Structural-Damage Detection by Distributed Piezoelectric Transducers and Tuned Electric Circuits
SSTRUCTURAL-DAMAGE DETECTION BY DISTRIBUTED PIEZOELECTRICTRANSDUCERS AND TUNED ELECTRIC CIRCUITS
F. dell’Isola, F. Vestroni, S. Vidoli
Universita` di Roma ‘‘La Sapienza,’’ Dip. Ingegneria Strutturale e Geotecnica,Rome, Italy
A novel technique for damage detection of structures is introduced and discussed. It is basedon purely electric measurements of the state variables of an electric network coupled to themain structure through a distributed set of piezoelectric patches. The constitutive parametersof this auxiliary network are optimized to increase the sensitivity of global measurements—as the frequency, response functions relative to selected electric degrees of freedom—withrespect to a given class of variations in the structural–mechanical properties. Because theproposed method is based on purely electric input and output measurements, it allows foraccurate results in the identification and localization of damages. Use of the electricfrequency-response function to identify the mechanical damage leads to nonconvex optimi-zation problems; therefore the proposed sensitivity-enhanced identification procedurebecomes computationally efficient if an a priori knowledge about the damage is available.
Keywords:
Frequency response, auxiliary systems, health monitoring, localization
INTRODUCTION
The problem of structural-health monitoring is one of the most urgentengineering tasks associated with the high requirements and the severe oper-ating conditions imposed on contemporary advanced structures. This subjecthas received considerable attention in recent years in the literature on aero-space, civil, and mechanical engineering. There is no general, universalapproach that could be used to effectively solve this problem in any case;the efficiency of the method depends essentially on the specific structureunder consideration, the availability of suitable experimental tests, and thetype of occurring damage. Many approaches are based on a direct inspectionof the structure in the vicinity of damage; to this class belong methods basedon acoustic and ultrasonic measurements, thermal emissions, radiography,and others. Many of them require in addition an a priori knowledge of thedomain where the damage occurred and can be used to detect damageson or close to the structure surface. An additional requirement that ofteneliminates effective approaches from practical engineering applications is
Address correspondence to F. dell’Isola, Universita` di Roma ‘‘La Sapienza,’’ Dip. IngegneriaStrutturale e Geotecnica, Via Eudossiana 18, 00184 Rome, Italy. E-mail: [email protected]
Research in Nondestructive Evaluation , 16: 101–118, 2005Copyright
American Society for Nondestructive TestingISSN: 0934-9847 print/1432-2110 onlineDOI: 10.1080/09349840591003302
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 he exact and complete knowledge of the state fields, as displacements, stres-ses, temperatures, in the whole domain of a damaged structure [1].To detect the occurrence of damaged zones, many approaches havebeen adopted for the choices of both the forcing inputs and the analysis ofthe system response. Some authors (see Refs. 2–4 and the references therein)used the information collected in the frequency-response functions; others,see for instance Ref. 5, prefer the use of the time signals; a last tendency isto use wavelet transforms to analyze both the time and frequency contentsof the signal [6]. The application of wave-propagation analysis for detectingstructural damages dates back to the late eighties and early nineties when theuse of stationary waves and of frequency-response functions was already astandard practice. The literature on the subject is extremely wide; a reviewof the system-identification methods can be found in Ref. 1.Thus, there is still the need for nondestructive methods enabling damagedetection and identification in complex structures; a favorable methodshould be based on simple, practicable measurements. Among other possibi-lities, the measurements of natural frequencies (see for instance Refs. 2–4)and local measurements of displacement, strain, or stress values in selectedsites of a structure belong to the simplest solutions worthy of interest. Unfor-tunately such a choice, in spite of significant advantages, is also associatedwith important difficulties. As a matter of fact damage often results in localmechanical changes of the structural parameters and its early detection isnecessary before the severity of damage could lead to abnormal functioningand structural failures. On the other hand these local changes, especially inthe early stage, often have little influence on the mechanical global charac-teristics, as the natural vibration frequencies. The detailed knowledge ofdisplacements or other fields carries information valuable in damage identi-fication and can notably improve the situation but; as it was already men-tioned, its actual achievement is either not possible or too expensive (e.g.,measurements of displacement fields in complex structures). Even in the caseof few lumped external forces, when the determination of the external workis easily achieved through displacement measurements is selected points ofthe structure, both the eigenfrequencies and the extrernal work representglobal quantities and the information concerning the local changes of struc-tural characteristics is deeply ‘‘hidden’’ and not directly available. This iswhy usually global measurements are less sensitive to local moderate varia-tions—such the ones resulting from damages—of structural characteristics.To avoid the discussed difficulties, the basic problem to be solved is howto increase the sensitivity of selected global measurements up to a levelenabling effective damage detection and identification; this problem isaddressed with different perspectives in Refs. 7–10. In particular, the ideasproposed in Ref. 9 follow the observation that, by a proper modification ofenergy distribution in a structure, one can significantly magnify the effectsof damage; in the performed theoretical and experimental investigationson simple structural elements as beams and plates, the effect of local
