Vibrations of Crystal Plates with Surface Structures for Resonator and Sensor Applications

Transcription

1 Uiversity of Nebraska - Licol DigitalCommos@Uiversity of Nebraska - Licol Egieerig Mechaics Dissertatios & Theses Mechaical & Materials Egieerig, Departmet of - Vibratios of Crystal Plates with Surface Structures for Resoator ad Sesor Applicatios Na Liu Uiversity of Nebraska-Licol, liua@huskers.ul.edu Follow this ad additioal works at: Part of the Egieerig Mechaics Commos Liu, Na, "Vibratios of Crystal Plates with Surface Structures for Resoator ad Sesor Applicatios" (). Egieerig Mechaics Dissertatios & Theses.. This Article is brought to you for free ad ope access by the Mechaical & Materials Egieerig, Departmet of at DigitalCommos@Uiversity of Nebraska - Licol. It has bee accepted for iclusio i Egieerig Mechaics Dissertatios & Theses by a authorized admiistrator of DigitalCommos@Uiversity of Nebraska - Licol.

2 VIBRATIONS OF CRYSTAL PLATES WITH SURFACE STRUCTURES FOR RESONATOR AND SENSOR APPLICATIONS by Na Liu A DISSERTATION Preseted to the Faculty of The Graduate College at the Uiversity of Nebraska I Partial Fulfillmet of Requiremets For the Degree of Doctor of Philosophy Major: Egieerig Uder the Supervisio of Professor Jiashi Yag Licol, Nebraska November,

3 VIBRATIONS OF CRYSTAL PLATES WITH SURFACE STRUCTURES FOR RESONATOR AND SENSOR APPLICATIONS Na Liu, Ph.D. Uiversity of Nebraska, Adviser: Jiashi Yag This dissertatio is maily o the theoretical aalysis of vibratig crystal plates for acoustic wave resoator ad sesor applicatios. The frequecy ad mode effects of differet surface structures o either or both sides of the crystal plates are the mai cocers i this dissertatio. These effects are fudametal to the improvemet of existig acoustic wave devices, or to the desig of ew acoustic wave devices, especially ew sesors based o these effects. At first, two-dimesioal equatios of motio for a aisotropic crystal plate with two thi films o its surfaces are derived by reductio from the three-dimesioal equatios of aisotropic elasticity ad joiig separate equatios for the crystal plate ad the thi films through iterface cotiuity coditios. The thi films possess multiphysical effects, icludig iertia, stiffess, itrisic stress, electric ad magetic coupligs. The equatios derived are used to aalyze the perturbatio of the resoat frequecies of the crystal plate uder the ifluece of the surface films. After that, two relatively more complicated but really existig problems i acoustic wave devices are aalyzed. Oe is the effects of a surface film with ouiform thickess, either stepped or gradually varyig. The other is a crystal plate carryig a array of thi films which may be periodic or operiodic. I additio to the effects of surface films, vibratios of crystal plates with arrays of surface attached fibers i

4 extesioal or flexural motios ad surface attached particles are ivestigated to explore the possibility of usig crystal resoators for fiber ad particle characterizatios. Fially, the effects of material property variatio of thi film piezoelectric actuators are also studied.

5 ACKNOWLEDGEMENTS Firstly, I would like to take this opportuity to express my deepest gratitude to my Ph.D. supervisor, Dr. Jiashi Yag. Dr. Yag is a accomplished expert i theoretical piezoelectricity ad piezoelectric/acoustic wave devices. I feel very lucky to be able to work with Dr. Yag for four years. His guidace ot oly just remaied i the professioal research, but also exteded to may issues i life. I am greatly idebted to Dr. Yag s igeious ideas ad selfless help throughout my Ph.D. study. I would also like to express my sicere appreciatio to Dr. Zhaoya Zhag, Dr. Li Ta, Dr. Lixia Gu, ad Dr. Mig Ha for servig as my supervisory committee members. I particularly would like to thak Dr. Zhag ad Dr. Gu for takig your valuable time to read my dissertatio. My special thaks also go to those professors I worked with durig our summer visits to Chia i ad. They are Prof. Weiqiu Che of Zhejiag Uiversity, Prof. Yuatai Hu of Huazhog Uiversity of Sciece ad Techology, Prof. Feg Ji of Xi a Jiaotog Uiversity, Prof. Ji Wag of Nigbo Uiversity, ad Prof. Jixi Liu of Shijiazhuag Tiedao Uiversity. I would also like to thak Dr. Mig Ha for offerig me a opportuity to participate i his iterdiscipliary research. It was a valuable experiece for me to use the mechaics kowledge I leared from books to aalyze practical problems i fiber-optic sesig applicatios. I also thak Dr. Joseph A. Turer ad Dr. Jeffrey E. Shield for lettig me idepedetly teach the udergraduate course Statics for five semesters, which will also remai to be oe of my most valuable experieces.

6 Fially, I would like to dedicate this dissertatio to my parets, Jia Liu ad Yigtao Ya, for their edless love ad support. Without their costat ecouragemet ad support, I would ot have goe this far.

7 v Cotets Itroductio. Quartz Crystal Resoator ad its Sesor Applicatios..... Quartz Crystal Resoators..... Quartz Crystal Resoator Based Sesors Curret State of Kowledge Objectives of this Dissertatio Crystal Plates with Multiphysical Films Effects of Nouiform Films..... Effects of Film Arrays Effects of Fiber Arrays Effects of Particles Thi Film Piezoelectric Actuators....4 Outlie of this Dissertatio... Mechaics of Piezoelectric Plates 5. Three-Dimesioal Equatios of Piezoelectricity Two-Dimesioal Equatios for Piezoelectric Plates Zero-Order Theory for Extesio..... First-Order Theory for Extesio, Flexure, ad Thickess-Shear.... Quartz ad Other Piezoelectric Materials... 6

8 vi Crystal Plates with Multiphysical Films. Itroductio.... Two-Dimesioal Equatios for a Crystal Plate with Surface Films..... Equatios for the Crystal Plate..... Equatios for the Films Equatios for the Crystal Plate with Surface Films Reductio to Extesio ad Elemetary Flexure Biasig Fields Caused by the Multiphysical Films Frequecy Shift of the Fudametal Thickess-Shear Mode Examples of Applicatio Electric Field Sesig Magetic Field Sesig Electrode Stress Reductio to Pure Thickess-Shear of a Quartz Plate with a Mass Film o Top Summary Effects of Nouiform Films 5 4. Itroductio Films with Stepped Thickess Structure Goverig Equatios Free Vibratio Solutio Numerical Results... 6

9 vii 4. Films with Gradually Varyig Thickess Structure Goverig Equatios Free Vibratio Solutio Numerical Results Summary Effects of Film Arrays Periodic Arrays Itroductio Goverig Equatios Free Vibratio Aalysis Numerical Results Noperiodic Arrays Itroductio Goverig Equatios Fourier Series Solutio Numerical Results Summary Effects of Fiber Arrays Thickess-Stretch of Plate ad Extesio of Fibers Itroductio...

10 viii 6.. Plate Thickess-Stretch Motio Rod Extesio Plate-Rod Iteractio ad Frequecy Equatio Frequecy-Depedet Equivalet Mass Layer Approximate Frequecy Solutio Special Cases Thickess-Shear of Plate ad Flexure of Fibers Itroductio Plate Thickess-Shear Motio Beam Flexure Plate-Beam Iteractio ad Frequecy Equatio Frequecy-Depedet Effective Mass Layer Approximate Frequecy Solutio Special Cases Summary... 7 Effects of Particles 5 7. Itroductio Crystal Plate Particle Boudary Coditios ad Frequecy Equatio Approximate Frequecy Solutio Discussio ad Numerical Results...

11 ix 7.7 Summary Thi Film Piezoelectric Actuators 7 8. Itroductio Equatios for Piezoelectric Actuators Equatios for Elastic Plates Equatios for Elastic Plates with Piezoelectric Actuators Reductio to Elemetary Flexure without Shear Deformatio ad Rotatory Iertia Numerical Results ad Discussio Summary Coclusios ad Future Work Coclusios Crystal Plates with Multiphysical Films Effects of Nouiform Films Effects of Film Arrays Effects of Fiber Arrays Effects of Particles Piezoelectric Actuators Future Work Frequecy shift compesatio by surface films Thermal Effects of surface films... 6

12 x 9.. Fiite Elemet Aalysis Experimets Refereces 65

13 xi List of Figures. Crystal oscillator.... Quratz crystal resoator.... Thickess-shear mode of a quartz plate Gas sesor based o QCM A piezoelectric plate (a) Bulk of quartz. (b) AT-Cut quartz plate A crystal plate with asymmetric surface films..... Frequecy shift due to electric field Curvature due to electric field Frequecy shift due to magetic field Curvature due to magetic field Frequecy shift due to idetical electrode stress Curvature due to electrode stress of uequal thickess Top ad side views of a quartz plate with a ouiform film of stepped thickess Trapped modes for Case, uiform film, a =5 mm, R =.5% Trapped modes for Case (), ouiform film, a =.5 mm, R =%, mm, () a =5 () R =.5% Trapped modes for Case (), ouiform film, a mm, =.5 mm, R =4%, () a =5 () R =.5% The first mode from Cases -()... 6

14 xii 4.6 The secod mode from Cases -() The third mode from Cases -() The fourth mode from Cases () ad () The fifth mode from Cases () ad () A quartz plate with a ouiform film Distributio of the thickess-shear displacemet of the first six trapped modes i the order of icreasig frequecy. (a):, ) = (,), (b):, ) = (,), (c): ( (, ) = (,), (d):, ) = (,), (e):, ) = (, ), (f):, ) = (,)...7 ( ( ( ( 4. Effects of R o the first mode, =4,444.4 m -. f (,) =,67,7,,586,98, ad,555,5 Hz whe R =%, 5%, ad 7%. (a): x depedece; (b): x depedece Effects of o the first mode, R =5%. f (,) =,58,685,,586,98, ad,595,58 Hz whe =,5, 4,444.4, ad, m -. (a): x depedece; (b): x depedece A quartz plate for QCMs ad coordiate system The three modes show i a sigle QCM (a) f ; (b) f ; (c) f x depedece (a) ad x depedece (b) of the three modes i a array. Solid lie: the st mode. Dash-dot lie: the d mode. Dash lie: the rd mode Effects of R o vibratio distributio of the st mode alog x (a) ad x (b) i a array. Dash-dot lie: R =%. Solid lie: R =5%. Dash lie: R =7% Vibratio distributio of the first mode i a array A operiodic array of QCMs Vibratio distributio i a array of two QCMs with oe thicker tha the other...9

15 xiii 5.8 Vibratio distributio i a array of two QCMs with oe loger tha the other Vibratio distributio i a array of four QCMs A crystal plate carryig a micro-rod array A sigle rod ad coordiate system Normalized effective mass ratio versus frequecy Frequecy shift versus rod legth. Solid lie: from Eq. (6.4). Dotted lie: from Eq. (6.9) A crystal plate with a micro-beam array Notatio ad coordiate system for beam bedig Normalized effective mass ratio versus frequecy Frequecy shift versus beam legth. Solid lie: from Equatio (6.6). Dotted lie: from Equatio (6.66) A fiite particle o a crystal resoator: otatio ad free-body diagram Effective mass versus frequecy ( mr / I ) Shear stress uder a piezoelectric actuator Schemes for modulatig actuatio stress. (a) Segmetatio. (b) Multiple electrodes. (c) Varyig thickess. (d) Nouiform polig Piezoelectric actuators o a elastic plate ad coordiate system Actuatio stress T uder a homogeeous actuator. k =k =. x =.5 m. (a) terms i the series. (b) terms i the series Actuatio stress T uder a homogeeous actuator. (a) k =k =. m -. (b) k =k =9.6 m (a) Material property variatio. (b) Actuatio stress T... 54

16 xiv 8.7 (a) Actuatio stress T. (b) Deformed plate. k =k =.46 m (a) Actuatio stress T. (b) Deformed plate. k =k =94.48 m (a) Actuatio stress T. (b) Deformed plate. k =k =57.8 m

17 xv List of Tables. Compact matrix otatio Resoat frequecies showig covergece Effective mass... 5

18 Chapter Itroductio. Quartz Crystal Resoator ad its Sesor Applicatios.. Quartz Crystal Resoators A piece of crystal, like every other elastic body, has a series of resoat frequecies. However, due to its uique material properties, the vibratig crystal is extesively used i electroic devices as a reliable frequecy provider. The most familiar example may be the quartz crystal watches or clocks for time keepig, i which the elapse of time is measured by coutig crystal vibratio cycles. Vibratig crystals also provide frequecy stadards for telecommuicatio devices i which frequecy selectios ad operatios are eeded. The crystals for these applicatios are called resoators. I additio to resoat frequecies, the speed of a propagatig wave i a crystal ca also be used as a referece. I fact, resoat frequecies ad wave speeds are basic properties of a elastic

19 body. Figure. shows a crystal resoator i a circuit which is used to provide a stable frequecy. This circuit is kow as a oscillator []. Tuig Voltage Crystal Resoator Amplifier Output Frequecy Figure.: Crystal oscillator. Most crystals are strogly aisotropic ad as a cosequece ofte possess piezoelectric effect. For a piezoelectric resoator, the operatig mode or wave ca be directly excited electrically, which meas that the resoator ca be easily itegrated ito a circuit just as show i Figure.. The acoustic wave device ofte employs a crystal to provide or work with frequecies i the rage about 6 ~ 9 Hz. A crystal resoator is a typical acoustic wave device. I fact, vibratio modes or waves i a crystal ca be classified ito two types. The first type is bulk acoustic waves (BAW), which are all over the crystal. The secod type is surface acoustic waves (SAW), which ca oly propagate ear the crystal boudary because the particle displacemets dissipate quickly iside the crystal. Both bulk ad surface waves have bee used for acoustic wave devices. This dissertatio is oly cocered with the bulk acoustic wave devices. Quartz is the most commo material for BAW devices. Resoators made of quartz are called quartz crystal resoators (QCRs). Figure. shows the structure of a typical quartz crystal resoator, which cosists of a

20 circular quartz plate ad two metallic electrodes (usually made of gold) deposited o the top ad bottom surfaces []. Drivig voltage is applied o the electrodes i order to excite the desiged thickess vibratio of the quartz plate. Besides quartz, crystals of the lagasite family are researched for a possible replacemet of quartz i the future. TOP VIEW Quartz Plate SIDE VIEW Metallic Electrodes Figure.: Quartz crystal resoator. Bulk acoustic waves i a crystal ca be classified as high- ad low-frequecy modes. Take a crystal plate as a example. The extesio ad flexure modes of the plate belog to the low-frequecy modes. The modes are ofte see i traditioal structural egieerig, whose frequecies deped strogly o the legth, width, ad thickess of the plate. The typical frequecy is ofte less tha MHz. O the cotrary, the high-frequecy modes, e.g., thickess-shear ad thickessstretch, are modes whose frequecies are solely determied by the plate thickess (the smallest dimesio). I fact, the shape of quartz crystal resoators is ofte a thi plate. Quartz crystal resoators work i high frequecy modes ad provide frequecies i the rage from tes to hudreds of mega hertz. Figure. shows the fudametal thickessshear mode of a quartz plate. The arrows show the directio of particle displacemet. The

21 4 fudametal mode oly has oe ode alog the thickess ad is atisymmetric about x. The secod thickess-shear mode has two odes ad is symmetric about x, ad so o. x x Figure.: Thickess-shear mode of a quartz plate. Whe a crystal plate is i pure thickess-shear vibratio, motios of material particles are parallel to the surfaces of the plate. Particle velocities vary alog the plate thickess directio oly ad do ot have i-plae variatios (see Figure.). A typical quartz plate for resoator applicatios has a thickess of a few teths of a millimeter, a diameter of a few millimeters, ad a fudametal thickess-shear frequecy of the order of a few to tes of MHz. Except for oe case i Chapter 6, crystal resoators aalyzed i this dissertatio operate with thickess-shear modes... Quartz Crystal Resoator Based Sesors The resoat frequecies of quartz crystal resoators ca be affected by may evirometal effects such as temperature chage, biasig fields. The quartz crystal resoators are also sesitive to mechaical effects, such as pressure ad acceleratios, etc. People always wat to make a resoator which is immue to all the outside iflueces. But o the cotrary, for sesor applicatios, these effects ca be used to make various

22 5 acoustic wave sesors icludig thermometers, electric or magetic field sesors, force ad pressure sesors, accelerometers ad gyroscopes [-6]. If a thi film is deposited o the upper surface of a quartz plate, the resoat frequecies of the quartz plate become lower tha that of the same quartz plate without the film. Sauerbrey [7] showed that this frequecy shift is proportioal to the mass of the film deposited o its surface. Based o this discovery, quartz crystal resoators have bee widely used for moitorig thi-film depositio ad mass sesig [8-5]. This kid of device is called a quartz crystal microbalace (QCM). The mass sesitivity of QCM is due to the iertial effect of the thi films or mass layers o the plate surfaces, which teds to lower the resoat frequecies. If QCM is put i liquid, iteractio betwee fluid ad the plate surface also results i lowered resoat frequecies, thus QCM ca be used to detect the fluid viscosity ad desity [6-]. If the film is made of special material which adsorbs certai gas molecules or particles i the air, it is possible to use QCM to measure the desity of certai molecules or particles i the air whe the accumulatio of these molecules or particles o the film is eough to produce additioal iertial effect o the frequecy of the resoator [,]. The schematic drawig of a gas sesor is show i Figure.4. Based o the same idea, QCM is widely used as chemical ad biochemical sesors [-].

23 6 Chemically Selective Film Crystal Resoator Figure.4: Gas sesor based o QCM. The gas ad biochemical sesors itroduced above are already commercially available. People are also motivated to build a gas or chemical/biochemical sesor, which is able to detect differet target items at the same time by moutig multiple fuctioal films o a sigle crystal plate to form a QCM array [-5]. I fact, Chapter 5 i this dissertatio is devoted to this topic. It is ievitable that itrisic stress is itroduced i the thi film durig the depositio process. This itrisic film stress exerts shear force o the quartz plate surface ad thus causes a frequecy shift of the QCM. By detectig the frequecy shift, the QCM ca be used to measure the itrisic stress i the thi film [6-8]. Besides, QCM is also reported beig used to study the tribology i thi films [9,4].. Curret State of Kowledge Due to the aisotropy of crystals, the three-dimesioal theory of aisotropic elasticity or piezoelectricity, which is used to aalyze acoustic wave devices, usually presets cosiderable mathematical challeges. Exact solutios ca be foud i rare situatios

24 7 oly. I most cases, to obtai results useful for device applicatios, two approaches are effective. Oe is to develop approximate, low-dimesioal structural theories to simplify the problems so that theoretical aalyses are possible. This icludes two-dimesioal theories of plates ad shells, oe-dimesioal theories of beams ad curved bars, ad zero-dimesioal theories of parallelepipeds. Aother approach is to use umerical methods, such as the fiite elemet method. The approximate two-dimesioal theory for plates was first itroduced by Midli [4-49]. The thickess-shear vibratio of a crystal plate, which is the workig mode for quartz crystal resoators, is well described by the two-dimesioal equatios. I [44,46,47] the effects of surface films, like fully or partially covered electrodes, are also studied. A systematic derivatio of the two-dimesioal equatios for piezoelectric plates ca be foud i [5]. I fact, Readers may fid that Midli s work laid the theoretic foudatio of this dissertatio. The two-dimesioal equatios for the crystal plates are preseted i detail i the followig chapter. More sophisticated models, cosiderig both the iertial ad stiffess effects of the mass layer, were developed i [5,5], but these oe-dimesioal models oly deped o the plate thickess coordiate. There have bee a few attempts cosiderig the i-plae coordiates depedece of the thickess-shear modes [,4,5], but these results are ot systematic ad also theoretically rough. I fact, the iclusio of i-plae depedece of the thickess-shear modes ot oly causes eergy-trappig pheomea, but also proves the limitatio of the Sauerbrey equatio. The i-plae depedece is expected to be well treated by the Midli s twodimesioal plate equatios. There have bee a series of papers about ouiform mass

25 8 layers or ouiform electrodes o quartz crystal resoators [54-59], however, besides the thickess coordiate, these aalyses are depedig o oly oe i-plae coordiate. Therefore, as a importat part of this dissertatio, we expad the i-plae depedece from oe coordiate to two. I additio to stiffess ad iertia, the crystal resoator is also affected by may other iflueces due to the sesitivities of surface films. The effect of itrisic stresses i surface films was studied i [6]. The thermal effect was discussed i [6,6] for resoator applicatios. If the films are made of materials that are sesitive to electric or magetic fields, their ifluece o the crystal resoator was studied i [6-65]. It is also possible to use surface piezoelectric films to maipulate frequecy shifts i a crystal plate if the device is i acceleratig motio [66]. I fact, either electric/magetic field, temperature, or acceleratio produces extra stresses ad strais i the resoator, also kow as biasig fields. The theory for small fields superposed o a bias [67] ad oliear piezoelectric theory [68] are eeded to study the effects of biasig fields. Recall the QCM array itroduced earlier i this chapter, if there is a matrix of small fuctioal films deposited o a sigle crystal resoator, these films ca be treated as film array. Oe-dimesioal arrays of resoators [69], QCMs [-5] ad trasducers [7] were studied. It is worth poitig out that i [-5,69,7], the electrodes or mass layers o the quartz crystal resoators are all assumed uiform. Therefore, as aother importat part of this dissertatio, the film arrays are expaded to two-dimesios ad the variatio of film thickess is also icluded. I additio to periodic arrays, the vibratio behavior of operiodic arrays of QCMs is also studied.

