Vl N 7-79 hp://xg/4/ns4 Naual Sene ansen espnse mullayee hllw ylne usng vaus hees geneale hemelasy Dau S Masha aha M enku * Khale lsba Depamen Mahemas Fauly Sene Kng bul nvesy Jeah Sau aba; * Cespnng uh: enku@kaueusa; Depamen Mahemas Fauly Sene Kaelshekh nvesy Ka l-shekh gyp; Depamen Mahemas Fauly pple Sene mm l-qua nvesy Hly Makkah Sau aba eeve 7 Mah ; evse pl ; aepe pl SC he pesen pape eals wh hemelas pblems nely lng hllw ylne mpse w een maeals wh axal symmey he meum s an-ee wh neglgble by es an wh nenal an exenal hea geneans he gvenng equans een hees he geneale hemelasy ae wen n ems splaemen an empeaue nemen he exa slun he pblem usng een hees geneale hemelasy has been eue he analyal expessns splaemens empeaue an sesses ae un n nal ms an a numeal example has been aken suss he ee he elaxan mes Fnally he esuls have been llusae gaphally n he espnses een hees Keyws: Mullayee Hllw Cylne; Geneale hees hemelasy; elaxan mes INODCION he gvenng equans splaemen an empeaue els n he lnea ynamal hey lassal hemelasy nss he uple paal eenal equan mn an Fue s law hea nun equan he equan splaemen el s nlle by a wave ype hypebl equan wheeas ha he empeaue el s a paabl usn ype equan hs amuns he emak ha he lassal hemelasy pes a ne spee pemnanly elas subanes bu an nne spee pemnanly hemal subanes whh ae uple gehe hs means ha a pa evey slun he equans exens nny [] mulae he hey uple hemelasy name as C-D hey elmnae he paax nheen n he lassal unuple hey hemelasy ha he elas hanges have n ee n he empeaue u he lassal ynamal uple hey hemelasy sll base n a paabl hea equan whh pes an nne spee he ppagan hea wave nas he physal as Geneale hees hemelasy have been evelpe ha ae ee m hs paax an Shulman [] -S hey nue he hey geneale hemelasy base n a new law hea nun by npang a lux ae em an nvlve a hypebl ype hea ansp equan alle he geneale hemelasy wh ne elaxan me he -S hey was exene by Dhalwal an Shee [] he ase ansp mea nqueness he slun he geneale hemelasy wh ne elaxan me une a vaey nns was pve by Ignaak [4] an Shee an Dhalwal [] espevely Geneale hey hemelasy wh w elaxan me paamees has als been ppse ase n a geneale nequaly hemynams Geen an nsay [] evelpe he hey hemelasy wh w elaxan me paamees name as G- hey he G- hey esn vlae he Fue s law hea nun when he by une nsean has a ene symmey In hs hey bh he equans mn an hea nun ae hypebl bu he equan mn s me an es m ha he lassal ynamal uple hey hemelasy he axsyme mullayee hllw ylne pblems have been susse by sme eseahes n he unuple uple an geneale hemelasy n he een yeas Jane an ee [7] nsee he hemelasy mullayee ylnes subjee knwn empeaues a an-ee bunaes by usng aplae ansm an he ne eene meh Kanl Cpygh Ses OPN CCSS
7 D S Masha e al / Naual Sene 7-79 [8] sue he ee seay-sae empeaue an pessue gaen n mpun ylnes e gehe by shnk Shee an nwa [9] susse he pblem an nnely lng elas ula ylne whse nne an ue suaes ae subje knwn empeaue an an ee Yang an Chen [] susse he ansen espnse ne-mensnal quas-sa uple hemelasy pblems an nnely lng annula ylne mpse w een maeals ee [] slve he w-mensnal quas-sa uple hemelas pblem nely lng hllw ylne mpse w een maeals wh axal symmey Chen e al [] susse als he ansen espnse ne-mensnal quas sa uple an unuple hemelasy pblems mullayee hllw ylne llam e al [4] slve he pblem an nne by wh a ula ylnal hle n a hamn el n he nex he geneale hey hemelasy In a een ale enku e al [] pesene he sa benng espnse a smply suppe unnally gae eangula plae subjee a hugh-hehkness empeaue el une he ee vaus hees geneale hemelasy wh elaxan mes In he pesen ale he analyal expessns splaemens empeaue an sesses nely lng hllw ylne mpse w een maeals wh axal symmey ae un n nal ms Numeal examples have been aken suss he ee he elaxan mes Fnally he esuls have been llusae gaphally n he eenes beween he een geneale hees hemelasy FOMION OF H POM hugh hs aea eseah we nse he llwng bunay value pblem We eal wh a pblem nely lng hllw ylne mpse hee layee w een maeals wh axal symmey he lengh he mullayee hllw ylne s an he nne an ue a he ylne ae ene by an espevely see Fgue We assume ha he hllw ylne s aken be heae suenly a he nne an ue suae