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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 PIEZOCERAMIC DEVICES WITH PASSIVE LOADS A. Agneni, L. Balis Crema and S. Sgubini Dipartimento di Ingegneria Aerospaziale e Astronautica, Universita" degli Studi di Roma ‘‘La Sapienza’’, 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 either with resistors or with inductors and resistors. Both the circuits can be represented by mechanical models: the first one is equivalent to viscoelastic models, whereas the second one is a single-degree-of-freedom (sdof) system, which is similar to a vibration absorber when added to the host structure. Since the device stiffness is frequency dependent, an iterative procedure ought to be adopted in order to get the eigenvalues, but completely acceptable numerical results can be obtained by an approximated technique, which uses the stiffness calculated at the desired angular frequency. Besides the mechanical models mentioned before can be profitably introduced into finite element codes (also commercially available), which allow one to get information on the modal parameters. For a structure with piezo devices shunted with resistive loads, if either the impulse response or the frequency response function is derived, the natural frequencies and damping ratios can be achieved by least-squares techniques without any bias, as demonstrated by studying a sdof system. The possibility provided by this approach has been shown by means of numerical examples carried out on beams, cantilevered and simply supported, modelled by finite element codes. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Piezoceramic materials can be used in order to increase the damping characteristics and then to reduce the amplitude of structure vibrations, and that is especially attractive in the aerospace 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 device is loaded with a simple resistor or when the load is formed by a resistor and an inductor, in order 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 ThreeParameter Model [3–5]. By comparing the stiffness of the viscoelastic models with the one of a piezo device, it is possible to derive the relationships linking the mechanical values with the mechanical and electrical characteristics of the two models. It is possible to take advantage of this similarity, in fact the viscoelastic models can be introduced not only into a 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 be evaluated}with negligible errors}by the eigenvalues obtained from the structure when the piezo element behaviour is provided by a simple complex number (different for each frequency) 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.
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function obtained from the structure [7]. This last approach can also adopt the usual fitting techniques, normally present in the identification commercial codes, practically without any bias in the results, as theoretically and numerically shown for a single-degreeof-freedom (sdof) system. The piezo device with the second load (resistor and inductor) is similar, although not equal}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 frequencydependent complex stiffness into finite element models and a completely mechanical system is achieved, as said before for the first case, not only by using a general mathematical 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 modal parameters: natural frequencies and damping ratios) from simple structures, as a cantilever or a simply supported beam, modelled by finite elements and with ideally bonded 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 a piezoelectric device with a load, whose impedance is ZL: 2 VL % ksh ðoÞ ¼ ksc 1 þ k ð1Þ V where ksc is the value of the stiffness when the piezo is short circuited, V is the voltage generated by the piezo device, VL is the voltage measured across the load when the capacity of the piezo is CpS ; i.e. the capacitance taken at constant strain. At last k%2 is related to the electromechanical coupling factor kij by k%2 ¼ k2ij =ð1 k2ij Þ: 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 of the Thevenin theorem, the piezo device can be also considered as a voltage generator with a capacitive impedance in series. When the dynamic stiffness is given in terms of the short circuit one, as in equation (1), the capacitive impedance is the one when the circuit is open and it is equal to ZOC ¼ 1=joCps and therefore the circuit closed on a load is a potential divider, whose voltage ratio is VL ZL : ð2Þ ¼ V ZL þ Zoc 2.1. RESISTOR LOADS When the piezo is loaded with a resistor (R ), the dynamic stiffness may be written as follows [4–6]: 1 r %2 ksh ðrÞ ¼ ksc 1 þ k%2 1 þ j k ð3Þ 1 þ r2 1 þ r2 where r ¼ oRCpS 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 the complex stiffness is maximum, oM ¼ ðRCpS Þ1 : It is possible to show [3] that the stiffness}normalised with respect to ksc and reduced to proper rational functions, as in equation (3)}except for a constant, is always formed by a Cauchy 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
Equation (3) can be also written in terms of the loss factor ZðrÞ : " # I k ðrÞ sh ¼ kR ksh ðrÞ ¼ kR sh ðrÞ 1 þ j R sh ðrÞ½1 þ jZðrÞ ksh ðrÞ
1099
ð4Þ
which can be easily derived: ZðrÞ ¼
k%2 r 1 þ r2 ð1 þ k%2 Þ
:
The maximum of ZðrÞ; for a resistive load, is reached at the angular frequency: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 k2ij Z @ A: oM ¼ RCpS
ð5Þ
ð6Þ
Z So the relaxation time results equal t1 Z ¼ oM : 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 also pointed out in [2]) and the Three-Parameter Model. Hereafter, a short description of these mechanical elements is presented.
2.1.1. Zener–Biot model Figure 1 shows this model, which is formed by a Maxwell damper with a stiffness in parallel [11–13]. The stiffness, complex and frequency dependent, is given by the following relationship: K1 K2 þ jom ðK1 þ K2 Þ : ð7Þ kc ðoÞ ¼ K1 þ jom The similarity with ksh ðrÞ is evident if kc ðoÞ is written in terms of its real and imaginary parts as proper rational functions: " # " # 1 oðm=K1 Þ kc ðoÞ ¼ ðK1 þ K2 Þ K1 þ jK1 : ð8Þ 1 þ o2 ðm=K1 Þ2 1 þ o2 ðm=K1 Þ2 Comparing equation (3) with equation (8), the mechanical elements of the Zener–Biot model can be written in terms of the mechanical and electric characteristics of the piezodevice: K1 ¼ ksc k%2 K2 ¼ ksc m ¼ ksc k%2 t:
Figure 1. Zener–Biot model.
ð9Þ
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A. AGNENI ET AL.
Equation (8) can be also represented as follows: I R kc ðoÞ ¼ kR c ðoÞ þ jkc ðoÞ ¼ kc ðoÞ½1 þ jZðoÞ:
ð10Þ
The study of the mechanical model can help to understand the behaviour of the piezo device. Following [11], the relaxed elastic modulus of the piezo device is given by kR c 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 made up of the stiffness K2 only. On the contrary the unrelaxed modulus is equal to kR c jo!þ1 ¼ ðK1 þ K2 Þ; i.e. for very high angular frequencies the damper behaves as a perfect stiff member and the system is reduced to the parallel of the springs K1 and K2. The imaginary part of the stiffness is equal to zero in both cases. Bearing in mind the relationships in equation (9), for a piezoelectric element the relaxed R %2 and the unrelaxed moduli are equal to kR sh ! ksc and ksh ! ksc ð1 þ k Þ; that is the stiffness R for the electric circuit open ksh ! 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 of speaking of natural frequencies in the range 04o4 þ 1; it is necessary to consider only ðpÞ those values satisfying the condition o5oðpÞ n ; for which the model was adopted, on being the intrinsic natural frequency of the piezo device. Therefore, the relationships gained before for the relaxed and unrelaxed moduli are valid for o5oM and oM 5o5oðpÞ n ; respectively. 2.1.2. Three-parameter model This model, presented in Fig. 2, has the following complex stiffness [11–13]: K1 ðK2 þ jomÞ : kc ðoÞ ¼ ðK1 þ K2 Þ þ jom The real and imaginary parts of the dynamic stiffness can be written as follows: " # K12 1 kc ðoÞ ¼ K1 K1 þ K2 1 þ o2 ðm=ðK1 þ K2 ÞÞ2 " # o m=ðK1 þ K2 Þ K12 þj : K1 þ K2 1 þ o2 m=ðK1 þ K2 Þ 2
ð11Þ
ð12Þ
As a consequence the mechanical parameters, in terms of the ones relevant to the piezo device, are K1 ¼ ksc 1 þ k%2 1 þ k%2 K2 ¼ ksc ð13Þ k%2 2 ksc 1 þ k%2 m¼ t: k%2 2.2. RESISTOR AND INDUCTOR LOADS When the load is formed by a resistor (R) and an inductor (L) in series, the voltage measured across these two components, normalised with respect to the one generated by the piezoceramic element, is VL R þ joL : ð14Þ ¼ R þ joL þ 1=joCpS V
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DAMPING BY PIEZOCERAMIC DEVICES
Figure 2. Three-parameter model.
