Simply Supported Plate

Simulation of Multiphysic-Problemsusing Comsol Multiphysics

Dominik M. Brunner

Term Paper

reference number: 08-060

Mentoring:

Florian Bachmann

Prof. Paolo Ermanni

18.12.2008

Simulation of Multiphysic-Problems using Comsol

Multiphysics

Dominik M. Brunner

Submitted to ETH Zurich as a Term Paper

December 2008

Abstract

In this thesis a passive shunt damping simulation of a at plate with Comsol Multi-

physics is presented. The geometry of the propfan blade was simplied to a at plate

in order to reduce the complexity of the model at this early stage of the project. A

couple of static analyses were conducted and a comparison with the analytical results

performed so as to validate the Comsol simulations. In this model the piezoceramic

was integrated to investigate the inuence of the boundary conditions on the static

deformation. A modal analysis was carried out with Comsol and compared to the

analytical solution. Then the models for the modeling of an open and a closed circuit

piezoelectric system were set up and the eigenfrequencies determined. Using those

results the analytical solution for the investigated resistive shunt and the resonant

RL shunt were obtained. In a further Comsol model the piezoelectric element was

coupled with a passive damping shunt by using the 'SPICE circuit editor' and the

optimal shunt damping parameters were identied. Apart from small deviations a

good correlation with the analytical solution was noted. Furthermore a model for

the uid-structure model in Comsol is presented but was not possible to complete

within the given time frame.

iv

Simulation von Multiphysic-Problemstellungen mit

Comsol Multiphysics

Dominik M. Brunner

Semesterarbeit

Dezember 2008

Zusammenfassung

In dieser Semesterarbeit wird die Modellierung eines 'passive damping shunts' mit

Comsol Multiphysics vorgestellt. In diesem frühen Stadium des Projektes wurde

die komplexe Geometrie des Rotorblattes auf eine ebene Platte reduziert. Einige

statischen Analysen wurden durchgeführt und die Ergebnisse mit den analytis-

chen Berechnungen verglichen, um die Simulationen von Comsol zu verizieren.

In diesem Modell wurde das Piezoelement in die Struktur integriert, um den Ein-

uss der Randbedingungen auf die statischen Verschiebungen zu bestimmen. In der

Modalanalyse wurde wie bei der statischen Analyse ein einfaches Comsol Modell mit

den analytischen Ergebnissen verglichen. Dann wurden die Modelle für den oenen

und geschlossenen piezoelektrischen Schaltkreis erstellt und die Eigenfrequenzen bes-

timmt. Mit diesen Resultaten wurden die analytischen Ergebnisse nach Hagood und

von Flotow für den untersuchten 'resistive shunt' und den 'resonant RL shunt' bes-

timmt. In einem weiteren Comsol Modell wurde das Piezoelement mit dem 'passive

damping shunt' unter Verwendung des 'SPICE circuit editor' verbunden und die op-

timalen Dämpfungsparameter bestimmt. Abgesehen von einer kleinen Abweichung

war eine gute Übereinstimmung mit den analytischen Berechnungen festzustellen.

Darüber hinaus wurde ein Modell für die Fluid-Struktur Interaktion erstellt, jedoch

nicht fertiggestellt da dies den Rahmen dieser Arbeit sprengen würde.

vi

Contents

Abstract iii

Zusammenfassung v

1 Introduction 1

1.1 DREAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 COMSOL Multiphysics . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Fundamentals of piezoelectricity . . . . . . . . . . . . . . . . . . . . . 3

1.4 Passive shunt damping . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Resistive shunt . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.3 Resonant RL shunt . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Static analysis of a at plate 10

2.1 Analytical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 FEM calculations with Comsol . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Modal analysis 14

3.1 Analytical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 FEM calculations with Comsol . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Eigenfrequency analysis . . . . . . . . . . . . . . . . . . . . . 15

3.2.1.1 Aluminum plate . . . . . . . . . . . . . . . . . . . . 15

3.2.1.2 GFRP plate . . . . . . . . . . . . . . . . . . . . . . . 15

vii

Contents viii

3.2.1.3 Open circuit system . . . . . . . . . . . . . . . . . . 15

3.2.1.4 Closed circuit system . . . . . . . . . . . . . . . . . . 16

3.2.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2.1 Pure resistive shunt . . . . . . . . . . . . . . . . . . 16

3.2.2.2 Resonant RL shunt . . . . . . . . . . . . . . . . . . . 19

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Eigenfrequency analysis . . . . . . . . . . . . . . . . . . . . . 19

3.3.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Fluid-structure interaction 23

