Structural Vibration Reduction by Switch Shunting of ... · Structural Vibration Reduction by Switch Shunting of Piezoelectric Elements: Modeling and Optimization J. DUCARNE,O.THOMAS*

Structural Vibration Reduction by Switch Shunting ofPiezoelectric Elements: Modeling and Optimization

J. DUCARNE, O. THOMAS* AND J.-F. DEU

Structural Mechanics and Coupled Systems Laboratory, Cnam, case 353, 2 rue Conte, 75003 Paris, France

ABSTRACT: This work deals with the reduction of structural vibrations by means of syn-chronized switch damping techniques on piezoelectric elements. Piezoelectric patches areattached to the vibrating structure and connected to an electrical circuit that includes aswitch. The latter allows to continuously switch the piezoelectric elements from an open-circuitstate to a specific electric impedance, synchronously with the mechanical oscillations. Thepresent study focuses on two goals: (i) the quantification of the added damping, (ii) theoptimization of the electric circuit parameters, carried out on a one degree of freedommodel. The free and forced responses of one mode of the mechanical structure are studiedin detail. The precise time response of the system is obtained with semi-analytical models forthe two cases where the electrical impedance is a simple resistance (synchronized switchdamping on short circuit) or a resistance in series with an inductance (synchronized switchdamping on inductor). The damping added by the device is estimated. In all cases, the mainresult of the study is that the piezoelectric coupling factor is the only parameter to optimizeand has to be maximized in order to maximize the added damping. An optimal value of theelectric circuit quality factor is obtained when using an inductance, for free and forcedresponse.

Key Words: vibration reduction, piezoelectric, shunt, switch, vibration control.

INTRODUCTION

STRUCTURAL vibration reduction has a wide range ofapplications, from large consumer products like

embarked electronic cards, computer hard disks orsport gear, to systems from the automotive, aircraftand space industries. Reducing vibrations can be neces-sary either to protect systems from failure and fatigue,or to increase the users auditory comfort, by reducing atthe source the radiated noise. An efficient and widelyused way of reducing vibrations is to use piezoelectricmaterials connected to an appropriate electrical device.The ability of these materials to convert mechanicalenergy into electrical energy and conversely can beused in two complementary ways. Either the mechanicalenergy is transferred into the electrical device in which itis dissipated, or the electrical device is designed so that itcreates an effect which is continuously opposed to themechanical oscillations.A large amount of piezoelectric-based vibration

reduction techniques have emerged in the last two dec-ades that fall into three main families: active, passive,and semi-passive. In the first one, a control device is

connected to piezoelectric elements bonded on the struc-ture, used either as sensors or as actuators. The principleis to create in real time an action on the mechanicalstructure that is opposed to the oscillations. It oftenrequires high performance digital signal processors andbulky power amplifiers to drive the actuators, that arenot suitable for many applications. Moreover, sinceenergy is injected, the system could be potentiallyunstable. On the other hand, the passive techniques,usually known as ‘shunts’, consists in connecting thepiezoelectric elements to a passive electrical circuit thatdissipates energy. However, even if excellent vibrationreduction performances can be achieved with resonant(RL) shunts, their main drawback is that they have to befinely tuned on a particular resonance, so that they arevery sensitive to a change, even small, in the mechanicalstructure characteristics (Hagood and Von Flotow,1991; Caruso, 2001; Ducarne et al., 2010). Moreover,the tuning requires a very large inductor (of severalHenrys) for usual mechanical low frequencies, impossi-ble to be realized without the help of externally poweredelectronic components.

To overcome those difficulties, several semi-passive(also called semi-active) approaches have been pro-posed. In a first family, some authors reduce the draw-backs of classical shunt techniques by using adaptiveinductor tuning (Wang et al., 1996; Davis and

*Author to whom correspondence should be addressed.E-mail: [email protected] 1 and 2 appear in color online: http://jim.sagepub.com

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 21—May 2010 797

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Lesieutre, 1998) or by considering electrical networks(Dell’Isola et al., 2004) to damp several modes of amechanical structure. In a second family, non-linearapproaches, commonly known as ‘switch’ techniques,have been proposed. Their principle is to continuouslychange the electrical circuit impedance synchronouslywith the mechanical oscillations. Two main families oftechniques have been proposed, that differs by theswitching time strategies as well as the electricalimpedance: the ‘state switching’ technique, introducedby Clark (2000) and the ‘synchronized switch damping’(SSD) techniques, introduced by Guyomar et al.(Richard et al., 1999; Richard et al., 2000; Petit et al.,2004; Lefeuvre et al., 2006) and by Corr and Clark(2003).This article deals with the SSD techniques. An electric

circuit including a switch is connected to the piezoelec-tric elements. The switch is left open most of the timeand is closed every time the structure reaches a maxi-mum of amplitude, for a short time, long enough toobtain an opposition of the voltage imposed at thepiezoelectric elements. Thanks to the circuit, the voltagehas the same effect on the structure than a force thatchanges of sign at each oscillation and thus opposes themotion. In this SSD family, three techniques have beenproposed, depending on the type of electrical circuitused. In the first one (denoted synchronized switchdamping on short circuit, or SSDS (Richard et al.,1999)), the electrical circuit includes only a resistance;in the second one (denoted synchronized switch damp-ing on inductor, or SSDI (Richard et al., 2000)), aninductor is added; in the third one (denoted synchro-nized switch damping on voltage, SSDV (Petit et al.,2004; Lefeuvre et al., 2006)), a constant voltage sourceis added. With the first two methods (SSDS and SSDI),even if some electronic components need to be powered(a small amount of energy is necessary for monitoringthe oscillations and switching), no energy is injected inthe system, so that the whole system (elastic mechanicalstructure together with the electrical circuit) energy bal-ance is passive. This explains the name ‘semi-passive’that usually denotes this family of techniques. As aconsequence, the system has the advantage of beingunconditionnaly stable. Moreover, because of the syn-chronization of the electric circuit to the structural oscil-lations, no precise tuning of the electric parameters onthe mechanical frequency characteristics (which is oneof the disadvantages of resonant shunts) is needed, sothat the system is very robust to changes in the mechan-ical structure. Finally, the inductor value for SSDI is inpractice very small, so that standard passive electroniccomponents can be used. With the SSDV technique, allthe above-mentioned characteristics are kept, exceptthat it falls into the active device family, since the vol-tage source injects energy in the system, which leads topotential instabilities (Lallart et al., 2008).

Several studies (Corr and Clark, 2003; Niederbergerand Morari, 2005; Guyomar and Badel, 2006; Lallartet al., 2008; Makihara et al., 2008) show that the opti-mization of the SSD electrical circuit (i.e., the choice ofthe electric components and the switch timing) can befairly complex, especially for the cases of multimodalproblems and wideband excitation. The goal of thisarticle is to propose a full system optimization, restrictedto the case of a one degree of freedom (dof) mechanicalstructure as well as an ideal switching strategy, based onideal electronic components. This reduction simplifiesthe study and enables a parametric study, based onoriginal analytical or semi-analytical expressions. Evenif this model is not very realistic, because it does nottake into account the energy transfer between modesas well as the unavoidable electronic components imper-fections, the obtained optimization results give a solidbasis for further studies. Both the cases of free andforced vibrations, in the case of a sine forcing, areaddressed.

In a first part of this article, a general electro-mechan-ical model of an elastic structure with piezoelectric ele-ments coupled to an electric circuit is presented, takinginto account both the direct and converse effect. Amodal approach is used to derive the discretized equa-tions of motion, obtained by expanding the unknowndisplacement onto the basis of vibration modes of thestructure with piezoelectric elements short circuited. Thefree electric charge constitutes an additional dof in thissystem that characterizes the electric circuit state. In thesecond part of this article, this model is restricted to onlyone mechanical dof, which is one of the modal coordi-nates, and the charge. The free time response of thesystem is precisely simulated by considering successivelyall switch operations with recurrence analytical rela-tions. The forced response is obtained in a similar waywith semi-analytical computation, for excitation at reso-nance and out of resonance. In both cases, some resultsabout the optimization of the electric parameters of theswitch as well as the estimation of the vibration reduc-tion are obtained, in both semi-passive cases SSDS andSSDI. The case of SSDV, which fall in the active family,is here left apart and kept for further studies.

ELECTROMECHANICAL MODEL

FE Formulation

We consider the case of an arbitrary elastic structurewith a single piezoelectric element (Figure 1). The finiteelement formulation proposed in (Thomas et al., 2009) isused. This formulation uses a standard discretization ofthe mechanical dofs and, provided some less restrictiveassumptions, takes into account global electricalvariables (the potential difference and free charge on

798 J. DUCARNE ET AL.

the electrodes). The complete electromechanical formu-lation, electrical circuit not included, then writes:

Mm 0

0 0

� �€U€V

� �þ

Km Kc

�KTc C

� �UV

� �¼

FQ

� �: ð1Þ

The unknowns in that system are the vector U of themechanical dofs (of size N) and V the electrical potentialdifference between the piezoelectric element’s electrodes.F is the vector of nodal forces (size N) and Q is the freeelectrical charge present on one of the electrodes (whilethe other electrode receives an opposite free electricalcharge (Thomas et al., 2009)). Mm and Km are themechanical mass and stiffness matrices, of size N�N.The mass and (elastic) stiffness of the piezoelectric ele-ment are included in these matrices. C is the blockedcapacity1 of the piezoelectric element. Finally, Kc is theelectromechanical coupling column vector of size N,which is characteristic of the two piezoelectric effects:

. KcV represents the nodal forces related to the con-verse piezoelectric effect when a voltage V is appliedto the electrodes;

. �KTc U is the free electric charge flowing in the circuit

at constant potential when the mechanical displace-ment is U (direct effect).

