ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no. 2, february

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1 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february Torsional Vibrations of Circular Elastic Plates wit Tickness Steps Min K. Kang, Rui Huang, and Terence Knowles, Member, IEEE Abstract Tis paper presents a teoretical study of torsional vibrations in isotropic elastic plates. Te exact solutions for torsional vibrations in circular and annular plates are first reviewed. Ten, an approximate metod is developed to analyze torsional vibrations of circular plates wit tickness steps. Te metod is based on an approximate plate teory for torsional vibrations derived from te variational principle following Mindlin s series expansion metod. Approximate solutions for te zerot- and first-order torsional modes in te circular plate wit one tickness step are presented. It is found tat, witin a narrow frequency range, te first-order torsional modes can be trapped in te inner region were te tickness exceeds tat of te outer region. Te mode sapes clearly sow tat bot te displacement and te stress amplitudes decay exponentially away from te tickness step. Te existence and te number of te trapped first-order torsional modes in a circular mesa on an infinite plate are determined as functions of te normalized geometric parameters, wic may serve as a guide for designing distributed torsionalmode resonators for sensing applications. Comparisons between te teoretical predictions and experimental measurements sow close agreements in te resonance frequencies of trapped torsional modes. I. Introduction Torsional vibrations of circular plates and cylindrical rods ave been studied for many years. Recent developments of torsional-mode sensors and actuators in microand nanoelectromecanicalsystems ave renewed interests in tis topic 1 5. Previous studies of torsional vibrations ave focused largely on cylindrical rods or safts, for wic a one-dimensional matematical model based on te strengt-of-materials approac 6 as been widely used in practice. However, it as been sown tat te one-dimensional model is only accurate at low frequencies, and tree-dimensional analysis is required at ig frequencies 7. In particular, safts wit stepped cross sections are common in engineering structures. Several tree-dimensional metods ave been developed to analyze torsional vibrations of stepped safts Jonson et al. 11 sowed tat, witin a certain frequency range, torsional modes can be trapped in te central section of stepped solid cylinders wit a sligtly larger diameter suc tat te vibration amplitude decays exponentially wit te Manuscript received July 1, 5; accepted July 14, 5. M. K. Kang and R. Huang are wit te Researc Center for Mecanics of Solids, Structures and Materials, Department of Aerospace Engineering and Engineering Mecanics, Te University of Texas, Austin, TX ruiuang@mail.utexas.edu. T. Knowles is wit Texzec, Inc., Round Rock, TX distance from te central section. Suc trapped modes may find interesting applications in te design of resonators, transducers, and sensors. In a parallel development, trapped vibrations in plates ave been studied for many decades to improve te performance of quartz crystal resonators. By mass loading of quartz crystal plates wit electrodes, tickness-sear vibrations can be trapped under te electrodes wit amplitude decreasing exponentially away from te electrodes in te unplated region It also as been observed tat, by decreasing te plate tickness from center to edge or contouring, te tickness-sear vibrations can be confined in te center portion of te plate, wic improves te resonator performance by reducing edge leakage due to boundary mismatc and mode conversion Muc less attention as been paid to torsional vibrations of plates. By using Love s tin plate teory, Onoe 1 analyzed te contour vibrations of circular plates, including torsional modes Onoe used