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- 1. Passive Vibration AttenuationViscoelastic Damping, Shunt Piezoelectric Patches, and Periodic StructuresMohammad Tawfik
- 2. Piezoelectric Materials and Structures Piezoelectric Structures: A part of The Smart Structure FamilyContents1. Piezoelectric Materials and Structures ............................................................................................... 4 1.1. Piezoelectric Structures: A part of The Smart Structure Family .................................................. 4 1.2. Classification of Piezoelectric Structures ..................................................................................... 5 1.2.1. Structures with Surface-Bonded Piezoelectric Patches ........................................................ 5 1.2.2. Structures with Embedded Piezoelectric Laminas ................................................................ 6 1.2.3. Structures with Piezoelectric Fibres...................................................................................... 6 1.3. Applications of Piezoelectric Structures in Control ..................................................................... 8 1.3.1. Piezoelectric Sensor/Actuator Modeling .............................................................................. 8 1.3.2. Self-Sensing Piezoelectric Actuators ..................................................................................... 9 1.3.3. Passively Shunted Piezoelectrics......................................................................................... 10 1.4. Modelling of Piezoelectric Structures ........................................................................................ 15 1.4.1. The Electromechanical coupling of Piezoelectric Material ................................................. 15 1.4.2. Simplified 1-D model........................................................................................................... 15 1.4.3. A Bar with Piezoelectric Patches ......................................................................................... 17 1.5. Finite Element Modelling of Plates with Piezoelectric Actuators .............................................. 22 1.5.1. Displacement Function ....................................................................................................... 24 1.5.2. Strain-Displacement Relation ............................................................................................. 26 1.5.3. Constitutive Relations of Piezoelectric Lamina ................................................................... 27 1.5.4. Stiffness and Mass Matrices of The Element ...................................................................... 28 1.6. Performance Characteristics of a Plate with Shunted Piezoelectric Patches ............................ 30 1.6.1. Overview ............................................................................................................................. 30 1.6.2. Experimental Setup ............................................................................................................. 30 1.6.3. Synthetic Inductor ............................................................................................................... 31 1.6.4. Performance Characteristics ............................................................................................... 32 1.7. Appendices................................................................................................................................. 44 1.7.1. References and Bibliography ..................................................................................... 44 1.7.2. Constitutive model for 1-3 composites............................................................................... 52 1.7.3. Constitutive model for Active Fibre Composites ................................................................ 57Passive Vibration Attenuation 2
- 3. Piezoelectric Materials and Structures Piezoelectric Structures: A part of The Smart Structure FamilyThis book presents an introduction of different techniques used in passive vibration attenuation. Theaim of the book is to give the reader the most important tools needed to understand more details ofthe different subjects that may be found in other literature. The author prepared the book based onlecture notes prepared for a graduate course taught in Cairo University, thus, the book is a step bystep approach to the subjects discussed supported by simple computer based examples thatdemonstrate the different topics. It is intended that a reader can read through the book and learnwithout the extra support of an instructor or other literature.Passive Vibration Attenuation 3
- 4. Piezoelectric Materials and Structures Piezoelectric Structures: A part of The Smart Structure Family1. Piezoelectric Materials and Structures1.1. Piezoelectric Structures: A part of The Smart Structure FamilyPiezoelectric materials belong to a family of engineering materials that is characterized by having thecapability of transforming electric energy into strain energy and vice versa. The piezoelectricmaterials were first reported in the late 19th century, and all the research that was performed on itwas regarding their capability of generating electric charges on their surface when mechanical loadsare applied on them, that is known as the forward piezoelectric action or the sensing action.The piezoelectric materials also exhibit what is known as the reverse action, that is, when thepiezoelectric material is subjected to an electric field, they undergo mechanical deformations. This isalso known as the actuator action.Due to those two characteristics of the piezoelectric materials, they have been the centre ofattention of different researches that were concerned with the sensing and the control of motion ofstructures. The piezoelectric materials were embedded into the structures and connected withmonitoring device to detect motions in those structures. Those sensors were very useful especially indetecting damages in structures or indicating excessive vibration in different locations. On the otherhand, the actuators were placed on structures to impose controlled deformations as those utilizedby aircraft to modify the aerodynamic shape of the airfoils or for vibration control of differentstructural elements. A new type of piezoelectric materials was also introduced by embeddingpiezoelectric fibres in a matrix material to enhance the control or sensing characteristicsThe applications and theory of piezoelectric materials have been described in many review articles 1-5 . Wada et al.1, presented one of the earliest review articles. In that article, a classification ofadaptive structures is presented dividing these structures into 5 groups as shown in Figure .1. These 1groups include sensory structures incorporating sensors to monitor the dynamics or the health ofstructures, adaptive structures with attached or embedded actuator elements that influence thedynamics or the shape of the structure; controlled structures involving both sensors and actuatorstogether with a controller; active structures with control elements acting as structural elements; andfinally, intelligent structures which are active structures with learning elements. 1Figure .1. Adaptive structures framework as suggested by Wada et al. 1Crawley presented an overview of the general trends in the applications of intelligent structures 2and classified the requirements of an intelligent structure into four main categories; actuators,sensors, control methodologies, and controller hardware. Rao and Sunar focused their review on 3 the application of piezoelectric sensors and actuators to structure control. Park and Baz4 reviewedPassive Vibration Attenuation 4
