DAMPING BY PIEZOCERAMIC DEVICES WITH PASSIVE LOADS

Mechanical Systems and Signal Processing (2003) 17(5), 1097–1114

doi:10.1006/mssp.2002.1520

DAMPING BY PIEZOCERAMICDEVICESWITH PASSIVELOADS

A. Agneni, L. Balis Crema and S. Sgubini

Dipartimento di Ingegneria Aerospaziale e Astronautica, Universit "aa degli Studi di Roma ‘‘LaSapienza’’, Via Eudossiana, 16-00184 Rome, Italy. E-mail: [email protected]

(Received 30 May 2001, accepted after revisions 5 August 2002)

The paper deals with the behaviour of structures with piezoceramic devices, loaded eitherwith resistors or with inductors and resistors. Both the circuits can be represented bymechanical models: the first one is equivalent to viscoelastic models, whereas the secondone is a single-degree-of-freedom (sdof) system, which is similar to a vibration absorberwhen added to the host structure. Since the device stiffness is frequency dependent, aniterative procedure ought to be adopted in order to get the eigenvalues, but completelyacceptable numerical results can be obtained by an approximated technique, which uses thestiffness calculated at the desired angular frequency. Besides the mechanical modelsmentioned before can be profitably introduced into finite element codes (also commerciallyavailable), which allow one to get information on the modal parameters. For a structurewith piezo devices shunted with resistive loads, if either the impulse response or thefrequency response function is derived, the natural frequencies and damping ratios can beachieved by least-squares techniques without any bias, as demonstrated by studying a sdofsystem. The possibility provided by this approach has been shown by means of numericalexamples carried out on beams, cantilevered and simply supported, modelled by finiteelement codes.

# 2003 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Piezoceramic materials can be used in order to increase the damping characteristics andthen to reduce the amplitude of structure vibrations, and that is especially attractive in theaerospace field. This effect is gained by either active or passive techniques. In this paper,the passive approach is considered, it can be achieved either when the piezoceramic deviceis loaded with a simple resistor or when the load is formed by a resistor and an inductor, inorder to form}with the inherent capacitor of the device}a resonant circuit [1].

The first type of load allows one to model the piezo device as a viscoelastic system,whose dynamic stiffness (complex stiffness) can be written as the Zener–Biot model [2, 3],which has been adopted in this paper to analyse the vibrating structures, or as the Three-Parameter Model [3–5]. By comparing the stiffness of the viscoelastic models with the oneof a piezo device, it is possible to derive the relationships linking the mechanical valueswith the mechanical and electrical characteristics of the two models. It is possible to takeadvantage of this similarity, in fact the viscoelastic models can be introduced not only intoa lumped mechanical system, but also into a finite element description of a structure [4–7].Actually the damping ratios, introduced by the piezo with resistor loads, can beevaluated}with negligible errors}by the eigenvalues obtained from the structure whenthe piezo element behaviour is provided by a simple complex number (different for eachfrequency) instead of the complex stiffness function [5] or by fitting the frequency response

0888–3270/03/+$30.00/0 # 2003 Elsevier Science Ltd. All rights reserved.

A. AGNENI ET AL.1098

function obtained from the structure [7]. This last approach can also adopt the usualfitting techniques, normally present in the identification commercial codes, practicallywithout any bias in the results, as theoretically and numerically shown for a single-degree-of-freedom (sdof) system.

The piezo device with the second load (resistor and inductor) is similar, although notequal}as pointed out in [1, 8]}to a vibration absorber, when added to a structure.

Therefore, this mechanical approach permits}in both cases}to introduce a frequency-dependent complex stiffness into finite element models and a completely mechanicalsystem is achieved, as said before for the first case, not only by using a generalmathematical code (MATLAB) [4], but also a commercial code (MSC/NASTRAN) [5–7].The simplicity of the proposed approach in obtaining FRFs (and in turn the modalparameters: natural frequencies and damping ratios) from simple structures, as acantilever or a simply supported beam, modelled by finite elements and with ideallybonded shunted piezo devices on one or more elements, is shown.

2. THEORETICAL BASIS

From the relationships in [1], it is possible to derive the complex stiffness of apiezoelectric device with a load, whose impedance is ZL:

kshðoÞ ¼ ksc 1þ %kk2 VL

V

� �� ð1Þ

where ksc is the value of the stiffness when the piezo is short circuited, V is the voltagegenerated by the piezo device, VL is the voltage measured across the load when thecapacity of the piezo is CS

p ; i.e. the capacitance taken at constant strain. At last %kk2 is relatedto the electromechanical coupling factor kij by %kk2 ¼ k2

ij=ð1� k2ijÞ: The piezoceramic device,

as well known, can be represented as a charge generator with in parallel a capacitance,when the element works far below its first mechanical resonant frequency [9]. By means ofthe Thevenin theorem, the piezo device can be also considered as a voltage generator witha capacitive impedance in series. When the dynamic stiffness is given in terms of the shortcircuit one, as in equation (1), the capacitive impedance is the one when the circuit is openand it is equal to ZOC ¼ 1=joCs

p and therefore the circuit closed on a load is a potentialdivider, whose voltage ratio is