In the case of several piezoelectric elements linked inparallel or series, one can fall back on the same systemby using the charge flowing in the circuit and the poten-tial difference at its ends as unknowns and eliminatingthe other electrical unknowns (Thomas et al., 2009).

Modal Expansion and Dimensionless Form

In order to obtain a reduced order model, themechanical equations are expanded on the short-circuiteigenmode basis. First, we introduce the N modal fre-quencies and shapes (fr, (r), solutions of the short-circuit problem:

Km � ð2�frÞ2Mm

� �(r ¼ 0: ð2Þ

It is worth remarking that the above short circuit pro-blem, which is Equation (1) with V¼ 0 and restricted tothe mechanical variables, is purely elastic and can besolved by any finite element code. The modal shapesare scaled so that:

8r ¼ 1, . . . ,N , (Tr Mm(r ¼ mr, ð3Þ

where mr is the arbitrary modal mass of the r-th mode.We define the following dimensionless variables,

denoted by an overbar:

�U ¼U

u0, �t ¼

t

T0, �V ¼

V

V0, �Q ¼

Q

Q0: ð4Þ

The scaling of vibration amplitude u0 and time T0 canbe chosen arbitrarily. One may chose for instance u0equal to one of the characteristic dimensions of thestructure (its thickness for instance) and T0 equal tothe first mode period. The dimensionless displacementvector is written as:

�Uð �tÞ ¼XNr¼1

(r qrð�tÞ, ð5Þ

where qr is the r-th dimensionless modal coordinate.Now, by injecting U¼ u0 �U in (1), and multiplying theresult by(T

r for each r¼1, . . . ,N, one obtains the follow-ing set of equations, thanks to the orthogonality of themodes with relation to Km and Mm:

€qr þ !2r qr þ !rkr �V ¼ Fr, 8r ¼ 1, . . . ,N, (6a)

�V� �Q�PNr¼1

!rkrqr ¼ 0; (6b)

8<:where the dimensionless angular frequency !r anddimensionless modal force Fr naturally appear:

!r ¼ 2�T0fr Fr ¼T20

u0mr(T

r F:

Other parameters have to be set to obtain the abovesimplified expression: the voltage scaling V0 is chosen toobtain symmetrical coefficients !rkr in Equation (6a) and

1The ratio between Q and V when the mechanical displacement is set to zero(U� 0).

SSDS SSDI

Piezoelectric patch

Elastic structure

Switch

Ext. forcing

L

RV

R

I = − Q

Figure 1. Structure with switched shunts.

Switch Shunting of Piezoelectric Elements 799

(6b) and the scaling of charge Q0 is chosen to eliminate Cfrom the electrical equation, which is possible with:

Q0 ¼ CV0, V0 ¼u0T0

ffiffiffiffiffiffimr

C

r: ð7Þ

One has to chose the same modal mass mr�m for allmodes in order to obtain a coherent dimensionlessmodal coupling factor kr, defined by:

kr ¼(T

r Kc

2�frffiffiffiffiffiffiffiffimCp : ð8Þ

As the system studied is a switched shunt where theelectrical circuit remains open for some time, it is con-venient to write the electromechanical coupling with thecharge (piecewise contant) instead of the voltage (con-stantly varying). A viscous modal damping term �r isalso added. Finally, to simplify the various equations,all the dimensionless parameters are now written with-out an overbar. Equations (6a) and (6b) then rewrite:

€qr þ 2�r!r _qr þ !2r qr

þ!rkrXNi¼1

!ikiqi þ !rkrQ ¼ Fr, 8r ¼ 1, . . . ,N ð9aÞ

QþXNi¼1

!ikiqi ¼ V: ð9bÞ

8>>>>>><>>>>>>:A sum term in each mechanical equation of the system(9a) couples the modes with one another, which is aconsequence of writing the coupling with the charge.This term is similar to an added stiffness, which appearsin open-circuit condition (Q¼ 0) (Thomas et al., 2009).An electric circuit will be added later and will introducean additional relation between charge and voltage(‘Working Principle’ section). The same formulationthan Equations (9a) and (9b) can also be obtainedwith analytical formulations of the elastic structure(Ducarne et al., 2010), so that the whole study of thisarticle can be applied to any elastic structure.

System Reduction and Coupling Factor

In the following we assume that the system’s responseis dominated by the r-th mode, which leads to neglectingall the others: 8i 6¼ r, qi¼ 0. The mechanical equations ofthe short-circuit problem (V¼ 0) then writes:

€qr þ 2�r!r _qr þ !2r qr ¼ Fr,

while the open-circuit (Q¼ 0) problem writes:

€qr þ 2�r!r _qr þ !2r ð1þ k2r Þqr ¼ Fr:

Therefore, the natural angular frequency in short circuitis !r while in open-circuit it is !r, defined by:

!r ¼ !r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ k2r

q: ð10Þ

The open-circuit condition induces an added stiffness,which raises the resonance frequency; this can be usedto define the effective piezoelectric coupling factorkeff widely used in the litterature (ANSI/IEEE, 1988),which is here equal to the absolute value of the modalcoupling factor of Equation (8):

jkrj ¼ keff ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!r

2� !2

r

!2r

s: ð11Þ

In a case where the other modes would be taken intoaccount, the open-circuit problem would write differ-ently and the resonance frequency would be slightly dif-ferent, with then jkrj. keff (Thomas et al., 2009). In allcases, another difference between kr and the effectivecoupling factor is that the sign of the voltage relatedto one mode shape is taken into account by kr, whichcan be negative, while keff is always positive.

SSD MODEL

Working Principle

In this section, the response of the structure with thepiezoelectric element connected to a SSD electric circuitand around the r-th resonance is investigated. We studyboth cases of SSDS, where the switch is connected to anelectrical resistance only and SSDI where the switch isconnected to a resonant circuit made of an inductanceand a resistance (Richard et al., 2000) (Figure 1). Oneobtains the following equations stemming from themechanical equation (9a) where u(t)� qr(t) denotes theonly remaining modal coordinate, and from the electri-cal R and RL impedances:

Mechanical part: €uþ 2�r!r _uþ !r2uþ kr!rQ¼ Fr, ð12aÞ

Switch open: _Q¼ 0, ð12bÞ

SSDS closed: �e _QþQþ kr!ru¼ 0, ð12cÞ

SSDI closed:1

!2e

€Qþ2�e!e

_QþQþ kr!ru¼ 0, ð12dÞ

Voltage: Qþ kr!ru¼ V: ð12eÞ

In the above equations, �e stands for the time constantof the SSDS shunt, linked to resistance R. �e and !e

denotes the damping factor (linked to resistance R)and the angular frequency (linked to inductance L) ofthe SSDI shunt. These values are dimensionless by virtueof the choice of T0 and the electric capacity of thepiezoelectric element and write:

�e ¼RC

T0; !e ¼

T0ffiffiffiffiffiffiffiLCp ; �e ¼

R

2

ffiffiffiffiC

L

r: ð13Þ

The switch command strategy is now explained. Mostof the time, the switch is open: no current flows ( _Q ¼ 0)


and the free electric charge Q on the piezoelectric ele-ments is constant (Equation (12b)). In this case, thestructure behaves like a single dof oscillator at theopen-circuit angular frequency !r (Equation (12a),Figure 2(a) and (b)). The piezoelectric converse effecthas the effect of a constant force �kr!rQ on the struc-ture, proportional to the charge.The idea behind switch damping is to use this con-

verse piezoelectric effect to reduce the motion of thestructure. Every time the displacement reaches a maxi-mum, the switch is closed for a brief time Te, very smallcompared to the mechanical period of the structure.

. For SSDS, the charge then behaves like a first-ordersystem (Equation (12c), Figure 2(d)) and Te is chosenlarge enough for the charge to reach an equilibrium.In any case, Te is very small as compared to thesystem mechanical period 2�=!r.

. For SSDI, the charge behaves like a second-ordersystem (Equation (12d), Figure 12(e)) and Te is chosentobe exactlyhalf a systemelectrical period,whichallowsan important change in the charge value between aswitch closing and the following switch opening.

After that, the switch opens again and the charge remainsconstant until the next switch. In most cases, the electriccharge changes of sign synchronously with the

oscillations, and its effect on the mechanical structure istherefore similar to a piecewise constant force �kr!rQalways opposed to the velocity of the structure(Figure 2(a) and (c)), allowing the reduction of its motion.