5 General conclusion and outlook 25

Bibliography 27

Appendix 28

A Piezoelectric Equations 28

A.1 Charge-Strain Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.2 Field-Strain Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.3 Charge-Stress Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.4 Field-Stress Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

B Material Properties 30

List of Figures

1.1 DREAM project outline[1] . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Piezoelectric eect[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Passive piezoelectric shunt damping techniques[8] . . . . . . . . . . . . 6

1.4 Resistive shunt [7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Resonant RL shunt [7] . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 GFRP plate and piezoelectric element . . . . . . . . . . . . . . . . . . 11

2.2 Static deformation of GFRP fabric and PIC 255 . . . . . . . . . . . . 13

3.1 Model geometry settings . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 SPICE circuit editor . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Electric boundary conditions for SPICE connection . . . . . . . . . . 18

3.4 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . 24

ix

List of Tables

2.1 Static deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Analytical and simulated eigenfrequencies of an aluminum plate . . . 19

3.2 Comparison open/closed circuit system . . . . . . . . . . . . . . . . . 20

3.3 Undamped tip displacement . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Optimal shunting parameters for maximal damping . . . . . . . . . . 21

3.5 Tip deformation at rst eigenfrequency . . . . . . . . . . . . . . . . . 21

x

Chapter 1

Introduction

1.1 DREAM

The acronym DREAM stands for valiDation of Radical Engine Architecture

systeMs and entitles an EU project coordinated by Rolls-Royce[1]. The project

is about developing new engine technologies and concepts for aerospace application.

The main goals of the program are to reduce fuel consumption, CO2 emissions, air

pollution and to keep the noise levels within an acceptable range. The project fo-

cuses on engine technology integration and validation, conducted on component and

system levels. It's emphasis lies in testing alternative fuels, passive and active con-

trol of aerodynamics and vibrations, contra-rotating open rotors with variable pitch,

innovative engine structures with added functionality and other new turbo machin-

ery. The participating Swiss consortium consists of the Swiss Federal Laboratories

for Materials Testing and Research (EMPA), the Ecole Polytechnique Fédérale de

Lausanne (EPFL) and the Swiss Federal Institute of Technology Zurich (ETHZ),

which are looking at three concepts of passive vibration damping. As shown in g-

ure 1.1, this is part of SP3 Direct Drive Open Rotor Snecma Aero & acoustics of

direct open rotor. The ETH Zurich-ST research contribution for DREAM is carried

out by Florian Bachmann within his dissertation. It includes the simulation of the

passive, shunted damping of propfan blades using piezoceramics and the experimen-

tal validation. Previous research on this topic was performed by B. Bratschi within

his master's thesis at ETH Zurich. In his thesis he tried to solve the uid-structure

1

1.2. COMSOL Multiphysics 2

interaction with ANSYS. The coupled eld simulations with the propfan blade could

not be conducted, since, due to a bug in the ANSYS licensing system. Thus it was

decided to solve the problem with Comsol Multiphysics which is accomplished in

this thesis. It was agreed to reduce the complexity of the propfan blade and to model

a at plate in the beginning. The main goals of this term paper were to model a

shunted damping system in Comsol and to integrate the piezoelectric element and

the electric circuit in the uid-structure interaction.

Figure 1.1: DREAM project outline[1]

1.2 COMSOL Multiphysics

Comsol Multiphysics is a nite element software package for a variety of engineering

and physics applications. Apart from nite element calculation the implementation

of coupled or multiphysic phenomena is possible in a simple manner. Comsol is based

on the partial dierential equation (PDE) toolbox of MATLAB and an interface

between the programs exists. Several application-specic modules are available for

COMSOL Multiphysics:

AC/DC Module

1.3. Fundamentals of piezoelectricity 3

Acoustics Module

CAD Import Module

Chemical Engineering Module

Earth Science Module

Heat Transfer Module

Material Library

MEMS Module

RF Module

Structural Mechanics Module

1.3 Fundamentals of piezoelectricity

Piezoelectricity is dened as a change in electric polarization with a change in applied

mechanical stress (direct piezoelectric eect), or the change in strain of a free crystal

when the applied electrical eld varies (inverse piezoelectric eect) [2]. The direct

eect can be used to convert mechanical into electrical energy, and vice versa for

the indirect eect. These eects are displayed in gure 1.2(b), (c) and (d), (e)

respectively.

Figure 1.2: Piezoelectric eect[2]

The tension along the direction of the polarization or the compression perpen-

dicular to the polarization generates a voltage with polarity opposite to that of the

poling voltage. The direct eect is also used in sensing applications, where defor-

mations of the material results in a change of voltage that can be detected. The

1.3. Fundamentals of piezoelectricity 4

inverse eect is used in actuation applications, such as in motors and devices that

precisely control positioning, and in generating sonic and ultrasonic signals. Fur-

thermore this property can be taken advantage of for damping purposes, where the

produced energy is dissipated in an electric circuit.