26 9. Objectives of this Dissertatio Quartz crystal resoators ad sesors usually have certai surface structures which i may cases are surface attached thi films. These thi films may be metal electrodes for drivig the resoators electrically. They also may be mass layers deposited o the surface for mass sesig. If the surface films are made of chemically or biologically selective materials, they ca be used to attract ad detect certai molecules. This dissertatio is devoted to the theoretical aalysis of differet effects produced by surface films ad other surface structures o the crystal resoator. There are six objectives i this dissertatio... Crystal Plates with Multiphysical Films For the sake of sesor applicatios based o quartz crystal resoators, our first objective is set o the aalysis of surface films which possess multiple physical properties, like piezoelectricity, piezomagetism, ad residual stress. These multiphysical films may be used for electric or magetic field sesors. First of all, we eed to expad the existig two-dimesioal equatios for crystal plates with surface films. This is doe by first derivig two-dimesioal equatios for the crystal plate ad the surface films separately, ad the joiig them usig the iterface cotiuity coditios betwee the crystal plate ad the surface films to obtai equatios for the etire structure. The fial equatios are more geeral tha existig equatios i the literature. I additio to the film iertia ad stiffess, we also iclude the electric ad

27 magetic coupligs as well as itrisic stress i the film. The obtaied equatios lay the theoretical foudatio for aalyses i the rest of the dissertatio. The curret equatios are used to study frequecy shifts of thickess-shear modes i crystal plates caused by various evirometal effects. Whe a effect is actig aloe, its frequecy effect is studied for sesor applicatios... Effects of Nouiform Films Exact thickess-shear modes ca oly exist i ubouded crystal plates without edge effects, ad the films o the crystal plates have to be uiform. I real devices, however, due to the fiite size of the device, pure thickess-shear modes caot exist because of edge effects. I additio, i most cases the films o the crystal plates oly cover the plate surfaces partially ad sometimes the films have ouiform thickess. Therefore, i real devices, the operatig modes i the crystal plates have slow, i-plae variatios. These modes have bee referred to as essetially thickess-shear modes or trasversely varyig thickess modes. Whe the films have ouiform thickess, the differetial equatios goverig the thickess-shear vibratio of the crystal plates carryig the films have spatially varyig coefficiets. I this case theoretical aalysis faces difficult mathematical obstacles ad kow results are few ad scattered. As a importat objective of this dissertatio, we use the two-dimesioal plate equatios to study the i-plae mode variatios of crystal plate with ouiform films. The gradually varyig thickess of the film is described by stackig mass layers with piecewise stepped thickesses. This method allows us to obtai aalytical results showig the basic effects of ouiform films. The accuracy of this approach ca be improved by

28 icreasig the umber of piecewise layers. For the special ad useful case of films with a quadratic thickess variatio, we preset a solutio takig the gradually varyig thickess ito cosideratio directly without approximatig it by a stepped thickess variatio... Effects of Film Arrays The multifuctioal QCM metioed previously ca be cosidered as moolithic arrays of mechaically iteractig vibratio elemets i thickess-shear modes that vary from oe elemet to aother. The arrays may be periodic or operiodic. These complicated structures preset difficult mathematical problems whose solutios have rarely bee obtaied. As oe objective of this dissertatio, we study the behavior of the twodimesioal periodic array of quartz crystal microbalaces usig the two-dimesioal plate equatios. We also aalyze the oe-dimesioal operiodic array usig the threedimesioal equatios of aisotropic elasticity. With some commo approximatios we are able to obtai revealig solutios showig the basic behaviors of resoator or sesor arrays...4 Effects of Fiber Arrays We also study the effects of fiber arrays o the resoat frequecies of crystal plate resoators. To the best of our kowledge, these effects have ever bee studied experimetally or theoretically. Whe the crystal plates are i thickess-stretch or thickess-shear vibratios, the fibers stadig o the crystal surfaces udergo extesioal or flexural vibratios,

29 respectively. Their effects o the plate resoat frequecies deped o the geometric ad physical parameters of the fiber arrays i a relatively complicated maer, which may be explored for fiber characterizatio. Fiber arrays i flexural motios followig the crystal plate surfaces may be cosidered for use as ultrasoic brushes...5 Effects of Particles The ext objective is to study the frequecy effects of discrete particles o a crystal plate. Like surface films, particles also cause frequecy shift of the crystal plate, however the mechaics behid it are far more complicated tha films. If the particles are dese, they may be approximately cosidered as a mass layer o the crystal plate ad their frequecy effect ca be well predicted by the Sauerbrey equatio. However, if the particles are sparse, the effect of each idividual particle has to be cosidered. Based o differet applicatios, the idividual particle is also modeled differetly. I some cases, the particle may be simply put as a poit mass fixed o the plate; i other cases, the rotatig ad slidig motio of the particle eeds to be icluded; moreover, sometimes eve the elastic deformatio of the particle caot be igored. I biochemical sesig, the particles are immersed i fluids, which makes the problem far more complicated. The theoretical aalyses of the effects of particles o QCM are scarcely reported. I this dissertatio, we oly cosider the case i which the particles are elastically attached to the quartz crystal resoator ad are allowed to roll without slidig o its surface. As a prelimiary study, oly the most basic effects ad behaviors of dilute particles are ivestigated.

30 ..6 Thi Film Piezoelectric Actuators The last objective is about the effects of ohomogeeous piezoelectric films o a elastic plate. The result is useful for piezoelectric actuator desig. Other tha a vibratio problem, we study the static problem of surface actuators o a elastic plate. The actuators are actually thi piezoelectric films. Therefore the actuator is also a kid of surface structure. We wat to study the effects of the material property variatio of thi film piezoelectric actuators o the distributio of the actuatio shear stress. The actuatio shear stress is ofte cocetrated at the edges of the actuator. This cocetratio is uwated. A system of two-dimesioal equatios for the flexure ad shear of a elastic plate with symmetric piezoelectric actuators o the plate surfaces is derived. The effects of material property variatio o the actuatio stress are examied through a example..4 Outlie of this Dissertatio The theories ad equatios used i this dissertatio are preseted i Chapter. The followig six chapters are the mai body of this dissertatio, each chapter covers oe of the six objectives of this dissertatio. Fially, i Chapter 9, the overall coclusios are draw ad possible future work followed by this dissertatio is suggested. The six mai chapters share a commo article structure. At the begiig of each chapter, a more detailed backgroud itroductio is preseted, which is followed by the derivatio of the goverig equatios, after that the theoretical solutio to the goverig equatios is elaborated. I the ed, umerical examples are icluded to support the

31 4 theoretical work ad also illustrate importat graphic results which are difficult or impossible to be obtaied from mathematic expressios oly.

32 5 Chapter Mechaics of Piezoelectric Plates At begiig of this chapter, the geeral three-dimesioal theory of piezoelectricity is itroduced. Sice crystal resoators are desiged to work withi the liear rage of deformatio, oly the liearized equatios of piezoelectricity are preseted i this chapter. However, those liearized partial differetial equatios are still very difficult to be solved aalytically except for several cases with certai simplificatios, some of which will be discussed i this dissertatio. For device desig purpose, people tur to the fiite elemet method to solve the equatios umerically ad obtai a good approximate solutio. I fact, fiite elemet method has become a powerful tool alog with the developmet of computer techology, but it still has its disadvatages, firstly it is difficult to capture the useful workig modes of the resoator amog a large amout of umerical results; secodly, it is also difficult to aalyze the role each parameter of resoator plays if fiite elemet method is used.

33 6 I order to deal with the deficiecies of the fiite elemet method, we tur to the low-dimesioal theories to describe structures like beams, plates ad shells. I fact, Midli ad his studets established a hierarchy of low-dimesioal theories for the vibratios of piezoelectric structures. For example, the order of the three-dimesioal equatios ca be lowered to oe-dimesioal equatios to describe beams, ad lowered to two-dimesioal equatios for plates ad shells. Although the low-dimesioal approach is still a approximate method, it gives very accurate solutios for the piezoelectric structures cosidered i this dissertatio. The zero-order ad first-order twodimesioal equatios for piezoelectric plates are preseted i Sectio.. At the ed of this chapter, i Sectio., a brief itroductio is give about the quartz ad some other piezoelectric materials used i this dissertatio.. Three-Dimesioal Equatios of Piezoelectricity I this sectio, we summarize the three-dimesioal equatios i liearized theory of piezoelectricity [5,7]. Cosider a piezoelectric body i a three-dimesioal domai. Equatio (.) is the equatio of motio of the body i idex form, where a comma followed by a idex deotes a partial derivative with respect to the coordiate associated with the idex ad a superposed dot meas time derivative. Idex i, j, k rage over, ad. The summatio covetio for repeated idices is employed. T ij deotes stress tesor, ρ deotes mass desity, f j is compoet of body force vector ad u j is compoet of displacemet. T ij, i f j u j. (.)

34 7 The quasi-static Maxwell s equatio for the electric field iside the body is show i (.) where D i is the compoet of electric displacemet. This equatio is also referred to as equatio of charge, D i,. (.) i Equatios (.)-(.4) are the costitutive equatios for piezoelectric materials. The material costats iclude elastic costats c ijkl, piezoelectric costats e kij, electric permittivity costats ε ik. Besides, S ij is strai tesor ad E i is electric field itesity. T ij c S e E, (.) ijkl kl kij k D i e S E. (.4) ijk jk ik k Equatio (.5) shows the liearized strai-displacemet relatio, ad (.6) is equatio of electric filed potetial ϕ, S ij u u ). (.5) ( i, j j, i E i, i. (.6) I summary, (.)-(.6) are the three-dimesioal equatios of liear piezoelectricity. Substitutio of (.5)-(.6) ito (.)-(.4) ad the (.)-(.4) ito (.)-(.) yields the geeral goverig equatios for u i ad ϕ, c ijkl u. f u, (.7) k li ekij, ki j j e. (.8) kiju i, jk ij, ij The costitutive equatios (.)-(.4) are ofte writte i compact matrix otatio as, T p c S e E, (.9) pq q kp k D e S E. (.) i iq q ik k

35 8 This is doe by replacig the double idices ij, kl i (.) ad (.4) with sigle idices p ad q which rage from to 6 accordig to the Table. below. Table.: Compact matrix otatio ij or kl p or q The strai tesor i compact matrix otatio are defied as S S 4 S S,, S S S 5, S S, S S 6 S. (.) Besides the goverig equatios listed i (.)-(.6), the piezoelectric body is also subjected to boudary coditios, which ca be further classified ito two types, i.e. mechaical ad electrical boudary coditios. Let us discuss the mechaical boudary coditios first. For a tractio-free surface of the body, the boudary coditio is, (.) i T ij where i deotes the compoets of the uit ormal to the surface. For a displacemetfree surface, the boudary coditio becomes, u. (.) j As for the electrical boudary coditios, if the surface is electroded, the electric potetial should be specified For the rest of the surface without electrode,. (.4). (.5) i D i If composite materials are cosidered, the cotiuity coditios across the iterface betwee two differet materials should be writte as

36 9 itij itij, ui ui, i Di i Di,. (.6). Two-Dimesioal Equatios for Piezoelectric Plates It is very difficult to solve the geeral three-dimesioal equatios (.)-(.6) directly, but these equatios ca be lowered to two-dimesioal equatios by usig a power series expasio approach. I this way, the three-dimesioal equatios are reduced to twodimesioal plate equatios formed by power series of fuctios with two i-plae coordiates of the plate. Cosider a piezoelectric plate (see Figure.) with x ad x axes i the mid-plae of the plate ad x alog the thickess directio. The thickess of the plate is h. x Piezoelectric plate h x h x Figure.: A piezoelectric plate. At begiig, the displacemet ad electric potetials ca be writte i power series forms by (.7) ad (.8). ( ) ui ( x, x, x, t) xui ( x, x, t), (.7) ( ) ( x, x, x, t) x ( x, x, t). (.8)

37 Substitutio of equatio (.7) i (.5) gives the correspodig power series expasio for strai, S ij ( ) ij x S, (.9) ad substitutio of (.8) i (.6) yields power series expasio for electric field, E i ( ) i x E, (.) where the two-dimesioal fuctios for the strai ad electric field of order are S ( ) ( ) ( ) ij [ u j, i ui, j i j j i ( ) ( ) ( )( u u )], (.) E ( ) i ( ) ( ), i i ( ). (.) The two-dimesioal fuctios of stress ad electric displacemet of order are defied by takig momets of the stress ad electric displacemet of order over the thickess of the plate, T ( ) ij h h ( ) T ij x dx, Di Di x dx. (.) h h By substitutig the three-dimesioal costitutive relatios (.) ad (.4) i (.) gives the plate costitutive equatios, T ( ) ( m) ( m) ij Bm ( cijkl Skl ekij Ek m ), (.4) D ( ) ( m) ( m) i Bm ( eijks jk ije j m ), (.5) where h m m h ( m ), m eve, Bm x x dx (.6) h, m odd. Through a procedure of variatioal formulatio, the plate equatio of motio ad the equatio of charge are give i (.7) ad (.8) [5], the idex a = ad but ot.

38 T ( ) aj, a ( ) j ( ) j m T F B u, (.7) m ( m) j ( ) ( ) ( ) D D D, (.8) a, a where the combiatio of surface ad body load is defied by (.9). The surface electric charge of order is defied by (.). F ( ) j h h ( ) x T j fi x dx h h, (.9) D ( ) x D h h. (.) I summary, equatios (.), (.) ad (.4)-(.) are the goverig equatios used to determie the two-dimesioal power fuctios () u i ad (). The umber of these two-dimesioal fuctios is ifiitive, however i practice, we always trucate the series i (.7) ad (.8). For example, if oly extesioal displacemet of the plate is cosidered, the oly ozero fuctios kept i the equatios are u, u ad,. If the extesio ad flexure with thickess-shear iside the plate are cosidered, the ozero fuctios eed to be determied are,. u, u, u, u, u ad.. Zero-Order Theory for Extesio For a thi piezoelectric plate, the i-plae extesio is its major motio. The zero-order theory uses u ad u to describe the extesioal displacemets of thi plate, the result shows eough accuracy for may applicatios. The equatios used i the zero-order theory are preseted i this sectio. At first, the displacemet compoets ad electric potetial are approximated by the trucated power series,

39 u i i ui u x, (.) x. (.) u ad u represet i-plae extesioal displacemets of the plate, u is the deflectio of the flexure, extesio due to Poisso s effect, u represets the thickess stretch or cotractio accompayig u ad u describe thickess-shear motio. I zeroorder theory, we are maily iterested i fuctios, u, u, u ad u ad u, so the rest of the displacemet u i (.), are elimiated through a stress relaxatio procedure, i.e. the approximate stress coditio that the stress resultat T. I order j to do so, the material costats c ijkl, e kij, ad ε ik eed to be modified to γ rs, ψ ir ad ζ ij as show i (.8), the process is described i [5]. Those stress resultats which survive i the zero-order equatios are two i-plae ormal stress resultats T ad T, oe iplae shear stress resultat T where T h Tdx h h h, T Tdx, T Tdx. (.) We list below the goverig equatios i power series discussed at begiig of this sectio but with the survivig ukow fuctios. See Referece [5] for detailed derivatio. (.4) is the equatio of motio i zero-order theory. The trucated equatios of charge are show i (.5). h h T, a, b ad, (.4) ab, a Fb h ub a D, D a, a D D. (.5) D, a (.6) ad (.7) are the plate costitutive equatios. Note that (.6)-(.8) are writte i compact matrix otatio.

40 T r h( S E ), r, s,, 5, (.6) rs s ir i D i h( S E ), is s ij j D i h ije j, (.7) where rs rs rvcvwcws c c, r, s,,5, ks ks kwcwvcvs e e, v, w,4,6, (.8) kj kj kvcvwe jw e, j, k,,. The strai-displacemet relatios are give i (.9), ad the equatios of electric potetial are show i (.4). u, S, u, S, S 5 u, u,, (.9) E, E, E. (.4),, It is worth metioig that (.5) is for the case that the upper ad lower surfaces of the piezoelectric plate are ot electroded. For the case of electroded plate, the values of ad i (.) eed to be specified o the two electroded surfaces of the plate... First-Order Theory for Extesio, Flexure ad Thickess-Shear I additio to the extesio of the crystal plate, sometimes the flexure of the plate ( u ) also eeds to be cosidered. The flexure motio of the plate is coupled with thickessshear ( u ad u ), which may or may ot be igored. I order to do so, we eed to use the first-order plate theory. This time the compoets of displacemet u i are approximated by the first three terms of a power series i x : i i ui () ui u u x x. (.4)

41 4 The electric field potetial ϕ is still approximated as show i (.). I (.4) u ad u are the i-plae extesioal displacemets, u represets the flexure deflectio, u ad u represet the thickess-shear motio, u ad () u represet the thickess stretch or cotractio accompayig extesio ad flexure due to Poisso s effect, () u ad () u represet the secod-order symmetric thickess-shear motio. I the firstorder theory, we are maily iterested i u, u, u, u, ad u, so the rest displacemet fuctios i (.4) are elimiated through a -step stress relaxatio procedure [5]. First step, let the stress resultat T for thi plate, which will modify c ijkl, e kij, ad ε ik i the zero-order costitutive equatio to c ijkl, e kij, ad ik as show i (.46). Secod step, use two correctio factors κ ad κ to modify the two zero-order strais S ad S to S ( S ad ), i this way c ijkl, e kij are further modified to c ijkl ad e kij i (.45). The purpose of the secod step is to make sure that the fudametal thickess-shear frequecy obtaied by the first-order theory is equal to that obtaied from the three-dimesioal equatios. Third step, let T for thi plate, so that the c ijkl, e kij j ad ε ik i the first-order costitutive equatio will be modified i the same way as show i (.8). The survivig stress fuctios are five zero-order stress resultats: three i-plae stress resultats T, T, T as show i (.); two out of plae stress resultats T ad T ; three first-order stress momets: T is the i-plae twistig momet. T ad T are two bedig momets ad T h Tdx h h h, T Tdx, (.4)

42 5 T h xtdx h h h h h, T xtdx, T xtdx. The goverig equatios with the survivig ukows are listed below. See Referece [5] for detailed derivatio. Equatios i (.4) are the first-order plate equatios of motio, T, a, b ad, ab, a Fb h ub T F h u, (.4) a, a T ab, a h Tb Fb u b. The equatios of charge for the case of uelectroded plate still take the form of (.5). The zero-order costitutive equatios are show i (.44), T ij D i h( c S e E ), (.44) ijkl kl kij h( e S E ), ikl kl ij j k where c c kc4 c5 kc6 c c kc4 c5 kc6 c pq (.45) kc4 kc4 k c44 kc45 kkc46 c 5 c5 kc54 c55 kc56 kc6 kc6 kkc64 kc65 k c66 e iq e e e e e e k e k e k e e e e k e 6 k e k e 6 6 ad c ijkl c c, ijkl ijckl c e kij e e, (.46) kij kcij c

43 6 e e. ij ij i j c (.47) are the first-order costitutive equatios, ad the material costats γ rs, ψ ir, ad ζ ij are still defied by (.8), T r D i ) h ( ( rsss kr Ek ), r, s,, 5, (.47) ) h ( ( isss ije j ), Equatios i (-48) are the strai-displacemet relatios, ad the equatios of electric potetial are give i (.49). u, S, u, S, S 5 u, u,, S 4 u, u, S 6 u, u, (.48) u, S, u, S, S 5 u, u,,, E,, E, E, (.49), E, E, E.,. Quartz ad Other Piezoelectric Materials As metioed i Chapter, most of the commercial crystal resoators are made of quartz. Quartz is crystal of silico dioxide. I crystallography, quartz belogs to class of trigoal crystal system [7]. There exists large quatity of quartz o earth. Quartz also ca be artificially sythesized to very high quality. Besides piezoelectricity, quartz has very low dampig. It has low solubility ad is comparably hard but ot brittle. It is also

44 7 easy to be cut ito differet shapes []. All above make the quartz the best material for resoators. Z X Y (a) Z x θ=5.5º x θ x X (b) Y Figure.: (a) Bulk of quartz. (b) AT-Cut quartz plate. Figure redraw based o Fig. A of Itroductio to Quartz Crystal Microbalace by S. L. Hellstrom,

45 8 The thickess-shear modes of a quartz plate are ofte employed for the high frequecy resoators. A particular cut of a crystal plate refers to the orietatio of the plate whe it is take out of a bulk crystal. Quartz plates of differet cuts exhibit differet aisotropies i coordiates ormal ad parallel to the plate surfaces. Figure. shows a bulk of quartz ad a widely used AT-cut plate for the thickess-shear resoator. I fact, the AT-cut quartz plate is a special case of rotated Y-cut quartz plates (cuts with differet θ agles). Other examples of rotated Y-cuts iclude BT-cut (θ=-45 ) ad Y-cut (θ= ). The elastic, piezoelectric ad dielectric costats of AT-cut quartz with respect to the plate coordiate system ( x, x, ad x ) are give i (.5). The desity of quartz is 649 kg/m [5,7] [ c pq] N/m, [ e ip ] C/m, (.5) [ ij ] C/(V m). Poled ferroelectric ceramics, commoly called piezoelectric ceramics, have a much stroger piezoelectric effect tha quartz. The piezoelectricity comes from the polarizatio i the ceramics due to applied loads. They are ofte used as trasducers ad actuators. For a plate of PZT-5H (desity 75 kg/m ) with the coordiate system as

46 9 show i Figure., the PZT-5H is poled i x directio, i.e. the polarizatio directio is alog x. The correspodig material costats are [5,7] [..5. c pq] N/m, [ e ip ] C/m, (.5).55 [ ij ] C/(V m). The crystal class (6mm) of the hexagoal crystal system also processes piezoelectricity due to polarizatio. The material costats matrices are similar to (.5). For example, zic oxide (ZO) belogs to this class ad is used i Chapter 6.