An important remark is that the displacement max-imum corresponds also to a maximum of voltage(Equation (12e), Figure 2(f)), which may be used experi-mentally to synchronize the switch (Richard et al.,2007). Also, the fact that the switch closes at a maximumof voltage allows important charge variations, whichwill have a stronger effect on the structure.

Main Assumption

The main assumption in the models used in this articleis that the evolution of the electric charge (when the cir-cuit is closed) is very fast as compared to the mechanicalevolution. It can be written as:

Te �2�

!r: ð14Þ

As a consequence, the closed-circuit durations are verysmall as compared to those in open circuit. When thecircuit is open, no charge flows ( _Q ¼ 0), and from thepoint of view of the structure, the above assumptionmeans that the charge is piecewise constant, with a con-stant value for every open-circuit period (Figure 2(c)).

0 2 4 6 8 10

Dis

plac

emen

t

0 2 4 6 8 10

Fre

e el

ectr

ical

cha

rge

Switch times

0 2 4 6 8 10−5

0

5

Vol

tage

(a) (b)

(c) (d) SSDS (e) SSDI

(f)

un

tn + Tetntn + Tetn

Qn

tn+1tn

Qn

Qn–1

DQ

XDQ

Qn–1

unun

Figure 2. Typical time evolution of (a) displacement for free response u(t) with (b) details on one mechanical half-period (length �/!r); (c) freeelectric charge evolution Q(t), with details on one electrical evolution step (of very short length Te<<�/!r) for SSDS (d) and SSDI (e); (f) evolutionof voltage V(t) with SSDI.


In order to describe the effect of the charge on thestructure, only two parameters are necessary:

. tn the time at which the switch closes for the n-th time;

. and Qn the charge reached after the n-th switch.

The main consequence of the present assumption, whichwill be used all throughout the article, is that the closed-circuit phases (between tn and tnþTe) are not includedin the final time evolution of u(t) and Q(t), which areonly the concatenation of several open circuit phases,with a constant Q that changes at each tn.

Charge Evolution

Even if the closed-circuit phases are not included inthe final time evolution of u(t) and Q(t), one has to detailthe evolution when the circuit is closed to analyze thesuccessive values of Qn. In Figure 2(d) and (e) are shownthe charge evolutions during one closed-circuit phase inthe case of SSDS or SSDI with a very fine timescalecompared to Figure 2(c).The evolution of the charge while the circuit is closed

can be simplified by taking two facts into account:

. the system is designed so that the circuit closes whenthe speed _u of the structure cancels and changesdirection;

. the mechanical evolution is very slow compared tothe electrical evolution.

One can then make the hypothesis that _u remains zeroduring the closed-circuit evolution. Therefore, u remainsconstant and equal to its initial value u(tn) during aclosed-circuit phase:

8t 2 ½tn, tn þ Te�, _uðtÞ ¼ 0 and uðtÞ ¼ uðtnÞ:

CASE OF SSDSFor SSDS, the evolution between tn and tnþTe

depends on Equation (12c) with u constant:

�e _QþQþ kr!ruðtnÞ ¼ 0, ð15Þ

and the initial value (at t¼ tn) is Qn�1. The chargebehaves like a decreasing exponential (Figure 2(d))with the charge converging towards a steady-statevalue of �kr!ru(tn); the switch is held closed for a timeTe long enough for this value to be reached. When theswitch opens, the value of the charge is:

Qn ¼ �kr!ruðtnÞ: ð16Þ

CASE OF SSDIFor SSDI, the evolution between tn and tnþTe

depends on Equation (12d) with u constant:

1

!2e

€Qþ2�e!e

_QþQþ kr!ruðtnÞ ¼ 0, ð17Þ

The charge behaves like a damped harmonic oscillator(Figure 2(e)) of angular frequency !e, damping factor �e,and oscillates around the equilibrium value, which is�kr!ru(tn). The initial value (at t¼ tn) is Qn�1. The evo-lution ends after a time Te chosen so that exactly onlyhalf an oscillation occurs (Te¼ p/!e). The final valuedepends on u and the initial value of the charge Qn�1:

Qn ¼ �kr!ruðtnÞ � X Qn�1 þ kr!ruðtnÞ½ �, ð18Þ

where the overshoot factor (which role is described onFigure 2(e)) is related to the damping factor of the elec-trical circuit:

X ¼ exp��effiffiffiffiffiffiffiffiffiffiffiffiffi1� �2e

p : ð19Þ

FREE RESPONSE WITH SSD

The Case of No Mechanical Damping

We now model the free response of the structurewith SSD vibration reduction and without mechanicaldamping. The charge is modeled with the previousassumptions. Between time tnþTe and tnþ1 is an open-circuit phase, the charge is constant with value Qn; u(t)is therefore solution of Equation (12a) with �r¼ 0:

€uþ !2r uþ kr!rQn ¼ 0: ð20Þ

For the first mechanical step, at t0¼ 0, the initial condi-tions are u(0)¼ u0, Q0¼ 0 and _uð0Þ ¼ 0. For the othersteps, they are obtained by continuity, with in particular_uðtn þ TeÞ ¼ 0. Other initial conditions at t¼ 0,for instance with a non-zero velocity, could be treatedin a similar manner, which would result in an offsetin the timing of the following switches. The globalresult would be similar; hence only the case _uð0Þ ¼ 0 isconsidered for the sake of simplicity. The solution ofEquation (20) with the initial condition of zero initialvelocity is the sum of a constant (offset) part and anoscillating part:

uðtÞ��tnþTe5t5tnþ1

¼ �un þ ~un cosð!rðt� tnÞÞ, ð21Þ

with un the oscillations amplitude (as described inFigure 2(b)) and the offset �un depending of the freecharge:

�un ¼ �kr!r

!2r

Qn: ð22Þ

un will be calculated later. One can remark that aftertnþTe, the next time at which the velocity cancels ishalf a mechanical open-circuit period: tnþ1 ¼ tn þ Teþ

�=!r. As Te is neglected, tnþ1 � tn ¼ �=!r; this resultcan be used in a recurrence relation in order to show


that in the case of free reponse with SSD, the n-th switchoccurs at:

tn ¼n�

!r: ð23Þ

The typical shape for this result is presented inFigure 2(a), with a zoom on the mechanical evolutionfor half an open-circuit period (b). The continuity of uallows to write a relation between u just before and justafter the n-th switch: u(tnþ1þTe)¼ u(tnþ1). Then, usingthis relation together with Equation (21) and neglectingTe, one obtains:

�unþ1þ ~unþ1 cosð!rðtnþ1� tnþ1ÞÞ ¼ �unþ ~un cosð!rðtnþ1� tnÞÞ,

which, with Equation (23), simplifies in:

�unþ1 þ ~unþ1 ¼ �un � ~un: ð24Þ

SSDS MODELTaking into account the contuinuity of mechanical

displacement (Equation (24)), the effect of the charge(Equation (22)) and the value of charge after everyswitch (Equation (16)), one obtains a recurrence relationwith �un and un:

�unþ1 ¼k2r!

2r

!2r

uðtnþ1Þ ¼k2r

1þ k2rð �un � ~unÞ,

~unþ1 ¼ �un � ~un � �unþ1 ¼1

1þ k2rð �un � ~unÞ:

ð25Þ

SSDI MODELTaking into account equations (24), (22) and the value

of the charge now depending on the damping factor ofthe circuit (Equation (18)), one obtains:

�unþ1 ¼k2r!

2r

!2r

ð �un � ~unÞð1þ XÞ � X �un

¼ �ð1þ XÞk2r1þ k2r

�un þk2r � X

1þ k2r~un,

~unþ1 ¼ �un � ~un � �unþ1 ¼1þ X

1þ k2r�un �

1� k2rX

1þ k2r~un:

ð26Þ

DECAY RATEThe values of ( �un)n2N and (un)n2N can be written with

the recurrence relations (25) for SSDS and (26) for SSDIin a matrix form:

un ¼ Aun�1 ¼ Anu0, ð27Þ

with un¼ ( ~un �un)T, the initial condition being u0¼ (u0;

0)T, and the transfer matrix A depending on the electri-cal parameters:

ASSDS¼

1

1þ k2r

�1 1

�k2r k2r

� �, ð28Þ

ASSDI¼

1

1þ k2r

Xk2r � 1 1þ X

�k2r ð1þ XÞ k2r � X

!: ð29Þ

The values of ( �un)n2N, ( ~un)n2N can then be written bymultipliying An and the initial vector. In order to sim-plify this study, one can introduce �1 and �2, the eigen-values of A. With P denoting the matrix of eigenvectorsof A, one can write:

An¼ P�1

�1 00 �2

� �P

� �n

¼ P�1�n1 00 �n2

� �P: ð30Þ

Taking this expression of An into account and Equation(27), ( �un)n2N, ( ~un)n2N can be written as a sum of twogeometric sequences:

�un ¼ �a1�n1 þ �a2�

n2

~un ¼ ~a1�n1 þ ~a2�

n2

�, ð31Þ

As a consequence of (31), by sorting j�1j � j�2j and con-sidering Equation (21), the displacement is bounded by:

juðtnÞj � j �unj þ j ~unj

� j �a1j þ j ~a1j þ ðj �a2j þ j ~a2jÞj�2j

n

j�1jn

� �j�1j

n

� aj�1jn:

with a¼ j �a1jþj ~a1j þ j �a2j þ j ~a2j. The above inequalitiesshow that u(t) is bounded by a decaying exponentialfunction of decay rate �, defined by:

ae��tn ¼ aj�1jn: ð32Þ

Using Equation (23), one obtains:

� ¼ �!r

�lnðj�1jÞ: ð33Þ

We will see in the next sections that for realistic valuesof the parameters, j�1j< 1, so that � is positive.Moreover, it increases when j�1j decreases. As a conse-quence, if one wants to damp the vibrations, the decayrate has to be increased, which is obtained by minimizingthe modulus of the eigenvalue j�1j of A of greatestmodulus. Since the time evolution frequency of u(t) isvery close to !r, it is convenient to define the damping

factor �SSDtot of the free response of the structure with SSD

and without mechanical damping by:

�SSDtot ¼�

!r¼ �

lnðj�1jÞ

�: ð34Þ


Several simulations have been carried out and haveshown that the above-defined equivalent dampingfactor, even if it is based on an inequality, allows agood representation of the motion amplitude decay(Figure 3), thus being a excellent parameter to measurethe SSD performance in free oscillations.

OPTIMIZATION OF SSDSIn the case of SSDS, from Equation (28) one obtains:

j�1j ¼1� k2r1þ k2r

, �2 ¼ 0:

The damping factor is then (Equation (34)):

�SSDStot ¼ �

1

�ln

1� k2r1þ k2r

� �: ð35Þ

One can notice that in this model, the damping factor(plotted in Figure 7 among other values) does notdepend on �e and thus on the electric resistance.One has just to ensure that the resistance is smallenough so that the electric time constant (as seen inEquation (15)) is much smaller than the mechanicalperiod of the system, so that Equation (14) is verified.

In order to increase �SSDtot and thus the damping effect

brought by the SSDS, Equation (35) and Figure 7 showsthat kr has to be maximized. An interesting result is thatthe modal coupling factor kr is the only free parameterto work on. This can be done by optimizing the piezo-electric element associated to the structure, using forinstance a finite element reduction method (Senechalet al., 2009; Thomas et al., 2009; Senechal et al., 2010)or an analytical method (Ducarne et al., 2007; Ducarneet al., 2010).

OPTIMIZATION OF SSDIIn the case of SSDI, �1 and �2 depend on �e, the

damping factor of the electric circuit. Figure 4(a)shows the eigenvalues moduli j�1j and j�2j as a functionof �e. One can observe that the eigenvalues are complexconjugates for low values of �e and distinct real for highvalues of �e. The optimal value �opte of �e is obtainedwhen the eigenvalues of the matrix defined byEquation (29) are equal and real:

j�1j ¼ j�2j ¼1� jkrj

1þ jkrj, ð36Þ

which leads to the optimal value of the overshoot factorand finally of �e:

Xopt ¼ð1� jkrjÞ

2

ð1þ jkrjÞ2, �opte ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðXoptÞ

2

lnðXoptÞ2þ �2

s, ð37Þ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(a) (b)

xe

Eig

enva

lues

mod

uli

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

kr

l2

l1

xeopt

x eopt

Figure 4. (a) Moduli of eigenvalues of ASSDI for kr¼ 0.2 (‘—’: �1, ‘- -’: �2). (b) Optimal electrical damping �opte as function of kr.

0 5 10 15 20−1

−0.5

0

0.5

1

Dis

plac

emen

t

Time

Figure 3. Simulation of time evolution with SSDS and decayingexponential envelope from expression (32), with parameter aadjusted by hand.


The total damping factor of the free response of thestructure with SSDI is then:

�SSDItot ¼ �

1

�ln

1� jkrj

1þ jkrj

� �: ð38Þ

Figure 4(b) shows the evolution of �opte as a function ofjkrj. Figure 7 shows the value of equivalent dampingobtained (among other values).Figure 5 shows the time evolution of u(t) for three

different values of �e and confirms the above results.For �e 5 �opte a beating phenomenon appears (relatedto the complex conjugate eigenvalues of A). What hap-pens is that an important charge (electrical energy) isstored in the piezoelectric element and relaunches thestructure’s vibrations (Equation (18) with �e low, X isalmost 1). Experimental studies of the influence of theelectrical resistance on the free response (Onoda et al.,2003) mention that phenomenon. For �e 4 �opte , thegreatest eigenvalue modulus increases and the decay isslower because the charge stored is too low; the optimalvalue of �e ensures the greatest decay rate.In order to fully optimize the system, we have to first

maximize kr in (38) by optimizing the piezoelectric ele-ment, and then to choose the appropriate �e withEquation (37) or Figure 4(b). In the same manner thanfor SSDS, the result will only depend of kr (Figure 7).One can note that !e, related to the inductance L, has noinfluence on the system behavior in this model: one hasjust to chose !e as high as possible to ensure that theelectric period is much smaller than the mechanicalperiod, so that Equation (14) is verified.

Effect of Mechanical Damping

The previous results have been obtained byneglecting the mechanical damping �r. Taking it

into account, the solution of (20) writes, for one open-circuit phase:

uðtÞ��tn5t5tnþ1

¼ �un þ ~une��r!r cosð ~!rðt� tnÞÞ, ð39Þ

with

~!r ¼ !r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2r

q, ð40Þ

the pseudo angular frequency of the oscillations. Thevalue of u at the end of an open-circuit phase, justbefore the switch, is now:

uðtnþ1Þ ¼ �un � Xr ~un, ð41Þ

where

Xr ¼ exp��rffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2r

p ; ð42Þ

is the overshoot factor of the mechanical oscillator. Therecurrence relation (27) is now rewritten to take intoaccount the mechanical damping:

un ¼ ABun�1 ¼ ðABÞnu0, B ¼

Xr 00 1

� �, ð43Þ

where the B matrix represents the mechanical losses.With analogous arguments than those of sections‘Optimization of SSDS’ and ‘Optimization of SSDI’,we can obtain the eigenvalues �1 and �2 of matrix AB.Following Equation (32) and remarking that the switchoccurs now at each half pseudo-period, so thattn � tn�1 ¼ �= ~!r, Equation (33) is replaced by:

� ¼ �~!r

�lnðj�1jÞ ¼ �

!r


p�

lnðj�1jÞ: ð44Þ

0 10 20−1

−0.5

0

0.5

1

Time

kr = 0.2, xe = 0.05(a) (b) (c)

0 10 20−1

−0.5

0

0.5

1

Time

kr = 0.2, xe = 0.25

0 10 20−1

−0.5

0

0.5

1

Time

kr = 0.2, xe = 0.5

Figure 5. Time evolution of displacement u(t) (‘—’) and �un (‘- -’) with SSDI, for three values of �e . (a): low �e; (b): optimal �e; (c): high �e .


Consequently, considering Equation (34), the totaldamping factor with SSD and mechanical damping is:

�SSDtot ¼ �


p�

lnðj�1jÞ: ð45Þ

OPTIMIZATIONIn the case of SSDS, the eigenvalues of AB are:

j�1j ¼jXr � k2r j

1þ k2r, and �2 ¼ 0: ð46Þ

The damping ratio of the free response then writes:

�SSDStot ¼ �


p�

lnXr � k2r1þ k2r

� �: ð47Þ

For the SSDI system, the eigenvalues �1 and �2 followthe same behavior than in the case of zero mechanicaldamping �r, whatever be its value (Figure 6(a)). It isshown that the mechanical damping keeps the overallshape of the curves and only stretches them to theright bottom corner. With the same procedure than inthe case of zero mechanical damping, one shows that theoptimal value �opte of �e is increased as a function of �r. Itdepends on kr and �r, with a complicated analyticalexpression, written in the appendix together with thecorresponding common value of �1 and �2. However,this analytical expression for �opte is not necessary,since an excellent approximation of its value is obtainedby adding the optimal value obtained without damping(Equation 37) to the mechanical damping �r:

�opte j�r 6¼0 ’ �opte j�r¼0 þ �r: ð48Þ

This result has not been proved analytically: it is only areading of Figure 6, where �opte � �r is displayed as afunction of coupling factor kr, and that shows that�opte � �r is manly a function of kr. This figure showsalso the existence of a particular value of kr, abovewhich �opte decreases as a function of kr. This point willbe explained in the following.

PERFORMANCE FOR LOW MECHANICALDAMPING

In both cases of SSDS and SSDI, we can define theadded damping, which is the difference between the totaldamping, including the effect of SSD, with the open-circuit damping: �add¼ �tot� �r. Figure 7 shows that�add is almost independant of the mechanical damping�r, so that it is very close to the total damping in the caseof no mechanical damping, defined by Equations (35)and (38). In other word, a good estimation of the totaldamping ratio with mechanical damping (�r 6¼ 0) of thestructure’s free response with SSD can be calculatedwith the mechanical damping ratio �r added to thetotal damping ratio without mechanical damping(�r¼ 0), the latter being defined by Equations (35)and (38):

�SSDStot ’�

SSDStot

��r¼0þ�r, �

SSDItot ’�

SSDItot

��r¼0þ�r: ð49Þ

This property is illustrated on Figure 7, for �r� 2%. Forhigher values of mechanical damping, the qualitativebehavior is kept, with slightly quantitative differences.

