FACTA UNIVERITATI eies: Woing nd Living Envionmentl Potetion Vol. 5 N o 8 pp. 9-57 COMPONENT OF THE MECHANICAL IPLACEMENT OF THE PIEZOCERAMIC RING POINT UC 5. Miln Đ. Rdmnović gn. Mnčić Zon M. Petušić Gon Z. tnčić Fulty of Eletoni Engineeing Univesity of Niš ei Astt. In tis ppe te omponents of te menil displements of te pieoemi ing points in te dil nd tiness dietion of osilltion e detemined. Menil displements e otined s te funtion of ing tiness nd te intenl-extenl dius tio. Te influene of ny dimension nges of te ing on te font fom of its displements is disussed. Teoetil nlyses sow tt te poposed metod is simple esy to lulte nd espeilly suitle fo te study of oupled vitions. Key Wods: Menil displements Pieoemi ing Resonne fequeny Vitionl modes. INTROUCTION Te initil ondition fo designing distinguisle ultsoni sndwi tnsdue is full undestnding nd eognition of its esonnt fequenies s well s te wveguide mode []. One of ll te esonnt modes is lwys pedominnt in te woing esonnt fequeny. Te pesene of te unwnted psite modes wi n exist t some fequeny lose to woing fequeny n signifintly edue te qulity of ultsoni lening nd o welding. In poweful industil ultsound pplitions fo te elition of ig-fequeny osilltions in tiness extension o longitudinl mode pieoeleti emis pieoemis of diffeent spes s exittion pts of omplex nd omposite ultsoni tnsdues e used. It is neessy to e quinted wit tei omplete pieoeleti teistis s well s elstiity popeties. Tese emi elements e metlied on tei min sufes in te tiness dietion wi is t te sme time te dietion of tei polition. Metlied sufes lso seve s eleti exittion ess points. Te tiness nd dil osilltions of te ing wi e mutully oupled e povoed. Reeived Noveme 6 8
5 M. Đ. RAMANOVIĆ.. MANČIĆ Z. M. PETRUŠIĆ G. Z. TANČIĆ Te existene of ot tiness nd dil esonnt mode uses te sufe of te pieoemi tnsdue usully to osillte nonunifomly. Teefoe it is neessy to pefom omptive nlysis of evey pieoemi tnsdue of speifi geometi spe wit ote tnsdues of simil spe nd fo nlogous pplition in ode to optimie te emitting sufe nd me it moe effiient []. Using te suggested teoetil model xisymmetil viting modes of pieoemi ing s te ultsoni tnsdue e nlyed in tis ppe nd te effets of ll its dimension vitions []. Te sptil distiution of te menil displements of te ing is detemined y tee-dimensionl ppo sed on wve font speding long oodinte xis in te dil nd tiness dietion ming it possile to pedit dynmi evio of te ing [].. MOEL OF EXTERNAL BEHAVIOR OF THE PIEZOCERAMIC RING Pieoemi elements wi e te sujet of te nlysis in tis ppe e pieoemi ings polied oss tiness tt is polied pllel to te -xis wit n oute dius inne dius nd tiness nd wit ompletely metlied iulinged plne sufes supplied wit ltenting exittion voltge. imensions of te ing nd pol-ylindil oodinte system wit its oigin in te ing ente e defined in Figue. Evey ing sufe is loded wit ousti impedne Z i wee v i nd F i e veloities nd foes on tose ontou sufes P i i. I V v F F v F v F v v v F F v 55 piln pot netwo me` F V I v F Fig.. Loded pieoemi ing: geomety nd dimensions; ing s 5-ess netwo Pieoemi mteils nd teefoe te pieoemi ings too e teied y tensos of tei elsti pieoeleti nd dieleti onstnts. Te most fequently used set of onstitutive pieoeleti equtions of te fou existing ones pesents tensos of menil stesses T nd n eleti field E inside te mteil in funtion of te tenso of eltive menil defomtions dilttions nd dieleti displement. In se of te pieoemi ing fom Figue wi will e nlyed in tis ppe omponents of te eleti field E nd E θ e equl to eo on two plne sufes euse tey e