The complex stiffness is still given by equation (1), whereas the real and the imaginary parts}written in terms of proper rational functions}are given by ( " #) 2 S ½1 o ðLC Þ p %2 kR sh ðoÞ ¼ ksc 1 þ k 1 ½1 o2 ðLCpS Þ2 þ ½oðRCpS Þ2 " # ð15Þ ½oðRCpS Þ I 2 % ksh ðoÞ ¼ ksc k : ½1 o2 ðLCpS Þ2 þ ½oðRCpS Þ2 As one can see, the previous functions represent a couple of Hilbert transforms and therefore the ksh ðtÞ is causal and real, as ksh ðoÞ is Hermitian. It is easy to show that the loss factor is given by the following relationship: ZðoÞ ¼
k2ij ½oðRCpS Þ ð1 o2 ðLCpS Þ ½1 k2ij o2 ðLCpS Þ þ ½oðRCpS Þ2
:
ð16Þ
2.3. TUNING When the load is a resistor, its value is chosen in such a way that the maximum of the loss factor is coincident with the peak of the mode (kth ) that is to be damped ðonðkÞ Þ: If instead the maximum of the imaginary part of the complex stiffness is used, small errors qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi are introduced because the two angular frequencies only differ in the term 1 k2ij : This operation is commonly called ‘tuning’, but strictly speaking no tuning is possible with such a load, in fact it does not form with the piezo inherent capacitor an electric oscillating system. The device, from the electrical point of view, is a high pass filter [4]. On the contrary, the resistive and inductive load, along with the inherent capacity of the piezo device, is a resonant circuit, which can be tuned choosing the inductor in such a way that the inductive impedance is equal to the capacitive one at the desired angular ðkÞ frequency}commonly one qffiffiffiffiffiffiffiffiffi ffi of the mode of interest ðon Þ}so that the electric resonant ðkÞ frequency oe ¼ 1= LCpS is equal to on : The system practically behaves as a vibration absorber, so a detuning yields an increase of the vibration amplitudes. This problem can be overcome by adopting those values of R and L which optimise the frequency response function of the structure with piezo devices [1], similar to that done for the mechanical systems [14]. 2.4. SDOF SYSTEM An sdof system (Fig. 3) is described, in the frequency domain, by the following equation of motion: ½Ms o2 þ joCs þ Ks þ ksh ðoÞ XðoÞ ¼ FðoÞ:
ð17Þ
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Figure 3. sdof system with a frequency-dependent stiffness ksh ðoÞ due to a piezo device.
The mass of the piezoceramic elements could be considered in Ms. The causality of the dynamic stiffness yields the causality of the whole mechanical system, as shown for similar cases in [13]. At last the FRF, relative to equation (17), is given by the following relationship: HðoÞ ¼
XðoÞ ¼ ½Ms o2 þ joCs þ Ks þ ksh ðoÞ1 : FðoÞ
2.4.1. Resistive load The FRF relative to the case of the Zener–Biot model is given by ½K1 þ jom HðoÞ ¼ jo3 ðMs mÞ o2 ðMs K1 þ Cs mÞ þ jo½mðKs0 þ K1 Þ þ K1 Cs þ ðKs0 þ K1 Þ
ð18Þ
ð19Þ
where Ks0 ¼ ðKs þ K2 Þ: Introducing equation (9) into equation (19), this last relationship becomes ½1 þ jot : ð20Þ HR ðoÞ ¼ 3 2 jo ðMs tÞ o ðMs þ Cs tÞ þ jo½tðKs þ Koc Þ þ Cs þ ðKs þ ksc Þ 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 piezo device has its terminals connected in short circuit, oEn ; or open, oD n : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKs þ ksc Þ oEn ¼ Ms ð21Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK s þ koc Þ D on ¼ Ms it reduces to HR ðoÞ ¼
1 ½1 þ jot : 2 E 2 Ms jo3 t o2 ð1 þ 2ss tÞ þ jo½tðoD n Þ þ 2ss þ ðon Þ
ð22Þ
It is noteworthy that when a multi-degree-of-freedom (mdof) system is considered, the physical quantities in equation (20) become modal, as also pointed out in [1], and the angular frequencies oEn and oD n }relative to the mode considered}take account of the structure constraints and piezo positions. Thus the FRF in equation (22) represents, for each mode, every structure whose damping is increased by means of a piezo device with resistive loads, and wherever the piezos are positioned and whichever the constraints are.
DAMPING BY PIEZOCERAMIC DEVICES
1103
If the damping of the structure is neglected, the previous relationship, written in terms of the Laplace variable s, becomes: 1 ½1 þ st HR ðsÞ ¼ : ð23Þ 2 3 2 E 2 Ms ts þ s þ st ðoD n Þ þ ðon Þ So the modal damping, introduced by a piezo device, could be identified by evaluating the roots of the polynomial [1]: 2 E 2 ts3 þ s2 þ tðoD n Þ s þ ðon Þ ¼ 0:
ð24Þ
Obviously it is necessary to know t; which can be chosen to be t ¼ ðRCpS Þ ¼ ðoM Þ1 ¼ ðon Þ1 ; so as to fit the peak of the mode to be damped ðon Þ with the maximum of kIsh ðoÞ: For different values of the resistor R, damping ratios corresponding to the relative t0 s can be 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 the system by oEn and oD n : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks þ kR sh ðoM Þ on ¼ : ð25Þ 2 %2 Introducing kR sh ðoM Þ ¼ ksc ð1 þ k =2Þ into equation (25), the following relationship is derived: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ððKs þ ksc Þ=Ms Þ þ ððKs þ koc Þ=Ms Þ ðoEn Þ2 þ ðoD nÞ ¼ on ¼ : ð26Þ 2 2 Besides, considering the relationships among the coefficients of equation (24) with the roots of the cubic, bearing in mind that one of them is real ðsr Þ and the other two must be complex and conjugate ðl ¼ sl jod Þ; since the polynomial is relative to an oscillating system, the decay rate for the real root and the damping ratio, introduced by the piezo devices, are given by ðoEn Þ2 on 2 ðoE Þ2 ðoD nÞ B ¼ 12 n 2 :l 2 ðoEn Þ þ ðoD nÞ
sr ¼
ð27Þ
Thus the previous modal parameters can be obtained only by the knowledge of the two angular frequencies oEn and oD n: In addition the simplified version of the relation HR(s), equation (23), can be expanded in partial fractions by the Heaviside procedure: Rr Rl Rl þ þ : ð28Þ HR ðsÞ ¼ ðs þ sr Þ ðs þ lÞ ðs þ l Þ The residues, relative to the real (Rr) and the complex and conjugate roots ðRl and Rl Þ; are given by the following relationships: 1 ðð1=tÞ sr Þ Rr ¼ Ms s2r sr ðl þ l Þ þ ll ð29Þ R* l 1 ðð1=tÞ lÞ ¼ : Rl ¼ Ms 2jod ðsr lÞ 2j Only Rl has been reported because equation (22) is Hermitian and therefore Rl is complex and conjugate with respect to Rl : Rl ¼ R l : It is straightforward to derive the
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impulse response function from equation (28):
t
hR ðtÞ ¼ Rr esr t þ Rl elt þ R l el :
ð30Þ
Actually, the values of oEn and oD n are very close and so it is possible to say that on ffi oEn ; 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 sr 4sl : 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, the response of the structure is practically equal to the one of an sdof system with constant parameters and viscous damping: HR ðsÞ ffi
Rl Rl þ : ðs þ lÞ ðs þ l Þ
ð31Þ
The differences are present especially at the lowest frequencies, where the spectrum of the damped function has its maximum; in any case, in the frequency domain, the contribution of ðRr esl t Þ at the peaks provided by the other two functions is generally small and negligible. The IR derived from equation (31) is a damped sinusoidal function: hR ðtÞ ffi R* l esl t sinðod tÞ:
ð32Þ
Also in this case, the differences between the relationships in equations (30) and (32) are especially present in the initial points, where the damped non-oscillating function gives its higher contribution. Therefore the modal parameters of a vibrating structure with piezo devices can be derived, with negligible errors, from experimental data by usual, also commercial, identification codes.