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 tructural stiffness variations is enlarged by a factor of 10. This task can beachieved by coupling the main structure to auxiliary systems whoseparameters are tuned according to suitable optimal conditions; to this endflexible supports, concentrated masses, discrete vibration absorbers withvariable position and mechanical characteristics have been used and tested.In this article we propose to substitute these mechanical auxiliary systemswith purely electric devices: the coupling between the mechanical (mainstructure) and the electric (auxiliary electric system) subsystems is assuredby piezoelectric patches glued along the structure. Indeed in the past fewyears some novel electromechanical integrated systems have been introduced(see, e.g., Ref. 11) based on the concept of electric analog of a given structureand aimed to control its mechanical vibrations. These structures can belabeled as piezoelectro-mechanical (PEM) because the control of mechanicalvibrations is achieved through an electric net connecting a set of distributedpiezoelectric patches. Thus, a PEM structure is constituted by a structuralmember to be controlled, a set of actuators uniformly distributed on theconsidered structural member, and a suitable electric circuit including aselements the piezoelectric transducers and completed by optimally insertedimpedances. Preliminary encouraging results are reported in Ref. 12 wherea comparison between the presently proposed technique and a standardapproach, measuring the structural eigenfrequencies, is also provided.Because in the process of increasing the sensitivity we are led to use theelectric frequency-response function, the identification procedure involvesthe simultaneous global minimization (with respect to mechanical-damageparameters) and global maximization (with respect to the electric sensi-tivity-enhancing parameters of a nonconvex function. If an a priori coarseknowledge about the damage—especially on its localization—is availablethat sufficiently restricts the range of damage parameters, then the proposedsensitivity-enhanced procedure is computationally efficient; otherwise,the possible existence of multiple optimal points could require heavy com-putational efforts.
SENSITIVITY AND EFFECTIVENESS OF DAMAGE-EVALUATIONFUNCTIONS
Our approach is actually based on a parametric identification; let usbriefly summarize the analogies between the system-identification and thedamage-detection techniques. Usually in system identification methods,one measures the response O (cid:1) of a given system (i.e., with given actualvalues p (cid:1) of parameters) to a forcing input I (cid:1) : I (cid:1) ! p (cid:1) ! O (cid:1) ; I (cid:1) ! p ! O ð Þ Then, being able to compute the response O of the same system with a genericvalue p of its parameters, one seeks for the value p (cid:1) P that fits O to O (cid:1) STRUCTURAL-DAMAGE DETECTION 103
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 etter than others. Here P means the space of admissible parameters values.Thus, it is rather natural to formulate the system-identification problems asminimization problems of suitable functions: indeed, consider the functional E ð p ; I (cid:1) ; O (cid:1) Þ , mapping an admissible value of system parameters p II, andthe actual inputs I (cid:1) and system responses O (cid:1) , into real positive numbers suchthat E ð p (cid:1) ; I (cid:1) ; O (cid:1) Þ ¼ ; E ð p ; I (cid:1) ; O (cid:1) Þ > ; for p p (cid:1) ð Þ Clearly the identification of the actual value p (cid:1) is equivalent to find the globalminimum of E ( (cid:2) ; I (cid:1) ; O (cid:1) ) in P .In a very similar way, in damage-detection problems one measures theresponses O and O (cid:1) of the same system in two different instants character-ized by possibly different values p and p (cid:1) of the parameters.; the subscript 0and (cid:1) respectively mean the undamaged and actual, possibly damaged, situ-ation. The problem is to identify the variation D p (cid:1) : ¼ p (cid:1) (cid:3) p of the systemparameters to which the variation of the system response O (cid:1) (cid:3) O is amen-able. To this end one must be able to compute the variation O (cid:3) O of thesystem response associated to a generic variation D p : ¼ p (cid:3) p of the systemparameters. I (cid:1) (cid:3)! p (cid:3)! O ; I (cid:1) (cid:3)! p + D p (cid:1) (cid:3)! O (cid:1) ; I (cid:1) (cid:3)! p + D p (cid:3)! O ð Þ Hence also the damage-detection problems can be reformulated as minimi-zation problems of suitable functions, the choice of a distinguished functionbeing the selective criterion among different methods. Note that in this pro-cess the identification of the actual values ð p ; p þ D p (cid:1) Þ of the system para-meters is not strictly necessary because only the variations from a fixedreference state, namely the undamaged one, can be compared.The measured responses ð O ; O (cid:1) Þ can be, in general, modal quantities ortime histories of the system-state variables in selected sites of the structure;more often, measurements of the frequency response functions of these sitesare used. Thus, for instance, the well-established technique of damageidentification through eigenfrequencies measurements can be regarded asthe minimization problem for the function summing the squares of the differ-ences between the positions of the peaks of the frequency-response functions O and O þ D O (cid:1) while disregarding any other information about them.Much better results can be achieved considering functions that weigh moreinformation contained in the frequency-response functions, for instance,their values in several fixed frequencies. As remarked in the Introduction, thiscan be a difficult or expensive task when dealing with mechanical measure-ments, particularly in complex structures. However, let us explicitly remarkthat the same task is easily achieved when dealing with measurements onelectric systems.