- 5. Piezoelectric Materials and Structures Classification of Piezoelectric Structuresthe state of the art of the applications and development of active constrained layer damping (ACLD)technique. In their paper, a broad variety of applications and configurations of ACLD are showntogether with a variety of analysis methods.In a very comprehensive review, Benjeddou5 presented the different methods and the number ofpapers published in the area of vibration suppression using hybrid active-passive techniques(Figure .2). He classified the available literature according to two criteria; the modelling technique, 1and type of structural elements used. With the aid of sketches and tables, he was able to present aclear picture of the accomplishments, trends, and gaps in the development of active-passive controltechniques. 14 12 10 Number of Papers 8 Beams 6 Plates Shells 4 2 0 1993 1994 1995 1996 1997 1998 Ye ar of PublicationFigure .2. Number of papers published on hybrid active-passive damping treatments of structural elements. 1 5(Benjeddou )1.2. Classification of Piezoelectric Structures 1.2.1. Structures with Surface-Bonded Piezoelectric PatchesThis type of structures is the most common one among all piezoelectric structures. A patch ofpiezoelectric material is usually bonded to the surface of the structure usually for the purpose ofsensing motion or controlling motion. When the base structure vibrates, the bonded piezoelectricpatch will move simultaneously producing electric charges on the surface. Those charges arecollected by a conductive layer, usually of silver, and then allowed to pass through an electricconductor to a measuring device. This sequence is the sensing sequence.When electric potential is applied to the surface of the piezoelectric material it undergoes strain. Asit is bonded to the surface of a structure, it will simultaneously strain causing the whole structure tomove. This sequence is what is known as the actuating sequence.In both cases, the bonding material, usually epoxy, should withstand the sheer stresses that aregenerated between the piezoelectric patch and the surface of the structure. Figure .3 presents a 1sketch for a typical piezoelectric sensor-actuator-controller configuration.Passive Vibration Attenuation 5
- 6. Piezoelectric Materials and Structures Classification of Piezoelectric StructuresFigure .3. Non-Collocated sensor and actuator. 1 1.2.2. Structures with Embedded Piezoelectric LaminasIn many applications, piezoelectric patches/laminas are embedded under the surface of thestructure. This usually is needed in structures where the applications are sensitive to the outersurface shape like in aircraft. In this case, the piezoelectric material has less bending authority sinceit becomes nearer to the neutral surface, nevertheless, the sheer stresses that were concentrated onone surface are distributed on two. This definitely reduces the requirements on the bondingmaterial. 1.2.3. Structures with Piezoelectric FibresPiezoelectric materials have the highest coupling factor between the strain/stress in one directionand the electric potential/charge on surfaces in the same direction. In the previously mentionedconfigurations, the coupling is between stress/strain in the plain of the structure and the electriccharges/potential on the surfaces parallel to it. It was suggested to embed piezoelectric fibers in thedirection parallel to the application of the loads, 1-3 composites, or parallel to the direction of thestrain, MFC and AFC.The main penalties that are imposed by using piezoelectric sensors and actuators is that they arerelatively heavy and that the control action they offer is always equal in the two planar directionswhich restricts the control applications. The active fibre composites concept was introduced tominimize or eliminate both the above-mentioned back draws of the piezoelectric sensors andactuators.Passive Vibration Attenuation 6
- 7. Piezoelectric Materials and Structures Classification of Piezoelectric Structures1-3 PiezocompositesFigure .4. A sketch for 1-3 composites 1The modelling of the 1-3 piezocomposites drew much of the research attention due to theirapparent efficiency as sensors and actuators especially in the sound applications59,60. Theformulation of the constitutive equations of the piezoelectric fibre composites in general hasimposed a challenge on the researchers in the mechanics of materials field. Models have beendeveloped using three-dimensional finite element analysis were proposed61 and gave accurateresults compared to analytical models. Other models were proposed to calculate the effectivematerial properties such as the method of cells proposed by Aboudi62 which is an extension to theoriginal63 and modified64 method of cells.Smith and Auld65 presented a formulation for the constitutive equations of the 1-3 composites thatare suited for the thickness mode oscillations. Their model presented the composite materialparameters in terms of the volume fraction and the material properties of the constituentpiezoelectric ceramic and matrix polymer that is more or less a formulation similar to theconventional composite material constitutive equations (See Appendix A).Avellaneda and Swart66,67 studied the effect of the Poissons ratio of the piezocomposite material onits performance as a hydrophone. In the course of their study, they introduced the hydrostaticelectromechanical coupling coefficient and the hydrostatic figure of merit with a great emphasis onthe effect of the polymer matrix Poissons ration. They showed that the reduction of the matrixPoissons ratio greatly affects the performance and sensitivity of the overall hydrostatic sensor.Shields et al.68 developed a model for the use of the active piezoelectric-damping composites(APDC), which is based on the use of 1-3 composites. They applied their model for the attenuation ofacoustic transmission through a thin plate into an acoustic cavity using active control methods. Theresults obtained from their finite element model were validated with an experiment that verified theaccuracy of the model. They concluded that the use of hard matrix material for the APDC results inhigher sound level attenuation. Another important result was the ability to use APDC in theattenuation of low frequency vibrations.Passive Vibration Attenuation 7
- 8. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in ControlPiezoelectric Fibre CompositesFigure .5. a sketch of active fiber composites with interdigitated electrodes 1Recently, the attention was drawn toward applying the active fibre composites in the planardirection (Error! Reference source not found.). This configuration allows the control of bending andin-plane vibration and torsion (due to the non-orthotropic piezoelectric effect) simultaneously.Bent69 and Bent and Hagood70,71 introduced a constitutive model for active fibre composites (SeeAppendix B for more details about the constitutive equations), and applied it with the interdigitatedelectrodes71 which was introduced earlier by Hagood et al.72. Applying the interdigitated electrodesto piezoelectric fibre composites allowed the use of the higher electromechanical couplingcoefficient d33 which in turn provided higher control authority in the plane of actuation.The piezoelectric fibre composites have not yet been introduced to many applications, though,McGowan et al.73 have introduced the concept of using the active composite for the twist control ofrotor blades, and Goddu et al.74 applied it to the control of sound radiation from a cylindrical shell.Bent and Pizzochero75 studied the