Phase Velocities of Three-Dimensional and Axis-Symmetrical Elastic Waves in Isotropic Cylindrical Shell

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1 Intentionl ounl of Teoeticl nd Mteticl Pysics, (6): 96- DOI:.593/j.ijtp.6.4 Pse Velocities of Tee-Diensionl nd Axis-Syeticl Elstic Wves in Isotopic Cylindicl Sell S. L. Ile nov *, A. A. Klescev Sint-Petesug Stte vy Tecnicl nivesity, Sint-Petesug, Lotsnsy st., 3, 98, Russi Astct Bsed on use of Deye s potentils one cn find te diect solution of te pole of definition of te ccteistic equtions fo wve nues of tee-diensionl nd xis-syeticl (flexul nd longitudinl) elstic wves in steel nd luiniu cylindicl sells of vious ticness. Keywods Pse Velocity, Deye s Potentil, Flexul Wve, Longitudinl Wve, Tee-Diensionl Pole, Axis-Sy et icl Pole. Intoduction Te Deye s potentils e used fo te fist tie fo study of te tee-diensionl flexul wves. Te ppe sows te existing of te tee-diensionl nd xis-syet icl flexul wves in cylindicl sell (conty to te cylindicl ). Te ppe deonsttes te clculted vlues of te pse velocities of xis-syeticl nd tee-diensionl flexul wves, wile te pse velocities of te non-eo fos of tee syeticl flexul wves e clculted fo te fist tie. Figue. Te defotion of cylindicl sell fo flexul () nd longitudinl () xis- syeticl wves * Coesponding uto: ils@le.u (S. L. Ilenov) Pulised online t ttp://jounl.spu.og/ijtp Copyigt Scientific & Acdeic Pulising. All Rigts Reseved. Te Tee-Diensionl Flexul Wves in Cylindicl Sell Te fist pt of te ticle sustntites te effectiveness of usge of Deye s potentils[ 3] fo studying of tee-diensionl flexul wves in cylindicl sell. In contst to te te flexul wve in cylindicl sell cn e tee-diensionl nd two-diensionl(xis-sye ticl). Te defotion of cylindicl sell in te popgtion of xis-syeticl flexul (а) nd longitudinl () wves in it is sceticlly sown in fig.. Let s stt wit exintion of tee-diensionl fle xu l wve in isotopic cylindicl sell. In tis cse te se teticl pptus is used s in te study of flexu l wves in [4], ut te nue of unnown quntities nd te nue of oundy conditions incese wit te ccount of te second (intenl) oundy sufce. ow te expnsions of te potentils Ф, V, [5 5] te te fo: i Φ e cos φ A ( ) B( ) i V e cos φ C( ) D( ) () i e sin φ E( ) F( ), wee ( ) ( ) ( ) - e te n s cylindicl function A, B, C, D, E, F - e te unnown coefficients of expnding, wic e clculted sing on te following pysicl

2 97 Intentionl ounl of Teoeticl nd Mteticl Pysics, (6): 96- oundy conditions on te extenl ( ) nd inside ( ) sufces of te sell: tee e no nol nd tngent tensions on te ot oundies of te sell. We desie te nlyticl fo of oundy conditions s following: ( ) (), (3), (4) If we sustitute () in te oundy conditions () - (4), we ll get te deteinnt of six ode[5 5]:, (5) wee [ [ [ ] { [ ]} ( )( ) { ( )( ) } 3 [ ] { [ ]} { ( ) } 4 ( ) 5 ( ) 6 [ [

3 S. L. Ilenov et l.: Pse Velocities of Tee-Diensionl nd Axis-Syeticl 98 Elstic Wves in Isotopic Cylindicl Sell 3 ( ){ [ ( ) ( ) ] ( ) [ ( ) ( ) ]} { ( )( ) ( )( ) ( ) ( )} 4 ( ){ [ ( ) ( ) ] ( ) [ ( ) ( ) ]} { ( )( ) ( ) ( ) ( ) ( )} 5 ( )( ) ( ) 6 ( )( ) ( ) 3 [ ( ) ( ) 3 [ ( ) ( ) 33 { ( ) [ ( ) ] ( ) ( )} 34 { ( ) [ ( ) ] ( ) ( )} 35 ( ) 36 ( ) 4 [ ( ) ( ) 4 [ ( ) ( ) 43 { ( ) [ ( ) ] ( ) ( )} 44 { ( ) [ ( ) ] ( ) ( )} 45 ( ) 46 ( ) 5 i ( ) 5 i ( ) 53 { ( )( ) 5 ( ) ( )} 54 { ( )( ) 5 ( ) ( )} 55 [ ( ) ( ) 56 [ ( ) ( ) 6 i ( ) 6 i ( ) 63 { ( )( ) 5 ( ) ( )} 64 { ( )( ) 5 ( ) ( )} 65 [ ( ) ( ) [ ( ) ( ) 66 Equting te deteinnt (5) eo nd opening i, we eceive te ccteistic eqution fo wve nues of fo of tee-diensionl flexu l wves in isotopic cylindicl sell of ny (ut constnt) ticness. Te decision of te ccteistic eqution fo steel nd lu in iu sells of vious ticness is suitted in fig. nd 3, tus extenl dius ws ccepted equl,, nd dius ccepted two enings:,99 (continuous cuve) nd,8 (dotted cuve). Te enings of velocities longitudinl (С ), ltel (С ) nd Rely s wve (С R ) e sown on te digs. Te ening coesponds to eo fo of flexul wve,