Components of te Menil isplements of te Pieoemi Ring Points 5 metlied nd it is lso ssumed tt tey e negligile equl to eo eveywee inside te mteil. Besides tt due to xil symmety only symmeti dil nd tiness modes of osilltion e exited. Aodingly ll vlues e independent of te ngle θ so te displement u θ is equl to eo. Te poposed model of pieoemi ing is otined wit te ssumption tt oodinte xes nd e dietions of pue unoupled modes of wve popgtion wit menil displements in te dil nd tiness dietion u u t nd u u t. Wit tese ssumptions te most fequently used set of onstitutive equtions wi desie te osilltion of pieoemi ing is edued to te following system of equtions in pol-ylindil oodinte system []: T T T θθ E θθ θθ θθ θθ / ε wee ij e te oeffiients of te elsti onstnt tenso; ε is te dieleti onstnt of te ing in ompessed stte; ij e elements of te pieoeleti onstnt tenso ij. Reltions etween omponents of te eltive defomtions tenso nd menil displement veto u e te following pq if p q: u / θθ u / u /. Ptil diffeentil equtions tt desie te osilltions of pieoemi ing in te dil nd tiness dietion e []: T T T Tθθ u ρ t u ρ t By By sustituting in one gets ptil diffeentil equtions of osilltion in te dil nd tiness dietion in te following fom u u u u u ρ t u ρ t Assuming tt te wves e moni e jωt omponents of te menil displement in te dil nd tiness dietion e solutions of pevious equtions nd tey e pesented toug two otogonl wve funtions:
5 M. Đ. RAMANOVIĆ.. MANČIĆ Z. M. PETRUŠIĆ G. Z. TANČIĆ u u [ A J B Y ] e jωt [ C sin os ] e jωt wee ω / v ω / v v / ρ nd v / ρ e wve teisti eigenvlue numes nd pse veloities of two unoupled wves in te dil nd tiness dietion espetively; ω is ngul fequeny; ρ is density of te pieoemi; J nd Y e Bessel s funtions of te fist ode of te fist nd seond ind espetively. Te onsequene of oosing su otogonl funtions fo displement is tt oundy onditions nnot e fulfilled t evey point on extenl sufes extly ut only ppoximtely.. NUMERICAL REULT AN ICUION In ode to fute nlye te ntue of te esonnt modes of te pieoemi ings it is neessy to detemine in detil te omponents of te menil displements of te pieoemi ing points in te dil nd tiness dietion. isplements u nd u in te nd dietion e ledy pesented toug two otogonl wve funtions wee if it is ssumed tt te pieoemi ing is menilly isolted one sould pply oundy onditions on te extenl sufes tt eflet tis stin stte witout menil stess. ine te equtions in do not fulfill tese oundy onditions tee pplies te ondition tt te integl of te menil stess on evey extenl sufe is equl to eo wi oesponds to te ondition tt te esulting foe on tose ontou sufes is equl to eo sed on wi te unnown onstnts A B C nd n e detemined: P P 5 P T dp T dp P T dp T dp. By sustituting te onstitutive equtions in into te oundy onditions defined y te equtions in 5 nd using te nown eltions in n eqution system is otined wit wi unnown onstnts A B C nd n e detemined. By solving tis system wit te ondition of sinusoidl exittion e jωt one gets: wee: 5 9 9 A B B C C 6 8
Components of te Menil isplements of te Pieoemi Ring Points 5. os ] [ ] [ sin 7 6 5 5 9 8 7 6 5 Y Y J J Y Y J J Y Y J J 7 Bsed on pevious omputtions te fist to e detemined e dil displements of te ing points in te funtion of / fo five ings wit te sme oute dius nd diffeent inne dii /;.;.;.6;.8 nd ll tt in te se of two diffeent tiness vlues fo te ing 5mm nd mm. Figue pesents tese dil displements of points etween te inne nd oute ylindil ontou ing sufe in te funtion of / wee y omping oesponding gps one my see te influene of te inese of te inne ing opening nd its tiness on te dil displement spe fo te fist fou esonnt modes. Te pesented menil displements e nomlied wit dil displement of te points on te intenl ylindil sufe fo te fist mode of te PZT8 ing wit dimensions of 8mm 5mm 5mm [5] wile te distnes in te dil dietion e nomlied wit n oute dius. If fo exmple one oseves te fist few lowest modes of tis ing /. one my notie tt fo te fist nd te tid dil mode te intenl nd extenl ylindil sufes move in te sme dietions wile fo te seond mode wi is lso dil te oseved sufes move in opposite dietions wi mens tt fo tis esonne se wit vlue /.7 tee exists iul line tt epesents te wve node fo wi te displement is u et. Te dietions of movement of te ited sufes e pesented in ppopite gps. Wit te inese of te inne dius tt is te inese of / tee e lineied displements of te fist nd te seond dil mode of te emi element wit tiness of 5mm nd ll fou modes of emi elements wit tiness of mm wile in ote ses displements mostly eep te lssi teoetil fom of Bessel s funtions.
