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Thèse de doctorat/ PhD Thesis Citation APA:

Mokrani, B. (2015). Piezoelectric shunt damping of rotationally periodic structures (Unpublished doctoral dissertation). Université libre de Bruxelles, Ecole polytechnique de Bruxelles – Electromécanicien, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/209112/4/b6f8b434-eb55-4020-b677-bc8a56ee298b.txt

(English version below)

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ULB

U

niversité

libre

de

B

ruxelles

École Polytechnique de Bruxelles

Piezoelectric Shunt

Dumping of Rotationally Periodic Structures

Bilal MOKRANI

Thesis submitted in candidature

degree of Doctor in Engineering Sciences January 2015

Active Structures Laboratory

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ULB

U

niversité

libre

de

B

ruxelles

3 École Polytechnique de Bruxelles

Piezoelectric Shunt

Damping of Rotationally Periodic Structures

Bilal MOKRANI

Active Structures Laboratory

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Jury

Supervisor: Prof. André Premnont (ULB)

President: Prof. Michel Kiimaert (ULB)

Secretary: Prof. Patrick Hendrick (ULB)

Membres:

Prof. Michel Géradin (Université de Liège)

Prof. Peter Hagedorn (Technische Universitàt Darmstadt)

Dr. Régis Viguié (SAFRAN Techspace Aero S.A.)

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Remerciements

Tout d’abord, je tiens à remercier mon superviseur le Professeur André Preumont, directeur du Laboratoire des Structures Actives de l’ULB, pour m’avoir accueilli dans son laboratoire, accordé sa confiance et son soutien pendant plus de 6 ans, et offert l’opportunité de travailler dans des domaines variés et intéressants; ses idées et ses conseils m’ont été d’une grande aide. Je remercie également tout mes col lègues et anciens collègues du Laboratoire des Structures Actives pour leur aide, leurs encouragements et l’ambiance chaleureuse qui s’est instaurée pendant toutes ces années.

Je remercie le Professeur Marc Mignolet de l’Arizona State University, pour m’avoir accueilli dans son laboratoire et pom ses précieux conseils dans le domaine des structures à symétrie cyclique et les problèmes de désaccordage; je tiens également à remercier Raghavendra Murthy pour son accueil chaleureux et son aide précieuse à la réalisation des modèles désaccordés.

Je remercie également l’équipe de SAFRAN Techspace Aero, pour les nombreuses discussions chaleureuses et fructueuses. Je remercie Régis Viguié et Damien Verhelst pour leurs conseils et leur présence.

Je tiens à remercier tous les collègues qui ont participé à ce projet: Mohamed Abu Gammar, Renaud Bastaits, José Perez-Buron, Gonçalo Rodrigues, et Elodie Rom- née, qui ont, chacun à sa façon, contribué à l’avancement de ce projet. Je remercie tout particulièrement loan Burda, Mihaita Horodinca et Iulian Romanescu qui ont chacun à leur tour, pris en charge la réalisation des différents dispositifs expérimen taux. L’aboutissement de ces travaux doit beaucoup à leurs talents.

Je remercie tous mes amis et ma famille jxjur leur soutien. Je remercie partic ulièrement mon frère Ahcene et ma soeur Hanane pour leur soutien indéfectible. Un grand merci à mes beaux parents pour leur présence et leur bienveillance. Je tiens à remercier infiniment mes parents pour leur amoiur, leur patience et tous leurs sacrifices. Je remercie également ma femme Maria pour son amour, son soutien et

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sa patience pendant toutes ces années. Enfin, je remercie mes petits anges Anis et Amine pour tout le bonheur qu’ils m’apportent.

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Abstract

New materials and new fabrication techniques in turbomachinery lead to monolithic structures with extremely low dumping which may be responsible for severe vibra tions Eind possible high-cycle fatigue problems. To solve this, various techniques of dumping enhancement are under investigation. The présent work is focused on piezoelectric shunt dumping.

This thesis considers the RL shunt dumping of rotationally periodic structures using an array of piezoelectric patches, with an appUcation to a bladed drum représentative of those used in turbomachinery. Due to the periodicity and the cyclic symmetry of the structure, the blade modes occur by families with very close résonance frequen- cies, and harmonie shape in the circumferential direction; the proposed RL shunt approaches take advantage of these two features.

When a family of modes is targeted for dumping, the piezoelectric patches are shunted independently on identical RL circuits, and tuned roughly on the aver age value of the résonance frequencies of the targeted modes. This independent configuration offers a damping solution effective on the whole family of modes, but it requires the use of synthetic inductors, which is a serions drawback for rotating machines.

When a spécifie mode with n nodal diameters has been identified as critical and is t£irgeted for damping, one can take advantage of its harmonie shape to organize the piezoelectric patches in two parallel loops. This parallel approach reduces considerably the demand on the inductors of the tuned inductive shunt, as compared to independent loops, and offers a practical solution for a fully passive intégration of the inductive shunt in a rotating structure.

Various methods are investigated nmnerically and experimentally on a cantilever beam, a bladed rail, a circular plate, and a bladed drmn. The influence of blade mistuning is also investigated.

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Glossary

List of acronyms

ASL Active Structures Laboratory

BLING BLaded rING

BLISK BLaded dISK

BLUM BLaded drUM

CAD Computer Aided Design

DVA Dynamic Vibration Absorber

FRF Frequency Response Fonction

PZT Lead-Zirconate-Titanate

RL Résistive and Inductive shunt

SSD State Switch Dumping

SSDI Synchronized Switch Dumping on Inductor

SSDNC Synchronized Switch Dumping on Négative Capacitance

SSDS Synchronized Switch Dumping on Short

SSDV Synchronized Switch Dumping on Voltage

ULB Université Libre de Bruxelles

ZZENF Zig-Zag shaped Excitation line in the Nodal diameter versus FVequency diagram List of symbols a bj C, Gstatic ds3, dsi 6{x)

Vector of modal coordinates (in Chapter 2); overshoot in the step response of the RLC circuit (in Chapter 3);

Influence vector corresponding to the transducer

Piezoelectric capacitance of the transducer when it is blocked and when it is left free

Static capacitance of the transducer, when it is attached to the structure

Piezoelectric constants Dirac fonction of x

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<^0 A,' F fi G{s) fc2 K Ka Kf 3Kf iKl, **‘n L Xj{s) M fJ-Mi n N Ü LÜq Ufij fij, ^fîj P Pns(^)î Qnsi^) Q, Qj r.e R s ^ns {t)i Cns{t) <t{w), CT(.) t Kronecker symbol Elongation of transducer j Force amplitude

Force applied by the transducer/actuator to the structure Mode shapes matrix and its vectors

Transfer function (or transmissibility matrix) Electromechanical coupling factor

Stifïhess matrix (it can be also a scalar)

Stiflhess of the transducer with short-circuited électrodes Global generalized electromechanical coupling factor of ail the transducers, corresponding to mode i