te term tangential modes. Te result was used by Meitzler in an ultrasonic tecnique for determining elastic constants of glass wafers. For tick plates tickness comparable to radius, exact solutions are available for bot solid circular plates and annular plates, wic are essentially te same as te exact solutions to te classical Pocammer equation for cylindrical rods and ollow cylinders 3, 4. For plates wit nonuniform tickness, an exact analytical solution generally is not possible 5. Tis paper develops an approximate metod for analyzing torsional vibrations of circular plates wit tickness varying in te radial direction as steps Fig. 1c. Suc plates are of interest because torsional vibrations may be trapped near te steps, similar to tat in stepped solid cylinders 11 and trapping of tickness-sear vibrations in contoured plates Recent experiments by Knowles et al. 6 ave observed trapped torsional modes in bot stepped and contoured aluminum plates, wic may be used to design sensors wit improved performance in te presence of liquids. Tis paper is organized as follows. Section II reviews te general tree-dimensional teory and exact solutions for torsional vibrations of circular and annular plates wit uniform tickness Fig. 1a and b. Section III develops an approximate plate teory from variational principle. In Section IV, an approximate metod is developed for torsional vibrations of plates wit tickness steps Fig. 1c, and approximate solutions are obtained for te zerot- and first-order modes. Trapped first-order modes are identified. A diagram is constructed predicting te existence and te number of trapped torsional modes in a circular mesa on /$. c 6 IEEE

2 35 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february 6 σ z = µ u z, u σ r = µ r u. 3 r For plates wit uniform tickness, 1 can be solved by separation of variables. Assume a armonic solution, namely: u r, z, t =P rqze iωt, 4 were ω is te angular frequency of vibration. Substitution of 4 into 1 leads to: were: d P dr + 1 r dp β dr + 1r P =, 5 d Q dz + k Q =, 6 β + k = ρω µ. 7 For β, k, te general solutions to 5 and 6 are: P r =AJ 1 βr+by 1 βr, 8 Qz =C sinkz+dcoskz, 9 Fig. 1. Scematics of te plates. a A uniform circular plate. b An annular plate. c A circular plate wit a tickness step. an infinite plate. Te teoretical results are compared to experimental measurements. Section V concludes wit remarks on potential applications of te teoretical results. II. General Teory and Exact Solutions Fig. 1 illustrates te geometries of te plates considered in tis paper. Assuming axisymmetric motion and isotropic, linear elastic materials, te momentum equation for te torsional motion is: u r + 1 r u r u r + u z = ρ u µ t, 1 were r,, z are te cylindrical coordinates as defined in Fig. 1, t is te time, u is te torsional displacement, ρ is te mass density, and µ is te sear modulus. Te stress associated wit te torsional motion as two nonzero sear components, wic relate to te torsional displacement by: were J 1 and Y 1 are te first-order Bessel functions of te first and second kinds, respectively, and te coefficients A, B, C, D must be determined from boundary conditions. An alternative form of te solution 8 is: P r =A H 1 1 βr+b H 1 βr, 1 were H 1 1 and H 1 are te first-order Hankel functions of te first and second kinds. Due to teir asymptotic beavior, te Hankel functions sometimes offer convenience for pysical interpretations of te solution. Wen β =, te solution 8 is replaced by a special solution: P r =A r + B r. 11 Similarly, wen k =, te solution 9 is replaced by: Qz =C z + D. 1 A complete solution for free vibrations can be obtained by a superposition of all possible solutions in te form of 4 wit te value of β or k determined from te boundary condition. For forced vibrations, a particular solution satisfying te forcing boundary conditions must be included in te superposition. Tis paper focuses on free vibrations only.

