Optimization Computational Model for Piezoelectric Energy - - PowerPoint PPT Presentation

Agostinho Matos, José Guedes, K. Jayachandran, Hélder Rodrigues Contact: ago.matoz@gmail.com

11/09/2014 Instituto Superior Técnico

Optimization Computational Model for Piezoelectric Energy Harvesters Considering Material Piezoelectric Microstructure

Motivation

Nowadays there are many sources of free energy: a) Natural Energy – wind, waves, solar, etc b) Human Technology – engines, industrial machines, etc Many of the energy sources cause mechanical

• vibrations. A piezoelectric material can convert

vibrations to power Real world applications can have various types of loadings

Motivation

Applications & More...

Motivation

A piezofiber composite plate

• f 2.2 𝑑𝑛3 produces 120

mW Now in 2014 it can be done 1.73e10 computations per mWh. To deliver power it is not enough... It is necessary to deliver the required power...

Piezoelectric Constitutive Equations & Others

𝑇 = 𝑇𝐹 𝑈 + 𝑒 𝑈 𝐹𝑙 𝐸 = 𝑒 𝑈 + 𝜁𝑈 𝐹𝑙 The electric current goint out the electrode (𝑇𝜚) is: 𝐽 = −𝑅𝑓 𝑅𝑓 = −𝑜𝑗𝐸𝑗𝑒𝑇

𝑇𝜚

For a Resistor, the harvested power: 𝑄

𝑏 = 1 2 𝑆 𝐽 2

Piezoelectric Problem Equations

Constitutive Equations 𝑈

𝑘𝑗,𝑘 = 𝜍𝑣𝑗

𝐸𝑗,𝑗 = 0 𝑇𝑗𝑘 =

𝑣𝑗,𝑘+𝑣𝑘,𝑗 2

; 𝐹𝑗 = −𝜚,𝑗 Electric Machine Equations, for a Resistor V=RI Boundary Conditions: 𝜚 = 𝜚 𝑝𝑜 𝑇𝜚 (electroded part) 𝐸

𝑘𝑜𝑘 = 0 𝑝𝑜 𝑇𝐸 (not electrodes)

𝑈𝑗𝑘𝑜𝑗 = 𝑢𝑘 𝑝𝑜 𝑇𝑈 𝑣𝑗 = 𝑣𝑗 𝑝𝑜 𝑇𝑣 𝑇 = 𝑇𝜚 ∪ 𝑇𝐸 = 𝑇𝑣 ∪ 𝑇𝑈

Piezoelectric Harvester Setup

Longitudinal Generator Transverse Generator Unimorph Cantilever Bimorph Cantilever

i) Yellow and Vi surfaces are electrodes; ii) Dark blue is substrate and light blue is a piezoelectric iii) Orange vector P indicates polarization or z-direction

Non-Ressonance Results

The electrical power of one resistance is 𝑄

𝑏

For the bimorph similar expressions to unimorph;

Harvester 𝑄

𝑏

Loading

Longitudinal Generator 1 2 𝑆 𝑥𝑒 3,3 𝜏𝑚𝑞𝐵

2

Pressure Transverse Generator 1 2 𝑆 𝑥𝑒 3,2 𝜏𝑢𝑞𝐵

2

Pressure Cantilever Unimorph 1 2 𝑆 𝑥𝑒 3,2 𝜏𝑏𝑞𝐵

2

Tip Bending Moment

Piezo Materials

Piezo Materials : PZT-5H and BaTiO3 - are transversely isotropic (IEEE format)

𝑻𝑭 in 1e- 12 m^2/N S11 S12 S13 S33 S44 S66 PZT-5H 16.5 -4.78 -8.45 20.7 43.5 42.6 BaTiO3 7.38 -1.39 -4.41 13.1 16.4 7.46 d in 1e-12 C/N d31 d33 d15 PZT-5H

• 274

593 741 BaTiO3

• 33.7

93.9 561 𝜻𝑼 in 8.85e- 12 F/m 𝜻𝟐𝟐 𝜻𝟒𝟒 PZT-5H

• 274

593 BaTiO3

• 33.7

93.9

For substrate it is used Brass

FEM Validation

It is compared the power results of the developed equations and ANSYS FEM results; power relative error is inferior to 8.5%

Configuration 𝑸𝒃𝟏 (pw) 𝑸𝒃𝑼𝒊𝒇𝒑𝒔𝒛_𝟏 (𝒒𝒙) |RE (%)| L.G. 3.92e-3 3.92e-3 0.00 T.G. 5.05e-4 5.05e-4 0.00 Unimorph 3.23e-4 3.52e-4 8.24 Bimorph Series 4.79e-4 5.14e-4 6.81 Bimorph Parallel 1.92e-3 2.06e-3 6.80