47 Chapter Crystal Plates with Multiphysical Films I this chapter, we cosider a crystal plate with surface films that are sesitive to differet physical effects, such as iertia, stiffess, itrisic stress, piezoelectric couplig, ad piezomagetic couplig. These multiphysical effects are itegrated ito the twodimesioal plate equatios (discussed i Chapter ), i order to obtai the geeralized equatios, which will the be used to study the thickess-shear vibratios of a rotated Y- cut quartz plate with multiphysical films. Frequecy shifts due to the multiphysical films are calculated ad examied i order to ivestigate the potetial sesor applicatio of usig a quartz resoator with sesitive surface film to detect electric or magetic field, or estimate the itrisic stress i thi films [7].

48 . Itroductio Crystal resoators are ofte used i harsh eviromets or o objects i largeacceleratio motio. For these applicatios, high frequecy stability of the crystal resoators agaist temperature chage or acceleratio is desired [74]. O the cotrary, for sesor applicatios like quartz crystal microbalaces (QCMs), high sesitivity to differet evirometal effects is desired. Such evirometal sesitivity may come from surface films o the QCM. I fact, it is ievitable that there exist itrisic stresses [6,6] i fuctioal films of a QCM due to maufacturig processes. Whe surface films have piezoelectric or piezomagetic coupligs [6,64], they will chage their shapes if ambiet electric or magetic fields appear. Both the itrisic stresses ad multiphysical coupligs i the films will ultimately shift the resoat frequecies of a crystal resoator through the stresses ad strais they produce. These stresses ad strais are called iitial or biasig fields i the resoators. Whe the biasig fields are preset, the resoator frequecies are slightly differet from those without biasig fields. The behavior of crystal resoators with the presece of biasig fields is govered by the theory for small fields superposed o a bias [67] which eeds to be derived from a oliear theory [68]. Due to the complexity of these theories, the effects of biasig fields i resoators due to surface films are relatively less studied. I [6] ad [6], the film itrisic stress ad thermal expasio were treated separately for resoator applicatios. Recetly it has bee show through simple oe-dimesioal aalyses, that plate crystal resoators structurally itegrated with piezoelectric/piezomagetic films

49 possessig strog piezoelectric/piezomagetic coupligs may be cosidered for electric/magetic field sesig [65]. It is also possible to use surface piezoelectric films to maipulate frequecy shifts i crystal resoators caused by acceleratios [66]. A crystal plate with a layer of a differet material is also useful to compesate frequecy shift due to temperature [6]. I this chapter, we explore further the ideas of electric/magetic field sesig ad frequecy maipulatio i [65,66]. I order to predict behaviors of real devices of fiite sizes, we aalyze the more realistic situatio of fiite, two-dimesioal plates for frequecy shifts caused by multiphysical effects i surface films. For this purpose we expad the first-order, two-dimesioal equatios for coupled extesio, thickess-shear ad flexural vibratios of a crystal plate with elastic surface films [44] to iclude itrisic stresses ad piezoelectric/piezomagetic coupligs i the films. Several applicable examples are preseted. Fially, the two-dimesioal equatios are reduced to describe the pure thickessshear vibratios of a quartz plate with a sigle mass film o top of the plate. The reduced equatio will be used i the followig two chapters.. Two-Dimesioal Equatios for a Crystal Plate with Surface Films I this sectio, the two-dimesioal equatios for crystal plates with films of multiphysical effects are geeralized.

50 .. Equatios for the Crystal Plate First we use first-order theory discussed i Chapter to describe the coupled extesioal, flexural ad thickess-shear vibratios of a quartz crystal plate aloe without surface films. Quartz is a material with very weak piezoelectric couplig. The very small piezoelectric couplig i quartz is eglected i frequecy aalysis of quartz devices. I this way, the crystal plate ca be cosidered as a elastic plate. Cosider the crystal plate of thickess h i Figure., x ad x are i the middle plae. x is alog the plate ormal. Idices i, j, k, l rage over,, ad a, b, c, d over, but skip. h x Films Crystal plate x h h x h Figure.: A crystal plate with asymmetric surface films. The compoets of displacemet u i are approximated by the first two terms of a power series i x : u i u x, x, t) x u ( x, x, t), (.) i ( i ad the ozero stress-resultats Tij ad stress momets Tab are T h ij T ij h dx, T xtabdx. (.) ab h h The stress-equatios of motio, Tij, i u j, are approximated by the followig five equatios: T, i, j,,, (.) ij, i Fj h u j

51 4 T ab, a Tb Fb h ub, a, b,, (.4) where a dot deotes time derivative, F j ad F b are surface loads defied by F j T j h T j h, Fb ht b h ht b h, (.5) ad is the mass desity. The stress-strai relatio, Tij cijkl Skl, which ca be cosidered as a modificatio of equatio (.) with piezoelectric couplig term dropped, is approximated by T hg S, ij ijkl kl T h S ab abcd cd, (.6) where g m ijkl i j k l ijkl ijkl ijkl ij kl, (ot summed) abcd abcd ab j kcd ( jk ). g g c c c c c c c c, (.7) g ijkl is obtaied from the stress relaxatio T. They are the modified ito * g ijkl by the shear correctio factors ad [4,44]. abcd is obtaied from the stress relaxatio T. I (.7), m ad are give by j m cos ( ij ), cos ( kl ). (.8) The values of the a will be provided later. The strai-displacemet relatio is give i idex form by S ij ) ) ( ( ( ui, j u j, i jui iu j ), Sab ( ua, b ub, a ), (.9) i which ij is the Kroecker symbol.

52 5.. Equatios for the Films Physical quatities associated with the upper ad lower films will be desigated by primes ad double primes, respectively. The film thickess, h ad h, are usually much thier tha the crystal plate. Therefore the films are modeled by the zero-order theory. The, the equatios for the upper film are T F, (.) jb ab, a j h u j F j T j h T j h, (.) T h( S e E h H), (.) ab ab abcd cd kab k kab k ad for the lower film are, ) ( Sab ( ua, b ub, a ), (.) T F, (.4) jb ab, a j h u j F j T j h T j h (.5) T h ( S e E h H ), (.6) ab ab abcd cd kab k kab k ) ( Sab ( ua, b ub, a ), (.7) where a, b, c, d rage over,, but skip. jb cb whe j c ad whe j. jb (.) ad (.6) are more geeral tha the correspodig oes i [44] due to the iclusio of the thi-film itrisic stress ab, the electric field E k, ad the magetic field H k. Take the upper film as a example, abcd is the thi-film elastic costat, e kab ad h kab are thi-film piezoelectric ad piezomagetic costats. They are obtaied from the stress relaxatio T j =. We assume that the itrisic stress ab, the strai S cd, the

53 electric field E k, ad the magetic field effects o the resoat frequecies are of iterest. ab, E k ad 6 H k are all ifiitesimal. Oly their first-order H k are cosidered kow ad they act as loads o the structure. For the lower film the situatio is similar... Equatios for the Crystal Plate with Surface Films The equatios of the crystal plate ad the surface films are joied together usig the followig iterface cotiuity coditios of tractios ad displacemets: T j h T j h, j h T j h T, (.8) u u h u, i i ia a u u h u. (.9) i i ia a From (.5) ad (.8), ad from (.5) ad (.8), we obtai F j T j h T j h, (.) F b htb h htb h. (.) From the sum ad differece of (.) ad (.5), T F, (.) j h T j h j Fj Fj where ht ht F h( F F ), (.) b h b h b b b F, (.4) j T j h T j h F, (.5) b htb h htb h are the loads o the outer surfaces of the films. Hece, from (.)-(.), F F j b F ( F F ), (.6) j j j F h( F F ). (.7) b b b

54 7 Equatios (.6) ad (.7) show that the surface loads of the crystal plate Fb are expressed i terms of the surface loads F j ad F j ad differece of the stress equatios of motio (.) ad (.4), Fj ad of the films. From the sum F j F j, (.8) jb( Tab Tab ), a h u j h u j F b F b ( Tab Tab ), a h ub h ub. (.9) Substitutig these expressios ito (.6) ad (.7), ad usig (.9), we have F j F, (.) j jb( Tab Tab ), a hrs u j h RD jaua F b F h, (.) b ( Tab Tab ), a h RS ub h RDub where we have deoted R S ( h h ) h, ( h h ) h. (.) R D R S ad R D are the mass ratios of the sum ad differece of the films to the crystal plate. Next by substitutig (.) ad (.) ito stress equatios of crystal plates (.) ad (.4), we arrive at the followig five stress equatios of motio for the crystal plate carryig the films: T, F h( R ) u h R u, (.) ij i j S j D jb b T ab, a T b Fb h ( RS ) u b h RDu b, (.4) where the total resultats or momets, which come from the cotributios of both the plate ad films, are T T ( T T ), (.5) ij ij ia jb ab ab T T h( T T ). (.6) ab ab ab ab

55 8 By substitutig the costitutive relatios for the crystal plate (.6) ad the costitutive relatios for the two thi-films (.) ad (.6) ito (.5) ad (.6), ad the usig the strai-displacemet relatios (.) ad (.7). Also by oticig the extesioal displacemets of the two films ca be expressed i terms of u i ad u a by meas of (.9) ad usig strai-displacemet relatio for the plate (.9), we arrive at the followig costitutive relatios for the plate with the two films: ij ijkl kl ia jb S abcd cd D abcd T hg S h ( S h S ) hn, (.7) cd ij where we have defied S D ( abcdscd h abcdscd h abcds ) cd T ab h hmab, (.8) N ij ad h h Nij ia jb ( ab e kabe k h kabh k ) ia jb ( ab e kabe k h kabh k ) (.9) h h M h( e E h H) h ( e E h H ), (.4) ab ab kab k kab k ab kab k kab k D ( h h h, ( h h h. (.4) S abcd abcd abcd) abcd abcd abcd ) M ab describe the cotributios from the films to the extesio ad bedig of the plate. By substitutig the strai-displacemet relatios (.9) ito (.7) ad (.8) ad the (.7) ad (.8) ito (.) ad (.4), we have g ijkl ( u k, li ( R S k u ) u l, i j ) jb D S abcd hr u ja u c, da a, h jb D abcd u c, da h F j N ij, i (.4) h abcduc, da g h ( RS bkl ) u ( u b k, l k hr u l Dub. ) h S abcd u c, da h D abcd u c, da h Fb M ab, a (.4) Equatio (.4) ad (.4) are the overall displacemet equatios of motio for the five displacemets, ui ad u a.

56 9 Fially, the shear correctio factors ad are determied by requirig the fudametal thickess-shear frequecies of a ifiite plate calculated from the approximate plate equatios ad the exact three-dimesioal equatios to be equal. For a plate of moocliic crystals which icludes rotated Y-cut quartz as a special case, the correctio factors are [44] where ca{ CaRS [ CaRS C ( )]} 4 a RS RD a, (.44) g [ R R ( R )] aa S D S c c C c j c { c j, c c j j. [( c c ) 4c / ] }, (.45). Reductio to Extesio ad Elemetary Flexure We ow begi to fid the frequecy shift of the crystal plate due to the biasig field caused by the multiphysical films. Our first task is to determie this static biasig field. We kow equatios (.) ad (.4) or (.4) ad (.4) are coupled equatios for the extesio u a, flexure u, ad thickess-shear plate is at static state, ot i motio, to ab, E k, thickess shear u a, u, ad ua of the crystal plate. If we cosider the u a are the biasig deformatios due H k i the upper film ad similar fields i the lower film. For thi plates the u a is usually very small ad ca be elimiated, resultig i a simpler theory for coupled extesio ad elemetary flexure without thickess shear. Such a theory will be sufficiet for our eeds ad may other applicatios. This simplified

57 4 theory is reduced from (.) ad (.4) as follows. First, we rewrite (.) ad (.4) as separate equatios for u a, u, ad u a : T, (.46) ab, a Fb h( RS ) ub h RDub T u, (.47) a, a F h( RS ) T T F, (.48) ab, a b b where, as oe of the two approximatios eeded for the reductio to elemetary flexure, we have eglected the rotatory iertia terms o the right-had side of (.4). Equatio (.48) ow provides the usual shear force-bedig momet relatio i the elemetary theory for flexure. Solvig (.48) for (.47) gives the followig equatio for elemetary flexure: T b, ad substitutig the resultig expressio ito T u. (.49) ab, ab Fb, b F h( RS ) Aother approximatio eeded for the reductio to elemetary flexure is that the plate shear strais S a vaish, amely, This implies, through (.9), that ) ( S a ( u, a ua ). (.5) u a, a u, (.5) S ab, ab u. (.5) With (.5) we ca write the extesioal equatio (.46) as T. (.5) ab, a Fb h( RS ) ub h RDu, b

58 4 (.5) ad (.49) are three equatios for u a ad u. The extesio u a ad the flexure u are coupled o the right-had side of (.5) whe the two films are differet. Sice S ad a S is ot preset i (.7), from (.7) ad (.9) we have ab abcd cd The geeral equatios of motio for S abcd cd D abcd T hg S h( S h S ) hn, (.54) h h Nab ( ab e kabe k h kabh k ) ( ab e kabe k h kabh k ). (.55) h h u a ad cd ab u are obtaied by substitutig (.54) ad (.8) ito (.5) ad (.49). S ab ad S ab are ow give by (.9) ad (.5), respectively, i terms of u a ad u..4 Biasig Fields Caused by the Multiphysical Films Cosider the static deformatio of a fiite plate with tractio-free surfaces all aroud. The top ad bottom surface loads F j ad F b. The edge of the plate is geometrically smooth (without corers) ad is free from ay mechaical resultats. I this case, T ad T satisfy the goverig equatios i (.5) ad (.49) as well as all ab ab boudary coditios. We cosider the case of uiform itrisic stresses ad electric/magetic fields i the films. I this case correspodig costat plate extesioal strais N ab ad M ab are costats. The the S ab ad flexural strais biasig fields, are determied by settig (.54) ad (.8) to zero S ab, i.e. the S D T hg S h( S h S ) hn, (.56) ab abcd cd abcd cd abcd cd ab

59 4 ) S D ( T ab h abcdscd h abcdscd h abcdscd hmab. (.57) For elemetary flexure S. a S is determied from the stress relaxatio coditio T. S is determied from the stress relaxatio coditios T. The j j S ij ad S ij are completely kow. I the relatively simple case of cylidrical deformatios of plates with u ad / x, from (.56) ad (.57) we obtai S D S S S M h( ) N, (.58) D S * h( ) h( )( g ) * S D S ( g ) M h N. (.59) D S * h ( ) h ( )( g ) D (.58) ad (.59) show that causes couplig betwee extesio ad bedig. For symmetric films it vaishes..5 Frequecy Shift of the Fudametal Thickess-Shear Mode I this sectio, we cosider thickess-shear vibratios of a rotated Y-cut quartz plate with the presece of biasig fields, S ij ad S ij, caused by the itrisic stresses ad electric/magetic fields i the surface films. The thickess-shear vibratio is a icremetal motio superposed o these biasig fields. For the most widely used fudametal thickess-shear mode, the frequecy shift caused by the biasig fields is give by [75,76]:

60 4 S ( c S c S c S c S ) c66 h ( c65 S5, c56s, c56s, ), c 66 (.6) where is ormalized by the fudametal thickess-shear frequecy ( ) of the crystal plate aloe whe the films are ot preset c. (.6) h 66 The third-order elastic costats, such as c 66 ad c 66, are for oliear material behavior. They have six idices i the tesor otatio ad three idices uder the compact matrix otatio [77]. The first-order plate strais S ij have cotributios oly whe they are ihomogeeous. I the special case of cylidrical deformatios, (.6) reduces to c c c ( ) S. (.6) c66 c66c.6 Examples of Applicatio I this sectio, we use the equatios derived i the above to study frequecy shifts i a resoator caused by electric/magetic fields ad electrode stresses. Cosider a crystal plate of Y-cut quartz which is a special case of rotated Y-cuts whe the agle of rotatio θ is zero i Figure.. The liear material costats ca be foud i [77]. The third-order elastic costats are from [78]. The followig relatios amog the third-order elastic costats exist ad are eeded [79] c c ( c (c c c c c ) / 4, ) / 4, c 66 ( c c ) /. (.6)

61 44 For the plate thickess we choose h= mm. The fudametal thickess-shear mode of the plate aloe without the surface films is / =.94 6 Hz..6. Electric Field Sesig Cosider the above quartz plate with oly oe surface film of PZT-5H poled i the thickess directio ( h ). Polarized ceramics have much stroger piezoelectric couplig tha quartz. Whe the plate is placed uder a exteral electric field i the thickess directio, the ceramic film teds to expad or cotract through piezoelectric couplig, but the quartz plate does ot do so because its e =e =e =. Istead, the quartz plate exteds ad beds uder the actio of the ceramic film. We solve (.56) ad (.57) o a computer for S ab ad S ab, ad use the stress relaxatio coditios to fid the other plate strais. The we calculate the frequecy shift usig (.6) ad plot the results i Figure.. Figure.: Frequecy shift due to electric field. A liear relatioship betwee the frequecy shift ad the electric field is predicted, which is ideal for electric field sesig. This liearity is also a cosequece of the theory

62 45 employed, i.e., the biasig fields are assume to be small ad are obtaied by the liear theory of elasticity, ad oly the first-order effect of the biasig fields o the icremetal thickess-shear vibratio is cosidered. For large biasig fields, the oliear theory of elasticity is eeded to determie the biasig fields, ad the secod- ad higher-order effects of the biasig fields eed to be cosidered [8]. The a oliear relatioship betwee the frequecy shift ad the biasig fields will be predicted. Such a calculatio requires the kowledge of the fourth-order material costats, which presetly are ot available. Therefore, the rage of the liear output caot be determied from the preset aalysis. For a moderate electric field of 5 V/m or V/mm, the relative frequecy shift is of the order of -6 which is measurable i crystal resoators whose frequecy shifts are typically described by ppm (parts per millio). Figure. shows that thicker piezoelectric films imply higher sesitivity as expected. These results agree with the [65] for cylidrical motios. Figure.: Curvature due to electric field. Sice the ceramic film is o oe side of the crystal plate oly, the electric field also causes bedig of the crystal plate. The curvatures of the middle plae of the crystal

63 plate are show i Figure. whe h / h.. The curvature i the x directio is smaller because c is larger tha c for Y-cut quartz plates Magetic Field Sesig Whe a Y-cut quartz plate carries oly oe piezomagetic film ( h ) of CoFe O 4 whose material costats ca be foud i [8], the aalysis is similar. Quartz does ot respod to magetic fields directly. Through the extesio or cotractio of the piezomagetic film, the quartz plate exteds ad beds, resultig i frequecy shifts ad curvatures. The results are show i Figures.4 ad.5, respectively, showig that the structure ca fuctio as a possible magetic field sesor. These also agree with the results of the cylidrical motios cosidered i [65]. Figure.4: Frequecy shift due to magetic field.