3 kang et al.: torsional vibrations of stepped elastic plates 351 A. Free Vibrations of a Circular Plate First consider a circular plate of radius a and tickness wit traction-free boundaries at all surfaces and edges Fig. 1a. Te boundary condition requires tat: dp dr P = r atr = a, 13 dq = dz atz =and. 14 In addition, it is implied tat te displacement at te center of te plate r = is finite. Consequently, any singular terms wit respect to r must be discarded from te solution. Terefore, te solutions become: { P m r = A r m = A m J 1 β m r m =1,,, 15 were β m = s m /a and s m is te mt nonzero root to J s =, and: Q n z =D z cos k n z, 16 were k n =nπ/ forn =, 1,,... Combining 15 and 16, one obtains te complete solution for torsional vibrations of te circular plate, namely: u r, z, t = A n r cos k n z e iωnt were: + n= B mn J 1 β m rcosk n z e, iωmnt 17 m=1 ω n = nπ µ ρ, 18 µ n π ω mn = ρ + s m a. 19 Solution 17 consists of all te possible torsional modes tat satisfy te boundary conditions for te circular plate, were 18 and 19 give te corresponding resonant frequency of eac mode. Fig. sows te spectrum for torsional vibrations of circular plates, were te frequency is normalized by te first cut-off frequency, ω 1 =π/ µ/ρ, and plotted against te ratio between te tickness and te radius of te plate. Te spectrum includes two limiting cases. To te rigt end of te spectrum, te first several modes wit te lowest frequencies correspond to te case wit β =. In tis case, te displacement is proportional to te r-coordinate as given by te first term in te bracket of 17, and te resonance frequencies ω n are independent of te radius of te plate. Tese modes are identical to tose predicted by te one-dimensional model for long cylindrical rods. To te left end of te spectrum, te first several modes wit te lowest frequencies now correspond to te case wit Fig.. Spectrum for torsional vibrations of uniform circular plates. Te circles are numerical results from Zou et al. 7. k =n =, for wic te displacement is independent of z and te resonance frequency is independent of te tickness. Tese modes are identical to tose predicted by Onoe 1 for tin circular plates /a 1. Between te two limits, te frequencies from different groups are interwoven. Te spectrum clearly sows wen te onedimensional model or te tin-plate approximations can be used and wen te exact tree-dimensional analysis is required. Zou et al. 7 recently conducted a treedimensional vibration analysis using te Cebysev-Ritz metod and predicted te first 1 resonance frequencies for a circular plate wit tickness ratio /a =.4, as sown by te circles in Fig.. Te excellent agreement confirms te accuracy of te Cebysev-Ritz metod. B. Free Vibrations of Annular Plates Next consider an annular plate wit outer radius a, inner radius b, andtickness Fig. 1b. Bot inner and outer edges are traction free. In addition to te boundary conditions 13 and 14, te traction-free condition at te inner edge requires tat: dp dr P =atr = b. r Consequently, 15 becomes: A r m = P m r = A m Y β m a J 1 β m r J β m a Y 1 β m r m =1,,, 1 were β m = s m /a wit s m being te mt root to te following equation: J sy ηs J ηsy s =, and η = b/a. Wen te inner radius b =, 1 recovers 15.

4 35 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february 6 were te frequencies ω n and ω mn take te same form as in 18 and 19. Te frequency spectrum for te annular plates is similar to Fig., but te radial wave number β m varies wit η, te ratio between te inner radius and te outer radius. Fig. 3 sows te first five radial wave numbers obtained from as functions of te ratio. Te left end of te figure corresponds to te case of solid circular plates η =. Toward te rigt end, te wave numbers rise rapidly, wic correspond to tin-walled tubes. A similar result was obtained by Clark 3 for ollow cylinders. Following similar procedures, exact solutions for torsional vibrations of circular and annular plates wit oter boundary conditions can be obtained. For plates wit nonuniform tickness, owever, exact solutions generally are not available 5. Consider a circular plate wit a tickness step Fig. 1c, wic as tickness 1 at te inner region <r<bandtickness 1 at te outer region b < r < a. Te general solution for torsional