2.4.2. Resonant circuit For this case, the complex stiffness ksh ðoÞ; relative to a resistive and inductive load, has to be introduced into equation (18). For an sdof system or for one mode of an mdof structure, since as said before the load forms a resonant circuit with the intrinsic capacitance of the piezo device, the FRF is also given by equation (18), where ksh ðoÞ; the one whose real and imaginary parts are given in equation (15). If instead of HðoÞ; the non-dimensional displacement (x/xst) is considered, the following relationship is obtained [1]: x ðd2 þ g2 Þ þ d2 rg ð33Þ ¼ xst ð1 þ g2 Þðd2 þ g2 þ d2 rgÞ þ Kij2 ðg2 þ d2 rgÞ where Kij is the generalised electromechanical coupling coeffcient: ksc ðoD Þ2 ðoEn Þ2 Kij2 ¼ k%2 ¼ n K þ ksc ðoEn Þ2
ð34Þ
and d¼
oe ; ðoEn Þ
g¼
jo ; ðoEn Þ
r ¼ RCpS on :
ð35Þ
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DAMPING BY PIEZOCERAMIC DEVICES
Since the optimal values, derived from [1], can be put as a function of the angular frequencies oEn and oD n [6]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðoD Þ2 ðoEn Þ2 ðoD Þ dopt ¼ 1 þ n ¼ nE 2 E ðon Þ ðon Þ ð36Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D E pffiffiffi ðon Þ ðon Þ ropt ffi 2 ðoEn Þ 2 ðoD Þ n 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 range through g: Obviously, if the optimised data are not used, the values of R and L are known along with CpS ; thus for each mode, equation (33) can be used, once Kij2 ; 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 can be obtained by adding to the mass and stiffness matrices of the structure the ones of the piezo devices. The piezo stiffness matrix, ksh ðoÞ; is gained by the frequency-dependent elastic modulus, which for the resistive load is given by the following relationship: ( " #) 1 jðoRCpS Þ 2 Esh ðoÞ ¼ Esc 1 þ k% 1 ð37Þ 1 þ ðoRCpS Þ2 whereas the elastic modulus, when a resistive–inductive load is used, is given by 8 2 39 > > 2 S S = < 1 o LC Þ jðoRC p p 7 26 % Esh ðoÞ ¼ Esc 1 þ k 41 5 : 2 > > : 1 o2 LC S þðoRC S Þ2 ; p
ð38Þ
p
The FRF matrix can be achieved by inverting the dynamic matrix: % &1 HðoÞ ¼ Mo2 þ joCs þ Ks þ ksh ðoÞ :
ð39Þ
The matrix M takes account of the mass matrix of the structure and the one of the piezo device, on the contrary Cs and Ks are the damping and the stiffness matrix of the structure only.
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 parameter mk ¼ Ks =ksc ; Cs ¼ 0; whilst ksh ðoÞ is the stiffness of a Zener–Biot model. The natural frequencies of the system, for several mk, are reported in Table 1, whilst the relative damping 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 Gauss filter [16] (superscript GH), and}in the frequency domain}by a least-squares fitting of the FRF with the residue/pole (i.e. a polynomial ratio) model (superscript FRF ). In this last case, for the reason said in the theoretical basis, the fitting has been carried out by a model with two complex conjugate poles, as a common sdof system with a dashpot in the place of the Zener–Biot element. The superscript a indicates that the data have been
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Table 1 Natural frequencies (in Hz) of the sdof system with variable stiffness with the parameter mk mk fnc fnGH fnFRF fna
5 103
101 2
1.3296 10 1.3296 102 1.3296 102 1.3280 102
5 101 2
1.3868 10 1.3868 102 1.3868 102 1.3855 102
1 2
1.6058 10 1.6038 102 1.6058 102 1.6050 102
5 2
1.8435 10 1.8434 102 1.8435 102 1.8429 102
10 2
3.1560 10 3.1560 102 3.1560 102 3.1559 102
4.2620 102 4.2620 102 4.2620 102 4.2620 102
Table 2 Damping ratio of the sdof system with variable stiffness with the parameter mk Mk zc zGH zFRF za
5 103
101
5 101
1
5
10
3.3519 102 3.3519 102 3.3519 102 3.2471 102
3.0720 102 3.0719 102 3.0720 102 2.9834 102
2.2725 102 2.2753 102 2.2725 102 2.2232 102
1.7146 102 1.7146 102 1.7146 102 1.6862 102
5.7829 103 5.7829 103 5.7829 103 5.7498 103
3.1627 103 3.1627 103 3.1627 103 3.1527 103
achieved by the relation in equation (26) and by the second one of equation (27). The results 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 considered for the values gained by fittings in the time and in the frequency domain. On the contrary, a maximum relative error of e ¼ 0:12 (%) is obtained for the natural frequency, estimated by equation (26), when mk ¼ 5 103 : A maximum error of e ¼ 3:13 (%) on the damping factor, gained by the second one of equation (27), was instead achieved for mk ¼ 101 ; that is when it is pre-ponderant the stiffness of the piezo device with respect to the one of the structure. The contribution due to the real root is practically immaterial. As mentioned before, the damping factor achieved through oEn and oD n is the one introduced by the piezo device, whereas the roots of the cubic polynomial provide the total damping. In fact, if a viscous damper is considered in the sdof system ðmk ¼ 5 and zs ¼ 7:5646 103 Þ; the identified modal parameters are shown in the following table (Table 3). Obviously, for a generic resistor ðR% Þ chosen by the experimenter, the relative time t% ¼ R% CpS is known, besides as CpS is the inherent capacity of the piezo and the angular frequencies oEn and oD n can be numerically or experimentally identified, % ¼ t% 1 is achieved from the the damping ratio relative to the angular frequency o cubic roots of the polynomial. This technique permits to plot damping ratios vs frequencies. The eigenvalues of a structure with a piezo device have to be obtained by an iterative procedure. The complex stiffness of a loaded piezo device is frequency dependent, thus the stiffness matrix of the whole finite element model of the structure is frequency dependent as well. But, damping ratios}for a generic frequency and for the one of a mode}could be estimated with immaterial errors [5], if a complex stiffness matrix, constant for each considered angular frequency, is added to the structure stiffness matrix in order to take account of the piezo device. The piezo stiffness matrix is built starting from the values of the real and the imaginary parts of its elastic modulus, whose behaviour with frequency is obviously equal to the one in equation (3). When a resistor load is considered, the
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DAMPING BY PIEZOCERAMIC DEVICES
Table 3 Natural frequencies and damping ratios achieved with different approaches fnc 3.1562 102
zc
fnFRF
zFRF
fna
za
1.3391 102
3.1560 102
1.3391 102
3.1559 102
5.7450 103
components of the elastic modulus become R 2 % Esh ðoÞ ¼ Esc 1 þ k 1
1 1 þ r2
r I Esh ðoÞ ¼ jEsc k%2 1 þ r2
ð40Þ
while the loss factor is ZðoÞ ¼
k2ij r 1 þ 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 the method mentioned above. The beam was divided into 10 finite elements, with piezoelectric patches, whose coupling factor was kij; ¼ 0:38; Esc ¼ 63 GPa; CpS ¼ 0:156 mF and the density is equal to 7500 kg m3. In order to simulate the beam in [1], four piezo elements were ideally bonded on the top and on the bottom of the first two finite elements near the fixed end. The surface dimensions of the piezo patch were equal to the ones of the single finite element of the beam and the thickness was equal to 0.25 mm. The results, obtained by a home-made MATLAB finite element model, were in agreement with the ones in [1]. The same technique}or a similar one, which provided kR sh ðoÞ 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 beam supported at its ends. The real and the imaginary parts and the loss factors for different non-dimensional frequency r, obtained dividing the angular frequency by oM ; are reported in Table 4. The damping ratio behaviour, for the cantilever beam, is shown in Fig. 4. The solid line represents 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 a Fortran program), and with (+) for bar elements (the complex stiffness matrices were provided by the MATLAB model and then introduced into the MSC/NASTRAN by a Fortran program). The damping ratios gained with the different methods are practically equal. Very small differences (Fig. 5) have been instead found when the damping ratios have been derived for the first bending mode of a beam simply supported at its ends; the piezo devices were considered ideally bonded on the top and on the bottom of the middle finite elements (5th and 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 also obtained from the FRFs, achieved by a MATLAB program, by a least-squares fitting of the curve.
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Table 4 Real and imaginary parts of the piezo device elastic modulus and loss factor vs non-dimensional frequency r 10
R½Esh ðoÞ 10 I½Esh ðoÞ 109 Z 102
7
0.1
0.3
0.5
1.0
2.0
3.0
10.0
6.3105 1.0527 1.6682
6.3878 2.9264 4.5812
6.5126 4.2530 6.5304
6.8316 5.3163 7.7819
7.1506 4.2530 5.9478
7.2569 3.1898 4.3955
7.3527 1.0527 1.4317
× 10-3
6
Damping ratio ζ
5
4
3
2
1 0
34.8
69.6
104.4
139.2
174.1
208.9
243.7
278.5
313.3
348.1
Frequency (Hz)
Figure 4. Damping ratios, relative to the first bending mode of the cantilever beam. The piezo devices were in the 1st and 2nd finite elements from the clamped end.