64 47 Figure.5: Curvature due to magetic field..6. Electrode Stress Plate quartz resoators are usually exited by electrodes o the top ad bottom of the plates. The electrodes ofte carry itrisic stresses due to their maufacturig processes. These electrode stresses are of the order of MPa [8]. Their effects o resoator frequecy stability is a importat issue i resoator desig. As a example, cosider a Y-cut quartz plate with idetical electrodes of gold with itrisic stresses of MPa i both the x ad x directios. The frequecy shift calculated from (.6) is show i Figure.6. It is of the order of -5, quite sigificat.

65 48 Figure.6: Frequecy shift due to idetical electrode stress. Aother importat aspect of the electrodes is their thickess. I resoator maufacturig, oe electrode is pre-deposited with a pre-determied thickess. The the thickess of electrode o the other side of the plate is modulated accordig to the desired frequecy of the resoator. This ormally results i a crystal plate with electrodes of uequal thickesses. It is kow that electrodes with differet thickesses cause udesirable mode coupligs ad affect resoator performace [44,8]. As a umerical example, cosider a Y-cut quartz plate with electrodes of differet thickesses. h / h. ad h / h.. The correspodig curvatures are calculated from (.56) ad (.57) ad are show i Figure.7. If sufficiet iformatio about the plate strais ad curvatures are measured experimetally, the electrode itrisic stresses ad/or their thickesses ca be calculated from measured data usig (.56) ad (.57).

66 49 Figure.7: Curvature due to electrode stress of uequal thickess..7 Reductio to Pure Thickess-Shear of a Quartz Plate with a Mass Film o Top We begi from equatio (.4), which maily describes the thickess-shear vibratio of a crystal plate with two differet multiphysical films o top ad bottom surfaces of the plate. If we igore the extesio ad flexure motio of the plate, i.e. u, ad oly k oe sigle film o top of the plate is cosidered, is reduced to R S R R h h, equatio (.4) D h abcduc, da g kbklul h ( R) u b. h S abcd u c, da h Fb M ab, a (.64) If oly the iertia effect of the film is cosidered, equatio (.4) which describes the multiphysical effects of the film, becomes M. The first equatio of (.4) also S becomes because the stiffess of the film is igored too. If there are o tractios abcd ab

67 5 o the outer surfaces of the quartz plate, F (see equatio (.5)). Take all the above ito accout, equatio (.64) is further reduced to b ) ( h abcduc, da g kbklul h ( R) u b. (.65) Now let us cosider the crystal plate is made of rotated Y-cut quartz. It is doig a thickess-shear vibratio with particle displacemet alog x ad the frequecy is ω. Use equatios (.7) ad (.8) based o rotated Y-cut quartz ad drop the time expoetial of u, we ca rewrite equatio (.65) i compact matrix otatio (see Sectio.) as, ) h ( ( u, 55u,) k c66u h ( R) u. (.66) I equatio (.66), k is the correctio factor. The fudametal thickess-shear frequecy (also referred to as cutoff frequecy) is obtaied by lettig u to be a costat i (.66), k c [ h ( R)]. (.67) 66 The same fudametal thickess-shear frequecy ca be also obtaied from the threedimesioal equatio [46], c [4h ( R) ]. (.68) By equatig (.67) ad (.68), we determie the correctio factor k as 66 k R. (.69) ( R) Sice typical R is below, k ca be approximated by k ( R). (.7) By substitutig (.7) i (.67) we have the fudametal thickess-shear frequecy as, c66 R c66 ( R). (.7) 4h R 4h

68 5 If the film is ot preset, which meas R=, (.7) is reduced to c66. (.7) 4h Equatio (.66) alog with (.7)-(.7) is used to study the thickess-shear vibratio of quartz plate boded with fiite-size film o top i the followig chapters..8 Summary Two-dimesioal equatios for coupled extesio, flexure ad thickess motios of aisotropic crystal plates with multiphysical surface films are derived, the reduced to coupled extesio ad elemetary flexure without shear, ad used i the aalysis of frequecy shifts i crystal resoators. It is show that a crystal plate with a piezoelectric/piezomagetic film may be used as a sesor for electric/magetic fields. Whe the film/plate thickess ratio is / ad the electric field is of the order of V/mm, the relative frequecy shift is of the order of a few ppm. Whe the magetic field is of the order of A/mm, the relative frequecy shift is of the order of -5. These agree with some previous aalyses i simpler situatios, ad are detectable frequecy shifts i crystal resoators. The equatios derived are also useful i the aalysis of the effects of electrode stress i crystal resoators. Furthermore, the two-dimesioal plate equatio for a pure thickess-shear vibratig quartz plate with a mass film o top was obtaied. This approximate equatio is used i Chapter 4 ad 5.

69 5 Chapter 4 Effects of Nouiform Films I this chapter, we study the effects of the ouiform thickess of a thi mass film o the resoace of a quartz crystal resoator. The plate equatio for thickess-shear vibratio of a rotated Y-cut quartz plate with surface film (see Sectio.7) is used i this chapter. A elliptical mass film with stepped thickess deposited o the top surface of a quartz plate is cosidered i Sectio 4. [84]. The resoat vibratio frequecies ad modes correspodig to this structure are obtaied. The effects of the mass film ouiformity are examied. Results show that the vibratios of some modes are cofied uder the mass film, this pheomeo is called eergy trappig. It is also show that a mass film thicker at the ceter teds to trap more modes uder the mass film ad push the distributio of the thickess-shear displacemet toward the ceter. Next, i Sectio 4., we study the effects of a mass film with gradually varyig thickess. A theoretical aalysis is performed o thickess-shear vibratio of a quartz plate with its upper surface deposited with a mass film havig gradually varyig

70 5 thickess [85]. Resoat vibratio frequecies ad modes are obtaied. The effects of the mass film ouiformity are examied. It shows similar results as from stepped thickess film, there exist trapped modes whose motio is maily uder the film. Whe the film becomes thicker, the modes are pushed toward the ceter. 4. Itroductio As itroduced i Chapter, the iertial effect of a thi surface film is to lower the resoat frequecies of a quartz crystal resoator [86,87]. This effect aloe ca be used to measure the thi-film mass desity ad thickess. Researchers also developed more sophisticated models cosiderig both the iertial ad stiffess effects of the mass layer [5,5], but these oe-dimesioal models oly iclude the plate thickess coordiate oly, without i-plae variatios or boudary effects. Behaviors of real devices are more complicated for several reasos ad ca deviate, sometimes cosiderably, from the results predicted by thickess-coordiate models. For example, due to the eergytrappig pheomeo, the thickess-shear vibratio is ot uiform ad is maily uder the mass layer ad decays expoetially away from the mass layer edge. The i-plae variatios of the thickess-shear modes also cause deviatio from the Sauerbrey equatio used to predict the frequecy shift. There have bee a few attempts cosiderig the iplae variatios of thickess-shear modes [,4,5], but overall theoretical results are few ad scattered. Recetly, it has bee poited out that the fuctioal film o a quartz crystal microbalace is sometimes ouiform ad little is kow about its implicatios [88]. While there have bee a few published results o ouiform mass layers or ouiform

71 54 electrodes o quartz crystal resoators either for mass sesig [54,55] or for eergy trappig i resoator applicatios [56-59], these aalyses are all for strip resoators with modes ad film (or electrode) thickess variatios depedig o oe i-plae coordiate oly. For a accurate uderstadig of the effects of ouiform mass films, a more sophisticated aalysis with the film thickess ad mode variatio depedig o both of the i-plae coordiates is performed i this chapter. 4. Films with Stepped Thickess We first cosider a elliptical mass film with stepped thickess. The umber of steps i the film thickess variatio is arbitrary. Therefore the model ca approximate a gradually varyig film thickess to ay desired accuracy usig a sufficietly large umber of steps. A elliptic shaped film is chose because it is kow to be most compatible with the distributio of the thickess-shear vibratio ad therefore is optimal i the sese of [47,89]. Due to material aisotropy of the quartz plate, a elliptical film allows a exact aalysis based o the plate equatio while a circular film does ot. 4.. Structure For our purpose it is sufficiet to cosider a ubouded plate of AT-cut quartz as show i Figure 4.. The plate has a thickess h ad a mass desity ρ. There is a thi, elliptical film with stepped thickess o the top of the crystal plate. The desity of the film is ρ'. Its varyig thickess is h' which is piecewise costat betwee two eighborig ellipses. The film is assumed to be very thi. Oly its iertia will be cosidered, its stiffess will be eglected [86,87].

72 55 x x x h Quartz x Figure 4.: Top ad side views of a quartz plate with a ouiform film of stepped thickess. Cosider the case of N ellipses. The Nth ellipse represets the boudary of the film. Begiig from the ceter, the th ellipse is described by ( a x x ( ) ( ) ) / 55 ( a ) ( ) The semi-major ad semi-mior axes are a / 55 ad, =,,, N. (4.) () a, respectively. We also (N ) deote a ad a. For AT-cut quartz, accordig to (.8) the two modified elastic costats are N/m ad N/m. The ratio betwee the major ad mior axes is.6 which is very close to the optimal electrode shape ( ) ( ) determied i [47,89]. We eed to aalyze N+ regios of ( a, a ) with =,,, N+. For a frequecy aalysis, we will eglect the small piezoelectric couplig of quartz as usual.

73 Goverig Equatios I geeral, thickess-shear vibratio may be coupled to flexural motio i a quartz plate. This couplig depeds o the plate dimesios ad is strog oly for certai aspect ratios (legth/thickess) of the plate [4]. For thi plates, the couplig to flexure is less likely to happe. For our purpose we assume that the couplig to flexure has bee avoided through desig ad we will cosider pure fudametal thickess-shear vibratio with oly oe odal poit i thickess directio. So the displacemet field is approximated by [4]: u x, x, x, t) x ( x, x, t), u, u, (4.) ( where is the plate thickess-shear displacemet (labeled as u i Chapter ), also idicates the rotatio of a lie elemet iitially ormal to the middle plae of the plate. We try to determie the resoat frequecy of the quartz plate uder the ifluece of the ouiform film. The plate equatio (.66) for the pure thickess-shear vibratio with a mass film discussed i Sectio.7 is used here. For the th regio, the goverig equatio for is where ( ) ( ) 55 h ( ) c66 ( R ) x x, (4.) ( ) ( ) ( ) ( R ), R ( ) h, (4.4) h ( ) s, ss s 55. (4.5) s 55

74 57 (4.5) shows ad 55 ca also be defied by the elastic compliaces s pq of quartz. From (4.) we ca determie the so called cutoff frequecy of the fudametal thickess-shear mode idepedet of x ad x by assumig is a costat, ( ) ( ) ( ) ( ) ) R, (4.6) ( ) ( c h ( R where the approximatio is for small ) c 4h () R. Above the cutoff frequecy, the displacemet field is oscillatory. Below the cutoff frequecy, the displacemet field is decayig. We cosider the case whe the film is thicker i the middle as show i Figure 4. with R > R () ( N ) > > R =. (4.7) We are iterested i the eergy-trappig modes for which the correspodig resoat frequecies are withi the iterval of. (4.8) ( ) ( N ) I this frequecy rage, is oscillatory i the ier most cetral regio uder the film ad decays outside the film. I a aular regio of the film maybe oscillatory or decayig depedig o whether is above or below the cutoff frequecy of the particular aular regio. 4.. Free Vibratio Solutio I the x, ) plae, we itroduce a ew coordiate system, ) by ( x ( x / x. (4.9) 55, I this ew coordiate system, the ellipses i (4.) are trasformed ito circles described by

75 58 ( a ( ) ) ( ) ( a ). (4.) (4.) becomes ( ) ( ) h ( ) c ( R ). (4.) We the itroduce a polar coordiate system ( r, ) defied by r cos, r si. (4.) We are oly iterested i modes that are idepedet of. The (4.) becomes where r r r ( ( ) ), (4.) ( ( ) ) ( R ( ) ) 55 h ( ( ) ) c (4.4) To be specific we cosider the case whe is withi ( ) () ( N ) ( N). (4.5) I this case the thickess-shear motio is oscillatory everywhere uder the film ad decays outside the film. Uder (4.5), for =,,,, N, we have ( ( ) ) >. For =N+, ( ( N ) ) < ad we deote ( N ) ) = (. The, i differet regios, geeral solutios to (4.) ca be writte as AJ( r), r a, (4.6) ( ) ( ) ( ) ( ) ( ) ( ) B J ( r) C Y ( r), a r a,, N, (4.7) (N ) DK ( r), a r, (4.8) where A, () B, () C ad D are udetermied costats. J ad Y are the zero-order Bessel fuctios of the first ad secod kids, respectively. They oscillate ad decay

76 59 accordig to r / for large r. K is the zero-order modified Bessel fuctio of the secod kid which decays expoetially at ifiity. At the iterfaces betwee eighborig regios both the stress ad displacemet i radius directio are cotiuous, so the ad its ormal derivative with respect to r should be cotiuous at the iterfaces [77]: ) ( ) ( ) ( () () () () a Y C a J B a AJ, (4.9) ) ( ) ( ) ( () () () () () () a Y C a J B a J A, (4.),,,..., ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N a Y C a J B a Y C a J B (4.),,,..., ), ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N a Y C a J B a Y C a J B (4.) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N N N N N N N a DK a Y C a J B, (4.) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N N N N N N N N N a K D a Y C a J B, (4.4) where a prime represets the differetiatio with respect to the etire argumet, r () or r. (4.9)-(4.4) represet N liear homogeeous equatios for A ad () B, () C ad D. For otrivial solutios of the udetermied costats, the determiat of the coefficiet matrix must vaish, which gives the frequecy equatio that determies the resoat frequecies. This will be doe umerically o a computer. The followig idetities are used to simplify the umerical calculatio:,,, K K I I Y Y J J, (4.5) where I is the zero-order modified Bessel fuctio of the first kid which is eeded whe is i a differet rage tha (4.5).

77 Numerical Results As a umerical example, we cosider a AT-cut quartz plate with a typical resoator thickess of h=. mm. To examie the effects of the film thickess variatio, we calculate three examples ad compare the results. Case is a plate with a uiform film. Case () is for a plate with a film whose thickess has oe sudde chage (step). Case () is similar to Case (), with oe sudde chage i the thickess by a differet amout. The specific parameters of the three cases are: Uiform film: a =5 mm, R =.5%. The cutoff frequecies of the cetral ad outer regios are 8,78,497 Hz ad 8,7,88 Hz. () Relatively weak ouiform film: a =.5 mm, R =%, () a =5 mm, () R =.5%. From the cetral to the outer regios, the cutoff frequecies are 8,4,6 Hz, 8,78,497 Hz, ad 8,7,88 Hz. () Relatively strog ouiform film: a =.5 mm, R =4%, () a =5 mm, () R =.5%. The cutoff frequecies are 7,97,4 Hz, 8,78,497 Hz, ad 8,7,88 Hz. The oly differece betwee Cases () ad () is R. I real applicatios R is usually less tha. A exaggerated value of R is chose to show its umerical effects more clearly. The frequecies ad mode shapes of the trapped modes for the three cases are show i Figures respectively. These modes are essetially uder the film (eergy-trappig). The modes are ormalized by their maxima. I Figure 4. for Case, there are four trapped modes. They are all oscillatory uder the film ad decay outside it. The four modes are with zero, oe, two ad three odal poits ad slowly icreasig frequecies. Ideally, oly the first mode without odal

78 6 poits aloe is eeded for mass sesig. The frequecies of the four modes are 8,8,967 Hz, 8,7,98 Hz, 8,48,456 Hz, ad 8,6,4 Hz, respectively. They are slowly icreasig. Figure 4.: Trapped modes for Case, uiform film, a =5 mm, R =.5%. I Figure 4. for Case (), five trapped modes are foud. They ca be classified ito two categories. The first mode without a odal poit aloe represets oe category. It is oscillatory uder the cetral circular regio r a where the film is thick, ad decays i both the aular regio a () where the film is thi ad the outer r a regio () r a where there is o film. The four other modes with oe, two, three, ad four odal poits are all oscillatory uder the film everywhere ( r () a ) o matter the film is thi or thick, ad decay outside the film. The frequecies of the modes are 8,55,6 Hz, 8,95,89 Hz, 8,,59 Hz, 8,9,4 Hz, ad 8,6,65 Hz, respectively. The first three frequecies are slightly lower tha those i Case because of the additioal mass of the thick cetral layer.

79 6 Figure 4.: Trapped modes for Case (), ouiform film, () () a =.5 mm, R =%, a =5 mm, R =.5%. I Figure 4.4 for Case (), five trapped modes are foud. They also fall ito two categories. The first two modes with zero ad oe odal poit are oscillatory uder the cetral circular regio r a ad decay outside it. The three other modes with two, three ad four odal poits are oscillatory everywhere uder the film whe () r a ad decay outside the film. The frequecies of the modes are 7,989, Hz, 8,55,574 Hz, 8,8,798 Hz, 8,64,7 Hz, ad 8,9,4 Hz, respectively. Figure 4.4: Trapped modes for Case (), ouiform film, () () a =.5 mm, R =4%, a =5 mm, R =.5%.

80 6 I Figures we plot the same mode from differet cases i the same figure. Clearly, as the ceter of the film becomes thicker, the same mode is pushed toward the ceter. Figure 4.5: The first mode from Cases -(). Figure 4.6: The secod mode from Cases -().

81 64 Figure 4.7: The third mode from Cases -(). Figure 4.8: The fourth mode from Cases () ad ().

82 65 Figure 4.9: The fifth mode from Cases () ad (). 4. Films with Gradually Varyig Thickess I this sectio we begi to cosider the case of a film with a cotiuously varyig thickess o a quartz crystal resoator, which is a more realistic situatio tha the stepped thickess variatio cosidered i Sectio Structure Cosider a AT-cut quartz plate as show i Figure 4.. The plate has a uiform thickess h ad a mass desity ρ. There is a film with a varyig thickess o the top of the crystal plate. The desity of the film is. Its varyig thickess is h.

83 66 Nouiform film Figure 4.: A quartz plate with a ouiform film. 4.. Goverig Equatios We cosider the pure thickess-shear vibratio of the plate based o the same displacemet approximatio as i (4.). The goverig plate equatio of (.66) is writte as 55 h c66 ( R) x x, (4.6) where is the plate thickess-shear displacemet, also the rotatio of a lie elemet iitially ormal to the middle plae of the plate, ad ( R), h R. (4.7) h Idetities i (4.5) also apply here. For small R, (4.6) ca be approximately rewritte as where 55 [ ( )] R x x, (4.8) c66. (4.9) 4h is the frequecy of the fudametal thickess-shear mode of the quartz plate whe the film is ot preset. I fact, sice R is very small, the differece betwee (4.6) ad

84 67 (4.8) is of the order of ( ) R, which is a higher-order ifiitesimal because is also small. We cosider the case whe the film thickess is varyig accordig to h h[ f ( x ) f ( )], (4.) x where h is the film ceter thickess, f ad f are slowly growig fuctios so that the film is thick i the ceter ad thi away from the ceter. Correspodig to (4.), from (4.7) we have where R R f ( x ) f ( )], (4.) [ x R h. (4.) h Substitutio of (4.) ito (4.8) gives where we have deoted 55 [ ( )] R f f x x, (4.) R. (4.4) is the frequecy of the fudametal thickess-shear mode of a crystal plate with a film of a uiform thickess h. We are iterested i the eergy-trappig modes with the correspodig frequecy withi. (4.5) These modes are large ear the plate ceter ad decay rapidly away from the ceter.