vibrations of suc plates sould consist of two parts. For te inner region, te displacement takes te form of 17; for te outer region, te displacement takes te form of 3. Te two parts are coupled troug te joint boundary at r = b, were te continuity conditions for te displacement and te sear traction ave to be applied. However, exact solutions satisfying te continuity conditions cannot be obtained analytically. Te remainder of tis paper develops an approximate plate teory, from wic approximate solutions to torsional vibrations of circular plates wit tickness steps are obtained. III. Approximate Plate Teory Fig. 3. Radial wave number for torsional vibrations of annular plates versus te ratio between inner and outer radius. Wit 16 uncanged, te complete solution for torsional vibrations of te annular plate is: u r, z, t = n= + A n r cos k n z e iωnt B mn Y β m a J 1 β m r m=1 J β m a Y 1 β m r cos k n z e iωmnt, 3 A general procedure for deducing approximate equations for elastic plates from te tree-dimensional teory of elasticity was first introduced by Mindlin 8 based on te series expansion metods and te variational metod. Te procedure as been used to derive approximate plate teories for bot elastic and piezoelectric crystal plates wit uniform 9 31 and nonuniform tickness 18, 19, 3. Here we follow te same procedure to derive approximate equations for torsional vibrations of isotropic elastic plates. Te approximate equations ten will be used to deduce an approximate solution to torsional vibrations of circular plates wit tickness steps in te next section. Considering torsional motion of a linear elastic continuum of volume V bounded by a surface S, te variational principle leads to: σr dt V r + σ z z + σ r ρ u r t δu dv =, 4 dt t n r σ r n z σ z δu ds =, 5 S were t is te traction at boundary S, and n is te outward normal at te boundary. For a circular plate Fig. 1a, te boundary S consists of two surfaces at z =andand te edge face at r = a. For an annular plate Fig. 1b, anoter edge at r = b adds to te boundary. Following Mindlin 8, te displacement may be expanded into a series of functions of te tickness coordinate. Various function series ave been used to develop approximate plate teories. For torsional vibrations, noticing te form of exact solutions in 17 and 3, we expand te displacement into a cosine series: u r, z, t = n= u n r, tcos nπ z. 6 Te ortogonality of te cosine series leads to: u n r, t = δ n u r, z, tcos nπ z dz, 7 were δ mn =1ifm = n and δ mn =oterwise. Substituting 6 into 4 and integrating over te tickness of te plate, we obtain tat: dt A n= σ n r,r + r σn r + nπ σn z + 1 F n 1+δ n ρü n δu n da =, 8

5 kang et al.: torsional vibrations of stepped elastic plates 353 were A is a plane parallel to te surface of te plate, and: σ n r σ n z = 1 = 1 σ r cos nπ z dz, 9 σ z sin nπ z dz, 3 F n = 1 n σ z σ z. 31 In a similar manner, substituting 6 into 5, we obtain tat: dt C n= t n n r σ n r δu n ds =, 3 were C is te contour of te plate edges, and: t n = 1 t cos nπ z dz. 33 In deriving 3, te integral over te two surfaces of te plate as been set zero as te surface traction is specified troug te definition of F n in 31. For 8 and 3 to be true for arbitrary variations, we ave, in A: σ n r,r + r σn r + nπ σn z + 1 F n and te boundary condition on C: t n = n r σ n r = 1+δ n ρü n, 34 or un =û n, 35 were û n is any specified displacement at te edge. As a result, te tree-dimensional problem described by 4 and 5 as been transformed to a system of twodimensional equations. Te variables, u n, σ n r, σn z, F n, t n are te nt-order, two-dimensional components of displacement, sear stresses, face traction, and edge traction, respectively. By inserting 6 into and 3 and ten into 9 and 3, we obtain tat: σ n r σ n z = µ1+δ n = µnπ u n r un, 36 r u n. 37 Substitution of 36 and 37 into 34 leads to: µ u n r + 1 r u n r un r nπ n µ u + δ n F n = ρü n, 38 wic is te nt-order displacement equation of torsional motion. Eac specific term in te infinite series expansion of te displacement in 6 can be obtained by solving 38 wit associated boundary conditions at te edges given by 35. For circular plates and annular plates wit tractionfree surfaces F n =, te infinite series expansion is identical to te exact