Generalized electromechanical coupling factor of transducer j corresponding to mode i

GeneraUzed electromechanical coupling factor of transducer j, respe- ctively, for the sine and for the cosine modes with n nodal diameters Generalized electromechanical coupling factor of ail the transducers for the sine and the cosine modes of n nodal diameters

Inductance

Eigenvalue of the total admittance matrix Ytot Mass matrix (it can be also a scalar)

Mass ratio: fx = m2/mi Modal mass of mode i

Number of nodal diameters (in Chapter 4-8); or munber of sÜces of the PZT stack(in Chapter 2-3)

Number of sectors Angular speed

Electrical résonance frequency

Résonance frequency of mode i, respectively, when ail the transducers are short-circuited, when they are open-circuited, and when only the

transducer is open-circuited Number of transducers

Generalized forces in modal coordinates Electrical charge vector and its component Polar coordinates

Résistance

Laplace variable: s = ju>-, or number of rotating forces (in Chapter 4) Modal amplitudes of the sine and the cosine modes of nnodal diameters and s nodal circles

Cumulative Root Mean Square RMS; and RMS value Time variable

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V

V, Vj

Wn,s,

W{r,e,t)

Y, ^truct Viot

FVequency Response Function of the blades Strain energy

Electrical voltage vector and its component

Sine and cosine mode shapes of n nodal diameters and s nodal circles Time response of the disk, at coordinates (r, 6)

Displacement, velocity and accélération of the degree of freedom Electrical damping coefficient

Mechanical damping coefficient, corresponding to mode i Admittance matrix of the electrical shunt

Admittance matrix of the structure Total admittance matrix. _{^truct "i"}

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Chapter 1 Introduction

Reducing the fuel consumption is a major issue in the air transport industry; this calls for innovative solutions in the design and operation of aircrafts, one of the main axes of improvement being the réduction of weight by increasing its functional efficiency. This can be achieved in varions ways, such as the use of new materials, of lightweight structural design and of smart structures. This study is concerned with a new design of a bladed drum, called BLUM®, Fig.1.1. It is a new design of the rotor of low-pressure compressors manufactured by SAFRAN Techspace Aero; it consists of a rotor drum on which the blades are attached by friction welding; its main advantage is the considérable weight savings comjîared to the classical design, but it is obtained at the expense of an extremely small inhérent damping, of the order of ^ ~ 10“'*.

Figure 1.1: The BLUM® made of 3 blade wheels. Left: low pressure compresser, integrating the BLUM®; Right: exploded CAD view of the low pressure compresser.

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2 1 Introduction

In this thesis, we investigate the use of piezoelectric shunt dumping to reduce the vibration level in the blades of the BLUM®.

1.1 Vibration of Turbomachinery Components

Axial timbomachines can be classified into two main categories depending upon whether the fluid/gas is supplying energy to the rotor (turbines), or the rotor is supplying energy to the gas (compressors). For both categories, the machine is aimed to generate a mechanical work in order to drive a secondary System which can be the fan blades of a jet engine, the rotor of an helicopter, the shaft of an electrical generator, or something else.

Low pressure

Figure 1.2: Jet engine turbofan: RoUs Royce Trent 900.

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1.1 Vibration of Turbomachinery Components 3

it is mixed with the fuel and burned; the resuit is a very hot compressed gas. The highly energetic gas is then expanded through the turbine which produces the rest of the thrust and provides the mechanical work that drives the engine rotor, including the compresser stages and the fan blades rotor. The performances of turbomachines dépend essentially on the compression rate provided by the compresser modules. An example of a turbofan is presented in Fig. 1.2.

1.1.1 Blisks

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4 1 Introduction

In conventional compresser bladed disks, the blades are attached to the disk using a root fixing feature, Fig.l.3.b. This fixing solution offers an inhérent damping to the blades and facilitâtes the maintenance of the engine and the replacement of damaged blades; however, it involves a considérable mass which increases the centrifugal loads applied to the disk, and decreases the energy efficiency of the engine. To reduce this parasitic mass, new designs of bladed disks, referred to as blisks, are being developed since three décades (Fig.l.3.a), where the blades and the disk are made in a single piece. This solution reduces considerably the mass of the engine rotors and the number of pièces, but also complicates the maintenance of the engine and increases the costs. Further weight saving of the rotors is possible by using bladed rings,

blings, so as the disk is replaced by a reinforced ring. Bladed drums, or blums,

involving blade fixing roots may also be made lighter by manufacturing them in a single piece.

1.1.2 Fatigue

In spite of the weight lowering provided by the blisk configuration, preserving the dynamic behavior and the strength of the disk and the blades is still a challenging task for the designers. Indeed, under normal operating conditions of the engine, while the disk is mainly stressed with static centrifugal loadings and température gradients, the blades are subjected to high static stresses induced by the high pres sure and température of the flow, and also to d}Tiamic stresses due to the dynamic variation of the fiow pressure (due to the presence of upstream stator vanes). This dynamic loads may excites the résonance of the blades, leading to high levels of vibration, responsible of high cycle fatigue and early failure of the engine (see e.g. Fig. 1.4).

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1.2 Damping of Turbomacbinery Blades 5

1.2 Damping of Turbomachinery Blades

Blades axe the most critical engine parts suffering from vibrations, that can lead to high cycle fatigue and to the failmre of the engine, unless they are damped. There are three main phenomena producing damping in a turbomachinery bladed disk: the high pressure/density of the flowing fluid, the inhérent damping of the material, and the friction produced at the joints of the engine. In monolithic blisk configmrations, there is no friction damping possibility and only the aerodynamic damping may prevent the vibration of the blades. However, some self-excited and self-sustained vibrations due to aerodynamic instabihties may occur; these vibrations may be reduced or avoided by increasing the inhérent damping of the blades.

1.2.1 Friction damping

The most classical way to increase the damping of tmrbomachinery blades is to incor- porate friction devices: (i) at the root of the blades, e.g. Fig.1.5; or (ü) between the blades themselves; and/or (ii) at the shrouds. The idea is to croate a contact smrface between the vibrating parts, allowing the dissipation of energy (Grifïin, 1990). Since dry friction dampers are nonUnear Systems, the damping performance dépends upon the level of vibration and the contact area between the parts (which is also a design parameter).

Figure 1.5: FViction damping device introduced at the blade root; from (Yeo and Goodman, 2006

).

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6 1 Introduction

(a) Friction ring dampær

(b) Friction fingers damper

Figure 1.6: Friction damping solutions for monolithic blisk: (a) Friction ring device (Lax- alde et al., 2007). (b) Friction fingers mechanism (Stangeland et al., 2007).

1.2.2 Viscoelastic treatment

Viscoelastic materials are extensively used for damping in many mechanical engineer ing applications; they are also used in turbomachinery. Fig. 1.7 shows an example of the use of viscoelastic material, introduced at the interface between the blade root and the disk. For monolithic blisks, viscoelastic coating may also be a damping candidate, however high température conditions limits its utilization.

