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 of 2.2 𝑐𝑚3 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

𝑆 = 𝑆𝐸 𝑇 + 𝑑 𝑇 𝐸𝑘 𝐷 = 𝑑 𝑇 + 𝜀𝑇 𝐸𝑘

The electric current goint out the electrode (𝑆𝜙) is:

𝐼 = −𝑄𝑒

𝑄𝑒 = −𝑛𝑖𝐷𝑖𝑑𝑆𝑆𝜙

For a Resistor, the harvested power: 𝑃𝑎 =1

2𝑅 𝐼 2

Piezoelectric Problem Equations

Constitutive Equations𝑇𝑗𝑖,𝑗 = 𝜌𝑢𝑖

𝐷𝑖,𝑖 = 0

𝑆𝑖𝑗 =𝑢𝑖,𝑗+𝑢𝑗,𝑖

2 ; 𝐸𝑖 = −𝜙,𝑖

Electric Machine Equations, for a Resistor V=RI

Boundary Conditions:

𝜙 = 𝜙 𝑜𝑛 𝑆𝜙 (electroded part)

𝐷𝑗𝑛𝑗 = 0 𝑜𝑛 𝑆𝐷 (not electrodes)

𝑇𝑖𝑗𝑛𝑖 = 𝑡𝑗 𝑜𝑛 𝑆𝑇

𝑢𝑖 = 𝑢𝑖 𝑜𝑛 𝑆𝑢

𝑆 = 𝑆𝜙 ∪ 𝑆𝐷 = 𝑆𝑢 ∪ 𝑆𝑇

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 𝑃𝑎

For the bimorph similar expressions to unimorph;

Harvester 𝑃𝑎 Loading

Longitudinal Generator 1

2𝑅 𝑤𝑑 3,3 𝜎𝑙𝑝𝐴

2 Pressure

Transverse Generator 1

2𝑅 𝑤𝑑 3,2 𝜎𝑡𝑝𝐴

2 Pressure

Cantilever Unimorph 1

2𝑅 𝑤𝑑 3,2 𝜎𝑎𝑝𝐴

2 Tip Bending Moment

Piezo Materials

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

𝜺𝑻 in 8.85e-

12 F/m 𝜺𝟏𝟏 𝜺𝟑𝟑

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 𝑷𝒂𝟎

(pw)

𝑷𝒂𝑻𝒉𝒆𝒐𝒓𝒚_𝟎

(𝒑𝒘)

|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 𝑃𝑎

The design variables : (𝜙, 𝜃, 𝜓) [313] for each piezoelectric material layer

Constraints: (𝜙, 𝜃, 𝜓) 𝜖 [−180, 180] degrees

Optimization method: simulated annealing

Setup Loadings – L.G. And T.G

Maximizing 𝑃𝑎 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

𝑷𝒂𝟎

(pw)

Time

(min) 𝑵𝒆𝒗𝒂𝒍

𝝓𝒎𝒂𝒙

(deg)

𝜽𝒎𝒂𝒙

(deg)

𝝍𝒎𝒂𝒙

(deg)

𝑷𝒂𝒎𝒂𝒙

(pw)

𝑷𝒂𝒎𝒂𝒙

𝑷𝒂𝟎

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 0 -40 3.33e-2 1.0

Configura

tion Plus

Loading

Condition

Shear

Load

(MPa)

Piezo

Mat

𝑷𝒂𝟎

(pw)

Time

(min) 𝑵𝒆𝒗𝒂𝒍

𝝓𝒎𝒂𝒙

(deg)

𝜽𝒎𝒂𝒙

(deg)

𝝍𝒎𝒂𝒙

(deg)

𝑷𝒂𝒎𝒂𝒙

(pw)

𝑷𝒂𝒎𝒂𝒙

𝑷𝒂𝟎

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 𝑑𝑖𝑗;

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

overall material properties based in

the microstructure

3D grains reconstruction

Homogenization & Future Work The homogenization calculates overall material properties of a composite microstructure

Optimizing overall material d33 varying material orientation increases |d33| 114%

? Questions ?

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

Bimorph Series and Parallel Connections