In the following paragraph, the governing equations of linear piezoelectricity are

introduced according to the IEEE Standard on Piezoelectricity[3],

[D

S

]=

[εT

dT

d

sE

] [E

T

]

where [D] ([C/m2]) denotes the vector of dielectric displacement, [S] ([−]) the

vector of strains, [E] ([V/m]) the vector of electrical eld and the vector of mechan-

ical stress[4]. The four complete forms of the equations can be found in Appendix

A. In piezoelectric materials stress and strain are related through the mechanical

compliance matrix[sE], where ()E denotes a measurement under constant electric

eld. The compliance matrix can be written as

[sE] =

sE11 sE12 sE13 0 0 0

sE12 sE22 sE23 0 0 0

sE13 sE23 sE33 0 0 0

0 0 0 sE44 0 0

0 0 0 0 sE55 0

0 0 0 0 0 sE66

.

The permittivity matrix

εT =

εT1 0 0

0 εT2 0

0 0 εT3

relates the electrical eld and the electrical displacement, where the superscript

()T stands for values that are taken under constant stress. The subscript ()t denotes

the matrix transpose. For εT either the dimensionless, relative values εT/ε0, [−] are

1.4. Passive shunt damping 5

considered, or the values εT ,[Fm

]. The matrix

d =

0 0 0 0 0 d15 0

0 0 0 0 d15 0 0

d31 d31 d33 d33 0 0 0

couples the mechanical and electrical equations.

1.4 Passive shunt damping

1.4.1 Introduction

In lightweight structures the integration of piezoceramic materials for passive shunt

damping is becoming popular, as it is an eective way to suppress vibrations[8].

The piezoelectric device transforms mechanical deformation energy into electrical

energy. This electric potential is then connected to an electrical impedance, which

is dissipating energy during vibration and thereby damps the overall system. There

exist various ways to build resonant shunt damping circuits. They consist of stan-

dard electrical elements as resistors, capacitors, inductors, operational ampliers and

switches. An overview is depicted in gure 1.3. The advantage of a passive shunt is

that the structure can be integrated into the system together with the piezoceramic

itself. No sensor or actuator devices or power supplies are needed and therefore the

system is operating completely autonomously.


Figure 1.3: Passive piezoelectric shunt damping techniques[8]

1.4.2 Resistive shunt

In the case of a resistive shunt, the electrical inductance comprises only a resistor

as displayed in gure 1.4.

Figure 1.4: Resistive shunt [7]


Upon loading the structure with a force Tj the inicted strain Sj results in an

electrical potential dierence Vi. According to the work of N. W. Hagood and A.

von Flotow[7], the maximum loss factor is

ηResmax jj =k2ij

2

√1− k2

ij,

with a nondimensional frequency of

ρi = RijCSPiω =

√1− k2

ij.

The square of the coecient kij denotes the percentage of mechanical strain energy

that is converted into electrical energy and indicates the eciency of the energy

transduction. The coecient is dened as

kij =dij√sEjjε

Ti

with the force in the j th direction and the applied electrical eld in the ith direction.

The electromechanical coupling coecient

CSPi = (1− k2

ij)CTPi

is the capacity of the material at constant strain, where

CTPi =

AiεTi

L3

denotes the capacity under constant stress. Given the material parameters and the

geometry of the structure, the resistor for optimal damping at the frequency ω is

R =L

ωAεT√

1− d2

sEεT

.

1.4.3 Resonant RL shunt

A resonant RL shunt is similar to a pure resistive shunt. The dierence is that the

piezoelectric's inherent capacitance is connected in parallel with a shunting resistor


R and a shunting inductance L, thus creating a resonant circuit. The circuit is then

tuned to maximize the energy dissipation of the resistor and the modal damping for

the desired mode.