solutions in 17 and 3. For circular plates wit tickness steps Fig. 1c, specific displacement terms for te inner and outer regions can be obtained from 38 separately, and te continuity condition at te step can be approximated by 35. Te procedure is presented in Section IV. IV. Approximate Solution for Stepped Plates Fig. 1c sketces te geometry of te plate under consideration. In general, all orders of te displacement in te series expansion must be considered for bot te inner and outer regions of te plate, wic leads to an infinite number of coupled equations. Suc analysis, owever, is practically impossible. Instead, as generally understood, at one resonance frequency, one particular displacement component dominates te oters. Consequently, it is possible to analyze a single mode approximately by neglecting te coupling wit te oter modes. Similar approaces ave been widely used in te truncation procedures in te development of approximate plate teories 8 3. Te present study focuses on te vibration of te inner region, wic may be trapped by te step. As an approximation, te displacement components of various orders in te inner region are considered separately. Te displacement of te outer region is expanded into a series to ensure te displacement continuity at te junction, wic will be furter truncated into a finite number of terms. Terefore, for te nt-order torsional vibration of te inner region, te displacement of te stepped plate takes te form of 39 see next page. Solving 38 for te inner and outer regions separately, we obtain tat: were: u n 1 u m = = An 1 J 1 β n 1 r A m J 1 β m r β n 1 = ρ µ ω β m = ρ µ ω e iωt, 4 + B m Y 1 β m r e iωt, 41 nπ, 4 1 mπ. 43 Te corresponding in-plane sear stresses are: σ n r1 = 1+δ n µa n 1 βn 1 J β n 1 r e iωt, σ m r = 1+δ m µβ m A m J β m r + B m Y β m r e iωt

6 354 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february 6 u n 1 r, tcos nπ z, r < b and <z< 1 ; u r, z, t = 1 u m r, tcos mπ z, b < r < a and <z<. m= 39 At te junction r = b, bot te displacement and te traction are required to be continuous. In te case of 1 >, part of te edge of te inner region is traction free wit unspecified displacement. Tus, te continuity of te displacement is only required for <z<,wic, by 7, leads to: u m = δ m u n 1 cos nπ z cos mπ z dz However, te traction at te edge of te inner region is fully specified by te traction-free part and te continuous part, wic requires tat: 1 1 σ n r1 = m= δ m σ m r cos mπ z cos nπ z1 dz. 47 Eq. 46 and 47 represent te approximate continuity conditions at te tickness step. Substitution of 4, 41, 44, and 45 into 46 and 47 leads to: J 1 β m b were: m= A m +Y 1 β m b B m =Λ mn J 1 β n 1 b 1 + δ m Λ mn β m J β m b A m + Y β m b B m =1+δ n β n 1 1 J β n 1 b Λ mn = δ m A n 1, 48 A n 1, 49 cos mπ z cos nπ z1 dz. 5 In addition, a traction-free condition at te outer edge r = a requires tat: J β m a A m + Y β m a B m =. 51 Eq. 48, 49, and 51 form a linear system of infinite degrees m =, 1,,... In practice, only a finite subset of te equations may be used to obtain approximate solutions. Let m take values from to M. Te linear system takes te form: Θ v =, 5 wereθisasquarematrixofsizem +1 and v is a vector consisting of all te coefficients, A n 1, Am,andB m.for nontrivial solutions, te determinant of te matrix vanises, namely: detθ =, 53 wic gives te frequency equation for free vibrations of te stepped plate. Te standard procedures of linear analysis ten can be used to calculate te resonance frequencies and te corresponding mode sapes. Similarly, for 1 <, te continuity conditions 46 and 47 become: u n 1 = δ 1 n 1 σ m r = δ n 1 m= And 48 and 49 become: m= u m cos mπ z cos nπ z1 dz, 54 σ n r1 cos nπ z 1 cos Λ mn J 1 β m b A m + Y 1 β m b 1 + δ m β m J β m b A m + Y =1+δ n Λ mn β n 1 1 J β n 1 b were: Λ mn = δ n 1 1 mπ z dz. B m 55 = J 1 β n 1 b A n 1, 56 β m b B m A n 1, 57 cos mπ z cos nπ z1 dz. 58 For 1 =, te two types of continuity conditions converge, and te solution reduces to tat for nt-order torsional mode in uniform circular plates. A. Zerot-Order Modes n = Consider first zerot-order torsional vibrations in te inner region of a stepped plate, for wic n =andte displacement is independent of te tickness coordinate. Suc modes also are called radial modes in tin circular plates. Te displacement in te outer regionin generalconsists of an infinite series expansion as in 39 in order to satisfy te continuity conditions at te step. Our calculations sow tat, for te zerot-order modes at frequencies below te first cut-off frequency, te first term m = in te expansion dominates, and te effect of additional