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 ossible Choices for the Function E In the system identification, a crucial role is played by the choice of thefunction E ð p ; I (cid:1) ; O (cid:1) Þ to be minimized over the space P of admissible para-meters. Here the procedure for parametric identification introduced in Ref.13 is applied. To this aim, let the equation D ð x ; p (cid:1) Þ O (cid:1) ¼ I (cid:1) ð Þ describe the relation between the forcing vector I (cid:1) and the resulting responsevector O (cid:1) in a given experiment. In the frequency-response function vector O (cid:1) and in the forcing vector I (cid:1) both the mechanical and electrical compo-nents, that is, all the degrees of freedom of the electro-mechanical system,are listed. The matrix D ð x ; p (cid:1) Þ , depending on the frequency and on theactual value of the parameters p (cid:1) , can be obtained by finite-elements proce-dures or, in the case of simple structures, in exact form by spectral-elementsprocedures [14]; see the Appendix for further details. Because not allthe degrees of freedom of the system are observed, it is useful to considerthe following partition of O (cid:1) and I (cid:1) : O (cid:1) ¼ m (cid:1) ; n (cid:1) f g T ; I (cid:1) ¼ g (cid:1) ; h (cid:1) f g T ð Þ into the measured, m (cid:1) , and nonmeasured, n (cid:1) , components of the responseand into their dual quantities g (cid:1) and h (cid:1) . Accordingly Eq. (4) is reduced to ~ DD ð x ; p (cid:1) Þ m (cid:1) ¼ g (cid:1) (cid:3) H ð x ; p (cid:1) Þ h (cid:1) ð Þ with D ð x ; p (cid:1) Þ ¼ D mm D mn D nm D nn (cid:1) (cid:2) ; ~ DD ð x ; p (cid:1) Þ : ¼ D mm (cid:3) D mn D (cid:3) nn D nm ð Þ and H ð x ; p (cid:1) Þ : ¼ D mn D (cid:3) nn : Let us remark that even if the functional depen-dence of the matrix D over the parameter p could be linear, the reduction fromEq. (4) to Eq. (6) via Eq. (7) leads to a nonlinear dependence of the matrix ~ DD over the parameter p ; this fact, namely the impossibility of measuring all thedegrees of freedom of the system, will lead to nonconvex problems.Hence, the functional to be minimized over the admissible parametersspace P to identify the actual values p (cid:1) may be chosen as follows: E ð p ; m (cid:1) ; g (cid:1) ; h (cid:1) Þ ¼ X Kk ¼ ~ DD ð x k ; p Þ m (cid:1) ð x k Þ (cid:3) g (cid:1) ð x k Þ þ H ð x k ; p Þ h (cid:1) ð x k Þ (cid:3)(cid:3) (cid:3)(cid:3) ð Þ The sum over a set of frequencies x k is crucial to include, in the functional,information over a large frequency bandwidth. Clearly, because of Eq. (6),the function (8) vanishes when p ¼ p (cid:1) ; moreover E is continuous with STRUCTURAL-DAMAGE DETECTION 105
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 espect to the parameter vector p because it involves continuous functions of p . When p p (cid:1) , the balance equation (6) is not satisfied and the functionweighs the so-called unbalanced generalized forces. Required Properties of E : Main Difficulties and Proposed Solutions The previous considerations allow us to highlight the main obstacles con-nected to damage-detection or system-identification problems and eventu-ally to prompt efficient shortcuts. The main difficulties are as follows: a)both the responses and the applied forcing inputs, because of practical impe-diments, are restricted to a low number of sites: this limits the effectiveness ofthe functional delegate to ascertain the difference between the undamagedand damaged responses; and b) the measured responses represent globalknowledge in the sense that combines local contributions from the overallstructure; thus is usually difficult to extract local informations on the actualvalues of the parameters.From a mathematical viewpoint, this is tantamount to say that thefunctional E ( (cid:2) ; I (cid:1) ; O (cid:1) ) can manifest a low sensitivity to variations of theparameters, that is, E ð p ; I (cid:1) ; O (cid:1) Þ (cid:3) E ð p (cid:1) ; I (cid:1) ; O (cid:1) Þj j < e for p P ð Þ with e a positive number measuring the experimental sensitivity.To overcome the depicted drawbacks, it is here proposed to couple themain structure (ms), whose mechanical properties have to be detected, withan auxiliary electric circuit (aec). The coupling between the two subsystems(ms and aec) is ensured by distributing an array of piezoelectric transducersalong the structure. Therefore, for a suitable choice of the auxiliary electriccircuit the electric response to any kind of forcing inputs is influenced bythe mechanical constitutive properties, and one can detect structuraldamages through purely electric inputs and measurements. To this aim afundamental hypothesis, which is not specific to dealing with electric com-ponents, is the perfect knowledge of the electric constitutive parameters.The proposed augmented system resolves some of the aforementioneddifficulties:1. It allows us to easily measure the frequency-response functions of severalsites or, to be more exact, of several degrees of freedom of the electric cir-cuit; indeed, because the piezoelectric patches are supposed to be uni-formly distributed along the structure, the associated electric degrees offreedom are in one-to-one correspondence with the structural regionswhere they are glued.2. A proper choice of the function E ð p ; I (cid:1) ; O (cid:1) Þ satisfies the conditions (2).The chosen function will weigh the frequency-response amplitudes inseveral fixed values x k of the frequency.