THE CASE OF HIGH MECHANICAL DAMPINGIn the case of high mechanical damping ratios �r,

above 2%, it is observed that �SSD becomes infinite fora particular value of the coupling factor kr, that depends

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xe

Eig

enva

lues

mod

uli

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

kr

xr= 0

xr= 0.1

xr= 0.5

xr= 0, 0.001, 0.01

xr=0.1

xr=0.5

xeopt

x eopt –

x r

(a) (b)

Figure 6. (a) Moduli of eigenvalues of ASSDIB for kr¼ 0.2 and various values of �r (‘—’: �1, ‘- -’: �2). (b) Optimal electrical damping �opte as

functions of kr with mechanical damping.


on �r. This can be seen on Figure 8, where the maximaare finite because of the x-axis discretization.Considering Equations (46) and (47), the numerator of�1 becomes zero for kr ¼

ffiffiffiffiffiXr

p, which explains the infi-

nite value of �SSDS for this particular value of kr. It canalso be shown by symbolic manipulations on Equations(56) and (58) that, in the case of SSDI, the same parti-cular value of kr ¼

ffiffiffiffiffiXr

pleads to an infinite �SSDI. In

fact, this property is observed for any values of �r,even low. However, in this latter case, the infinitevalues of �SSD occurs for kr very close to 1, which isout of the range of common practical values of kr.To understand the meaning of an infinite value of

�SSDI, the system’s time evolution for a high dampingfactor (�r¼ 0.5) and for three values of the couplingcoefficient, namely kr 5

ffiffiffiffiffiXr

p, kr ¼

ffiffiffiffiffiXr

pand kr 4

ffiffiffiffiffiXr

p,

is shown on Figure 9. Three situations can be observed.

. Below the coupling factor critical valueffiffiffiffiffiXr

p, the

behavior is the same than that explained in thesection ‘Performance for Low Mechanical Damping’.The total (or added) damping factor increases with kr(Figure 8) and Figure 9(a) shows that the SSDI has alittle damping effect.

. If the coupling factor equals its critical value, thetotal damping is theoretically infinite. This can beexplained by observing in Figure 9(b) that the SSDIfully stops the system’s oscillations. After the firstpseudo-period, all the initial mechanical energy hasbeen dissipated by the SSDI and the system remainsat its equilibrium position, without any oscillations.

. Above the coupling factor critical valueffiffiffiffiffiXr

p, the

total damping factor decreases with kr, which meansthat the SSDI efficiency decreases. This can beexplained by Figure 9(c), where it is observed that

0 0.2 0.4 0.6 0.8 110−4

10−3

10−2

10−1

100

kr

x aS dS dDS

, xaS dS dD

I

0 0.05 0.1 0.15 0.2

10−4

10−3

10−2

10−1

kr

x aS dS dDS

, xaS dS dD

I

xaSdSdDI

xaSdSdDS

xaSdSdDS

xaSdSdDI

Figure 7. Added damping as a function of kr, in the case of SSDS and SSDI, for various low mechanical damping values (�r2 {0,0.001, 0.01,0.02}). All curves are merged.

0 0.2 0.4 0.6 0.8 110−4

10−3

10−2

10−1

100

101

kr

0 0.2 0.4 0.6 0.8 110−4

10−3

10−2

10−1

100

101(a) (b)

kr

xr = 0, 0.01

xr = 0.1xr = 0.5

xr = 0, 0.01

xr = 0.1xr = 0.5

x aS dS dDS

x aS dS dDI

Figure 8. Added damping as a function of kr, in the case of SSDS (a) and SSDI (b), for various high mechanical damping values (�r2 {0, 0.01,0.1, 0.5}).


the switch has an erratic behavior in this case. Thecharge (proportional to �u) changes of value synchro-nously with the oscillations, but it does not change ofsign. The SSDI thus not produces any efficient vibra-tion reduction. It only changes the oscillation pattern.

The above results show that for high mechanical damp-ing, increasing the coupling factor above the particularvalue

ffiffiffiffiffiXr

pdoes not lead to an increase of the system’s

performances. However, this is not a problem in practicefor the present application of vibration reduction, sincemechanical systems of high damping ratios are naturallydamped and an additional vibration reduction device isoften not necessary. This is also why only the case ofmechanically underdamped systems (with �r< 1) hasbeen considered in this study. For overdamped systems,no oscillations occurs, Equations (21) and (39) do notapply and have to be replaced by decreasing exponentialfunctions, thus leading to another model, beyond thescope of the present article.

FORCED RESPONSE WITH SSD

Time Response of the System

We study the structure with SSDS or SSDI in forcedoscillations around the r-th mode. The assumptionsmade in the section ‘SSD Model’ are kept. The modalforcing is harmonic and writes FrðtÞ ¼ ~Fr cos�t and theproblem is defined by Equations (12b)–(12d) withEquation (12a) becoming in that case:

€uþ 2�r!r _uþ !2r uþ kr!rQn ¼ ~Fr cosð�tÞ: ð50Þ

The switch strategy is exactly the same than the onedescribed in the section ‘SSD Model’, namely that theswitch will close every time the velocity cancels( _uðtnÞ ¼ 0) and remains closed a very short time Te.

Between tnþTe and tnþ1, the switch is open and thesolution of Equation (50) writes:

uðtÞ��tnþTe5t5tnþ1

¼ �un þ uTðtÞ þ uFðtÞ, ð51Þ

where �un, uT(t), and uF(t) are respectively the static, tran-sient, and forced solution of Equation (50).

. �un is the static response of the system to the freeelectric charge Qn applied; like in Equation (20),it writes:

�un ¼ �kr!r

!2r

Qn:

. uT(t) is the transient reponse; it writes:

uTðtÞ ¼ e��r!rðt�tnÞ ~un cosð ~!rðt� tnÞÞ þ ~u0n sinð ~!rðt� tnÞÞ� �

,

ð52Þ

with ~! defined by Equation (40); ~un and ~u0n depend onthe initial conditions at tn.

. uF(t) is the forced, steady-state response of the systemand writes:

uFðtÞ ¼ ~Fr!2r ��2

d ð�Þcos�t�

2�r!r�

d ð�Þsin�t

� �, ð53Þ

with d ð�Þ ¼ ð!2r ��2Þ

2þ 4�2r!

2r�

2. uF(t) alone is thesteady-state reponse of the system in open-circuitcondition, which can be used later as a reference foramplitude comparison.

The values of ðtn,Qn, �un, ~un, ~u0nÞn2N are obtained bysuccessive computation:

(1) First �u0, ~u0, ~u00 are identified from the initial condi-tions (t0¼ 0, Q0¼ 0, u(0)¼ 0, _u ¼ 0);

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

TimeD

ispl

acem

ent

kr = 0.403 − xr = 0.5

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

Time

Dis

plac

emen

t

kr = 0.8 − xr = 0.5

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

Time

Dis

plac

emen

t

kr = 0.2 − xr = 0.5(a) (b) (c)

Figure 9. Time evolution of displacement u(t) (‘- -’: without SSDI; ‘—’: with SSDI) and �un (light ‘- -’), for high damping factor (�r¼ 0.5), the optimal

electrical damping factor (�e ¼ �opte ) and three values of kr: kr 5

ffiffiffiffiffiXr

pðaÞ, kr ¼

ffiffiffiffiffiXr

p(b) and kr 4

ffiffiffiffiffiXr

p(c), where

ffiffiffiffiffiXr

p¼ 0:403 with �r¼ 0.5

(Equation (42)).


(2) Then the next zero of _u is numerically found,2 occur-ring at time t1;

(3) The charge evolution happens instantly, as describedat section 2.3 and the new value of the charge Q1 isderived from equations (16) for SSDS or (18) forSSDI;

(4) By continuity of u and _u the values of �u1, ~u1, ~u01 areidentified. The next switch time t2 is computed byfinding the next time at which _u ¼ 0, going back tostep 2.

An important difference with the free response is thatthe zeros of �unþ uT(t)þ uF(t) cannot be computed ana-lytically; there is no way to predict tn other than tocompute the intermediate values and switching timescould be irregular. In order to analyze the behavior ofthe system, the time response will be computed for agiven time interval.

