PiezoelectricShuntDampingofRotationallyPeriodicStructures UniversitélibredeBruxelles

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Université libre de Bruxelles

É c o l e

P o l y t e c h n i q u e

d e

B r u x e l l e s

Piezoelectric Shunt

Damping of Rotationally Periodic Structures

Bilal MOKRANI

Thesis submitted in candidature for the

degree of Doctor in Engineering Sciences January 2015

Active Structures Laboratory

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Jury

Supervisor: Prof. André Preumont (ULB)

President: Prof. Michel Kinnaert (ULB)

Secretary: Prof. Patrick Hendrick (ULB)

Membres:

Prof. Michel Géradin (Université de Liège)

Prof. Peter Hagedorn (Technische Universität 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 colcol-lègues 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 pour 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 Ioan 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 pour 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 amour, 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.

Le travail réalisé dans cette thèse s’inscrit dans le cadre du projet "Skywin - HM+: Avions plus Intelligents", ﬁnancé par la Région Wallonne.

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Abstract

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

This thesis considers the RL shunt damping of rotationally periodic structures using an array of piezoelectric patches, with an application to a bladed drum representative 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 resonance frequen-cies, and harmonic shape in the circumferential direction; the proposed RL shunt approaches take advantage of these two features.

When a family of modes is targeted for damping, the piezoelectric patches are shunted independently on identical RL circuits, and tuned roughly on the aver-age value of the resonance 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 serious drawback for rotating machines.

When a specific mode with n nodal diameters has been identified as critical and is targeted for damping, one can take advantage of its harmonic shape to organize the piezoelectric patches in two parallel loops. This parallel approach reduces con-siderably the demand on the inductors of the tuned inductive shunt, as compared to independent loops, and offers a practical solution for a fully passive integration of the inductive shunt in a rotating structure.

Various methods are investigated numerically and experimentally on a cantilever beam, a bladed rail, a circular plate, and a bladed drum. 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 Function

PZT Lead-Zirconate-Titanate

RL Resistive and Inductive shunt

SSD State Switch Damping

SSDI Synchronized Switch Damping on Inductor

SSDNC Synchronized Switch Damping on Negative Capacitance

SSDS Synchronized Switch Damping on Short

SSDV Synchronized Switch Damping on Voltage

ULB Université Libre de Bruxelles

ZZENF Zig-Zag shaped Excitation line in the Nodal diameter versus Frequency

diagram

List of symbols

α Vector of modal coordinates (in Chapter 2); overshoot in the step

response of the RLC circuit (in Chapter 3);

b_j Inﬂuence vector corresponding to the jth transducer

C, CS _{Piezoelectric capacitance of the transducer when it is blocked and}

when it is left free

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

d33, d31 Piezoelectric constants

δ(x) Dirac function of x

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δ_ij Kronecker symbol

Δj Elongation of transducer j

F Force amplitude

f_j Force applied by the jth transducer/actuator to the structure

Φ, φi Mode shapes matrix and its vectors

G(s) Transfer function (or transmissibility matrix)

k2 Electromechanical coupling factor

K Stiﬀness matrix (it can be also a scalar)

K_a Stiﬀness of the transducer with short-circuited electrodes

K_i2 Global generalized electromechanical coupling factor of all the

transducers, corresponding to mode i j_K2

i Generalized electromechanical coupling factor of transducer j

corresponding to mode i j_K2

n,s, jKn,c2 Generalized electromechanical coupling factor of transducer j,

respe-ctively, for the sine and for the cosine modes with n nodal diameters

K_n2 Generalized electromechanical coupling factor of all the transducers

for the sine and the cosine modes of n nodal diameters

L Inductance

λj(s) Eigenvalue of the total admittance matrix Ytot

M Mass matrix (it can be also a scalar)

μ Mass ratio: μ = m₂/m₁

μ_i Modal mass of mode i

n Number of nodal diameters (in Chapter 4-8); or number of slices

of the PZT stack(in Chapter 2-3)

N Number of sectors

Ω Angular speed

ω_e Electrical resonance frequency

ω_i, Ω_i,l_Ω

i Resonance frequency of mode i, respectively, when all the transducers

are short-circuited, when they are open-circuited, and when only the

lth transducer is open-circuited

p Number of transducers

pns(t), qns(t) Generalized forces in modal coordinates

Q, Q_j Electrical charge vector and its component

r, θ Polar coordinates

R Resistance

s Laplace variable: s = jω; or number of rotating forces (in Chapter 4)

s_ns(t), c_ns(t) Modal amplitudes of the sine and the cosine modes of n nodal diameters

and s nodal circles

σ(ω), σ_(·) Cumulative Root Mean Square RMS; and RMS value

t Time variable

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T (jω) Frequency Response Function of the blades