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1.2 Damping of Turbomachinery Blades 7

1.2.3 Other solutions

There exist also other solutions for the damping of turbomachinery blades, such as; electromagnetic dampers acting on the blades through actively controlled magnets (Hoffman, 1996); impact dampers integrated into the blade tip (Duffy et al., 2004); and damping materials integrated into hollow blades (Motherwell, 2005). Despite the eflBciency of these solutions, they are complex and difficult to implement, as compared to friction damping solutions.

1.2.4 Piezoelectric shunts

Piezoelectric materials offer many advantages for structural control, including energy efBciency and easy intégration. In the last two décades, research hâve been conducted on piezoelectric shunt damping techniques, but ail these techniques are faced with many issues making their applications very limited. Despite these issues, many academie studies propose to use piezoelectric shunts for the damping of tur bomachinery blisks.

Tang and Wang (1999) propose to attach one piezoelectric patch on each blade of the blisk. The patches are then shunted on tuned passive and active circuits in order to increase the damping of the blade modes. Despite the elegance of the proposed solution, it is very difficult to implement it in a real blisk, unless the patches are integrated within the blade (to avoid any interaction with the flow), and a power source is available.

Kauffman and Lesieutre (2012) propose also to integrate the patches in the blades, and then detuning the résonance frequency of the blades by switching the state of the électrodes (open or closed) when the excitation frequency (a multiple of the ro tation speed of the engine) approaches the résonance frequency of the blade. Once again, the problem of the location of the piezoelectric patches and the complexity of the switching logic (based on the knowledge of the rotation speed) are serions issues.

Many other studies attempt to use the developments in piezoelectric shunt damping techniques for the damping of turbomachinery blades and jet engine fans (see e.g. Sénéchal, 2007; Zhou et al., 2014).

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8

1 Introduction

1.3 Motivations

The use of friction dumping Systems in turbomachinery bladed disks is a technol- ogy whose maturity and efficiency hâve been demonstrated and confirmed during several décades; however, the new design of bladed disks in monolithic blisk con figurations renders impossible the use of classical friction dumping solutions. The need of Systems increasing the blade dumping of monolithic blisks, and the inability of using classical dumping technologies are the main motivations of the présent work. This thesis has been reaJized within em industrial project involving SAFRAN Techspace Aero as industrial partner. The aim of the project is to develop a passive piezoelectric shunt device for the dumping of the blade modes of the BLUM®.

1.4 Outlines

The thesis is organized in the following way:

• Chapter 2 is devoted to the dérivation of the governing équations of a structure equipped with a set of piezoelectric transducers. The optimal tuning of the electrical circuits are also derived, for various types of linear shunts.

• Chapter 3 investigates the Synchronized Switch Dumping on Inductor tech nique (SSDI), and compares it to the linear R shunt, and RL shunt; the study is supported by simulations and experiments conducted on a cantilever beam. The use of a négative capacitance, to enhance the dumping performances, is also investigated.

• Chapter 4 describes the résonance conditions of axisymmetric and rotationally periodic structures, excited with rotating forces.

• Chapter 5 investigates the linear RL shunt when it is applied to a bladed rail (periodic structure). The study is conducted numerically and experimentally.

• Chapter 6 extends the implémentation of the linear RL shunt for the dumping of the blade modes of a bladed drum. The implémentation considers several piezoelectric patches located on the inner side of the blade support rim, and shunted independently to linear RL circuits. The numerical and experimental results are presented. •

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1.5 References 9

only two inductors. The proposed strategy is validated numerically and experimentally on a circular plate, and validated nmnericaJly on the bladed drum.

• Chapter 8 analyzes the effect of the blade mistuning on the proposed parallel and indépendant RL shunt; the simulation and experimental results are also presented.

I. 5 References

Australian Transport Safety Biueau. Examination of a failed compressor blisk. Technical Report 39/01, Ref. BE/200100016, Occurance 200102263, September 2002.

J. F. Cortequisse and A. Lhoest. One-piece bladed drum of an axial tiubomachine compressor, Jime 24 2010. US Patent App. 12/580,070.

K. P. Duffy, G.V. Brown, and R.L. Bagley. Self-timing impact damper for rotating blades, December 7 2004. US Patent 6,827,551.

•T. H. Griffin. A review of friction dumping of turbine blade vibration. International

Journal of Turbo and Jet Engines, 7(3-4):297-308, 1990.

.T. Hoffman. Magnetic dumping System to limit blade tip vibrations in turbomachines, February 13 1996. US Patent 5,490,759.

.1. L. Kauffman and G. A. Lesieutre. Piezoelectric-based vibration réduction of turbomachinery bladed disks via résonance frequency detuning. AIAA journal, 50(5):1137-1144, 2012.

D. Laxalde. Etudes d’amortisseurs non-linéaires appliqués aux roues aubagés et aux

systèmes multi-étage. PhD thesis, Ex;ole Centrale de Lyon, 2010.

D. Laxalde, F. Thouverez, J. J. Sinon, and J. P. Lombard. Qualitative analysis of forced response of blisks with friction ring dampers. European Journal of Mechan-

ics & Solids, 26(4):676-687, 2007.

A. Motherwell. Hollow component with internai dumping, December 27 2005. US Patent 6,979,180.

N. Nguyen. Blading System and method for controlling structural vibrations, Au- gust 15 2000. US Patent 6,102,664.

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10

References

A. Sénéchal. Réduction de vibrations de structure complexe par shunts piézoélec triques: Application aux turbomachines. PhD thesis, Conservatoire National des

Arts et Métiers, 2007.

M.L. Stangeland, R.E. Berenson, G.A. Davis, and E.J. Krieg. Turbine blisk rim friction finger damper, May 15 2007. US Patent RE39,630.

J. Tang and K. W. Wang. Vibration control of rotationally periodic structures using passive piezoelectric shunt networks and active compensation. Journal of vibration

and acoustics, 121(3):379-390, 1999.

J. C. Williams and E. A. Starke Jr. Progress in structural materials for aerospace Systems. Acta Materialia, 51(19):5775-5799, 2003.

M. Wlasowski. Reduced monobloc multistage drum of axial compressor, Septem- ber 15 2011. US Patent 20,110,223,013, 2011.

S. Yeo and P.J. Goodman. Vibration damper for a gas turbine, July 19 2006. EP Patent 1,249,576.

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Chapter 2 Linear Passive Shunts with

Discrète Piezoelectric

Transducers Array

In this thesis, we study the use of shunted piezoelectric transducers for the dumping of rotationally periodic structures. When only one vibration mode is targeted and its shape is known, we propose to lay a set of identical piezoelectric transducers symmetricaUy on the structure, such that each transducer bas the same authority over that mode; the polarization of the transducers is then arranged in such a way that they generate in-phase electrical charges when the structure vibrâtes according to the targeted mode. With these considérations, the behavior of the System becomes similar to that of a single degree of freedom System equipped with a set of piezoelectric transducers (Fig.2.1).