Figure 1.5: Resonant RL shunt [7]

The optimal circuit damping is found to be using an electrical damping ratio ropt

and an optimal tuning parameter δopt of

ropt =√

2K2ij(

1 +K2ij

) and δopt =√

1 +K2ij

which in general are dened as

r = RiCSpiω

En and δ =

ωeωn,

where ωe = 1/√LiCS

pi denotes the electrical resonant frequency. The generalized

electromechanical coupling coecient Kij can be described as

K2ij =

(ωDn)2 − (ωEn )2(ωEn )2 ,

where ωDn and ωEn represent the natural frequencies of the open and the shorted

circuit respectively. These two frequencies can either be obtained through experi-

ments or by using the following analytical equations[8]:

ωDn =

√√√√K +KE

p

1−k2ij

M

ωEn =

√(K +KE

p

)M

,

where M denotes the gure of merit. The optimal shunt inductance and the

optimal resistor for a given frequency ωn can then be calculated as

1.5. Thesis outline 9

Ropt =

√2

K2ij

(1+K2ij)

CSpiω

En

and Lopt =1

ω2nCpi

(1 +K2

ij

) .1.5 Thesis outline

In chapter 1 there are given some background information on the project DREAM,

followed by a short introduction on the fundamentals of piezoelectricity. Then the

principle of passive shunt damping is explained. In the following chapter a static

analysis of a at plate was done, including a comparison between the analytical

calculation and the simulation using Comsol Multiphysics. Subsequently a piezo-

electric element is included in the model, setting up the needed boundary condi-

tions. In chapter 3 an analytical modal analysis and a Comsol simulation of a

simply supported plate was done likewise. A model including the piezoelectric ele-

ment was built and an eigenfrequency analysis of an open and closed circuit system

conducted. Based on these models and the theoretical foundations, two models for

passive shunt damping were created and analyzed. In chapter 4 the Comsol model

of a Fluid-Structure interaction is presented. In the last chapter the conclusion and

a summary are given, as well as future prospects on how the project could continue.

Chapter 2

Static analysis of a at plate

2.1 Analytical calculations

The rst analytical calculation was performed with a glass bre reinforced plastic

(GFRP) beam with the length l = 200mm[6]. The beam was xed on one side and

loaded with a force Fz = 10N on the other side. The tip displacement is determined

by

wzmax =Fzl

3

3EIy.

The used material data can be found in Appendix B.

2.2 FEM calculations with Comsol

For the following simulations a plate with the dimensions 200mm ·60mm ·2mm was

used. Unless noted otherwise these dimensions were used and are further referred

to as a plate. The material data of the GFRP fabric was implemented in the

Comsol material library. A 3D model of the structural mechanics module was chosen,

selecting the 'solid, stress-strain' feature for static analysis. After setting up the

geometry, a new coordinate system was created and rotated, so that the new z-axis

was looking in the according direction of the GFRP fabric.

In the subdomain settings, the previous implemented material data for GFRP

fabric was selected. The adequate material model for the bre orientation was

applied corresponding to the library data.

10

2.2. FEM calculations with Comsol 11

Within the boundary settings, one side was considered to be xed, whereas the

other side was loaded with a force Fx = 10N .

For the free mesh parameters a predened coarse mesh was applied before solving

the model.

In a second approach the 'piezoelectric eects' of the 'structural mechanics model'

were selected. For this model the same plate was used again. In addition a piezo-

electric element with the dimensions 50mm · 30mm · 0.2mm was integrated.

Figure 2.1: GFRP plate and piezoelectric element

To achieve a maximum electrical potential dierence, the piezoceramic was placed

at the xed end of the GFRP plate. At this point the greatest strains occur during

an excitation at the rst bending mode. Prior to choosing the subdomain settings,

a user dened material for the piezoelectric material PIC 255 (see Appendix B) was

applied in the materials library for further use. In the Comsol 'piezoelectric eects'

module, it is only possible to use decoupled, isotropic or anisotropic materials. The

orthotropic material data for the GFRP plate was implemented in the anisotropic

form. Furthermore a new coordinate system was set up to link the structure with

the according material direction.

In the subdomain settings the piezoelectric material model was selected for the

piezoceramic PIC 255. Similarly for the GFRP fabric the material model and the

2.3. Results 12

according coordinate system were applied. The poling direction for the Comsol

'piezoelectric eects' is the 3 direction as within the IEEE standards.

Within the boundary condition the same clamping and the loading conditions

were set up as in the previous model. To realise the electrode layers of the piezo-

electric element, a second Comsol module for 'AC/DC - statics' was used. The

dependent variables were set to be the same in both modules. Apart from the two

electrodes, all other faces were set inactive within this module. The upper and the

lower face of the piezoceramic are marked in gure 2.1. For these two faces shell el-

ements made of copper were used. Supplementary the electric boundary conditions

were set up. All faces of the piezoceramic were assumed 'Zero charge/Symmetry'

except the upper and lower face. These were set to be 'Ground' and 'Continuity'

respectively.

For the free mesh parameters a predened extra coarse mesh was applied before

solving the model for static deformation.

To get an estimation of the impact of the additional material of the piezoceramic

at the highest loaded point, another model was created without any piezoelectric

eects, selecting the GFRP fabric for both geometries.

