This document introduces viscoelastic materials and classical damping models including the Maxwell, Kelvin-Voigt, and Zener models. It examines the creep, relaxation, and dynamic properties of each model. The Maxwell model describes the material as a spring and dashpot in series. The Kelvin-Voigt model uses a spring and dashpot in parallel. The Zener model combines features of the Maxwell and Kelvin-Voigt models. Graphs show how the storage modulus and loss factor vary with frequency for each model.

1. Viscoelastic Damping
Mohammad Tawfik
Cairo University
Aerospace Engineering Department
2 February, 2014

2. Introduction
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
Viscoelastic Damping
2

3. 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:
  s  d
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:

  Es s  Cd  d
According to this relation, we may write:
s 

Es
d  

Cd
dt
According to this, the total strain may be expressed as:
Viscoelastic Damping
3

4. Classical Models


Es


Cd
dt
Or




Es


Cd
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 stressstrain 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:




Es



Cd
zero
Solving the differential equation, we get:


Cd
t
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:
0


Es


Cd
When solved, the above relation gives the stress time relation as:
Viscoelastic Damping
4

5. Classical Models
   0e tE
s
Cd
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:
   0e jt
Thus, the strain response is presented as:
   0 e jt
Substituting in the differential equation, we get:
o 
Es Cd j
o
Es  jCd
Giving:
o
C d E s  2  E s C d j

o
2
2
Es   2Cd
2
2
Separating the real and imaginary parts, we get:
2
 C d 2 E s 2
Es Cd  
 2
 o
o  
j 2
2
2
E s   2Cd
E s   2Cd 


Where, the storage modulus is:
Cd Es 2
2
2
Es   2Cd
2
E' 
And the loss modulus becomes:
Es Cd 
2
2
E s   2Cd
2
E" 
The loss modulus, defines as the ratio between the storage and loss moduli, may be given as:

Es
Cd 
Now, the stress strain relation may be expressed as:
 o  E 1  j  o
Viscoelastic Damping
5

6. Classical Models
Where the complex modulus is given by:
E *  E 1  j 
1
0.9
0.8
Modulus
0.7
0.6
E
0.5
u
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Frequency
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:
  s d
Viscoelastic Damping
6

7. Classical Models

  Es s  Cd  d
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:


1  e
Es
 E s t Cd

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:
  Es 0
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:
   0 e jt
   0e jt
Resulting in the relation:
  Es  jCd  o
14
12
Modulus
10
8
6
E
4
u
2
0
0
2
4
6
8
10
Frequency
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:
Viscoelastic Damping
7

8. 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:
   s  1
Figure 5. Schematic for a viscoelastic element using the Zener model
Stress-Strain relation, according to the zener model, may be written as:

  Es  s  E p  1  Cd  1
From which we may write in Laplace domain:
s 

Es
, 1 

E p  sCd
Or:


Es

 E  sCd  Es 

 p
 E E  sC  
E p  sCd
d 
 s p

Back to time domain, we get:
Es E p  sCd   E p  sCd  Es 
From which we get the differential equation:


Es E p  Es Cd   E p  Es   Cd 
Or:


E  E    
Viscoelastic Damping
8

9. Classical Models
Studying Zener Model characteristics, we get for the creep:

E  E    0
Giving:

0
E

e t 
Es
And for the relaxation, we get:

E    
Giving:
   0  E 0 1  e t  
While for the storage modulus and loss factor we get:
E o  jE o   o  jo
Rearranging, we get:
1  j
1   2  j    
o  E
o  E
o
1  j
1   2 2
Or:
 1   2 j     
 o  E
 1   2 2  1   2 2  o



Or simply:
 o  E 1  j  o
Viscoelastic Damping
9

10. Classical Models
2
1.8
1.6
Modulus
1.4
1.2
1
0.8
0.6
E
0.4
u
0.2
0
0
1
2
3
4
Frequency
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:
   o e i t
   o e it
And the differential equation


E  E    
Which give:
 oeit   oieit  E oeit  E oieit
Expanding the complex exponentials, we get:
 o cost  i sin t    oi cost  i sin t 
 E o cost  i sin t   E oi cost  i sin t 
Equating the real and imaginary parts:
 o cos t   o sin t  E o cos t  E o sin t
 o sin t   o cos t  E o sin t  E o cos t
Viscoelastic Damping
10

11. Classical Models
 o sin t  E ' o sin t  E '' o cos t
 Total   e   d


elastic
dissipativ e
 dissipativ e  E  o cos t
''