A first example (Fig. 6) shows the FRF for the cantilever beam, limited to the first two modes. In this case a modal damping equal to z ¼ 0:001 has been added in order to take into 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 frequency range, the second mode presents a damping ratio higher than 1.0 103. The FRF gained when a resistive and inductive load with the optimal values chosen so as to fit oð1Þ n is presented in Fig. 7. In this case, since the piezo circuit behaves as a vibration absorber, the mode on which it is 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 effect is present in the other modes; the identified damping ratio relative to the second mode is
1109
DAMPING BY PIEZOCERAMIC DEVICES × 10
4.5
-3
4
3.5
Damping ratio ζ
3
2.5
2
1.5
1
0.5
0
86.6
173.2
259.7
346.3
432.9
519.5
606.1
692.6
779.2
865.8
Frequency (Hz)
Figure 5. Damping ratios of the simply supported beam. The piezo devices were positioned in the 5th and 6th finite elements.
10 -1
10
-2
-3
H(ω)
10
-4
10
10-5
-6
10
0
200
400
600
800
1000
1200
1400
1600
Angular frequency (rad/s)
Figure 6. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded with a resistor ‘tuned’ on the first bending mode.
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A. AGNENI ET AL.
Table 5 Natural frequencies and damping ratios for the first two modes of the cantilever beam (piezo with resistive load) Mode
Natural frequency (Hz)
Damping ratio
34.814 207.18
7.6532 103 2.3527 103
1st 2nd
-1
10
10
-2
H(ω)
10-3
10
-4
-5
10
10 -6
10
-7
0
200
400
600
800
1000
1200
1400
1600
Angular frequency (rad/s)
Figure 7. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded with a 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 with resistive and inductive load) Mode 1st split 2nd
Natural frequency (Hz) 32:717 36:760 347.83
Damping ratio 6:9264 102 4:8808 102 1.0089 103
practically equal to the structure mode only. The FRFs, achieved for the simply supported beam, are instead shown in Figs 8 and 9. The modal parameters identified from a least-squares fitting of the previous curves are reported 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 of damping is introduced into the second mode, because the piezo devices, positioned on the 5th and 6th finite elements, provide a low strain energy for that mode. On the contrary, for
1111
DAMPING BY PIEZOCERAMIC DEVICES 100
10-1
H(ω)
10-2
-3
10
10-4
-5
10
0
500
1000
1500
2000
2500
3000
3500
Angular frequency (rad/s)
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.
101
10
0
-1
H(ω)
10
-2
10
10-3
-4
10
10-5
0
500
1000
1500
2000
2500
3000
3500
Angular frequency (rad/s)
Figure 9. FRF magnitude of the supported beam, measured at its middle point. The piezo devices, at the 5th and 6th finite elements, have been loaded with a resistor and an inductor, whose optimal values were relative to the first bending mode.
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A. AGNENI ET AL.
Table 7 Natural frequencies and damping ratios for the first two modes of the simply supported beam (piezo with resistive load) Mode 1st 2nd
Natural frequency (Hz)
Damping ratio
86.576 347.83
5.0721 103 1.2071 103
Table 8 Natural frequencies and damping ratios for the first modes of the simply supported beam (piezo with resistive and inductive load) Mode 1st split 2nd
Natural frequency (Hz) 82:009 90:030 347.87
Damping ratio 5:7681 102 3:5312 102 1.0025 103
a resistive and inductive load, tuned on the first bending mode, the behaviour is similar to the one obtained for the cantilever beam (Table 8). As regards the commercial code MSC/NASTRAN, it is possible to introduce a frequency-dependent function, which represents the elastic properties of a viscoelastic material added to the structure, and then to derive the frequency response functions. Since that code only needs the frequency function mentioned above, it is possible to get the FRFs not only for the case of piezo with a resistive load, which actually behaves as a viscoelastic material, but also when the tuning circuit is considered [7]. Therefore, the damping introduced by piezoelectric elements with passive loads can be obtained, with completely acceptable errors, by using finite element models, made by the experimenters or by commercially available codes. The proposed approach, applied to simple structures, can be easily employed for complex aerospace structures, as carried out in [7], where the dynamic behaviour of a small glider wing has been studied. Besides, no difficulties have been encountered in obtaining, through a finite element code, the multimodal damping [17], achievable by the electrical circuit proposed and experimentally tested in [18].
4. CONCLUSIONS
Piezoceramic devices are more and more applied, especially in the aerospace field, for their peculiar characteristics: they, if used as passive dampers, are small, lightweight and efficient in comparison with the relative mechanical systems. Damping can be increased by using resistive loads, and the complex stiffness of the loaded piezo device results to be similar to the one presented by classic viscoelastic models. Unlike these last ones, it is very easy to put the maximum of the loss factor curve of the piezo device on the desired frequency}generally a natural frequency of a mode}in order to introduce the maximum damping into the structure.
DAMPING BY PIEZOCERAMIC DEVICES
1113
On the other hand, piezoceramic patches, applied to a structure with inductors and resistors as loads, practically behave as vibration absorbers and so, in many cases, they can profitably substitute these last mechanical elements. In particular the resistive and inductive load permits a very fine tuning on a structure mode. Therefore it works as a very effective damper on the tuned mode, with no influence on the others, and unlike the viscoelastic behaviour. In the paper, it has been shown how both the systems can be introduced into a mechanical structure, and evaluated by finite element models. Because of the frequency-dependent stiffness of the piezo devices, the overall eigenvalues ought to be derived by an iterative procedure but, for each frequency of interest, the relative complex constant value of the dynamic stiffness can be considered. The available results by this procedure are completely acceptable. Similar results can be gained by a different approach, based on finite element models, but requiring the evaluation of the frequency response function matrix. In conclusion, the mechanical models}for both the loads}are useful not only for simple structures, as the one presented in the paper, but also for more general systems, as the aerospace structures, that can be described by finite element models, in which the dynamic stiffness of the piezo elements is added to the stiffness of the host structure. The numerical examples show how good results can be achieved by this procedure and how simple is the use of the mechanical models of piezo devices in the study of structures when finite element models are employed.
ACKNOWLEDGEMENTS
This work has been supported by the MURST grants: Tecniche per Aumentare lo Smorzamento in Sistemi Spaziali and Metodi Attivi e Passivi per Aumentare lo Smorzamento in Strutture Spaziali.
REFERENCES 1. N. H. Hagood and A. von Flotow 1989 WRDC-TR-89-3116 Proceedings of Damping 89, West Palm Beach, Vol. 3, ICC-1/ICC-31. Damping of structural vibration with piezoelectric materials and passive electrical networks. 2. J. B. Aldrich, N. H. Hagood, A. von Flotow and D. W. Vos 1993 Proceedings of the Damping ‘93 San Francisco, Vol. 1, BAC-1/BAC-14. Design of passive piezoelectric damping for space structures. 3. A. Agneni, L. Balis Crema and S. Sgubini 1999 IAF 50th International Astronautical Congress, Amsterdam. Complex stiffness identification in shunted piezoelectric absorbers. 4. A. Agneni and S. Sgubini 1999 Atti del XV Congresso Nazionale A.I.D.A.A, Torino, Vol. 2, 1345–1355. Piezoceramic devices as viscoelastic systems. 5. A. Agneni, A. Paolozzi and S. Sgubini 2000 European COST F3, Proceedings of the Conference on System Identification & Structural Health Monitoring, Madrid, Vol. 2, 591–599. Piezoceramic devices modelled as mechanical systems in finite element codes. 6. A. Agneni and A. Paolozzi 2000 Proceedings of Smart Structures and Devices Conference, SPIE’s Smart Materials and MEMS, Melbourne. On the use of finite element codes for structures with passive piezoceramic devices. 7. A. Agneni, G. Coppotelli, F. Mastroddi and S. Sgubini 2001 IFASD 2001, International Forum on Aeroelasticity and Structural Vibrations, Madrid. Damping of aeroelastic vibrations by shunted piezoelectric devices. 8. A. M. Rivas McGowan 1999 CEAS/AIAA/ICASE/NASA Langley International Forum on Aeroelasticity and Structural Dynamics, Williamsburg, 553–571. An examination of applying shunted piezoelectrics to reduce aeroelastic response. 9. L. E. Kinsler and A. R. Frey 1962 Fundamentals of Acoustics, 337–343. New York: John Wiley & Sons.