V Strain energy

V, Vj Electrical voltage vector and its component

W_n,s, W_n,s∗ Sine and cosine mode shapes of n nodal diameters and s nodal circles

W (r, θ, t) Time response of the disk, at coordinates (r, θ)

x_i, ˙x_i, ¨x_i Displacement, velocity and acceleration of the ith degree of freedom

ξe Electrical damping coeﬃcient

ξ_i Mechanical damping coeﬃcient, corresponding to mode i

Y_s Admittance matrix of the electrical shunt

Y_struct Admittance matrix of the structure

Ytot Total admittance matrix: Ytot= Ystruct+ Ys

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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 reduction of weight by increasing its functional eﬃciency. This can be achieved in various 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 considerable weight savings compared to the classical design, but it is obtained at the expense of an extremely small inherent damping, of the

order of ξ 10−4.

BLUM

Figure 1.1: The BLUMmade of 3 blade wheels. Left: low pressure compressor, integrating

the BLUM; Right: exploded CAD view of the low pressure compressor.

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2 1 Introduction

In this thesis, we investigate the use of piezoelectric shunt damping to reduce the

vibration level in the blades of the BLUM.

1.1 Vibration of Turbomachinery Components

Axial turbomachines 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.

Combustion chambers Turbine High pressure compressor Low pressure compressor Rotor shaft Fan blades

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

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1.1 Vibration of Turbomachinery Components 3

it is mixed with the fuel and burned; the result 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 compressor stages and the fan blades rotor. The performances of turbomachines depend essentially on the compression rate provided by the compressor modules. An

example of a turbofan is presented in Fig.1.2.

1.1.1 Blisks

Figure 1.3: (a) Conventional disk with blade ﬁxings compared to a blisk and a bling. (b)

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4 1 Introduction

In conventional compressor bladed disks, the blades are attached to the disk using a

root fixing feature, Fig.1.3.b. This fixing solution offers an inherent damping to the

blades and facilitates the maintenance of the engine and the replacement of damaged blades; however, it involves a considerable 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 decades (Fig.1.3.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 pieces, 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 temperature gradients, the blades are subjected to high static stresses induced by the high pres-sure and temperature of the flow, and also to dynamic stresses due to the dynamic variation of the flow pressure (due to the presence of upstream stator vanes). This dynamic loads may excites the resonance 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).

Figure 1.4: Blade failure of a compressor blisk, reported to high cycle fatigue of the blades,

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1.2 Damping of Turbomachinery Blades 5

1.2 Damping of Turbomachinery Blades

Blades are the most critical engine parts suffering from vibrations, that can lead to high cycle fatigue and to the failure 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 inherent damping of the material, and the friction produced at the joints of the engine. In monolithic blisk configurations, 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 instabilities may occur; these vibrations may be reduced or avoided by increasing the inherent damping of the blades.

1.2.1 Friction damping

The most classical way to increase the damping of turbomachinery blades is to

incor-porate friction devices: (i) at the root of the blades, e.g. Fig.1.5; or (ii) between the

blades themselves; and/or (ii) at the shrouds. The idea is to create a contact surface

between the vibrating parts, allowing the dissipation of energy (Griﬃn,1990). Since

dry friction dampers are nonlinear systems, the damping performance depends upon the level of vibration and the contact area between the parts (which is also a design parameter). Contact surface Blade Disk Blade root Friction damper

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

2006).

In classical bladed disk conﬁgurations, it is easy to incorporate the friction dampers, at the roots of the blades or between the blades. In blisk conﬁgurations, since there are no joints between the blades and the disk, the friction may be introduced in the

structure using friction rings as shown in Fig.1.6.a (Laxalde,2010), or using friction

ﬁnger mechanisms as shown in Fig.1.6.b (Stangeland et al., 2007), to name only

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6 1 Introduction

Blade

Disk Groove

Ring

(a) Friction ring damper Disk Blade Friction fingers (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 temperature conditions limits its utilization.

