

4/14/1

2/1* )

41(

) 1()/(

B

BB

eWW

eeWW

iimD

W

wti

B

t

B

)2/()2/(/

2/

2/

2/2/2)(

0

2

BB

wti

B

B

B

BBB

eC

eC

energy

dxdenergy

eCeeWCCWEnergy

losseswithoutsassemblylayerdconstraineofnumberwave

3-calculate loss factor of CLD

Energy

d

B

C

Energy/dx

2

4- For 3 layers CLD



231

22

31

2h

hhh

Wh

h

h

UUx

If U3=0 a-

hWhU

hWhU

Wh

h

h

U

2

21

22

1

Quasi-static Equilibrium

Longitudinal load on layer2=shear load

hWhU

geomtryfrom

UhU

hhG

bddbh

21

1111

1

1

1

1

;

*

)()*(



111

2211

11211

211

*

**

*

hlet

Wh

h

hh

G

WG

hh

G

hh

GhWhh

iWh

h

h

G

WiW

eeWiW

eWW

Wh

h

h

G

B

B

iWtiB

wti

B

B

3

221

*

3*

0

3*

)(

0

221

*

*

solution n propagatio one

*

It has a solution;

equationsatisfiesitthatcheck

WG

h

ih

B

B

)(

*1

21

2*2



2

2v G' h Energy dissipated per unit length

Energy in bending waves

)*sin(

)*cos(*

)*sin(W

)*cos(W

)*sin(W

0

2*

0

0

0

0

0

0

)(

0

*

tWW

tWW

velocityAngulertW

velocitylineartW

wtW

eWW

B

BB

B

B

wti B

00

0

3*

0

2*

)cos(*

)sin(**

WFWpower

tWDFshear

tWDWDmoment

t

tt

22

.

23

2

2

23*

3*2

0

223*23*0

2

1

)/(

2

''

2

-2

/2*

sin*cos*

g

gDh

WD

Gh

WD

powerEnergy

DWWDDW

tVlayerconst

ot

V

ot

tott



1

0g

when Max. is

*

constr

21

2*

optimum

constr

g

parametershearG

g

NOTE

2

1

22

31

111

U

tionlong.vibrain CLD0 W

,

h

U

h

hW

h

U

if

gofmeaningphysicaltheiswhat

h

21/*

21

0*

hG

oe

h

G

Also



1*

21

/* 21

e

o

hG

o

h

G

eU

U

eUU

22

2*

B

22*

21

2g

2

1

*

e

eB

e

let

g

G

h

lengthwaveshear

lengthgwavebending

b- if U3=0

0 F

0

&

31

333111

31

22

31

F

UhUhhavetoorderin

dependentareUUwhere

Wh

h

h

UU

Wh

hU

h

UU

hhwhere

UU

2

1

2

31

1

3

13

333111

3311

/1

0



Whh

U

hWhU

31

3

31

231

2

21

3

31

*

*

layer topeqm.of

11

111

11

GU

GhU

h

But

Wh

h

hh

G

GWhh

2231

31

*

31

31

31

231

)(*

Follow same procedure as case of U3=0 to get

2

3

1

22

1

11

)1/)(/(

g

g

ggDh t

Vconstr

Summary

1- Longitudinal vibration

-to find optimum length of constraining layers



(Following plunket &lec. paper)

lengthticcharactersEhh

GB

B

Loptimum

1**

28.3

221

*

2-comparing between CLD &un CLD

Energy dissipated in un CLD<<< CLD

Tension shear

3-transiverse vibration

A -definition

-specific damping

-loss factor

-loss factor &damping ratio selection

B-CLD with U3=0

*shear parameter g=1 for optimum

*g=ratio of bending to shear wave length

---optimum is ensured if there is balance between shear and bending

C-U3=0

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