Response with SSDS

For many simulations with different parametersvalues, the displacement time evolution with SSDS isfound to be converging towards a steady-state response(for instance in the case of Figure 10) that is periodic ofangular frequency � (the excitation frequency). This dis-placement is not exactly sinusoidal and odd harmonicsof the forcing frequency appear; this is related to theeffect of the charge, which is a square signal at the exci-tation frequency, without constant component. It mustbe precised here that the periodic nature of the signal hasbeen assessed by observing the Fourier spectrum of thedisplacement. In order to evaluate the response of thesystem as a function of excitation frequency �, the rootmean square (RMS) value uRMS of the time evolution of

u(t) in steady state is numerically evaluated and plottedas a function of forcing frequency �, for various �around resonance !r. u

RMS is defined by:

uRMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

Z T

0

u2ðtÞdt

s, ð54Þ

and computed with T sufficiently large so that the steadystate is attained. Figure 11 is obtained. It is similar to afrequency response function plot, but it takes intoaccount all the spectral components of u(t).

The maximum amplitude is obtained at the open-circuit resonance frequency. To evaluate the efficiencyof the system, we define the attenuation brought by the

SSDS, denoted as ASSDSdB , as the difference, in dB,

between the system’s peak amplitude with SSDS andthe system’s open-circuit amplitude at resonance

(Figure 11). It is observed that ASSDSdB depends only on

kr and �r and is plotted in Figure 11. Simulations showthat a change of !r (for instance, by adding stiffness tothe structure), with all others parameters equal, onlychanges the timescale and the attenuation broughtremains identical. Several other simulations show thatthe amplitude with and without switch depends linearlyon ~Fr, and so that the attenuation is independent of ~Fr.It is found that kr has to be as high as possible in orderto maximize the attenuation. Finally, as in the case ofthe free response of SSDS (section ‘Optimization ofSSDS’), no optimal value of the electric resistance isobtained: one has just to ensure that R is small enoughso that the electric time constant is much lower that themechanical period of the system (Equation (14)).

Response with SSDI

Considering the system’s response with SSDI,we observed that for low values of �e, the system’s2Using for instance the Matlab fzero algorithm (The MathWorks, Inc.).

0 100 200 300−20

−15

−10

−5

0

5

10

15

20

t

u(t

)

−20 −10 0 10 20−20

−15

−10

−5

0

5

10

15

20

u(t )

du/d

t (t)

0 5 10 15 20−50

0

50

100

150

w

u(w

) (d

B)

Figure 10. Time evolution of mechanical displacement u(t) with SSDS, phase space (transient in gray), and spectrum with SSDS (open-circuitspectrum in dashed gray) for kr¼ 0.2 and �r¼ 0.1%; system forced at open-circuit resonance (W ¼ !r ).


displacement time evolution does not stabilize in aperiodic steady-state signal. An example is given inFigure 12(left), where the trajectory in the phase spaceis typical of a chaotic system. It must be precised that noverification of the chaotic nature of u(t) has been madeyet (e.g., by computing Lyapunov exponents). Forhigher values of �e, the system response stabilizes itselfin a periodic regime of frequency �, after a short tran-sient (Figure 12(mid, right)). The effect of the charge,apparent on the spectra, is more important than in thecase of SSDS, with a lower amplitude of the response,but relatively higher amplitude of the odd harmonics.In a similar way than in the case of SSDS (section

‘Response with SSDS’), we can plot the RMS valueuRMS of u(t) as a function of excitation frequency �and define the attenuation brought by the switching

ASSDIdB (Figure 13). Then, the evolution of ASSDI

dB as a

function of �e can be obtained, as shown onFigure 13(b). One can observe that above a criticalvalue of �e, the system response is stable and periodic

and that ASSDIdB decreases when �e increases. This critical

value can be chosen as an optimal value �opte for �e.However, as the transition between stabilized andunsteady response is very sharp, one may want toavoid being too close to the critical value.

Figure 14(a) shows the evolution of �opte as a functionof kr, for various values of the mechanical dampingfactor �r. For low �r values (lower than 0.01), it isobserved that �opte depends only on kr, since all curvesare merged. This value for forced response is approxi-mately half the optimal value for the free response,shown on the same graph for reference. Above�r¼ 0.01, �opte also depends on �r.

Finally, the attenuation ASSDIdB obtained with optimal

electric damping (�e ¼ �opte ) as a function of kr for

various values of �r is shown on Figure 14(b). Onceagain, kr has to be maximized in order to obtainthe best possible performance. Moreover, the more themechanical system is naturally damped, the less theSSDI is efficient. As in the case of the free response(section ‘Optimization of SSDI’), no optimal value ofthe electric inductance L, is obtained: one has just toensure that L is small enough so that the electric timeconstant is much lower that the mechanical period of thesystem (Equation (14)). In order to optimize the system,R has to be chosen after L to obtain the correct value of�e, using Figure 14(a).

Comparison with Known Results

R AND RL SHUNTINGThe efficiency of SSD devices is compared to classic

resistive (R) and resonant (RL) shunt techniques inFigure 15. The shunt results are taken from (Ducarneet al., 2007; Ducarne et al., 2010), where the shunt elec-tric parameters (the R and L values) have been optimallyset to maximize the vibration reduction, in a wayinspired by the results of Hagood and Von Flotow,(1991). Those optimal values are different from thoseof the SSD devices, especially in the RL-shunt case forthe L value, which has to be tuned to the mechanicalresonance, thus leading to very large L of several hun-dreds of Henrys for the lowest resonance frequencies.Inspite of showing better performances, SSD techniquesdo not present the major drawback of R and RL shunttechniques that require a fine tuning of the electric

0.5 1 1.5−10

0

10

20

30

40

50

60(a) (b)

AdBSSDS

W

Am

plitu

de (

dB R

MS

)

Without switchingSSDS

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

xr = 50%

xr = 10%

xr = 1%

xr = 0.1%

xr = 0.01%

kr

Atte

nuat

ion

AdBS

SD

S (

dB)

Figure 11. (a) RMS value uRMS of time response u(t) as a function of excitation frequency W, for kr¼ 0.2 and �r¼ 0.1%. (b) ASSDSdB as a function of

kr, for different mechanical damping factors �r.


parameters on the mechanical resonance frequencies,thus leading to performances very sensitive to changesin the tuning, even very small.The higher performance of SSD techniques obtained

here, as compared to those of classical R and RL shunts,can be qualitatively explained as follows. Because of thesingle harmonic excitation signal, for each frequency,the SSDI device acts as a non-linear oscillator (squarewaveform), synchronizing itself and acting against the

structure’s motion. The electric damping factor �e ofthis oscillator can be relatively low, and the compara-tively high charge stored in the piezoelectric elementhave a strong vibration reducing effect at each fre-quency. When one optimizes the RL shunt, which hasa single tuning frequency, things are different. In fact,if one were to choose a zero electric damping factor,the attenuation obtained would be infinite at theopen-circuit resonance frequency, where a mechanical

0 100 200 300−10

−5

0

5

10

t

u(t

)

−10 −5 0 5 10−10

−5

0

5

10

u(t )

du/d

t(t)

xe = 0.06

0 100 200 300−10

−5

0

5

10

t

u(t

)

−10 −5 0 5 10−10

−5

0

5

10

u(t )

du/d

t(t)

xe = 0.0603

0 100 200 300−10

−5

0

5

10

t

u(t

)

−10 −5 0 5 10−10

−5

0

5

10

u(t )

du/d

t(t)

xe = 0.12

0 5 10 15 20−50

0

50

100

150

w

u(w

) (d

B)

Peak reduction : −40.2 dB

0 5 10 15 20−50

0

50

100

150

w

u(w

) (d

B)


0 5 10 15 20−50

0

50

100

150

w

u(w

) (d

B)


Figure 12. Time evolution of mechanical displacement u(t) with SSDI, phase spaces with SSDI and compared spectra with and without SSDI,for kr¼ 0.2 and �r¼ 0.1%. (left): System forced near resonance (� ’ !r ) with slightly overcritical damping �e; (center): optimal damping �e;(right): large damping �e .


antiresonance would appear. However, in that case, tworesonances of the coupled, linear system would appearsideways (Hagood and Von Flotow, 1991; Ducarneet al., 2010) and would raise the level of the vibratoryresponse at neighboring frequencies. Therefore, in orderto reduce the vibration level at all frequencies, a higherdamping factor has to be chosen, lowering the chargeavailable in the piezoelectric element to counter thevibrations, thus smoothing the RL shunt performancesaround the open-circuit natural frequency.

OTHER SSD MODELSTwo major characteristics of our model, that con-

trasts with previous results of the literature that alsoconsider the harmonically forced response of a one dofmechanical system with a SSD device (Guyomar et al.,2008), can be emphasized. First, no assumption about

the periodicity of the oscillations is made: our model cansimulate responses which are not periodic, with irregularswitching times. This is the case, for example, if theelectrical damping factor is chosen below the optimalvalue �opte (Figure 12(left)). Second, our model cantake into account any value of the excitation frequency,especially at resonance, but also far from the resonance(Figure 13(a)).