2.3 Results

The analytical calculations of the beam model resulted in a total tip displacement of

25.64mm, whereas the Comsol model without the piezoelectric element calculated

a displacement of 25.57mm. The dierence between the two results was explained

by the plate used in Comsol opposing to the perfect beam that was assumed in the

analytical calculations. The output of the model with the piezoelectric element is

depicted in gure 2.2. The left gure represents the earthed upper face, the right

gure the face with the 'Continuity' boundary condition.

2.3. Results 13

Figure 2.2: Static deformation of GFRP fabric and PIC 255

In the following table 2.1 the total tip deformations of the three dierent models

are given. It was observed that by adding another small piece of GFRP near the

clamping, the deformation was reduced by 12%. A far higher reduction of the tip

displacement was noted by adding the piezoceramic. Also the inuence of the electric

boundary conditions was observed in this model.

model total tip deformation [mm]

GFRP plate 25.6

GFRP plate and piezoelectric element PIC 255 20.8

GFRP plate and piezoelectric element GFRP 23.5

Table 2.1: Static deformation

Chapter 3

Modal analysis

3.1 Analytical calculations

An eigenfrequency analysis of an aluminum plate was carried out. The plate was

assumed simply supported along all edges and it had the same dimensions of 200mm·

60mm · 2mm as the previous studied plate. The material constants can be found in

Appendix B. Starting from the classical plate equation[5] for thin plates

pz = D

(∂4

∂x4+ 2

∂4

∂x2∂y2+

∂4

∂y4

)w0

with the plate's bending stiness D dened as

D =Et3

12 (1− ν2),

the following equation is derived:

w2mn =

π4D

ρh

(m2

a2+n2

b2

)2

.

The value a/b denotes the length to width ratio of the plate.

14


3.2 FEM calculations with Comsol

3.2.1 Eigenfrequency analysis

3.2.1.1 Aluminum plate

For the eigenfrequency analysis of an aluminum plate the structural mechanics mod-

ule for 'solid, stress-strain' was used. The material was picked from the library and

all boundaries were chosen to be free. No damping was assumed as in the analytical

calculations. After creating a predened coarse mesh the model was solved for an

eigenfrequency analysis.

3.2.1.2 GFRP plate

Another eigenfrequency analysis was carried out for a GFRP plate. The material

was changed from aluminum to GFRP and a xation as depicted in gure 3.1 was

added. At this stage an undamped eigenfrequency analysis was conducted. Then

a structural damping with a loss factor of 0.01 was applied for the GFRP plate for

this and all following models unless noted otherwise.

3.2.1.3 Open circuit system

The eigenfrequencies of the open circuit and the closed circuit system were required,

to calculate the optimal shunting parameters. The basis for the model used in this

section was the model from chapter 2, where the static deformation of a GFRP

plate and a piezoelectric element PIC 255 was studied. In gure 3.1 the geometry

settings are shown. In the boundary settings all parameter remained the same,

in the subdomain settings the structural damping was added. The eigenfrequency

analysis was then carried out using an extra coarse mesh.


Figure 3.1: Model geometry settings

3.2.1.4 Closed circuit system

In the 'AC/DC - statics' module all faces of the piezoceramic were set to active

in order to change the existing model of an open circuit system to a closed circuit

system. The thereby applied copper shell elements connected the upper and lower

electrode. Another way of realizing the closed circuit system was investigated: In-

stead of adding more shell elements to the structure, the electric boundary conditions

of these faces were changed from 'Zero charge/Symmetry' to 'Ground'.

3.2.2 Frequency response

3.2.2.1 Pure resistive shunt

The model in this section is based on the open circuit system model in section

3.2.1.3. At rst the value of the resistor was calculated according to the theory in

chapter 1, to maximize the obtained damping. From the open and closed circuit

system it was known that the eigenfrequency was approximately 33 [Hz] . In Comsol

the 'SPICE circuit editor' was used to simulate the electric circuit and to connect

the two electrodes of the piezoelectric element with the resistor. SPICE is a general-

purpose circuit simulation program for nonlinear dc, nonlinear transient, and linear

ac analyses. It originates from UC Berkeley[10]. The circuits may contain resistors,


capacitors, inductors, voltage and current sources, switches and diodes. In the

SPICE editor the following commands were implemented:

The potential at node 0 was set to 0 by default. A resistor R0 was created to

connect the nodes 0 and 1 and another resistor R1 to connect the nodes 1 and 2. A

subcircuit was created labeling node 1 top and node 2 bottom. At the end, the

circuit was connected with the Comsol le. On conrmation, Comsol created the

according constants, global expressions and equations. The Comsol SPICE circuit

editor is depicted in gure 3.2.