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 . It sensibly reduces the value e of the experimental sensitivity, theright-hand term in Eq. (9). Indeed the electric voltages (or currents) canbe measured within tolerances much lower than the typical tolerancesof structural eigenfrequencies.4. Moreover, the constitutive parameters of the auxiliary circuit can be tunedto enhance the sensitivity of the chosen function E , namely the left-handterm in Eq. (9).This last property of the proposed auxiliary circuit deserves more atten-tion and, as will be shown in the following, is very useful: the set P of theconstitutive parameters of interest is the disjoint union of both the mechan-ical parameters of the main subsystem ( P ms ) and the electric parameters ofthe auxiliary subsystem ( P aec ). Suppose a further partition of PP ¼ P [ P ; P \ P ¼ Ø ð Þ into the set P of parameters to be identified and the set P of the parametersthat can be used to maximize the sensitivity of the function with respect tothe parameters in P . The actual value p (cid:1) of all the system parameters inthe experiment can be consequently written as p (cid:1) ¼ p (cid:1) ; p (cid:1) f g ; p (cid:1) is theactual value of the parameters to be identified and p (cid:1) is the actual valueof the remaining parameters. Because of the previous hypotheses, the para-meters in the set P are supposed perfectly known. If at least some of themcan be easily controlled (for instance the electric parameters in the circuit),then one can use these to enhance the sensitivity of the function E . In Fig. 1we exemplify this enhancement procedure by showing a possible plot andcontour plot for the function E : if a value p (cid:1) exists for which the restrictedfunction E ð(cid:2) ; p (cid:1) ; I (cid:1) ; O (cid:1) Þ resolves more sharply the actual value p (cid:1) then p (cid:1) has to be chosen in the experiment to increase the detection sensitivity.The entire process for the identification of the actual unknown parameters FIGURE 1.
Min–max process associated with the use of an auxiliary subsystem in damage detection.
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 p (cid:1) and p (cid:1) ) would consist of a sequence of maximization and minimizationproblems according to the following scheme: p ; k þ ¼ p to achieve max p P E ð p k ; p ; I (cid:1) ; O (cid:1) Þ p ; k þ ¼ p to achieve min p P E ð p ; p ; k þ ; I (cid:1) ; O (cid:1) Þ ð Þ The overall procedure is illustrated in Fig. 1 where the different maximiza-tion (AB, CD) and minimization (BC, DE) steps are drawn.The choice P ¼ P ms ; P ¼ P aec compels the use of the electric auxiliaryparameters to maximize the functional sensitivity to the mechanical para-meters but, obviously, it is not the only one conceivable. We explicitlyremark that the scheme (11) does not converge in every region of the para-meters and for every damage-detection function, several locally optimalpoints can in general exist; however it converges if the region is restrictedto be close enough to the global optimal solution. For this reason to avoidproblems with multiple optima, the use of a coarse estimation of the actualdamage parameters is compelled. DESCRIPTION OF THE OVERALL SYSTEM
The proposed method for structural-damage identification relies on thecoupling of the main structure with an auxiliary electric network. The energyis transformed from the mechanical to the electric form by means of a set ofpiezoelectric patches distributed along the structure. These kind of electro-mechanical devices has been initially proposed (e.g., Ref. 15) to controlstructural vibrations. In Fig. 2 a sketch of a PEM structure is shown; the piezo-electric patches and the passive electric network connecting them constitutean auxiliary system. More precisely, from the electric viewpoint, the piezo-electric patches behave as capacitors in parallel with current generators (thecurrent being proportional to the time rate of local mechanical strain); fromthe mechanical viewpoint, the same patches are local stiffeners at the end
FIGURE 2.
