ActiveandPassiveVibrationIsolationandDampingviaShuntedTransducers Universit´eLibredeBruxelles

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Universit´ e Libre de Bruxelles

F a c u l t ´e d e s S c i e n c e s A p p l i q u ´e e s

Active and Passive Vibration Isolation and Damping via Shunted Transducers

Bruno de Marneffe

k

V

I f

c

k

f

c

I

noisy side quiet side

f

d

Force Sensor

F

^Electric_circuit

Electromagnetic

transducer

noisy side

quiet side

Control- ler

f

d

Thesis submitted in candidature for the

degree of Doctor in Engineering Sciences 14 December 2007

Active Structures Laboratory

Department of Mechanical Engineering and Robotics

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Jury

President : Prof. Alain Delchambre (ULB) Supervisor : Prof. Andr´e Preumont (ULB)

Members :

Prof. Stephen J. Elliott (ISVR - Southampton) Prof. Johan Gyselinck (ULB)

Dr. Stanislaw Pietrzko (EMPA - Switzerland) Prof. Paul Sas (KUL - Leuven)

iii

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Remerciements

Je voudrais tout d’abord remercier le professeur André Preumont, directeur du Laboratoire des Structures Actives de l’ULB et promoteur de cette thèse, pour m’avoir accueilli au sein de son service et m’avoir permis, pendant plus de quatre ans, de travailler dans des domaines variés et intéressants; ses idées et ses conseils m’ont été d’une grande aide. Je remercie également tous mes collègues et anciens collègues pour leur aide, leurs encouragements et l’ambiance chaleureuse qui s’est instaurée pendant toutes ces années.

Je remercie tout particulièrement Iulian Romanescu et Mihaita Horodinca qui ont, chacun à leur tour, pris en charge la réalisation des différents dispositifs expérimentaux. L’aboutissement de ces travaux, et particulièrement de la plate- forme de Stewart, doit beaucoup à leurs talents de mécanicien.

Ce travail est la continuation directe de travaux de recherches effectués avant mon arrivée à l’ULB: je dois beaucoup à mes prédécesseurs qui ont balisé la voie

à suivre. Mes remerciements vont particulièrement à Frédéric Bossens qui m’a, le premier, appris les rudiments du contrôle de structures. J’exprime également ma gratitude à Michel Osée, du département BEAMS de l’ULB, et à Jean-Philippe Verschueren, de Micromega-Dynamics S.A., pour leur patience et leur aide lors de la mise en oeuvre des différents circuits électroniques. Merci aussi à Arnaud Der- aemaeker pour son aide lors de la modélisation de structures piézoélectriques, et à Mohamed El Ouni, Thomas Lemaˆıtre (stagiaire ESTACA) et Samuel Veillerette qui ont tous trois participé à la préparation du nouveau vol parabolique. Je tiens de même à souligner la disponibilité de Gillian Lucy, du département d’anglais de l’ULB, qui a patiemment relu ce manuscrit: ses commentaires m’ont permis d’en corriger de nombreuses fautes d’anglais.

Merci enfin `a ma famille et `a Nadine pour leur soutien et leurs encouragements.

Au cours de ce travail j’ai été supporté par le Pôle d’Attraction Inter-Universitaire IUAP 5 (Advanced Mechatronics Systems), par l’ESA dans le cadre du pro- gramme PRODEX (C90147) et par le Sixième Programme Cadre de l’UE avec le projet CASSEM (Composite and Adaptative Structures: Simulations, Experi- mentation and Modelling). J’ai également bénéficié du soutien indirect de projets de l’UE (InMAR: Intelligent Materials for Active Noise reduction) et de l’ESA (LSSP: Low Stiffness Stewart Platform et SSPA: Smart Structures For Payloads And Antennae).

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Abstract

Many different active control techniques can be used to control the vibrations of a mechanical structure: they however require at least a sensitive signal amplifier (for the sensor), a power amplifier (for the actuator) and an analog or digital filter (for the controller). The use of all these electronic devices may be impractical in many applications and has motivated the use of the so-called shunt circuits, in which an electrical circuit is directly connected to a transducer embedded in the structure. The transducer acts as an energy converter: it transforms mechanical (vibrational) energy into electrical energy, which is in turn dissipated in the shunt circuit. No separate sensor is required, and only one, generally simple electronic circuit is used. The stability of the shunted structure is guaranteed if the electric circuit is passive, i.e., if it is made of passive components such as resistors and inductors.

This thesis compares the performances of the shunt circuits with those of clas- sical active control systems. It successively considers the use of piezoelectric transducers and that of electromagnetic (moving-coil) transducers:

• In a first part, several damping techniques are applied on a benchmark truss structure equipped with a piezoelectric stack transducer. A unified formulation is found and experimentally verified for an active control law, the Integral Force Feedback (IFF), and for various passive shunt circuits (resistive and resistive-inductive). The use of the so-called “negative capac- itance” shunt is also investigated in detail. Two different implementations are discussed: they are shown to have very different stability limits and performances.

• In a second part, vibration isolation with electromagnetic (moving-coil) transducers is introduced. The effects of an inductive-resistive shunt circuit are studied in detail; an equivalent mechanical representation is found. The performances are compared with those of resonant shunts and with those of an active isolation technique. Next, the construction of a six-axis isolator based on a Stewart Platform is presented: the key parameters and the main limitations of the system are highlighted.

vii

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Introduction

1.1 Vibration damping of smart structures

The importance of the structural damping ratio ξ is highlighted in Fig. 1.1 (Preumont, 2006), which shows, on an example with a single degree-of-freedom (d.o.f), the influence of ξ on (i) the amplification of the structural response near the resonance frequency and (ii) the number N of cycles necessary to reduce the amplitude of the impulse response by 50%. Typical damping values encountered in various fields of structural engineering are also indicated in the figure. Notice the very low values for space structures: these are due to the absence of aero- dynamic and gravity-induced friction forces as well as the use of stiff, bonded joints (as opposed to bolted joints) that prevent the dissipation of vibrational en- ergy (Nye et al., 1996). The situation is even worse when the application requires cryogenic temperatures (such as an InfraRed telescope that needs to be cooled down), because material damping ratios decrease considerably with temperature;

as an example, aluminium at 40K was found to have a damping ratio as low as 10

⁻⁴

%, i.e. 2% only of its nominal value at room temperature (Peng et al., 2004).