Optimization Algorithm

The objective function : Max 𝑄

𝑏

The design variables : (𝜚, 𝜄, 𝜔) [313] for each piezoelectric material layer Constraints: (𝜚, 𝜄, 𝜔) 𝜗 [−180, 180] degrees Optimization method: simulated annealing

Setup Loadings – L.G. And T.G

Maximizing 𝑄

𝑏 is the same as maximizing piezoelectric

constants

Max d in 1e-12 C/N d31 d33 d34 d35 BaTio3 186 224 166 561 PZT 5H 274 593 48.5 741

10 MPa 10 MPa

Load Cases P: Load Cases PS:

All the loadings are harmonic 1Hz Load Cases for Longitudinal & Transverse Generators:

10 or 40 MPa Shear

Results– L.G. And T.G

Configura tion Plus Loading Condition Shear Load (MPa) Piezo Mat 𝑸𝒃𝟏 (pw) Time (min) 𝑶𝒇𝒘𝒃𝒎 𝝔𝒏𝒃𝒚 (deg) 𝜾𝒏𝒃𝒚 (deg) 𝝎𝒏𝒃𝒚 (deg) 𝑸𝒃𝒏𝒃𝒚 (pw) 𝑸𝒃𝒏𝒃𝒚 𝑸𝒃𝟏 P.1 – L.G.

• BaTiO3

3.92e-3 46.2 253

• 70

50

• 115

2.15e-2 5.5 P.2 – L.G.

• PZT-5H

1.56e-1 46.7 253 90 180 130 1.56e-1 1.0 P.3 – T.G.

• BaTiO3

5.05e-4 36.5 190

• 120
• 125

5 1.45e-2 28.7 P.4 – T.G.

• PZT-5H

3.33e-2 47.4 253

• 10
• 40

3.33e-2 1.0 Configura tion Plus Loading Condition Shear Load (MPa) Piezo Mat 𝑸𝒃𝟏 (pw) Time (min) 𝑶𝒇𝒘𝒃𝒎 𝝔𝒏𝒃𝒚 (deg) 𝜾𝒏𝒃𝒚 (deg) 𝝎𝒏𝒃𝒚 (deg) 𝑸𝒃𝒏𝒃𝒚 (pw) 𝑸𝒃𝒏𝒃𝒚 𝑸𝒃𝟏 PS.1 – L.G. 10 BaTiO3 3.92e-3 45.0 235 50 55 50 6.35e-2 16.2 PS.2 – L.G. 10 PZT-5H 1.56e-1 49.1 253

• 80

180

• 40

1.56e-1 1.0 PS.3 – T.G. 10 BaTiO3 5.05e-4 48.8 253

• 180

55

• 35

3.53e-2 70.0 PS.4 – T.G. 10 PZT-5H 3.33e-2 48.6 253

• 145

180

• 110

3.33e-2 1.0 PS.5 – L.G. 40 BaTiO3 3.92e-3 47.7 253

• 140
• 55
• 135

3.35e-1 85.4 PS.6 – L.G. 40 PZT-5H 1.56e-1 48.8 253 65 20

• 45

1.58e-1 1.0 PS.7 – T.G. 40 BaTiO3 5.05e-4 26.5 145 160 50 130 2.25e-1 445.7 PS.8 – T.G. 40 PZT-5H 3.33e-2 48.5 253

• 180

40 50 5.34e-2 1.6

Conclusion & Future Work

Non-ressonance with a resistance connected what is desired to increase in the case of a constant stress loading is the piezoelectric constants 𝑒𝑗𝑘; It is necessary to investigate if in ressonance the power will increase too as for out of ressonance When choosing a piezoelectric material for a specific application the loading type must be accounted The piezo material can be modelled as a polycrystallyne one

Homogenization & Future Work

A piezoelectric material has a crystalline microstructure. Each crystal or grain has its own orientation with its grain boundaries; the 3D orientation of each single crystal can be knowed using X-ray diffraction contrast tomography; Homogenization theory allows to calculate

• verall material properties based in

the microstructure

3D grains reconstruction

Homogenization & Future Work

The homogenization calculates

• verall

material properties of a composite microstructure Optimizing overall material d33 varying material

• rientation increases |d33| 114%

? Questions ?

Acknowledgements: This work is supported by the Project FCT PT DC/EME-PME /120630/2010

Bimorph Series and Parallel Connections