VL

V

� �¼

ZL

ZL þ Zoc

� �: ð2Þ

2.1. RESISTOR LOADS

When the piezo is loaded with a resistor (R ), the dynamic stiffness may be written asfollows [4–6]:

kshðrÞ ¼ ksc 1þ %kk2 1�1

1þ r2

� �� þ j %kk2 r

1þ r2

� �� ð3Þ

where r ¼ oRCSp is the non-dimensional angular frequency. It is convenient to normalise

the angular frequency with respect to the value for which the imaginary part of thecomplex stiffness is maximum, oM ¼ ðRCS

p Þ�1:

It is possible to show [3] that the stiffness}normalised with respect to ksc and reduced toproper rational functions, as in equation (3)}except for a constant, is always formed by aCauchy pulse and its Hilbert transform [10]. For this reason its impulse response is causal,in addition, because ksh(o) is Hermitian, the relative time function is real.

DAMPING BY PIEZOCERAMIC DEVICES 1099

Equation (3) can be also written in terms of the loss factor ZðrÞ :

kshðrÞ ¼ kRshðrÞ 1þ j

kIshðrÞ

kRshðrÞ

" #¼ kR

shðrÞ 1þ jZðrÞ½ � ð4Þ

which can be easily derived:

ZðrÞ ¼%kk2r

1þ r2ð1þ %kk2Þ: ð5Þ

The maximum of ZðrÞ; for a resistive load, is reached at the angular frequency:

oZM ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

ij

qRCS

p

0@

1A: ð6Þ

So the relaxation time results equal t�1Z ¼ oZ

M :The dynamic stiffness in equation (3) is equivalent to the ones relative to models which

are commonly used to represent viscoelastic materials: the Zener–Biot model (as alsopointed out in [2]) and the Three-Parameter Model. Hereafter, a short description of thesemechanical elements is presented.

2.1.1. Zener–Biot model

Figure 1 shows this model, which is formed by a Maxwell damper with a stiffness inparallel [11–13].

The stiffness, complex and frequency dependent, is given by the following relationship:

kcðoÞ ¼K1K2 þ jom ðK1 þ K2Þ

K1 þ jom: ð7Þ

The similarity with kshðrÞ is evident if kcðoÞ is written in terms of its real and imaginaryparts as proper rational functions:

kcðoÞ ¼ ðK1 þ K2Þ � K11

1þ o2ðm=K1Þ2

" #þ jK1

oðm=K1Þ

1þ o2ðm=K1Þ2

" #: ð8Þ

Comparing equation (3) with equation (8), the mechanical elements of the Zener–Biotmodel can be written in terms of the mechanical and electric characteristics of thepiezodevice:

K1 ¼ ksc %kk2

K2 ¼ ksc

m ¼ ksc %kk2t: ð9Þ

Figure 1. Zener–Biot model.


Equation (8) can be also represented as follows:

kcðoÞ ¼ kRc ðoÞ þ jkI

c ðoÞ ¼ kRc ðoÞ 1þ jZðoÞ½ �: ð10Þ

The study of the mechanical model can help to understand the behaviour of the piezodevice. Following [11], the relaxed elastic modulus of the piezo device is given bykRc ðoÞjo!0 ¼ K2; that is for very low angular frequencies, the damper moves following the

displacement without any reaction on the stiffness K1; the system behaves as it were madeup of the stiffness K2 only. On the contrary the unrelaxed modulus is equal to kR

c jo!þ1 ¼ðK1 þ K2Þ; i.e. for very high angular frequencies the damper behaves as a perfect stiffmember and the system is reduced to the parallel of the springs K1 and K2. The imaginarypart of the stiffness is equal to zero in both cases.

Bearing in mind the relationships in equation (9), for a piezoelectric element the relaxedand the unrelaxed moduli are equal to kR

sh ! ksc and kRsh ! kscð1þ %kk2Þ; that is the stiffness

for the electric circuit open kRsh ! koc: Actually, for the piezo device, it is not possible to

follow the same procedure as the mechanical element, in fact the lower limit is still valid,but the upper limit ðo ! þ1Þ is beyond the validity of the piezo device model. Instead ofspeaking of natural frequencies in the range 04o4þ1; it is necessary to consider onlythose values satisfying the condition o5oðpÞ

n ; for which the model was adopted, oðpÞn being

the intrinsic natural frequency of the piezo device. Therefore, the relationships gainedbefore for the relaxed and unrelaxed moduli are valid for o5oM andoM5o5oðpÞ

n ;respectively.