Smart Materials

Up to some levels, the structural damping ratio can be raised by the use of Visco- Elastic Materials (VEMs): see e.g. Nye et al. (1996) for interesting examples of such passively damped space structures. Another possibility consists in imple- menting an active control system including sensors, actuators and the appropriate electronics; structures with such systems are said to be smart or adaptive because they can adapt to minimize the impact of external perturbations. Many different kinds of sensors and actuators are commercially available; see e.g. Janocha (1999) for a good review and explanations. The first part of this thesis considers the

1

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2 1 Introduction

Soil Radiation

0.001 0.005 0.01 0.05 0.1 ø

N=110

N=22 N=11

N=2 N=1 dB=40

dB=54

dB=34

dB=20 dB=14

dB

0

Space Structures

Mechanical Structures

Civil Engineering

h(t)

! t

Dynamic amplification

Impulse response

50%

Figure 1.1: Dynamic amplification at resonance (in dB) and number of cycles N to reduce the amplitude of the impulse response by 50% as a function of the damping ratio ξ (the damping scale is logarithmic).

use of piezoelectric materials, which are able to convert an electrical signal into a deformation (and vice-versa). Active control with piezoelectric elements has several advantages over passive damping with Visco-Elastic Materials (VEMs):

• The characteristics of VEMs are known to vary rapidly with temperature.

Piezos, by contrast, are suitable in a much larger temperature range. For example, Bronowicki et al. (1996) found that the transfer function between piezo actuators and sensors embedded in a truss was nearly constant over the range ±100

^◦

C. • The same applies to the bandwidth: while VEMs are typically efficient only in a limited frequency range, piezos can be actuated from (almost) DC to hundreds of kHz.

• Introducing passive damping with VEMs into structures where stresses and

strains are very small, such as in space structures, is a very challenging

problem, while piezoceramics have a virtually unlimited resolution. It re-

quires, of course, appropriate electronics and sensitive enough sensors.

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1.1 Vibration damping of smart structures 3

It is often found that active systems, in spite of all the necessary equipment, introduce less mass than a passive solution made of viscoelastic material. The main drawbacks of piezos are their nonlinearities (e.g. hysteresis), their limited stroke (a few micrometers) and the high voltages (up to kilovolts) required for the actuation. The last two problems are of lesser concern in space, where only very small actuation forces and strains are needed; Wada (1993) pointed out that a large strain capability is required only to survive the high dynamic strains imposed during the launch of the structure into space.

Active trusses and collocated control

1 2

3 4

5 6

7 8

9 10

11 12

13 14

15 16

17 18

19 20

x y

q

Detail of an active member

Force transducer

f d

Piezoelectric linear actuator Active member

p

2

p

1

Figure 1.2: Schematic representation of an active truss. The active struts consist of a linear piezoelectric transducer aligned with a force sensor.

The concept of active trusses is quite natural: it consists in replacing one or several passive bars by active members or active struts (Anderson et al., 1990).

Piezoelectric transducers are ideally suited for this purpose, because of their high stiffness; other types of transducers based e.g. on electrostrictive materials can also be used but they are not investigated here. An example of such active strut is shown in Fig. 1.2, which schematizes a uniaxial piezoelectric actuator (acting along its main axis) aligned with a force sensor, and its insertion into an active truss.

An important feature of this active strut is the collocation between the actuator

and the sensor. An actuator/sensor pair is said to be collocated if it is physically

located at the same place and energetically conjugated, such as force and dis-

placement or velocity, or torque and angle (Preumont, 2002). The properties of

collocated systems are remarkable; in particular, the stability of the control loop

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4 1 Introduction

is guaranteed when certain simple, specific controllers are used

¹

. Such controllers include the so-called “Positive Position Feedback” or PPF (Goh and Caughey, 1985; Fanson and Caughey, 1990), the “Direct Velocity Feedback” or DVF (Balas, 1979) and the “Integral Force Feedback” or IFF (Preumont et al., 1992). Other controllers such as LQG, H

₂

or H

_∞

may be more efficient, but they are also model-dependent, more complex to implement, and their stability is not guar- anteed. Properties of collocated systems are extensively discussed in Preumont (2002); note that:

1. Only the stability of the closed-loop system is guaranteed (the closed-loop performances are not).

2. The guaranteed stability only holds as long as ideal equipment is assumed:

in practice imperfections of the actuator/sensor pair or of the electronics (such as non-linearities or a limited bandwidth) might make the system unstable, in spite of the collocation.

Note finally that the use of these simple controllers does not exclude the parallel use of more complex controllers such as LQG; see e.g. Aubrun (1980) or Preumont (2002, chap. 13) for more information.

Shunt damping

Force sensor

f

Piezoelectric linear transducer

Charge amplifier

Controller Voltage

amplifier

Electric circuit

a) b)

V V

V

_f

c p

V

p

I

p

Figure 1.3: (a) Active control with a separate sensor/actuator pair; (b) shunt damping with a piezoelectric transducer.