The aim of this chapter is to formulate the behavior of an arbitrary flexible struc ture equipped with a set of piezoelectric transducers. The formulation is limited to the situation where the transducers are identical and hâve an equal authority over the targeted mode, which is représentative of oin problem. We consider only the linear résistive R and resistive-inductive RL shunts. Then, we dérivé the charac- teristic équation of the shimted structure for two configurations of the transducers:

(i) when they are shunted independently to identical circuits; (ii) and when they

are connected in parallel and shunted to a single circuit. Previous results are used to deduce the optimal tuning of the electrical parameters. This chapter can be considered as a generaUzation of Chapter 4 of (de Marneffe, 2007).

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12 2 Linear Passive Shunts with Discrète Piezoelectric Transducers Array

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2.1 Introduction 13

2.1 Introduction

Piezoelectric materials hâve been used extensively as sensors and actuators in many engineering fields. Their merits of being energy efficient and easily intégrable, en- coinaged researchers to develop varions schemes for vibration control and structural damping. Major research in structural dumping was conducted during the 80’s, in the Framework of large space structures programs. Along these years, many active control schemes based on piezoelectric transducers hâve been developed; the most successful of them are the techniques based on collocated actuator-sensor pairs, such as the Positive Position Feedback (Fanson and Caughey, 1990), the Direct velocity Feedback (Balas, 1979), and the Integr£il Force Feedback (Preumont et al., 1992), to cite only the most popular.

Meanwhile, interests went also to purely passive vibration damping techniques based on shunted piezoelectric materials. This has been initiated by the proof of concept demonstrated by (Forwaxd, 1979): He demonstrated the damping capability of a piezoelectric patch when it is shimted to an inductor L. (Hagood and von Flo- tow, 1991) provided thorough analytical formulation of the linear résistive R and resistive-inductive RL shunts. Their results triggered a huge number of research in the field of piezoelectric shunt damping (Benjeddou, 2000). This research, focused on linear passive RL shunts, can be placed into 4 main categories, depending upon the number of the transducers and the number of targeted modes:

• Single mode daunping with a single transducer: The structure is equipped with a single piezo; the challenge is to find a simple passive circuit able to dis- sipate the transformed electrical energy, and to dérivé the optimal tuning of the electrical components. (Hagood and von Flotow, 1991) showed how the behavior of the R shunt can be assimilated to that of a viscoelastic material, while the RL shunt can be assimilated to the Den Hartog Dynamic Vibra tion Absorber (cfr. Appendbc B); they derived the optimal tuning of the R shunt and the sériés RL shunt. Based on these results, (Wu, 1996) proposed to connect the resistor and the inductor in parallel. However, (Caruso, 2001) demonstrated the similarity between the sériés and the parallel RL shunts: they hâve the same performances and they require almost the same inductor. One should notice that most of research is focused on RL shunt because of its performance, very superior than the R shunt. Other nonlinear passive shunts appears after, they are discussed in the next chapter. •

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circuits to the transducer, in such a way that the résonance frequencies of the electrical network are identical to the résonance frequencies of the targeted modes (see e.g. Hollkamp, 1994, Wu, 1998 and Fleming et al., 2002). Many successful designs hâve been proposed in this context, however, complex circuits requiring very fine tuning are needed.

• Single mode damping with several transducers: When only one mode is damped with many transducers, the tuning of the electrical circuits is different from what it is when only a single transducer is used. The transducers can be shunted independently to linear circuits in a decentralized control architecture, or combined together, where varions architectures are possible. This chapter formulâtes the shunt damping of a single mode with a set of piezoelectric transducers; two configurations are considered: (i) when the transducers are shunted independently on linear circuits; and (ii) when they are mounted in a single parallel loop, and shunted on a single linear circuit.

• Multi-mode damping with several transducers: in this category, two main architecture can be encountered: (i) Independent (decentralized) shunts, such that each transducer is shunted independently to a passive circuit; the circuits can be a single RL branch tuned on a single mode, or multi-branch

RL circuit tuned on many modes; (ii) Coupled (centralized) shimt, where

the transducers are combined between them through electrical networks. The coupling between the transducers can be done in different ways: by simply connecting the RL branches between adjacent transducers, or alternatively by employing more complex networks involving many RL branches (see e.g. deirisola et al., 2004, Maurini et al., 2004, Bisegna et al., 2006); the aim is always to maximize the energy absorption of the circuit. (Tang and Wang, 1999) proposed an intuitive and élégant architecture of coupled piezoelectric shunts for a rotationally periodic structure, where the resulting circuit is a ro-

tationally periodic network which exhibits the same dynamics as the structure:

either for the mode shapes and the résonance frequencies.

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2.2 Governing équations 15

2.2 Governing équations

Consider the linear structure of Fig.2.2 equipped with p identical linear piezoelec- tric transducers working according to the dss mode. Each transducer consists of a stacking of n identical piezoelectric dises polarized through their thicknesses.

The structure is assumed undamped and is represented by its mass and stifhiess matrices M and K respectively. These matrices are computed without considering the stiffness and the mass of the transducers. Using the Lagrange formulation, the governing équations of the structure are given by:

P

Mx + Kx = -J2 bjfj (2.1) i=i

where fj are the forces applied by the transducers to the structure and bj are the influence vectors indicating the degrees of freedom where the transducers act on the structure. Each piezoelectric transducer j of Fig.2.2 is governed by the following constitutive équations:

f Qj 1

r C(1 -

ndssKa

\ fj J —ndssKa Ka

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16 2 Linear Passive Shunts with Discrète Piezoelectric Dransducers Array

is the stiffness of the transducer with short-circuited électrodes (1^- = 0), C is the free electrical capacitance {fj = 0), and ^33 is the piezoelectric constant, k is the electromechanical coupling factor; it measures the abihty of the transducer to convert mechanical energy into electrical energy and vice versa. It can be expressed as:

■n?d“^Ka

C (2.3)

Because of the electromechanical coupling, the mechanical stiffness of the transducer dépends on the electrical boundary conditions and the electrical capacitance dépends on the mechanical boundary conditions:

• The stiffness of the transducer in open circuit {Qj = 0) is related to that in short-circuit (V^ = 0) by:

^1

fj 'Qi=o

(1

- fc

2

)

(2.4)

• The capacitance of a blocked transducer (Aj = 0) is related to that of a free transducer {fj = 0) by:

<^^ = &I

_Vj

a,=o = C(1-A:") (2.5) Finally, by substituting Eq.(2.2) into Eq.(2.1), one gets the governing équations of the full electromechanical System:

P P Mx + {K + KaY, bj Vj (2.6) j = l j=l < Qi ' ■C(l-A;2) 0 0 ndssKabl VI Va Q2 . = 0 C{l-k^) ■ 0 nd33Kab2 Qp 0 0 ■ C(1 - fc2) nd33Kab^ X (2.7) 2.2.1 Modal basis

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When ail the transducers are short-circuited, i.e Vj= 0, the structure obeys:

+ K + Ka £ bjbJj X = 0

where s is the Laplace variable.