different factors affecting the manufacturing and performance ofthe active fiber composites. They demonstrated their effectiveness with applications to helicopterrotor blade harmonic control, tail buffet load alleviation, and torpedo silencing.1.3. Applications of Piezoelectric Structures in Control 1.3.1. Piezoelectric Sensor/Actuator ModelingIn a review paper, dedicated to piezoelectric sensors and actuators, Chee et al. presented a 6classification of the different mathematical models that simulate the dynamics of these controlelements. Linear as well as non-linear piezoelectric constitutive equations have been discussed.Emphasis has also been placed on PZT ceramics, PVDF layers, piezoelectric rod 1-3 composites,piezoelectric fibre composites, and inter-digitated electrode piezocomposites. Crawley and de Luis7 presented an analytical model for the piezoelectric sensors and actuators thatare either surface-bonded or embedded in the structure. The model is limited to Euler-Bernaullibeams and ignored the variation of strain in the piezoelectric material by assuming relatively thinpiezoelectric layers (Figure .6). It was concluded that the use of segmented actuators is more 1effective than the use of continuous ones.Hagood et al. presented a derivation of the equations of motion of an arbitrary elastic structure 8with piezoelectric elements coupled with passive electronics. They used a very important conceptwhen applying their equations, that is; the electro-dynamics of the piezoelectric material are ignoredwhen the material is used as an actuator and the effect of the piezoelectric material on the structurePassive Vibration Attenuation 8
- 9. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in Controlis ignored when it is used as a sensor. They developed a state-space model and applied it to beamsusing the Rayleigh-Ritz formulation. Their results were verified experimentally. The conceptsdeveloped in that paper are limited only to the case of thin actuators and sensors. But in the case ofthick piezoceramic patches attached on the surface of thin plates, ignoring the effect of thepiezoceramic actuator/sensor on the dynamics of the plate would certainly produce inadequateresults. 7Figure .6. Strain distribution in a beam with piezoelectric material: (a) surface attached (b) embedded . 1 Koshigoe and Murdock9 introduced a formulation for the sensor/actuator associated with platedynamics together with a shunted active/passive circuit. They then solved the equations of motionin the modal coordinates presenting a simplified analytical formulation for plates with piezoelectricelements. Their model is verified experimentally on a plate using an accelerometer as a sensor andsurface bonded PZT patches as actuators.In a most recent study, Vel and Batra 0 presented an analytical method for the analysis of laminated 1plates with segmented actuators and sensors. The Eshelby-Stroh formulation is used for the case ofplain-strain problem. The inter-laminar stresses for different boundary conditions are presented. 1.3.2. Self-Sensing Piezoelectric ActuatorsThe concept of self-sensing piezoelectric actuators is based on the simple use of one piezoelectricelement as sensor and actuator simultaneously instead of two separate elements. That conceptachieves two important goals; first, the reduction of the weight of the piezoelectric elementsinvolved in the structure. Second, it achieves a truly collocated sensor/actuator arrangement whichis preferred in control applications as it ensures the stability of the control system (see Figure .7, 1Figure .8, and Figure .9). 1 1Figure .7. Non-Collocated sensor and actuator. 1 Figure .8. Collocated sensor and actuator. 1Passive Vibration Attenuation 9
- 10. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in ControlFigure .9. Self-sensing piezoelectric actuator. 1Dosch et al.11 introduced a formulation for the self-sensing piezoactuator as a special case ofcollocated sensor/actuators pair. They suggested implementing a complementary circuit(Figure .10) to the piezoelectric sensor-actuator to enable the measurement of the sensor potential 1separately. They presented two different configurations for measuring the strain and the rate ofstrain. They verified the accuracy of their model with an experiment on the suppression of thevibration of a cantilever beam. (a) (b) Figure .10. A sketch of the circuit suggested by Dosch and Inman11 to measure the (a) rate of change of 1piezoelectric voltage and (b) voltage. Anderson et al.12 used similar models for the analysis of the behaviour of a self-sensingpiezoactuator. They converted the equations into state-space model and applied the model to acantilevered beam. They concluded experimentally validated the predictions of the models anddemonstrated the effectiveness of the self-sensing piezoelectric actuators. Vipperman and Clark 3 1extended their analysis toward the implementation of an adaptive controller. They used a hybridanalogue and digital compensator and implemented the model on a cantilever beam. Their resultswere verified experimentally.Dongi et al. 4 implemented the concept of self-sensing piezoactuators to the suppression of panel 1flutter. They used the principle of virtual work to derive a finite element model which is based on thevon Karman non-linear strain-displacement relation for a plate. They used different controlstrategies to ensure high robustness properties. 1.3.3. Passively Shunted PiezoelectricsThe concept of passive shunting is a simple one. As the piezoelectric material can be viewed as atransformer of energy, from mechanical to electric energy and vice versa, a part of the electricenergy generated by that transformer could be allowed to flow in a circuit that is connected to theelectrodes of the piezoelectric patch. The dissipation characteristics of the shunt circuit would,naturally, be determined by the electric components involved.Passive Vibration Attenuation 10
- 11. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in ControlThe most widely used shunt circuit is that consisting of an inductance and a resistance. That circuitwhen connected to the piezoelectric patch, acting like a capacitance, would create and RLC circuitwhich has dynamic characteristics analogous to mass-spring-damper system. If the resonancefrequency of the circuit is tuned to some frequency value, the circuit will draw a large value ofcurrent from the attached piezoelectric patch at that frequency, that current will be dissipated in theresistance in the form of heat energy; thus, the electromechanical system loses some of its energythrough that dissipation process.The concept of using the piezoelectric material as a member element of an electric circuit that hasdynamically designed characteristics was introduced as early as 1922 by Cady 5 for the radio 1applications. In a review article about shunted piezoelectric elements, Lesieutre16 presented aclassification of the shunted circuits into inductive, resistive, capacitive, and switched circuits. Heemphasized that the inductive circuits which include an inductor and a resistance in parallel with thepiezo-capacitor (Figure .11) are the most widely used circuits in damping as they are analogous to 1the mechanical vibration absorber. Inductive Resistive Capacitive SwitchedFigure .11. Configurations of the different shunt circuit. 1Hagood and von Flotow 7 presented a quantitative analysis of piezo-shunting with passive networks. 1They introduced a non-dimensional model that indicates that the damping effect of shunted circuitresembles that of viscoelastic materials (Figure .12, Figure .13, and Figure .14). They applied their 1 1 1model to a cantilever beam and verified the accuracy of the model experimentally. A drawback ofthe model stems from the fact that the piezoelectric patch with the shunt circuit is assumed to dampvibration even if it was placed symmetrically on a vibration node, thus contradicting the basicproperties of the piezoelectric patches as integral elements (Figure .15). 1Figure .12. Mechanical (physical) model of the piezoelectric patch with shunted circuit. 1 7 1Figure .13. Analogous spring-mass-damper model as suggested by Hagood and von Flotow . 1Passive Vibration Attenuation 11