Blade Disk Viscoelastic material Blade root

Figure 1.7: Viscoelastic-based blade damping device; the viscoelastic material is placed at

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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 eﬃciency 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 en-ergy eﬃciency and easy integration. In the last two decades, research have been conducted on piezoelectric shunt damping techniques, but all these techniques are faced with many issues making their applications very limited. Despite these issues, many academic 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 pro-posed 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 resonance frequency of the blades by switching the state of the electrodes (open or closed) when the excitation frequency (a multiple of the ro-tation speed of the engine) approaches the resonance 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 serious 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 damping systems in turbomachinery bladed disks is a technol-ogy whose maturity and efficiency have been demonstrated and confirmed during several decades; however, the new design of bladed disks in monolithic blisk con-figurations renders impossible the use of classical friction damping solutions. The need of systems increasing the blade damping of monolithic blisks, and the inability of using classical damping technologies are the main motivations of the present work. This thesis has been realized within an industrial project involving SAFRAN Techspace Aero as industrial partner. The aim of the project is to develop a passive

piezoelec-tric shunt device for the damping of the blade modes of the BLUM.

1.4 Outlines

The thesis is organized in the following way:

• Chapter 2 is devoted to the derivation of the governing equations 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 Damping 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 negative capacitance, to enhance the damping performances, is also investigated.

• Chapter 4 describes the resonance 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 implementation of the linear RL shunt for the damping

of the blade modes of a bladed drum. The implementation 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.

• Chapter 7 provides a practical solution to reduce the size of the inductances,

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1.5 References 9

only two inductors. The proposed strategy is validated numerically and exper-imentally on a circular plate, and validated numerically on the bladed drum.

• Chapter 8 analyzes the eﬀect of the blade mistuning on the proposed parallel

and independent RL shunt; the simulation and experimental results are also presented.

1.5 References

Australian Transport Safety Bureau. 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 turbomachine compressor, June 24 2010. US Patent App. 12/580,070.

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

J. H. Griﬃn. A review of friction damping of turbine blade vibration. International

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

J. Hoﬀman. Magnetic damping system to limit blade tip vibrations in turboma-chines, February 13 1996. US Patent 5,490,759.

J. L. Kauﬀman and G. A. Lesieutre. Piezoelectric-based vibration reduction of

turbomachinery bladed disks via resonance 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, Ecole Centrale de Lyon, 2010.

D. Laxalde, F. Thouverez, J. J. Sinou, 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 internal damping, 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

Discrete Piezoelectric

Transducers Array

In this thesis, we study the use of shunted piezoelectric transducers for the damping 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 symmetrically on the structure, such that each transducer has 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 vibrates according to the targeted mode. With these considerations, the behavior of the system be-comes 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 have an equal authority over the targeted mode, which is representative of our problem. We consider only the linear resistive R and resistive-inductive RL shunts. Then, we derive the charac-teristic equation of the shunted 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 generalization of Chapter 4 of (de Marneﬀe,2007).

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12 2 Linear Passive Shunts with Discrete Piezoelectric Transducers Array Polarization

M

. . .

V1 V2 Vp K (a) (b) (c) (d) Piezo patch

Figure 2.1: Equivalence between an axisymmetric structure, vibrating according to its

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2.1 Introduction 13

2.1 Introduction

Piezoelectric materials have been used extensively as sensors and actuators in many engineering ﬁelds. Their merits of being energy eﬃcient and easily integrable, en-couraged researchers to develop various schemes for vibration control and structural damping. Major research in structural damping 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 have 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 Integral 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 (Forward, 1979): He demonstrated the damping capability of a

piezoelectric patch when it is shunted to an inductor L. (Hagood and von

Flo-tow, 1991) provided thorough analytical formulation of the linear resistive 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 damping 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 derive 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. Appendix B); they derived the optimal tuning of the R

shunt and the series 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 series and the parallel RL shunts: they have 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.

• Multi-mode damping with a single transducer: aiming to control many

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14 2 Linear Passive Shunts with Discrete Piezoelectric Transducers Array

circuits to the transducer, in such a way that the resonance frequencies of the electrical network are identical to the resonance frequencies of the

tar-geted modes (see e.g. Hollkamp, 1994, Wu, 1998 and Fleming et al., 2002).

Many successful designs have 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 various architectures are possible. This chapter formulates 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) shunt, 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. dell’Isola 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 elegant 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 resonance frequencies.

For all the categories mentioned above, the need of huge inductors is an issue which

may be overcome using synthetic inductors based on Antoniou circuits (Antoniou,

1969). In many practical applications, e.g. turbomachinery, this solution is not

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2.2 Governing equations 15

2.2 Governing equations

Consider the linear structure of Fig.2.2 equipped with p identical linear

piezoelec-tric transducers working according to the d33 mode. Each transducer consists of a

stacking of n identical piezoelectric discs polarized through their thicknesses.