A typical active control implementation requires at least a sensitive signal am- plifier (for the sensor), a power amplifier (for the actuator) and an analog or digital filter (for the controller), as illustrated in Fig. 1.3a with the active strut

1it also requires that the control architecture be decentralized, i.e. that the feedback path include only one actuator/sensor pair, and be thus independent of other sensors or actuators possibly placed on the structure.

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1.2 Vibration isolation 5

described in the previous section. The use of all these electronic devices may be impractical in many applications and has motivated the use of shunt circuits (For- ward, 1979; Hagood and von Flotow, 1991; Hollkamp, 1994), in which no sensor is used: instead, an electrical circuit is connected to the electrodes of the trans- ducer (Fig. 1.3b). In this configuration, the piezo acts as an energy converter:

it transforms mechanical (vibrational) energy into electrical energy, which is in turn dissipated in the shunt circuit. No separate sensor is required, and only one (generally simple) electronic circuit is used.

The stability of the shunted structure is guaranteed if the electric circuit is pas- sive, i.e., if it is made of passive components such as resistors and inductors; when the circuit is active, as in chapter 5, care must be taken that the shunt does not destabilize the system.

1.2 Vibration isolation

n 1

Transmissibility

Noisy side e.g. attitude actuator (RWA)

6 d.o.f.

Isolation

Transmits low frequency attitude control torque Attenuates high frequency

disturbances

(a) (b)

Quiet side (e.g. optics or attitude sensors)

Figure 1.4: (a) Principles of a vibration isolation device placed between the “noisy side” and the “quiet side” of the structure, and (b) isolation objectives.

It is important to distinguish between vibration damping and vibration isolation.

As shown in Fig. 1.1, the damping of a structural mode consists in reducing the response of the structure near the corresponding natural frequency: the effects of damping are very narrow-band and hardly noticeable far from the resonance frequencies. Isolation, on the other hand, consists in reducing the vibration trans- mission from one part of the structure (sometimes called “noisy side”) to the other (“quiet side”): the reduction of the transmission generally occurs in a large fre- quency region. Fig. 1.4a schematizes a situation in which both sides (“noisy”

and “quiet”) are separated by an isolation device. The quiet side contains the

payload, and the noisy side includes the attitude control actuators (Reaction

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6 1 Introduction

k

V

I f

c

k

f

c

I

noisy side quiet side

f

d

Force Sensor

F

^Electric

circuit

Electromagnetic

transducer

noisy side

quiet side

a) b)

Control- ler

f

d

Figure 1.5: Single d.o.f. isolation systems with moving-coil transducers. (a) Ac- tive system with force feedback; (b) shunted system.

Wheel Assembly, RWA). The role of the isolator is twofold (Fig. 1.4b): it should (i) totally isolate the two bodies beyond the cut-off frequency ω

_n

of the attitude control, and (ii) transmit the positioning commands (torque etc.) below ω

_n

. The isolator can be passive; in its simplest form it consists of a spring and a viscous damper positioned in parallel. This system, however, involves a fun- damental trade-off, described in chapter 6, which motivated the use of active systems. Many different kinds of actuators (hydraulic, pneumatic, ...) can be used to this end, depending on the application, and a wide variety of control laws have been proposed in the literature. Previous work developed at the ULB (Abu Hanieh, 2003; Preumont et al., 2007), aiming at space applications, used a force feedback control law with electromagnetic (moving-coil) transducers, as shown in Fig. 1.5a.

In this work, a different approach is used, in which there is no feedback loop;

instead, the moving-coil transducer is shunted by an electrical circuit (Fig. 1.5b).

Just as with shunted piezoelectric systems, the transducer directly converts me-

chanical (vibration) energy into electrical energy, and no sensor is needed. Sta-

bility is once again guaranteed if the electrical circuit is passive. Although

shunted voice-coils have been used many times to introduce damping in struc-

tures (e.g. Behrens et al., 2005, or Paulitsch et al., 2007), we are not aware of

any isolation system employing shunted voice-coils.

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1.3 Outline 7

1.3 Outline

The first part of this thesis deals with the damping of a truss structure equipped with a piezoelectric stack transducer. It consists of four chapters:

• Chapter 2 introduces the general equations that govern such structures, and the experimental benchmark structure used in this work. A state-space model is built and updated so as to fit the experimental measurements.

• Chapter 3 introduces active damping with Integral Force Feedback (IFF);

numerical and experimental results are presented.

• Chapter 4 describes the use of passive electric shunts: resistive and resistive- inductive circuits are considered. An analytical formulation is developed and numerically validated.

• Chapter 5 investigates the use of an active electric shunt, namely the neg- ative capacitance shunt. Two different implementations are discussed, and stability is studied with Nyquist plots. Experimental results are presented, which compare and summarize the performances of the different passive and active shunts considered in this study.

The second part of this work deals with vibration isolation with shunted electro- magnetic (moving-coil) transducers. It consists of two chapters:

• Chapter 6 investigates single-axis isolation. The performances of single-pole (RL) and resonant (RLC) shunt circuits are analyzed and compared with those of the Integral Force Feedback; experimental results are presented.

• Chapter 7 extends the results to multi-axis isolation. The construction of a multi-axis active isolator is presented; it is based on the Stewart Platform architecture and on a decentralized Integral Force Feedback control law. In a second step, modifications are introduced on this prototype in such a way that it can be used with passive shunt circuits.

1.4 References

A. Abu Hanieh. Active Isolation and Damping of Vibrations Via Stewart Plat- form. PhD thesis, Universit´e Libre de Bruxelles, 2003.

E.H. Anderson, D.M. Moore, and J.L. Fanson. Development of an active truss element for control of precision structures. Optical Engineering, 29(11):1333–

1341, Nov. 1990.

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8 References

J.N. Aubrun. Theory of the control of structures by low-authority controllers. J.