Figure 3.2: SPICE circuit editor

The upper and lower face's electrical boundary conditions of the piezoelectric

element were then changed to 'oating potential'. The group index was changed to

top for the upper face and the value for the chargeQ0 was changed to sim_X1_top_q

according to the equations of the SPICE circuit. The same settings were performed

with the lower face for the parameter bottom. In the following gure (gure 3.3)

the electric boundary settings for the top face are shown.


Figure 3.3: Electric boundary conditions for SPICE connection

Several steps were then used to obtain a frequency response for a pure resistive

shunt with maximum damping of the rst eigenmode.

At rst a value for the resistor R1 was chosen and an eigenfrequency analysis

conducted. Because Comsol set the frequency of the SPICE circuit to 1000 [Hz]

by default, this value was iteratively set to the resulting eigenfrequency within the

constants. This way the eigenfrequency for a predened resistance was obtained.

A second model was then produced based on the rst one in order to conduct the

harmonic analysis. A force F = 0.1[N ] was applied on one tip point of the plate as

shown in gure 3.1. Thereby the bending and torsional eigenmodes were excitated.

The frequency denition of the SPICE circuit was moved from the constants to

the global equations. There it was coupled with the frequency 'smpz3d' of the

structure, which was being varied during the harmonic analysis. The evaluation of

the deformation was then conducted at the opposing tip point of the excitation,

see gure 3.1. There both types of natural oscillations were detectable at a phase

shift of π2. Having two models, one for the eigenfrequency analysis and one for the

harmonic analysis, the value of the resistor R1 was varied. This way Roptcomsol could

be obtained with a minimal deformation when the plate was excitated with the rst

eigenfrequency. After Roptcomsol was found, a frequency response for the frequency

domain 0− 200 [Hz] was performed.

Another model was created leaving all parameters identical but omitting the

structural damping.

3.3. Results 19

3.2.2.2 Resonant RL shunt

The procedure for the resonant RL shunt was the same as described for the pure

resistive shunt. Therefore only the changes that were made are listed in this section,

assuming that all other parameters stayed identical. The analytical values for the

resistor and the inductor were calculated in order to search for the optimal values

of the Comsol simulation nearby. In the SPICE editor the inductor L1 was added

between the nodes 1 and 2.

3.3 Results

3.3.1 Eigenfrequency analysis

In the following table (table 3.1) the eigenfrequencies of the analytical calculations

(section 3.1) and the Comsol simulation (section 3.2.1.1) were contrasted:

m/n Analytical eigenfrequency [Hz] Simulated eigenfrequency [Hz]

1/0 122.3 122.3

2/0 489.2 489.2

3/0 1100.6 1100.6

0/1 1358.8 1358.7

1/1 1481.1 1480.9

2/1 1848.0 1847.5

Table 3.1: Analytical and simulated eigenfrequencies of an aluminum plate

Apart from small deviation of less then 0.1% the corresponding frequencies

matched very well. The Comsol simulation of the eigenfrequencies for this simple

case was valid.

The results of the open (section 3.2.1.3) and closed circuit systems (section

3.2.1.4) are presented in table 3.2. When the mesh size was varied, small changes

in the resulting eigenfrequencies were noticed with a maximum deviation of 1.5%.

Both alternatives of modeling the closed circuit system yielded exactly the same

3.3. Results 20

results. The thickness of the applied copper shell elements showed no inuence on

the solution.

open circuit system closed circuit system

eigenfrequency [Hz] 33.210 32.617

tip deformation [mm] 23.52 24.10

Table 3.2: Comparison open/closed circuit system

Using the presented frequencies the generalized electromechanical coupling coef-

cient Kij can be calculated:

Kij =

√(33.212 − 32.6172)

32.6172= 0.192.

3.3.2 Frequency response

To check whether the damping of the added resonant RL shunt was working, the

model of the simple plate (3.2.1.2) and the model of resonant RL shunt were com-

pared. In both cases the structural damping of the plate had been set to zero. When

excitated with the rst eigenfrequency, the resulting displacement was expected to

go to innity for the simple plate. The resistive shunt was expected to contribute

some damping. The ndings are given in table 3.3. In the undamped case the

displacement was going to innity. Due to the fact that the eigenfrequencies were

only determined with nite accuracy the amplitude of the oscillations increased but

stopped at some point. The displacements of the resonant RL shunt are three orders

of magnitude smaller. Clearly the simulated deformations of the plate are far beyond

the material limits, but the inuence of the shunt damping could be demonstrated.

tip displacement [mm]

simple plate 1.6 · 106

resonant RL shunt 3085

Table 3.3: Undamped tip displacement

3.3. Results 21

Table 3.4 comprises the optimal shunting parameters of the analytical calcula-

tions and the Comsol simulations.