2.1.2. Three-parameter model

This model, presented in Fig. 2, has the following complex stiffness [11–13]:

kcðoÞ ¼K1ðK2 þ jomÞ

ðK1 þ K2Þ þ jom: ð11Þ

The real and imaginary parts of the dynamic stiffness can be written as follows:

kcðoÞ ¼ K1 �K2

1

K1 þ K2

1

1þ o2ðm=ðK1 þ K2ÞÞ2

" #

þ jK2

1

K1 þ K2

o m=ðK1 þ K2Þ� �

1þ o2 m=ðK1 þ K2Þ� �2

" #: ð12Þ

As a consequence the mechanical parameters, in terms of the ones relevant to the piezodevice, are

K1 ¼ ksc 1þ %kk2� �

K2 ¼ ksc1þ %kk2� �

%kk2

m ¼ksc 1þ %kk2� �2

%kk2t:

ð13Þ

2.2. RESISTOR AND INDUCTOR LOADS

When the load is formed by a resistor (R) and an inductor (L) in series, the voltagemeasured across these two components, normalised with respect to the one generated bythe piezoceramic element, is

VL

V

� �¼

Rþ joLRþ joLþ 1=joCS

p

: ð14Þ

Figure 2. Three-parameter model.


The complex stiffness is still given by equation (1), whereas the real and the imaginaryparts}written in terms of proper rational functions}are given by

kRshðoÞ ¼ ksc 1þ %kk2 1�

½1� o2ðLCSp Þ�


2 þ ½oðRCSp Þ�

2

" #( )

kIshðoÞ ¼ ksc %kk

2½oðRCS

p Þ�


2 þ ½oðRCSp Þ�

2

" #:

ð15Þ

As one can see, the previous functions represent a couple of Hilbert transforms andtherefore the kshðtÞ is causal and real, as kshðoÞ is Hermitian.

It is easy to show that the loss factor is given by the following relationship:

ZðoÞ ¼k2ij½oðRC

Sp Þ�

ð1� o2ðLCSp Þ� ½1� k2

ij � o2ðLCSp Þ� þ ½oðRCS

p Þ�2: ð16Þ

2.3. TUNING

When the load is a resistor, its value is chosen in such a way that the maximum of theloss factor is coincident with the peak of the mode (kth ) that is to be damped ðoðkÞ

n Þ: Ifinstead the maximum of the imaginary part of the complex stiffness is used, small errors

are introduced because the two angular frequencies only differ in the termffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� k2

ij :q

This

operation is commonly called ‘tuning’, but strictly speaking no tuning is possible with sucha load, in fact it does not form with the piezo inherent capacitor an electric oscillatingsystem. The device, from the electrical point of view, is a high pass filter [4]. On thecontrary, the resistive and inductive load, along with the inherent capacity of the piezodevice, is a resonant circuit, which can be tuned choosing the inductor in such a way thatthe inductive impedance is equal to the capacitive one at the desired angularfrequency}commonly one of the mode of interest ðoðkÞ

n Þ}so that the electric resonant

frequency oe ¼ 1=ffiffiffiffiffiffiffiffiffiffiLCS

p

qis equal to oðkÞ

n : The system practically behaves as a vibration

absorber, so a detuning yields an increase of the vibration amplitudes. This problem canbe overcome by adopting those values of R and L which optimise the frequency responsefunction of the structure with piezo devices [1], similar to that done for the mechanicalsystems [14].

2.4. SDOF SYSTEM

An sdof system (Fig. 3) is described, in the frequency domain, by the following equationof motion:

½�Mso2 þ joCs þ Ks þ kshðoÞ�XðoÞ ¼ FðoÞ: ð17Þ

Figure 3. sdof system with a frequency-dependent stiffness kshðoÞ due to a piezo device.


The mass of the piezoceramic elements could be considered in Ms. The causality of thedynamic stiffness yields the causality of the whole mechanical system, as shown for similarcases in [13]. At last the FRF, relative to equation (17), is given by the followingrelationship:

HðoÞ ¼XðoÞFðoÞ

¼ ½�Mso2 þ joCs þ Ks þ kshðoÞ��1: ð18Þ

2.4.1. Resistive load

The FRF relative to the case of the Zener–Biot model is given by

HðoÞ ¼½K1 þ jom�

�jo3ðMsmÞ � o2ðMsK1 þ CsmÞ þ jo½mðK 0s þ K1Þ þ K1Cs� þ ðK 0

s þ K1Þð19Þ

where K 0s ¼ ðKs þ K2Þ: Introducing equation (9) into equation (19), this last relationship

becomes

HRðoÞ ¼½1þ jot�

�jo3ðMstÞ � o2ðMs þ CstÞ þ jo½tðKs þ KocÞ þ Cs� þ ðKs þ kscÞ: ð20Þ

Considering in the previous relationship ss ¼ Cs=2Ms

� �; that is the decay rate due to the

damping of the structure, and the natural angular frequencies of the system when the piezodevice has its terminals connected in short circuit, oE

n ; or open, oDn :

oEn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKs þ kscÞ

Ms

r

oDn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKs þ kocÞ

Ms

r ð21Þ

it reduces to

HRðoÞ ¼1

Ms

� �½1þ jot�

�jo3t� o2ð1þ 2sstÞ þ jo½tðoDn Þ

2 þ 2ss� þ ðoEn Þ

2: ð22Þ

It is noteworthy that when a multi-degree-of-freedom (mdof) system is considered, thephysical quantities in equation (20) become modal, as also pointed out in [1], and theangular frequencies oE

n andoDn }relative to the mode considered}take account of the

structure constraints and piezo positions. Thus the FRF in equation (22) represents, foreach mode, every structure whose damping is increased by means of a piezo device withresistive loads, and wherever the piezos are positioned and whichever the constraints are.


If the damping of the structure is neglected, the previous relationship, written in termsof the Laplace variable s, becomes:

HRðsÞ ¼1

Ms

� �½1þ st�

ts3 þ s2 þ st ðoDn Þ

2 þ ðoEn Þ

2: ð23Þ

So the modal damping, introduced by a piezo device, could be identified by evaluatingthe roots of the polynomial [1]:

ts3 þ s2 þ tðoDn Þ

2sþ ðoEn Þ

2 ¼ 0: ð24Þ

Obviously it is necessary to know t; which can be chosen to be t ¼ ðRCSp Þ ¼ ðoMÞ�1 ¼

ðonÞ�1; so as to fit the peak of the mode to be damped ðonÞ with the maximum of kI

shðoÞ:For different values of the resistor R, damping ratios corresponding to the relative t0s canbe derived, in this way the damping ratio behaviour vs frequencies can be plotted.

For o ¼ oM and ss ¼ 0; it is possible to estimate the natural angular frequency of thesystem by oE

n and oDn :

on ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKs þ kR

shðoMÞ2

s: ð25Þ

Introducing kRshðoMÞ ¼ kscð1þ %kk2=2Þ into equation (25), the following relationship is

derived:

on ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððKs þ kscÞ=MsÞ þ ððKs þ kocÞ=MsÞ

2

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðoE

n Þ2 þ ðoD

n Þ2

2:

sð26Þ

Besides, considering the relationships among the coefficients of equation (24) with theroots of the cubic, bearing in mind that one of them is real ðsrÞ and the other two must becomplex and conjugate ðl ¼ sl � jodÞ; since the polynomial is relative to an oscillatingsystem, the decay rate for the real root and the damping ratio, introduced by the piezodevices, are given by

sr ¼ðoE

n Þ2

on

B ¼ 12

ðoEn Þ

2 � ðoDn Þ

2

ðoEn Þ

2 þ ðoDn Þ

2:l

ð27Þ

Thus the previous modal parameters can be obtained only by the knowledge of the twoangular frequencies oE

n andoDn :

In addition the simplified version of the relation HR(s), equation (23), can be expandedin partial fractions by the Heaviside procedure:

HRðsÞ ¼Rr

ðsþ srÞþ

Rl

ðsþ lÞþ

Rl

ðsþ l Þ: ð28Þ

The residues, relative to the real (Rr) and the complex and conjugate roots ðRl andRl Þ;are given by the following relationships:

Rr ¼1

Ms

ðð1=tÞ � srÞ

s2r � srðlþ l Þ þ ll

Rl ¼1

Ms

ðð1=tÞ � lÞ2jodðsr � lÞ

¼*RRl

2j:

ð29Þ

Only Rl has been reported because equation (22) is Hermitian and therefore Rl is com-plex and conjugate with respect to Rl : Rl ¼ R

l: It is straightforward to derive the


impulse response function from equation (28):

hRðtÞ ¼ Rre�srt þ Rle

�lt þ R le

�l t : ð30Þ

Actually, the values of oEn andoD

n are very close and so it is possible to say that on ffioE

n ; as a consequence sr is in the order of oEn ; in contrast, with the same approximation

sl ffi zoEn : Thus, when the damping introduced by the piezo devices ðzÞ is small enough,

the following relation is valid sr4sl: On the other hand, ð1=tÞ ¼ on ffi oEn and therefore,

under these conditions, Rr is very small (in the order of zero). As a consequence, theresponse of the structure is practically equal to the one of an sdof system with constantparameters and viscous damping:

HRðsÞ ffiRl

ðsþ lÞþ

Rl

ðsþ l Þ: ð31Þ

The differences are present especially at the lowest frequencies, where the spectrum ofthe damped function has its maximum; in any case, in the frequency domain, thecontribution of ðRre

�sltÞ at the peaks provided by the other two functions isgenerally small and negligible. The IR derived from equation (31) is a damped sinusoidalfunction:

hRðtÞ ffi *RRle�slt sinðodtÞ: ð32Þ

Also in this case, the differences between the relationships in equations (30) and (32) areespecially present in the initial points, where the damped non-oscillating function gives itshigher contribution.

