Cantilevered Piezoelectric Energy Harvester – Modeling and Simulation

XIANGZHEN SUN

IntroductionResearch Topic: Harvesting Energy from ambient

Energy Source: Mechanical Vibration

Material Selection: Piezoelectric materials

Piezoelectric effect:

1. Generate an AC (alternating current) voltage when subjected to mechanical stress or vibration.

2. Vibrate when subjected to an AC voltage, or both.

Usage: Radio wave transmission, remote sensor, transducer, powering small electronic components

Model Description

Electromechanical Modeling of cantilevered piezoelectric harvesterPZT: composite of piezoelectric and ceramic, improved mechanical properties, high temperature resistant Substructure: protection, vibration magnitude restraintVibration decomposition: small rotation and longitude translation Circuit Model: voltage source + load resistance

Modeling Methods 1. Coupled Single Degree of Freedom (SDOF)

Transduction Mechanism of vibration – to – electric:

Electromagnetic, Electrostatic, Piezoelectric

• Magnetic seismic mass motion, w.r.t., Generator Housing

• 2 degree of freedom coupled

• Issue: Contradict to experimental result, can not be

governed by merely one equation

𝑚�̈�+𝑐 �̇�+𝑘𝑧=−𝑚�̈�

Modeling Methods 2. Improved SDOF Energy Harvester Model

Core Idea: Motion Decomposition

Transverse Vibrations

Longitude Vibrations

Governing Equation:

Two correction factors, correcting the base excitation amplitude with a single mode dependent of the distributed mass.

𝑚�̈�+𝑐 �̇�+𝑘𝑧=−𝜇1𝑚 �̈�𝑚 �̈�+𝑐 �̇�+𝑘𝑧=− 𝜅1𝑚�̈�

Modeling Methods Continue …

Pros:

As indicated, correction coefficient, representive of

the excitation amplitude, w.r.t., distributed mass,

has the right tendency.

Cons:

• Highly dependent on the accuracy of distributed

mass

• It is extremely hard to estimate distributed mass

accurately.

• Susceptible to estimation error

Modeling Methods 3. Single-Mode Equation based on Distributed Parameter Solution

Core Idea: Instead of using ODE or ODE groups to approximate vibrations, this method makes use of the PDE w.r.t. length ‘x’ & time ‘t’ to govern the motion of vibration

Vibration states, determined by separation of variable (SOP):

Pros: Simplified PDE problem

Shape function along the cantilevered beam:

(x, t) W(x)e j tw

1cosh( ) cos( )

( ) {[cosh( x) cos( x)] [sinh( x) sin( x)]}sinh( ) sin( )

b b

b b

L LW x C

L L

Modeling Methods Continue …

Time – average power generated:

2 2 2 2 231

2334(1 bL )

b h e AP RR

Modeling Methods Distributed Parameter Electromechanical Model, with dimensionless modes

Still, SOP is used here to simplify PDE equation, however, the way to separate variables is far more advanced:

(represented by an absolutely and uniformly convergent series of the eigenfunctions)

This time, eigenvalues are governed by the following eigenfunction:

1

( , ) (x) (t)rel r rr

w x t

1( ) cosh( ) cos( ) (sinh( ) sin( )r r r rr rx x x x x

mL L L L L

Modeling Methods Continue …

If we compare the eigenfuntions in these two successive methods:

(1)

(2)

We can see the form of eigenfunctions are similar, but the previous one only have one mode

, the latter one has numerous.

Modeling Methods Continue …

Transcendental Trigonometry Equation of Characteristic:

How to solve it?

solve( )………………………………….. only one solution in a single region

fzero( )…………………………………… solution is randomly provided in a selected region

I used “variable steps method”:

//Pseudo codes//

Equ = cos(x) * cosh( x ) + 1;

cos( )cosh( ) 1 0

Modeling Methods Continue …

x0 = 1.8; h = 0.001; h1 = 3; count = 0; error = 0.01;

For x = x0 – h1 : h : x0 + h1

count = count + 1;

If ( Abs (Equ) < = error )

X( i ) = x

If ( 50 < x < 120 )

x0 = x0 + 5

If ( x >= 120 )

x0 = x0 + 10

Modeling Methods Dimensionless Frequency Number from equation:

We can see the solution of λ became repetitive from the 75th one, so I can use 75 modes to

approximate the dimensionless modes

cos( ) cosh( ) 1 0

Modeling Methods To support my idea of using 75 modes to approximate the dimensionless modes, let’s have a look at the Dimensionless quotient in the eigenfunction formula:

, where

We can see from the figure that the

dimensionless quotient starts to slightly

fluctuate around 1 after the 4th value

There is no need to solve infinite solutions!

sinh( ) sin( )cosh( ) cos( )

r rr

r r

Modeling Methods Mass Normalized Eigenfunction:

Observation:

fluctuation starts to be stable when length

of the beam exceeds 40 mm

Modeling Methods How to get to the Power vs. frequency plot …

Observation:

When load resistance increases the natural

frequency of the system does not shift with it.

Several resonant frequency appears : not single

2 21

20

2 21

2( )1

2

wr r

r r r rj t

cr r

r r r r c

jmjv t

jjY ej

Modeling Methods Simulation result, compared with single mode result:

Thank you