Analytical calculation Comsol simulation

Pure resistive shunt Ropt = 44.3kΩ Ropt = 55.5kΩ

Resonant RL shunt Ropt = 12.5kΩ Lopt = 224H Ropt = 11kΩ Lopt = 265H

Table 3.4: Optimal shunting parameters for maximal damping

Between the analytical and the Comsol solutions dierences of up to 20% oc-

curred. The reason for this deviation was not identied so far. One way to check the

results would be to conduct some experiments. Especially for the generalized elec-

tromechanical coupling coecient Kij this should be accomplished. This parameter

was used for calculating the analytical results, although it was obtained through

simulations.

For the following simulations the optimal shunting parameters were used. The

maximal deformations of the plate, when excitated with the rst eigenfrequency, are

presented in table 3.5.

tip deformation [mm]

plate without piezoceramic 24.8

pure resistive shunt 7.5

resonant RL shunt 1.5

Table 3.5: Tip deformation at rst eigenfrequency

The tip deformation was reduced signicantly by including the piezoelectric el-

ement and the pure resistive shunt. With the resonant RL shunt an even higher

damping was achieved as it was expected from the theory. In gure 3.4 the frequency

response of the models without the piezoceramic, with resistive shunt damping and

with resonant RL shunt damping is depicted.

3.3. Results 22

Figure 3.4: Frequency response

Due to the increased stiness of the structure, the eigenfrequencies changed to

higher values when the piezoelectric element was added. The damping of the two

shunts at the rst eigenfrequency is clearly visible.

Chapter 4

Fluid-structure interaction

The next step towards a comprehensive simulation of the overall system was to

couple a simple plate with a uid ow. The Comsol module for 'Fluid-Structure

Interaction' was used in which all parameters are already linked. The module com-

prises three parts: 'Solid, Stress-Strain', 'Incompressible Navier-Stokes' and 'Moving

Mesh'. The rst part is to describe the structural deformations which are solved

using an elastic formulation and a nonlinear geometry formulation to allow large

deformations. The uid ow is described by the Navier-Stokes equations, solving

for the velocity eld and the pressure. The motion of the deformed mesh is modeled

using Winslow smoothing[11]. The used geometry is depicted in gure 4.1. Around

the GFRP plate a cube with the dimensions 200mm · 200mm · 300mm was con-

structed to contain the uid ow. For both geometries and the three parts of the

module, the subdomain and boundary settings were applied:

Within the subdomain settings of the part 'Incompressible Navier-Stokes', the

plate was selected inactive (solid domain) and the material parameters for the uid

implemented (uid domain). Accordingly in part 'Solid, Stress-Strain' the material

parameters of the GFRP plate were implemented (solid domain) and the uid do-

main selected inactive (uid domain). In the part 'Moving Mesh' the uid and solid

23

Chapter 4. Fluid-structure interaction 24

domains were dened.

Figure 4.1: Fluid-structure interaction

In the boundary settings of the part 'Incompressible Navier-Stokes' all faces of

the surrounding cube were set to 'Wall - No slip'. The reverse side was changed to

'Outlet - Pressure p0 = 0, no viscous stress', and the front side to 'Inlet - Velocity'

with the velocity uo. In the part 'Solid, Stress-Strain' for all faces 'Fluid loads' were

selected except the lower surface which was xed to the surroundings. In the part

'Moving Mesh' all active boundaries were selected to be xed for the surrounding

cube, all other boundaries were set to 'Structural displacement'.

The presented model did not converge in a reasonable time. Either the utilized

computational power and mesh size were too low or the used solver was inappropriate

for this problem. Further investigations were abandoned due to the time limitations

of this term paper.

Chapter 5

General conclusion and outlook

The static analysis of a at plate was conducted. It was shown that displacements

simulated by Comsol matched the analytical calculations. Then a Comsol model

for the piezoceramics was set up and run. The eects of the boundary conditions

were noted and the inuence of the piezoelectric material on the tip deformation

displayed.

A modal analysis of a simply supported plate was performed and the Comsol

model was compared to the analytical results. Here too the results corresponded.

Furthermore the models of the open and closed circuit systems were set up and the

electromechanical coupling coecient was calculated using these ndings.