PEM structure: host structure, piezo patches, electric circuit, and forcing and sensing systems.
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 oints of which two opposite couples proportional to the electric voltage areapplied.The time evolution of these electromechanical systems are described bythe functions u ; u i ; i ¼ ; ; . . . ; N meaning respectively the displacement fields and the flux linkages (i.e., timeintegrals of the electric potentials) of the N nodes in the circuit. Here U and F means two suitable functional spaces respectively; to fix ideas one couldset U ¼ H ð D (cid:4) T ; IR M Þ and F ¼ H ð T ; IR Þ ; with D being the referencedomain of the structure and T the time axis. The evolution equations readas follows: A ð u Þ þ € uu (cid:3) g i ð x Þ _ uu i ¼ F b ij u j þ € uu i þ d ij _ uu j þ c ij G ð _ uu Þj x j ¼ l i ; i ¼ ; ; . . . ; N (cid:4) ð Þ where A is a linear self-adjoint differential operator, _ uu i the electric potentialin the i th of the N electric nodes, and ( F and l i the applied loads and theapplied current acting on the i th electric node. If the circuit is realized usingonly resistances, inductances, and transformers, the matrices b ij and d ij areguaranteed to be symmetric and positive defined. In Eq. (12) the superposeddot means time differentiation and summation over repeated indices. The lin-ear differential operator G , the matrix c ij , and the N -component vector g i ð x Þ account for the piezoelectric couplings. Here and in what follows theinvolved physical quantities have been made dimensionless following thestandard practice in designing experimental setups; as reference quantitiesthe beam length, the first natural period of the uncoupled mechanical sys-tem, and the electric flux linkage (cid:1) uu ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M B = C N ; p where M B is the beam massper unit length and C N the capacitance of the electric network per unitlength, have been chosen (see, for instance, Ref. 16).Once the number of piezoelectric patches is fixed, one can attempt anoptimization procedure on the coefficients b ij and d ij to maximize the sensi-tivity of the overall system to the even local changes of the mechanicalconstitutive parameters (in this case reflected in alterations of the operator A ). Different circuit topologies are possible to connect the piezoelectricpatches among them, the selected one being indicated either by optimalconditions found for b ij and d ij or simply by simplicity requests. For instancethe simplest one is realized by choosing a circuit with only one degree offreedom as done in Fig. 3.In this case the optimization procedure, to enhance the sensitivity tolocal changes of mechanical stiffness, is applied to the scalar parameters b and d , meaning respectively the impedance and resistance in the circuit. Itis useful to remark that this circuital scheme is a simple generalization ofthe 1-degree-of-freedom (DOF) shunt circuit by Hagood and von Flotow[17], a standard technique for vibration control. STRUCTURAL-DAMAGE DETECTION 109
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 n the Conclusions, the auxiliary electric circuit is chosen instead as theanalog circuit of the structure to be detected, namely an Euler–Bernoullibeam; thus the network is governed, in the homogenized limit, by afourth-order space derivative (see, e.g., Ref. 18 for details).The discussed procedure of damage identification is here applied to therelevant case of a simply supported Euler–Bernoulli beam. This choice repre-sents the simplest experimental setup that can be conceived to prove thefeasibility of the proposed sensitivity-enhancement method.
Transmission Line as Auxiliary System
As the auxiliary electric circuit connecting the piezoelectric patches dis-tributed along the beam the fourth-order transmission line synthesized in Ref.18 has been chosen. Two moduli of this connection are shown in Fig. 4; inthis case an inductance and a transformer are needed in each module;however, through the use of multiple channel date-acquisition boards, thesecircuits can also be simulated by computer.Once an homogenization procedure has been carried out, the governingequations for the proposed augmented electromechanical system, in case of
FIGURE 4.
Euler beam coupled to its electric analog circuit.
FIGURE 3.
Simplest auxiliary electric circuit: 1-degree-of-freedom shunt circuit.