4 99 Intentionl ounl of Teoeticl nd Mteticl Pysics, (6): 96- wic velocity spies siptoticly to velocity of Rely s wve С R, Λ - lengt of longitudinl wve in te sell s teil Λ c f, wee f - fequency of wve in H. In fig. te pse velocities of tee-diensionl flexul wves in steel sells, in fig. 3 - in lu in iu sells e sown. 6 C,/s ,99,8 C C C R /Λ Figue. Pse velocities of tee-diensionl flexul wves in steel sells 7 C,/s ,99,8 C C C R /Λ Figue 3. Pse velocities of tee-diensionl flexul wves in luinu sells Let s tun to xis-syeticl longitudinl nd fle xu l wves. In ccodnce wit[5 5] in xis-syeticl cse te oundy conditions () - (4) ecoe siple: te condition (3) disppes nd te condition () tes te following fo (in tis cse te index o ): ( ) (6), Te deteinnt of te fout ode deived fo oundy conditions tes te fo[5]* : wee (7) ( ) ( ) [ ( ) ( ) ( ) ( ) [ ( ) ( ) 3 i ( ) 4 i ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) [ ( ) ( ) 3 i ( ) 4 i ( ) i ( ) * In te pesent wo te deivtives of cylindicl functions t dil coodinte e ed te following wy: ( ) ( ) ( ) ( ) in te wo[5] diffeently: ( ) ( ) ( ) ( ) ( ) ( ). Bot ppoces e copetent. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 i ( ) 4 i ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Equting te deteinnt (7) eo nd opening it, we get te ccteistic eqution fo wve nues of flexul nd longitudinl xis-syeticl wves. Te deteinnt fo xis-syeticl tosion wves e epesented in[5]: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), (8)

5 S. L. Ilenov et l.: Pse Velocities of Tee-Diensionl nd Axis-Syeticl Elstic Wves in Isotopic Cylindicl Sell wee ( ), - is te desied wve nue of xis-syeticl tosion wve in te sell. Equting te deteinnt (8) eo nd opening it, we get te ccteistic eqution fo wve nues of xis-syeticl tosion wves. 3. Te Results of ueicl Expeient fo Deteintion of Pse Velocities of Elstic Wves Te second pt of te ticle investigtes te esults of nueicl expeient fo deteintion of pse velocities of elstic wves (xis-syeticl nd tee-diensionl) in cylindicl sell nd nlyses te clculted dependences. Te esults of clcultions of pse velocities e epesented in te fig.4. 6 C,/s C C C R /Λ Figue 4. Pse velocities of xis-syeticl flexul, longitudinl nd tosion wves in steel cylindicl sell One cn find te following designtions in te pictue: te cuve - is te pse velocity of eo ode of fle xu l wve te cuve - is te pse velocity of eo ode of longitudinl wve te cuves 3, 4, 5 - e te pse velocities of noneo odes of longitudinl o of flexul wves te stigt line 6 - is te pse velocity of eo ode of tosion wve te cuve 7 - is te pse velocity of te fist ode of tosion wve: Λ c f. In ode to define te influence of extenl nd intenl liquid ediu s on dispesion cuves of pse velocities of sell to te deteinnt of te sixt ode (5) e dded two coluns nd two lines nd it convets into te deteinnt of te eigt ode, nd deteinnt of te fout ode into te deteinnt of te sixt ode. Te potentils of sound wves (in extenl envionent) nd in te filling of te sell expnd y te cylindicl functions te following wy: ( ) Φ G H ( γ ) cos e i Φ K ( γ ) cos e, wee: ( ω c3 ) γ ( ω c4 ) 3 () γ c nd c4 - sound velocities in extenl nd intenl envionents coespondingly. Te coponent of wve vecto lengtwise of xis, s in te sell in te foce of Snellius lw. In tis cse te oundy conditions () fo nol tensions on te ot sufces of te sell tnsfos: (9) ( )( ) ( ) i, wee: ρ - te solidity of te extenl envionent. () ( )( ) ( ) i, () wee: ρ - te solidity of te intenl envionent. Two oe oundy conditions e dded: te nol coponents of displceent vecto e continuous in te ot odes of te sell: Φ A A Φ (3)