85 Free Vibratio Solutio As a partial differetial equatio, (4.) is separable. Let The from (4.) we ca obtai x, x ) F ( x ) F ( ). (4.6) ( x F 55 F,, F F R R f ( x ) F f ( x ) F,, (4.7) where the separatio costats ad must satisfy the followig equatio which gives the resoat frequecies oce ad are determied:. (4.8) To be specific, we cosider the case whe x) x, f( x) f ( x. (4.9) Correspodigly, F 55 F,, F F R x R x F F,. (4.4) Equatios similar to (4.4) were also ecoutered i the study of cotoured resoators with varyig thickesses [9-9]. (4.4) is mathematically aalogous to Schrodiger s equatio for a quatum mechaical harmoic oscillator [94]. The two equatios i (4.4) have the same structure. Cosider (4.4) first. Itroduce the followig chage of variable: The it ca be verified that whe x. (4.4) A (4.4) is trasformed ito / 4 A, (4.4) R

86 69 d d F( F( ) ), (4.4) where A or. (4.44) A (4.4) is the well-kow Weber s equatio [9,9]. Let Thus, equatio (4.4) becomes / F ( ) H( ) e. (4.45) d d H dh d H. (4.46) For odd itegers of λ, the solutio of (4.46) is the Hermite polyomials of differet orders, i.e. [94], where [94,95], H( ) H ( ),,,,, (4.47) H H H H.,, 4 8,, (4.48) (4.4) ca be treated similarly. Substitutig (4.4), (4.), (4.9), (4.44) ad a similar solutio from (4.4) ito (4.8) gives the resoat frequecies as 55 R. (4.49) A A

87 Numerical Results As a umerical example, cosider a AT-cut quartz resoator with h mm, ad R =5%. I real applicatios R is usually less tha. A exaggerated value of R is chose to show its umerical effects more clearly. I this case the thickess-shear mode without a film has a frequecy of f /( ) =,654,68 Hz. The frequecy whe the plate is with a uiform mass layer of R =5% is f /( ) =,569,77 Hz. =4,444.4 m - is used. For this value of, the film thickess decreases to zero whe its radius is 5 mm, a practically reasoable size. Calculatios show that i this case the operatig mode of, ) =(,) i fact essetially decays to zero before it ( reaches the film boudary ad thus does ot really feel the film boudary. Figure 4. shows the first six modes i the iterval of f, f ) i the order of ( icreasig frequecy., ) =(,), (,), (,), (,), (,), ad (,). f (,) =,586,98, ( f (,) =,6,9, (,) f =,66,4, f (,) =,67,6, f (,) =,6,98, f (,) =,64,84 Hz. The displacemets of all these modes are essetially trapped uder the film. The modes are with differet umbers of odal lies i the x ad/or x directios. Roughly, higher order modes have higher frequecies. (a) (b)

88 7 (c) (d) (e) (f) Figure 4.: Distributio of the thickess-shear displacemet of the first six trapped modes i the order of icreasig frequecy. (a): (, ) = (,), (b): (, ) = (,), (c): (, ) = (,), (d):, ) = (,), (e):, ) = (, ), (f):, ) = (,). ( ( ( Figures 4. ad 4. show the effects of R ad o the most useful mode i devices with, ) =(,). Cosiderig the i-plae material aisotropy of the plate, ( both the mode variatios alog the x ad/or x directios are show separately. It is see that whe R becomes larger or the film becomes thicker, the frequecies become lower ad the modes are pushed toward the ceter (Figure 4.). Whe becomes larger, i.e., the film thickess variatio is more rapid or the film becomes smaller, the frequecies become higher ad modes are also pushed toward the ceter (Figure 4.).

89 7 (a) (b) Figure 4.: Effects of R o the first mode, =4,444.4 m -. f (,) =,67,7,,586,98, ad,555,5 Hz whe R =%, 5%, ad 7%. (a): x depedece; (b): x depedece.

90 7 (a) (b) Figure 4.: Effects of o the first mode, R =5%. f (,) =,58,685,,586,98, ad,595,58 Hz whe =,5, 4,444.4, ad, m -. (a): x depedece; (b): x depedece. 4.4 Summary Either the ouiform film is modeled by stepped thickess films or by a gradually varyig two-dimesioal fuctio, the coclusios are similar. A ouiform film,

91 74 thicker at the ceter, teds to trap more modes with odal poits uder the film ad push the modes toward the ceter. I the regio where the film is gradually gettig thier, the vibratio may be reduced sigificatly from oscillatory to decayig. The frequecies of the modes are slightly lowered whe the film becomes thicker i the ceter. These effects cause cosiderable complicatios i mass sesig. They also have implicatios i the actual operatio of electrically forced vibratios of a quartz crystal resoator. If a trapped mode with a odal poit is uder the drivig electrodes, the chage of sig of the shear deformatio across the odal poit causes charge cacellatio o the electrodes ad affects the impedace of the device.

92 75 Chapter 5 Effects of Film Arrays I this chapter, we cocetrate o the effects of periodic or operiodic film array deposited o a sigle quartz plate. This study is based o the desig of multiple quartz crystal microbalaces (QCMs) located o a sigle quartz plate. Each quartz crystal microbalace is deposited with a differet fuctioal film, thus the overall structure ca work as a multifuctioal sesor. The iteractios betwee the QCMs are to be miimized by desig. The purpose of the study i this chapter is to uderstad the vibratio modes ad frequecies of a quartz plate uder the ifluece of a mass film array deposited o its upper surface. This study is the iitial step towards a practical desig of such kid of multifuctioal sesor. I Sectio 5., we first cocetrate o the free thickess-shear vibratios of a two-dimesioal, periodic array of QCMs modeled by mass films with gradually varyig thickess [96]. A theoretical aalysis is performed usig the two-dimesioal equatio for quartz plate with surface film derived i Sectio.7. It is show that mathematically

93 76 the problem is govered by Mathieu s equatio with a spatially varyig coefficiet. A periodic solutio for resoat frequecies ad modes is obtaied with which the effects of the films are examied. Results show that the vibratio may be trapped or o-trapped uder the film. The trapped modes decay differetly i the two i-plae directios of the plate. The mode shapes ad the decay rate of the trapped modes are sesitive to the film thickess. Next, we focus o the effects of a operiodic film array i Sectio 5. [97]. The two-dimesioal equatio for quartz plate with surface film previously used is ot able to capture the complicated boudary coditios imposed by the operiodic films o the crystal plate. We tur to use the three-dimesioal equatios of aisotropic elasticity to study thickess-shear (TSh) ad thickess-twist (TT) vibratios of a quartz plate with a operiodic oe-dimesioal array of surface films. A aalytical solutio is obtaied for the free vibratio eigevalue problem usig Fourier series from which the resoat frequecies ad mode shapes are calculated. 5. Periodic Arrays 5.. Itroductio Oe-dimesioal arrays of resoators [69], QCMs [-5] ad trasducers [7] were aalyzed usig equatios for piezoelectric plates. I [-5,69,7] the electrodes or mass layers o the quartz crystal resoators are uiform. I this sectio, we study the more geeral situatio of a moolithic array of QCMs. The curret aalysis is differet from the array aalyses i [-5,69,7] i two aspects. Oe aspect is that the mass films are of cotiuously varyig thickess. The other is that we cosider a two-dimesioal array.

94 Goverig Equatios Cosider a AT-cut quartz plate as show i Figure 5.. A periodic array of rectagular QCMs is o top surface of the plate. Oly ie of them are show. The plate has a uiform thickess h ad a mass desity ρ. There is a idetical thi mass film with a slowly varyig thickess o the top of each QCM. The desity of the mass film is. Its varyig thickess is h ( x, x ). The specific form of this fuctio will be give later. The film is assumed to be very thi. Oly its iertia will be cosidered. Its stiffess will be eglected [86,87]. Figure 5.: A quartz plate for QCMs ad coordiate system. For a frequecy aalysis, we eglect the small piezoelectric coupligs of quartz as usual. As show i equatio (4.), the pure thickess-shear vibratio of quartz plate ca be described by u x, x, x, t) x ( x, x, t), u, u, (5.) ( where x, x, ) is the plate fudametal thickess displacemet. Through a similar ( t procedure as illustrated i Sectio 4.., the goverig plate equatio for is writte as 55 [ ( )] R x x, (5.)

95 78 where R h h is the mass ratio betwee the overall film array ad quartz plate, besides c66, (5.) h is the frequecy of the fudametal thickess-shear mode of the quartz plate whe the film array is ot preset. The film array thickess is writte as h h [ f ( x ) f ( )], (5.4) x where h is the film array ceter thickess at each QCM, f ad f are small ad slowly growig periodic fuctios so that the film array is thick at the ceter ad thi at its edges at each QCM. Similar as show i Sectio 4.., the plate equatio ca be writte as 55 [ ( )] R f f x x, (5.5) where R h h. R is the frequecy of the fudametal thickessshear mode of the quartz plate with a film of a uiform thickess h. We are maily iterested i the eergy-trapped modes with the frequecy withi. (5.6) 5.. Free Vibratio Aalysis As a partial differetial equatio, (5.5) is separable. Let x, x ) F ( x ) F ( ). (5.7) ( x The from (5.5), by the stadard procedure of separatio of variables, we obtai

96 79 F 55 F,, F F R R f ( x ) F f ( x ) F,, (5.8) where the separatio costats ad must satisfy the followig equatio which gives the resoat frequecies oce ad are determied:. (5.9) To describe a film array with a periodically varyig thickess, we choose h x h cos a h x cos a x cos 4 4 a h x cos 4 4 a, (5.) where a ad a are the legth ad the width of a idividual QCM, respectively. I a typical QCM, e.g., the oe at the origi with x a ad x a, the film described by (5.) assumes its maximal thickess h at the ceter (,) ad it vaishes at the four corers of the rectagle. Therefore (5.) describes a film thick at the ceter ad thi at the edges. I other QCMs of the array, we have the same ouiform film because of the periodicity of (5.). From (5.) ad (5.4) we idetify x x f cos cos, f. (5.) 4 4 a 4 4 a The (5.8) ca be writte as x F, R cos F, (5.) a x 55F, R cos F, (5.) a where for coveiece we have itroduced ad which satisfy the followig equatio from (5.9):

97 8 R. (5.4) (5.) ad (5.) are the well-kow Mathieu s equatio [98]. For the periodic array i Figure 5., we are iterested i periodic solutios of (5.) ad (5.) that are eve fuctios of x ad x. Therefore, accordig to Fourier series, we let cos x a A A F, cos x a B B F, (5.5) where A ad B (=,,, ) are udetermied coefficiets. (5.) ad (5.) ca be solved i the same way. We focus o (5.) i the followig. Substitutig (5.5) ito (5.), usig the relevat trigoometric idetity to covert the product terms ito sums, we obtai. cos 4 ] ) ( [ 4 cos 4 ] ) ( [ 4 x a A R A a A R a x A R A a A R A R A (5.6) Multiplyig both sides of (5.6) by ) / cos( x a m with m=,,,, itegratig the resultig expressio over a period (-a,a ), usig the orthogoally of the trigoometric fuctios, we arrive at the followig recurrece relatios which are liear homogeeous equatios for A m :.,,4,, 4 ] ) ( [ 4, 4 ] ) ( [, 4 m A R A a m A R A R A a A R A R A m m m (5.7)

98 8 For otrivial solutios of A m, the determiat of the coefficiet matrix of (5.7) must vaish, which gives the frequecy equatio that determies. The correspodig otrivial A m obtaied from (5.7) determies F through (5.5). I a similar way, ad B m ca be obtaied by solvig (5.). The ca be obtaied from (5.4). These will be doe umerically o a computer Numerical Results For umerical examples we fix h= mm ad a =a = cm which are typical for a QCM. We cosider the case of R =5%. I this case f Hz ad f Hz. Table 5. shows the results of umerical tests usig three, five, or te terms i the Fourier series. Three resoat frequecies, f, f, ad f, are foud withi. It ca be see that whe usig five ad te terms the frequecies are already idistiguishable. Numerical tests also show that whe usig te terms i the series the modes correspodig to the three frequecies also become stable. This is as expected because, as to be see later, the behaviors of the modes we are iterested i are relatively simple, with oly a few i-plae oscillatios ad therefore ca be well approximated by a few terms i the Fourier series. Our calculatios below will be based o te terms i the series. Table 5.: Resoat frequecies showig covergece. Number of terms i the Fourier Series f (Hz) f (Hz) f (Hz)

99 8 (a) (b) (c) Figure 5.: The three modes show i a sigle QCM (a) f ; (b) f ; (c) f. For the three modes foud, Figure 5. shows their vibratio distributio i a sigle QCM, e.g., the oe withi x a ad x a. The first mode is with the frequecy f, the lowest frequecy. The vibratio is large ear the ceter ad decays to almost zero ear the edges. Therefore this is a well trapped mode. For this mode eighborig QCMs have little iteractio which is the desired situatio for a QCM array. The secod mode is with f. It is well trapped alog x ad is early zero at x a. However, alog x it chages its sig with two odal poits ad has a large amplitude ear x a. Therefore, for this mode, i the x directio, the vibratio leaks out of the mass layer ad eighborig QCMs begi to iteract sigificatly. Similarly, the third

100 8 mode with f is well trapped alog x but has two odal poits alog x ad a large amplitude ear x a. To see the variatios of the three modes foud more clearly, i Figure 5. we plot their x depedece ad x depedece separately for a array of QCMs. Clearly, the first mode is trapped i both directios, the secod mode is trapped i the x directio oly, ad the third mode is trapped i the x directio oly. We ote that the modes decay faster i the x directio tha i the x directio. Therefore rectagular QCMs with a a are more reasoable tha square QCMs for device miiaturizatio. Similarly, elliptical QCMs are more reasoable tha circular oes. I additio, Figure 5. also shows that eve for the first mode there is still some vibratio left betwee eighborig QCMs. (a)

101 84 (b) Figure 5.: x depedece (a) ad x depedece (b) of the three modes i a array. Solid lie: the st mode. Dash-dot lie: the d mode. Dash lie: the rd mode. Figure 5.4 shows the effect of the mass ratio R o the vibratio distributio of the first mode for a array of QCMs. A large R represets a mass layer thicker at the ceter. The figure shows that a thicker mass layer is associated with fast decay of the vibratio amplitude from the ceter ad smaller vibratio amplitude betwee eighborig QCMs or less iteractios amog them. (a)

102 85 (b) Figure 5.4: Effects of R o vibratio distributio of the st mode alog x (a) ad x (b) i a array. Dash-dot lie: R =%. Solid lie: R =5%. Dash lie: R =7% For a more visual presetatio, i Figure 5.5 we show the vibratio distributio of the first mode i a array of QCMs. Figure 5.5: Vibratio distributio of the first mode i a array.

103 86 5. Noperiodic Arrays 5.. Itroductio I this sectio we study the thickess-shear (TSh) ad thickess-twist (TT) vibratios of a moolithic, operiodic array of QCMs. Istead of usig the approximate plate equatio, we will use the exact three dimesioal equatios of aisotropic elasticity with the omissio of the small elastic costat c 56. We will use a global Fourier series ad hece do ot eed to break the array ito regios with or without films. A sigle QCM is treated this way i [99], we expad the result to oe-dimesioal array of QCMs. We wat to study the free vibratio eigevalue problem so that we ca examie the basic vibratio characteristics of a QCM array icludig resoat frequecies ad vibratio modes. 5.. Goverig Equatios A example of rotated Y-cut quartz plate with operiodic film array is show i Figure 5.6. The x directio is determied from x ad x by the right-had rule. The plate is ubouded i the x directio ad does ot vary alog x. Figure 5.6 shows a cross sectio. It carries a array of K differet films at the top surface. The case of K=5 is show. The k th layer is withi ak x bk. It has a desity k ad thickess h k. The films are assumed to be very thi. Their iertia will be cosidered but their stiffess will be eglected [86,87]. For free vibratio frequecy aalysis the small piezoelectric couplig of quartz is eglected as usual ad a elastic aalysis is sufficiet [45,49].

104 87 x h k x x =a k x =b k h x =c Figure 5.6: A operiodic array of QCMs. We cosider the case whe the displacemet of the quartz plate ca be approximated as u u( x, x, t), u u, (5.8) where the oly ozero displacemet compoet u is depedet o both x ad x, which form a plae ormal i x directio. The motio described by (5.8) is called shearhorizotal or atiplae motio. For moocliic crystals, this kid of motio is allowed by the liear theory of aisotropic elasticity. I acoustic wave devices, the correspodig modes are called thickess-twist modes i geeral ad iclude thickess-shear (u oly deped o x ) ad face-shear (u oly deped o x ) modes as special cases. From (.5) ad (.9) the oly two ozero strai compoets are: S S u, S S u. (5.9) 5, 6, For Y-cut quartz, the correspodig ozero stress ca be writte as T c u c u, T c u c u, (5.) 55, 56, 56, 66, Based o (.), without body force, the equatio of motio is T T u. (5.),, The goverig equatio to be satisfied by u is obtaied by substitutig (5.) ito (5.): c u c u c u u. (5.) 66, 55, 56,

105 88 For the plate i Figure 5.6, the boudary coditio at the plate top surface is T khk u, x h, ak x bk, k,,, K,, x h, elsewhere. (5.) I ak x b, the boudary coditio i (5.) represets Newto s secod law applied k to the film. Similarly, the boudary coditio at the plate bottom surface is simply The boudary coditios at the left ad right edges are T. (5.4) T, x, c, x h. (5.5) Due to the c 56 term with mixed derivatives, (5.) is ot separable ad thus presets cosiderable mathematical challeges. I spite of the importace of the problem, kow theoretical results are limited to propagatig waves i ubouded plates. For the practically more useful case of fiite rectagular plates there are little theoretical results. The fact that the films covers oly part of the plate surface ad the related piecewise boudary coditio i (5.) makes the problem eve more challegig. Therefore for quite some time there has bee little theoretical progress o this topic. For AT-cut quartz 9 plates, c 55 =68.8, c 56 =.5, ad c 66 =9. N/m (see Equatio.5). c 56 is very small compared to c 55 ad c 66. Therefore, we will make the usual approximatio of eglectig the small c 56 []. 5.. Fourier Series Solutio Cosider free vibratios of the quartz plate. Let u ( x, x, t) u ( x, x )exp( it). (5.6) We costruct the followig solutio from separatio of variables:

106 89 u A cos( x ) B si( x ) m x [ Am cos( mx) Bm si( mx)]cos, m c (5.7) where A, B, A m ad B m are udetermied costats, ad c55 m c55 h m m, m,,,, c66 c66 c 4h s c66 c, (5.8) h c 66 s. (5.9) (5.7) satisfies the goverig equatio (5.) ad boudary coditio at two edges (5.5) whe the small c 56 is eglected. s is the resoat frequecy of the fudametal thickess-shear mode i a ubouded quartz plate whe the film array is ot preset. Quartz plate devices are usually with large legth/thickess ratios, i.e., c>>h. I this case, for a m that is ot large, m is positive. We are iterested i thickess-shear ad thickess-twist modes with o more tha a few odal poits alog the x directio for which a large m is ot eeded. I the case whe a large m is ideed eeded, we ca redefie m with a mius sig ad chage the sie ad cosie fuctios i (5.7) ito hyperbolic sie ad hyperbolic cosie fuctios. To apply the boudary coditios at the plate top ad bottom, from (5.) ad (5.7) we have T c u c A si( x ) c B cos( x ) 66, m x 66 [ m m si( m ) m m cos( m )]cos. m c c A x B x (5.) Substitutio both of (5.7) ad (5.) back ito boudary coditios (5.) ad (5.4) gives

107 9 c A si( h) c B cos( h) m x c [ Am m si( mh) Bm m cos( mh)]cos c 66 m ak x bk, k,,, K,, elsewhere, khk { A cos( h) B si( h) m x [ Am cos( mh) Bm si( mh)]cos }, m c (5.) c A si( h) c B cos( h) m x c A h B h c 66 m [ m m si( m ) m m cos( m )]cos. (5.) We multiply both sides of (5.) by cos( x / c) with =,,, ad itegrate the resultig expressio over (,c) to obtai cc A si( h) cc B cos( h) [ A cos( h) B si( h)] C m [ A cos( h) B si( h)] C,, m m m m m (5.) c c 66[ A si( h ) B cos( h )] [ A cos( h) B si( h)] C m [ A cos( h) B si( h)] C,,,,, m m m m m (5.4) where K C h b a k k k ( k k ) K c mbk mak Cm khk si si Cm, k m c c. (5.5) x mx C h dx C K b k m k k cos cos m, ak k c c m,,,,. Similarly, from (5.) we have c A si( h) c B cos( h),, (5.6) 66 66

108 A si( h) B cos( h),,,,. (5.7) (5.), (5.4), (5.6) ad (5.7) are liear homogeeous equatios for A, B, A m ad B m. For otrivial solutios the determiat of the coefficiet matrix has to vaish, which 9 determies the resoat frequecies. The otrivial solutios of A, B, A m ad B m determie the correspodig modes. This is a complicated eigevalue problem because the eigevalue or the resoat frequecy is preset i every m. To simplify these equatios a little, we solve (5.6) ad (5.7) for A ad A : A B cot( h),, A B cot( h),,,,. (5.8) Substitutio of (5.8) ito (5.) ad (5.4) results i liear equatios for B ad B m oly Numerical Results We also itroduce khk s s( R), R Max{ R k }, Rk. (5.9) k h R k is the mass ratio betwee the k th film ad the plate. s is the resoat frequecy of the fudametal thickess-shear mode i a ubouded quartz plate fully covered by a uiform-thickess film with the largest mass ratio. The modes with the resoat frequecies withi s are determied sice we are iterested i x distributio of s u at the top of the plate surface while the quartz plate is maily doig a fudametal thickess-shear vibratio. I Figure 5.7 we show the results of a array of two QCMs with the oe o the right a little thicker tha the oe o the left. The exact dimesios are a.75 mm,