(2.8)

When the transducers are charge-driven instead of voltage-driven, Vj can be eliminated from Eq.(2.6) and Eq.(2.7), leading to

Ms^ +K + Ka

(1 - U

£ bjb] I ^ =

ndssKg

£ bjQj (2.9)

thus, if the transducers are open-circuited, i.e. Qj = 0, the structure obeys:

(

2

.

10

)

which is équivalent to Eq.(2.8) where the short-circuited stifîness Kg of the stand-alone transducer is replaced by the open-circuited stiffness Kg/{1 — fc^).

• Finally, when ail the transducers are short-circuited except the transducer is open-circuited, i.e. V} = 0 and j ^ l, the new équation of the System is:

+ K + jJ^^bibJ + è "" = 0 (2.11)

The solutions of these eigenvalue problems hâve three different sets of natural fre- quencies and mode shapes, we refer to the natural frequencies: (i) when ail the trans ducers are open-circuited by Cli, (ii) when ail the transducers are short-circuited by Wi, (iii) and finally, when ail the transducers are short-circuited except one by *f2i, where the superscript refers to the transducer which is open-circuited.

Note that, for the various electrical boundary conditions, the variation of the mode shapes is very marginal and they are supposed nnchanged for simplicity.

2.2.2 Modal coordinates

The System of Eq.(2.6) can be projected into its modal coordinates using the trans formation: X = where $ = ((^i,..., <^„) is the mode shapes matrix of the structure obtained by solving the eigenvalue problem of Eq.(2.8):

{K^KgŸ^bjb])-JlM

\ J=i )

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The mode shapes are orthogoneil with respect to the mass and the stifïness matrices;

= diag{fj,i} (2.13)

^'^{K + ATa è = diaginiijf} (2.14) j=i

where Wt are the natural frequencies with short-circuited transducers and fii are the modal masses. Using the transformation: x = $a, Eq.(2.6) can be rewritten in modal coordinates as:

P P

M$â + {K + Ka^^ bjbj)^a = nd^^Ka (2-15)

j=i i=i

which gives, after left multiplication by the use of the orthogonality relations (2.13) and (2.14), and the Laplace transformation:

X “ fid^Kd <t>i<l>ï \

/q(s2 + Jf) ) ^bjVj (2.16)

Substituting x into Eq.(2.7), the electrical charges in the transducers obey:

'

Qi

' ■

Cn C12

• •

C\p

Q

2

.

. =

C21

C22 • •

C

2

p

<

V2

. Qp . _

Cpi Cp

2 • ■ ^pp . . ^p, (2.17) with Cji = ôjidl - fc2) (2.18)

after using the equahty Ck^ = n^d^^Ka, Sji is the Kronecker symbol: ôji = 1 if

j = l, and Sji = 0 if J / Z.

Equation (2.17) can be considered as the dynamic electrical capacitance of the structure such that its off-diagonal terms represent the inter-transducers coupling provided mechanically by the structure, while each diagonal term Cu represents the capacitance of the transducer measured when ail the other transducers are short-circuited;

Cu = ^ = C{1 - A:'

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Finally, by introducing the définition of the modal fraction of strain energy:

l

t'i = Kg{bf4>if

(HLjf

(2.20)

Eiq.(2.19) can be rewritten as:

Cu

Qi

Vi

= C(1 - fc2) + Ck‘

_{{Çc^V^^Z+i)}}

(

2 .

21 )

‘ui is the ratio between the strain energy in the transducer and the total strain

energy of the structure when it vibrâtes according to mode i.

2.2.3 Generalized electromechanical coupling factor

The définition of the electromechanical coupling factor of the standalone transducer is generalized for a spécifie mode i when the transducer is integrated into a multi- mode structure. It is referred to as the generalized electromechanical coupling factor, and is given by:

Kf =

fi?

(

2

.

22

)

where flj are the natural frequencies of the structure when the transducer is open- circuited, and Wj are those when the électrodes are short-circuited.

When the structure is equipped with p transducers and not only one, Eq.(2.22) is adapted; for each transducer, a generalized electromechanical coupling factor is defined:

=

'fi? (2.23)

where 'fii are the natural frequencies of the structure when ail the transducers are short-circuited except the f''*, while Wj are the natural frequencies when ail the trans ducers are short-circuited.

One should notice that the pôles of Qi/Vi are at Ui, while the zéros are at ^fii (i.e. Qi = 0). Therefore, in practical situations, the electromechanical coupling factor of each transducer can be easily estimated by measuring its impédance when the other transducers are short-circuited.

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In the literature, the définition

‘Kf = ‘nf-ca?

uf l —k^ Vi (2.25)

is often used instead of that of Eq.(2.23). The différence between the two définitions is marginal for small k and however, the définition of Eq.(2.25) is simpler.

When the transducers are used simultaneously, we define the global generahzed elec- tromechanical coupling factor as the couphng factor of the équivalent transducer:

n? (2.26)

where fii and Ui are the natural frequencies of the structure when ail the transducers are open-circuited and when they are short-circuited respectively. In a different way, if one considers the définition of Eq.(2.25), one finds easily:

= = (2-27)

j=l j=l

Here, Ki is a direct metric of the total amount of strain energy which can be trans- formed into electrical energy by the p transducers. This définition is useful for the tuning of the resistor R of the J?L-shunt.

2.3 Electrical admittance

2.3.1 Admittcince of a single transducer

Referring to Eq.(2.19), the electrical admittance of each transducer is defined when ail the other transducers are short-circuited. It is given by:

yM -

lv,.=o “

V,

k‘^Ak^J2

r«2/,.,2.

r(w+i)

V^iv,=o Vi

or by introducing ‘Kf from Eq.(2.25),

(2.28)

(2.29)

When only the transducer is shunted to a circuit of admittance ^V^hunti the Kirchhoff low gives:

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2.3 Electrical admittance 21

Thus, the chaxacteristic équation is obtained by expressing the equality between the admittance of the structure and that of the passive shunt,

r,(s) = -'nhu„t(s) (2.30)

Therefore, for different type of shunts, to identify the optimal tuning of the shunt admittance *V^hunt(^)> one should first solve analytically Eq.(2.30), and then select the optimal values of the shunt which provides highest values of damping.

