Viscoelastic Damping lecture notes

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

42. <Bibliography
Bibliography
A dynamic function formulation for the response of a beam structure to a moving mass. Foda, M. A.
and Abduljabbar, Z. 1998. 1998, Journal of Sound and Vibration, Vol. 210, pp. 295-306.
A HYSTERESIS MODEL FOR THE FIELD-DEPENDENT DAMPING FORCE OF A MAGNETORHEOLOGICAL
DAMPER. LEE, S.-B. CHOI AND S.-K. 2001. 2001, Journal of Sound and Vibration, p. 9.
A Review of Power Harvesting from Vibration Using Piezoelectric Materials. Sodano, Henry A.,
Inman, Daniel J. and Park, Gyuhae. 2004. 2004, The Shock and Vibration Digest, Vol. 36, pp. 197205. DOI: 10.1177/0583102404043275.
A Review of Power Harvesting Using Piezoelectric Materials (2003-2006). Anton, Steven R. and
Sodano, Henry A. 2007. 3, 2007, Smart Materials and Structures, Vol. 16, pp. R1-R21. DOI:
10.1088/0964-1726/16/3/R01.
A Review of the State of the Art in Magnetorheological Fluid Technologies - Part I: MR fluids and MR
fluid models. Goncalves, Fernando D., Koo, Jeong-Hoi and Ahmadian, Mehdi. 2006. 3, 2006, The
Shoack and Vibration Dijest, Vol. 38, pp. 203-219. DOI: 10.1177/0583102406065099.
A.V. Srinivasan, D. Michael McFarland. 2001. Smart Structures analysis and design. Cambridge :
Press Syndicate Of The University Of Cambridge, 2001.
Aero-Thermo-Mechanical Characteristics od Shape Memory Alloy Hybrid Composite Panels with
Geometric Imperfections. Ibrahim, H., Tawfik, M. and Negm, H. 2008. Cairo : s.n., 2008. 13th
International Conference on Applied Mechanics and Mechanical Engineering. 27-29 May 2008, Cairo,
Egypt.
Application of the theory of impulsive parametric excitation and new treatments of general
parametric excitation problems. Hsu, C. S. and Cheng, W. H. 1973. 1973, Journal of Applied
Mechanics, Vol. 40.
Benedetti, G. A. 1973. Transverse vibration and stability of a beam subject to moving mass loads.
Civil Engineering, Arizona State University. 1973. PhD Dissertation.
Coussot, Philippe. 2005. RHEOMETRY OF PASTES,SUSPENSIONS AND GRANULAR MATERIALS. s.l. :
John Wiley & Sons, Inc., 2005.
Dave, Dr. 2004. Dr. Dave's Do It Yourself MR Fluid. s.l. : Lord Corporation, 2004.
Dynamic behavior of beam structures carrying moving masses. Saigal, S. 1986. 1986, Journal of
Applied Mechanics, Vol. 53, pp. 222-224.
Dynamic deflection of cracked beam with moving mass. Parhi, D. R. and Behera, A. K. 1997. 1997,
Proceeding of the Institution of Mechanical engineering, Vol. 211, pp. 77-87.
Viscoelastic Damping
42