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 onvanishing forces only at the boundary, can be written as ð a u Þ þ € uu (cid:3) c _ uu ¼ bu IV þ € uu þ d _ uu þ c _ uu ¼ (cid:4) for each regular interval in [0, 1]. Here, and in what follows, a prime meansthe space derivative and all the quantities are dimensionless. We explicitlyremark that the number of piezoelectric patches chosen should be largeenough to assure that the continuum model (13) is applicable; a rule ofthumb is to distribute at least four patches in the smallest wavelength con-sidered. Of course, if one deals with a limited number of piezoelectricpatches, the discrete model would be more reliable, but the continuummodel (13) has the advantage of immediately showing the main features ofthis electro-mechanical system. For instance, the comparison of Eq. (13a)with Eq. (13b) shows that PEM systems are based on internal resonancephenomena; to see this, simply choose a to be constant along the beamand b ¼ a .Recalling that _ uu physically represents the electric potential, note that theconstitutive relation for the dimensionless bending moment in this case reads M ¼ a u (cid:3) c _ uu ð Þ where a is the bending stiffness and c the coefficient resulting from the piezo-electric coupling. In a similar way, also the electric bending moment that is,the electric current expending power on _ uu , has two different contributions,namely l ¼ bu þ c _ uu ð Þ The system is subjected to the following boundary conditions: u ð ; t Þ ¼ u ð ; t Þ ¼ ; M ð ; t Þ ¼ M ð ; t Þ ¼ u ð ; t Þ ¼ u ð ; t Þ ¼ ; l ð ; t Þ ¼ l ð t Þ ; l ð ; t Þ ¼ l ð t Þ ð Þ These correspond to a simply supported beam and to an electricallygrounded transmission line that is subjected to a nonvanishing value of theelectric bending moment at its edges, l and l (Fig. 4); the electric signalused for testing are ideal Dirac deltas in all the following simulations.Because of Eqs. (14) and (15), the essential boundary conditions in Eq. (16)are expressed in terms of kinematical fields: u ð ; t Þ ¼ u ð ; t Þ ¼ ; u ð ; t Þ ¼ l ð t Þ = b ; u ð ; t Þ ¼ l ð t Þ = b ð Þ It has been shown [11] that the proposed circuital scheme guarantees amulti-modal coupling between the beam and the electric system for a propervalue of the line inductance; in other words, the tuning of the line inductanceallows for the simultaneous internal resonance of all the structural modes
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Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 ith all the electric modes. This circumstance generalizes the technique ofHagood to the case of multimodal control and leads to the most efficientcoupling between the mechanical and electric subsystems. Moreover, in thiscondition the system reveals to be very effective to electrically sense thestructural damage.
Detection and Localization of Structural Stiffness Reduction
With reference to the described system (Fig. 4), we identify the damagedprofile of the dimensionless bending stiffness a by means of purely electricmeasurements (the flux linkages u ; u ) and purely electric forcing inputs(the Dirac deltas l and l ) in the auxiliary network. Indeed, only the follow-ing quantities m (cid:1) ¼ f u ; u g > ; g (cid:1) ¼ f l ; l g > ; h (cid:1) ¼ ð Þ are measured and contribute to the functional E to be minimized, namely thefunctional chosen in Eq. (8). For the dimensionless bending stiffness a ð s Þ along the beam span, we seek for solutions in the form a ð s Þ ¼ a ; for s
62 ð x (cid:3) E ; x þ E Þ a d ; for s x (cid:3) E ; x þ E Þ (cid:4) ð Þ This hypothesis reduces the space of parameters P to a finite dimensionalspace and hints to a purely locally ð E << Þ damaged state of the beam;indeed d ; (cid:5) is the percentage loss with respect to the undamaged state, x E ; (cid:3) E Þ the abscissa where the damaged zone is centered, and 2 E therange of the damaged zone that is assumed to be known; refer to the grayshaded zone in Fig. 4. Thus, setting P ¼ fð d ; x Þ = d ; Þ ; x E ; (cid:3) E Þg ; P ¼ f b = b > g ð Þ means to seek for the correct value p (cid:1) ¼ f d (cid:1) ; x (cid:1) g of the unknown constitut-ive mechanical parameters, using the electric parameter b , constant withrespect to the beam abscissa s , to increase the functional sensitivity. Becauseof the simplicity of the geometry, the system is divided into three parts, eachwith constant parameters, and the spectral element method [14] is used tocompute the exact form of the matrix ~ DD ð w k ; p Þ for all the needed iteratesof the maximization–minimization process.The dimensionless quantities used in the numerical simulations are a ¼ ; c ¼ = ; E ¼ = ; d (cid:1) ¼ : ; x (cid:1) ¼ :
8, and d ¼
0. All of them corre-spond to technically feasible conditions and materials.To understand how the electric P -parameter b can affect the sensitivityof the function with respect to the P parameters, d and x , the locus of all thepoints satisfying E ð d ; x ; b Þ ¼ e is drawn in Fig. 5, with the experimental sen-sitivity fixed to the value e ¼ b ranges from 0.8 to 1.2. Note that the