2.3.2 Admittance matrix of the trïmsducers array

In a similar way, from Eq.(2.17), we define the electrical admittance matrix of the structure equipped with p transducers as the relationship between the voltage vector

V = {1^1,..., Vp}^ and the current vector I = sQ =

s{Qi,...,Qp)^'-I = sQ^ Estruct(s)V^ (2.31)

where l^truct(s) is, in this case, apxp matrix. When aU the transducers are shimted independently, the Kirchhoff low yields:

l^truct (S)17(S) = -Eshunt(s)V^(s)

[Estruct(s) + Fshunt(s)] V{s) = rtot(s)F(s) = 0, and V(s) ^ 0 (2.32) Here, l^hunt is a diagonal matrix and its element corresponds to the admittance of the circuit shunted to the transducer. The pôles of the independently-shimted structure are found by solving the équation

Etot(s)V^(s) = 0, andK(s)/0 (2.33)

They are the solution of:

det[Ftot(s)]= nAj(s)-0 (2.34)

j=i

det[ytot(s)] is the chaxacteristic équation of the System, while Xj{s) axe the eigen- values of the matrix Itot(s). Therefore, to identify the pôles of the shunted System in function of the shimt parameters, one should, first, find the analytical formulas of Aj(s), and then, solve Aj(s) = 0. However, in a general case, when the transducers are different and placed arbitrarily with respect to the targeted mode shape, it is very difRcult to find an analytic formulation of Aj (s) because the matrix Ttot(s) does not corne into a spécial form.

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22 2 Linear Passive Shunts with Discrète Piezoelectric 'B'ansducers Array

transducers, the matrix VtotCs) becomes a symmetric Toeplitz matrix in the vicin- ity of the targeted mode frequency, and the computation of its eigenvalues \j{s) with an analytical solver gives simple formulas. In the case where the number of transducers p is even, the matrix itot(s) becomes circulant and the analytical for mulas of its eigenvalues \j{s)exist (cfr. Appendix C). This situation is common for axisymmetric structures (see e.g. Tang and Wang, 1999).

2.4 Shunt of a single d.o.f. System

Without loss of generality, to illustrate the situation of the structure when the modal amplitudes are equal for ail the transducers, let us consider the one degree of freedom System of Fig.2.3. The aim is to identify the optimal values of the electrical components for the various types of shunt: R and RL. The system consists of p identical linear transducers placed in parallel to the spring K and supporting a mass

M. The dynamic behavior of this system is an approximation of the behavior of an

axisymmetric structiure equipped with p = 2n transducers and vibrating according to its mode with a shape of n spatial harmonies (nodal diameters), Fig.2.1.

(a)

Figure 2.3: One degree of freedom System equipped with p identical linear transducers: (a) Parallel RL shunt; (b) sériés RL shunt; (c) R shunt.

The displacement of M and the electrical charges in the transducers Qj are governed by:

p Mx + {pKa + K)x = ndssKa Y, Vi

j=l

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2.4 Shunt of a single d.o.f. System 23 and ' Qi ' ■ C(1 - fc2) 0 0 ndi^Ka

^

•• • Q2 > = 0 C(1 - fc2) . 0 nd^Ka < Qp 0 0 ■ C{1 — fc2) ndssKa _X - (2.36)

Following the same steps as in section 2.3, and by eliminating x in Eq.(2.36), one gets the electrical admittance of the System as:

Fstruct(®) — S

■ Cil (s)

Cl

2

(s) • • Clp(s)

C2l(s)

C22(S) • • C2p(s)

Cpi(s)

Cp2(s) ■ •

Cpp{s) such that 7ij(s) = C^cSij(l — k^) + k‘

+

1

(2.37) (2.38)

Sij is the Kronecker symbol, Wj is the natural frequency of the System when ail the transducers are short-circuited, and

U = Ka pKa + K

is the modal fraction of strain energy of the transducer. The electromechanical coupling factor of each transducer is given by:

_

1 (1 - fc2) + ‘uk^

which can be approximated for small and V, by

(l-fc2)

(2.39)

(2.40)

Finally, the global electromechanical coupling factor of ail the transducers when they are used simultaneously is defined as:

Z=1

(2.41)

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2.4.1 Independent loops

An obvions way to use a set of p transducers for the damping is to connect them indep>endently to the shunt circuits. In this case, the Kirchhoff law yields:

[ntruct(s) + nhunt(s)] = rtot(s)F - 0

where Vtot(s) is, in this case, a circulant matrix with the following shape:

Yi{s) Y2{s) ■■ Y2{s)

>^2(s) Tl (s) ..■■ Y2{s)

^2(S) Y2{s) ■■■■ n(s)

such that:

Kl (s) =

sc(l-k^ +

+ Ys(s)

\ s /üJi + l J

and l^(s) is the diagonal term of the shunt matrix

l^hunt(s)-(2.42)

(2.43)

(2.44)

(2.45)

The pôles of the independently-shunted transducers System are found by solving the équation:

P

det [Vtot(s)] = n = 0 (2.46)

j=i

and since Itot(s) is a circulant matrix, its eigenvalues Xj{s) are found analytically (see Appendix C):

Ai(s) = Yi{s) + (p-mis) =sc(l- + Y,{s) (2.47)

\ ^ /‘*^i + ^ /

Xj{s) = Vi(s) - Y2(s) = sC(l - fc2) + y^(s) , for j > 1 (2.48)

One can observe that Ai (s) = 0 is the characteristic équation of the structure when ail the transducers are shimted independently; while Aj(s) = 0, j > 1 are the charaic- teristic équations of the shunted System with blocked transducers (i.e. bjx = 0 and the transducers behave as pure capacitors without any couphng with the structure).

By substituting Eq.(2.40) and Eq.(2.41), one gets

Ai(s) = sC(l - fc2)

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2.4 Shuiit of a single d.o.f. System 25

The pôles of the shunted System are obtained by solving Ai (s) = 0. The opti mal values of the shunt components can be chosen to maximize the damping ratio. The solution of Ai (s) = 0 is demonstrated in (de Marneffe, 2007) for varions linear shunts, where the optimal values of the electrical components are derived. These are summarized in Table.2.2, at the end of this chapter, for a multi-mode structure.

One should notice that although the transducers are shunted independently, the electrical components of the shunts are tuned using the global electromechanical coupling factor = p ■ instead of the single transducer electromechanical cou- pling factor This cornes from the fact that ail the transducers are targeting the same mode at the same time and they must be tuned simultaneously in such a way that the optimal tuning of ail the shunts (acting together) results in the highest damping ratio.

2.4.2 Parallel loops

Another way to use a set of piezoelectric transducers is to connect them together, in a parallel or in a sériés loop, so that they act as a single transducer: the parallel shunt is more advantageous than the sériés because it results in a capacitance equal to the sum of the individual capacitance of ail the transducers, which, for the RL shunt, reduces the demand on the inductor. In this section we consider the trans ducers connected in parallel in order to dérivé the optimal timing for different type of shunts.

