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
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
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
,
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
,
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