Guidance and Control, 3(5):444–451, Sept.-oct. 1980.

M.J. Balas. Direct velocity feedback control of large space structures. AIAA Journal of Guidance and Control, 2(3):252–253, 1979.

S. Behrens, A.J. Fleming, and S.O.R. Moheimani. Passive vibration control via electromagnetic shunt damping. IEEE/ASME transactions on mechatronics, 10(1):118–122, Feb. 2005.

A.J. Bronowicki, L.J. McIntyre, R.S. Betros, and G.R. Dvorsky. Mechanical validation of smart structures. Smart Materials and Structures, 5:129–139, 1996.

J.L. Fanson and T.K. Caughey. Positive position feedback control for large space structures. AIAA Journal, 28(4):717–724, April 1990.

R.L. Forward. Electronic damping of vibrations in optical structures. Applied Optics, 18(5):690–697, March 1979.

C.J. Goh and T.K. Caughey. On the stability problem caused by finite actuator dynamics in the collocated control of large space structures. International Journal of Control, 41(3):787–802, 1985.

N.W. Hagood and A. von Flotow. Damping of structural vibrations with piezo- electric materials and passive electrical networks. Journal of Sound and Vibra- tion, 146(2):243–268, 1991.

J.J. Hollkamp. Multimodal passive vibration suppression with piezoelectric ma- terials and resonant shunts. Journal of Intelligent Material Systems and Struc- tures, 5:49–57, Jan. 1994.

H. Janocha. Adaptronics and Smart Structures: Basics, Materials, Design, and Applications. Springer, 1999. (Editor).

T.W. Nye, A.J. Bronowicki, R.A. Manning, and S.S. Simonian. Applications of robust damping treatments to advanced spacecraft structures. Advances in the Astronautical Sciences, 92:531–543, Feb. 1996.

C. Paulitsch, P. Gardonio, and S.J. Elliott. Active vibration damping using an inertial, electrodynamic actuator. ASME Journal of Vibration and Acoustics, 129:39–47, Feb. 2007.

C.Y. Peng, M. Levine, L. Shido, and R. Leland. Experimental observations

on material damping at cryogenic temperatures. In SPIE 49th International

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References 9

Symposium on Optical Science and Technology, Denver, Colorado, August 2-6, 2004., Aug. 2004. http://hdl.handle.net/2014/40006.

A. Preumont. Vibration Control of Active Structures: and Introduction. Kluwer, 2002. 2nd edition.

A. Preumont. Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems. Springer, 2006.

A. Preumont, J.P. Dufour, and C. Malekian. Active damping by a local force feedback with piezoelectric actuators. AIAA Journal of Guidance, Control and Dynamics, 15(2):390–395, March-April 1992.

A. Preumont, M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. De- raemaeker, F. Bossens, and A. Abu Hanieh. A six-axis single-stage active vibration isolator based on stewart platform. Journal of Sound and Vibration, 300:644–661, 2007.

B.K. Wada. Summary of precision actuators for space application. Technical

report, Jet Propulsion Laboratory (JPL), 1993. http://citeseer.ist.psu.edu

/344213.html.

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10 References

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Chapter 2

Piezoelectric structures and active trusses

This chapter introduces the general equations that govern piezoelectric structures:

it considers successively the constitutive equations of piezoelectric materials, the different actuation modes, the behavior of uniaxial “d

₃₃

” transducers and how they are embedded into a general piezoelectric structure. The benchmark truss structure used in this thesis and its state-space model are also presented.

2.1 Some early significant realizations

Most of the early work on the active damping of structures with piezoelements was done in the United States and focused on space structures. As early as 1979, R.L. Forward (Hughes Research Laboratories) demonstrated the feasibility of the technique: in proof-of-concept experiments, he increased the damping ratio of a bar in extension and of a membrane mirror prototype. He investigated passive (inductive) shunting as well as some very simple active control laws with several non-collocated actuator-sensor pairs (Forward, 1979). One year later he demonstrated the active damping of a composite mast with a more thorough theoretical development (Swigert and Forward, 1981; Forward, 1981).

Interest in the field then rose in many research departments, which started inves- tigations on beam and plate structures with PZT patches bonded on them (or embedded in them): see e.g. the work of Crawley and de Luis (1987), Burke and Hubbard (1987), Fanson and Caughey (1990) or Hanagaud et al. (1992). They tried to model analytically this new class of structure as well as the effects that active control might have on them; collocated controllers such as the Positive Position Feedback (PPF) were introduced (Goh and Caughey, 1985; Fanson and

11

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12 2 Piezoelectric structures and active trusses

Caughey, 1990).

However, all these experiments concerned ‘d

₃₁

’ piezoactuators, i.e. thin patches of piezoceramic (see Fig. 2.1b), and plate structures. Research on active trusses, by contrast, began in the late 80’s: see e.g. the work of Anderson et al. (1990), Fanson et al. (1989), Chen et al. (1989) or Bronowicki et al. (1996) who devel- oped active members made of piezo transducers and verified their compatibility with space applications: temperature dependance, linearity, power consumption, bandwidth, etc.

It was found during this period that collocated force or velocity feedback on the active strut can be used to tailor the strut mechanical impedance and thus maximize the energy dissipated in the active strut; the usefulness of force feedback to this end was stressed (Chen et al., 1989). Integral Force Feedback (IFF), in which the collocated force in the strut is integrated, was investigated at the ULB from 1988 on; experimental results underlined its efficiency as well as its guaranteed stability (Preumont et al., 1992).