7 kang et al.: torsional vibrations of stepped elastic plates 355 Fig. 4. Frequency spectrum of te zerot-order torsional vibrations of circular plates wit a tickness step 1 / =1.1, a/ = 1. Te open squares are te exact solutions for uniform circular plates, independent of te plate tickness. Te circle indicates an arbitrarily selected point for te plot of mode sapes in Fig. 5. terms is negligible. Fig. 4 sows te first tree resonance frequencies of te zerot-order modes varying wit te radius of te inner region. Te frequencies are normalized by te cut-off frequency, ω 1 =π/ 1 µ/ρ. Tetickness ratio is fixed as 1 / =1.1, and te outer radius is a/ = 1. At bot ends of te plot wit te radius ratio b/a being or 1, te stepped plate reduces to uniform circular plates wit tickness and 1, respectively. Te zerot-order resonance frequencies of a uniform circular plate are given by 19 wit n = and labeled in Fig. 4 as open squares. It is interesting to note tat, altoug te zerot-order resonance frequencies of a uniform circular plate are independent of its tickness and tus identical at bot ends of Fig. 4, te frequencies oscillate sligtly in between, wit te number of oscillations increasing wit te mode number and te amplitude of oscillation depending on te tickness of te step. Suc oscillation may be caused by te interactions between te torsional waves and te discontinuous boundary at te step. To ceck te continuity conditions at te step, Fig. 5 plots te mode sape corresponding to an arbitrarily selected point labeled as A b/a =. in Fig. 4. Te zerot-order two-dimensional components of te displacement and te traction are plotted, bot normalized by te maximum absolute values. It is noted tat te displacement is continuous across te step, but te stress by itself is not continuous. Te error is due to te approximation of te traction continuity condition specified by 47, wic leads to continuity of te total traction for te zerot-order modes, i.e., σ r1 1 = σ r at r = b. Suc approximation is reasonable for te zerot-order modes. Also noted is tat te variations of te displacement and te stress along te tickness direction are different for te inner and outer regions due to te tickness step. Te error, owever, is small wen te tickness ratio is close to one. Fig. 5. Mode sapes for te zerot-order torsional mode corresponding to point A b/a =. in Fig. 4, in a circular plate wit a tickness step 1 / =1.1, a/ = 1. Te dased line indicates te location of te tickness step. B. First-Order Modes n =1 Next consider te first-order modes wit n =1.Te focus is on te vibration modes near te first cut-off frequency. A convergence study wit up to five terms sows tat, in tis case, te second term m = 1 dominates in te series expansion 39 for te displacement of te outer region. Including oter terms produces negligible difference in te results. Fig. 6 sows te resonance frequencies varying wit te radius ratio b/a for a fixed tickness ratio 1 / =1.1 and outer radius a/ = 1. Te left end of te plot corresponds to a uniform circular plate of tickness, wic as a cut-off frequency, ω c =π/ µ/ρ, and te rigt end corresponds to a uniform plate of tickness 1 wit a lower cut-off frequency, ω c1 =π/ 1 µ/ρ 1 >. Te exact solutions for te uniform circular plates are calculated from 19 wit n = 1 and labeled as squares at bot ends. In between, te resonance frequencies cange continuously. No first-order modes can be found below te cut-off frequency ω c1.ofparticularinterest are te vibrational modes wit frequencies between te two cut-off frequencies. For tese modes, te radial wave number in te outer region, β 1, is imaginary, and te Bessel functions describing te distributions of te displacement and te stress asymptotically reduce to tose decaying exponentially from te step. Consequently, te vibration is trapped witin te inner region. Fig. 7 plots te mode sape associated wit te point A b/a =.