43. <Bibliography
Dynamic Modeling of Semi-Active ER/MR Fluid Dampers. Xiaojie Wang, Faramarz Gordaninejad.
2001. Reno, NV : s.n., 2001. Wang, X. and Gordaninejad, F., “Dynamic Modeling of Semi-Active
ER/MR Fluid Dampers,”. p. 10.
Dynamic Stability and Response of a beam subject to deflection dependent moving load. Katz, R., et
al. 1987. 1987, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 109, pp. 361-365.
Dynamic stability of a beam carrying moving masses. Nelson, H. D. and Conover, R. A. 1971. 1971,
Journal of Applied Mechanics, Vol. 38, pp. 1003-1006.
Dynamic stability of a beam excited by a sequence of moving mass particles. Makhertich, S. 2004.
2004, Journal of the Acoustical Society of America, Vol. 115, pp. 1416-1419.
Dynamic stability of a beam loaded be a sequence of moving mass particles. Benedetti, G. A. 1974.
1974, Journal of Applied Mechanics, Vol. 41, pp. 1069-1071.
Dynamic Stability of Stepped Beams Under Moving Loads. Aldraihem, O. J. and Baz, A. 2002. 2002,
Journal of Sound and Vibration, Vol. 250, pp. 835-848.
Dynamics and Stability of Gun-Barrels with Moving Bullets. Wagih, A., et al. 2008. Cairo : s.n., 2008.
13th International Conference on Applied Mechanics and Mechanical Engineering. 27-29 May 2008,
Cairo, Egypt.
Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets. Tawfik, M. 2008. 2008,
Advances in Acoustics and Vibration, Vol. 2008. Article ID 483857, 6 pages.
doi:10.1155/2008/483857.
—. Tawfik, Mohammad. 2008. Article ID 483857, 2008, Advances in Acoustics and Vibration, Vol.
2008, p. 6 pages. doi:10.1155/2008/483857.
Efficient computation of parametric instability regimes in systems with large number of degrees-offreedom. Kochupillai, J., Ganesan, N. and Padnamaphan, C. 2004. 2004, Finite elements in Analysis
and Design, Vol. 40, pp. 1123-1138.
Elmy, Amed O. 2007. Application of MR fluids in Vibration Damping. Engineering and Material
Science, German University in Cairo. 2007. BSc Thesis.
Energy conservation in Newmark-based time integration algorithms. Krenk, Steen. 2006. 2006,
Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 6110-6124.
Energy Harvesting from a Backpack Instrument with Piezoelectric Shoulder Straps. Granstorm,
Jonathan, et al. 2007. 5, 2007, Smart Materials and Structures, Vol. 16, pp. 1810-1820. DOI:
10.1088/0964-1726/16/5/036.
Experimental and Spectral Finite Element Study of Plates with Shunted Piezoelectric Patches. Tawfik,
M. and Baz, A. 2004. 2, 2004, International Journal of Acoustics and Vibration, Vol. 9, pp. 87-97.
Finite element analysis of elastic beams subjected to moving dynamic loads. Lin, Y. H. and
Trethewey, M. W. 1990. 1990, Journal of Vibration and Acoustics, Vol. 112, pp. 323-342.
Viscoelastic Damping
43

44. <Bibliography
Finite element vibration analysis of rotating Timoshenko beams. Rao, S. and Gupta, R. 2001. 2001,
Journal of Sound and Vibration, Vol. 242, pp. 103-124.
Finitel element analysis of an elastic beam structure subjected to a moving distributed mass train.
Rieker, J. R. and Trethewey, M. W. 1999. 1999, Mechanical Systems and Signal Processing, Vol. 13,
pp. 31-51.
G. Magnac, P. Meneroud, M.F. Six, G. Patient, R. Leletty, F. Claeyssen. CHARACTERISATION OF
MAGNETO-RHEOLOGICAL FLUIDS. Meylan, France : s.n.
G. Yang, a B.F. Spencer,a Jr., J.D. Carlsonb and M.K. Sainc. Large-scale MR fluid dampers: modeling,
and dynamic performance considerations. Indiana, USA : s.n.
Gravatt, John W. 2003. Magneto-Rheological Dampers for Super-Sport Motorcycle Applications. 8th
MAY 2003.
Guangqiang Yang, B.S., M.S. 2001. LARGE-SCALE MAGNETORHEOLOGICAL FLUID DAMPER FOR
VIBRATION. December 2001.
Impulsive parametric excitation: Theory. Hsu, C. S. 1972. 1972, Journal of Applied Mechanics, Vol.
39, pp. 551-558.
Instability of vibration of a mass that moves uniformly along a beam on a periodically
inhomogeneous foundation. Verichev, S.N. and Metrikine, A.V. 2003. 2003, Journal of Sound and
Vibration, Vol. 260, pp. 901-925.
Instability of vibration of a moving-train-and-rail coupling system. Zheng, D. Y. and Fan, S. C. 2002. 2,
2002, Journal of Sound and Vibration, Vol. 255, pp. 243-259.
Kallio, Marke. 2005. The elastic and damping properties of magnetorheological elastomers.
Vuorimiehentie : JULKAISIJA – UTGIVARE – PUBLISHER, 2005.
Kelly, S. G. 2000. Fundamentals Of Mechanical Vibrations. s.l. : Mc-Graw-Hill Higher Education, 2000.
Lai, W H Liao and C Y. 2002. Harmonic analysis of a magnetprheological da,per for vibration control.
Shatin, NT, Hong Kong : s.n., 5th April 2002.
Li Pang, G. M. Kamath, N. M. Werely. ANALYSIS AND TESTING OF A LINEAR STROKE
MAGNETORHEOLOGICAL DAMPER. Maryland, USA : s.n.
Linear Dynamics of an elastic beam under moving loads. Rao, G. V. 2000. 2000, Journal of Vibration
and Acoustics, Vol. 122, pp. 281-289.
MAGNETO-RHEOLOGICAL FLUID DAMPERS MODELING:NUMERICAL AND EXPERIMENTAL. N. Yasreb,
A. Ghazavi, M. M.Mashhad, A. Yousefi-koma. 2006. Montreal, QC, Canada : s.n., 2006. The 17th
IASTED International Conference MODELLING AND SIMULATION. p. 5.
Malkin, Alexander Ya. Rheology Fundamentals. Moscow : ChemTec Publishing.
Viscoelastic Damping
44