Therefore the modal parameters of a vibrating structure with piezo devices can bederived, with negligible errors, from experimental data by usual, also commercial,identification codes.

2.4.2. Resonant circuit

For this case, the complex stiffness kshðoÞ; relative to a resistive and inductive load,has to be introduced into equation (18). For an sdof system or for one mode of anmdof structure, since as said before the load forms a resonant circuit with theintrinsic capacitance of the piezo device, the FRF is also given by equation (18), wherekshðoÞ; the one whose real and imaginary parts are given in equation (15). If instead ofHðoÞ; the non-dimensional displacement (x/xst) is considered, the following relationship isobtained [1]:

x

xst

� �¼

ðd2 þ g2Þ þ d2rg

ð1þ g2Þðd2 þ g2 þ d2rgÞ þ K2ij ðg2 þ d2rgÞ

ð33Þ

where Kij is the generalised electromechanical coupling coeffcient:

K2ij ¼ %kk2 ksc

K þ ksc

� �¼

ðoDn Þ

2 � ðoEn Þ

2

ðoEn Þ

2ð34Þ

and

d ¼oe

ðoEn Þ; g ¼

joðoE

n Þ; r ¼ RCS

pon: ð35Þ


Since the optimal values, derived from [1], can be put as a function of the angularfrequencies oE

n and oDn [6]:

dopt ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

ðoDn Þ

2 � ðoEn Þ

2

ðoEn Þ

2

s¼

ðoDn Þ

ðoEn Þ

ropt ffiffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðoD

n Þ2 � ðoE

n Þ2

qðoD

n Þ2

ðoEn Þ

ð36Þ

and the reference angular frequency is oEn [1], the non-dimensional displacement in

equation (33), for the considered mode, can be evaluated for the desired frequency rangethrough g: Obviously, if the optimised data are not used, the values of R and L are knownalong with CS

p ; thus for each mode, equation (33) can be used, once K2ij ; r; d and g have been

estimated through oEn and oD

n and equation (35).

2.5. MDOF FREQUENCY RESPONSE FUNCTION

The complete frequency response function of a structure modelled by finite elements canbe obtained by adding to the mass and stiffness matrices of the structure the ones of thepiezo devices. The piezo stiffness matrix, kshðoÞ; is gained by the frequency-dependentelastic modulus, which for the resistive load is given by the following relationship:

EshðoÞ ¼ Esc 1þ %kk2 1�1� jðoRCS

p Þ

1þ ðoRCSp Þ

2

" #( )ð37Þ

whereas the elastic modulus, when a resistive–inductive load is used, is given by

EshðoÞ ¼ Esc 1þ %kk2 1�1� o2LCS

p

� �� jðoRCS

p Þ

1� o2LCSp

� �2þðoRCS

p Þ2

264

375

8><>:

9>=>;: ð38Þ

The FRF matrix can be achieved by inverting the dynamic matrix:

HðoÞ ¼ �Mo2 þ joCs þ Ks þ kshðoÞ% &�1

: ð39Þ

The matrix M takes account of the mass matrix of the structure and the one of the piezodevice, on the contrary Cs and Ks are the damping and the stiffness matrix of the structureonly.

3. NUMERICAL TESTS

In a first example, an sdof system (Fig. 3) has been considered, the system stiffness (Ks)is a function of the short circuit stiffness of the piezo device (ksc) through the parametermk ¼ Ks=ksc; Cs ¼ 0; whilst kshðoÞ is the stiffness of a Zener–Biot model. The naturalfrequencies of the system, for several mk, are reported in Table 1, whilst the relativedamping ratios are shown in Table 2.