E '' '
E  o 1  sin 2 t
'
E
  ( E ' o ) 2  ( E ' o sin t ) 2
  ( E ' o ) 2   e
d



2
2

2
  ( E ' o ) 2   e


2
 
2
  d    e  ( E ' o ) 2
 
 
Divide by ( E ' o ) 2 
2
2
 d   e 
 '    '  1
 E    E  
o 
o 


This equation represent ellipse with major diameter  2 * E ' o & minor diameter  2 *E ' o
Figure 7.
The Area in the curve exist only when   0
Golla-Hughes-McTavish (GHM) 1983
Simple mass+visco elastic material
Viscoelastic Damping
11

12. Classical Models

S 2  2n S 
 S    1   2
0
2
S  2n S  n 

2
n 2
let  Z  2

2
S  2n S  n
Z 00  2n Z 0  n Z  n 
2
2
Z 00  2n Z 0  n   Z 
Z  int ernal vaiable
substitute( Z )  in DE.of .system

 S 2  2n S 

n 2


 S    1   
2

 S 2  2 S   2    0
n

n
n 




2

S 2  2n S Z  0
2
n
 2
 00    2 n   Z   0
n
 S 2     
 00   1     Z  0
& . 00  n   2n  0  n   0
2
2
00
-     0
0    0     
 0     0


 


 


 
0
2  
1  00  0 2   0  - n 2
  0
n  


 

0  00
0  0

0
  
-     0


 


        
   00  
2   0  
  
0
    0
 0
   -  
2 

n   

n    




Complex stiffeners is given as
Viscoelastic Damping
12

13. Classical Models

S 2  2n S 
   1   2
2
S  2n S  n 

*

S 2  2n S 
G  G 1   2
2
S  2n S  n 

*

S 2  2n S 
   1   2
2
S  2n S  n 

 ,  , n are unknown prameters
*
mass supported on a visco elastic spring : 
S 2  2n S 
 S    1   2
0
2
S  2n S  n 

assume the internal variable Z
1
2
n 2
Z  internal DOF  2

2
S  2n S  n
S2 Z  2n SZ  n    
2
2
3
From equation 1&2
Viscoelastic Damping
13

14. Classical Models


S 2  2 n S
n 2
 S    1  
0
2
2 
2
n
S  2 n S   n 



S 2  2 n S
2
 S     
Z 0
2
2
n
 S 2           0
 S 2    1       0
in time domain
 00   1       0
eqn.  3  00  2 n  0   n    n   0
2

0



0


2
0  00  0 0   0     1 -    


 
0
 00   
 0   
2  
1    0 2 n    -  n 2

n 


 
0  00
0 0
 0
  


 
       1 -      0
   00  
2   0  
 

    
 0
   -  
2 

n  
n   



Viscoelastic Damping
14

15. Classical Models
 Visco elastic material has its own internal DOF =Z
 In general X&Z are vectors


S 2  2 n n S

  1   n   n 2
2

 S  2 n n S   n






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
 1   
-  

  
-       F 
  
    0 

   
 1       F
F    redused stiffnes matrix
    1
         
    1
consider
    1
 1
 1   
1
-  
 
1   
0 
Viscoelastic Damping
-   1

  1

15

16. Classical Models
compare energy terms : 
1 
1
  reduced        total 
2
2
 
1
 
   total  
2
 


strain energy of entire system  strain energy of primary DOF.
" " redused can be obtained also as follow : -       F 
 1   
  
-  
    0 


0     F 

  
 1
1   0 


 

  

 0

0 
0
  00  
 
0
2


n 

0 
 1   
2   0  

- 
n 


-  
F 

   0  
 
 

 redused  00  C redused  00   redused 
Visco Elastic Material Damping
*Golla-Hughes-McTavish (GHM) model
Stiffners complex modulas (longitudinal or sheer)

S 2  2 n S 
   1   2
2 
S  2 n S   n 



*
Viscoelastic Damping
16

17. Unconstrained Layer Damping
For structure& V E M-------------system dynamics


0


0  00
0


  
   00  
    0
n 2    


 0
 
     1
2   0   
    - 
n    

0
-       F 
  
    0 

**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!
Viscoelastic Damping
17

18. Constrained Layer Damping
K
EA  1  1
AL 2 1
  1 1  & M  6 1 2 
L 



Composite Bar
• The effect of each part of the bar may be added to the other part
linearly incorporating the effect of both materials
1
 1

 A  V AV 2
MC  B B
L
6
1
KC 
E B AB  EV AV
L
 1
1

1
2

Homework #9
• Use the datasheet of the DYAD606 viscoelastic material to calculate the
bar response with modulus of elasticity varying with frequency
Constrained Layer Damping
• 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.
Constrained Layer Damping
Viscoelastic Damping
18