112 F. dELL’ISOLA ET AL.
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 ightened regions in the plot correspond to values of b ’ ¼ a ; thiscondition, in case of an undamaged structure, would lead to a completemodal coupling of all the mechanical and electrical modes, because allthe mechanical and electrical eigenfrequencies of Eq. (13) would match,as previously discussed. In Fig. 5 the points internal to the surface representthe parameter values where the functional E ð d ; x ; b Þ is less than e or in otherwords the parameter values that are solution of the identification problemwithin a tolerance of e . Thus, the values over the axis P where the surfaceis shrunk correspond to optimal values of the parameter b because less vari-ance is allowed for the solutions in the P parameters. Figure 5 in itselfrepresents a comparison between the standard and the proposed enhancedevaluation procedure, the advantage of this last relying in the optimizationof the damage-detection functional sensitivity; this allows us to get a sharperresolution and as a consequences to obtain a damage characterization withinsmaller confidence ranges.In Figs. 6 and 7 for a value of the parameter b ¼ (cid:1) bb ¼
1, which resultsare close to the optimal one, the contour plots of the function E ð d ; x ; (cid:1) bb Þ FIGURE 5.
Points in P satisfying the condition E ð d ; x ; b Þ ¼ e . The parameter b ranges from 0.8 to 1.2 STRUCTURAL-DAMAGE DETECTION 113
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 ver the parameters space P ¼ ½ : ; (cid:5) (cid:4) ½ ; (cid:5) are drawn for two differentload conditions. There are several local minima in both cases but onlyone global minimum. However, in the two different load conditions, the FIGURE 6.
Level plot of log ð E ð d ; x ; (cid:1) bb ÞÞ on the space P for the load condition l ð x Þ ¼ ; l ð x Þ ¼ d ¼ : ; x ¼ : FIGURE 7.
Level plot of log ð E ð d ; x ; (cid:1) bb ÞÞ on the space P for the load condition l ð x Þ ¼ l ð x Þ ¼
114 F. dELL’ISOLA ET AL.
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 irections of maximum gradient near the global optimum are independent;this means that the two loads give independent information about the dam-age and can usefully be used in conjunction in the identification process.In both the load cases, disregarding any knowledge about these contourplots, the Nelder–Mead [19] simplex algorithm is successfully used froma set of four random points in P to find the global minimum for the givenvalue b ¼ (cid:1) bb of the electric parameter. On the other hand, because of thenonconvexity of the functional, the whole min–max process, described inEq. (11) and leading to the simultaneous identification of the damage para-meters and optimal tuning of the electric parameter b , always convergesonly if it is started in a convex neighborhood of the point ð d ¼ d (cid:1) ; x ¼ x (cid:1) Þ ; thus, the min–max procedure is efficient if at least an apriori estimate about the damage is given.The chosen damage level, d (cid:1) ¼ :
5, and its position, x (cid:1) ¼ :
8, lead tovariations D - of the first three natural frequencies ranging from 2 % to4 % , where the experimental sensitivity e - for natural frequencies measure-ments is about 1 % . On the other side the percentage variations D u of themeasured electric flux linkages occurring in the used function can rangefrom 0 to 100 % depending on the parameter b whereas the associatedexperimental sensitivity e u is at least 0.1 % . Therefore, the comparison ofthe ratios D -e - (cid:6) ! ; D ue u (cid:6) ! b ð Þ meaning the experimental confidences, shows an evident advantage of theproposed technique with respect to the standard method of damage detec-tion, where the variation of frequencies are measured. In particular whenthe electric subsystem is tuned with b ¼ D u = e u is about 10 ,which represents a very satisfactory result. CONCLUSIONS
It is shown how purely electric measurements of voltages in an auxiliaryelectric circuit allow for the detection of mechanical local damages of astructure. The coupled electro-mechanical system, with respect to thetechniques proposed in the literature, usually based on measurements ofmechanical eigenfrequencies, turns out to be: . More flexible, because the selected circuital topology can be easilyadapted to satisfy different optimal conditions; . More sensitive to local changes of mechanical parameters, because it isbased on purely electric measurements and the auxiliary circuit isoptimized to enhance the sensitivity; and
STRUCTURAL-DAMAGE DETECTION 115
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More easily tunable, because the electric parameters can be tuned in anadaptive way.The measurement of the frequency-response function of only two elec-tric degrees of freedom turned out to be sufficient to identify the level andlocalization of a concentrated damage. Further research efforts should bedevoted in determining a procedure of coarse localization of damage toavoid the difficulties caused by multiple optima. The development of a sen-sitivity-enhancement procedure based on the possibility of an independenttuning of all the inductances appearing in the auxiliary network couldbe another area of interest; in this way we expect to obtain additionalinformation on the position of the damaged zone while performing thesensitivity tuning.