The previous modal parameters have been estimated, from noiseless synthesised data,both in the time domain, by the method founded on the Hilbert transform and the Gaussfilter [16] (superscript GH), and}in the frequency domain}by a least-squares fitting ofthe FRF with the residue/pole (i.e. a polynomial ratio) model (superscript FRF ). In thislast case, for the reason said in the theoretical basis, the fitting has been carried out by amodel with two complex conjugate poles, as a common sdof system with a dashpot in theplace of the Zener–Biot element. The superscript a indicates that the data have been

Table 1

Natural frequencies (in Hz) of the sdof system with variable stiffness with the parameter mk

mk 5� 10�3 10�1 5� 10�1 1 5 10

f cn 1.3296� 102 1.3868� 102 1.6058� 102 1.8435� 102 3.1560� 102 4.2620� 102

f GHn 1.3296� 102 1.3868� 102 1.6038� 102 1.8434� 102 3.1560� 102 4.2620� 102

fFRFn 1.3296� 102 1.3868� 102 1.6058� 102 1.8435� 102 3.1560� 102 4.2620� 102

f an 1.3280� 102 1.3855� 102 1.6050� 102 1.8429� 102 3.1559� 102 4.2620� 102

Table 2

Damping ratio of the sdof system with variable stiffness with the parameter mk

Mk 5� 10�3 10�1 5� 10�1 1 5 10

zc 3.3519� 10�2 3.0720� 10�2 2.2725� 10�2 1.7146� 10�2 5.7829� 10�3 3.1627� 10�3

zGH 3.3519� 10�2 3.0719� 10�2 2.2753� 10�2 1.7146� 10�2 5.7829� 10�3 3.1627� 10�3

zFRF 3.3519� 10�2 3.0720� 10�2 2.2725� 10�2 1.7146� 10�2 5.7829� 10�3 3.1627� 10�3

za 3.2471� 10�2 2.9834� 10�2 2.2232� 10�2 1.6862� 10�2 5.7498� 10�3 3.1527� 10�3


achieved by the relation in equation (26) and by the second one of equation (27). Theresults have been compared with the ones gained by the solutions of the cubic polynomial(superscript c). The estimates are completely equal, if only five digits are consideredfor the values gained by fittings in the time and in the frequency domain. On thecontrary, a maximum relative error of e ¼ 0:12 (%) is obtained for the natural frequency,estimated by equation (26), when mk ¼ 5� 10�3: A maximum error of e ¼ 3:13 (%) onthe damping factor, gained by the second one of equation (27), was instead achievedfor mk ¼ 10�1; that is when it is pre-ponderant the stiffness of the piezo device withrespect to the one of the structure. The contribution due to the real root is practicallyimmaterial.

As mentioned before, the damping factor achieved through oEn and oD

n is the oneintroduced by the piezo device, whereas the roots of the cubic polynomial provide thetotal damping. In fact, if a viscous damper is considered in the sdof system ðmk ¼5 and zs ¼ 7:5646� 10�3Þ; the identified modal parameters are shown in the followingtable (Table 3).

Obviously, for a generic resistor ð %RRÞ chosen by the experimenter, the relative time%tt ¼ %RRCS

p is known, besides as CSp is the inherent capacity of the piezo and the

angular frequencies oEn and oD

n can be numerically or experimentally identified,the damping ratio relative to the angular frequency %oo ¼ %tt�1 is achieved from thecubic roots of the polynomial. This technique permits to plot damping ratios vsfrequencies.

The eigenvalues of a structure with a piezo device have to be obtained by an iterativeprocedure. The complex stiffness of a loaded piezo device is frequency dependent, thus thestiffness matrix of the whole finite element model of the structure is frequency dependentas well. But, damping ratios}for a generic frequency and for the one of a mode}could beestimated with immaterial errors [5], if a complex stiffness matrix, constant for eachconsidered angular frequency, is added to the structure stiffness matrix in order to takeaccount of the piezo device. The piezo stiffness matrix is built starting from the values ofthe real and the imaginary parts of its elastic modulus, whose behaviour with frequencyis obviously equal to the one in equation (3). When a resistor load is considered, the

Table 3

Natural frequencies and damping ratios achieved with different approaches

f cn zc fFRFn zFRF f an za

3.1562� 102 1.3391� 10�2 3.1560� 102 1.3391� 10�2 3.1559� 102 5.7450� 10�3


components of the elastic modulus become

ERshðoÞ ¼ Esc 1þ %kk2 1�

1

1þ r2

� ��

EIshðoÞ ¼ jEsc

%kk2 r1þ r2

� �� ð40Þ

while the loss factor is

ZðoÞ ¼k2ijr

1þ r2 � k2ij

: ð41Þ

The cantilever aluminium beam (29.30� 2.55� 0.317 cm3) in [1] has been considered,supposing zero the damping of the structure, in order to check the results obtained by themethod mentioned above. The beam was divided into 10 finite elements, with piezoelectricpatches, whose coupling factor was kij; ¼ 0:38; Esc ¼ 63 GPa; CS

p ¼ 0:156 mF and thedensity is equal to 7500 kgm�3. In order to simulate the beam in [1], four piezo elementswere ideally bonded on the top and on the bottom of the first two finite elements near thefixed end. The surface dimensions of the piezo patch were equal to the ones of the singlefinite element of the beam and the thickness was equal to 0.25mm.