. . .

f

1 Structure Piezoelectric transducer

f

2 É_j_{= b}T jx Tã=₂1_xT_Kx V =2 1 xçT_Kxç V1 V2 Vp fp f2 f1 fp f2 f1

Figure 2.2: Flexible structure equipped with p linear piezoelectric transducer.

The structure is assumed undamped and is represented by its mass and stiﬀness 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 equations of the structure are given by:

M ¨x + Kx =−

p

j=1

b_jf_j (2.1)

where f_j are the forces applied by the transducers to the structure and b_j 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 equations: Qj f_j = C(1− k2) nd33Ka −nd33Ka Ka Vj Δ_j (2.2)

such that Vj is the electrical voltage between the electrodes, Qj the electrical charge

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is the stiﬀness of the transducer with short-circuited electrodes (V_j = 0), C is the

free electrical capacitance (fj = 0), and d33 is the piezoelectric constant. k is the

electromechanical coupling factor; it measures the ability of the transducer to convert mechanical energy into electrical energy and vice versa. It can be expressed as:

k2 = n

2_d2 33Ka

C (2.3)

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

• The stiﬀness of the transducer in open circuit (Qj = 0) is related to that in

short-circuit (Vj = 0) by:

Δ_j

f_j Qj=0 = K_a

(1− k2₎ (2.4)

• The capacitance of a blocked transducer (Δj = 0) is related to that of a free

transducer (f_j = 0) by:

CS = Qj

V_jΔj=0 = C(1− k

2₎ _(2.5)

Finally, by substituting Eq.(2.2) into Eq.(2.1), one gets the governing equations of

the full electromechanical system:

M ¨x + (K + Ka p j=1 bjbTj)x = nd33Ka p j=1 bjVj (2.6) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Q₁ Q2 .. . Qp ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C(1− k2) 0 · · · 0 nd₃₃K_abT₁ 0 C(1− k2₎ _{· · ·} ₀ _nd 33KabT2 .. . . .. ... ... 0 0 · · · C(1 − k2) nd33KabTp ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V₁ V₂ .. . Vp x ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (2.7) 2.2.1 Modal basis

As described in the previous section, the stiﬀness of the structure depends on the boundary conditions. Therefore, one should deﬁne an eigenvalue problem of the

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

⎛ ⎝M s2+ K + K_a p j=1 b_jbT_j ⎞ ⎠x = 0 (2.8)

where s is the Laplace variable.

• When the transducers are charge-driven instead of voltage-driven, Vj can be

eliminated from Eq.(2.6) and Eq.(2.7), leading to

⎛ ⎝_{M s}2_{+ K +} Ka (1− k2₎ p j=1 bjbTj ⎞ ⎠_{x =} nd33Ka C(1− k2₎ p j=1 bjQj (2.9)

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

⎛ ⎝M s2+ K + Ka (1− k2₎ p j=1 b_jbT_j ⎞ ⎠x = 0 (2.10)

which is equivalent to Eq.(2.8) where the short-circuited stiﬀness Ka of the

stand-alone transducer is replaced by the open-circuited stiﬀness K_a/(1− k2).

• Finally, when all the transducers are short-circuited except the lth _transducer

is open-circuited, i.e. Vj = 0 and j= l, the new equation of the system is:

⎛ ⎝_{M s}2_{+ K +} Ka (1− k2₎blb T l + Ka p j=1, j=l bjbTj ⎞ ⎠_{x = 0} _(2.11)

The solutions of these eigenvalue problems have three diﬀerent sets of natural fre-quencies and mode shapes, we refer to the natural frefre-quencies: (i) when all the

trans-ducers are open-circuited by Ωi, (ii) when all the transducers are short-circuited by

ω_i, (iii) and ﬁnally, when all the transducers are short-circuited except one by lΩ_i,

where the superscript refers to the lth 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 unchanged 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 = Φα, where Φ = (φ₁, ..., φ_n) is the mode shapes matrix of the structure

obtained by solving the eigenvalue problem of Eq.(2.8):

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The mode shapes are orthogonal with respect to the mass and the stiﬀness matrices:

ΦTM Φ = diag{μi} (2.13) ΦT(K + K_a p j=1 b_jbT_j)Φ = diag{μ_iω2_i} (2.14)

where ω_i are the natural frequencies with short-circuited transducers and μ_i are the

modal masses. Using the transformation: x = Φα, Eq.(2.6) can be rewritten in

modal coordinates as:

M Φ ¨α + (K + K_a p j=1 b_jbT_j)Φα = nd₃₃K_a p j=1 b_jV_j (2.15)

which gives, after left multiplication by ΦT, the use of the orthogonality relations

(2.13) and (2.14), and the Laplace transformation:

x = nd₃₃K_a i φiφTi μ_i(s2_{+ ω}2 i) _p j=1 b_jV_j (2.16)

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

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Q₁ Q₂ .. . Q_p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C₁₁ C₁₂ · · · C_1p C₂₁ C₂₂ · · · C_2p .. . ... . .. ... C_p1 C_p2 · · · C_pp ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ V₁ V₂ .. . V_p ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (2.17) with Cjl= δjlC(1− k2) + Ck2 i Ka bT_jφiφTi bl μ_i(s2_{+ ω}2 i) (2.18)

after using the equality Ck2 = n2d2₃₃K_a; δ_jl is the Kronecker symbol: δ_jl = 1 if

j = l, and δ_jl= 0 if j = l.

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 Cll represents

the capacitance of the lth transducer measured when all the other transducers are

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Finally, by introducing the deﬁnition of the modal fraction of strain energy: l_ν i = Ka(bT_l φi)2 μ_iω2 i (2.20)

Eq.(2.19) can be rewritten as:

C_ll= Ql V_l = C(1− k 2_{) + Ck}2 i l_ν i (s2_/ω2 i + 1) (2.21) l_ν

i is the ratio between the strain energy in the lth transducer and the total strain

energy of the structure when it vibrates according to mode i. 2.2.3 Generalized electromechanical coupling factor

The deﬁnition of the electromechanical coupling factor of the standalone transducer is generalized for a speciﬁc 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:

K_i2 = Ω 2 i − ωi2 Ω2 i (2.22)

where Ω_i are the natural frequencies of the structure when the transducer is

open-circuited, and ω_i are those when the electrodes 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 factorlK_i

is deﬁned: l_K2 i = l_Ω2 i − ωi2 l_Ω2 i (2.23)

wherelΩ_i are the natural frequencies of the structure when all the transducers are

short-circuited except the lth, while ωiare the natural frequencies when all the

trans-ducers are short-circuited.

One should notice that the poles of Ql/Vl are at ωi, while the zeros are at lΩi

(i.e. Q_l = 0). Therefore, in practical situations, the electromechanical coupling

factor of each transducer can be easily estimated by measuring its impedance when the other transducers are short-circuited.

Moreover, based on the assumption of unchanged mode shapes and using the

or-thogonality relations (2.13) and (2.14), one can relatelKi and lνi:

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In the literature, the deﬁnition l_K2 i = l_Ω2 i − ω2i ω_i2 k2 1− k2 l_ν i (2.25)

is often used instead of that of Eq.(2.23). The diﬀerence between the two deﬁnitions

is marginal for small k andlν_i, however, the deﬁnition of Eq.(2.25) is simpler.

When the transducers are used simultaneously, we deﬁne the global generalized elec-tromechanical coupling factor as the coupling factor of the equivalent transducer:

K_i2 = Ω 2 i − ωi2 Ω2 i (2.26)

where Ωiand ωiare the natural frequencies of the structure when all the transducers

are open-circuited and when they are short-circuited respectively. In a diﬀerent way,

if one considers the deﬁnition of Eq.(2.25), one ﬁnds easily:

K_i2 = k 2 1− k2 p j=1 j_ν i = p j=1 j_K2 i (2.27)

Here, K_i is a direct metric of the total amount of strain energy which can be

trans-formed into electrical energy by the p transducers. This deﬁnition is useful for the tuning of the resistor R of the RL-shunt.

2.3 Electrical admittance

2.3.1 Admittance of a single transducer

Referring to Eq.(2.19), the electrical admittance of each transducer is deﬁned when

all the other transducers are short-circuited. It is given by:

Y_l(s) = Il V_lVj=0 = sQ_l V_l Vj=0 = sC 1− k2+ k2 i l_ν i (s2_/ω2 i + 1) (2.28)

or by introducinglK_i2 from Eq.(2.25),

Y_l(s) = sC(1− k2) 1 + i l_K2 i (s2_/ω2 i + 1) (2.29)

When only the lth _{transducer is shunted to a circuit of admittance} l_Y

shunt, the

Kirchhoﬀ low gives:

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2.3 Electrical admittance 21