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10. A. D. Poularikis (ed.) 1995. In The Transforms and Applications Handbook, Chapter 7, 463–629. Boca Raton, FL: CRC-IEEE Press. (L. S. Hahn) Hilbert transform. 11. B. J. Lazan 1968 Damping of Materials and Members in Structural Mechanics, 73–74. Oxford: Pergamon Press. 12. I. Perepechko 1975 Acoustic Methods of Investigating Polymers, pp. 39–44. Moscow: MIR. 13. A. Agneni, F. Mastroddi and S. Sgubini 1997 Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Roma, Vol. 3, 111–122. Causality of damping models. 14. J. P. Den Hartog 1985 Mechanical Vibrations, 93–105. New York: Dover. 15. A. Agneni and F. Mastroddi 1995 Atti del XIII Congresso Nazionale AIDAA, Roma, Vol. 2, 775–786. System vibrations with hysteretic damping: frequency and time domain behavior. 16. A. Agneni, 2000 Mechanical Systems and Signal Processing 14, 193–204. On the use of the gauss filter in modal parameter estimation. 17. A. Agneni and S. Sgubini 2001 Atti del XVI Congresso Nazionale AIDAA, Palermo, CD file 132. Multimodal damping by piezoceramic devices with passive loads. 18. J. J. Hollkamp 1994 Journal of Intelligent Material Systems and Structures 5, 49–57. Multimodal passive vibration suppression with piezoelectric materials and resonant shunts.
DAMPING BY PIEZOCERAMIC DEVICES WITH PASSIVE LOADS A. Agneni, L. Balis Crema and S. Sgubini Dipartimento di Ingegneria Aerospaziale e Astronautica, Universita" degli Studi di Roma ‘‘La Sapienza’’, 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 either with resistors or with inductors and resistors. Both the circuits can be represented by mechanical models: the first one is equivalent to viscoelastic models, whereas the second one is a single-degree-of-freedom (sdof) system, which is similar to a vibration absorber when added to the host structure. Since the device stiffness is frequency dependent, an iterative procedure ought to be adopted in order to get the eigenvalues, but completely acceptable numerical results can be obtained by an approximated technique, which uses the stiffness calculated at the desired angular frequency. Besides the mechanical models mentioned before can be profitably introduced into finite element codes (also commercially available), which allow one to get information on the modal parameters. For a structure with piezo devices shunted with resistive loads, if either the impulse response or the frequency response function is derived, the natural frequencies and damping ratios can be achieved by least-squares techniques without any bias, as demonstrated by studying a sdof system. The possibility provided by this approach has been shown by means of numerical examples carried out on beams, cantilevered and simply supported, modelled by finite element codes. # 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Piezoceramic materials can be used in order to increase the damping characteristics and then to reduce the amplitude of structure vibrations, and that is especially attractive in the aerospace 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 device is loaded with a simple resistor or when the load is formed by a resistor and an inductor, in order 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 ThreeParameter Model [3–5]. By comparing the stiffness of the viscoelastic models with the one of a piezo device, it is possible to derive the relationships linking the mechanical values with the mechanical and electrical characteristics of the two models. It is possible to take advantage of this similarity, in fact the viscoelastic models can be introduced not only into a 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 be evaluated}with negligible errors}by the eigenvalues obtained from the structure when the piezo element behaviour is provided by a simple complex number (different for each frequency) 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.
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function obtained from the structure [7]. This last approach can also adopt the usual fitting techniques, normally present in the identification commercial codes, practically without any bias in the results, as theoretically and numerically shown for a single-degreeof-freedom (sdof) system. The piezo device with the second load (resistor and inductor) is similar, although not equal}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 frequencydependent complex stiffness into finite element models and a completely mechanical system is achieved, as said before for the first case, not only by using a general mathematical 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 modal parameters: natural frequencies and damping ratios) from simple structures, as a cantilever or a simply supported beam, modelled by finite elements and with ideally bonded 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 a piezoelectric device with a load, whose impedance is ZL: 2 VL % ksh ðoÞ ¼ ksc 1 þ k ð1Þ V where ksc is the value of the stiffness when the piezo is short circuited, V is the voltage generated by the piezo device, VL is the voltage measured across the load when the capacity of the piezo is CpS ; i.e. the capacitance taken at constant strain. At last k%2 is related to the electromechanical coupling factor kij by k%2 ¼ k2ij =ð1 k2ij Þ: 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 of the Thevenin theorem, the piezo device can be also considered as a voltage generator with a capacitive impedance in series. When the dynamic stiffness is given in terms of the short circuit one, as in equation (1), the capacitive impedance is the one when the circuit is open and it is equal to ZOC ¼ 1=joCps and therefore the circuit closed on a load is a potential divider, whose voltage ratio is VL ZL : ð2Þ ¼ V ZL þ Zoc 2.1. RESISTOR LOADS When the piezo is loaded with a resistor (R ), the dynamic stiffness may be written as follows [4–6]: 1 r %2 ksh ðrÞ ¼ ksc 1 þ k%2 1 þ j k ð3Þ 1 þ r2 1 þ r2 where r ¼ oRCpS 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 the complex stiffness is maximum, oM ¼ ðRCpS Þ1 : It is possible to show [3] that the stiffness}normalised with respect to ksc and reduced to proper rational functions, as in equation (3)}except for a constant, is always formed by a Cauchy 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
Equation (3) can be also written in terms of the loss factor ZðrÞ : " # I k ðrÞ sh ¼ kR ksh ðrÞ ¼ kR sh ðrÞ 1 þ j R sh ðrÞ½1 þ jZðrÞ ksh ðrÞ
1099
ð4Þ
which can be easily derived: ZðrÞ ¼
k%2 r 1 þ r2 ð1 þ k%2 Þ
:
The maximum of ZðrÞ; for a resistive load, is reached at the angular frequency: 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 k2ij Z @ A: oM ¼ RCpS
ð5Þ
ð6Þ
Z So the relaxation time results equal t1 Z ¼ oM : 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 also pointed out in [2]) and the Three-Parameter Model. Hereafter, a short description of these mechanical elements is presented.
2.1.1. Zener–Biot model Figure 1 shows this model, which is formed by a Maxwell damper with a stiffness in parallel [11–13]. The stiffness, complex and frequency dependent, is given by the following relationship: K1 K2 þ jom ðK1 þ K2 Þ : ð7Þ kc ðoÞ ¼ K1 þ jom The similarity with ksh ðrÞ is evident if kc ðoÞ is written in terms of its real and imaginary parts as proper rational functions: " # " # 1 oðm=K1 Þ kc ðoÞ ¼ ðK1 þ K2 Þ K1 þ jK1 : ð8Þ 1 þ o2 ðm=K1 Þ2 1 þ o2 ðm=K1 Þ2 Comparing equation (3) with equation (8), the mechanical elements of the Zener–Biot model can be written in terms of the mechanical and electric characteristics of the piezodevice: K1 ¼ ksc k%2 K2 ¼ ksc m ¼ ksc k%2 t:
Figure 1. Zener–Biot model.
ð9Þ
1100
A. AGNENI ET AL.
Equation (8) can be also represented as follows: I R kc ðoÞ ¼ kR c ðoÞ þ jkc ðoÞ ¼ kc ðoÞ½1 þ jZðoÞ:
ð10Þ
The study of the mechanical model can help to understand the behaviour of the piezo device. Following [11], the relaxed elastic modulus of the piezo device is given by kR c 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 made up of the stiffness K2 only. On the contrary the unrelaxed modulus is equal to kR c jo!þ1 ¼ ðK1 þ K2 Þ; i.e. for very high angular frequencies the damper behaves as a perfect stiff member and the system is reduced to the parallel of the springs K1 and K2. The imaginary part of the stiffness is equal to zero in both cases. Bearing in mind the relationships in equation (9), for a piezoelectric element the relaxed R %2 and the unrelaxed moduli are equal to kR sh ! ksc and ksh ! ksc ð1 þ k Þ; that is the stiffness R for the electric circuit open ksh ! 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 of speaking of natural frequencies in the range 04o4 þ 1; it is necessary to consider only ðpÞ those values satisfying the condition o5oðpÞ n ; for which the model was adopted, on being the intrinsic natural frequency of the piezo device. Therefore, the relationships gained before for the relaxed and unrelaxed moduli are valid for o5oM and oM 5o5oðpÞ n ; respectively. 2.1.2. Three-parameter model This model, presented in Fig. 2, has the following complex stiffness [11–13]: K1 ðK2 þ jomÞ : kc ðoÞ ¼ ðK1 þ K2 Þ þ jom The real and imaginary parts of the dynamic stiffness can be written as follows: " # K12 1 kc ðoÞ ¼ K1 K1 þ K2 1 þ o2 ðm=ðK1 þ K2 ÞÞ2 " # o m=ðK1 þ K2 Þ K12 þj : K1 þ K2 1 þ o2 m=ðK1 þ K2 Þ 2
ð11Þ
ð12Þ
As a consequence the mechanical parameters, in terms of the ones relevant to the piezo device, are K1 ¼ ksc 1 þ k%2 1 þ k%2 K2 ¼ ksc ð13Þ k%2 2 ksc 1 þ k%2 m¼ t: k%2 2.2. RESISTOR AND INDUCTOR LOADS When the load is formed by a resistor (R) and an inductor (L) in series, the voltage measured across these two components, normalised with respect to the one generated by the piezoceramic element, is VL R þ joL : ð14Þ ¼ R þ joL þ 1=joCpS V
1101
DAMPING BY PIEZOCERAMIC DEVICES
Figure 2. Three-parameter model.