In a second step, proof-of-concept set-ups were actually tested in space: see e.g.

the ACTEX experiment (Nye et al., 1999), which implemented several active control laws on a secondary payload riding on a spacecraft, the CFIE experiment (Loix et al., 1997), which actively damped a piezoelectric plate embarked in the space shuttle, or the CASTOR experiment (Bousquet et al., 1997), which introduced various damping technologies on a truss mock-up on board the MIR space station. In spite of these experimental demonstrations, however, we are not aware of any actual space structure implementing active control with piezoelectric transducers.

2.2 Piezoelectric material

2.2.1 Constitutive equations

The piezoelectric constitutive equations were standardized in 1988 by the IEEE association (IEEE Std., 1988). Assuming linear characteristics and constant tem- perature, they can be written in tensorial form:

T

_ij

= c

^E_ijkl

S

_kl

− e

_kij

E

_k

(2.1)

D

_i

= e

_ikl

S

_kl

+ ε

^S_ik

E

_k

(2.2)

where the usual summation convention on repeated indices has been used and

i, j, k, l take a value from 1 to 3. T

_ij

and S

_kl

are the stress and strain ten-

sors, respectively, while D

_i

and E

_k

represent the electrical displacement and

electric field vectors. c

^E_ijkl

are the elastic constants under constant electric field

(Hooke’s tensor, N/m

²

), ε

^S_ik

the dielectric constants under constant strain (in

Coulomb/(V olt.m) or F arad/m) and e

_kij

(in Coulomb/m

²

or N ewton/(V olt.m))

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2.2 Piezoelectric material 13

are the material constants that effectively couple the mechanical and electrical properties of the material.

Thanks to the many symmetries of the mechanical tensors, an easier matrix notation can be used instead of the tensorial one. Introducing the stress and strain vectors

¹

as:

T =

 

 

 

 

 T

₁₁

T

₂₂

T

₃₃

T

₂₃

T

₃₁

T

₁₂

 

 

 

 



and S =

 

 

 

 

 S

₁₁

S

₂₂

S

₃₃

2S

₂₃

2S

₃₁

2S

₁₂

 

 

 

 



(2.3)

respectively, Eq. 2.1 and 2.2 can be rewritten in the more compact matrix form:

T = [c

^E

]S − [e]

^t

E (2.4)

D = [e]S + [ε

^S

]E (2.5)

where [c

^E

], [e] and [ε

^S

] are (6 × 6), (3 × 6) and (3 × 3) matrices, respectively, and [.]

^t

represents the matrix transposed. In this notation, S and E have been chosen as the two (vectorial) independent variables, and (T , D) as the dependent variables. This choice is not unique: for example, T and E are often chosen as the dependent variables instead. Eq. 2.4 and 2.5 become in this case:

S = [s

^E

]T + [d]

^t

E (2.6)

D = [d] T + [ε

^T

]E (2.7)

where [s

^E

] = [c

^E

]

⁻¹

is the compliance matrix under constant electric field, [d] is another (3 × 6) coupling matrix in Coulomb/N ewton or m/V olt and the super- script [.

^T

] has been added on ε to emphasize the fact that it has been measured under constant stress: it is thus different from ε

^S

. The importance of the electri- cal and mechanical boundary conditions on the material properties is underlined in the following. Some useful relations between the matrices are:

[e] = [d][c

^E

] (2.8)

c

^D

= c

^E

+ e

^t

{ε

^S

}

⁻¹

e (2.9) s

^D

= s

^E

− d

^t

{ε

^T

}

⁻¹

d (2.10)

ε

^S

= ε

^T

− d[c

^E

]d

^t

(2.11)

where the [.

^D

] superscript means constant-charge (Q = 0) boundary conditions.

See e.g. the IEEE standards (1988) for a complete list of all the possible relations.

1notice the doubling of the non-diagonal terms ofSkl: the last three components ofSrepre- sent shearangles.

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14 2 Piezoelectric structures and active trusses

2.2.2 Piezoelectric modes of actuation

Thanks to many crystal symmetries, the material coupling matrices [d] and [e]

have few non-zero components (Cady, 1946). Eq. 2.6 and 2.7 can be developed explicitly; for PZT (Lead Zirconate Titanate) ceramics or PVDF (Polyvinylidene Difluoride) polymer, they are:

Actuation:

 

 

 

 

 S

₁₁

S

₂₂

S

₃₃

2S

₂₃

2S

₃₁

2S

₁₂

 

 

 

 



=



 



s

₁₁

s

₁₂

s

₁₃

0 0 0 s

₁₂

s

₂₂

s

₂₃

0 0 0 s

₁₃

s

₂₃

s

₃₃

0 0 0

0 0 0 s

₄₄

0 0

0 0 0 0 s

₅₅

0 0 0 0 0 0 s

₆₆



 



| {z }

compliance

 

 

 

 

 T

₁₁

T

₂₂

T

₃₃

T

₂₃

T

₃₁

T

₁₂

 

 

 

 

 +



 



0 0 d

₃₁

0 0 d

₃₂

0 0 d

₃₃

0 d

₂₄

0 d

₁₅

0 0

0 0 0



 



| {z } coupling

 

 E

₁

E

₂

E

₃

 



(2.12) Sensing:

 

 D

₁

D

₂

D

₃

 

 =



 0 0 0 0 d

₁₅

0 0 0 0 d

₂₄

0 0

d

₃₁

d

₃₂

d

₃₃

0 0 0





| {z }

coupling

 

 

 

 

 T

₁₁

T

₂₂

T

₃₃

T

₂₃

T

₃₁

T

₁₂

 

 

 

 

 +



 ε

^T₁

0 0 0 ε

^T₂

0 0 0 ε

^T₃





| {z } permittivity

 

 E

₁

E

₂

E

₃

 

 (2.13)

where, by convention, the coordinate direction 3 coincides with the polarization direction of the ceramic. PZT materials are isotropic in the plane, and thus have d

₃₁

= d

₃₂

and d

₂₄

= d

₁₅

. By contrast, the d

₃₁

and d

₃₂

coefficients of PVDF can be made different, which allows a certain amount of decoupling between the directions 1 and 2; PVDF materials also have d

₂₄

= d

₁₅

= 0.