109 b 8.75 mm, a.5 mm, b 6.5 mm, c mm ad h. 5 9 mm. To show the results more clearly we use films thicker tha those i ormal applicatios of QCMs. Twety-oe terms of the series are used i the calculatio. The first resoat frequecy calculated from usig ietee ad twety-oe terms are =9958 rad/s ad 9955 rad/s, respectively, which is cosidered sufficietly accurate. If twety-four or more terms are used i the series, the m i Equatio (5.8) will become egative. Six modes are foud withi s s. The first mode has vibratio uder the thicker film o the right oly. This mode has o odal poits (zeros) alog the x directio. The thicker film has more iertia ad lowers the cutoff frequecy s of the crystal plate more tha the thier film. Therefore the vibratio occurs uder the thicker film first. At a higher frequecy the secod mode appears which has vibratio uder the thier film o the left oly, which does ot have a odal poit. The third ad fourth modes have vibratio uder the thicker ad thier films, respectively. Each mode has oe odal poit. Similarly, the fifth ad six modes have two odal poits each. Oce a odal poit appears, the charge o the drivig electrodes due to piezoelectric couplig will cacel with each other at least to some degree. This will reduce the electrical sigals ad is udesirable i geeral. x R =8% R =% x x =a x =b x =a x =b h x =c

110 rad/s rad/s rad/s rad/s rad/s rad/s Figure 5.7: Vibratio distributio i a array of two QCMs with oe thicker tha the other. Figure 5.8 shows the result for a array of two QCMs with the oe o the right slightly loger tha the oe o the left. The geometric parameters are a.5 b 7.5 mm, a mm, b 7.5 mm, c mm ad h. 5 mm, mm. Agai twety-oe terms of the series are used i the calculatio. Seve modes are foud withi s s. Except the sixth mode, the other six modes i Figure 5.8 are similar to the six modes i

111 94 Figure 5.7. Roughly a loger film has more mass ad affects the vibratio i a way similar to a thicker mass layer. x R =8% R =8% x x =a x =b x =a x =b h x =c rad/s rad/s rad/s rad/s

112 rad/s rad/s rad/s Figure 5.8: Vibratio distributio i a array of two QCMs with oe loger tha the other. I Figure 5.9 we have a array of four QCMs amog which two are idetical (the first ad the third from the left), oe is slightly thicker (the third from the left), ad oe is slightly loger (the fourth from the left). They are with a. 75 mm, b mm, a.75 mm, b mm, a. 75 mm, b mm, a mm, b4 4.5 mm, c 45mm ad h. 5mm. This is a loger plate ad forty-ie terms i the series are used i the calculatio to esure sufficiet accuracy. Twelve modes are foud withi s s. It ca be see that the plate ca vibrate i complicated maers. The vibratio may be uder oe or two films, with differet umber of odal poits.

113 96 R =8% R =% R =8% R 4 =8% x x =a x =a x =a x =a 4 x x =b x =b x =b x =b 4 h x =c rad/s rad/s rad/s rad/s rad/s rad/s

114 rad/s rad/s rad/s 847.8rad/s rad/s rad/s Figure 5.9: Vibratio distributio i a array of four QCMs. 5. Summary For a periodic array of QCMs with ouiform mass films, the solutio to the twodimesioal plate equatio is obtaied by solvig the Mathieu s equatio with a spatially varyig periodic coefficiet whose periodic solutio ca be obtaied by Fourier series. Numerical results show that there exist both trapped ad o-trapped modes. The trapped

115 98 modes decay differetly alog x ad x. Therefore rectagular ad elliptical QCMs are better desigs tha square ad circular oes. The trapped modes decay faster for mass layers thicker at the ceter. For a operiodic array of QCMs, the exact three-dimesioal equatios of aisotropic elasticity are used to study the effects of oe-dimesioal film array. For ATcut quartz, the small elastic costat c 56 is omitted. Solutio is obtaied by Fourier series. Results show the vibratio may be maily trapped i some of the QCMs but ot i others. Specifically, the vibratio teds to occur uder thicker or loger mass layers first. The umber of odal poits alog the x directio may also vary.

116 99 Chapter 6 Effects of Fiber Arrays I this chapter we study the case of usig a crystal resoator to carry a array of microfibers or aofibers mouted o top of the resoator. By detectig the frequecy shift caused by these mouted fibers, it is possible to characterize the geometric/physical properties of these tiy fibers. We cosider two types of vibratio modes of the crystal resoator: thickess-stretch (TSt) ad thickess-shear (TSh). The two cases are elaborated i detail i the followig two sectios. I sectio 6., we aalyze thickess-stretch vibratios of a plate of hexagoal crystal carryig a array of fibers with their bottoms fixed to the top surface of the plate []. The fibers udergo logitudial vibratios whe the crystal plate is i thickessstretch motio. The plate is modeled by the theory of aisotropic elasticity. The fibers are modeled by the oe-dimesioal structural theory for extesioal vibratio of rods. A frequecy equatio is obtaied ad solved usig perturbatio method. The effect of the fiber array o the resoat frequecies of the crystal plate is examied. The results are

117 potetially useful for usig thickess-stretch modes of hexagoal crystal plates for measurig the geometric ad physical parameters of the fibers. I sectio 6., we study thickess-shear vibratios of a rotated Y-cut quartz crystal plate carryig a array of fibers with their bottoms fixed to the top surface of the plate []. The fibers udergo flexural vibratios whe the plate is i thickess-shear motio. The quartz plate is also modeled by the theory of aisotropic elasticity. The fibers are modeled by the Euler-Beroulli theory for beam bedig. A frequecy equatio that determies the resoat frequecies of the structure is derived. A aalytical solutio o fiber-iduced frequecy shift is obtaied usig a perturbatio procedure. It is also show that the frequecy shift may be used to characterize geometric/physical properties of the fiber array. 6. Thickess-Stretch of Plate ad Extesio of Fibers 6.. Itroductio Recetly, due to the extesive effort o micro- ad ao-techologies, various micro- or ao-scale fiber arrays ca be made usig differet techiques [-7]. These ew structures have great potetials for ew devices icludig dyamic tuig of surface wettig, dry adhesives that mimic gecko foot fibrillars, efficiet microeedles i drug delivery, substrates for sesig cell respose ad MEMS actuators. There is a strog eed to measure the geometric ad physical parameters of these small fiber arrays. This sectio describes the possibility to use a thickess-stretch mode crystal resoator ad extesioal

118 vibratios of microfibers to measure the properties of the micro or eve aofibers. I terms of mechaics, the microfibers may be more suitably called micro-rods whe they are i extesioal motio. To demostrate this idea, we costruct a theoretical model of a plate of hexagoal crystals carryig a array of micro-rods with their bottoms fixed to the top surface of the plate (see Figure 6.). Whe the crystal plate is i thickess-stretch motio, the rods udergo extesioal vibratios. The axial forces at the bottoms of the rods exert a ormal force o the plate surface ad thereby affectig the resoat frequecies of the plate. For a typical crystal plate, its thickess is of the order of mm. We cosider small rods whose effect o the resoat frequecies of the plate is a small perturbatio which will be determied by a theoretical aalysis. We ca show that, through the rod array-frequecy effect, iformatio about the rod array ca be extracted. 6.. Plate Thickess-Stretch Motio x P h x Figure 6.: A crystal plate carryig a micro-rod array. The six-fold axis of the hexagoal crystal plate is idicated by a arrow labeled with P i Figure 6.. As metioed at the ed of Sectio., the matrices of the elastic costats for hexagoal crystals ad polarized ceramics (trasversely isotropic) are similar, so the

119 aalysis below is also valid for plates of polarized ceramics. We are cosiderig a frequecy effect which is mechaical i ature. To illustrate the mai idea, a elastic aalysis is sufficiet. Therefore, for the hexagoal crystal plate, we use the equatios of aisotropic elasticity ad eglect piezoelectric couplig which is ecessary i a electrically forced vibratio aalysis. I free vibratio frequecy aalysis, the piezoelectric stiffeig effect also causes a frequecy shift whe there are o rods o the plate, but compared to the rod-iduced small frequecy perturbatio, this piezoelectric stiffeig effect is a higher-order small effect ad is igored i the aalysis. Cosider the followig displacemet field, u u x )exp( i t), u u. (6.) ( Sice we are maily iterested i the effects caused by the micro-rods, we assume the crystal plate vibrates i pure thickess-stretch motio. From (.) ad (.5), for hexagoal crystal, the otrivial compoets of the strai ad stress tesors are S u,, (6.) T T c, T cu,, (6.) u, where the time-harmoic factor has bee dropped for coveiece. From (.) the relevat equatio of motio is T, cu, u. (6.4) The geeral solutio to (6.4) ad the correspodig expressio for the stress compoet eeded i the relevat boudary ad cotiuity coditios are where B is a udetermied costat, ad u T Bcos ( x ), (6.5) h c h Bsi ( x ), (6.6)

120 c. (6.7) We ote that the tractio-free boudary coditio T at the plate bottom x h is already satisfied by (6.6). 6.. Rod Extesio z L x Figure 6.: A sigle rod ad coordiate system. For the rods, we use the oe-dimesioal theory for logitudial motios [8,9]. Correspodig to the thickess-stretch vibratio of the plate, for steady-state timeharmoic motios, all rods vibrate i phase. For a typical rod (see Figure 6.), let the logitudial displacemet be w( z)exp( i t), ad the axial force be P. We have P Aw, (6.8) P EAw, (6.9) where E is the Youg s modulus, is the mass desity of the rod, ad A is the area of the rod cross sectio. A prime idicates a derivative with respect to z. A superimposed dot is a time derivative. It ca be foud i a straightforward maer that the solutio to

121 4 (6.8) that satisfies the free ed coditio at the top of the rods, i.e., P ( L) EAw( L), is give by w Df ( z, E,, L, ), (6.) where D is a udetermied costat ad f cos( z L),. E (6.) From (6.9) ad (6.) we calculate the axial force at the rod bottom as P( z ) EADf (, E,, L, ). (6.) 6..4 Plate-Rod Iteractio ad Frequecy Equatio Let the umber desity of the rods per uit area of the plate surface be N. The cotiuity coditios of the displacemet ad stress at x h are u ( x T ( x h) w( z ), h) NP( z ). (6.) Strictly speakig, the bottom of a rod exerts a small ad essetially cocetrated ormal force o the plate surface. This force produces complicated local stress ad strai fields. The resultat of the local stress field is equal ad opposite to the axial force i the rod at its bottom. We are maily iterested i the effect of this resultat o the thickess-stretch vibratio frequecy of the plate rather tha the details of the local stress field. Therefore, i (6.), we have assumed that the rods are small ad N is sufficietly large. I this case there are eough rods per uit area of the plate surface. The a average of the ormal forces o the plate surface ca be calculated. Substitutig (6.5), (6.6), (6.) ad (6.) ito (6.), we obtai

122 5 Bcosh Df (, E,, L, ), cb si h NDEAf (, E,, L, ). (6.4) (6.4) is a system of liear homogeeous equatios for B ad D. For otrivial solutios the determiat of the coefficiet matrix has to vaish. This yields the followig equatio that determies the resoat frequecies: where ξ is related to ω through (6.7). NEAf (, E,, L, ) tah, (6.5) c f (, E,, L, ) 6..5 Frequecy-Depedet Equivalet Mass Layer For compariso, cosider a simpler problem i which there is a thi mass layer with desity ad thickess h attached to the top surface of the crystal plate at x h. I this case, accordig to Newto s secod law, the boudary coditio at x h is T h u ad the frequecy equatio ca be easily obtaied as tah Rh, (6.6) where R ( h) /(h) is the mass ratio betwee the mass layer ad the crystal plate. I our case of a rod array o a crystal plate, the frequecy equatio (6.5) ca be writte i the same form as (6.6), with NEAf (, E,, L, ) NEA si( L) R. (6.7) c h f (, E,, L, ) c hcos( ) L (6.7) may be viewed as a effective mass ratio betwee the rod array ad the crystal plate. Equivaletly, R h is the effective mass of the rod array over uit area of the crystal plate surface. Clearly, R depeds o EA, ad L which are the material ad

123 6 geometric parameters of the rods. R also depeds o N which is a array property. We ote that R is frequecy depedet through i (6.) ad i (6.7). Whe f (, E,, L, ) =, i.e., the rod bottoms have o displacemets, R becomes ifiite. I this case f (, E,, L, ) = determies a series of resoat frequecies for a rod with fixed bottoms ad free top. We deote the resoat frequecies by m=,,,. These frequecies will be eeded later. They are give by [9] m, with ( ) E,,,... (6.8) L 9 For a umerical example we cosider a ZO plate with c =.6 N/m, ρ=5665 kg/m, ad h=.5 mm. For the rods we also cosider ZO with =5665 kg/m, 8 E=4 GPa [], diameter D=. μm, L= μm, ad N= /cm. I this case rad/s ad.597 rad/s. We plot the frequecy depedece of R i Figure 6. where R is ormalized by R NAL/( ) which is the h static mass ratio betwee the rod array ad the crystal plate. Figure 6. shows that R is strogly frequecy depedet. At the low frequecy limit R / R approaches oe which is the static mass ratio. R may be smaller or larger tha oe, ad may eve become egative. This is as expected for frequecy depedet masses as also see i the study of particle- QCR iteractios [] ad metacomposites [] of materials with iteral structures ad/or iteral degrees of freedom. I the problem we are aalyzig the frequecy depedece of the effective mass layer is due to the extesibility of the rods. Near the rod resoat frequecies (oly the first two are show i the frequecy rage i Figure 6.), R becomes large ad is i fact ubouded because there is o dampig i the idealized elastic structure we are aalyzig.

124 7 Figure 6.: Normalized effective mass ratio versus frequecy Approximate Frequecy Solutio To exhibit the relatio betwee the resoat frequecies ad the rod array parameters more explicitly, we look for a perturbatio solutio of (6.5) i the case of small R whe the rods represet small effects o the crystal plate. For the zero-order solutio, we simply eglect the rods ad set the right-had side of (6.5) to zero. This results i two sets of resoat frequecies: si cos c h, h /,,4,6,,, (6.9) h c h, h /,,,5,,, (6.) h where the superscript idicates that they are the zero-order approximatio i the perturbatio procedure. Through (6.5), it ca be idetified that (6.9) ad (6.) represet modes atisymmetric ad symmetric about x, respectively. From a device poit of view we are oly iterested i the symmetric modes i (6.) which ca be coveietly

125 8 excited by a electric field i the plate thickess directio. Therefore, i the followig, we focus o the symmetric modes. For the first-order perturbatio, we write [] where R, we obtai where h,,,5,, (6.) is a small perturbatio. Substitutig (6.) ito (6.5), for small ad small NEAf (, E,, L, ) R, (6.) c f (, E,, L, ). (6.) c h ( ) (6.) implies the followig frequecy shift through (6.) ad (6.7): where NEA si( L) R, (6.4) c ( ) hcos( L). (6.5) E We ote that the right-had side of (6.4) does ot deped o the ukow frequecy ow ad istead it is a fuctio of, which is kow. I the above derivatio, sice R has bee assumed small, the special case whe a particular plate resoat frequecy is close to a particular beam resoat frequecy f (, E,, L, ) approaches zero ad R becomes large. m has to be excluded. I such a case The frequecy shift i (6.4) is a fuctio of the parameters of the rod array. This provides the theoretical foudatio for determiig the parameters of the array through

126 9 frequecy measuremets. The procedure is as follows: are kow theoretically. are measured from experimets. The are kow, ad (6.4) becomes a system of equatios of the parameters of the rod array whe several frequecies are used. These equatios are relatively complicated trascedetal equatios. It does ot seem to be easy to measure all of the parameters of the array at the same time. A relatively simple situatio is whe some of the parameters of the array are already kow ad oly oe or two parameters are left to be determied. The it may be coveiet to solve (6.4) Special Cases I the special case whe the resoat frequecies of the rods are much higher tha the fudametal thickess-stretch frequecy of the ZO plate, the rods follow the surface of the plate i the so-called quasi-static maer. I this special case we show that our geeral result i the above aalysis reduces to a expressio similar to the well-kow Sauerbrey equatio for a quartz plate carryig a mass film i thickess-shear vibratio. From (6.5) we write the plate surface displacemet as Bcos h ad the plate surface acceleratio as ( ) Bcos h. I quasi-static motios, the rod acceleratio is take to be the same as the plate surface acceleratio, i.e., w w w u E E w Bcos h, w ( E w( L). ) Bcos h, (6.6) (6.6) determies w Bcos( h)[ ( ) ( z Lz) ], (6.7) E or

127 f ( E ) ( z Lz). (6.8) Substitutig (6.8) ito (6.), we obtai NAL R, (6.9) h where R is the static mass ratio betwee the rods ad the plate. I Figure 6.4 we plot the frequecy shifts predicted by (6.4) ad (6.9) for the fudametal thickess-stretch mode with = by solid ad dotted lies, respectively. Equatio (6.9) (dotted lie) predicts a simple liear relatio, ideal for sesor applicatio. This is true for relatively short rods whose first resoace is much higher tha the fudametal thickess-stretch frequecy of the plate, i.e.,. As the rods become loger, whe L.79 4 m ad the first resoat frequecy of the rod approaches the fudametal thickess-stretch frequecy of the plate, i.e.,, (6.9) is o loger valid ad (6.4) predicts a jump discotiuity. Sice we did ot iclude dampig, the discotiuity is sigular which does ot matter for the applicatio we are cosiderig. Near the jump, (6.4) is ot valid either but after the jump (6.4) is valid ad (6.9) is ot. Therefore (6.4) is more geeral tha (6.9). Whe the rod legth is further icreased, whe L m ad the secod resoat frequecy of the rod coicides with the fudametal plate thickess-stretch frequecy, i.e.,, the secod discotiuity appears.

128 Figure 6.4: Frequecy shift versus rod legth. Solid lie: from Eq. (6.4). Dotted lie: from Eq. (6.9). Aother special case is whe the rods are very log (L= ). I this case the boudary coditio at L= is that the disturbace propagatig alog the rods due to the vibratio of the crystal plate is outgoig for large z (radiatio), i.e., w Cexp( iz), (6.) which is subject to w( ) Bcosh. (6.) Equatios (6.) ad (6.) determie w Bcos( h)exp( iz), (6.) or f exp( i z). (6.) Substitutio of (6.) ito (6.) ad the ito (6.4) gives ina E. (6.4) c

129 (6.4) is imagiary, represetig a damped motio of the crystal plate due to the radiatio of eergy (radiatio dampig). 6. Thickess-Shear of Plate ad Flexure of Fibers 6.. Itroductio I this sectio, we put the fiber array o top surface of a rotated Y-cut quartz crystal resoator vibratig i a thickess-shear mode to ivestigate the possibility of characterizig the geometric/physical properties of these fibers from their frequecy effect o the quartz crystal resoator. A theoretical model of a quartz plate carryig a array of microfibers with their bottoms fixed to the top surface of the plate (see Figure 6.5) is costructed. Whe the crystal plate is i thickess-shear motio, the fibers udergo flexural vibratios. I terms of mechaics, these microfibers may be more suitably called microbeams whe they are i flexural motio. The shear forces at the bottoms of the beams exert a drag o the plate surface ad thereby affectig the resoat frequecies of the plate. For a typical quartz crystal resoator, the plate thickess is of the order of mm. We cosider small beams whose effect o the resoat frequecies of the plate is a small perturbatio which ca be determied by a theoretical aalysis. It is show that iformatio about the beam array ca be extracted through its frequecy effects.