APPENDIX: SPECTRAL ELEMENT ASSOCIATED WITH EQ. (13)
The spectral-element (SE) method, as formulated for instance in Ref. 5, isapplied to the Eqs. (13); this method is applied when the coefficients of a PDEare piecewise constants and is based on the partition of the domain of inter-est into suitable subdomains. Each of these becomes a spectral element andthe considered PDE is solved, representing its solution in terms of its eigen-functions in each element. According to the choice (19) of the dimensionlessbending stiffness a ð s Þ , we divide the support in three parts with constantsparameters, namely the intervals ½ ; x (cid:3) E Þ ; ð x (cid:3) E ; x þ E Þ and ð x þ E ; (cid:5) . Ineach of them, the Fourier transform of Eqs. (13) reads as follows: a h u IV (cid:3) x u (cid:3) i xcu ¼ ; bu IV (cid:3) x u þ i xdu þ i xc u ¼ ; (cid:4) h ¼ ; ; ð A1 Þ where a h is the constant value of the dimensionless bending stiffness in the h th interval. The SE method seeks for solutions ð u ð s ; x Þ and u ð s ; x Þ of Eqs.(A1) in the form: u ð s ; x Þ ¼ X Jj ¼ u hj exp k hj ð x Þ s (cid:6) (cid:7) ; u ð s ; x Þ ¼ X Jj ¼ u hj exp k hj ð x Þ s (cid:6) (cid:7) ; ð A2 Þ where the k hj ð x Þ functions are the J solution ( J ¼ h th interval:det a h k (cid:3) x (cid:3) i xc k i xc k b k (cid:3) x þ i xd (cid:1) (cid:2) ¼ ba h k þ ½ i dxa h (cid:3) ð b þ a h þ c Þ x (cid:5) k þ x (cid:3) i dx : ð A3 Þ
116 F. dELL’ISOLA ET AL.
Downloaded By: [Università degli Studi di Roma La Sapienza] At: 13:24 14 June 2010 ecause the dispersion relations are quadratic in k , they are easilysolvable, to get f k hj ð x Þg ¼ f ; (cid:3) ; i ; (cid:3) i g(cid:4) ð b þ a h þ c Þ x (cid:3) i dxa h (cid:7) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð b þ a h þ c Þ x (cid:3) i dxa h (cid:5) (cid:3) ba h ð x (cid:3) i dx Þ q ba h ð A4 Þ Accordingly the coefficients u hj and u hj are then chosen to satisfy theboundary conditions of Eq. (16), and the continuity conditions and the jumpconditions at the boundaries of the three intervals. The resulting linear systemof equations reads as Eq. (4) once the coefficients u hj and u hj are groupedinto the output vector O while the boundary applied forces and currentsare grouped into the input vector I .Note that, when using the SE method, the dimension of the resultingsystem (4) is relatively small; for instance in our case (two second gradientequations over three intervals with constant parameters) we have2 (cid:4) (cid:4) (3 þ ¼
16 degrees of freedoms. Using a FE approach to the sameproblem would lead to a much bigger problem to get a similar precision.However in SE approach the functional dependence of the dynamic stiffnessmatrix D SE ð x ; p Þ with respect to the frequency x is transcendent—throughthe exponential functions in Eq. (A2) and the k hj ð x Þ functions in Eq. (A4)—whereas in the FE method is polynomial: D FE ð x ; p Þ ¼ K ð p Þ (cid:3) x M ð p Þ þ I x C ð p Þ ð A5 Þ where K , M , and C are the stiffness, mass, and damping matrices respect-ively.In the process of minimizing the damage-detection functional, we needto evaluate E ð p Þ in many points of the parameters space P ; for each one ofthese evaluations, the dynamic stiffness matrix D SE ð x ; p Þ or D FE ð x ; p Þ mustbe assembled. Because, generally, the assembly of D FE ð x ; p Þ is faster thanthe assembly D FE ð x ; p Þ , the SE method should be preferred. However, thereare procedures where the damage-detection functional only accounts for theresonant frequencies [i.e., the singular frequencies of D ð x ; p Þ (cid:3) ]; in thesecases the relatively ease and velocity of the SE assembly are lost becausethe resonant frequencies are found as the roots of a transcendent function,namely the determinant of D SE ð x ; p Þ , whereas in the FE case (26) they arefound by standard eigenvalue problems. REFERENCES
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