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

Y_l(s) =−lY_shunt(s) (2.30)

Therefore, for diﬀerent type of shunts, to identify the optimal tuning of the shunt

admittance lY_shunt(s), 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 transducers 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 ={V1, ..., Vp}T and the current vector I = sQ = s{Q1, ..., Qp}T:

I = sQ = Ystruct(s)V (2.31)

where Y_struct(s) is, in this case, a p×p matrix. When all the transducers are shunted

independently, the Kirchhoﬀ low yields:

Y_struct(s)V (s) =−Y_shunt(s)V (s)

[Y_struct(s) + Y_shunt(s)] V (s) = Y_tot(s)V (s) = 0, and V (s)= 0 (2.32)

Here, Yshunt is a diagonal matrix and its lth element corresponds to the admittance

of the circuit shunted to the lth transducer. The poles of the independently-shunted

structure are found by solving the equation

Ytot(s)V (s) = 0, and V (s)= 0 (2.33)

They are the solution of:

det [Y_tot(s)] =

p

j=1

λ_j(s) = 0 (2.34)

det [Y_tot(s)] is the characteristic equation of the system, while λ_j(s) are the

eigen-values of the matrix Ytot(s). Therefore, to identify the poles of the shunted system

in function of the shunt parameters, one should, ﬁrst, ﬁnd the analytical formulas of

λ_j(s), and then, solve λ_j(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 diﬃcult to ﬁnd an analytic formulation of λ_j(s) because the matrix Y_tot(s) does

not come into a special form.

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transducers, the matrix Y_tot(s) becomes a symmetric Toeplitz matrix in the

vicin-ity of the targeted mode frequency, and the computation of its eigenvalues λj(s)

with an analytical solver gives simple formulas. In the case where the number of

transducers p is even, the matrix Y_tot(s) becomes circulant and the analytical

for-mulas of its eigenvalues λj(s) exist (cfr. AppendixC). 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 all 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 structure equipped with p = 2n transducers and vibrating according

to its mode with a shape of n spatial harmonics (nodal diameters), Fig.2.1.

M

. . .

V1 V2 Vp fp f2 f1 K Shunts (a) (b) R L R L R (c)

Figure 2.3: One degree of freedom system equipped with p identical linear transducers: (a)

Parallel RL shunt; (b) series RL shunt; (c) R shunt.

The displacement of M and the electrical charges in the transducers Q_jare governed

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2.4 Shunt of a single d.o.f. system 23 and ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Q₁ Q2 .. . Qp ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C(1− k2₎ ₀ _{· · ·} ₀ _nd 33Ka 0 C(1− k2) · · · 0 nd33Ka .. . . .. ... ... 0 0 · · · C(1 − k2) nd33Ka ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V₁ V2 .. . Vp 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:

Y_struct(s) = s ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ C11(s) C12(s) · · · C1p(s) C₂₁(s) C₂₂(s) · · · C_2p(s) .. . ... . . . ... C_p1(s) C_p2(s) · · · C_pp(s) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (2.37) such that C_ij(s) = C δ_ij(1− k2) + k2 l_ν s2_/ω2 1+ 1 (2.38)

δ_ij is the Kronecker symbol, ω2₁ is the natural frequency of the system when all the

transducers are short-circuited, and

l_{ν =} Ka

pK_a+ K

is the modal fraction of strain energy of the lth transducer. The electromechanical

coupling factor of each transducer is given by: l_K2

1 =

l_νk2

(1− k2_{) +}l_νk2 (2.39)

which can be approximated for small k2 and lν, by

l_K2 1

k2

(1− k2₎

l_ν _(2.40)

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

K₁2= p l=1 l_K2 1 = p·lK12 (2.41) Note that K2

1 can be deduced from Eq.(2.26) by measuring the natural frequency

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2.4.1 Independent loops

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

[Ystruct(s) + Yshunt(s)] V = Ytot(s)V = 0 (2.42)

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

Ytot(s) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Y₁(s) Y₂(s) · · · Y₂(s) Y2(s) Y1(s) · · · Y2(s) .. . ... . .. ... Y₂(s) Y₂(s) · · · Y₁(s) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (2.43) such that: Y1(s) = sC 1− k2+ k2 l_ν s2_/ω2 1+ 1 + Ys(s) (2.44) Y₂(s) = sCk2 l_ν s2_/ω2 1 + 1 (2.45)

and Ys(s) is the diagonal term of the shunt matrix Yshunt(s).