The results, obtained by a home-made MATLAB finite element model, were inagreement with the ones in [1]. The same technique}or a similar one, which providedkRshðoÞ and ZðoÞ to the finite element code for each considered angular frequency}has been

used with the MSC/NASTRAN code for the same cantilever beam and for a beamsupported at its ends. The real and the imaginary parts and the loss factors for differentnon-dimensional frequency r, obtained dividing the angular frequency by oM ; arereported in Table 4.

The damping ratio behaviour, for the cantilever beam, is shown in Fig. 4. The solid linerepresents the values provided by MATLAB, whilst the ones achieved by the MSC/NASTRAN code have been reported: with asterisks (*) for brick elements, with circles (0)for bar elements (the complex stiffness matrices of the piezo devices were gained by aFortran program), and with (+) for bar elements (the complex stiffness matrices wereprovided by the MATLAB model and then introduced into the MSC/NASTRAN by aFortran program).

The damping ratios gained with the different methods are practically equal. Very smalldifferences (Fig. 5) have been instead found when the damping ratios have been derivedfor the first bending mode of a beam simply supported at its ends; the piezo devices wereconsidered ideally bonded on the top and on the bottom of the middle finite elements (5thand 6th). The different approaches gave results in the same order of magnitude.

The modal damping, both for the resistive and for resistive–inductive load, can be alsoobtained from the FRFs, achieved by a MATLAB program, by a least-squares fitting ofthe curve.

0 34.8 69.6 104.4 139.2 174.1 208.9 243.7 278.5 313.3 348.11

2

3

4

5

6

7× 10-3

Frequency (Hz)

Dam

ping

rat

io ζ

Figure 4. Damping ratios, relative to the first bending mode of the cantilever beam. The piezo devices were inthe 1st and 2nd finite elements from the clamped end.

Table 4

Real and imaginary parts of the piezo device elastic modulus and loss factor vs non-dimensionalfrequency

r 0.1 0.3 0.5 1.0 2.0 3.0 10.0

R½EshðoÞ� � 1010 6.3105 6.3878 6.5126 6.8316 7.1506 7.2569 7.3527I½EshðoÞ� � 109 1.0527 2.9264 4.2530 5.3163 4.2530 3.1898 1.0527Z� 10�2 1.6682 4.5812 6.5304 7.7819 5.9478 4.3955 1.4317


A first example (Fig. 6) shows the FRF for the cantilever beam, limited to the first twomodes. In this case a modal damping equal to z ¼ 0:001 has been added in order to takeinto account the structure damping, whereas the piezo devices, positioned as said before,had the resistor load such as oM ¼ oð1Þ

n ; i.e. ‘tuned’ on the first bending mode.The estimated values of the modal parameters are reported in Table 5.It is worth noting that the first mode, unless for structure damping, is completely in

agreement with the damping ratio obtained through the eigenvalues at the peak frequency(Fig. 4). Besides, since the damping due to a resistive load broadens on a wide frequencyrange, the second mode presents a damping ratio higher than 1.0� 10�3. The FRF gainedwhen a resistive and inductive load with the optimal values chosen so as to fit oð1Þ

n ispresented in Fig. 7.

In this case, since the piezo circuit behaves as a vibration absorber, the mode on which itis tuned splits into two modes, whose modal parameters are reported in Table 6.

Obviously, this circuit introduces a quite high damping on the tuned mode, but no effectis present in the other modes; the identified damping ratio relative to the second mode is

0 86.6 173.2 259.7 346.3 432.9 519.5 606.1 692.6 779.2 865.80.5

1

1.5

2

2.5

3

3.5

4

4.5× 10

-3

Frequency (Hz)

Dam

ping

rat

io ζ

Figure 5. Damping ratios of the simply supported beam. The piezo devices were positioned in the 5th and 6thfinite elements.

0 200 400 600 800 1000 1200 1400 160010-6

10-5

10-4

10-3

10-2

10-1

Angular frequency (rad/s)

H(ω

)

Figure 6. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded witha resistor ‘tuned’ on the first bending mode.


Table 5

Natural frequencies and damping ratios for the first two modes of the cantilever beam (piezowith resistive load)

Mode Natural frequency (Hz) Damping ratio

1st 34.814 7.6532� 10�3

2nd 207.18 2.3527� 10�3

0 200 400 600 800 1000 1200 1400 160010-7

10-6

10-5

10-4

10-3

10-2

10-1


H(ω

)

Figure 7. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded witha resistor and an inductor, gained from the optimal values for the first bending mode.

Table 6

Natural frequencies and damping ratios for the first modes of the cantilever beam (piezo withresistive and inductive load)


1st split32:71736:760

�6:9264� 10�2

4:8808� 10�2

�2nd 347.83 1.0089� 10�3


practically equal to the structure mode only. The FRFs, achieved for the simply supportedbeam, are instead shown in Figs 8 and 9.