45. <Bibliography
Moving-Loads-Induced Instability in Stepped Tubes. Aldrihem, O. J. and Baz, A. 2004. 2004, Journal
of Vibration and Control, Vol. 10, pp. 3-23.
Non-Linear dyanmics of an elastic beam under moving loads. Wayou, A. N. Y., Tchoukuegno, R. and
Woafo, P. 2004. 2004, Journal of Sound and Vibration, Vol. 273, pp. 1101-1108.
Nonlinear Panel Flutter with Temperature Effects of Functionally Graded Material Panels with
Temperature-Dependent Material Properties. Ibrahim, H. H., Tawfik, M. and Al-Ajmi, M. 2,
Computational Mechanics, Vol. 41. DOI:10.1007/s00466-007-0188-4.
Norris, James A. Behavior of Magneto-Rheological Fluids Subject to Impact and Schock Loading.
OPTIMAL DESIGN OF MR DAMPERS. Henri GAVIN, Jesse HOAGG and Mark DOBOSSY. 2001. Seattle
WA : s.n., 2001. U.S.-Japan Workshop on Smart Structures for Improved Seismic Performance in
Urban Regions. p. 12.
P.Edwards. Mass-Spring-Damper Systems.
Phenomenological Model of a Magnetorheological Damper. B.F. Spencer Jr., S.J. Dyke, M.K. Sain
and J.D. Carlson. 1996. 1996, ASCE Journal of Engineering Mechanics, p. 23.
Phenomenological Model of a Magnetorheological. Spenser, Jr., B. F. 1996. 1996, Journal of
Engineering Mechanics, Vol. 23.
Poynor, James. Innovative Designs for Magneto-Rheological Dampers.
Random Response of Shape Memory Alloy Hybrid Composite Plates Subject To Thermo-Acoustic
Loads. Ibrahim, H. H., Tawfik, M. and Negm, H. M. 2008. 3, 2008, Journal of Aircraft, Vol. 44. DOI:
10.2514/1.32843.
Rashaida, Ali A. 2005. FLOW OF A NON-NEWTONIAN BINGHAM PLASTIC FLUID OVER A ROTATING
DISK. 2005.
Response of periodically stiffened shells to moving projectile propelled by internal pressure wave.
Ruzzene, M. and Baz, A. 2006. 2006, Mechanics of Advanced Materials ans Structures, Vol. 13, pp.
267-284.
Semi-analytic solution in time domain for non-uniform multi-span Bernoulli-Euler beams transversed
by moving loads. Martinez-Castro, A. E., Museros, P. and Castillo-Linares, A. 2006. 2006, Journal of
Sound and Vibration, Vol. 294, pp. 278-297.
Singh, V. P. 1997. Mechanical vibrations. Delhi : DHANPAT RAI & CO., 1997.
Solution of the moving mass problem using complex eigenfunction expansions. Lee, K. Y. and
Renshaw, A. A. 2000. 2000, Journal of Applied Mechanics, Vol. 67, pp. 823-827.
Stochastic Finite Element Analysis of the Free Vibration of Functionally Graded Material Plates.
Shaker, A., et al. 2008. 3, 2008, Computational Mechanics, Vol. 44, pp. 707-714. DOI
10.1007/s00466-007-0226-2.
Viscoelastic Damping
45