The poles of the independently-shunted transducers system are found by solving the equation: det [Y_tot(s)] = p j=1 λ_j(s) = 0 (2.46)

and since Ytot(s) is a circulant matrix, its eigenvalues λj(s) are found analytically

(see AppendixC): λ1(s) = Y1(s) + (p− 1)Y2(s) = sC 1− k2+ pk2 l_ν s2_/ω2 1+ 1 + Ys(s) (2.47) λj(s) = Y1(s)− Y2(s) = sC(1− k2) + Ys(s) , for j > 1 (2.48)

One can observe that λ1(s) = 0 is the characteristic equation of the structure when

all the transducers are shunted independently; while λj(s) = 0, j > 1 are the

charac-teristic equations of the shunted system with blocked transducers (i.e. bT_jx = 0 and

the transducers behave as pure capacitors without any coupling with the structure).

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

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2.4 Shunt of a single d.o.f. system 25

The poles of the shunted system are obtained by solving λ₁(s) = 0. The

opti-mal values of the shunt components can be chosen to maximize the damping ratio.

The solution of λ1(s) = 0 is demonstrated in (de Marneﬀe, 2007) for various 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 K₁2 = p·lK₁2 instead of the single transducer electromechanical

cou-pling factor lK₁2. This comes from the fact that all 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 all 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 series loop, so that they act as a single transducer: the parallel shunt is more advantageous than the series because it results in a capacitance equal to the sum of the individual capacitance of all 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 derive the optimal tuning for diﬀerent type of shunts.

When shunted in parallel, the transducers act like a single transducer and

gen-erate a total electrical charge Q_t equal to the sum of the charges generated by each

transducer: Q_t= p j=1 Q_j (2.50)

Since the transducers are identical and their voltages V_j are identical too, the

re-sulting voltage V_t is then

Vt= V1 = 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

M ¨x + (K + pK_a)x = p(nd₃₃K_a)V_t (2.52)

Q_t= pC(1− k2)V_t+ p(nd₃₃K_a)x (2.53)

The admittance of the system Y_struct is then obtained by eliminating x in Eq.(2.53):

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where ν is, in this case, the total modal strain energy in all the transducers:

ν =

p

j=1 j_ν

Finally, the poles of the system when it is shunted to an admittance Ys(s) are

obtained by solving the scalar equation

Ytot(s) = psC(1− k2) 1 + K 2 1 s2_/ω2 1+ 1 + Ys(s) = 0 (2.55)

The optimal tuning of the diﬀerent shunts, which maximizes the modal damping, 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 equivalent open-circuit stiﬀness

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

and Eq.(2.49), the characteristic equations 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 shunt is the increase of the apparent capacitance of the shunted transducers which

reduces, for an RL shunt, the demand on inductors by p2_{: only 1 inductor is}

ne-cessitated instead of p inductors, with a size p times smaller than those required by an independent RL shunt. This constitutes a signiﬁcant advantage which will be exploited later.

2.5 Shunt of a multi-mode structure

All 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 iden-tify the poles of the structure when it is shunted independently is the analytical

derivation of the characteristic equation from the matrix Ytot(s) (Eq.2.33). Since

the matrix Y_tot(s) is always symmetric, it would be possible to ﬁnd an analytical

solution of its eigenvalues λ_j(s). However, although this solution exists, a more

complex mathematical formulation of the problem is needed to derive analytically the diﬀerent equations; but this is not in the scope of this thesis.

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2.5 Shunt of a multi-mode structure 27

. . .

Structure f f Shunts R R R L L (a) (b) (c)

Figure 2.4: Diﬀerent type of linear shunts: a) linear R shunt; b) linear series RL shunt; c)

linear parallel RL shunt.

over a speciﬁc mode. Therefore, for the purpose of simplicity, for the targeted mode

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

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

bT_jφ_iφT_i b_k> 0,

and

bT_jφ_i = bT_l φ_i,

where bT_jφ_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 have the same modal fraction of strain energy, and they act in phase, then

bT_jφ_i = bT_l φ_i (2.56)

When the structure is excited according to its mode i, the response is dominated

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28 2 Linear Passive Shunts with Discrete Piezoelectric Transducers Array Ci2 !1 Ò1 !2 Ò2 !3 Ò3 !i Òi CS_=C(1àk2₎ V Q V Q ! ! (a) (b) Cstatic 0 0

Figure 2.5: (a) Capacitance Q/V of a piezoelectric transducer embedded in a ﬁctitious 3

mode structure; (b) the approximation around ω_i if the modal density is low.