130 Figure 6.5: A crystal plate with a micro-beam array. 6.. Plate Thickess-Shear Motio Sice the small piezoelectric couplig of quartz ca be eglected, we use the equatios of aisotropic elasticity for the quartz plate. Cosider the followig displacemet field that describes the thickess-shear motio [] of iterest: u u x )exp( i t), u u. (6.5) ( Sice we are maily iterested i the effects caused by the micro-beams, it is coveiet for us to assume the crystal plate is i pure thickess-shear motio. The otrivial compoets of the strai ad stress tesors are S u, (6.6), T c u, T c u, (6.7) 56, 66, where the time-harmoic factor has bee dropped for simplicity. The relevat equatio of motio is T, c66u, u. (6.8) The geeral solutio to Equatio (6.8) ad the correspodig expressio for the stress compoet eeded i the relevat boudary ad cotiuity coditios are

131 where B is a udetermied costat, ad T 4 u Bcos ( x h), (6.9) c Bsi ( x ), (6.4) 66 h c. (6.4) We ote that the tractio-free boudary coditio T at x h is already satisfied by Equatio (6.4) Beam Flexure Figure 6.6: Notatio ad coordiate system for beam bedig. For the flexural motio of the beams, we use the Euler-Beroulli theory of bedig [8,9]. Correspodig to the thickess-shear vibratio of the plate, for time-harmoic motios, all beams vibrate i phase. Followig the otatio i Referece [8] (see Figure 6.6), for a typical beam i its ow coordiate system, let the deflectio curve be

132 5 v (x), the rotatio or slope be θ, the bedig momet be M, ad the shear force be V. We have [8,9] EIv Av, (6.4) v, M EIv, V EIv, (6.4) where E is the Youg s modulus, is the mass desity, ad I ad A are the momet of iertia ad the area of the beam cross sectio. A prime idicates a derivative with respect to x. For time-harmoic motios with a frequecy ω, it ca be foud i a straightforward maer that the solutio to Equatio (6.4) that satisfies M (L) =, V (L) =, ad = is where C is a udetermied costat ad v Cf ( x, EI, A, L, ), (6.44) f cos( x L) cosh( x L) [si ( x L) sih ( x L)], A EI / 4, sihl si L. coshl cosl (6.45) From Equatios (6.4) ad (6.44) we calculate the shear force at the beam bottom as V( ) CEIf (, EI, A, L, ). (6.46) which will be eeded i the cotiuity coditio ext Plate-Beam Iteractio ad Frequecy Equatio Let the umber desity of the beams per uit area of the crystal surface be N. The cotiuity coditios of the displacemet ad shear stress at x h betwee the top surface of the plate ad the bottoms of the beams are

133 6 u ( x T ( x h) v( x ), h) NV ( x ). (6.47) Substitutig Equatios (6.9), (6.4), (6.44) ad (6.46) ito Equatio (6.47), we obtai Bcosh Cf (, EI, A, L, ), c Bsi h NCEIf (, EI, A, L, 66 ). (6.48) Equatio (6.48) is a system of liear homogeeous equatios for B ad C. For otrivial solutios the determiat of the coefficiet matrix has to vaish. This yields the followig frequecy equatio that determies the resoat frequecies of the plate carryig the beams: NEIf (, EI, A, L, ) tah, (6.49) c f (, EI, A, L, ) 66 where ξ is related to ω through Equatio (6.4), ad f (, EI, A, L, ) cos( L) cosh( L) [si( L) sih( L)], f (, EI, A, L, ) si( L) sih( L) [ cos( L) cosh( L)]. (6.5) Therefore Equatio (6.49) is a equatio for ω Frequecy-Depedet Effective Mass Layer We ca write the right-had side of (6.49) i the same form as the frequecy equatio for a uiform mass layer o a quartz crystal resoator [44] tah Rh, (6.5) where R is the mass ratio betwee the mass layer ad the quartz crystal resoator. I our case of a beam array,

134 7 NEIf (, EI, A, L, ) R (h) c f (, EI, A, L, ) NEI hc si( L) sih( L) [ cos( L) cosh( L)], cos( L) cosh( L) [si( L) sih( L)] (6.5) which may be called the effective mass ratio betwee the beam array ad the crystal plate or, equivaletly, R h is the effective mass of the beam array over uit area of the surface of the crystal plate. Clearly, R depeds o EI, A ad L which are the material ad geometric parameters of the beams. R also depeds o N which is a array property. R is frequecy depedet through i Equatio (6.45) ad i Equatio (6.4). We ote that whe f (, EI, A, L, ) =, i.e., the beam bottoms have o displacemets, R becomes ifiite. f (, EI, A, L, ) = determies a series of resoat frequecies for a beam with rigidly fixed bottom ad free top. We deote these resoat frequecies by with m=,,,. m For a umerical example we cosider a AT-cut quartz plate with c = 9. 9 N/m, = 649 kg/m, h.5mm. For the beams we cosider ZO with 66 56kg / m, E 4GPa [], diameter D. μm, L= μm, ad 8 N /cm. For the catilever beams m are give i [9]: where, for the first two resoaces, we have EI D E m m m, m=,,,, (6.5) A 4.875, L 4.694, (6.54) L ,.474 rad/s. (6.55)

135 8 Figure 6.7: Normalized effective mass ratio versus frequecy. R We plot the frequecy depedece of R i Figure 6.7 where R is ormalized by NAL/( ) which is the static mass ratio betwee the beams ad the crystal plate. h Figure 6.7 shows that R is strogly frequecy depedet. At the low frequecy limit R / R approaches oe which is the static mass ratio. R ca be smaller or larger tha oe, ad ca eve become egative. This is as expected for frequecy depedet effective masses as ofte see i the study of the so-called metacomposites [] of materials with iteral structures ad/or iteral degrees of freedom. I the problem we are aalyzig the frequecy depedece of the effective mass layer is due to the flexibility of the beams. Near the beam resoat frequecies m (oly the first two are show i the frequecy rage i Figure 6.7), R is ubouded because there is o dampig i the idealized elastic structure we are aalyzig.

136 Approximate Frequecy Solutio To exhibit the relatio betwee the resoat frequecies ad the beam array parameters more explicitly, we look for a perturbatio solutio of Equatio (6.5) i the case of small R whe the beams represet small effects o the quartz plate. For the zero-order solutio, we simply eglect the beams ad set the right-had side of Equatio (6.5) to zero. This results i two sets of resoat frequecies []: si cos c h, h /,,4,6,,, (6.56) h 66 c h, h /,,,5,,, (6.57) h 66 where the superscript idicates that they are the zero-order approximatio i the perturbatio procedure. Through Equatio (6.9), it ca be idetified that Equatios (6.56) ad (6.57) represet modes symmetric ad atisymmetric about x, respectively. From a device poit of view, we are oly iterested i the atisymmetric modes i Equatio (6.57) which ca be coveietly excited by a electric field i the plate thickess directio. Therefore, i the followig, we focus o the atisymmetric modes. For the first-order perturbatio, we write [77] where small h,,,5,, (6.58) is a small perturbatio. Substitutig Equatio (6.58) ito Equatio (6.5), for ad small R, we obtai R 66 c (. h NEIf (, EI, A, L, ) hc 66 ) f (, EI, A, L,, ) (6.59)

137 Equatio (6.59) implies the followig relative frequecy shift through Equatios (6.58) ad (6.4): NEI ( ) ( ) hc 66 si( cos( R L) sih( L) cosh( L) L) [ cos( L) cosh( L)], [si( L) sih( L)] (6.6) / 4 A( ) sih L si L, ( EI. (6.6) cosh L cos L ) We ote that, differet from Equatio (6.5), the right-had-side of Equatio (6.6) does ot deped o the ukow frequecy ow ad istead it is a fuctio of which is kow. I the above derivatio, sice R has bee assumed to be small, the special case whe a particular plate resoat frequecy is close to a particular beam resoat frequecy has to be excluded. I such a case f (, EI, A, L, ) approaches zero ad m R becomes large. The frequecy shift i Equatio (6.6) is a fuctio of the parameters of the beam array. This provides the theoretical foudatio for determiig the parameters of the beam array through frequecy measuremets. The procedure is as follows: are kow theoretically. will be measured by experimets. The are kow ad the Equatio (6.6) becomes a system of equatios of the beam array parameters whe several frequecies are used. These equatios are relatively complicated trascedetal equatios. It does ot seem to be easy to measure all of the parameters of the array at the same time. A relatively simpler situatio is that some of the parameters of the array are already kow ad oly oe or two parameters are left to be determied. The it may be coveiet to solve Equatio (6.6).

138 6..7 Special Cases I the special case whe the beams are so small that their resoat frequecies are much higher tha the fudametal thickess-shear frequecy of the plate, the beams follow the motio of the surface of the crystal plate quasi-statically. I this case a approximatio of Equatio (6.5) ca be obtaied as follows. From Equatio (6.9) we write the zero-order uperturbed plate surface displacemet as Bcos h ad the correspodig plate surface acceleratio as ( ) Bcos h. By quasi-static motios we mea that the beam acceleratio is approximately equal to the plate surface acceleratio: EIv Av EIv A( v Bcos h, ) Bcos h,, M ( L), V ( L). (6.6) Equatio (6.6) determies A v ( 4 EI Bcos ) Bcos h, 4 h[ L ( x L) 4 4xL ] (6.6) or f A ( 4 EI 4 4 ) [ L ( x L) 4xL ]. (6.64) Substitutios of Equatio (6.6) ito Equatios (6.49) ad (6.59) lead to ta h R h, (6.65) NAL h R, (6.66) respectively. R is the static mass ratio betwee the beams ad the plate. Equatio (6.66) is the well-kow Sauerbrey equatio for mass-iduced frequecy shift i a quartz crystal resoator [7]. I the quasi-static case the beam iertia lowers the plate resoat frequecies but the beam elasticity has disappeared.

139 Figure 6.8: Frequecy shift versus beam legth. Solid lie: from Equatio (6.6). Dotted lie: from Equatio (6.66). I Figure 6.8 we plot the frequecy shifts predicted by Equatios (6.6) ad (6.66) for the fudametal thickess-shear mode with = by solid ad dotted lies, respectively. 7 For the plate we are cosiderig, =.4 rad/s. Equatio (6.66) predicts a simple liear relatio (dotted lie), ideal for sesor applicatio. This is true for relatively short beams whose first resoace is higher tha the fudametal thickess-shear frequecy of the plate, i.e.,. As the beam becomes loger, whe L m ad the first resoat frequecy of the beam approaches the fudametal thickess-shear frequecy of the plate, i.e.,, Equatio (6.66) is o loger valid. The result from Equatio (6.6) deviates from that of Equatio (6.66) ad predicts a jump discotiuity. Whe the beam legth is further icreased, whe L.9 5 m ad the secod resoat frequecy of the beam coicides with the fudametal plate thickess-shear frequecy, i.e.,, the secod discotiuity appears.

140 Aother special case is whe the beams are very log (L= ). I this case the boudary coditios at L are dropped. Istead we require the vibratio of the beam to be decayig or outgoig for large x (radiatio), i.e., v x ix C e Ce, (6.67) which is subject to v( ) Bcosh,. (6.68) Equatios (6.67) ad (6.68) determie or x i x i v Bcosh[ ie e ], (6.69) f x ix ie e. (6.7) Substitutio of (6.7) ito (6.59) gives EI A / 4 NA ( i ) h. (6.7) Equatio (6.7) is complex. Its real part is the frequecy shift which is egative as expected. Its imagiary part describes a damped motio due to the radiatio of eergy. 6. Summary I Sectio 6., a frequecy equatio is derived for free thickess-stretch vibratios of a plate of hexagoal crystals loaded with a array of micro-rods i extesioal motio. A approximate expressio for frequecy shifts i the crystal plate due to the rods is obtaied whe the rods represet small effects o the vibratios of the crystal plate. Based o the frequecy equatio, a frequecy-depedet effective mass ratio is itroduced with which

141 4 the array is equivalet to a homogeeous mass layer. The effective mass ratio may be positive ad egative alog with the frequecy chage. It becomes sigular whe the resoat frequecies of the rods aloe coicide with the resoat frequecies of the plate aloe. This work provides the theoretical foudatio for usig frequecy shifts of the crystal plate to characterize the physical ad geometric parameters of the rod array. I Sectio 6., the problem is solved i a similar maer. At first a frequecy equatio is derived for free thickess-shear vibratios of a rotated Y-cut quartz plate loaded with a array of micro-beams i flexural motio. The a approximate expressio for frequecy shifts i the quartz plate due to the beams is obtaied whe the beams represet small effects o the vibratios of the plate. Based o the frequecy equatio, a frequecy-depedet effective mass ratio is itroduced with which the beam array is equivalet to a homogeeous mass layer. The effective mass ratio also ca be positive ad egative alog with the frequecy. It is theoretically proved that it is possible to use frequecy shifts of the plate to measure the physical/geometric parameters of the beam array. The structure aalyzed may also be cosidered as a ultrasoic brush for cleaig a surface.

142 5 Chapter 7 Effects of Particles I this chapter, we study thickess-shear (TSh) vibratio of a rotated Y-cut quartz crystal resoator carryig fiite-size circular particles that have a rotatioal degree of freedom ad rotatory iertia [4]. The particles are elastically attached to the quartz crystal resoator ad are allowed to roll without slidig o its surface. A aalytical solutio o particle-iduced frequecy shifts i the quartz crystal resoator is obtaied usig equatios of aisotropic elasticity. Examiatios of the frequecy shifts show that while it ca be used to characterize geometric/physical properties of the particles, the frequecy shifts ca have relatively complicated behaviors that cause deviatios from the Sauerbrey equatio i mass sesig. A frequecy-depedet effective particle mass is itroduced to classify ad characterize differet aspects of the particle iduced frequecy shifts.

143 6 7. Itroductio Quartz crystal microbalace (QCM) is the most commoly kow applicatio of quartz crystal resoator as a sesor for differet measurads it cotacts by desig. Besides, a ew method is proposed to use quartz crystal resoator to characterize the geometric/physical properties of micro- or ao-fibers i Chapter 6. I fact, researchers have bee costatly explorig the use of quartz crystal resoators for ew sesor applicatios. Basically, a plate quartz crystal resoator i thickess-shear motio provides a platform for probig materials ad structures i geeral. A quartz crystal resoator is sometimes loaded with small particles. This ofte appears as a udesirable effect due to cotamiatio. But it ca be also used for measurig particle properties. Particles ca be sparse due to cotamiatio or i the iitial depositio stage i a mass sesor. They ca also be dese ad evetually form a thi film o the quartz crystal resoator. While these particles are usually small compared with the thickess of a quartz plate, they are ot always so. Quartz crystal resoators are made thier ad thier ow. I biosesig, cells ad other biological particles are ofte loaded o a quartz crystal resoator. The thickess of the quartz plate may ot be several times larger tha the cell diameter. Whe the particle size is ot ifiitesimal compared to the plate thickess, the rotatioal degree of freedom ad the rotatory iertia of the particles may have effects that caot be eglected. As a specific example of rigid spherical particles, polystyree microspheres ca be purchased from Polyscieces, Ic. The diameters of these microspheres rage from a few μms to tes of μms. They may or may ot be small compared to the thickess of a quartz crystal resoator, which is

144 7 typically a few hudred micrometers. This chapter attempts to study the frequecy effects of particles through a theoretical aalysis. While the mass ad shear stiffess effects of a relatively thick film o a quartz crystal resoator have bee studied [5]. There are just a few reported theoretical results discussig the effects of discrete particles o a quartz crystal resoator [,6,7]. I these papers, oly greatly simplified two-particle models were used, oe represets the resoator ad the other a particle o it. I additio, the particles are assumed oly have traslatioal motio, without particle rotatioal degree of freedom ad rotatory iertia. Therefore a ew model is eeded i order to reveal the rotatioal effects of the particles. I this chapter we cosider fiite, circular particles with rotatioal degree of freedom ad rotatory iertia. As usual, we first aalyze the crystal plate ad the particles separately, ad the apply boudary coditios which will lead to the frequecy equatio. The frequecy equatio is solved approximately by a perturbatio procedure, leadig to the frequecy shifts produced by the particles. 7. Crystal Plate Cosider the crystal plate (see Figure 7.) has the followig displacemet field. Sice we are maily iterested i the effects caused by the particles, it is coveiet for us to assume the crystal plate is doig a pure thickess-shear motio. u u x )exp( i t), u u. (7.) (

145 8 M θ, F f T x r U, V u h h x Figure 7.: A fiite particle o a crystal resoator: otatio ad free-body diagram. Similar to (6.5)-(6.8) i Sectio 6.., the otrivial compoets of the strai ad stress tesors are S u, (7.), T c u, T c u, (7.) 56, 66, where the time-harmoic factor has bee dropped for simplicity. The relevat equatio of motio is T, c66u, u. (7.4) The geeral solutio to (7.4) ad the correspodig expressio for the stress compoet eeded i the relevat boudary coditios are T u A cosx A si x, (7.5) c A si x A cos ), (7.6) 66 ( x where A ad A are udetermied costats, ad c. (7.7) 66

146 9 7. Particle The particles are represeted by rigid ad circular cyliders or spheres with the same radius r ad mass m (see Figure 7.). The momet of iertia about the ceter of mass is I. For spheres I mr / 5 ad for cyliders I mr /. The liear displacemet ad velocity of the ceter are U ad V U, respectively. The agular displacemet ad agular velocity are θ ad. We cosider the case whe the particles are rollig without slippig o the plate. This ca at least partially accout for the complicatio i biological mass sesors i which the additioal mass o the quartz crystal resoator may ot be perfectly or rigidly attached to its surface [88,8]. I additio to the frictioal force F betwee the particles ad plate surface, the particles ca also iteract with the plate surface elastically with a effective force f [,6,7] ad a effective momet M. The particles are assumed to be sparse, without iteractios amog themselves. Accordig to the otatio ad free-body diagram i Figure 7., the equatio of motio i the x directio ad the momet equatio about the mass ceter for the particles are F f mv, (7.8) M ( F f ) r I. (7.9) The particle-plate iteractio force f ad momet M are described by liear ad agular sprigs with f k( U u), (7.) M, (7.)

147 where u u ( h ) is the upper surface displacemet of the crystal plate. The o-slip coditio of the particles for rollig without slippig with respect to the movig plate surface is give by or, upo differetiatio with respect to time, From (7.) ad (7.), we have U u r( ), (7.) V u r( ). (7.) f kr. (7.4) Substitutig (7.) ito (7.8), usig u u ad for time-harmoic motios, we obtai F f m u m r. (7.5) Similarly, substitutio of (7.) ito (7.9) ad usig gives ( F f ) r I. (7.6) Elimiatig from ( ), we obtai kr I F f, (7.7) kr F f meu, (7.8) where m m e ( ) (7.9) m r I is the effective mass of the particles. The reaso for defiig the effective mass this way will become clearer i (7.8).

148 7.4 Boudary Coditios ad Frequecy Equatio The lower surface of the plate is tractio free. At the upper surface of the plate, let the particle umber desity be N. The the shear stress at the plate upper surface is completely determied by the iteractio forces betwee the particles ad the plate surface. Therefore the boudary coditios at the top ad bottom surfaces of the plate are T ( h) N( F ), (7.) f T ( h). (7.) With the use of (7.6) ad (7.8), we ca write (7.) ad (7.) as two liear ad homogeeous equatios for A ad A : c 66 ( A si h A A si h A cosh. cosh) Nm e ( A cosh A si h), (7.) For otrivial solutios, the determiat of the coefficiet matrix of (7.) has to vaish, which gives the followig frequecy equatio that determies the resoat frequecy : Nme tah. (7.) c I the special case whe the particles are perfectly fixed to the plate surface without relative motio or rollig, the frequecy equatio ca be reduced from (7.) by settig m e =m Approximate Frequecy Solutio Next, we look for a perturbatio solutio of (7.) i the case of sparse ad light particles represetig small effects o the quartz crystal resoator. I this case Nm e is very small.