An interesting result has thus been obtained: loweringthe resistance may render the system unstable, evenreduced to one single mode. This contrasts with previousstudies where the system’s response is assumed to bealways periodic. In particular, this assumption is madeby the LGEF3 team in (Guyomar et al., 2008), which

0.5 1 1.5−10

0

10

20

30

40

50

60(a) (b)

AdBSSDI

Ω

Am

plitu

de (

dB R

MS

)

Without switchingSSDI

0 0.2 0.4 0.6 0.8 1

30

35

40

45

50

xe

AdB

xeopt

Figure 13. (a) RMS value uRMS of time response u(t) as a function of excitation frequency W . (b) Attenuation ASSDIdB as a function of �e . In both

cases, kr¼ 0.2 and �r¼ 0.1%.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1(a) (b)

kr

Opt

imal

dam

ping

xe

Free responsexr = 0

xr = 0.1

xr = 0.5

xr = 0, 0.001, 0.01

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

xr = 50%

xr = 10%

xr = 1%

xr = 0.1%

xr = 0.01%

kr

Atte

nuat

ion

AdBS

SD

I (dB

)

Figure 14. (a) �opte as a function of kr; (b) ASSDI

dB as a function of kr, for different mechanical damping factors �r.

3LGEF means Laboratoire de Genie Electrique et Ferroelectricite, INSA, Lyon,France.


allows obtaining the following analytical expression forthe attenuation obtained with SSDI:

ALGEFdB ¼ 20 log 1þ k2r

1

2�r

4

�

1þ X

1� X

ð55Þ

where X is the overshoot factor (Equation (19)).

Figure 16 shows ASSDIdB as a function of �e, obtained

with the LGEF model, together with the present study

model. An excellent match is obtain for �e 4 �opte .

However, the LGEF model shows that by lowering �ebelow �opte , one can arbitrarily achieve any attenuation

even with very low coupling, which is put in the wrongby the present model. The present study hence givesus a maximum possible performance with SSDI.Experimental studies of the influence of �e on the freeresponse also mention problems with low values, andpossible remedies (Onoda et al., 2003).

CONCLUSIONS

In this article, a one dof reduction of the completeelectromechanical model of a structure with piezoelectricelements coupled to SSDS and SSDI electric circuits isproposed. By restricting the analysis to one mechanicalmode only, the system free response has been analyti-cally obtained. A similar analysis has been conducted toobtain the forced response of the structure subjected to aharmonic forcing of any frequency. Table 1 summarizesa few results, recalled in the following.

In the case of a free response, analytical formulas havebeen obtained in both cases of SSDS and SSDI. A majorresult is that the total damping is the sum of the mechan-ical damping and the so-called added damping, whichdepends only on kr. We recall here that the modal cou-pling coefficient kr is very close to the traditional effec-tive coupling coefficient jkrj. keff (Thomas et al., 2009).For a structure with a coupling coefficient kr¼ 0.2 and astructural damping factor �r¼ 0.1%, one obtains 2.5%of added damping with SSDS and 10% with SSDI.

In the case of a forced response, similar results havebeen obtained, except that no analytical expressions areavailable. In both cases of SSDI and SSDS, it has beenfound that the system stabilizes in a periodic steady stateof same frequency than the one of the forcing. Someunstable response have been found for low values ofthe electric resistance in SSDI. Frequency responsecurves have been plotted: they have a shape similar toresonance curves, with a maximum of amplitude at theopen-circuit resonance frequency. The attenuations, indB, brought by the SSD device have been obtainednumerically. They depend only on kr and �r. If kr¼ 0.2and �r¼ 0.1% the attenuation is expected to be of 30 dBfor SSDS and 46 dB for SSDI.

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

45

50

kr

Atte

nuat

ion

AdB

(dB

)

AdBSSDI

AdBRL

AdBSSDS

AdBR

Figure 15. Expected attenuation AdB as a function of coupling coef-ficient jkrj. keff for SSDI, SSDS, and simple resistive (R) and reso-nant (RL) shunts, for �r¼ 0.17%. ‘—’, ‘- -’: theory; ‘*’: experiments.(Results on R and RL shunt from Ducarne et al. (2007), Ducarneet al. (2009).)

0 0.1 0.2 0.3 0.4 0.530

35

40

45

50

55

60

65

70

xe

AdB

AdB

AdB with LGEF model

Figure 16. Comparison between the predicted attenuation ASSDIdB as

a function of �e from the presented model (‘—’) and fromEquation (55) (‘- -’), for kr¼0.2 and �r¼ 0.1%.

Table 1. Summary of the main characteristics of SSDSand SSDI systems in free and forced response.Numerical values of nadd and AdB are obtained withkr^0.2, nr^0.1%.

Free response Forced response

SSDS �add ¼ �1� ln

1� k2r

1þ k2r

� �AdB¼ f(kr, �r)

�add¼ 2.5% AdB¼ 30 dB

SSDI �add ¼ �1� ln

1� jkr j

1þ jkr j

� �AdB¼ f(kr, �r)

�opte ¼ f ðkrÞ þ �r �opt

e ¼ f ðkrÞ

�add¼ 10% AdB¼ 46 dB


In the case of SSDI, an optimal value �opte of the elec-tric damping factor (linked to the the electric resistance,Equation (13)) has been found (which is different for thefree or forced response). In the free response case,�opte � �r depends on kr only, whereas in the forcedresponse case, �opte depends on kr only.We can now analyze what is the influence of the

systems parameter on its response. FollowingEquation (12), the systems parameters are !r, kr, �r, R(linked to �e, Equation (13)) for SSDS and (R,L),(linked to (�e,!e), Equation (13)) for SSDI. To beginwith, !r has no influence on the system’s vibrationreduction performances: the added damping in freeresponse and the attenuation in forced response do notdepend on !r. Moreover, resistance R in the case ofSSDS and inductance L in the case of SSDI are directlyrelated to the electric time constant, that has to be assmall as possible, much smaller than the mechanicalperiod, to ensure an efficient switching strategy.Provided Equation (14) is verified, their particularvalue has no influence on the system’s response.Consequently, the system’s response depends only on(1) coupling factor kr, (2) structural damping �r and inthe case of SSDI, (3) resistance R, linked to electricaldamping factor �e. In a practical case, �r is a problemdatum that cannot be changed. Moreover, the presentstudy proves that for SSDI, an optimal value of R hasbeen exhibited, which is a function of kr and �r only.Consequently, as a general conclusion, the only free

parameter that influences the performances of the SSDdevices is the coupling factor, that has to be maximizedin order to enhance the vibration attenuation. Then, forany value of kr, it is possible to find optimal values ofthe electric parameters of the circuit. We recall againthat resistance R in the case of SSDS and inductanceL in the case of SSDI have to be chosen so thatEquation (14) is verified. Then, for SSDI, the optimalvalue of R is related to �opte (Equation (13)). As a rule-of-thumb, one may use �opte ¼ kr þ �r for the free responseand �opte ¼ 0:4 kr for the forced response (Figure 4(b)and Figure 14).It is very important to remark that most structures

have more than one mode electromechanically coupledto the piezoelectric element. These modes will react to theeffect of the charge and even when forced at one reso-nance, other modes may be strongly excited, for instanceby harmonics of the charge signal (see Figure 12; theeffect of the charge is similar to an excitation force witha square signal). The motion of these modes is going toinfluence the piezoelectric voltage and the synchroniza-tion of the switch will become difficult. Experimentalstudies and simulations with several dofs made by theauthor (Ducarne et al., 2008; Ducarne, 2009) and foundin the litterature (Makihara et al., 2005) show that thiseffect is important. The actual performance obtainedwith real structures will therefore be lower than predicted

by the present model. This is why no experiments areprovided in the present article to verify the modelingresults, because in practice, elastic structures have manymodes, that would interact, and our one dof model wouldfail to predict the experimental results.

Another issue not taken into account in the presentmodel is that the electronic switching circuit may haveimperfections. Three strong assumptions would have tobe removed in order to get a better accuracy of themodel with regards to experimental studies, at the detri-ment of simplicity. The first assumption is that the beha-vior of the circuit is linear (equivalent to a resistance andinductance). In practice the transistors and diodes usedto switch the current can be better modeled by a voltagedrop instead of an internal resistance. The two otherassumptions are that the switch occurs instantly and atevery maximum of the voltage. Experimentally, theswitching time is finite and the switch may occur slightlybefore or after the voltage maximum ; sometimes localmaxima are also ignored by the synchronization circuit.The choice of inductance (related to the voltage inver-sion time Te) has a strong effect on energy transferbetween modes (Ducarne et al., 2008; Ducarne, 2009).The switch timing, if not perfect, is a very complex pro-blem and an elaborate electronic model may be needed ;experimentally the use of a realtime computer to synch-onize the switch also adds considerable complexity if onewants to analyze the detail of the time response.

However, this model allows us to come to someimportant conclusions, namely that the coupling factoris the most important factor for the performance ofthese shunts, and also that lowering the resistance mayrender the system unstable, even reduced to one singlemode. The present study hence gives us a maximumpossible performance with SSDI.