- 12. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in Control 7 1Figure .14. Analogous electrical model as presented by Hagood and von Flotow . 1Figure .15. Sketch to illustrate the dissipation argument. 1Different studies 8- 0 investigated the use of passively shunted piezoelectric patches for vibration 1 2 damping using the technique introduced earlier by Hagood and von Flotow17. Law et al.21 presenteda new method for analyzing the damping behaviour of resistor-shunted piezoelectric material. Theirmodel is based on the energy conversion rather than the mechanical approach that describes thebehaviour of the material as a change in the stiffness (Figure .16). Two equivalent models are 1proposed including: an electrical model (resistance, capacitance, electric sources), and a mechanicalmodel (force, spring, damper). A two-degree of freedom experiment was set up to test the accuracyof the model, and the experimental results were in good agreement with the predictions of themodel.Figure .16. The piezoelectric material is used as an energy converter. 1Tsai and Wang22, 3 applied the concept of using active and passive control to simultaneously damp 2the vibration of a beam using piezoelectric materials as shown in Figure .17. The objective of their 1study was to answer four questions namely; 1- Do the passive elements always complement theactive actions? 2- If the active and passive elements do not always complement each other, shouldthey be separated? 3- Should the active and passive control parameters be selected simultaneouslyor sequentially? 4- How should the bandwidth of the active passive piezoelectric network (APPN)affect the design? Tsai and Wang presented an analytical formulation for the problem and thecontrol low derivation which is then discretized using the Galerkin method. They concluded that thepassive shunt not only provided passive damping but also enhanced the active control authorityaround the tuned frequency.Passive Vibration Attenuation 12
- 13. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in ControlFigure .17. A sketch of hybrid control for a cantilever beam. 1The extension of using shunt circuits for damping multiple vibration modes was also investigated.Hollkamp 4 presented an extension to the analysis of single mode damping formulation to cover 2multiple-mode damping by introducing extra circuits in parallel to the initial shunt circuit. He showedthat the attempt to damp more than one mode resulted in less damping for each mode than whendamping each separately. Nevertheless, the damping of the multiple modes proved to be effective.Wu 5 also investigated the damping of multiple modes using a different configuration of shunt 2circuits in which sets of resistance and inductance or capacitance and inductance connected inparallel are connected together in series (Figure .18). These circuits were designed to provide 1infinite impedance (anti-resonance) at the design frequencies. (a) (b) 4 2 5 2 Figure .18. Circuit configurations as suggested by (a) Hollkamp and (b) Wu . 1Recently, different attempts for broadband vibration attenuation were introduced using “Negative-Capacitance” shunt circuits 6-28. The realization and application of the circuit in vibration damping 2was also introduced by different patents 9- 1. The method has proven effective in damping out 2 3vibrations over a broadband of frequencies.As a more practical application of the shunt circuit damping, McGowan 2 utilized shunt circuit in 3damping out the aeroelastic response of a wing below flutter speed. She developed the structuralmodel based on the typical section technique and the aerodynamic model based on Theodorsen’smethod. She concluded that the passive control methodology is effective for controlling the flutterof lightly damped structures. Also experimental and analytical study was performed to investigatethe effect of using passive shunt circuits for the control of flow induced vibration of turbomachineblades 3. The study concluded the effectiveness of that technique in the attenuation of blade 3vibration.Zhank et al. 4 presented another application for shunted piezoelectric material by applying it to 3damping the acoustic reflections from a rigid surface. They used a one-dimensional model toinvestigate the effectiveness of the model and they concluded that it is a promising application. Theyalso proposed the use of negative capacitance for the same application.Passive Vibration Attenuation 13
- 14. Piezoelectric Materials and Structures Applications of Piezoelectric Structures in Control Warkentin and Hagood35 attempted the enhancement of the shunt circuit sensitivity by introducingnon-linear electric elements. They investigated the use of diode and variable resistance elements inthe shunt circuits and applied their model to develop a one-dimensional electromechanical circuitmodel. They concluded that the nonlinear shunt networks had a potential for providing significantadvantages over conventional piezoelectric shunts for structural damping.Davis and Lesieutre 6 used a modal strain energy approach to predict the damping generated by 3shunted resistance. They introduced a variable that measures the contribution of the circuit to theenergy dissipation. This variable depends on the strain induced in the piezoelectric material. Then,they applied the finite element method to determine the effective strain energy. Finally, theypresented their results in terms of the conventional loss factor and confirmed their resultsexperimentally.Saravanos 7 presented an analytical solution of the problem of plate vibration with embedded 3piezoelectric elements shunted to resistance circuit (Figure .19). The study used the Ritz method to 1solve the resulting coupled electromechanical equations. The paper presents a very good startingpoint for further development of analytical or numerical methods for the analysis of plates with shunted piezoelectric elements. Saravanos and Christoforou38 developed a model to investigate theresponse of a plate under low-velocity impact. The analysis presented is more rigorous than thepreviously introduced methods as it includes explicitly the circuit dynamics into the equations, thus,avoiding the problem introduced ealier by Hagood and von Flotow 7. 1Figure .19. Shunted piezoelectric material with composite structures. 1Park and Inman 9 compared the results of shunting the piezoelectric elements with an R-L circuit 3connected either in parallel or in Series. They developed an analytical model to predict thebehaviour of a beam with a shunted circuit. The predictions of the model are verifiedexperimentally. They noted that the amount of energy dissipated in the series shunting case isdirectly dependent on the shunting resistance, while in the parallel case, the energy dissipateddepends on the inductance and capacitance as well.Recently, Caruso 0 presented a comparative theoretical and experimental study of different shunt 4circuits. He incorporated the structural damping in his analysis which did increase the complexity ofthe analysis. However, he modelled the piezoshunted system using the traditional approach as aviscoelastic material.Passive Vibration Attenuation 14