It was then displayed how to integrate and couple an electric network with a

piezoceramic element in Comsol for the use of passive shunt damping. This was

carried out for a pure resistive shunt and a resonant RL shunt and the optimal

shunting parameters were obtained. Both simulations showed some deviation from

the analytical calculations conducted by Hagood and von Flotow. This may be due

to the fact that the Comsol model was excitated and analysed in a single point. In

a more exact simulation the displacements of several points would have to averaged

to get better results. The plate's inherent damping might have an inuence on the

overall system as well. The main reason is undetermined at the moment and has to

be further investigated.

A model for the uid-structure interaction was built and the subdomain and

boundary conditions applied. The solver for the uid ow showed a very slow

25

Chapter 5. General conclusion and outlook 26

convergence. Resolving this problem was not possible within the time frame of the

thesis.

For the proceeding project DREAM several investigations need to be accom-

plished. The ndings of this work and the presented model have to be veried,

either by experimental validation or other simulations. Some optimization of the

simulation might be necessary, probably including the Matlab interface that is avail-

able in Comsol. In addition more powerful and complex shunts need to be simulated

to obtain an even higher damping. The model of the piezoelectric element and the

coupled electric network needs to be integrated in the model of the uid-structure

interaction. When these steps have been accomplished, it is possible to complete

the model by replacing the at plate with the spatial and complex geometry of the

propfan blade.

Bibliography

[1] http://www.ecare-sme.org/plus/download/DREAM.pdf

[2] G. Locatelli (2001), Piezo-actuated adaptive structures for vibration

damping and shape control - modeling and testing, Fortschritts-Berichte

VDI, Reihe 11, No. 303

[3] ANSI-IEEE 176 (1987) Standard on Piezoelectricity

[4] S.O. Reza Moheimani and A.J. Fleming (2006), Piezoelectric Transducers

for Vibration Control and Damping, Springer Verlag

[5] E. Mazza, Methods of Structural Analysis, Lecture Notes, ETH Zurich

[6] M. Sayir (2008), Ingenieurmechanik 1 und 2, Vieweg + Teubner in GWV

Fachverlage GmbH

[7] N. W. Hagood and A. von Flotow (1991), ), Damping of Structural Vibra-

tions with Piezoelectric Materials and Passive Electrical Networks,

Journal of Sound and Vibration, 146(2), 243-268

[8] S. O. R. Moheimani and A. J. Flemming (2006), ), Piezoelectric Transduc-

ers for Vibration Control and Damping, Springer Verlag

[9] ANSI-IEEE 177 (1966), Standard Denitions and Methods of Measure-

ment for Piezoelectric Vibrators

[10] http://bwrc.eecs.berkeley.edu/Classes/icbook/SPICE/

[11] Comsol Multiphysics Modeling Guide

27

Appendix A

Piezoelectric Equations

A.1 Charge-Strain Form[DS

]=[εT

dT

dsE

] [ET

]εT = εS + dcEdt

[Fm

]=[AsV m

]=[CVm

]sE = cE

−1= sD + dtε

T−1d

[m2

N

]d = esE = εTg

[mV

]=[CN

]A.2 Field-Strain Form[

ES

]=[βT

gt

−gsD

] [DT

]βT = εT

−1 [mF

]=[V mAs

]=[V mC

]sD = cD

−1= sE + dtε

T−1d

[m2

N

]g = εT

−1d

[V mN

]=[m2

As

]=[m2

C

]A.3 Charge-Stress Form[

DT

]=[εT

−et

ecE

] [ES

]εS = εT − dcEdt

[Fm

]=[AsV m

]=[CVm

]cE = sE

−1= cD − etεS

−1d

[Nm2

]e = dcE = εSh

[NVm

]=[Cm2

]

28

A.4. Field-Stress Form 29

A.4 Field-Stress Form[ET

]=[βS

−ht

−hcD

] [DS

]βS = εS

−1 [mF

]=[V mAs

]=[V mC

]cD = sD

−1= cE + etε

S−1e

[Nm2

]h = εS

−1d

[NAs

]=[NC

]

Appendix B

Material Properties

GFRP fabric (glass/epoxy fabric prepreg)

value unit

Density ρ = 2000[kgm3

]Young's modula E11 = 26 [GPa]

E22 = 26 [GPa]

E33 = 10 [GPa]

Shear modula G12 = 3.53 [GPa]

G23 = 3.70 [GPa]

G13 = 3.70 [GPa]

Poisson ratios υ12 = 0.112 [−]

υ23 = 0.311 [−]

υ13 = 0.112 [−]

Aluminum

value unit

Density ρ = 2700[kgm3

]Youngs modulus E = 70 [GPa]

Poisson ratio υ = 0.33 [−]

30

Appendix B. Material Properties 31

Piezoceramic PIC 255

Documents

Simply Supported Plate