Piezoelectric Shunt Damping

Damping with Piezoelectric MaterialsMohammad Tawfik

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Damping with Piezoelectric Material

Mohammad Tawfik



Objectives

• General Introduction to smart materials and structures

• Recognize the nature of piezoelectric material

• Understand the use of passive shunt circuits

• Dynamics of structures with shunt piezoelectric materials



Smart Structures



Smart Structures: What?

• Controlled change in properties– Change in mechanical properties– Change in geometry

• Energy Converters!

– Mechanical Electrical (Piezoelectric)

– Heat Mechanical (SMA)– Mechanical Heat (Viscoelastic)– Etc…



Smart Structure: Why?

• Vibration Damping

• Shape Control

• Noise Reduction

• Vibration/Damage Sensing

• Heat Sensing



Smart Structures: Classification

Wada, Fanson, and Crawly



Piezoelectric Materials



What is Piezoelectric Material?

• Piezoelectric Material is one that possesses the property of converting mechanical energy into electrical energy and vice versa.



Piezoelectric Materials

• Mechanical Stresses Electrical Potential Field : Sensor (Direct Effect)

• Electric Field Mechanical Strain : Actuator (Converse Effect)

Clark, Sounders, Gibbs, 1998



Conventional Setting

Conductive Pole



Piezoelectric Sensor

• When mechanical stresses are applied on the surface, electric charges are generated (sensor, direct effect).

• If those charges are collected on a conductor that is connected to a circuit, current is generated



Piezoelectric Actuator

• When electric potential (voltage) is applied to the surface of the piezoelectric material, mechanical strain is generated (actuator).

• If the piezoelectric material is bonded to a surface of a structure, it forces the structure to move with it.



Other types of Piezo!



1-3 Piezocomposites

T 3=cE

33S 3+e33E 3

D3=e33S 3+εS

33E3



Active Fiber Composites (AFC)

ceff

11=c

E11

+v pe31

2

(vC ε 33+vp ε

S33)

eeff

31=ε33e31

vC ε33+vpε

S33

εeff

33=ε33 ε

S33

(vC ε33+vp ε

S33)



Applications of Piezoelectric Materials in Vibration Control



Collocated Sensor/Actuator



Self-Sensing Actuator



Hybrid Control



Passive Damping / Shunted Piezoelectric Patches



Passively Shunted Networks

Resonant

Capacitive Switched

Resistive



Adaptive Structures

Wada, Fanson, and Crawly

Passive Networks



How does it work?



Shunted Piezoelectric Material (Physical)



Shunted Piezoelectric Material (Physical)

•Mechanical energy is converted to electrical energy through piezoelectric effect•Electric charge is driven by potential difference through the circuit•Energy is dissipated in the resistance



Shunted Piezoelectric Material (Electric)



Shunted Piezoelectric Material (Energy)



Mechanical Impedance / Viscoelastic Analogy

Z11RES

=1−k31

2

1+iρ3

Z11RSP=1−k 31

2 δ2

γ 2+δ 2rγ+δ2

Resistor Shunt

R-L Shunt

r=RC ωn (dissipation tuning parameter )

γ=sωn

( complex non-dimensional frequency )

δ=ωe

ωn

( resonant shunted piezoelectric frequency tuning parameter )



Viscoelastic analogy

• The model of the shunted piezoelectric patches, in many researches, is reduced to an equivalent of a viscoelastic patch.

• But Piezoelectric patches are elements that respond to the total strain rather than the local strain!



The Problem With Viscoelastic Analogy!

Base structure



Modeling of Piezoelectric Structures



Constitutive Relations

• The piezoelectric effect appears in the stress strain relations of the piezoelectric material in the form of an extra electric term

• Similarly, the mechanical effect appears in the electric relations

S=s11T +d 31E

D=d 31T 1+¿33E



Constitutive Relations

• ‘S’ (capital s) is the strain• ‘T’ is the stress (N/m2)• ‘E’ is the electric field (Volt/m)• ‘s’ (small s) is the compliance; 1/stiffness

(m2/N)• ‘D’ is the electric displacement, charge per

unit area (Coulomb/m)• Electric permittivity (Farade/m)¿



The Electromechanical Coupling

• d31 is called the electromechanical coupling factor (m/Volt)