- 15. Piezoelectric Materials and Structures Modelling of Piezoelectric Structures1.4. Modelling of Piezoelectric Structures 1.4.1. The Electromechanical coupling of Piezoelectric MaterialThe behaviour of the piezoelectric material, as mentioned before, is characterized by the couplingbetween the mechanical and the electric states. The constitutive relations of piezoelectric materialare presented in many publications76-77. In general, piezoelectric material have 6 components ofmechanical stresses and strains 1, 2, 3, 4, 5, 6, 1, 2,3,4,5,6, respectively, where thecomponents with subscripts 1 through 3 are the normal components while the ones with subscripts4 through 6 are the sheer components. In addition, each surface of the piezoelectric material haveits electric field E and its electric displacement D. E and D are in the direction of the surface. Theconstitutive relation of the piezoelectric material may be written as:D d E d S E Where the components are: 1 1 D1 E1 2 2 D D2 , E E2 , 3, 3 D E 4 4 3 3 5 5 6 6 s11 E E s12 E s13 0 0 0 E E E 0 0 0 0 d15 0 s12 s11 s23 0 0 0 0 s E E E 0 0 0 s s 0 0d 0 0 d15 S E 13 23 33 E d 31 d 31 d 33 0 0 0 0 0 0 s44 0 0 0 0 0 0 E s 44 0 0 s66 E 0 0 0 0 1 0 0 0 1 0 0 0 3 ‘E’ is the electric field (Volt/m), ‘s’ (small s) is the compliance; 1/stiffness (m2/N), ‘D’ is the electricdisplacement, charge per unit area (Coulomb/m2), is the electric permittivity (Farade/m) or(Coulomb/mV), dij is called the electromechanical coupling factor (m/Volt). 1.4.2. Simplified 1-D modelLet’s focus our attention on the case of one dimensional case. The stresses and strains will be takenas the ones in the ‘1’ direction, while the electric field will be that in the ‘3’ direction. We may thenreduce all the matrices and vectors into scalar quantities.Passive Vibration Attenuation 15
- 16. Piezoelectric Materials and Structures Modelling of Piezoelectric StructuresRecall that the electric displacement is the charge per unit area QD AAnd that the rate of change of the charge is the current 1 ID A Idt AsWhere ‘s’ is the Laplace parameter. Also, the electric field is the electric potential difference per unitlength VE tSubstituting in the constitutive relations, we get d 311 s11 1 V t A 33 sI Ad 31s 1 V tIntroducing the electric capacitance, we getI Ad 31s 1 CsVWhich can also be presented as the electrical admittance (reciprocal of the impedance)I Ad 31s 1 YVNow, if you focus on the case of open circuit (no current or constant electric displacement), theequation above may be written as Ad 31sV 1 YWhich may be used into the strain equation to get 2 Asd 311 s11 1 1 tYOr d31 21 s111 1 s11D 1 33 s11 Passive Vibration Attenuation 16
- 17. Piezoelectric Materials and Structures Modelling of Piezoelectric Structures DWhich indicates that the effective structure compliance s11 will be less (higher stiffness). While forthe case of short circuit (zero impedance or constant electric field) s11 s . EOn the other hand, when no mechanical strain is applied on the structure, we get the electricrelations as d2 1 31 V Y SVI Y 33 s11 Indicating that the effective admittance is less (higher impedance) 1.4.3. A Bar with Piezoelectric PatchesNow, let us consider the case of a bar with piezoelectric patches attached to both upper and lowersurfaces. In the case when the problem is static, we may have the piezoelectric patch in either astate of open circuit or open circuit. This produces the simple relations for the bar displacementdifferential equationWith the boundary conditions at any side will beWhere the subscripts ‘s’ stands for structure and ‘p’ stands for piezoelectric patches. The modulus ofelasticity of the piezoelectric patches will be in the case of open circuit and inthe case of short circuit. Also is a given value for the displacement and ‘P’ is a given value for t heend load.If an electric potential is applied on the patch, the problem may be described by the samedifferential equation, however the boundary conditions at the end of the piezoelectric patch will beHowever, for the bar with the shunted piezoelectric patch, the equations may be found from theHamilton’s principle. First we need to rewrite the constitutive relations such that the stress and theelectric voltage are the primary variables.Passive Vibration Attenuation 17
- 18. Piezoelectric Materials and Structures Modelling of Piezoelectric StructuresNow, we get:Writing down the relation for the total potential energy of the structure with the piezoelectric patch,we get:Substituting with the constitutive relations, we get:Expanding and rearranging the terms,Applying the variation principles to obtain the first variationPassive Vibration Attenuation 18
- 19. Piezoelectric Materials and Structures Modelling of Piezoelectric StructuresAs for the kinetic energy,Applying the first variation,The external work exerted on the structure is through the circuit that is shunted to the piezoelectricpatch, thus, we may write:Finally, applying the Hamilton principle which states thatApplying for each term, we get:Passive Vibration Attenuation 19
- 20. Piezoelectric Materials and Structures Modelling of Piezoelectric StructuresFinally, we may sum up the three terms to get:Passive Vibration Attenuation 20
- 21. Piezoelectric Materials and Structures Modelling of Piezoelectric StructuresSeparating the equation above into two terms each multiplied by the variation of one of thevariables, we get the space equationPassive Vibration Attenuation 21
- 22. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric ActuatorsSubject to the boundary conditionsAs for the electric equationNow, we have obtained two coupled partial differential equations in the bar deflection and theelectric displacement as the primary variables. It can be shown, that in the case of harmonicvibration and absence of electric displacement, when the excitation frequency becomes equal tothat of the electric circuit natural frequency, the mechanical displacement amplitude will essentiallybecome zero; this is analogous to the problem of the vibration absorber.Similar derivation for the equation of motion of a beam with piezoelectric patches can be performedand a similar conclusion will be obtained for the vibration absorber analogy.1.5. Finite Element Modelling of Plates with Piezoelectric ActuatorsIt has to be noted that the previously presented literature presented a wide variety of methods toanalyze structures with bonded piezoelectric elements. The different methods were applicable inspecial cases but lacked the generality that can be introduced by numerical methods. However,those analytical approaches paved the way for the development of numerical methods that could beof more practical use. In the following, an introduction is presented to the finite element modelsused for modelling piezoelectric sensors and actuators for different applications.Benjeddou 1 presented a comprehensive survey of the available literature on the finite element 4modelling of structures with piezoelectric elements. In that survey, he showed the trend ofincreasing interest in the field of structural control with piezoelectric elements (Figure .20). The 1common assumptions that are used in the piezoelectric modelling, as pointed out in that paper,were; linear variation of electric potential through thickness, poling direction along the thickness andonly longitudinal stress or strain could be induced by monolithic piezoelectric materials, and only thetransverse components electric field and displacement are retained.Passive Vibration Attenuation 22