6 Intentionl ounl of Teoeticl nd Mteticl Pysics, (6): 96- Φ A A In xis-syeticl cse te condition (6) tnsfos: Φ ( ) i ( ) i (5) (6) (4) Te dded oundy conditions in xis-syeticl cse: Φ A Φ Φ A In xis-syeticl cse: Φ (7) (8) ( ) Φ G H ( γ ) e (9) Φ K ( ) e. () γ Te oundy conditions (), () nd (7), (8) will dd two lines to deteinnt, nd te u ltip lies of unnown coefficients G, K o G, K (xis-syeticl pole) will dd two coluns. In tis cse only two lines in ec of tese two coluns will e diffeent fo eo. If te sell odes te liquid only fo one side (nd fo te ote is still te vcuu) te deteinnt fo deteintion of wve nues will ve te sevent ode in tee-diensionl pole nd te fift ode (in xis-syeticl cse), tt is to deteinnts (5) и (7) one line nd one colun e dded coespondingly. Te esults pesented in te ticle e eceived in te conducting of scientific esec in te fewo of Stte contct P 4 fo 3 Apil FPP (Scientific nd scientific - pedgogicl pesonnel of innovtive Russi fo te yes 9-3). 4. Conclusions Bsed on use of Deye s potentils one cn find te diect solution of te pole of definition of te ccteistic eqution fo pse velocities of tee diensionl flexul wves. Te nueicl solution of tis eqution destintes to otin te dispesive cuves of pse velocities. Te dispesive dependencies of pse velocities of xis syeticl flexul, longitudinl nd tosion wves e epesented. REFERECES [] Deye P. Ann. Pys. 3, 755 (99). [] Foc V. A. Eletognetic Diffction nd Popgtion Poles (Sov. Rdio, Moscow, 97 Pegon Oxfod, 965). [3] Klescev A. A. Te Deye s potentils nd «te potentils of Deye s type» in te poles of diffction, dition nd speding of elstic wves.// Acousticl jounl.. V , P [4] Klescev A. A., Kluin I.I. Aout of te flexul wves in elstic cylindicl. // Poceedings of Sint-Petesug Stte vy Tecnicl nivesity. v P [5] Ilenov S. L., Klescev A. A. Pse velocities of flexul wves in isotopic cylindicl sell of ny ticness (stict decision) // Poceedings of te X Session of te Russin Acousticl Society. -, v., Moscow. GEOS. P [6] Klescev A. A. Diffction nd popgtion of wves in te elstic edius nd odies.// Sint-Petesug: Vls,. P. 56. [7] Klescev A. A. Te dispesive equtions of displceent vectos of diffeent odes of elstic wves in isotopic nd nisotopic cylindicl sells. // Poceedings of te XIII Session of te Russin Acousticl Society.- 3. М.: GEOS. P [8] Klescev A. A. Te sound diffction fo point sound souce in elstic cylindicl sells.// Acousticl jounl.. 4. V. 5.. P [9] Ilenov S. L., Klescev A. A. Sugilo K.A. Te pse velocities of elstic wves in s nd sells.// // Poceedings of te Regionl scientific confeence «Te sipuilding eduction nd science» 3». Pul.ouse of te Sint-Petesug Stte vy Tecnicl nivesity, 3. V.. P [] Ilenov S. L., Klescev A. A. Pysicl odel of sound dition unde ction of te tuulent pulstions. // Poceedings of te Scientific confeence «Bunov s lectues». Pul. ouse of te Kylov sipuilding esec institute. Sint-Petesug. 4. P. 3. [] Klescev A. A. Diffction, dition nd popgtion of elstic wves.// Sint-Petesug.:Pofpint, 6. P.6. [] Klescev A. A. Expeientl sctteing ccteistics of low fequency non-fixed sound signl fo elstic cylindicl sells in Fenel s one. // Poceedings of te XXI

7 S. L. Ilenov et l.: Pse Velocities of Tee-Diensionl nd Axis-Syeticl Elstic Wves in Isotopic Cylindicl Sell Session of te Russin Acousticl Society.-9. М.: GEOS.. P [3] Klescev A. A., Legus F.F., Mslov V. L. Wve pocess in te solids.//sint-petesug. Pul. ouse of te Sint-Petesug Stte vy Tecnicl nivesity,. P.6. [4] Klescev A. A. Sctteing of Low Fequency Pulsed Sound Signls fo Elstic Cylindicl Sells. // Acousticl Pysics.. V P [5] Klescev A. A. Acoustic Scttees.//Te second puliction. Sint-Petesug. Pi,. P. 67.