Andrzej

Tylikowski

Warsaw University of

Technology

The 9-th International Congress

on

SOUND AND VIBRATION

July 8-11, 2002, Orlando, FL.

Dynamics of

Dynamics of

Delaminated Elements

Delaminated Elements

of Smart Structures

of Smart Structures

Outline

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Motivations
Previous results
Effects of piezoactuator delamination
Problem formulation
Dynamic equations of a beam with piezoactuators
Steady state vibrations excited by piezoactuators
Results
Delamination in rotating shaft with thermoactive fibers
Problem formulation
Shape memeory alloys
Stability analysis
Results

Assumptions

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Linearity

Edge delamination of constant width

Delamination reduces effective
bending stiffness

Damping included

Previous Studies

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002



Study on the steady state vibrations of beams with delaminations

(1988) Mujumdar, Suryanarayan

 Influence of actuator delamination on vibration control

(1992) Sung Jin Kim, Jones J. D.

 Delamination in composite rotors (1997) Wettergren H. L.

 Flexural vibrations of rotating shafts with delaminations

(2002) Tylikowski A.

Constitutive Equations

of Piezoelectricity

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002









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

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















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

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













0

0

0

0

0

0

0

0

0

0

0

0

0

}

{

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

}

{

15

15

33

31

31

T

66

44

44

33

13

13

13

11

12

13

12

11

d

d

d

d

d

d

s

s

s

s

s

s

s

s

s

s

s

s

s

ij

ij

Actuator Delaminations

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Delamination

x

1

x

3

l

PZT

beam

x

2

1 2 3 4

Delamination

x

1

x

3

l

beam

x

2

1 2 3 4 5

x

4

One sided

Two sided

PZT

Delamination Effects

Delamination

x

1

x

3

l

PZT

beam

x

2

1 2 3 4

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Delamination Effects

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Delamination Effects

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Hybrid Rotating Shaft

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Hybrid Shaft Configuration

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Stability Definition

9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002



























,.)

(

,.)

(

t

w

w

t 0

0

0

0

Liapunov stability

0







,.)

(

lim

t

w

t

Liapunov asymptotic stability

SMA Properties

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Shaft Dynamics Equations

0

)

2

(

2

2

2

2

0

)

(

2

2

2

0

)

(

2

2

,

2

2

2

22

,

22

,

66

12

,

11

,

22

,

12

,

2

,

,

,

22

,

22

,

66

,

66

12

,

2

,

,

,

12

,

66

12

,

66

,

11

,

,















































































yy

yyyy

xxyy

xxxx

y

x

e

t

t

tt

y

yy

xx

xy

e

t

t

tt

x

xy

yy

xx

t

tt

w

R

w

R

a

w

d

w

d

d

w

d

v

R

a

u

R

a

v

w

w

v

w

w

R

a

v

a

v

a

u

a

a

w

v

v

w

v

w

R

a

v

a

a

u

a

u

a

u

u











9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

Stability Analysis



 

 











dD

w

R

w

d

w

d

w

w

d

w

d

v

u

a

R

w

v

u

a

R

w

v

a

u

a

w

v

w

v

u

w

w

v

v

u

u

V

y

xy

yy

yy

xx

xx

x

y

y

x

y

x

D

t

t

t

t

t

t

]

2

4

2

2

2

)

(

2

2

2

2

2

1

2

,

2

2

2

,

66

2

,

22

,

,

12

2

,

11

2

,

,

66

,

,

12

2

,

22

2

,

11

2

2

2

2

,

2

,

2

,

2

,

2

,

2

,



















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

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































9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

2

/

1

V

w 

Stability Analysis



 











dD

w

R

w

d

w

d

w

w

d

w

d

v

u

a

R

w

v

u

a

R

w

v

a

u

a

w

v

v

w

w

v

u

dt

dV

y

xy

yy

yy

xx

xx

x

y

y

x

y

x

D

t

t

t

]

4

2

2

)

1

(

2

2

,

2

2

2

,

66

2

,

22

,

,

12

2

,

11

2

,

,

66

,

,

12

2

,

22

2

,

11

2

2

2

2

,

2

,

2

,







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

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































9-th Congress on Sound and Vibrations , Orlando, July 8-11, 2002

2

















e

i

i









Damping ratio

Stability Analysis

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

2





dD

w

dD

w

d

w

d

w

w

d

w

d

D

D

xy

yy

yy

xx

xx













2

2

,

66

2

,

22

,

,

12

2

,

11

4

2



Bending stiffnesses dependent on x











dD

R

w

R

R

a

R

w

v

a

v

R

a

R

w

u

a

v

u

a

u

a

dt

dV

y

y

x

y

x

x

D

]

2

)

1

(

2

2

2

2

2

2

2

22

,

22

2

,

2

2

22

,

12

,

,

12

2

,

11



















































Stability Analysis

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Quadratic form with respect to

R

w

v

u

y

x

,

,

,

,







































2

2

2

22

22

12

22

2

2

22

12

12

12

11

)

1

(

R

R

a

a

a

a

R

a

a

a

a

a

Sylvester conditions

Results

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Results

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Conclusions

9-th Congress on Sound and Vibrations , Orlando, July 8-12, 2002

Developed dynamic models predict the response
of composite structures with interlayer separations

Increase of edge delamination length decreases
active control performance

Presented analytical models can be used to
fast introductory analysis of smart structures with
delaminations