- 23. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators 1 4Figure .20. Number of published papers involving finite element modeling of piezoelectric structures. (Benjeddou ) 1Tzou and Tseng 2 developed a finite element model for the sensors and actuators attached to the 4surface of plates and shells. The finite element they considered was for a thin piezoelectric solid withinternal degrees of freedom (DOF). Then, Hamilton principle is used to formulate the dynamicsproblem in the finite element form, and Guyan 3 reduction to condense the DOF’s associated with 4electrical potential. The time response of the system was calculated using the Wilson- method 4. 4Their results were obtained using two different control lows; constant gain velocity feedback andconstant amplitude velocity feedback, and the effect of the feedback gain was illustrated. Hwang and Park45 introduced a model for the plate elements with attached piezoelectric sensors andactuators. They used the classical plate theory and the Hamilton principle to develop their model.They introduced four-node quadrilateral non-conforming element. In their paper, they investigatedthe effect of different piezoelectric sensor/actuator configurations on the vibration control.Zhou et al. 6 extended the finite element model to cover nonlinear regimes using the von Karman 4non-linear strain-displacement relation and the principle of virtual work. The effects of aerodynamicand thermal loading were added as well. The controller was designed using the LQR method. Theequations of motion were transformed to the modal coordinates then cast into a state-space model.They concluded that the piezoelectric-based controller is effective in suppressing the panel flutter.Later, Oh et al. 7 presented a formulation for the post-buckling vibration of plates. Their model was 4developed using the layer-wise plate theory. In their study, they investigated the phenomena ofsnapthrough.Liu et al. 8 developed a finite element model for the control of laminated composite plates 4containing integrated piezoelectric sensors and actuators, rather than attached piezoelectricpatches. They built their model using the classical laminated composite plate theory and theprinciple of virtual displacement, then derived the equations for a four-node non-conformingelement. With the use of negative velocity feedback control scheme, they investigated the vibrationsuppression of a beam and a plate with different piezoelectric embedding configurations.Several attempts were made to develop finite element models that have higher accuracy byincreasing the polynomial order of the elements or by using higher order mechanical modelling toaccurately describe the mechanical behaviour of the structure. Further, higher order electricalmodels were used to accurately describe the non-linear electric field in the piezoelectric material.Bhattacharya et al. 9 developed a finite element model based on the Raleigh-Ritz principle to 4represent the dynamic behaviour of a laminated plate with piezoelectric layers. They used an eight-node isoparametric quadrilateral element with both structural and electrical degrees of freedom.They applied the first order shear deformation theory. In their results, they presented differentconfigurations of piezoelectric stacking, boundary conditions, and electric voltage application.Passive Vibration Attenuation 23
- 24. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators Hamdi et al.50 presented a finite element formulation for a beam element with piezoelectric laminasusing the Argyris’ natural mode method 1 for the first time. The method is characterized by being 5free of shear locking problem. They applied the model for the shape control of a beam. Theyconcluded that the proposed formulation is effective in reducing the computational effort with highaccuracy results. Zhou et al. 2 presented another development in the finite element models by 5introducing a higher order potential field that should accurately describe the field in thepiezoelectric elements. While Peng et al. 3 introduced the third order shear theory to their finite 5element model to increase the modelling accuracy.Kim and Moon 4,55 presented, for the first time, a finite element formulation for piezoelectric plate 5elements with passively shunted circuit elements that incorporated the electric circuit dynamics.They used the Hamilton’s principle to derive the non-linear finite element model. The electric degrees of freedom of an element were presented as one per node54 or one per element55. Theyapplied their model for the prediction of plate behaviour subjected to aerodynamic loading (panel-flutter). Their model was based on the von Karman non-linear strain-displacement relations. Theycompared the results obtained from an active control model using LQR method with those obtainedfrom a passive RL circuit. They concluded that, the suppression using the passive control in not morethan that obtained using active control. However, the need of controller, power supplies, andamplifiers for the active control case would reduce its efficiency compared to the passive elementsthat only require the addition of a resistance and an inductance.Saravanos 6, 7 presented a formulation for the finite element problem of a composite shell with 5 5 piezoelectric laminas. He proposed the “Mixed Piezoelectric Shell Theory” 56 (MPST) that utilizes thefirst order shear theory for the displacement and the discrete-layer approximation for the electricpotential. He used the Love assumption for shallow shells (radius is much larger than thickness). Themodel he developed was for an eight-node curvilinear shell element. The model is applied todifferent cases of composite layouts and geometric boundary conditions and concluded that themodel is accurate in predicting the dynamics of the shells. Further; he included a passively shuntedcircuit to damp out the vibration of the shells 7. Meanwhile, Chen et al. 8 presented a similar finite 5 5element formulation but for thin shell elements which presents a special case of the formulationpresented by Saravanos.Later Tawfik and Baz127 presented an experimental and finite element study of the vibration of plateswith piezoelectric patches shunted with LR circuits. The study introduced, for the first time, aspectral finite element model for the plate vibration and emphasised the effectiveness of theshunted piezoelectric patches in damping the vibration as well as localization effects when usingseveral ones. On the other hand, Tawfik128 presented the spectral finite element model andcompared it to the performance of different other models for plate vibration and confirmed,numerically, that 4 and 9-node C1 elements were adequate for the modelling of the problem.In the following subsections, the derivation procedure of the finite element model with any numberof nodes and shunted piezoelectric patches will be presented. 1.5.1. Displacement FunctionThe numerical construction of the propagation surfaces, which will be introduced later, requires highorder elements [ 7]. Thus, a 16-node element is considered (Figure .21), with 4 DOF per node 9 1which provides a full 7th order interpolation function.Passive Vibration Attenuation 24
- 25. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric ActuatorsFigure .21. Sketch of the 16-node element. 1The transverse displacement w(x,y), at any location x and y inside the plate element, is expressed by w( x, y) H w a (1)where H w is a 64 element row vector and {a} is the vector of unknown coefficients. For the plateelement under consideration, the bending degrees of freedom associated with each node are w w H a x w 1 w H w, x a2 H (2) y w, y 2 w H w, x , y a64 xy where Hw,i is the partial derivative of Hw with respect to i. Substituting the nodal coordinatesinto equation (13), the nodal bending displacement vector {wb} is obtained as follows, wb Tb a (3) w1 w1 H w 0,0 x H w, x 0,0 w1 y H w, y 0,0 where wb w [Tb ] (4) H w, x , y 0,0 2 & 1 xy 2 w16 H w, x , y a / 3,2b / 3 xy From equation (14), we can obtain a Tb 1wb (5)Substituting equation (16) into equation (12) givesPassive Vibration Attenuation 25