19. Constrained Layer Damping
Sheer Stresses
Viscoelastic Damping
19

20. Constrained Layer Damping
 

x h2
  E2
u
x
 2u


2
x
E2 h2
 2u
G *  u  u0 



x 2 E2 h2  h1 


Axial Displacement
• The axial displacement relation becomes:
 2u
G *  u  u0 



x 2 E2 h2  h1 


E2 h2 h1  2u
 u  u0
G * x 2
B*
 2u
 u  u0
x 2
• The axial displacement relation becomes:
• Solving:
B*u xx  u   0 x
 x 
 x 
u  a1Sh *   a1Ch *    0 x
B 
B 

 x 
B * Sh *  

 B 
u  0 x 

 l 
Ch *  

 2B  

Sheer Strain
Viscoelastic Damping
20

21. Constrained Layer Damping

u  u0 u   0 x

h1
h1
 x 
* 
B 
 
 l 
Ch * 
 2B 
 0 B * Sh
Lost Energy
G o2 B*
l/2
W  h1G   2 dx 
l / 2
h1Ch

*
2
2
l 2B   Sh x
l/2
2
*
B* dx
l / 2
Note that
l/2



B*
l
*
/ 2Sh x B dx  2 Sh l B  2
l
2
 ass 
A
W
(1 / 2) 2  0 h1 h2 l
2
 4
l
cos( / 2)
0
 0  sh( A) sin( / 2)  sin( ) cos( / 2) 

l 
ch( A)  cos( )



l
sin( / 2)
0
G *  G (cos  i sin  )  G cos(1  tan  )  G cos(1  i )
0 
h1 h2 E 2
G
 v  tan
Example
.01 * .01 *107
0 
 1' '
103
Loptimum  3.28 * 0  3.28' '
Viscoelastic Damping
21

22. Constrained Layer Damping
For unconstrained layer damping
d (W )  hh11  0
''
2
W  total energy dissipated
 hh11  0 L
''
1
2
In the constrained case
l/2
Wconstrained 
l/2
 d (W ) hh G  
''
1
l / 2
2
dx
l / 2
hh1G ' '
l/2

2
dx
Wconstrained
l / 2

Wunconstrained
hh11 '' L o 2
G*  G 'iG' '  G ' 1  i 
*  'i' '  ' 1  i 
G ' '  G '
Viscoelastic Damping
' '  '
22

23. Constrained Layer Damping
'
  poisson's ratio
21   
for VEM ,  0.5
G' 
G' 
'
3

G' '  G' 
' ' '

3
3
G' '
 1/ 3
' '
Wcon
1
Ratio 

 2 dx
2 
Wuncon 3l o  l / 2
l/2

2 h2  2 o  sh Asin / 2   sin cos / 2  

3 h1 G l sin 
cosh   cos 


o  sh Asin / 2   sin cos / 2  
  0.124
l sin 
cosh   cos 


v  1    45o
h2
1
h1
R
2
 104
G
2
* 1 * 104 * 0.124  1000
3
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)
Viscoelastic Damping
23

24. Constrained Layer Damping
2-A general base structure
3-longitudinal vibration
beam
2w
 max .ofVEM  d 2

2w
M   2

 max  cons tan t   0
For the beam:Energy dissipated
Viscoelastic Damping
24

25. Constrained Layer Damping
l/2
l/2
 d (W ) hh G  
Wconstrained 
''
2
1
l / 2
dx
l / 2

* 
 
  l 
h1ch * 
 2 
 0*sh
W  hh1G ' '  *
2
l/2

2
d Wxx
l / 2
2
sh2 (  / *)
dx
2
ch (l / *)
Exercise
Show that the above composite has
t t
r rh
* 3
 1  re rh  3(1  rr ) 2 e *
1 1
1  re rh
*


re  2  2 (1  i )  re (1  i )
1
1
*
where : 
rh 
h2
h1
Viscoelastic Damping
*
re 
2
1
25

26. Constrained Layer Damping
And show that:-


re rh 3  6rh  4r  2re rh  re rh
2

3
2
4

(1  re rh ) 1  4re rh  6re rh  4re rh  re rh
2
3
2
4
2

Take;-
rh 
h2
1
h1
re  3.585 * 10 4
  0.00502 2
  0.00519 2
Kinematics of CLD
h2
Wx
2
h
U   U 3  3 Wx
2
U   U1 
U   U A  (U 3  U1 )  (
Wx   
h1  h3
)Wx
2
U U A
h2
Viscoelastic Damping
26