According to Eq. 2.12, when an electric field E

₃

is applied parallel to the po- larization direction of a PZT material, the piezo transducer expands along its thickness (d

₃₃

) and shrinks in the in-plane directions, because the d

₃₁

and d

₃₂

coefficients are negative. By contrast, if an electric field E

₁

or E

₂

is applied perpendicularly to the polarization direction, a shear deformation (d

₁₅

or d

₂₄

) appears in the transducer. On a macroscopic scale, three different transduction modes are possible, all illustrated in Fig. 2.1:

• In the thickness or d

₃₃

mode, several thin slices of PZT are stacked together and separated by electrodes; the direction of expansion is parallel to the electric field.

• In the in-plane or d

₃₁

mode, a thin piezoelectric film is bonded on (or em-

bedded in) a plate structure and creates a bending moment. The direction

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2.2 Piezoelectric material 15

+

_

ÉL=nd33V

ÉL

ÉL=Ed₃₁L E=V=t

ÉL

í E₁ L₀

1

3

í=d₁₅E₁

ÉL=íL0

V

P P

V

L P

E t

Supporting structure

P V

d

₃₃

d

₃₁

d

15

a)

b)

c)

Figure 2.1: The three actuation modes (d

₃₃

, d

₃₁

and d

₁₅

) of PZT piezoelectric transducers. P indicates the direction of polarization.

of expansion is perpendicular to the electric field.

• In the shear or d

₁₅

mode, the electric field is applied perpendicular to the polarization direction. The transmission from shear actuation of the piezo to the structure (or vice-versa) requires a specific mechanical design; see e.g. Benjeddou et al. (1997) for more details.

This thesis deals exclusively with d

₃₃

transducers which, thanks to their shape and stiffness, can easily be embedded into truss-like structures.

2.2.3 Electromechanical coupling factor

Piezoelectric electromechanical coupling factors are material constants that mea- sure the effectiveness of the conversion of mechanical energy into electrical energy (and vice-versa); they play a vital role in shunt damping of piezoelectric struc- tures as described in chapters 4 and 5. Three different factors (one per actuation mode) are defined by:

• thickness mode: k

₃₃

= d

₃₃

/ q

s

^E₃₃

ε

^T₃

• in-plane mode: k

₃₁

= k

₃₂

= d

₃₁

/ q

s

^E₁₁

ε

^T₃

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16 2 Piezoelectric structures and active trusses

• shear mode: k

₁₅

= k

₂₄

= d

₁₅

/ q

s

^E₅₅

ε

^T₁

or, more synthetically, if the mechanical force is measured in the j

^th

direction and the electrical field in the i

^th

direction:

k

_ij

= d

_ij

/ q

s

_jj

ε

^T_i

(2.14)

(Hagood and von Flotow, 1991). PZT ceramics typically have k

₃₃

≈ k

₁₅

≈ 0.7 and k

₃₁

≈ 0.3. The electromechanical coupling factor can be interpreted as the ratio between the amount of energy that is converted during a quasi-static loading cycle and the maximal energy stored in the transducer during the same cycle; see e.g. Preumont (2006, p. 103) for a demonstration. Electromechanical coupling factors are most easily measured by means of an impedance measurement as described in chapter 4.

Because this work is devoted to structures with uniaxial d

₃₃

transducers, the short notation ‘k’ is used hereafter to denote the longitudinal coupling factor k

₃₃

.

2.3 Uniaxial (d

33

) piezoelectric transducer

+ _

Cross section:

Thickness:

# of disks in the stack:

Electric charge:

Capacitance:

A t

n l = nt

Q = nAD Electrode

Free piezoelectric expansion: Voltage driven:

Charge driven:

C = n

²

"A=l

î = nd

33C Q

î = d

₃₃

nV

ã

Figure 2.2: Piezoelectric linear transducer.

Consider the piezoelectric linear transducer of Fig. 2.2: in accordance to §2.2.2,

it is made of n identical slices of piezoceramic material stacked together, each of

them polarized through the thickness. If one assumes that the stress, the strain,

the electric field and the electric displacement are one-dimensional and parallel

to the direction of polarization, the constitutive equations (2.12) and (2.13) for

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2.3 Uniaxial (d

₃₃

) piezoelectric transducer 17

the piezoelectric material reduce to:

½ D S

¾

=

· ε

^T

d

₃₃

d

₃₃

s

^E

¸ ½ E T

¾

(2.15) where the subscripts [.]

₃

or [.]

₃₃

are implicitly assumed (D instead of D

₃

etc).

If all the electrical and mechanical quantities are uniformly distributed in the transducer, the global constitutive equations are obtained by integrating Eq. 2.15 over the volume of the transducer; using the notations of Figure 2.2, one finds:

½ Q

∆

¾

=

· C nd

₃₃

nd

₃₃

1/K

_a

¸ ½ V f

¾

(2.16) where Q = nAD is the total electric charge on the electrodes of the transducer,

∆ = Sl is the total extension (l = nt is the length of the transducer), f = AT is the total force and V is the voltage applied between the electrodes, resulting in an electric field E = V /t = nV /l. The capacitance of the transducer with no external load (f = 0) is C = ε

^T

An

²

/l, and K

_a

= A/s

^E

l is the stiffness with short-circuited electrodes (V = 0). Note that the electromechanical coupling factor k (§2.2.3) can be defined alternatively by

k

²

= d

₃₃²

s

^E

ε

^T

= n

²

d

₃₃²

K

_a

C (2.17)

