Large Amplitude Vibrations and Modal Sensing of Intelligent Thin Piezolaminated Structures

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18 Ιουλ 2012 (πριν από 5 χρόνια και 1 μήνα)

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WALLED
STRUCTURES
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Large Amplitude Vibrations and
Modal Sensing of Intelligent Thin
PiezolaminatedStructures
S. Lentzenand R. Schmidt
Insitutof General Mechanics, RWTH AachenUniversity
THIN
WALLED
STRUCTURES
EURODYN 2005, PARIS

MRT FOSD theory of shells
•Numerical procedure
•Modal array sensors
•Numerical examples
•Summary
Contents
THIN
WALLED
STRUCTURES
EURODYN 2005, PARIS
01
3ttt
VVV
=
+Θ⋅
%%%
()
012
2
33
α
βαβαβ
αβ
ε
εεε
=
+Θ⋅+Θ⋅
01
3
33
3
α
α
α
ε
εε
=
+Θ⋅
33
0
ε
=
(
)
2
ij
O
ε
ϑ
=
(
)
2
O
αβ
ω
ϑ
=
(
)
3
O
α
ω
ϑ
=
Assumptions:
•small strains:with
•small rotations about the normal:
•moderate rotation of the normal:
2
1
ϑ
Nonlinear FOSD theory of shells for small strains
and moderate rotations
Kinematichypothesis:
Green-Lagrange strain tensor:

THIN
WALLED
STRUCTURES
EURODYN 2005, PARIS
000
|3
b
αβ
αβ
αβ
ϕ
νν
=−
0000
||3
1
2
b
αβαββα
αβ
θ
ννν
⎛⎞
=+−
⎜⎟
⎝⎠
Strain-displacement relations
Tangential strains:
0000
1
2
αβαβ
α
β
ε
θϕϕ
=+
111000101
||
1
2
bbbb
λλλλ
α
βαββαλλ
λβλααβ
αββα
ε
ννϕϕϕνϕν
⎛⎞
=+−−++
⎜⎟
⎝⎠
21111
||
1
2
bbbb
λκλλ
αβλκλβλα
αβαβ
ενννν
⎛⎞
=−−
⎜⎟
⎝⎠
Transverse shear strains:
and
Linearisedtangential strains:
Linearisedrotations:
00110
3
1
2
λ
αα
αλ
εϕννϕ
⎛⎞
=++
⎜⎟
⎝⎠
α
111
3
|
1
2
λ
α
λ
α
ε
νν
=
u
uuuuuuu
000
3,
b
λ
α
λ
α
α
ϕ
νν
=+
and
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WALLED
STRUCTURES
EURODYN 2005, PARIS
{}
[]
{}
[
]
{
}
000
D
eE
εδ
=+
{}
[]
{}
[
]
{}
000
T
SceE
ε
=−
[
][][]
edc
=
[
]
[
]
[
]
TT
ecd=
Constitutive relations
Direct piezoelectric effect
Converse piezoelectric effect
with
and
and
{}{}{}{}
11
11
1
22
221
2
12
0012002
3
23
233
13
13
2
2
2
D
E
SDDEE
D
E
ε
σ
ε
σ
εε
τ
ε
τ
ε
τ
⎧⎫
⎧⎫
⎪⎪
⎪⎪
⎧⎫


⎪⎪
⎪⎪
⎪⎪
⎪⎪⎪⎪⎪⎪
====
⎨⎬⎨⎬⎨⎬⎨⎬
⎪⎪⎪⎪⎪⎪⎪⎪
⎩⎭
⎩⎭
⎪⎪⎪⎪
⎪⎪⎪⎪
⎩⎭
⎩⎭
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STRUCTURES
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State of Equilibrium
ie
WW
δ
δ
=
(
)
Q
iiiiiii
eiQ
VAA
WFARvdVvdADndA
σ
σ
δρδσδδϕ
=−−+−
∫∫∫
Further Assumptions
•Electric field is only existent in transverse direction
•Electric potential is homogeneous between an electrode pair
{
}
03
00
E
E
⇒=
%
{}{}{}{}
(
)
0
0
0000
TT
i
V
WSEDdV
δδεδ
=−