The complex stiffness is still given by equation (1), whereas the real and the imaginary parts}written in terms of proper rational functions}are given by ( " #) 2 S ½1 o ðLC Þ p %2 kR sh ðoÞ ¼ ksc 1 þ k 1 ½1 o2 ðLCpS Þ2 þ ½oðRCpS Þ2 " # ð15Þ ½oðRCpS Þ I 2 % ksh ðoÞ ¼ ksc k : ½1 o2 ðLCpS Þ2 þ ½oðRCpS Þ2 As one can see, the previous functions represent a couple of Hilbert transforms and therefore the ksh ðtÞ is causal and real, as ksh ðoÞ is Hermitian. It is easy to show that the loss factor is given by the following relationship: ZðoÞ ¼
k2ij ½oðRCpS Þ ð1 o2 ðLCpS Þ ½1 k2ij o2 ðLCpS Þ þ ½oðRCpS Þ2
:
ð16Þ
2.3. TUNING When the load is a resistor, its value is chosen in such a way that the maximum of the loss factor is coincident with the peak of the mode (kth ) that is to be damped ðonðkÞ Þ: If instead the maximum of the imaginary part of the complex stiffness is used, small errors qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi are introduced because the two angular frequencies only differ in the term 1 k2ij : This operation is commonly called ‘tuning’, but strictly speaking no tuning is possible with such a load, in fact it does not form with the piezo inherent capacitor an electric oscillating system. The device, from the electrical point of view, is a high pass filter [4]. On the contrary, the resistive and inductive load, along with the inherent capacity of the piezo device, is a resonant circuit, which can be tuned choosing the inductor in such a way that the inductive impedance is equal to the capacitive one at the desired angular ðkÞ frequency}commonly one qffiffiffiffiffiffiffiffiffi ffi of the mode of interest ðon Þ}so that the electric resonant ðkÞ frequency oe ¼ 1= LCpS is equal to on : The system practically behaves as a vibration absorber, so a detuning yields an increase of the vibration amplitudes. This problem can be overcome by adopting those values of R and L which optimise the frequency response function of the structure with piezo devices [1], similar to that done for the mechanical systems [14]. 2.4. SDOF SYSTEM An sdof system (Fig. 3) is described, in the frequency domain, by the following equation of motion: ½Ms o2 þ joCs þ Ks þ ksh ðoÞ XðoÞ ¼ FðoÞ:
ð17Þ
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A. AGNENI ET AL.
Figure 3. sdof system with a frequency-dependent stiffness ksh ðoÞ due to a piezo device.
The mass of the piezoceramic elements could be considered in Ms. The causality of the dynamic stiffness yields the causality of the whole mechanical system, as shown for similar cases in [13]. At last the FRF, relative to equation (17), is given by the following relationship: HðoÞ ¼
XðoÞ ¼ ½Ms o2 þ joCs þ Ks þ ksh ðoÞ1 : FðoÞ
2.4.1. Resistive load The FRF relative to the case of the Zener–Biot model is given by ½K1 þ jom HðoÞ ¼ jo3 ðMs mÞ o2 ðMs K1 þ Cs mÞ þ jo½mðKs0 þ K1 Þ þ K1 Cs þ ðKs0 þ K1 Þ
ð18Þ
ð19Þ
where Ks0 ¼ ðKs þ K2 Þ: Introducing equation (9) into equation (19), this last relationship becomes ½1 þ jot : ð20Þ HR ðoÞ ¼ 3 2 jo ðMs tÞ o ðMs þ Cs tÞ þ jo½tðKs þ Koc Þ þ Cs þ ðKs þ ksc Þ 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 piezo device has its terminals connected in short circuit, oEn ; or open, oD n : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKs þ ksc Þ oEn ¼ Ms ð21Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK s þ koc Þ D on ¼ Ms it reduces to HR ðoÞ ¼
1 ½1 þ jot : 2 E 2 Ms jo3 t o2 ð1 þ 2ss tÞ þ jo½tðoD n Þ þ 2ss þ ðon Þ
ð22Þ
It is noteworthy that when a multi-degree-of-freedom (mdof) system is considered, the physical quantities in equation (20) become modal, as also pointed out in [1], and the angular frequencies oEn and oD n }relative to the mode considered}take account of the structure constraints and piezo positions. Thus the FRF in equation (22) represents, for each mode, every structure whose damping is increased by means of a piezo device with resistive loads, and wherever the piezos are positioned and whichever the constraints are.
DAMPING BY PIEZOCERAMIC DEVICES
1103
If the damping of the structure is neglected, the previous relationship, written in terms of the Laplace variable s, becomes: 1 ½1 þ st HR ðsÞ ¼ : ð23Þ 2 3 2 E 2 Ms ts þ s þ st ðoD n Þ þ ðon Þ So the modal damping, introduced by a piezo device, could be identified by evaluating the roots of the polynomial [1]: 2 E 2 ts3 þ s2 þ tðoD n Þ s þ ðon Þ ¼ 0:
ð24Þ
Obviously it is necessary to know t; which can be chosen to be t ¼ ðRCpS Þ ¼ ðoM Þ1 ¼ ðon Þ1 ; so as to fit the peak of the mode to be damped ðon Þ with the maximum of kIsh ðoÞ: For different values of the resistor R, damping ratios corresponding to the relative t0 s can be 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 the system by oEn and oD n : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ks þ kR sh ðoM Þ on ¼ : ð25Þ 2 %2 Introducing kR sh ðoM Þ ¼ ksc ð1 þ k =2Þ into equation (25), the following relationship is derived: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ððKs þ ksc Þ=Ms Þ þ ððKs þ koc Þ=Ms Þ ðoEn Þ2 þ ðoD nÞ ¼ on ¼ : ð26Þ 2 2 Besides, considering the relationships among the coefficients of equation (24) with the roots of the cubic, bearing in mind that one of them is real ðsr Þ and the other two must be complex and conjugate ðl ¼ sl jod Þ; since the polynomial is relative to an oscillating system, the decay rate for the real root and the damping ratio, introduced by the piezo devices, are given by ðoEn Þ2 on 2 ðoE Þ2 ðoD nÞ B ¼ 12 n 2 :l 2 ðoEn Þ þ ðoD nÞ
sr ¼
ð27Þ
Thus the previous modal parameters can be obtained only by the knowledge of the two angular frequencies oEn and oD n: In addition the simplified version of the relation HR(s), equation (23), can be expanded in partial fractions by the Heaviside procedure: Rr Rl Rl þ þ : ð28Þ HR ðsÞ ¼ ðs þ sr Þ ðs þ lÞ ðs þ l Þ The residues, relative to the real (Rr) and the complex and conjugate roots ðRl and Rl Þ; are given by the following relationships: 1 ðð1=tÞ sr Þ Rr ¼ Ms s2r sr ðl þ l Þ þ ll ð29Þ R* l 1 ðð1=tÞ lÞ ¼ : Rl ¼ Ms 2jod ðsr lÞ 2j Only Rl has been reported because equation (22) is Hermitian and therefore Rl is complex and conjugate with respect to Rl : Rl ¼ R l : It is straightforward to derive the
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A. AGNENI ET AL.
impulse response function from equation (28):
t
hR ðtÞ ¼ Rr esr t þ Rl elt þ R l el :
ð30Þ
Actually, the values of oEn and oD n are very close and so it is possible to say that on ffi oEn ; 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 sr 4sl : 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, the response of the structure is practically equal to the one of an sdof system with constant parameters and viscous damping: HR ðsÞ ffi
Rl Rl þ : ðs þ lÞ ðs þ l Þ
ð31Þ
The differences are present especially at the lowest frequencies, where the spectrum of the damped function has its maximum; in any case, in the frequency domain, the contribution of ðRr esl t Þ at the peaks provided by the other two functions is generally small and negligible. The IR derived from equation (31) is a damped sinusoidal function: hR ðtÞ ffi R* l esl t sinðod tÞ:
ð32Þ
Also in this case, the differences between the relationships in equations (30) and (32) are especially present in the initial points, where the damped non-oscillating function gives its higher contribution. Therefore the modal parameters of a vibrating structure with piezo devices can be derived, with negligible errors, from experimental data by usual, also commercial, identification codes.