46. <Bibliography
Stochastic Finite Element Analysis of the Free Vibration of Laminated Composite Plates. Shaker, A., et
al. 2008. 4, 2008, Computational Mechanics, Vol. 41, pp. 493-501. DOI:10.1007/s00466-007-0205-7.
The Structural Response of Cylindrical Shells to Internal Moving Pressure and Mass. Baz, A., Saad
Eldin, K. and Elzahabi, A. 2004. Cairo : s.n., 2004. Proceeding of 11th International Conference on
Applied Mechanics and Mechanical Engineering (AMME). pp. 720-734.
The use of finite element techniques for calculating the dynamic response of structures to moving
loads. Wu, J-J, Whittaker, A. R. and Cartmell, M. P. 2000. 2000, Computer and Structures, Vol. 78,
pp. 789-799.
Thermal Buckling and Nonlinear Flutter Behavior of FunctionallyGraded Material Panels. Ibrahim, H.
H., Tawfik, M. and Al-Ajmi, M. 2007. 5, 2007, Journal of Aircraft, Vol. 44, pp. 1610-1618.
DOI:10.2514/1.27866.
Thermal post-buckling and aeroelastic behaviour of shape memory alloy reinforced plates. Tawfik,
M., Ro, J-J. and Mei, C. 2002. 2, 2002, Smart Materials and Structures, Vol. 11, pp. 297-307.
doi:10.1088/0964-1726/11/2/313.
Transient vibration analysis of high-speed feed drive system. Cheng, C. C. and Shiu, J. S. 2001. 3,
2001, Journal of Sound and Vibration, Vol. 239, pp. 489-504.
Vibration analysis of beams with general boundary conditions transveresed by a moving force. Abu
Hilal, M. and Zibdeh, H. S. 2000. 2, 2000, Journal of Sound and Vibration, Vol. 229, pp. 377-388.
Vibration analysis of continuous beam subjected to moving mass. Ichikawa, M., Miyakawa, Y. and
Matsuda, A. 2000. 3, 2000, Journal of Sound and Vibration, Vol. 230, pp. 493-506.
Vibration and stability of axially loadded beams on elastic foundation under moving harmonic loads.
Kim, Seong-Min. 2004. 2004, Engineering Structures, Vol. 26, pp. 95-105.
Vibration Attenuation in A Periodic Rotating Timoshenko Beam. Alaa El-Din, Maged and Tawfik,
Mohammad. 2007. Cairns : s.n., 2007. ICSV14.
Vibration Attenuation in Rotating Beams with Periodically Distributed Piezoelectric Controllers. Alaa
El-Din, Maged and Tawfik, Mohammad. July 2006. Vienna, Austria : s.n., July 2006. 13th
International Congress on Sound and Vibration.
Vibration Characteristics of Periodic Sandwich Beam. Badran, H., Tawfik, M. and Negm, H. 2008.
Cairo : s.n., 2008. 13th International Conference on Applied Mechanics and Mechanical Engineering.
27-29 May 2008, Cairo, Egypt.
Vibration Control of Tubes with Internally Moving Loads Using Active Constrained Damping. Ro, J-J,
Saad Eldin, K. and Baz, A. 1997. Dallas, TX : s.n., 1997. ASME Winter Annual Meeting.
Vibration Control Using Smart Fluids. Sims, Neil D., Stanway, Roger and Johnson, Andrew R. 1999.
3, 1999, The Shock and Vibration Dijest, Vol. 31, pp. 195-203. DOI:10.1177/058310249903100302.
Villarreal, Karla A. EFFECTS OF MR DAMPER PLACEMENT ON STRUCTURE VIBRATION PARAMETERS.
Tokyo, Japan : s.n.
Viscoelastic Damping
46

47. <Bibliography
Wave attenuation in periodic helicopter blades. Tawfik, M., Chung, J. and Baz, A. 2004. Amman,
Jordan : s.n., 2004. Jordan International Mechanical engieering Confernce.
What Makes a Good MR Fluid? Carlson, J. David. 2002. 4, 2002, Journal of Intelligent Material
Systems and Structures, Vol. 13, pp. 431-435. DOI: 10.1106/104538902028221.
Zhang, G Y Zhou and P Q. 2002. Investigation of the dynamic mechanical behavior of the doublebarreled configuration in a magnetorheological fluid damper. Hefei, Anhui, China : s.n., 5th April
2002.
Viscoelastic Damping
47