Alternative forms of Eq. 2.16 are e.g.:

½ Q f

¾

=

· C(1 − k

²

) nd

₃₃

K

_a

−nd

₃₃

K

_a

K

_a

¸ ½ V

∆

¾

(2.18)

or ½

V f

¾

= K

_a

C(1 − k

²

)

· 1/K

_a

−nd

₃₃

−nd

₃₃

C

¸ ½ Q

∆

¾

(2.19) from which two important relations can be deduced:

1. The capacitance of the transducer under constant-strain (denoted C

^S

) is:

C

^S

= Q V

¯ ¯

∆=0

= C(1 − k

²

) (2.20)

2. The stiffness of the transducer with open electrodes (Q = 0) is

∆ f

¯ ¯

Q=0

= K

_a

(1 − k

²

) (2.21)

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18 2 Piezoelectric structures and active trusses

Because k

²

≈ 0.5 for PZT, the stiffness as well as the capacitance depend sig- nificantly on the boundary conditions, which has important consequences for the different shunt mechanisms (chapters 4 and 5). Note that C

^S

is more a conve- nient mathematical coefficient than a ‘real’ physical parameter. Indeed, it cannot be measured directly: it is impossible to produce a perfect clamping that would guarantee a real constant-volume measurement, and other methods imply high- frequency measurements (see Eq. 4.8) which are not practical either. The IEEE standards (1988) recommend instead separate measurements of C and k, and then the use of Eq. 2.20 to obtain C

^S

.

2.4 Structure with a piezoelectric stack transducer

2.4.1 Governing equations

Consider the linear structure of Fig. 2.3, assumed undamped for simplicity, and equipped with a discrete, massless piezoelectric stack transducer as discussed in the previous section. A voltage V is applied across the electrodes of the transducer, and an electric charge Q flows onto them; there is a relation between Q and V that will be discussed in chapter 4.

Piezoelectric Transducer

Structure Q

Figure 2.3: General linear structure equipped with a piezoelectric stack trans- ducer.

The dynamic equations of the structure (without the piezo) are, in Laplace vari-

ables: ¡

M s

²

+ K ¢

x = F (2.22)

where K and M are the stiffness and mass matrices of the structure, obtained

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2.4 Structure with a piezoelectric stack transducer 19

e.g. by means of a Finite Element model, and F is the vector of external forces.

Here we consider that the only forces exerted on the structure come from the transducer:

F = bf (2.23)

where b is the projection vector relating the end displacements of the strut to the global coordinate system, and f is the force exerted by the piezo (Eq. 2.18 or 2.19). Similarly, the elongation ∆ of the transducer is linked to the structural displacement by:

∆ = b

^T

x (2.24)

The coupled equations governing the piezoelectric structure can be found by combining Eq. 2.22 to Eq. 2.24 with Eq. 2.18; they are

¡ M s

²

+ K + K

_a

bb

^T

¢

x = bK

_a

nd

₃₃

V (2.25)

C(1 − k

²

)V + nd

₃₃

K

_a

b

^T

x = Q (2.26) Note that the mass of the actuator can easily be added to the mass matrix M if necessary.

2.4.2 Various eigenvalues problems

Analyzing further Eq. 2.25 and 2.26, three different eigenvalue problems can be defined, corresponding respectively to the boundary conditions f = 0, V = 0 and Q = 0.

1. From Eq. 2.25, the eigenvalue problem when the axial stiffness of the actu- ator is cancelled, i.e. K

_a

= 0, is given by:

¡ M s

²

+ K ¢

x = 0 (2.27)

2. If V = 0, i.e. if the piezo is short-circuited, the structure obeys:

¡ M s

²

+ K + K

_a

bb

^T

¢

x = 0 (2.28)

3. Finally, if the structure is charge-driven instead of voltage-driven, V can be eliminated from Eq. 2.25 and 2.26; the new equation is:

M s

²

x + µ

K + K

_a

1 − k

²

bb

^T

¶

x = b K

_a

1 − k

²

nd

₃₃

Q

C (2.29)

and if the structure is open-circuited, i.e. if Q = 0, it obeys:

µ

M s

²

+ K + K

_a

1 − k

²

bb

^T

¶

x = 0 (2.30)

which is the same as Eq. 2.28 but with the short-circuit stiffness K

_a

replaced

by the open-circuit one K

_a

/(1 − k

²

).

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20 2 Piezoelectric structures and active trusses

The solutions of these eigenvalue problems are three different sets of natural frequencies; in this work the natural frequencies when K

_a

= 0 are called z

_i

(i = 1, . . . , n), those when the piezo is short-circuited are called ω

_i

and those when the piezo is open-circuited are called Ω

_i

.

2.4.3 Modal coordinates

The characteristic equations (2.25) and (2.26) can be transformed into modal coordinates according to x = Φα, where Φ = (φ

₁

, . . . , φ

_n

) is the matrix of the mode shapes, solutions of the eigenvalue problem (2.28). The mode shapes are normalized according to

Φ

^T

M Φ = diag(µ

_i

) (2.31) and

Φ

^T

(K + K

_a

bb

^T

)Φ = diag(µ

_i

ω

_i²

) (2.32) with ω

_i

the i

^th

natural frequency of the structure with short-circuited electrodes and µ

_i

the i

^th

modal mass. The important parameter

ν

_i

= φ

^T_i

(K

_a

bb

^T

)φ

_i

φ

^T_i

(K + K

_a

bb

^T

)φ

_i

= (φ

^T_i

b)

²

K

_a

µ

_i

ω

_i²

(2.33)

can then be defined; it represents the ratio between (twice) the strain energy in the actuator and (twice) the total strain energy when the structure vibrates according to mode i: it is the fraction of modal strain energy (Preumont et al., 1992). Physically, ν

_i

can be interpreted as a compound indicator of controllability and observability of mode i by the transducer. It is readily available in most Finite Element Analysis softwares.