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STRUCTURES
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Finite Element Implementation
[]
{
}
[
]
{
}
{
}
{
}
ie
M
qDqFF
++=
&&&
{
}
{
}
ie
QQ
=
{
}
{
}
ii
me
FF+
{
}
{
}
ii
me
QQ
+
(1)
(2)
Central Differences
Implicit Methods (Newmark, …)
{}{}
{}
[]
{
}
{}
TT
iT
q
WqK
δδδϕ
ϕ


Δ


Δ=


Δ


⎩⎭
{
}
(
)
tt
tt
fq
ϕ


=
{}
(
)
,
,,,,
tttit
tt
qfqqFMD
−Δ

=
from (2)
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STRUCTURES
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Finite Element Implementation, cont’d
[
]
*
1T
Tqqqq
K
KKKK
ϕ
ϕϕϕ

=−
[]
qqq
T
T
q
KK
K
KK
ϕ
ϕ
ϕϕ


=




tangential stiffness matrix
after static condensation of {
φ
}:
1T
qq
KKK
ϕ
ϕϕϕ

can be fully occupied e.g.:
for an electrode pair covering the complete structure
Solution: decoupled iterations
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STRUCTURES
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1
n
iikjk
i
g
α
δ
=
=

[]
{
}{}
j
j
Ge
α
=
[
]
ki
Gg=
with
modal sensor signal
linear combiner
sensors
structure
φ
1
φ
2
φ
n
α
1
α
2
α
n
Modal Array Sensor Principle
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STRUCTURES
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Cantilevered Beam with PZT Sensors
70
GP
a
E
=
0.3
ν
=
3
2700
kg
m
ρ
=
3
2840
kg
m
ρ
=
2G
P
a
E
=
10
31
2.2
1
0
m
V
d

=⋅
0.3
ν
=
11
33
1.062
1
0
F
m
δ

=⋅
1
2
10
1
0
0

x

5

x

1

m
m
0.1 mm
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Cantilevered Beam with PZT Sensors under
Pressure Load
load [10
3
N/m
2
]
0
2
4
6
8
10
tip displacement [cm]
01
linear
nonlinear
2
123
45678910
0
-100
-200
sensor voltage [V]
linear
nonlinear
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Gain Factor Distributions
-1
1
0
max
12
3
4
5
6
7
8
9
10
α/α[-]
mode 1
mode 3
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FFT Analysis of the Modal Signals
02004006008001000
frequency [Hz]
10-3
10-2
10-1
mode 3
78.84 Hz
493.8 Hz
10-1
100
101
mode 1
linear
nonlinear
linear
nonlinear
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STRUCTURES
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Cantilevered Cylindrical Shell
R
=
3
0
0
m
m
sensor
actuator
1
5
0
mm
210
G
P
a
E
=
0.3
ν
=
3
7800
k
g
m
ρ
=
3
7600
k
g
m
ρ
=
63
GPa
E
=
10
31
1.79
10
m
V
d

=⋅
0.3
ν
=
8
33
1.65
10
F
m
δ

=⋅
0.002
ξ
=
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Cantilevered Cylindrical Shell
step line load 666.67 N/m for 1 ms
hoop/radial tip displacement [mm]
time [s]
Balamuruganet al.present
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STRUCTURES
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Eigenfrequenciesof the Clamped Cylindrical Shell
mode 1
[Hz]
mode 2
[Hz]
mode 3
[Hz]
mode 4
[Hz]
mode 5
[Hz]
mode 6
[Hz]
Balamuruganet al.
(1 sensor)
7.1611.5922.6163.2677.40172.7
Linear
(1 sensor)
7.0511.3422.1862.0475.79169.0
linear
(5 sensors)
7.0511.3422.2362.0475.95169.2
nonlinear 800 N
(5 sensors)
6.9711.8723.0764.3578.92174.2
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Sensor Voltage due to an Additional Hoop Line Force
of 6.67.10-2
N/m
time [s]
sensor voltages [ΔV]
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FFT Analysis of the Modal Signals
due to an Additional Hoop Line Force of 6.67.10-2
N/m
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Summary

A geometrically nonlinear Finite Element has been
presented to analysepiezolaminatedshells
•Some issues concerning the FE implementation have been
discussed
•Several issues concerning modal sensing of geometrically
nonlinear vibrations with modal array sensors are discussed
by means of some numerical examples
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