Manipulating the Equations

D=QA

D=1A∫ Idt=

IAs

• The electric displacement is the charge per unit area:

• The rate of change of the charge is the current:

• The electric field is the electric potential per unit length: E=

Vt



Using those relations:

• Using the relations:

• Introducing the capacitance:

• Or the electrical admittance:

S=s11T +d 31

tV

I=Ad 31 sT 1+A∈33 s

tV

I=Ad 31 sT 1+CsV

I=Ad 31 sT 1+YV



For open circuit (I=0)

• We get:

• Using that into the strain relation:

• Using the expression for the electric admittance:

V=−Ad 31 s

YT 1

S=s11T−Asd 31

2

tYT 1

S=s11(1−d 31

2

¿33 s11)T 1



The electromechanical coupling factor

• Introducing the factor ‘k’:

• ‘k’ is called the electromechanical coupling factor (coefficient)

• ‘k’ presents the ratio between the mechanical energy and the electrical energy stored in the piezoelectric material.

• For the k13, the best conditions will give a value of 0.4

S=s11(1−k312 )T 1



Different Conditions

• With open circuit conditions, the stiffness of the piezoelectric material appears to be higher (less compliance)

• While for short circuit conditions, the stiffness appears to be lower (more compliance)

S=s11(1−k312 )T 1 == sDT 1

S=s11T=sET



Different Conditions

• Similar results could be obtained for the electric properties; electric properties are affected by the mechanical boundary conditions.



Damping of Structural Vibration with Piezoelectric Materials and

Passive Electrical NetworksN. W. HAGOOD AND A. VON

FLOTOWJournal of Sound and Vibration (1991)

146(2), 243-268



The Constitutive Relations for Piezoelectric Materials

• The constitutive relation is:

• Where:



Constitutive relation (cont’d)

• The electric permiativity:

• Electromechanicalcoupling:

• Mechanicalcompliance:



Electrical relation

• Into constitutive relations:

• Where the capacitance is:



The Electric admittance

• Introducing the electric admittance:

• Generally; with a shunt circuit:



The Electromechanical Model



Remember

• The electric admittance is the reciprocal of the electric impedance.

• Also, you may have up to three circuits:

Z EL=1

Y EL



From Constitutive Relations

• The voltage may be written as:

• Into the strain equation



The Electromechanical Compliance

• The Electromechanical Compliance

• Or

• Where

• Generally:



The Mi matrices



Uniaxial Loading Cases

• The compliance:• Introducing the

electromechanical coupling coefficient:

• The compliance becomes:

• For open circuit conditions




• For open circuit, the compliance becomes:

• A similar expressions for capacitance in case of zero stress is:




• Introducing the mechanical impedance:(Which is the reciprocal of the compliance)

• We may write the non-dimensional mechanical impedance:



The Complex Modulus

• Now, let’s reintroduce the complex modulus of the viscoelastic material:

• Where:



Resistive Shunt Example

• For the case or resistive shunting, the resistance and the capacitance are in parallel 1

Z EL=

1

ZD+

1

Z SU=Cs+

1R

=RCs+1

R




• Recall:

• Using the previous results:

• Simplifying:

Z jjME

=Z jjRES

=1−k ij

2

1−k ij2( RCs

1+RCs )

Z jjRES

=1−k ij

2

1+(1−k ij2 )RCs




• Substituting s=iω and

• Introducing the non-dimensional parameter ρ=RCω, we get:

• Finally:

Z jjRES=1−

k ij2

1+(1−k ij2 ) ρi

=1−k ij

2

1+(1−k ij2 )

2ρ2

+k ij

2 (1−k ij2) ρi

1+(1−k ij2 )

2ρ2

Z jjRES

=(1−k ij

2

1+(1−k ij2)

2ρ2 )(1+

k ij2 ρ

1+(1−k ij2 ) ρ2

i)




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10

E Current

Eta Current

E von Flotto

Eta von Flotto



Homework #11

1. Derive the equations for the RL shunt circuit.

2. Plot the frequency response of piezoelectric bar with a shunt circuit (R & RL)



RL Shunt Example

• For the case or RL shunting, the resistance and the inductance are in series and are in parallel with the capacitance

