melih papila, [email protected] assessment of axisymmetric piezoelectric composite plate...
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Melih Papila, [email protected]
Assessment of Axisymmetric Piezoelectric Composite Plate Configurations for Optimum
Volume Displacement
Melih PapilaMultidisciplinary & Structural Optimization Group
Interdisciplinary Microsystems Group
2
Melih Papila, [email protected]
Acknowledgement
Guigin WangQuentin Gallas
Dr. Bhavani SankarDr. Mark SheplakDr. Lou Cattafesta
SPONSOR: NASA Langley Research Center
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Melih Papila, [email protected]
Design Problem: Piezoelectric composite
driver
Synthetic Jets Sound
generating/receiving devices MEMS PZT Microphone
Displacement actuators
Maximum volume displacementMaximum natural frequency
Orifice
Cavity
Oscillating Piezo-Composite Diaphragm
Net Flow
Applications
Gallas (2002)
4
Melih Papila, [email protected]
Volume displacement
Electric field
(Voltage )
PZT layer expands/contra
cts
Plate bends
Lateral deflection
w(r)
rdrrwVol )(2
piezoceramic
shim
V
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Melih Papila, [email protected]
Configurations
Outer ring
piezoceramic
shim
V
piezoceramic
shim
V
Inner disc
Bimorph
piezoceramic
shim
Unimorph
piezoceramic
shim
V
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Melih Papila, [email protected]
Objective
Investigate trade off between volume displacement and natural frequency via Pareto Optimization
“BEAT THE EXISTING DESIGN”
Find the optimum dimensions of the shim and piezoelectric layers in order to achieve
optimum volume displacement
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Melih Papila, [email protected]
Outline
Design problem & objective Analysis tool and verification Optimization of Circular Piezo-composite
plateDesign variablesObjective function and ConstraintsResults
Trade-off via Pareto Optimization: Volume displacement versus Natural Frequency
Concluding Remarks
8
Melih Papila, [email protected]
Analysis: Natural Frequency
0wP
)(rwCeq
Meq 0w
eqeq
natMC
f2
1
Mass Equivalent
Compliance Equivalent
:
:
wM
wC
eq
eq
20
20
2
2
wM
C
w
eq
eq
Energy Kinetic
Energy Potential
analytic
al
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Melih Papila, [email protected]
Analysis: Volume displacement
rdrrwVol P 0)(2
V
P
)(rw
analytic
al
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Melih Papila, [email protected]
Analysis: Lateral deflection Classical lamination theory
Each layer is isotropic, linear elastic, constant thickness
Bonding line and electrode layer are neglected
Equilibrium equations Constitutive equations
including piezoelectric effect Boundary and interface
matching conditions
32)3(
21)2(
1)1(
)(
)(
0)(
)(
RrRrw
RrRrw
Rr rw
rw
piezoceramic
shim
(3)
(2)
(1)
R1
R2
R3
Wang et al. (2002)Prasad et al. (2002)
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Melih Papila, [email protected]
Analysis: Model verification
Using scanning laser vibrometer
Piezoceramic
(PZT-5A) Shim
(Brass) Elastic Modulus (Pa) 6.301010 8.961010 Poisson’s Ratio 0.33 0.32 Density (kg/m3) 7700 8700
Rel. Dielectric Constant 1750 -
31d (m/V) -1.7510-10 - Allowable stress (Pa) 20106 200106
V10 mm
11.5 mm
0.20 mm0.23 mm
Gallas (2002)
fnat (Hz)3505
3542
Test
Analysis
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Melih Papila, [email protected]
Analysis: Model verification
unimorph
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 5 10 15 20
r (mm)
w (
mm
)
abaqus
analytical
17.6 mm18.5 mm
0.12 mm0.08 mm
16.9 mm
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Melih Papila, [email protected]
Outline
Design problem & objective Analysis tool and verification Optimization of Circular Piezo-composite
plateDesign variablesObjective function and ConstraintsResults
Trade-off via Pareto Optimization: Volume displacement versus Natural Frequency
Concluding Remarks
14
Melih Papila, [email protected]
Design Variables
R1 : radius of the inner PZT layer(s)
R2 : inner radius of the outer ring PZT layer(s)
ts : thickness of the shim and PZT layer(s)
tp : thickness of the PZT layers
(R3 : radius of the shim – fixed)
Lateral deflection of the composite plate determines volume displacement and operational frequency limit
ts
R1
R2
R3
tp
tp
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Melih Papila, [email protected]
Objective Function:Maximum Volume
displacementElectric field
(Voltage )
PZT layer expands/contra
cts
Plate bends
Lateral deflection
w(r)
Large PZT coverageSmall ts
rdrrwVol
P 0)(2
maximize
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Melih Papila, [email protected]
Formulation & Implementation
UBxLBthat such
all 0
0lim
natff
Solved by MATLAB
Optimization Toolbox
)(,,, 21
x Maximizex
Volps ttRR
ts
R1
R2
R3
tp
tp
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Melih Papila, [email protected]
R3 =11.5 mm, tmin = 0.076 mm
Maximum volume displacement
Parameter
Baseline Design
Inner-outer optimum
(unimorph)
Inner-outer optimum (bimorph)
2R (mm) 11.5 11.5 11.5
1R (mm) 10.0 10.86 10.51
st (mm) 0.150 0.076 0.076
pt (mm) 0.080 0.192 0.105
fnat (Hz) 2114 2114 2114
Vol x1010 (m3) 52.251 131.5 187.4
maxV (V) 94.488 226.5 123.8
Frequency Limit + + + Allowable Stress - - - Variable Bounds - + +
tmin
R1
R2
tp
tp
1.05 1.42
Bimorph/unimorphAmount of PZT
Volume displacement
18
Melih Papila, [email protected]
R3 =11.5 mm Maximum volume displacement
effect of lower bound, tmin
Parameter
Optimum (bimorph)
0.076
Optimum (bimorph)
0.025
2R (mm) 11.50 11.50
1R (mm) 10.51 11.30
st (mm) 0.076 0.025
pt (mm) 0.105 0.135
fnat (Hz) 2114 2114
Vol x1010 (m3) 187.4 266.1
maxV (V) 123.8 159.1
Frequency Limit + + Strength Limit - - Variable Bounds + +
1.38 1.42
0.025/0.076Amount of PZT
Volume displacement
19
Melih Papila, [email protected]
Outline
Design problem & objective Analysis tool and verification Optimization of Circular Piezo-composite
plateDesign variablesObjective function and ConstraintsResults
Trade-off via Pareto Optimization: Volume displacement versus Natural Frequency
Concluding Remarks
20
Melih Papila, [email protected]
Methodology:Pareto
Optimization
In multi-objective optimization problem with conflicting objectives
Pareto optimal points: one objective cannot be improved without deterioration in one of the other objectives,
Construct a Pareto hypersurface
minimized
maxim
ized
Objective 1
maximized
Ob
jecti
ve 2
maxim
ized
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Melih Papila, [email protected]
50.0
70.0
90.0
110.0
130.0
150.0
1400 1600 1800 2000 2200 2400 2600
f (Hz.)