5 M. Đ. RAMANOVIĆ.. MANČIĆ Z. M. PETRUŠIĆ G. Z. TANČIĆ Fig.. Nomlied dil displements of te points etween ylindil sufes
Components of te Menil isplements of te Pieoemi Ring Points 55 Fig.. Nomlied tiness displements of te points etween iul-inged sufes
56 M. Đ. RAMANOVIĆ.. MANČIĆ Z. M. PETRUŠIĆ G. Z. TANČIĆ Figue sows te tiness displements of te points etween iul-inged oundy sufes fo te fist fou esonnt modes fo te ings wit te sme oute dius nd diffeent inne dii /;.;.6;.8 nd wit te sme lengt vlues s in te pevious nlysis of dil displements 5mm nd mm. Hee too te menil displements of te points in te tiness dietion e nomlied wit te tiness displement of te points on te iul-inged sufe fo te fist mode of te PZT8 ing wit dimensions 8mm 5mm 5mm wile te distnes in te tiness dietion e nomlied wit one lf of te tiness. It is ovious in ll te pesented ses tt te plin metlied sufes lwys move in opposite dietions. As expeted emi tiness s te getest influene on te tiness displement ppene wile te influene of te inne opening exists nd it is gete wit tie emis wile wit tinne emis it exists only fo lge inne dimetes. Wit tie emis ylindil ltel oundy sufes lso inese so te influene of te dil osilltions on tiness displements due to te pesene of gete ltel lod is lso gete.. CONCLUION Te osilltoy evio of pieoemi ings is essentilly diffeent in espet to pieoemi diss wit te sme oute dimete nd tiness. In tis ppe we detemined te menil displements of pieoemi ing in te tiness nd dil dietions. Tis metod n seve fo optiml seletion detemining of tnsdue spe depending on its dimensions i.e. tiness inne nd oute dimete tio in ode to design tnsdue wit te est vition teistis. ptil displement distiution of te PZT8 emi ing is defined to detemine te pysil ntue of evey esonnt mode o in ote wods to suessfully identify ll te dil nd tiness esonnt modes. Te deived menil displements e of te Bessel funtion spe in lmost ll te ses. On te sis of tis nlysis te optiml tiness nd dimete tio of te ing n e detemined in ode to ieve n dequte pefomne i.e. it is possile to pedit nd ieve moe omplex displements y vying te ing dimensions ming it possile to design moe sopistited ultsoni sndwi tnsdues. Unlie te omplete ultsoni sndwi tnsdue [] only pieoemi ings e nlyed in tis se nd euse of te eleti exittion of ny metlied sufes wit suitle eletodes wi epesents te menil lod fo te ing only te mesuement of displement in te dil dietion n e done wit suffiient uy. Tese mesuements deived using n nlog senso system wit n opti fie will e te sujet of te uto s fute inteest in te futue. REFERENCE. M. Rdmnović.Mnčić esign nd modeling of te powe ultsoni tnsdues Niš Fulty of Eletoni Engineeing... um.ue E.Moi Vition Pefomnes of Vious ped Pieoeleti Cemi Tnsdues fo Geneting Ultsoni Powe Jpnese Jounl of Applied Pysis Vol. pp. 99-98... Mnčić M.Rdmnović Pieoemi ing loded on e fe: tee-dimensionl ppo Jounl of Tenil Aoustis Vol. pp..-.7... Mnčić M.Rdmnović ynmis of n Ultsoni ndwi Tnsdue t Intentionl ymposium on Powe Eletonis Ee 99 Novi d ei pp. -9 999. 5. Five pieoeleti emis Bulletin 66/F Veniton Ltd. 976.
Components of te Menil isplements of te Pieoemi Ring Points 57 KOMPONENTE MEHANIČKIH POMERAJA TAČAKA PIEZOKERAMIČKOG PRTENA Miln Đ. Rdmnović gn. Mnčić Zon M. Petušić Gon Z. tnčić U ovom du odeđene su omponente meniči pomej tč pieoemičog psten u dijlnom i deljinsom pvu osilovnj. Meniči pomeji su doijeni u funiji deljine psten o i odnos unutšnjeg i spoljšnjeg pečni. isutovn je utij pomene dimenij psten n oli font njegovi pomej. Teoijse nlie pouju d je pedloženi metod jednostvn l počune i poseno pogodn nliu spegnuti viij. Ključne eči: Meniči pomeji Pieoemiči psten Reonntn fevenij Osiltoni mod