2.4.2. Resonant circuit For this case, the complex stiffness ksh ðoÞ; relative to a resistive and inductive load, has to be introduced into equation (18). For an sdof system or for one mode of an mdof structure, since as said before the load forms a resonant circuit with the intrinsic capacitance of the piezo device, the FRF is also given by equation (18), where ksh ðoÞ; the one whose real and imaginary parts are given in equation (15). If instead of HðoÞ; the non-dimensional displacement (x/xst) is considered, the following relationship is obtained [1]: x ðd2 þ g2 Þ þ d2 rg ð33Þ ¼ xst ð1 þ g2 Þðd2 þ g2 þ d2 rgÞ þ Kij2 ðg2 þ d2 rgÞ where Kij is the generalised electromechanical coupling coeffcient: ksc ðoD Þ2 ðoEn Þ2 Kij2 ¼ k%2 ¼ n K þ ksc ðoEn Þ2
ð34Þ
and d¼
oe ; ðoEn Þ
g¼
jo ; ðoEn Þ
r ¼ RCpS on :
ð35Þ
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DAMPING BY PIEZOCERAMIC DEVICES
Since the optimal values, derived from [1], can be put as a function of the angular frequencies oEn and oD n [6]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðoD Þ2 ðoEn Þ2 ðoD Þ dopt ¼ 1 þ n ¼ nE 2 E ðon Þ ðon Þ ð36Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 D E pffiffiffi ðon Þ ðon Þ ropt ffi 2 ðoEn Þ 2 ðoD Þ n 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 range through g: Obviously, if the optimised data are not used, the values of R and L are known along with CpS ; thus for each mode, equation (33) can be used, once Kij2 ; 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 can be obtained by adding to the mass and stiffness matrices of the structure the ones of the piezo devices. The piezo stiffness matrix, ksh ðoÞ; is gained by the frequency-dependent elastic modulus, which for the resistive load is given by the following relationship: ( " #) 1 jðoRCpS Þ 2 Esh ðoÞ ¼ Esc 1 þ k% 1 ð37Þ 1 þ ðoRCpS Þ2 whereas the elastic modulus, when a resistive–inductive load is used, is given by 8 2 39 > > 2 S S = < 1 o LC Þ jðoRC p p 7 26 % Esh ðoÞ ¼ Esc 1 þ k 41 5 : 2 > > : 1 o2 LC S þðoRC S Þ2 ; p
ð38Þ
p
The FRF matrix can be achieved by inverting the dynamic matrix: % &1 HðoÞ ¼ Mo2 þ joCs þ Ks þ ksh ðoÞ :
ð39Þ
The matrix M takes account of the mass matrix of the structure and the one of the piezo device, on the contrary Cs and Ks are the damping and the stiffness matrix of the structure only.
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 parameter mk ¼ Ks =ksc ; Cs ¼ 0; whilst ksh ðoÞ is the stiffness of a Zener–Biot model. The natural frequencies of the system, for several mk, are reported in Table 1, whilst the relative damping 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 Gauss filter [16] (superscript GH), and}in the frequency domain}by a least-squares fitting of the FRF with the residue/pole (i.e. a polynomial ratio) model (superscript FRF ). In this last case, for the reason said in the theoretical basis, the fitting has been carried out by a model with two complex conjugate poles, as a common sdof system with a dashpot in the place of the Zener–Biot element. The superscript a indicates that the data have been
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Table 1 Natural frequencies (in Hz) of the sdof system with variable stiffness with the parameter mk mk fnc fnGH fnFRF fna
5 103
101 2
1.3296 10 1.3296 102 1.3296 102 1.3280 102
5 101 2
1.3868 10 1.3868 102 1.3868 102 1.3855 102
1 2
1.6058 10 1.6038 102 1.6058 102 1.6050 102
5 2
1.8435 10 1.8434 102 1.8435 102 1.8429 102
10 2
3.1560 10 3.1560 102 3.1560 102 3.1559 102
4.2620 102 4.2620 102 4.2620 102 4.2620 102
Table 2 Damping ratio of the sdof system with variable stiffness with the parameter mk Mk zc zGH zFRF za
5 103
101
5 101
1
5
10
3.3519 102 3.3519 102 3.3519 102 3.2471 102
3.0720 102 3.0719 102 3.0720 102 2.9834 102
2.2725 102 2.2753 102 2.2725 102 2.2232 102
1.7146 102 1.7146 102 1.7146 102 1.6862 102
5.7829 103 5.7829 103 5.7829 103 5.7498 103
3.1627 103 3.1627 103 3.1627 103 3.1527 103
achieved by the relation in equation (26) and by the second one of equation (27). The results 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 considered for the values gained by fittings in the time and in the frequency domain. On the contrary, a maximum relative error of e ¼ 0:12 (%) is obtained for the natural frequency, estimated by equation (26), when mk ¼ 5 103 : A maximum error of e ¼ 3:13 (%) on the damping factor, gained by the second one of equation (27), was instead achieved for mk ¼ 101 ; that is when it is pre-ponderant the stiffness of the piezo device with respect to the one of the structure. The contribution due to the real root is practically immaterial. As mentioned before, the damping factor achieved through oEn and oD n is the one introduced by the piezo device, whereas the roots of the cubic polynomial provide the total damping. In fact, if a viscous damper is considered in the sdof system ðmk ¼ 5 and zs ¼ 7:5646 103 Þ; the identified modal parameters are shown in the following table (Table 3). Obviously, for a generic resistor ðR% Þ chosen by the experimenter, the relative time t% ¼ R% CpS is known, besides as CpS is the inherent capacity of the piezo and the angular frequencies oEn and oD n can be numerically or experimentally identified, % ¼ t% 1 is achieved from the the damping ratio relative to the angular frequency o cubic roots of the polynomial. This technique permits to plot damping ratios vs frequencies. The eigenvalues of a structure with a piezo device have to be obtained by an iterative procedure. The complex stiffness of a loaded piezo device is frequency dependent, thus the stiffness matrix of the whole finite element model of the structure is frequency dependent as well. But, damping ratios}for a generic frequency and for the one of a mode}could be estimated with immaterial errors [5], if a complex stiffness matrix, constant for each considered angular frequency, is added to the structure stiffness matrix in order to take account of the piezo device. The piezo stiffness matrix is built starting from the values of the real and the imaginary parts of its elastic modulus, whose behaviour with frequency is obviously equal to the one in equation (3). When a resistor load is considered, the
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DAMPING BY PIEZOCERAMIC DEVICES
Table 3 Natural frequencies and damping ratios achieved with different approaches fnc 3.1562 102
zc
fnFRF
zFRF
fna
za
1.3391 102
3.1560 102
1.3391 102
3.1559 102
5.7450 103
components of the elastic modulus become R 2 % Esh ðoÞ ¼ Esc 1 þ k 1
1 1 þ r2
r I Esh ðoÞ ¼ jEsc k%2 1 þ r2
ð40Þ
while the loss factor is ZðoÞ ¼
k2ij r 1 þ 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 the method mentioned above. The beam was divided into 10 finite elements, with piezoelectric patches, whose coupling factor was kij; ¼ 0:38; Esc ¼ 63 GPa; CpS ¼ 0:156 mF and the density is equal to 7500 kg m3. In order to simulate the beam in [1], four piezo elements were ideally bonded on the top and on the bottom of the first two finite elements near the fixed end. The surface dimensions of the piezo patch were equal to the ones of the single finite element of the beam and the thickness was equal to 0.25 mm. The results, obtained by a home-made MATLAB finite element model, were in agreement with the ones in [1]. The same technique}or a similar one, which provided kR sh ðoÞ 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 beam supported at its ends. The real and the imaginary parts and the loss factors for different non-dimensional frequency r, obtained dividing the angular frequency by oM ; are reported in Table 4. The damping ratio behaviour, for the cantilever beam, is shown in Fig. 4. The solid line represents 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 a Fortran program), and with (+) for bar elements (the complex stiffness matrices were provided by the MATLAB model and then introduced into the MSC/NASTRAN by a Fortran program). The damping ratios gained with the different methods are practically equal. Very small differences (Fig. 5) have been instead found when the damping ratios have been derived for the first bending mode of a beam simply supported at its ends; the piezo devices were considered ideally bonded on the top and on the bottom of the middle finite elements (5th and 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 also obtained from the FRFs, achieved by a MATLAB program, by a least-squares fitting of the curve.