Q*1
0^10
(m
^3)
(Hz.) natf
)(m x10 310Vol
R3 =11.5 mm : Pareto frontnatural frequency versus volume
displacement
baseline
Freq. , Vol. -
45% , 15%
Freq. , Vol.
23% , -15%
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Melih Papila, [email protected]
Analytical solutions allowed numerical optimization
Baseline designs were beaten and substantial improvement is predicted
Bimorph configuration without the outer ring offers the optimum performance
Minimum gauge for the layers is a limiting factor
Pareto optimization was used to understand tradeoff betweenMaximum volume displacementMaximum natural frequency
Concluding Remarks
23
Melih Papila, [email protected]
Prasad et al. (2002), “Two-Port Electroacoustic Model of an Axisymmetric piezoelectric Composite Plate,” 43rd AIAA Structures, Structural Dynamics and Materials Conference, Denver, CO.
Wang et al. (2002), “Analysis of a composite piezoelectric circular plate with initial stresses for MEMS,” ASME International Mchanical Engineering Congress, 2002, New Orleans, LA.
Gallas et al. (2003), “Optimization of Synthetic Jet Actuators,” AIAA Aerospace Sciences Meeting, 2003, Reno, NV.
Relevant work
THANK YOU…
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Melih Papila, [email protected]
R3 =11.5 mm Maximum volume displacement
effect of lower bound, tmin
Parameter
Optimum (unimorph)
0.076
Optimum (unimorph)
0.025
2R (mm) 11.50 11.50
1R (mm) 10.86 11.27
st (mm) 0.076 0.044
pt (mm) 0.192 0.224
fnat (Hz) 2114 2114
Vol x1010 (m3) 131.5 149.5
maxV (V) 226.5 264.2
Frequency Limit + + Strength Limit - + Variable Bounds + +
1.21 1.14
0.025/0.076Amount of PZT
Volume displacement
25
Melih Papila, [email protected]
Effect of oppositely polarized outer ring
R1
R2
R3
tpts
Ef Ef
R1=16.89, ts=0.081, tp=0.123
0.000
200.000
400.000
600.000
800.000
1000.000
1200.000
1400.000
16.500 17.000 17.500 18.000 18.500R2 (mm)
Qx1
0^10
(m
^3)
opposite polarization
same polarization
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Melih Papila, [email protected]
Results: R3 =18.5 mm Maximum volume displacement lower bound on t= 0.076 mm
Parameter
Baseline Design
Inner-outer optimum
(unimorph)
Inner-outer optimum (bimorph)
2R (mm) - 18.50 18.50
1R (mm) 12.50 16.84 16.36
st (mm) 0.100 0.076 0.076
pt (mm) 0.110 0.129 0.076
fnat (Hz) 632 632 673
Vol x1010 (m3) 561.3 995.3 1298.3
maxV (V) 129.9 152.2 90.3
Frequency Limit + + - Strength Limit - - - Variable Bounds - + +
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Melih Papila, [email protected]
Linear Theory – The piezoelectric term
The piezoelectric effect is added by using the following relation for generalized force resultants:
0P
r rP
N NA B
N N
0P
r rP
M MB D
M M
Where2
3111 12
3112 221
zPr
fPz
dQ QNE dz
dQ QN
23111 12
3112 221
zPr
fPz
dQ QME zdz
dQ QM
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Melih Papila, [email protected]
Piezoelectric Composite Plate Optimization Problem
Objective functionMaximum volume displacement
Design variablesShim Structural Variables : ThicknessPiezoelectric layer Variables : Radii and
thickness
Constraints Frequency LimitStrength LimitVariable Bounds
29
Melih Papila, [email protected]
Configurations
R1
R2
R3
tpts
Ef Ef
VAC
EfT = Ef = VAC /tp
positive
R1
R2
tp
tp
ts
Ef
-Ef
Ef
-Ef
VAC
EfB = -Ef = -VAC /tp
positive
EfT = Ef = VAC /tp