8 356 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february 6 Fig. 6. Frequency spectrum of te first-order torsional vibrations of circular plates wit a tickness step 1 / =1.1, a/ = 1. Te open squares are te exact solutions for uniform circular plates. Te circle indicates an arbitrarily selected point for te plot of mode sapes in Fig. 7. in Fig. 6. From 46 and 47, te approximate continuity conditions at te step r = b for te first-order modes are: u 1 =Λ 11u 1 1 and σ1 r1 1 =Λ 11 σ 1 r,were Λ 11 =.939. Terefore, te continuity conditions are approximately satisfied. As sown in Fig. 7, bot te displacement and te stress decay exponentially from te step, wic confirms te existence of trapped torsional modes in te stepped plate. Similar analyses can be conducted for iger-order modes e.g., n =, 3,..., and trapped torsional modes are expected to exist near eac cut-off frequency. C. Trapped Torsional Modes in Infinite Plates Te existence of trapped torsional modes in stepped plates offers possibilities to design an array of localized energy traps or resonant cavities on a large plate, eac serving as a torsional-mode resonator. Tese resonators are very sensitive to surface loading and may be used for a variety of sensing applications. Eac tickness step can be a circular mesa, for example, formed by macining, etcing, or bonding of a decal onto te surface of te plate. Te design parameters include te tickness and te radius of te mesa as well as te spacing between adjacent mesas. Te frequencies of te trapped modes depend on te mesa dimensions, and te spacing must be large enoug to avoid coupling between adjacent resonators. Eac mesa tus can be considered sitting on an infinite plate a isolated from te oters. In tis case, it is more convenient to use Hankel functions to describe te displacement outside te mesa, and by teir asymptotic beavior only te first Hankel function remains so tat te displacement at te infinite outer boundary vanises. Terefore, any possible modes must be trapped near te mesa. Fig. 8 sows te resonance frequencies for te first-order modes Fig. 7. Mode sapes of te first-order mode corresponding to point A of Fig. 6, sowing te caracteristic of a trapped torsional mode in a circular plate wit a tickness step 1 / =1.1, a/ = 1. Te dased line indicates te location of te step. varying wit te mesa radius for a fixed tickness ratio 1 / =1.1. It is found tat, depending on te mesa dimensions, tere may exist zero, one, or multiple trapped modes. Fig. 9 depicts a diagram for te number of trapped first-order torsional modes in a circular mesa wit te radius b/ and te tickness ratio 1 / as te coordinates. Te lines divide te plane into six regions, wit te number of trapped modes denoted in eac region. Te diagram is independent of te material properties as long as te plate is omogeneous, isotropic, and elastic. Te diagram may be read in two ways. For a circular mesa wit a given tickness, tere exists a critical radius, below wic no trapped mode exists; multiple trapped modes may exist wen te radius is large. It is often desirable to ave a single trapped mode, wic requires a mesa radius witin te window bounded by te lowest two lines in Fig. 9. Alternatively, if te mesa radius is fixed, tere exists a critical tickness, below wic no trapped modes can be found, and a single trapped mode exists wen te tickness is witin te window. Suc a diagram may serve as a guide for designing energy-trapped torsional-mode resonators. Te teoretical results from tis study are compared to experimental measurements. A brief description of te experimental procedure follows. More details will be presented elsewere 6. Circular mesas were macined in a 6 series cast aluminum plate, wit te mesa perimeters defined by removing metal in te form of a moat. Te moat was made wide enoug typically exceeding two

9 kang et al.: torsional vibrations of stepped elastic plates 357 TABLE I Dimensions of Six Samples Used in Experimental Measurements. A B C D E F Diameter, b mm Tickness, 1 mm Tickness, mm Tickness ratio, 1 / Ratio b/ Fig. 8. Te resonant frequencies of trapped first-order torsional modes in a circular mesa on an infinite plate 1 / =1.1. Fig. 1. Comparison of resonance frequencies of trapped torsional modes between experiments solid lines and predictions dased lines. Te dimensions of te samples A F are listed in Table I and indicatedinfig.9. Fig. 9. A diagram for te number of trapped first-order torsional modes in a circular mesa on an infinite plate. Te circles correspond to te samples A F used in experiments. or more wavelengts so tat te displacement, wic decayed exponentially away from te mesa perimeter, was