2.4.4 Placement of the active struts

Parallel to the development of adaptive or intelligent transducers, research has been conducted on the optimal placement of the active struts. Indeed, the struc- tures are generally so large that it would be computationally too intensive to test all the different possibilities. A wide variety of optimisation algorithms were pro- posed to this end in the literature; two popular examples are the Simulated An- nealing method (Chen et al., 1991) and the Genetic Algorithm method (e.g. Rao et al., 1991). See also Padula and Kincaid (1999) for a review of the different placement strategies.

Although these methods are effective, they fail to give a clear physical justification

for the choice of the struts placement. An alternative, more physical method has

been used (with some variations) e.g. by Fanson et al. (1989), Preumont et al.

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2.5 Experimental benchmark structure 21

(1992) or Bronowicki et al. (1999): it merely consists in placing the transducer in the struts with maximal fraction of modal strain energy ν

_i

, where i is the mode to be controlled. This method was chosen when designing the benchmark truss structure of this study. Lu, Utku, and Wada (1992) have considered another strategy, based on a pole placement technique; their method turned out to select the struts with the highest ν

_i

as well.

2.5 Experimental benchmark structure

2.5.1 Active strut

Piezo transducer (PI P010.30H)

Prestressing wire

a) b)

Figure 2.4: a) The piezoelectric stack actuator built for this work; b) (not to scale) another commercially available design (PI 840-30) and its collocated force sensor.

A piezoelectric stack transducer was specially built for this work; it is represented in Fig. 2.4a. It is made of a hollow cylindrical stack actuator (PI P-010.30H);

an internal Kevlar wire (φ = 1.1mm) exerts a 16 kg prestress. The prestress is necessary because such stack transducers cannot withstand traction forces.

Other commercially available transducers, such as that in Fig. 2.4b, introduce the prestress via an external envelope instead: it is stronger, but also stiffer, which tends to reduce the effective electromechanical coupling factor k as demonstrated in Preumont (2006, p. 110).

The main characteristics of the transducer, obtained from measurements, are

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22 2 Piezoelectric structures and active trusses

presented in Table 2.1. K

_a

could not be measured directly but was identified from model updating as explained below. The effective coupling factor k

²

of the transducer is a little higher with the prestress than without (0.36 vs. 0.325);

the reasons for this behavior are unclear (it may be due to nonlinearities of the ceramic).

Material type PIC-151

Dimensions (mm) L = 40, R

_i

= 2.5, R

_e

= 5

K

_a

≈ 90N/µm

C 122nF

k

²

0.36 n (number of discs) 60

Table 2.1: Identified stack actuator characteristics.

2.5.2 Active truss

The truss structure used in this work is depicted in Fig 2.5. It consists of 12 bays of 140 mm each, made of steel bars of 4 mm diameter connected with plastic joints; it is clamped at the bottom. It is equipped with two active struts (piezo transducer + collocated force sensor B&K 8200) as indicated in the figure. This truss was already considered in the experimental setup of Preumont et al. (1992), but in this work one of the piezoelectric transducers has been replaced by the new one presented in §2.5.1. The second transducer is an out-dated high-voltage Philips PXE-HPA1 piezo stack with a very low k factor; in this work it is used only as an excitation source.

2.5.3 Mode shapes and actuator placement

A Finite Element (FE) model of the truss has been constructed with the com- mercial software SAMCEF. The passive struts are modelled with beam elements, and the active ones are obtained by Guyan’s reduction of a separate model with piezoelectric volume elements (Fig. 2.6). The reduction is performed directly in SAMCEF; this procedure was necessary because SAMCEF’s libraries do not include any piezoelectric beam element. After reduction, the transducer model has only 12 mechanical variables (6 d.o.f. at its end points) and 1 electrical vari- able (voltage): it behaves like a beam element with uniaxial (d

₃₃

) piezoelectric transduction capabilities.

The first two mode shapes of the passive truss (i.e., when all the struts are

identical - no piezo yet) are shown in Fig. 2.7; the arrows indicate the approximate

direction of deformation. The fractions of modal strain energy ν

_i

are shown in

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2.5 Experimental benchmark structure 23

Strut 1

Strut 2 Strut 2:

Excitation

Strut 1:

damping + Measurements

Force sensor Piezoelectric

transducer

a)

b)

c)

Figure 2.5: a) truss structure used in the experiment; b) detail of an active strut;

c) disposition of the active struts (zoom).

12 mechanical d.o.f.

1 electrical d.o.f.

Figure 2.6: Full FE model of the stack with piezoelectric volume elements and

condensed structure with 12 mechanical d.o.f. (6 at each endpoint) and 1 elec-

trical d.o.f.

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24 2 Piezoelectric structures and active trusses

Mode 1 (17.8 Hz)

Mode 2 (21.7 Hz)

Maximum strain energy in strut 1

Maximum strain energy in strut 2

z

x y

strut 1 strut 2

Figure 2.7: Structural mode shapes when all the struts are passive and identi- cal. Top view is also shown; the arrows indicate the (approximate) direction of deformation.

Table 2.2 for the first six modes and the two struts. From the figure and the table one can see that strut 1 has a large influence on mode 1 and almost no influence on mode 2, and that the opposite occurs for strut 2. This result motivated the positions of the transducers in the actual truss.

ω

_i

/2π ν

_i

ActiveandPassiveVibrationIsolationandDampingviaShuntedTransducers Universit´eLibredeBruxelles

Universit´ e Libre de Bruxelles

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