The results of this article can now be used as a basis for aN dof numerical and experimental study. In order to simu-late the response of the system, the set of Equation (9a)willbe used, with a restriction to a few modes only; progres-sively themodel will be enriched, taking into accountmoreand more variables (Ducarne et al., 2008; Ducarne, 2009).For an experimental study, first the piezoelectric elementstype, size. and position have to be selected tomaximize themodal coupling with the targeted modes; then the switchcan be designed to obtain the required properties of impe-dance and synchronization. The high coupling is the key tohigh attenuations. Examples of piezoelectric patch geome-try and location optimizations to maximize the couplingfactor can be found in (Ducarne et al., 2010; Senechal et al.,2009; Senechal et al., 2010).

ACKNOWLEDGMENTS

This research is carried out under DGA contractnumber 05.43.063.00 470 7565 (INSA Lyon/LGEF,


CNAM/LMSSc, UCBL/LENAC), for which theauthors are grateful. They also wish to thank DanielGuyomar and his team of LGEF/INSA (Lyon,France) for their advice through a training of the firstauthor in their laboratory.

APPENDIX

Optimal values in the case of free response with SSDI

For maximizing the free response damping with SSDIand non-zero mechanical damping (section ‘Effectof Mechanical Damping’), the optimal values of theelectrical overshoot factor and the electrical dampingfactor are:

Xopt

¼Xrk

4r þð1þ4XrþX

2r Þk

2r þXr�2

ffiffiffiffiffiXr

pð1þXrÞð1þk

2r Þjkrj

ð1�k2rXrÞ2

,

ð56Þ

�opte ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðXoptÞ

2

lnðXoptÞ2þ �2

s: ð57Þ

The corresponding common value of the eigenvaluesof AB is:

�1 ¼ �2 ¼ð1þ XrX

optÞk2r � Xr � Xopt

2ð1þ k2r Þ: ð58Þ

It has been verified that without mechanical damping(�r¼ 0)Xr¼ 1), the above values leads to Equations(36) and (37).

REFERENCES

ANSI/IEEE Std 176-1987. 1988. IEEE Standard on Piezoelectricity.The Institute of Electrical and Electronics Engineers, Inc.New York, USA.

Caruso, G. 2001. ‘‘A Critical Analysis of Electric Shunt CircuitsEmployed in Piezoelectric Passive Vibration Damping,’’ SmartMaterials and Structures, 10:1051–1068.

Clark, W.W. 2000. ‘‘Vibration Control with State-switchingPiezoelectric Material,’’ Journal of Intelligent Material Systemsand Strutures, 11:263–271.

Corr, L.R. and Clark, W.W. 2003. ‘‘A Novel Semi-active MultimodalVibration Control Law for a Piezoceramic Actuator,’’ Journal ofVibration and Acoustics, 125:214–222.

Davis, C.L. and Lesieutre, G.A. 1998. ‘‘Actively Tuned SolidState Piezoelectric Vibration Absorber,’’ In: Proceedings ofSPIE: Smart Structures and Materials 1998: PassiveDamping and Isolation, Vol. 3327, San Diego, CA, USA, pp.169–182.

Dell’Isola, F., Maurini, C. and Porfiri, M. 2004. ‘‘Passive Damping ofBeam Vibrations Through Distributed Electric Networks andPiezoelectric Transducers: Prototype Design and ExperimentalValidation,’’ Smart Materials and Structures, 13:299–308.

Ducarne, J. 2009. ‘‘Modelisation et Optimisation de Dispositifs Non-lineaires d’amortissement de Structures par SystemesPiezoelectriques Commutes’’, PhD Thesis, ConservatoireNational des Arts et Metiers, (in French).

Ducarne, J., Thomas, O. and Deu, J.-F. 2007. ‘‘Optimisation deDispositif Passif d’attenuation de Vibration par ShuntPiezoelectrique,’’ In: Actes du 8eme Colloque National en Calculde Structures, Vol. 2, Giens, France, pp. 519–524.

Ducarne, J., Thomas, O. and Deu, J.-F. 2008. ‘‘VibrationReduction by Switch Shunting of Piezoelectric Elements:Nonlinear Energy Transfers Between Modes andOptimization,’’ In: Proceedings of the 19th InternationalConference on Adaptative Structures and Technologies, Ascona,Switzerland, October.

Ducarne, J., Thomas, O. and Deu, J.-F. 2010. ‘‘Performance andOptimization of Piezoelectric Shunts for Vibration Reduction,’’Journal of Sound and Vibration, (accepted).

Guyomar, D. and Badel, A. 2006. ‘‘Nonlinear Semi-passiveMultimodal Vibration Damping: An Efficient ProbabilisticApproach,’’ Journal of Sound and Vibration, 294:249–268.

Guyomar, D., Richard, T. and Richard, C. 2008. ‘‘Sound WaveTransmission Reduction Through a Plate Using PiezoelectricSynchronized Switch Damping Technique,’’ Journal ofIntelligent Material Systems and Structures, 19:791–803.

Hagood, N.W. and Von Flotow, A. 1991. ‘‘Damping of StructuralVibrations with Piezoelectric Materials and Passive ElectricalNetworks,’’ Journal of Sound and Vibration, 146:243–268.

Lallart, M., Badel, A. and Guyomar, D. 2008. ‘‘Nonlinear Semi-activeDamping Using Constant or Adaptive Voltage Sources:A Stability Analysis,’’ Journal of Intelligent Material Systemsand Structures, 19:1131–1142.

Lallart, M., Lefeuvre, E., Richard, C. and Guyomar, D. 2008. ‘‘Self-powered Circuit for Broadband, Multimodal PiezoelectricVibration Control,’’ Sensors and Actuators A, 143:377–382.

Lefeuvre, E., Badel, A., Petit, L., Richard, C. and Guyomar, D. 2006.‘‘Semi-passive Piezoelectric Structural Damping by SynchronizedSwitching on Voltage Sources,’’ Journal of Intelligent MaterialSystems and Structures, 17:653–660.

Makihara, K., Onoda, J. and Minesugi, K. 2005. ‘‘Low-energy-consumption Hybrid Vibration Suppression Based onan Energy-recycling Approach,’’ AIAA Journal, 43:1706–1715.

Makihara, K., Onoda, J. and Minesugi, K. 2008. ‘‘Performance ofSimple and Sophisticated Control in Energy-recycling Semi-active Vibration Suppression,’’ Journal of Vibration andControl, 14:417–436.

Niederberger, D. and Morari, M. 2005. ‘‘An Autonomous ShuntCircuit for Vibration Damping,’’ Smart Materials andStructures, 15:359–364.

Onoda, J., Makihara, K. and Minesugi, K. 2003. ‘‘Energy-recyclingSemi-active Method for Vibration Suppression with PiezoelectricTransducers,’’ AIAA Journal, 41:711–719.

Petit, L., Lefeuvre, E., Richard, C. and Guyomar, D. 2004.‘‘A Broadband Semi Passive Piezoelectric Technique forStructural Damping’’, In: Proceedings of SPIE Smart Structuresand Materials Conference: Passive Damping and Isolation,Vol. 5386, pp. 414–425.

Richard, C., Guyomar, D., Audigier, D. and Ching, G. 1999. ‘‘Semi-passive Damping Using Continuous Switching of a PiezoelectricDevice’’, In: Proceedings of SPIE Smart Structures and MaterialsConference: Passive Damping and Isolation, Vol. 3672,pp. 104–111.

Richard, C., Guyomar, D., Audigier, D. and Bassaler, H. 2000.‘‘Enhanced Semi-passive Damping Using Continuous Switchingof a Piezoelectric Device on an Inductor’’, In: Proceedings ofSPIE Smart Structures and Materials Conference: PassiveDamping and Isolation, Vol. 3989, pp. 288–299.

Richard, C., Guyomar, D. and Lefeuvre, E. 2007. ‘‘Self-poweredElectronic Breaker with Automatic Switching by DetectingMaxima or Minima of Potential Difference Between its PowerElectrodes’’, Technical Report, Organisation Mondiale de la


Propriete Intellectuelle 2007. Patent # PCT/FR2005/003000,Publication number: WO/2007/063194.

Senechal, A., Thomas, O., Deu, J.-F. and Jean, P. 2009. ‘‘Attenuationde Vibrations de Structures Complexes par ElementPieeoelectrique. Application a une aube de Turbomachine,’’In: Actes du 9eme Colloque National en Calcul de Structures,Giens, France. May (in French).

Senechal, A., Thomas, O., Deu, J.-F. and Jean, P. 2010. ‘‘Optimizationof Shunted Piezoelectric Patches for Complex StructureVibration Reduction - Application to a Turbojet Fan Blade,’’In: Proceedings of the 4th European Congress on ComputationalMechanics, Paris, France.

The MathWorks, Inc. Available at: http://www.mathworks.com.MATLAB Function Reference.

Thomas, O., Deu, J.-F. and Ducarne, J. 2009. ‘‘Dynamics of anElastic Structure with Piezoelectric Patches: Finite-elementFormulation and Electromechanical Coupling Coefficients,’’International Journal of Numerical Methods in Engineering,80:235–268.

Wang, K.W., Lai, J.S. and Yu, W.K. 1996. ‘‘An Energy-basedParametric Control Approach for Structural VibrationSuppression via Semi-active Piezoelectric Networks,’’ Journal ofVibration and Acoustics, 118:505–509.


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