27. Active Constrained layers damping



U   U   h 2W 
h2
(U 1  U 3 )  (
h1 h3
  h)W 
2 2
h2
U1 U 3 h
 W
h2
h2
Active Constrained layers damping
Viscoelastic Damping
27

28. Active Constrained layers damping
* U   U   0 
2
2
* 
h1h2 2
G*
   L/2

U
0
X
Solution procedure
*solve for U
*determine γ
W  G ' 'h2 2
Compute *
W   Wdx Compute *
*put in dimensionless form η
Viscoelastic Damping
28

29. Active Constrained layers damping
For ACLD
   passive  active
If controller fails---------------system still “fail-safe” because of passive damping

 p   0  ( p   d ) 0
t
Notes (viscoelastic)
if
   ' (1   i )
F   ' (1   i )    '    ' i 
if
   0 e iwt
 o  iw o e iwt  iw
' 0
F  
  Felastic  Fdamping
w
'
2 / 
Energy dissipated per cycle 

0
dx
Fd
dt 
dt
2 / 

0
 '

 o 2 dt
for   0 sin t
2 / 
Energy  W 

 '
0
potential Energy 
W
 2 
We

 2  0 2 cos 2 (t )dt 
2 '
2
2
  o    ' o
2
1 ' 2
  0  We
2
W

2We
W
 specific damping capacity
We
Viscoelastic Damping
29

30. Active Constrained layers damping
Viscous Damping
Fd  C o
2 / 
 C
W  dissipated energy 
o
 o dt
0
 C  o
2
2 / 
2
 cos
2
t dt
0


2
2
C 2  o  C o

Equivalent viscous damping to viscoelastic material
 '  C
C
 '

C  '
1
damping ratio   

Co
 2  '

2
'

2
at resonance   

2
2-Transeverse Vibration
Kinematics equation:
 
U1  U 3 h
 Wx
h2
h2
h
h1  h3
 h2
2
U=longitudinal deflection of base structure
Viscoelastic Damping
30

31. Active Constrained layers damping

=shear angle
Wx=slope of deflection line
h2   U 1  hW 
U 1  h2   hW 
U 1  h2   hW   1
=F 1h1U 1 x Force on top layer per unit width =
dF
 1h1U1  G *  shearstress  2
d
1h1 (h2   hW )  G *
 
G*
h

  W
1h1h2
h2
let
 1  1h1  longitudinal Rigidity
  
G*
h
  W
1
h2
NOTE
Bending in beam:
( ) *  Dt  Dt (1   i)
*
Equation of motion;
Dt W   m 2W  0
*
W  
m 2
Dt
*
W 0
W    B W  0
*4
Viscoelastic Damping
31

32. Active Constrained layers damping
where  B  bending wave number
W  W0 e i ( wt  B  )
*
one propagation solution
W
( D * t / m) 1 / 4
let
 *B 

W 1/ 2


  B (1  i )
4
( Dt / m)1 / 4 (1  i  )1 / 4
*
B
W  W0 e i ( wt  B  ) e (  B  / 4 )
W  W0 e (  B / 4 ) 
Energy  CW 2  CW0 e (  B / 2) 
2
d energy
2   B
 CW0 (
)e
dx
2
  B

2
d Energy/dx
  B

Energy
2
 2 d Energy/dx
C 
 constrained layers assembly
B
Energy
solution for  C ;
* put W  W0 e i ( wt  
* solve
 
*
B )
G*
h

  W 
 1 h2
h2
for 
* calculate d energy/dx   G'  v h2  2
Using the solution given in ''Damping of flexural waves by constrained layers ''
Journal of acoustic society of America, Vol 31, 7 pp952-962, 1959
Viscoelastic Damping
32

33. Active Constrained layers damping
W   i *3 BW0 e i B  e iWt
*
W   i B W
*3
 
G*
h

   B 3 iW
 1 h2
h2
 
ih B
W


G*
h2 1 

*2
( B  1 2 ) 

check that it satisfies equation
Summary
Loss factor for constrained layer damping during transverse vibration
WD dissipated Energy

W
Elastic Energy
WD
loss factor   
2 W
specifi damping 
2-for beam in bending
Equation of motion
Viscoelastic Damping
33