- 26. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators w( x, y) H w Tb wb N w wb 1 (6)where [Nw] is the shape function for bending given by N w H w Tb 1 (7)Similarly, the electric displacement associated with the piezoelectric patch could be written in theform D( x, y) H D b (8)where H D is a 16 element row vector with its terms resulting from the multiplication of two 3rdorder polynomials in both x and y-directions and {b} is the vector of unknown coefficients.Substituting the nodal coordinates into equation (19), we obtain the nodal electric displacementvector {wD} in terms of {b} and following the same procedure as for the mechanical degrees offreedom, we get, D( x, y) H D TD wD N D wD 1 (9)where [ND] is the shape function for electric displacement given by N D H D TD 1 (10) 1.5.2. Strain-Displacement RelationConsider the classical plate theory, for the strain vector {} can be written in terms of the lateraldeflections as follows x y z (11) xy where z is the vertical distance from the neutral plane and { } is the curvature vector which can bewritten as, 2w 2 2 x w 2 Cb { a } (12) y2 w 2 xy wherePassive Vibration Attenuation 26
- 27. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators Hw , xx Cb H w, yy (13) 2 H w, xy Substituting equation (17) into equation (23), gives Cb Tb 1{wb } Bb { wb } (14)where Bb Cb Tb 1 (15)Thus, the strain-nodal displacement relationship can be written as z{ } z Bb wb (16) 1.5.3. Constitutive Relations of Piezoelectric LaminaThe general form of the constitutive equation of the piezoelectric patch are written as follows x x E y Q e y T (17) xy e xy D E where, x , y , xy are the stress in the x-direction, stress in the y-direction, and the planar shearstress respectively; x , y , xy are the corresponding mechanical strains; D is the electricdisplacement (Culomb/m2), E is the electric field (Volt/m), e piezoelectric material constantrelating the stress to the electric field, is the material dielectric constant at constant stress(Farad/m), and Q is the mechanical stress-strain constitutive matrix at constant electric field. EQE is given by, E E 1 2 12 0 E QE E 0 1 12 2 E 0 0 21 where E is the Young’s modulus of elasticity at constant electric field, and is the Poisson’s ratio.Equation (28) can be rearranged as followsPassive Vibration Attenuation 27
- 28. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators x x E y Q ee T e y (18) xy e xy T E D x x D or y Q y eD (19) xy xy x E e y D Tand (20) xy 1where . 1.5.4. Stiffness and Mass Matrices of The ElementThe principal of virtual work states that U T W 0 (21)where is the total energy of the system, U is the strain energy, T is the kinetic energy, W is theexternal work done, and (.) denotes the first variation.The Potential EnergyThe variation of the mechanical and electrical potential energies is given by U dV D EdV T (22) V Vwhere V is the volume of the structure. Substituting equation (30) and (31) into equation (33) gives, U z T QD z eD dV D eT z D dV (23) V VSubstituting from equations (20) and (27), we get, U z B w Q z B w e N w dV T D b b b b D D V (24) N w e z B w N w dV T T D D b b D D VThe terms of the expansion of equation (35) can be recast as followsPassive Vibration Attenuation 28
- 29. Piezoelectric Materials and Structures Finite Element Modelling of Plates with Piezoelectric Actuators z B w Q B w dV w k w , 2 T D T b b b b b b b V z B w e N wD dV wb T kbD wD , T b b D V N V D wD T eT z Bb wb dV wD T k Db wb wD T kbD T wb ,and N V D wD T N D wD dV wD T k D wD ;where [kb] is bending stiffness matrix, [kbD] is bending displacement-electric displacement couplingmatrix, and [kD] is the electric stiffness matrix.The Kinetic EnergyThe variation of the kinetic energy T of the plate/piezo patch element is given by, 2w A T w h 2 dA t (25)where is the density/equivalent density and h is the thickness of the element. The above equationcan be rewritten in terms of nodal displacements as follows 2w dA h wb T N w T N w wb dA wb T mb wb A w h 2 t A (26)where [mb] is the element bending mass matrix.The external workThe variation of the external work done exerted by the shunt circuit is given by W DLqdA (27) Awhere A is the element area, L is the shunted inductance, and q is the charge flowing in the circuit.But, as the charge is the integral of the electric displacement over the element area; then equation(38) reduces to, W DdA LDdA (28) A ASubstituting from equation (20), gives W wD T N D T dA N D LwD dA (29) A APassive Vibration Attenuation 29
- 30. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric Pwhich can be recast in the following form, W wD T mD wD (30)where [mD] is the element electric mass matrix.Finally, the element equation of motion with no external forces can be written as mb 0 wb k b kbD wb 0 0 mD wD k Db k D wD 0 (31) 1.6. Performance Characteristics of a Plate with Shunted Piezoelectric Patches 1.6.1. OverviewIn sections .4 and .5, the foundations required to handle the problem of plates with shunted 1 1piezoelectric patches were laid. Different finite element models were developed to handle thedifferent aspects of the problems.This section presents experimental performance characteristics of a plate with shunted piezoelectricnetworks. The experiments aim at monitoring the modal parameters of the plate using scanninglaser vibrometer (Polytec PI –V2000, Auburn, MA).The modal parameters considered are the natural frequencies and mode shapes. Theseexperimental parameters are used to validate the predictions of the finite element model presentedin section .5. 1The experiments aim also at monitoring the frequency response of the plate when it is controlledfirst with only two shunted piezoelectric patches which are arranged in a non-periodic manner. Thenthe frequency response is monitored when the plate is provided with nine shunted piezoelectricpatches organized in periodic manner over the plate surface.The obtained results are compared with those recorded when patches are not shunted. Suchcomparisons are essential to quantify, in general, the passive damping imported to the plate due tothe shunting. For the case of the periodic arrangement, the experiments aim at demonstrating thelocalization effects when the patches are non-uniformly shunted. Finally, the propagation surfaces ofa plate with partial coverage with a piezoelectric patch are going to be presented as a naturalexpansion of the models developed earlier. 1.6.2. Experimental SetupAn experiment was set up and conducted on a square plate clamped from all sides. The aluminium(6061 alloy) plate has the following properties: modulus of elasticity (E) 71 GPa, Poisson’s ratio ()0.3, density () 2700 kg/m3, length 0.507 m, and thickness 1 mm. Symmetric piezoelectric squarepatches (model T110-H4E-602 Piezo Systems Inc.) were bonded on two positions of the plate. Thepiezoceramic properties are: modulus of elasticity (E) 68 GPa, Poisson’s ratio () 0.3, density ()7800 kg/m3, length 0.073 m, thickness 0.27 mm, dielectric constant ( ) 2.37*10-8 Farad/m, andpiezoelectric coefficient (d) -320*10-12 m/V.Passive Vibration Attenuation 30