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Table 4 Real and imaginary parts of the piezo device elastic modulus and loss factor vs non-dimensional frequency r 10
R½Esh ðoÞ 10 I½Esh ðoÞ 109 Z 102
7
0.1
0.3
0.5
1.0
2.0
3.0
10.0
6.3105 1.0527 1.6682
6.3878 2.9264 4.5812
6.5126 4.2530 6.5304
6.8316 5.3163 7.7819
7.1506 4.2530 5.9478
7.2569 3.1898 4.3955
7.3527 1.0527 1.4317
× 10-3
6
Damping ratio ζ
5
4
3
2
1 0
34.8
69.6
104.4
139.2
174.1
208.9
243.7
278.5
313.3
348.1
Frequency (Hz)
Figure 4. Damping ratios, relative to the first bending mode of the cantilever beam. The piezo devices were in the 1st and 2nd finite elements from the clamped end.
A first example (Fig. 6) shows the FRF for the cantilever beam, limited to the first two modes. In this case a modal damping equal to z ¼ 0:001 has been added in order to take into 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 frequency range, the second mode presents a damping ratio higher than 1.0 103. The FRF gained when a resistive and inductive load with the optimal values chosen so as to fit oð1Þ n is presented in Fig. 7. In this case, since the piezo circuit behaves as a vibration absorber, the mode on which it is 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 effect is present in the other modes; the identified damping ratio relative to the second mode is
1109
DAMPING BY PIEZOCERAMIC DEVICES × 10
4.5
-3
4
3.5
Damping ratio ζ
3
2.5
2
1.5
1
0.5
0
86.6
173.2
259.7
346.3
432.9
519.5
606.1
692.6
779.2
865.8
Frequency (Hz)
Figure 5. Damping ratios of the simply supported beam. The piezo devices were positioned in the 5th and 6th finite elements.
10 -1
10
-2
-3
H(ω)
10
-4
10
10-5
-6
10
0
200
400
600
800
1000
1200
1400
1600
Angular frequency (rad/s)
Figure 6. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded with a resistor ‘tuned’ on the first bending mode.
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A. AGNENI ET AL.
Table 5 Natural frequencies and damping ratios for the first two modes of the cantilever beam (piezo with resistive load) Mode
Natural frequency (Hz)
Damping ratio
34.814 207.18
7.6532 103 2.3527 103
1st 2nd
-1
10
10
-2
H(ω)
10-3
10
-4
-5
10
10 -6
10
-7
0
200
400
600
800
1000
1200
1400
1600
Angular frequency (rad/s)
Figure 7. FRF magnitude of the cantilever beam, measured at its tip. The piezo devices have been loaded with a 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 with resistive and inductive load) Mode 1st split 2nd
Natural frequency (Hz) 32:717 36:760 347.83
Damping ratio 6:9264 102 4:8808 102 1.0089 103
practically equal to the structure mode only. The FRFs, achieved for the simply supported beam, are instead shown in Figs 8 and 9. The modal parameters identified from a least-squares fitting of the previous curves are reported 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 of damping is introduced into the second mode, because the piezo devices, positioned on the 5th and 6th finite elements, provide a low strain energy for that mode. On the contrary, for
1111
DAMPING BY PIEZOCERAMIC DEVICES 100
10-1
H(ω)
10-2
-3
10
10-4
-5
10
0
500
1000
1500
2000
2500
3000
3500
Angular frequency (rad/s)
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.
101
10
0
-1
H(ω)
10
-2
10
10-3
-4
10
10-5
0
500
1000
1500
2000
2500
3000
3500
Angular frequency (rad/s)
Figure 9. FRF magnitude of the supported beam, measured at its middle point. The piezo devices, at the 5th and 6th finite elements, have been loaded with a resistor and an inductor, whose optimal values were relative to the first bending mode.
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A. AGNENI ET AL.
Table 7 Natural frequencies and damping ratios for the first two modes of the simply supported beam (piezo with resistive load) Mode 1st 2nd
Natural frequency (Hz)
Damping ratio
86.576 347.83
5.0721 103 1.2071 103
Table 8 Natural frequencies and damping ratios for the first modes of the simply supported beam (piezo with resistive and inductive load) Mode 1st split 2nd
Natural frequency (Hz) 82:009 90:030 347.87
Damping ratio 5:7681 102 3:5312 102 1.0025 103
a resistive and inductive load, tuned on the first bending mode, the behaviour is similar to the one obtained for the cantilever beam (Table 8). As regards the commercial code MSC/NASTRAN, it is possible to introduce a frequency-dependent function, which represents the elastic properties of a viscoelastic material added to the structure, and then to derive the frequency response functions. Since that code only needs the frequency function mentioned above, it is possible to get the FRFs not only for the case of piezo with a resistive load, which actually behaves as a viscoelastic material, but also when the tuning circuit is considered [7]. Therefore, the damping introduced by piezoelectric elements with passive loads can be obtained, with completely acceptable errors, by using finite element models, made by the experimenters or by commercially available codes. The proposed approach, applied to simple structures, can be easily employed for complex aerospace structures, as carried out in [7], where the dynamic behaviour of a small glider wing has been studied. Besides, no difficulties have been encountered in obtaining, through a finite element code, the multimodal damping [17], achievable by the electrical circuit proposed and experimentally tested in [18].
4. CONCLUSIONS
Piezoceramic devices are more and more applied, especially in the aerospace field, for their peculiar characteristics: they, if used as passive dampers, are small, lightweight and efficient in comparison with the relative mechanical systems. Damping can be increased by using resistive loads, and the complex stiffness of the loaded piezo device results to be similar to the one presented by classic viscoelastic models. Unlike these last ones, it is very easy to put the maximum of the loss factor curve of the piezo device on the desired frequency}generally a natural frequency of a mode}in order to introduce the maximum damping into the structure.
DAMPING BY PIEZOCERAMIC DEVICES
1113
On the other hand, piezoceramic patches, applied to a structure with inductors and resistors as loads, practically behave as vibration absorbers and so, in many cases, they can profitably substitute these last mechanical elements. In particular the resistive and inductive load permits a very fine tuning on a structure mode. Therefore it works as a very effective damper on the tuned mode, with no influence on the others, and unlike the viscoelastic behaviour. In the paper, it has been shown how both the systems can be introduced into a mechanical structure, and evaluated by finite element models. Because of the frequency-dependent stiffness of the piezo devices, the overall eigenvalues ought to be derived by an iterative procedure but, for each frequency of interest, the relative complex constant value of the dynamic stiffness can be considered. The available results by this procedure are completely acceptable. Similar results can be gained by a different approach, based on finite element models, but requiring the evaluation of the frequency response function matrix. In conclusion, the mechanical models}for both the loads}are useful not only for simple structures, as the one presented in the paper, but also for more general systems, as the aerospace structures, that can be described by finite element models, in which the dynamic stiffness of the piezo elements is added to the stiffness of the host structure. The numerical examples show how good results can be achieved by this procedure and how simple is the use of the mechanical models of piezo devices in the study of structures when finite element models are employed.
ACKNOWLEDGEMENTS
This work has been supported by the MURST grants: Tecniche per Aumentare lo Smorzamento in Sistemi Spaziali and Metodi Attivi e Passivi per Aumentare lo Smorzamento in Strutture Spaziali.
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10. A. D. Poularikis (ed.) 1995. In The Transforms and Applications Handbook, Chapter 7, 463–629. Boca Raton, FL: CRC-IEEE Press. (L. S. Hahn) Hilbert transform. 11. B. J. Lazan 1968 Damping of Materials and Members in Structural Mechanics, 73–74. Oxford: Pergamon Press. 12. I. Perepechko 1975 Acoustic Methods of Investigating Polymers, pp. 39–44. Moscow: MIR. 13. A. Agneni, F. Mastroddi and S. Sgubini 1997 Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Roma, Vol. 3, 111–122. Causality of damping models. 14. J. P. Den Hartog 1985 Mechanical Vibrations, 93–105. New York: Dover. 15. A. Agneni and F. Mastroddi 1995 Atti del XIII Congresso Nazionale AIDAA, Roma, Vol. 2, 775–786. System vibrations with hysteretic damping: frequency and time domain behavior. 16. A. Agneni, 2000 Mechanical Systems and Signal Processing 14, 193–204. On the use of the gauss filter in modal parameter estimation. 17. A. Agneni and S. Sgubini 2001 Atti del XVI Congresso Nazionale AIDAA, Palermo, CD file 132. Multimodal damping by piezoceramic devices with passive loads. 18. J. J. Hollkamp 1994 Journal of Intelligent Material Systems and Structures 5, 49–57. Multimodal passive vibration suppression with piezoelectric materials and resonant shunts.