approximated in the vicinity of ω_i by

C_jl= C δ_jl(1− k2) + k2Ka(b T j φiφTi bl) μ_i(s2_{+ ω}2 i) + k2K_a t>i (bT_j φ_tφT_tb_l) μ_tω2 t (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 deﬁnition oflνi and lKi2, and taking in account Eq.(2.56), it can be

written C_jl= C(1− k2) δ_jl+ l_K2 i (s2_/ω2 i + 1) + δ_jl t>i l_K2 t (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:

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2.6 Shunt performances 29

The meaning of C_i2 is illustrated in Fig.2.5. It satisﬁes the inequality:

C_static> C_i2> CS(1− k2)

The static capacitance of a single transducer C_static is given

Cstatic = C 1− k2+ k2 i l_ν i (2.62)

For the low frequency modes, C_i2 is close to C_static.

Equation (2.59) is similar to the characteristic equation 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 K_i2 = p·lK_i2. For the various types of

shunt, 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 shunted together in parallel, the resulting charge Q_t is equal

to the sum of the charges produced by all the transducers. Therefore, in a similar way to the single degree of freedom system, it is easy to determine the characteristic equation of the system with the transducers mounted in parallel:

Ytot(s) = spCi2+ spC(1− k2) p·lK_i2 s2_/ω2 i + 1 + Ys(s) (2.63)

This equation is also solved similarly to the characteristic equation of a multi-mode structure equipped with a single transducer of a free capacitance pC and a

general-ized electromechanical coupling factor K_i2 = p·lK_i2.

For both conﬁgurations of the transducers, independent loops and parallel loop,

the system has an equivalent electromechanical coupling factor K_i2 = p·lK_i2 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 equivalent capacitance is p times bigger than in the independent conﬁguration,

which reduces the demand on inductors by p2 when shunted to a RL circuit.

2.6 Shunt performances

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in function of the electrical components as shown in (de Marneﬀe,2007). Then the

optimal parameters leading to the highest values of the damping ratio are deduced for both conﬁgurations of the shunt: the independent loops and the parallel loop.

Table.2.1 summarizes the maximum achievable damping with the R and the RL

shunts.

Shunt parallel/independent loops

ξ_imax R Ωi− ωi 2ωi ≈ p l_K2 i 4 ≈ K_i2 4 series RL 1 2 Ω2_i − ω2_i ω_i2 ≈ √p l_K i 2 ≈ K_i 2 parallel RL 1 2 Ω2_i − ω2_i Ω2 i =√p l_K i 2 ≈ K_i 2

Table 2.1: Maximum attainable damping ratios with various passive shunts when p

trans-ducers are used. lK_i is the generalized electromechanical coupling factor of a single trans-ducer while K_i is the global generalized electromechanical coupling factor of the equivalent transducer deﬁned by Eq.(2.27). It is assumed that the transducers are identical with the same modal amplitude.

Table.2.2summarizes the optimal tuning of the electrical components of the R shunt,

the parallel RL shunt and the series RL shunt (Fig.2.4). The main diﬀerence

be-tween the current results and the results presented in (de Marneﬀe, 2007) is the

value of the generalized electromechanical coupling factor used for the tuning. In-deed, when p transducers are targeting the same mode, one should consider the

global electromechanical coupling factor Ki even if the transducers are shunted

in-dependently.

Finally, one should notice that the damping ratio generated by the R shunt in-creases linearly when the number of the transducers inin-creases, 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

p > 4

l_K2

i

,

PiezoelectricShuntDampingofRotationallyPeriodicStructures UniversitélibredeBruxelles

Université libre de Bruxelles

É c o l e

P o l y t e c h n i q u e

d e

B r u x e l l e s

Piezoelectric Shunt

Damping of Rotationally Periodic Structures

Bilal MOKRANI

Active Structures Laboratory

Jury

Remerciements

Abstract

Glossary

Contents

Chapter 1

Introduction

1.1

Vibration of Turbomachinery Components

1.2

Damping of Turbomachinery Blades

1.3

Motivations

1.4

Outlines

1.5

References

Chapter 2

Linear Passive Shunts with

Discrete Piezoelectric

Transducers Array

M

. . .

. . .

2.1

Introduction

2.2

Governing equations

. . .

. . .

f

f

2.3

Electrical admittance

2.4

Shunt of a single d.o.f. system

M

. . .

. . .

2.5

Shunt of a multi-mode structure

. . .

. . .

2.6

Shunt performances