When shunted in parallel, the transducers act like a single transducer and gen- erate a total electrical charge Qt equal to the smn of the charges generated by each

transducer: ^

Qt = J2Qi (2-50)

J=1

Since the transducers are identical and their voltages Vj are identical too, the re- sulting voltage Vt is then

Vt = Vi = V2 = ... = Vp (2.51) By substituting Eq.(2.50) and Eq.(2.51) into Eq.(2.35) and Eq.(2.36), the System with parallel shunted transducers obeys

Mx + {K + pKa)x = p{nd33Ka)Vt (2.52)

Qt = pC(l - k^)Vt+ p{nd^Ka)x (2.53) The admittance of the System V^truct is then obtained by ehminating xin Eq.(2.53):

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where 1/ is, in this case, the total modal strain energy in ail the transducers:

i=i

Finally, the pôles of the System when it is shunted to an admittance Ys{s) are obtained by solving the scalar équation

Ftot(s) = psC{l - fc2) ^1 +

j

+

n(s)

= 0 (2.55)

The optimal tuning of the different shunts, which maximizes the modal dumping, are summarized in Table.2.2. One can observe that, when shunted in parallel, the trans ducers act exactly like a single transducer with an équivalent open-circuit stiShess

Keq = pKa, and a free capacitance Ceq = pC. Moreover, by comparing Eq.(2.55)

and Eq.(2.49), the char^lcteristic équations of the System are identical except for the value of the electrical capacitance which is p times bigger when the transducers are mounted in parallel.

In conclusion, shunting the transducers independently or in parallel results in the same damping performances because the global electromechanical coupling factor is the same for both configurations. However, the main advantage of the parallel shimt is the increase of the apparent capacitance of the shunted transducers which reduces, for an RL shunt, the demand on inductors by p^: only 1 inductor is ne- cessitated instead of p inductors, with a size p times smaller than those required by an indépendant RL shunt. This constitutes a significant advantage which will be exploited later.

2.5 Shunt of a multi-mode structure

AU the results arising from the single degree of freedom System can be extended, without loss of generalities, to a multi-mode system. The main difficulty to identify the pôles of the structure when it is shunted independently is the analytical dérivation of the characteristic équation ffom the matrix Vtot(s) (Eq.2.33). Since the matrix Ttot(s) is always symmetric, it would be possible to find an analytical solution of its eigenvalues Aj(s). However, although this solution exists, a more complex mathematical formulation of the problem is needed to dérivé analyticaUy the different équations; but this is not in the scope of this thesis.

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2.5 Shunt of a multi-mode structure 27

Figure 2.4: Diflferent typ>e of linear shunts: a) linear R shunt; b) linear sériés RL shunt; c) linear parallel RL shunt.

over a spécifie mode. Therefore, for the piupose of simplicity, for the targeted mode

i, only the situation when the piezoelectric transducers act in phase and their modal

amphtudes are identical will be considered, i.e.:

bj(j>i(j)jbk > 0,

and

b]d>i = bj4,i,

where bj<f>i is the amplitude of mode i in the transducer j.

2.5.1 Independent loops

Let us consider again the System of Fig.2.2. If the transducers are mounted in such a way they hâve the same modal fraction of strain energy, and they act in phase, then

b](i>i = bj<j>i (2.56)

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Figure 2.5: (a) Capacitance Q/V of a piezoelectric transducer embedded in a fictitious 3 mode structure; (b) the approximation around Wj if the modal density is low.

approximated in the vicinity of Wj by

Cji — C - k^) +

Pi(s2 + uf) + k^KaJ2_t>i

{bj(j)t4>tbi)\

fHi^t J

(2.57)

If one neglects the off-diagonal terms of the quasi-static contribution of the high frequency modes [last component in Eq.(2.57)], the capacitance matrix is circulant and, using the définition oî‘i>i and ^Kf, and taking in account Eq.(2.56), it can be written

Cj, = C(1 - fc2)

E

(2-58)

This expression is very close to that obtained in section 2.4.1 for a single d.o.f. System. Following the same procedure, the eigenvalues are readily obtained:

Al (a) = sCi2 + sC{l -fc2)

j

+ y,(s) (2.59)

Aj(s) = sC(l - fc2) + y(s) , for j > 1 (2.60)

with

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2.6 Shunt performances 29

The meaning of Ci2 is illustrated in Fig.2.5. It satisfies the inequ8ility:

C'static > Ci2 > C^{1 — k^)

The static capacitance of a single transducer Cgtatic is given

Cstatic = C ^1

-For the low frequency modes, C,2 is close to

Cstatic-(2.62)

Equation (2.59) is similax to the characteristic équation of a multi-mode structure equipped with a single transducer such that its free capacitance is C and its gener- alized electromechanical coupling factor is = P • For the varions types of shimt, the optimal tuning of the electrical components are derived using Eq.(2.59); they are summarized in Table.2.2.

2.5.2 Parallel loop

If the transducers are shimted together in parallel, the resulting charge Qt is equal to the sum of the charges produced by ail the transducers. Therefore, in a similar way to the single degree of freedom System, it is easy to détermine the characteristic équation of the System with the transducers mounted in parallel:

Ftot(s) = SpCi2 + spC{l - fc^) + ^s(s) (2.63)

This équation is also solved similarly to the characteristic équation of a multi-mode structure equipped with a single transducer of a free capacitance pC and a generalized electromechanical coupling factor Kf = p •

For both configurations of the transducers, independent loops and parallel loop, the System has an équivalent electromechanical coupling factor Kf = p ■ ^Kf which must be considered for the tuning of the circuit components, whether the transducers are used independently or in parallel. As previously noted, with the parallel shunt, the équivalent capacitance is p times bigger than in the independent configmation, which reduces the demand on inductors by p^ when shimted to a RL circuit.

2.6 Shunt performances

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30 2 Linear Passive Shunts with Discrets Piezoelectric Transducers Array

in function of the electrical components as shown in (de Mameffe, 2007). Then the optimal parameters leading to the highest values of the damping ratio are deduced for both configmations of the shunt: the independent loops and the parallel loop. Table.2.1 summeurizes the maximum achievable damping with the R and the RL shunts.

Shunt parallel/independent loops_^max

R --- »—- w —-Qi-iJi 'AT? Kf 2ui ^4 4 sériés RL 1 iDl-wf ^ “='^2 2 parallel RL 2]l n? 2 ~ 2

Table 2.1; Maximum attainable damping ratios with varions passive shunts when p trans ducers are used. is the generalized electromechanical coupling factor of a single transducer while Ki is the global generalized electromechanical coupling factor of the équivalent transducer defined by Eki.(2.27). It is assumed that the transducers are identical with the same modal amplitude.

Table.2.2 summarizes the optimal tuning of the electrical components of the R shunt, the parallel RL shunt and the sériés RL shunt (Fig.2.4). The main différence be- tween the current results and the results presented in (de Marneffe, 2007) is the value of the generalized electromechanical couphng factor used for the tuning. In- deed, when p transducers are targeting the same mode, one should consider the global electromechanical coupling factor K{ even if the transducers are shimted in- dependently.