1

Z EL=

1

ZD+

1

Z SU=Cs+

1Ls+R

=LCs2+RCs+1

Ls+R

Z EL=LCs2+RCs

LCs2+RCs+1



RL Shunt Example

• Recall:

• Using the previous results:

• Simplifying:

Z jjME

=Z jjRSP

=1−k ij

2

1−k ij2( LCs2+RCs

LCs2+RCs+1)

Z jjRSP=

(1−k ij2 )(LCs2+RCs+1)

1+(1−k ij2 )(LCs2+RCs )



RL Shunt Example

• Ignoring (1-k2) in the denominator:

Z jjRSP=1−k ij

2

ωe2

ωn2

s2

ωn2+ωe

2

ωn2RC ωn

sωn

+ωe

2

ωn2

Z jjRSP=1−k ij

2 δ 2

γ2+δ2 rγ+δ 2



RL Shunt Example

Z jjRSP=

(1−k ij2 )( s

2

ωe2+RC

ωn

ωn

s+1)1+(1−k ij

2 )( s2

ωe2+RC

ωn

ωn

s)• Using the full

term:

Z jjRSP=

(1−k ij2 )( s

2

ωn2+ωe

2

ωn2RC

ωn

ωn

s+ωe

2

ωn2 )

ωe2

ωn2+(1−k ij

2)( s2

ωn2+ωe

2

ωn2RC

ωn

ωn

s)Z jjRSP=

(1−k ij2 )(γ 2+δ2rγ+δ2 )

δ2+(1−k ij

2 )(γ2+δ2 rγ )



RL Shunt Example: Hagood results



Comparing results



Modulus of Elasticity

• Recall that:

• And:

• Where:



Modulus of Elasticity

• Substituting:

• Getting the stiffness:

• Simplifying:

s jjSU=

A j

Z jjSU L j s

E jjSU

=Z jjSU L j s

A j

=Z jj

DZ jjME L j s

A j

E jjSU

=Z jj

ME

s jjD

=Z jj

ME

s jjE (1−k ij

2 )



Finite Element Model



Recall

• Recall the constitutive relations of Piezoelectric materials:

• Rearranging the terms:

S1=s11T 1+d 31 E3

D3=d 31T 1+¿33E3

T 1=1

s11

D S1−h31 D3

E3=−h31S 1+1

¿33SD3



Where:

h31=d 31

s11 ∈33(1−k312 )

¿33s =¿33(1−k 31

2 )s11D=s11(1−k 31

2 )



Potential Energy

• Writing the expression for the potential energy of the shunted piezoelectric material:

U=12∫0

l

S 1T 1 Adx+12∫0

l

D3E 3Adx

U=12∫0

l

S 1( S1

s11D−h31D3)Adx+ 1

2∫0

l

D3(−h31 D3+D3

¿33S )Adx



The interpolation functions

u ( x )=(1− xl )u1+( xl )u2=⌊N ( x ) ⌋{ue}

d ( x)=(1−xl )d 1+( xl )d 2=⌊N (x) ⌋ {d e}

S1 (x )=du ( x)dx

=(−1l )u1+(1

l )u2=⌊N x ( x)⌋ {ue }



The Stiffness Matrices

k e=1

s11D∫

0

l

{N x}⌊ N x ⌋ Adx

k D=1

¿33S ∫

0

l

{N }⌊N ⌋Adx

k eD=−h31∫0

l

{N x}⌊N ⌋Adx



Kinetic Energy

T=12∫0

l

ρ u2Adx

me= ρA∫0

l

{N }⌊N ⌋dx



External Work

W=∫0

l

VDbdx=∫0

l

(L I+RI )Dbdx=∫0

l

(LQ+RQ )Dbdx

¿∫0

l

A ( L D+R D )Dbdx

mD=AbL∫0

l

{N }⌊N ⌋dx cD=AbR∫0

l

{N }⌊N ⌋dx



Element Equation

[me 0

0 mD ]{ueD e

}+[0 00 cD ]{

ueDe

}+[ k e k eDkDe kD ]{ueDe

}={ f0 }

Piezoelectric Shunt Damping

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