The modal parameters identified from a least-squares fitting of the previous curves arereported in Table 7 for the resistive load, and in Table 8 for the resistive and inductive loads.

Unlike the previous example, relevant to the resistive load, a very small amount ofdamping is introduced into the second mode, because the piezo devices, positioned on the5th and 6th finite elements, provide a low strain energy for that mode. On the contrary, for

0 500 1000 1500 2000 2500 3000 350010-5

10-4

10-3

10-2

10-1

100


H(ω

)

Figure 8. FRF magnitude of the simply supported beam, measured at its middle point. The piezo devices(positioned at the 5th and 6th finite elements) have been loaded with a resistor ‘tuned’ on the first bending mode.

0 500 1000 1500 2000 2500 3000 350010-5

10-4

10-3

10-2

10-1

100

101


H(ω

)

Figure 9. FRF magnitude of the supported beam, measured at its middle point. The piezo devices, at the 5thand 6th finite elements, have been loaded with a resistor and an inductor, whose optimal values were relative tothe first bending mode.


Table 7

Natural frequencies and damping ratios for the first two modes of the simply supported beam(piezo with resistive load)


1st 86.576 5.0721� 10�3

2nd 347.83 1.2071� 10�3

Table 8

Natural frequencies and damping ratios for the first modes of the simply supported beam(piezo with resistive and inductive load)


1st split82:00990:030

�5:7681� 10�2

3:5312� 10�2

�2nd 347.87 1.0025� 10�3


a resistive and inductive load, tuned on the first bending mode, the behaviour is similar tothe one obtained for the cantilever beam (Table 8).

As regards the commercial code MSC/NASTRAN, it is possible to introduce afrequency-dependent function, which represents the elastic properties of a viscoelasticmaterial added to the structure, and then to derive the frequency response functions. Sincethat code only needs the frequency function mentioned above, it is possible to get theFRFs not only for the case of piezo with a resistive load, which actually behaves as aviscoelastic material, but also when the tuning circuit is considered [7]. Therefore, thedamping introduced by piezoelectric elements with passive loads can be obtained, withcompletely acceptable errors, by using finite element models, made by the experimenters orby commercially available codes.

The proposed approach, applied to simple structures, can be easily employed forcomplex aerospace structures, as carried out in [7], where the dynamic behaviour of asmall glider wing has been studied. Besides, no difficulties have been encountered inobtaining, through a finite element code, the multimodal damping [17], achievable by theelectrical circuit proposed and experimentally tested in [18].

4. CONCLUSIONS

Piezoceramic devices are more and more applied, especially in the aerospace field, fortheir peculiar characteristics: they, if used as passive dampers, are small, lightweight andefficient in comparison with the relative mechanical systems. Damping can be increased byusing resistive loads, and the complex stiffness of the loaded piezo device results to besimilar to the one presented by classic viscoelastic models. Unlike these last ones, it is veryeasy to put the maximum of the loss factor curve of the piezo device on the desiredfrequency}generally a natural frequency of a mode}in order to introduce the maximumdamping into the structure.


On the other hand, piezoceramic patches, applied to a structure with inductors andresistors as loads, practically behave as vibration absorbers and so, in many cases, they canprofitably substitute these last mechanical elements. In particular the resistive andinductive load permits a very fine tuning on a structure mode. Therefore it works as a veryeffective damper on the tuned mode, with no influence on the others, and unlike theviscoelastic behaviour. In the paper, it has been shown how both the systems can beintroduced into a mechanical structure, and evaluated by finite element models. Because ofthe frequency-dependent stiffness of the piezo devices, the overall eigenvalues ought to bederived by an iterative procedure but, for each frequency of interest, the relative complexconstant value of the dynamic stiffness can be considered. The available results by thisprocedure are completely acceptable. Similar results can be gained by a different approach,based on finite element models, but requiring the evaluation of the frequency responsefunction matrix.

In conclusion, the mechanical models}for both the loads}are useful not only forsimple structures, as the one presented in the paper, but also for more general systems, asthe aerospace structures, that can be described by finite element models, in which thedynamic stiffness of the piezo elements is added to the stiffness of the host structure. Thenumerical examples show how good results can be achieved by this procedure and howsimple is the use of the mechanical models of piezo devices in the study of structures whenfinite element models are employed.

ACKNOWLEDGEMENTS

This work has been supported by the MURST grants: Tecniche per Aumentare loSmorzamento in Sistemi Spaziali and Metodi Attivi e Passivi per Aumentare loSmorzamento in Strutture Spaziali.

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Multimodal passive vibration suppression with piezoelectric materials and resonant shunts.

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DAMPING BY PIEZOCERAMIC DEVICES WITH PASSIVE LOADS