Finally, one should notice that the damping ratio generated by the R shunt in- creases linearly when the number of the transducers increases, while the damping generated by a RL shunt is proportional to the square root of the number of trans ducers. From table.2.1, by equating the damping ratio provided by the R and the

RL shunts, one gets the number of transducers for which the purely R shunt becomes

more efficient than the RL shunt, it is given by

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2.7 Référencés 31

Shunt Independent shunt Parallel shunt

L/opt Ropt ^opt

exact n.a. füJÎ 1 n.a. 1 /W 1

V fij ntCt2 p\ fli fliCi2

approx. n.a. 1 ^iCstatic n.a. 1 1 P ^tC'static exact ûi 2ui 1 jf 1 3 C S ".* 1

Ciiüj pCi2Ü\ pCi2Üf\ fî?

S approx. 1 O ^ 1 1 oV^ 'Ki

^^^C'static WtOstatic P ^iCstatic P ^*^iCatatic

^ exact

■qS

1 1 1 1 1 1

Ci2UJ^ _{2yJÇï1-JtCi2} P Caul _{P2yJü'i-u:‘lCi2}

(h

ca approx. 1 1 1 1 1 1 1

^static 2 UJi C'statîc^ P ^^^static ^P\/P ^i^static^

Table 2.2: Optimal values of the electrical comp>onents R and L when p transducers are used for the damping of mode i. u), and (}{ are the natural frequencies of the structure when ail the transducers are short-circuited and when they are open-circiiited respectively. ‘Ki is the generaUzed electromechanical coupling factor of a single transducer and Cgtatic and

Ci2 are their capacitance defined by Ek}.(2.61) and Eq.(2.62), resi>ectively (adapted from

de Marneffe, 2007).

2.7 References

A. Antoniou. Réalisation of gyrators using operational amplifiers, and their use in RC-active-network synthesis. In Proceedings of the Institution of Electrical Engi-

neers, volume 116, pages 1838-1850. lET, 1969.

M. J. Balas. Direct velocity feedback control of large space structures. Journal of

Guidance, Control, and Dynamics, 2(3);252-253, 1979.

A. Benjeddou. Advances in piezoelectric finite element modeling of adaptive struc tural éléments: a survey. Computers & Structures, 76(l):347-363, 2000.

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32 Référencés

damping of piezoactuated beams. Journal of Sound and vibration, 289(4):908- 937, 2006.

G. Caruso. A critical analysis of electric shunt circuits employed in piezoelectric passive vibration damping. Smart Materials and Structures, 10(5):1059, 2001. B. de Marneffe. Active and passive vibration isolation and damping via shunted

transducers. PhD thesis, Université Libre de Bruxelles, 2007.

F. dell’Isola, C. Maurini, and M. Porfiri. Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation. Smart materials and Structures, 13(2) :299, 2004. J-L. Fanson and T-K. Caughey. Positive position feedback control for large space

structures. AIAA journal, 28(4):717-724, 1990.

A. J. Fleming, S. Behrens, and S. R. Moheimani. Optimization and implémentation of multimode piezoelectric shunt damping Systems. lEEE/ASME Transactions

on Mechatronics, 7(l):87-94, 2002.

R. L. Forward. Electronic damping of vibrations in optical structures. Applied

Optics, 18(5):690-697, 1979.

N. W. Hagood and A. von Flotow. Damping of structural vibrations with piezoelec tric materials and passive electrical networks. Journal of Sound and Vibration, 146(2):243-268, 1991.

J. J. Hollkamp. Multimodal passive vibration suppression with piezoelectric mate rials and résonant shunts. Journal of Intelligent Material Systems and Structures, 5(l):49-57, 1994.

C. Maurini, F. dell’Isola, and D. Del Vescovo. Comparison of piezoelectronic net works acting as distributed vibration absorbers. Mechanical Systems and Signal

Processing, 18(5):1243-1271, 2004.

A. Preumont, J-P. Dufour, and C. Malekian. Active damping by a local force feed back with piezoelectric actuators. Journal of guidance, control, and dynamics, 15 (2):390-395, 1992.

J. Tang and K. W. Wang. Vibration control of rotationally periodic structures using passive piezoelectric shunt networks and active compensation. Journal of vibration

and acoustics, 121(3):379-390, 1999.

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References 33

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(53)

Chapter 3 Semi-active Shunts

In this chapter we investigate a semi-active nonlinear shunt strategy referred to as

Synchronized Switch Damping on Inductor, SSDI. We demonstrate that the maxi

mum attainable damping with the SSDI is proportional to the square of the elec- tromechanical coupling factor k^. Consequently, we propose the use of a négative capacitance to increase and thus improve the performances of the SSDI. The an- alytical formulation is performed on a simple single degree of freedom System while the proposed SSDI enhancement is validated experimentaUy on a cantilever beam.

The objectives of this chapter are twofold: (i) investigate the non-linear synchro nized switch damping technique and describe its behavior anal3rtically; and (ii) com pare it to the classical linear shunts for an application in rotating turbomachinery components.

3.1 Introduction

The simplest dissipative circuit consists of a linear resistor R tuned properly on the targeted mode. It has moderate performances as compared to the résistive and inductive RL circuit: ~ k/2 against ~ fc^/4. Most practical implémen tations of the RL shunt require massive inductors when applied to low-frequency modes (L = \/Cüj/) and show a sharp décliné in effectiveness when the résonance

frequency of the structure detunes from that of the circuit: A déviation of ±3% in the electrical frequency is enough to reduce the attainable damping by 2. The loss of performance due to detuning and the demand of huge inductors of RL shunt motivated the quest for more robust passive techniques.

In a similar way as the variable-stiffness damping technique proposed by (Onoda et al., 1991), based on the feict that a piezoelectric transducer is stiffer in open cir

Disponible à / Available at permalink :

ULB

U

niversité

libre

de

B

ruxelles

École Polytechnique de Bruxelles

Piezoelectric Shunt

Dumping of Rotationally Periodic Structures

Bilal MOKRANI

Active Structures Laboratory

ULB

U

niversité

libre

de

B

ruxelles

3

École Polytechnique de Bruxelles

Piezoelectric Shunt

Damping of Rotationally Periodic Structures

Bilal MOKRANI

Active Structures Laboratory

Jury

Remerciements

Abstract

Glossary

V

V, Vj

W{r,e,t)

Contents

Chapter 1

Introduction

1.1

Vibration of Turbomachinery Components

1.2

Damping of Turbomachinery Blades

).

8

1.3 Motivations

1.4 Outlines

I. 5 References

10

Chapter 2

Linear Passive Shunts with

Discrète Piezoelectric

Transducers Array

(d)

2.1 Introduction

2.2

Governing équations

f Qj 1

r C(1 -

fj 'Qi=o

- fc

)

Vj

(2.8)

£ bjb] I ^ =

(

.

)

Qi

Cn C12

C\p

Q

.

C21

C

p

V2

Cpi Cp

(2.20)

Cu

Qi

Vi

{Çc^V^^Z+i)}

_Vj

_{{Çc^V^^Z+i)}}