34. Active Constrained layers damping
( ) *  Dt  Dt (1   i)
*
Equation of motion;
Dt W   m 2W  0
*
W  
m 2
Dt
*
W 0
W    B W  0
*4
1/ 4
 mW 2 
where  *B  bending wave number  
 D*

 t 

*
B
W 1/ 2


  B (1  i )
1/ 4
1/ 4
4
( Dt / m) (1  i  )
W  W0 e i ( wt  B  ) e (  B  / 4 )
W  W0 e (  B / 4 ) 
 B  wave number of constrained layer sassembly without losses


Energy  CW 2  C W0ei ( wt   B  ) e B / 2  Ce B / 2
2
denergy / dx Ce B / 2 ( B / 2)

 ( B / 2)
energy
Ce B / 2
3-calculate loss factor of CLD
C 
 2 d Energy/dx
B
Energy
4- For 3 layers CLD
Viscoelastic Damping
34

35. Active Constrained layers damping
 
U1  U 3 h
 Wx
h2
h2
h
h1  h3
 h2
2
If
U3=0 a-
 
U1 h
 W
h2 h2
U1  h2  hW
U   h2   hW
Quasi-static Equilibrium
Longitudinal load on layer2=shear load
 1
d   (db)


U 
  G *   h1 1  h1  1
 h1  1U 


from geomtry;
(h1 * b)
U 1  h2    hW 
Viscoelastic Damping
35

36. Active Constrained layers damping
1h1 h2   hW   G * 
1h1h2
 hh
   1 1 W  
G*
G*
G *
h
  
 W
1h1h2 h2
let
1h1   1
  
G*
h
  W 
 1 h2
h2
W  W0 e i ( wt  
*
B )
W   i *3 BW0 e i
one propagation solution
*
B
e iWt
W   i B W
*3
  
G*
h
   B 3 iW
 1 h2
h2
It has a solution;
 
ih B
W


G*
h2 1 

*2
 ( B  1 2 ) 
check that it satisfies equation
Viscoelastic Damping
36

37. Active Constrained layers damping
  G'  v h2 2 Energy dissipated per unit length
Energy in bending waves
W  W0 e i ( wt  
*
B )
W  W0 sin( wt   B * )
W 0  W0  cos(t   B * )  linear velocity
W 0   W0  B sin(t   B * )  Anguler velocity
W    B *W0 cos(t    * )
W     W0 sin(t    * )
*2
moment    Dt *W   Dt *  *2 W0 sin(t    )

  Dt *  *3W0 cos(t    )

W 0
power  FW 0  

shear  F  
 W 2 0   *3 Dt * cos2  Dt *   W 2 o sin 2  W0   *3 Dt
*3
2
Energy  power * 2 / 
 -2  *3 DtWo2

   h2G ' 'V  2

2
2  3 DtWo2

const .layer
(h 2 / Dt ) g
 V
1  g 2
Viscoelastic Damping
37

38. Active Constrained layers damping
g
G*
 shear parameter
 *2 1 2

constr
is Max. when
constr
0
g
g optimum  1
NOTE
 1  1h1
what is the physical meaning of g ,
if
W  0  CLD in long.vibration
 
U1  U 3 hW

h2
h2
 
U1
h2
Also
G*
 0
 1h2
   o e  G*/ 1h2 
  
Viscoelastic Damping
38

39. Active Constrained layers damping
U  U oe
Uo
e
U
 G*

 h
1 2



G * /  1 h2 

e  1


 1h2
G*
1
g  *2 2
 B e
e 
* 
B
let
g
2
g

2
2 2  e2
bendin gwave length
shear wave length
b- if U3=0
 
U1  U 3 h
 W
h2
h2
where U1 & U 3 are dependent
in order to have
1h1U1  3h3U 3   0
F1

F3
0
 1U1   3U 3   0
where
U3 
 
 1  1h1
 3  3h3
 1
U1
3
1  1 /  3
h
U1  W
h2
h2
Viscoelastic Damping
39

40. Active Constrained layers damping
 1   3 

U1  h2  hW
 

3



2 
 3h2
 3h
 U1 

W 

 
1   3
1   3


  1 

h1    eqm.of top layer
  
 1U1 
h1  G * 

 1U1   G * 
But
 1 3h2
h
   1 3 W  G*
1   3
1   3
  
G * ( 1   3 )
h
  W
 1h3h2
h2
Follow same procedure as case of U3=0 to get
 constr  V
( 1 h 2 / Dt )( g / 1  g ) 2
 1
g 
1



 3 1 g 
2
Summary
1- Longitudinal vibration
-to find optimum length of constraining layers
Viscoelastic Damping
40

41. Active Constrained layers damping
(Following plunket &lec. paper)
Loptimum
 3.28
B*
G*
1
B* 

h1h2 E2 characterstic length
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
Viscoelastic Damping
41

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