149 For the lowest-order (zero-order) solutio, we simply eglect the particles ad set the right-had side of (7.) to zero. This results i two sets of resoat frequecies: si cos c h, h /,,4,6,,, (7.4) h 66 c h, h /,,,5,,, (7.5) h 66 where the superscript idicates the zero-order solutio. It ca be idetified that (7.4) ad (7.5) represet modes symmetric ad atisymmetric about x, respectively. From a device poit of view we are oly iterested i the atisymmetric modes i (7.5) which ca be coveietly excited by a electric field i the plate thickess directio. Therefore, i the followig, we focus o the atisymmetric modes oly. For the firstorder perturbatio, we write where obtai h,,,5,, (7.6) is small. Substitutig (7.6) ito (7.), for small ad small Nm e, we e ( ) c66 Nm, (7.7) where m ( ). (7.7) implies the followig frequecy shift through (7.6) ad (7.7): e m e me m Nm me h m R, (7.8) where Nm R (7.9) h

150 is the mass ratio betwee the particles ad the crystal plate. We ote that the right-hadside of (7.8) does ot deped o the ukow frequecy ow. (7.8) shows that the frequecy shift is a fuctio of the geometric ad physical parameters of the crystal plate ad the particles. Therefore the frequecy shift ca be used to measure these parameters i a combiatio. 7.6 Discussio ad Numerical Results The differece betwee m e ad m is due to the fiite size, the rotatioal degree of freedom, ad the rotatory iertia of the particles. The followig observatios ca be made from (7.9): (i) Whe r=, from (7.9) we have m e m. I this case the rotatioal effects of the particles disappear ad (7.8) reduces to the classical result of R (the wellkow Sauerbrey equatio i a differet form). (ii) Whe I, (7.) from (7.9) we have m. I this case, from (7.8), we have F f. The et e iteractio force betwee the particles ad the plate surface is zero. The acceleratio of the mass ceter of the particle is zero. The plate does ot feel the particles. (7.) defies a frequecy as The (7.9) ca be writte as. (7.) I

151 4 m m e. (7.) m r I( ) (iii) Whe, the deomiator of (7.) is larger tha oe. We have m e m. I this case the particles appear lighter. (iv) Whe, there are two possibilities depedig o whether the deomiator of the right-had side of (7.) is positive or egative. Cosider the case whe it is positive first. The coditio that the deomiator of the right-had side of (7.) is positive, is equivalet to I m r. (7.) Whe the left- ad right-had sides of (7.) are equal, it defies aother frequecy ˆ by ˆ I mr. (7.4) Physically, ˆ is the frequecy of the particle whe it is rollig without slidig o a fixed surface with elastic costraits described by k ad. With (7.4), we ca write (7.) as ˆ which automatically implies that. I this case the deomiator of (7.) is smaller tha oe ad is positive. We have m e m. The particles appear heavier. (v) Whe ˆ, the deomiator of (7.) becomes egative ad m. I this case the particles appear to be with a egative mass ad raises the plate resoat frequecies. This aomaly was also observed ad explaied i [6] usig a two-particle model ad i [8] usig imperfectly boded mass layers. e We summarize some of the behaviors of m e below.

152 5 Table 7.: Effective mass. ˆ ˆ ˆ m e m Jump m m m e m e e We also plot m e / m versus Ω i Figure 7. by rearragig (7.), where the ormalized frequecy m e, (7.5) m mr I /. (7.6) Figure 7.: Effective mass versus frequecy ( mr / I ). The behavior show i Figure 7. is complicated. The familiar situatio of / m happes oly whe. I this case the rotatioal effects of the particles ca be eglected, the plate will feel oly the mass of the particles, ad the the Sauerbrey equatio applies. m e

153 6 I additio, Figure 7. shows that m e / m ca be larger or smaller tha oe, ad eve be egative. As a result, accordig to (7.8) the frequecy shifts iduced by these particles may be larger, smaller, or eve with the opposite sig whe compared to the predictio by the Sauerbrey equatio. 7.7 Summary Whe circular particles with a rotatioal degree of freedom ad rotatory iertia are allowed to roll without slidig o the surface of a quartz crystal resoator, it feels the particles through their effective mass m e. The effective mass may be the same as, larger tha, or smaller tha the true particle mass m, or may eve be egative alog with the chage of frequecy. As a cosequece, the frequecy shifts iduced by these particles may be the same, lager, smaller, or eve with the opposite sig whe compared to the classical predictio based o the Sauerbrey equatio. The rotatioal effects of the particles may be used to explai certai deviatios from the classical predictio whe the particles are ot very small or whe they are ot rigidly attached to the plate surface. These effects are qualitatively cosistet with the behaviors of the so called metamaterials with iteral degrees of freedom []. The results are obtaied from a simple theoretical model. We hope to see experimetal results for compariso from researchers with experimetal capabilities.

154 7 Chapter 8 Thi Film Piezoelectric Actuators A elastic plate with thi piezoelectric films boded o its two major surfaces is a typical smart structure used i actuatio. The useful deformatio of this smart structure is caused by the shear stress trasferrig from the films (also referred to as actuators) to the elastic plate. This iterfacial shear stress is iduced by extesio or cotractio of the piezoelectric film uder electric voltage applied o its two major surfaces. I this chapter, we study the effects of the material property variatio of thi film piezoelectric actuators o the actuatio shear stress. A system of two-dimesioal equatios for the flexure ad shear of a elastic plate with symmetric piezoelectric actuators o the plate surfaces is derived [9]. The equatios are reduced to the case of elemetary flexure without shear as a special case. It is show that the distributio of the actuatio stress depeds o the thickess ad material property variatios of the actuators, ad that actuators with varyig material properties ca be used to make modal actuators for producig a particular deformatio or excitig a particular vibratio mode.

155 8 The effects of material property variatio o the actuatio stress are examied through a example. Figure 8.: Shear stress uder a piezoelectric actuator. It is show that the iterfacial shear stress cocetrates at the two eds of the actuator [-] (See Figure 8.). I fact, the cocetratio of shear stress always exists for a film o a substrate whe the film expads or cotracts due to differet effects [4,5]. It is theoretically proved that edig the electrodes a short distace from the edge of actuator reduces the stress cocetratio. It is also show aalytically that actuators with ouiform thickess may also reduce the cocetratio of shear stress [6]. However, the above results are based o simple models. It is very difficult to get a aalytical solutio whe a more complicated model is employed. Fiite elemet aalyses of a piezoelectric actuator o a substrate were reported i [7-]. 8. Itroductio I geeral, the shear stress betwee the piezoelectric films ad the elastic plate is cocetrated ear the edges of the piezoelectric films. I order to produce a particular

156 9 deformatio of the elastic plate, a specific distributio of the actuatio stress is eeded. I the case of dyamic problems, to excite a particular vibratio mode, the specific actuator required is called a modal actuator []. There are several kow ways of modulatig the actuatio stress (see Figure 8.). Oe is to use several piezoelectric actuators of differet sizes ad shapes at proper locatios (segmetatio, see Figure 8.(a)) [,]. Aother way is to use proper electrode cofiguratios (see Figure 8.(b)) [,,,4,5], which icludes the so called iter-digital trasducers (IDTs) [6,7]. These two techiques have bee widely used ad relatively thoroughly studied. There are two other ways of modulatig the actuatio stress which have ot received much attetio. Oe way is to use piezoelectric films of ouiform thickess [6,8] (see Figure 8.(c)). Aother way is to use piezoelectric films with ouiform material properties (see Figure 8. (d)) [,9-4]. Figure 8.: Schemes for modulatig actuatio stress. (a) Segmetatio. (b) Multiple electrodes. (c) Varyig thickess. (d) Nouiform polig. Recetly, the rapid developmet of material maufacturig techologies allows the productio of piezoelectric materials with ohomogeeous material properties for actuator applicatios [4-49]. However, mechaics aalyses o ohomogeeous

157 4 actuators [4,4] are all based o simple oe-dimesioal models to qualitatively demostrate the basic ideas. Therefore these models are iadequate for geeral aalysis of plate structures with ohomogeeous actuators. I this chapter we systematically derive geeral two-dimesioal equatios for elastic plates with ohomogeeous piezoelectric actuators with varyig material properties i the maer of [44,,5-5] by joiig separate equatios of the piezoelectric actuators ad the elastic plate through their iterface cotiuity coditios. The equatios obtaied are used to examie the effects of the actuator material property variatio o the actuatio stress through a few examples. 8. Equatios for Piezoelectric Actuators Cosider the elastic plate with two thi-film piezoelectric actuators, as show i Figure 8.. We are iterested i the plate bedig produced by the actuators. The two actuators at the plate top ad bottom are idetical, with the same ceramics poled i the x directio. However, at this poit, for clarity we deote their thickesses by h' ad h" differetly which will be made the same later. h' ad h" may be slowly varyig ad are much smaller tha the plate thickess h. The actuators may be fully or partially electroded. Whe the two actuators are uder opposite voltages, oe exteds ad the other cotracts or vice versa, thereby producig bedig of the elastic plate.

158 4 Figure 8.: Piezoelectric actuators o a elastic plate ad coordiate system. We summarize the equatios for the extesioal motio [] of the upper piezoelectric film below. The displacemet u i ad electrostatic potetial are approximated by u u ( x, t), a a b u u ( x, t), b x b ( x, t), (8.) where a, b =, ad t is time. x i are parallel to x i, but the origi of the x i system is at the middle plae of the piezoelectric film. u a is the extesioal displacemet. We ote that i geeral the approximatio of i (8.) may have a term [5] which is idepedet of x. Sice does ot cotribute to the mai actuatig electric field E, it is ot icluded i (8.) [,5,5]. The equatios of motio of the film take the form [,5] T ab, a F F b hu hu, b, (8.) where is the actuator mass desity. The thi film extesioal ad i-plae resultats ad surface loads are deoted by h h ab ab, j [ j ] h h T T dx F T. (8.)

159 4 I (8.) it has bee assumed that the piezoelectric film is very thi ad does ot resist bedig. Therefore the surface ormal load is resposible for the motio of the film i the x directio. I a electroded regio the electric potetial is o more tha a fuctio of time ad is related to the applied voltage. For uelectroded regio the followig equatio of electrostatics is eeded to determie [,5,5] where a D D, (8.4) D, a h ( ) Da x Da dx, h D x D] h [ h. (8.5) Costitutive equatios for (8.) ad (8.4) are [,5,5] T r D D a h( S h rs s s h( S s ab, b, r ), ), (8.6) where r, s =,, 6 uder the compact matrix otatio [77] ad ) ( Sab ( ua, b ub, a ). (8.7) For ceramics poled i the x directio [,5,5], we have p p p E E E c c c c ( c ) / c, p p p E E E [ rs ] c c, c c ( c ) / c, p c p E 66 c 66 c 66, (8.8) p E E e e e c / c, [ ks ], p p k,,, e e (8.9)

160 4 p p S E p e 5 c 44 kj p S E p e c [ ], /, /, (8.) which may be fuctios of x b for ohomogeeous actuators. Substitutio of (8.7)-(8.) i (8.6) yields T T T h[ c u hc p h[ c u p p 66,, [ u, c p c p u u,, u, ], e p e p ], ], (8.) D h[ e u ], D p p aa, h p a, a. (8.) Substitutio of (8.) ito (8.) yields the equatios for the displacemet u a : [ h( c u c u e )] [ hc ( u u )] F hu, p p p p,,, 66,,, [ hc ( u u )] [ h( c u c u e )] F hu p p p p ( 66,,,,,, ). (8.) With substitutio of (8.) ito (8.4), we obtai the followig equatio for uelectroded regio: i the h p p p, a he u a, a h,, a (8.4) where we have take D because the plate dielectric costat p of polarized ceramic is usually much larger tha the dielectric costats of air ad the material of the elastic plate.

161 44 8. Equatios for Elastic Plates The elastic plate show i Figure 8. has flexure deformatio coupled with shear. The equatios of motio for the elastic ca be writte as [4] T T a, a ba, b F T a hu F a, h u a, (8.5) where T x T dx,,, ( ) ab h h ab F T ( h) T ( h), F h[ T ( h) T ( h)]. b b b (8.6) We ote that i [4] the equatios correspodig to (8.5) have a modified mass desity which is ecessary for the correct behavior of high-frequecy thickess-shear waves. We are iterested i low-frequecy flexural waves oly, for this case the mass correctio factor (see Sectio..) is very close to oe. Therefore we ca simply set it to oe here. The costitutive relatios for a orthotropic elastic plate ca be writte as T T T T T hc hc ( u ( u,, h ( u h ( u h 66( u,,, u u u ), ), u,, ), ), u, ), (8.7) (8.8) where

162 45 c c / c, c c / c, c c c / c, c c, c c, c (8.9) The displacemet field i the elastic plate is approximately give by ( ) ( ) u x u, u u. (8.) a a 8.4 Equatios for Elastic Plates with Piezoelectric Actuators I this sectio we combie the equatios for thi piezoelectric actuators ad the equatios for elastic plates i 8. ad 8. ito a set of equatios for the whole structure. From (8.), the surface loads of upper actuator, we obtai the load o the top actuator as: F T h F T h. (8.) ( ), ( ) b b From the displacemet fields i (8.) ad (8.) the cotiuity of the displacemet betwee the top actuator ad the elastic plate ca be writte as u hu, u u. (8.) a a Similarly, for the bottom actuator we have F ( ), ( ) b T b h F T h, (8.) u hu, u u. (8.4) a a From (8.) ad (8.), the equatios of motio for the top ad bottom actuators (8.) ca be writte as T ab, a T T b ( h) hu ( h) hu, b, (8.5)

163 46 T T ab, a T b ( h) h u ( h) h u. b, (8.6) From the surface loads of the elastic plate i (8.6), ad ( ) we have F b F hu h, (8.7) u h[ T ab, a hu T h u ]. (8.8) b Substitutio of (8.7) ad (8.8) ito the equatios of motio of the plate i (8.5) yields ab, a b T a, a ( T ba hu ht ba hu ht ba ), b T hu a h u, a h[ hu a hu a ]. (8.9) With the cotiuity coditios i (8.) ad (8.4), we ca write (8.9) as T T a, a ba, b mu ˆ T a, Iu ˆ a, (8.) where we have deoted T ba T ba mˆ h h h, Iˆ h ht ba hh ht ba, hh. (8.) T ab has the physical meaig of total plate momets cosistig of cotributios from the bedig of the elastic plate ad the extesio of the actuators. m is the mass per uite area of the plate with the actuators. I is the rotatory iertia. We assume that the applied voltages o the top ad bottom actuators are of equal magitude ad opposite sigs ad deote. (8.)

164 47 We also assume that the top ad bottom actuators are idetical with the same material ad the same thickess variatio. The, for costitutive relatios, from (8.), (8.8), (8.) ad (8.4) we obtai T T T ˆ u, ˆ u T, ˆ ˆ u ˆ 66, u, ( u, p 4hhe p 4hhe u, ),,, (8.) where ˆ ˆ h h 4hh 4hh p c, p c, ˆ ˆ 66 p h 4hh c, p h 66 4hh c66. (8.4) Expressios for ( ) T a i terms of u ( ) ad u a are give by (8.7). Substitutio of (8.7) ad (8.) ito (8.) gives the displacemet equatios for the flexural motio with shears of the whole plate hc ( u u ) hc ( u u ) mu ˆ, 55,, 44,, [ ˆ u ˆ u 4 hhe ] [ ˆ ( u u )] hc ( u u ) Iu ˆ, p,,, 66,,, 55, [ ˆ ( u u )] [ ˆ u ˆ u 4 hhe p ] hc ( u u ) Iu ˆ. 66,,,,,, 44, (8.5) I the uelectroded portios of the actuators where ( ) is ukow, the electrostatic equatio i (8.4) is eeded which ow takes the followig form with (8.) : h p p p ( ), a he hua, a h, a. (8.6) At the boudary of a fiite plate with a uit ormal vector a ad a uit taget vector s b, we eed to prescribe [,5,5]:

165 48 ab b u ( ), a T or a a ab s b a T or u ( ) a sa, (8.7) T a or u. For electric boudary coditios we eed to prescribe [,5,5]: D or. (8.8) a a 8.5 Reductio to Elemetary Flexure without Shear Deformatio ad Rotatory Iertia If the elastic plate is very thi, the effects of rotatory iertia ad shear deformatios i (8.5) may be eglected by settig as below (see Sectio. for detail): Iˆ, u u u u. a, a a, a (8.9) Uder this simplicatio, (8.) ad (8.5)-(8.7) are reduced to, T T T ˆ u ˆ u ˆ,, ˆ u ˆ 66u,,, u, p 4hhe p 4hhe,, (8.4) T ˆ u ( ˆ ˆ ) u ˆ u ab, ab, 66,, ( ˆ ˆ ) u ( ˆ ˆ ) u 4 ˆ u,,,,,, 66,, [4 hhe ] [4 hhe ] mu ˆ, p p,, (8.4) h p p p ( ), a he hu, aa h, a, (8.4) u a T ab b or, (8.4) T as Ta or u, s

166 49 where Ta T ba, b, T as at ab sb. (8.44) I the special case whe the elastic plate is isotropic, the material costats satisfy ˆ ˆ ˆ ˆ. (8.45), ˆ Numerical Results ad Discussio To illustrate the mai idea of material property variatio i a relatively simple maer, p we cosider the special case whe the piezoelectric costat e is a fuctio of x ad x but all other geometric ad physical parameters are true costats. I this case the derivatives of ˆab ad ˆ 66 vaish. The goverig equatio (8.4) reduces to ˆ ( u u u ) [4 hhe ] [4 hhe ], (8.46) p p,,,,, where we have assumed that the elastic plate is isotropic ad dropped the iertial term for static problems. For a specific example, cosider a fully electroded rectagular plate withi x a ad x b uder a time-idepedet applied voltage V h. The plate is simply-supported with the followig boudary coditios: u, x, a, T ˆ u 4hhe, x, a, p, (8.47) u, x, b, T ˆ u 4hhe, x, b. p, (8.48) The solutio to this problem ca be obtaied by trigoometric series. Let m x x u Am si si. (8.49) a b m, where A m are to be determied. Formally A m are give by

167 5 m x x m x x Am u dx dx dx dx a b a b 4 mx x a b a b si si si si a b u si si dxdx. ab a b (8.5) To determie the remaiig itegral i (8.5) we multiply both sides of (8.46) by si( m x / a)si( x / b) ad itegrate the resultig equatio over x a ad x b. With itegratio by parts ad the use of (8.47) ad (8.48), we obtai b a mx x u si si dxdx a b b a p p mx x 4hh ( )si si e, e, dx dx a b m b p p x 4hh [ e ( x a)cosm e ( x )]si dx a b a p p mx 4hh [ e ( x b)cos e ( x )]si dx ] b a 4 4 ˆ m m ( ) ( ) ( ) ( ). a b a b (8.5) We are iterested i modal actuators. Therefore we cosider the case whe a b ( x, x ) ecosk( x )cosk ( x ), (8.5) e p where e is a costat. As a umerical example, suppose the actuators are made of PZT- 5H poled i x directio, ad the elastic plate is made of alumium alloy 66-T6. e is calculated from the homogeeous material costats of PZT-5H accordig to the formula for p e i (8.9). For geometric parameters cosider a=b= cm, h= cm, ad h'= mm. The two ier electrodes of the actuators i cotact with the elastic plate are grouded. The two outer electrodes at the top ad bottom are uder a voltage of V.

168 5 (a) (b) Figure 8.4: Actuatio stress T uder a homogeeous actuator. k =k =. x =.5 m. (a) terms i the series. (b) terms i the series. For completeess ad compariso we first review the case of homogeeous actuators with k =k = m - i (8.5). Figure 8.4 shows the distributio of the actuatio

169 5 stress T (x ) alog x =5 cm. The result for T is similar. Figure 8.4 (a) is with oe hudred terms i the series. Figure 8.4(b) is with two hudred terms i the series. Qualitatively the two figures are similar, with little stress i the middle ad cocetrated forces at the edges [,,,5]. Whe usig more terms i the series, the cocetrated edge forces keep icreasig. This is as expected because it is difficult for a trigoometric series to coverge to a discotiuity or sigularity. The oscillatios i the figure are also typical for a trigoometric series ear a discotiuity or sigularity. This situatio will disappear i the modal actuators to be discussed i the followig. I the rest of the calculatio we will keep two hudred terms i the series. (a)

170 5 (b) Figure 8.5: Actuatio stress T uder a homogeeous actuator. (a) k =k =. m -. (b) k =k =9.6 m -. I this case Figure 8.5 is for ohomogeeous actuators. I Figure 8.5(a) k =k =. m -. p e is reduced by oe third (e/) ear the edges whe compared to its value at the ceter. Correspodigly the cocetrated forces at the edges become smaller as compared to Figure 8.4(b). I Figure 8.5(b) k =k =9.6 m -. I this case p e is reduced by two thirds (e/) ear the edges ad the cocetrated forces at the edges become eve smaller whe compared to Figure 8.5(a).

171 54 (a) (b) Figure 8.6: (a) Material property variatio. (b) Actuatio stress T. I Figure 8.6 we cosider three specially desiged actuators for which p e vaishes at the edges. Figure 8.6(a) shows the variatio of p e alog x =5 cm. Figure 8.6(b) shows the correspodig actuatio stresses. For these actuators, there are o cocetrated forces at the actuator edges. The trigoometric series coverges well without oscillatios. The actuatio stress may have differet wavelegths for actuatig differet deformed shapes as modal actuators. (a) (b) Figure 8.7: (a) Actuatio stress T. (b) Deformed plate. k =k =.46 m -.

172 55 (a) (b) Figure 8.8: (a) Actuatio stress T. (b) Deformed plate. k =k =94.48 m -. (a) (b) Figure 8.9: (a) Actuatio stress T. (b) Deformed plate. k =k =57.8 m -. For the same three actuators as i Figure 8.6, for a differet view of the results, we plot i Figures 8.7, 8.8 ad 8.9 two-dimesioal distributios of the actuatio stress T ad the correspodig deformed shapes of the plate. Clearly, ohomogeeous actuators ca produce various distributios of the actuatio stress ad deform the plate ito differet shapes for differet eeds.