negligible at te outer edge of te moat and ence eac mesa was isolated acoustically. A noncontact electromagnetic acoustic transducer EMAT was used to generate oscillatory surface tractions. Torsional modes were excited wen te traction force was applied in te circumferential direction. Resonant frequencies were determined by adjusting te pulse train frequency for peak initial amplitude as observed wit an oscilloscope. Te system error of tis measurement is estimated to be 5%. A series of tests were conducted to measure te surface motion at resonances in order to confirm te observation of trapped torsional modes, for example, by using an absorbing stylus to determine te locations of displacement maxima and using a pick-up coil to confirm te circumferential motion. Table I lists te dimensions of six circular mesas used in experiments. Tese samples correspond to te points A F labeled in Fig. 9. Te diagram predicts tat samples C and E ave a single trapped torsional mode. Samples A, D, and F ave two trapped modes. Sample B as tree trapped modes. Fig. 1 sows te measured resonance frequencies in comparison wit te predictions by te approximate metoddevelopedintispaper.incalculatingtefre-

10 358 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 53, no., february 6 quencies, te sear modulus and te mass density of te cast aluminum are taken to be 6 GPa and 7 kg/m 3,respectively. Te agreement between te measurements and te predictions is excellent, except for Sample B, in wic only two trapped modes were observed but tree were predicted by te teory. It was noted in experiments tat spurious modes were generated and not trapped wen te step tickness was relatively large and te frequency was ig. Te validity of te approximate teory is tus limited to small step tickness wit frequencies close to te cut-off frequency. V. Conclusions Tis paper presents a teoretical study of torsional vibrations in isotropic elastic plates. In particular, an approximate metod is developed to analyze torsional vibrations in circular plates wit tickness steps. Approximate solutions are presented for te zerot- and first-order torsional modes. Of practical interest is te trapped firstorder mode, wic is teoretically predicted and confirmed by te mode sapes. Te number of trapped first-order torsional modes in a circular mesa on an infinite plate is determined as a function of te normalized geometric parameters, wic may serve as a guide for designing distributed torsional-mode resonators for sensing applications. Comparisons between te teoretical predictions and experimental measurements sow close agreements in te resonance frequencies of trapped torsional modes. Acknowledgment R. H. acknowledges College of Engineering at Te University of Texas, Austin, for financial support. References 1 H. G. Craigead, Nanoelectromecanical systems, Science, vol. 9, pp ,. A.M.Fennimore,T.D.Yuzvinsky,W.-Q.Han,M.S.Furer,J. Cumings, and A. Zettl, Rotational actuators based on carbon nanotubes, Nature, vol. 44, pp , 3. 3 P. A. Williams, S. J. Papadakis, A. M. Patel, M. R. Falvo, S. Wasburn, and R. 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11 kang et al.: torsional vibrations of stepped elastic plates 359 Min Kyoo Kang was born in Taegu, Sout Korea, in He received is B.S. degree in te Scool of Mecanical Engineering of Hanyang University in Sout Korea in and received M.S. degree in macine design and production engineering from te same university in. Currently e is a P.D. student in te department of Aerospace Engineering and Engineering Mecanics of Te University of Texas at Austin. His researc interests include vibrations and wave propagation in beams and plates, piezoelectric devices, and molecular dynamics simulations. Terence Knowles M received is B.S. degree in Applied Pysics from te University of London in 1965, and is M.S. in Pysics from De Paul University, Cicago, IL in 197. He is co-founder and president of Texzec Inc. at Round Rock, TX. His researc interests include vibrations and wave propagation in beams, rods, and plates, ultrasonic transducers, and sensors. He olds numerous patents in tese areas. Rui Huang received is B.S. degree in Teoretical and Applied Mecanics from te University of Science and Tecnology of Cina in 1994 and is P.D. degree in Civil and Environmental Engineering, wit specialty in Mecanics, Materials, and Structures, from Princeton University in 1. He is currently an assistant professor of Aerospace Engineering and Engineering Mecanics at te University of Texas at Austin. His researc interests include mecanics and materials in microelectromecanical systems MEMS, microelectronics, and emerging nanotecnologies.