- 31. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric PThe plate is excited using an electro-mechanical speaker (model TS-W26C, 350W Woofer, Pioneer,Japan) (Figure .22) driven by a power amplifier (model PA7E, Wilcoxon Research), and the resulting 1response is measured with an accelerometer (model 357C10, PCB, Depew, NY). The excitationfunction and the accelerometer output signal are processed using spectrum analyzer (model SR780,SRS, Sunnyvale, CA) (Figure .23). 1Figure .22. A picture of the speaker used to excite the plate. 1Figure .23. A picture of the Spectrum analyzer. 1 1.6.3. Synthetic InductorThe values of inductance required to create resonating shunt circuit for the damping purposes arealways higher than those available commercially. Thus, synthetic inductors are used instead. Severalversions of these synthetic inductors are used in the various resonating circuits employed instructural damping24,39. The version used in this study is sketched in Figures 5.3 and 5.4 as presentedby Chen125. This configuration was selected after proving to be more stable in maintaining theinductance value it is tuned to when compared to another design suggested in literature39.Figure .24. A schematic of the synthetic inductor circuit. 1Passive Vibration Attenuation 31
- 32. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric PFigure .25. Shunting network used in present study. 1To tune and measure the performance characteristics of the circuit, it was connected to acapacitance, to present the piezo-patch, and the frequency response of the circuit was measuredusing the spectrum analyzer (Figure .26). 1Figure .26. Connection sketch for the circuit performance analysis. 1For the experiment purpose, the synthetic inductor circuit was realized using a 1458 dual amplifier ICwith R1=R3=R4=10 k, C=10 nF, and R2=50 k potentiometer, while the resistance connected inseries with the inductor was a 10 k potentiometer. 1.6.4. Performance CharacteristicsNumerical vs. Analytical PredictionA case study for the verification of the prediction of the finite element model was considered for aplate with different boundary conditions. These conditions include clamped from all sides (CCCC),cantilever (CFFF), clamped from two opposite sides and free from the other two (CFCF), and simplysupported from all sides (SSSS). The plate aspect ratio is 1 and Poisson’s ratio is 0.3. The modelpredictions of the frequency parameter, L / DP , where L is the plate length for a square plate,for different modes for the four different boundary conditions using a 7x7 uniform mesh arepresented in Table .6.1. The predictions are compared with the analytical predictions presented by 1Leissa124 and the results obtained from a finite element model using traditional polynomialinterpolation functions. (Bogner-Fox-Schmidt (BFS) C1 conforming element125)The presented results demonstrate the high accuracy of the developed finite element model. Amaximum relative error of 2.82% was obtained for mode (1,1) for the case of CCCC plate.Table .6.1. Comparison of numerical and analytical results for the frequency parameter of the four different test cases 1(Poisson’s ratio = 0.3) Spectral BFS Mode # Analytical Frequency % Error Frequency % Error 1,1 19.75 19.85 0.53 19.33 -2.12 1,2 49.32 49.37 0.11 49.11 -0.42 SSSS 2,2 78.99 78.89 -0.13 77.84 -1.46 3,1 98.74 98.74 0.00 101.35 2.65Passive Vibration Attenuation 32
- 33. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric P 3,2 128.31 128.00 -0.24 127.48 -0.65 4,1 167.81 168.23 0.25 169.53 1.03 3,3 177.63 176.32 -0.74 174.23 -1.91 1,1 35.11 35.79 1.93 33.70 -4.02 1,2 72.93 72.88 -0.07 70.27 -3.65 2,2 107.52 106.84 -0.63 104.75 -2.58 CCCC 3,1 131.65 131.65 0.00 132.18 0.40 3,2 164.36 162.17 -1.34 159.87 -2.73 4,1 210.33 210.80 0.22 211.07 0.35 3,3 219.32 215.25 -1.86 213.68 -2.57 1 22.17 21.68 -2.20 20.11 -9.27 2 43.60 42.84 -1.74 44.67 2.45 CFCF 3 120.10 117.81 -1.91 120.16 0.05 4 136.90 136.62 -0.21 N/A 5 149.30 145.50 -2.55 146.28 -2.02 1 3.49 3.40 -2.70 3.40 -2.70 2 8.55 8.36 -2.23 8.88 3.88 CFFF 3 21.44 21.94 2.34 21.16 -1.31 4 27.46 27.17 -1.07 29.00 5.59 5 31.17 30.56 -1.95 31.87 2.24Experimental Results with Two Piezo-PatchesDifferent experiments were conducted on the plate setup described in section .6.2. Two 1piezoelectric patches were bonded to the plate as shown in Figure .27. All the results showed very 1high effectiveness of the proposed damping circuit in reducing the amplitude of vibration of thetargeted frequency.Figure .27. A sketch of the plate with dimensions. 1For the purpose of comparison of the numerical and experimental models, the numerical model ismodified to accommodate the effect of the flexible boundary conditions. Also, the material dampingratio was tuned for each mode for the purpose of matching the experimental results. Theexperiments were conducted by exciting the plate using the speaker and a sine sweep functiongenerated by the analyzer. The damping was applied by attaching the central PZT patch to thesynthetic inductor (Figure .24) in series with a resistance. 1Passive Vibration Attenuation 33
- 34. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric P Contour plots of the different modes of vibration of the plate together with picture generated by the laser vibrometer for the shape of the plate as being exited at a frequency equal to that of the natural frequency are presented in Figures 5.7 through 5.14. Numerical ExperimentalFigure .28 A contour plot of mode (1,1) and picture of the same mode . 1 Numerical ExperimentalFigure .29 A contour plot of mode (1,2) and picture of the same mode . 1 Numerical ExperimentalFigure .30 A contour plot of mode (2,2) and picture of the same mode . 1 Passive Vibration Attenuation 34
- 35. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric P Numerical ExperimentalFigure .31 A contour plot of mode (1,3) and picture of the same mode . 1 Experimental NumericalFigure .32 A contour plot of mode (3,2) and picture of the same mode . 1 Numerical ExperimentalFigure .33 A contour plot of mode (4,1) and picture of the same mode . 1 Passive Vibration Attenuation 35
- 36. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric P Numerical ExperimentalFigure .34 A contour plot of mode (3,3) and picture of the same mode . 1 Numerical ExperimentalFigure .35 A contour plot of mode (4,2) and picture of the same mode . 1 0 -5 -10 Amplitude (dB) -15 -20 Numer. Open Circuit Numer. Closed Circuit -25 -30 80 85 90 95 100 105 110 115 120 125 130 Frequency (Hz) (a) Passive Vibration Attenuation 36
- 37. Piezoelectric Materials and Structures Performance Characteristics of a Plate with Shunted Piezoelectric P 0 -5 -10 Amplitude (dB) -15 -20 Exp. Open Circuit Exp. Closed Circuit -25 -30 80 85 90 95 100 105 110 115 120 125 130 Frequency (Hz) (b)Figure .36. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode 1(3,1). 0 -5 -10 Amplitude (dB) -15 -20 Numer. Open Circuit -25 Numer. Closed Circuit -30 170 175 180 185 190 195 200 205 210 215 220 Frequency (Hz) (a) 0 -5 -10 Amplitude (dB) -15 -20 Exp. Open Circuit -25 Exp. Closed Circuit -30 170 175 180 185 190 195 200 205 210 215 220 Frequency (Hz) (b)Figure .37. Comparison of (a) numerical and (b) experimental results for damped and undamped cases around mode 1(3,3).The modes targeted for damping were the (3,1) mode at 111 Hz and the (3,3) mode at 195 Hz.Figures 5.15 and 5.16 present a comparison between the experimental results obtained with theaccelerometer placed at the centre of the plate with those predicted by the developed finiteelement model. Reduction in the vibration amplitude of 7 dB was obtained at mode (3,1) and 12 dBat mode (3,3). The numerical model predicted 6 dB at mode (3,1) and 8 dB at mode (3,3).Theobtained results indicate close agreement between the numerical prediction and the experimentalresults for modes (3,1) and (3,3) for open circuit cases.Passive Vibration Attenuation 37

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