une empeaue an espevely We ake n aun ha he by es ae absen an hen he unamenal equans he bunay value pblem n he nex he een hees geneale hemelasy n he ase quas-sa an be wen as: qulbum equans he ylne alng an ens: whee j ae he mpnens sess ens an ae he ylnal nae sysem Geneal hea nun equan n he nex geneale hem-elasy hees: n whh s he empeaue an s he uvy; s he ensy; an s he spe hea eeene empeaue; ae he mpnens san a nsan eman he mpnens he hemelas ens ae gven by ens; an ae he sen an h hemal elaxan mes; an ae he hemal n- 4 whee an ae Pssn s as; an ae Yung s mul; an an ae lnea hemal expansn eens Duhamel-Neumann s elans laye numbe k: Fgue he hee-laye hllw ylne an s nae sysem Cpygh Ses OPN CCSS
D S Masha e al / Naual Sene 7-79 Cpygh Ses OPN CCSS 7 7 7 8 whee s he s hemal elaxan me an ae he mpnens splaemen ve he elas nsans j ae gven by G 9 he ull sysem equans he een hees geneale hemelasy wll appea by he llwng nsuns: C-D hey -S hey G- hey SOION OF H POM Subsung qs-8 n qs- we ge he llwng sysem paal eenal equans: qs an epesen he equans equlbum he hllw ylne alng he an ens espevely whle q epesens he uple ansen hea nun equan he kh laye he axsymme hllw ylne he bunay an neae nns he pesen mpse hllw ylne ae gven by: unay Cnns a a a a Ineae Cnns k k 4 Inal Cnns a = slve he abve equans we nue he llwng mensnless quanes: * j j j j
D S Masha e al / Naual Sene 7-79 Cpygh Ses OPN CCSS 74 Ne ha he nex epesens he hemmehanal ppees laye Subsung he abve mensnless quanes n qs- we ge 7 8 he mensnless sesses ae als gven by 9 whee j j he slun qs-8 may be gven by usng he llwng subsuns an ha sasy he bunay nns gven n q: sn s s w u 4 whee he unns an ae gven espevely by: sng q n qs-7 ne an ge 9 8 7 4 w u w u w u whee 4
D S Masha e al / Naual Sene 7-79 7 7 8 9 he slun he abve sysem s-e eenal equans may be easly gven all hees he espnng sluns C-D -S an G- hees ae als bane m he geneal ne 4 NMIC XMPS In e llusae he esuls gaphally he gemey an hemelas nsans he w maeals he hllw ylne ae gven n able he ylne s mpse hee layes w sn maeals wh he same hkness eah laye ayes an have ppees he same maeal S he w neaes ae gven a = an = 7 espevely he vaus nn-mensnal paamees use ae: u u he numeal esuls ae ple n Fgues -7 he values an ae aken have he same value as he values an an an ae gven espevely n ems an he a he ue aus he ylne s lengh s gven by / In an he * s ppe smply m he mensnless elaxan mes Fgue llusaes he vaan mensnless empeaue hugh axal paamee value he mensnless me namely 8 an a he sen neae he mensnless aal en = 7 he mpuans wee ae u C-D -S an G- hees hemelasy Fgue shws he vaan mensnless aal sess hugh he axal paamee he values mensnless me an aal en ae hsen be 8 an 7 espevely he esuls wee alulae -S an G- hees In wha llws we es u aenn he esuls -S hey Fgues 484 an llusae espevely he vaan mensnless aal an axal splaemens u an u ; he mensnless empeaue ; an he mensnless sesses an hugh he aal en he mullayee hllw ylne een values he mensnless me 7 an 9 wh he elaxan me Smla esuls ae ple n Fgues 79 an 7 hugh he aal en he 7 able he gemey an maeal nsans a nely lng hllw ylne xal paamee N m N m N m N G m Maeal Maeal 8 8 wa 8 mk wa mk wa mk K 4 K 4 kg 9 9 m kj kg K 8 4 -S G- C-D - -4 - - - 4 empeaue Fgue Vaan mensnless empeaue hugh he axal en he hllw ylne vaus hemelasy hees = 8; = 7 Cpygh Ses OPN CCSS
7 D S Masha e al / Naual Sene 7-79 xal paamee 8 4 G- -S xal splaemen u 9 7 -S - -4 - - - 4 aal sess σ - 7 7 87 aal paamee Fgue Vaan mensnless aal sess hugh he axal en he hllw ylne -S an G- hees = 8; = 7 Fgue Vaan mensnless axal splaemen u hugh he aal en he hllw ylne een values he me paamee aal splaemen u 8 4 - -4 - -8 -S 7 7 87 aal paamee xal splaemen u 9 7 = = = -S - 7 7 87 aal paamee Fgue 4 Vaan mensnless aal splaemen u hugh he aal en he hllw ylne een values he me paamee Fgue 7 Vaan mensnless axal splaemen u hugh he aal en he hllw ylne een values he elaxan me 7 aal splaemen u 7 4 - - = = = -S -8 7 7 87 aal paamee empeaue 4 8 9 -S 7 7 87 aal paamee Fgue Vaan mensnless aal splaemen u hugh he aal en he hllw ylne een values he elaxan me 7 Fgue 8 Vaan mensnless empeaue hugh he aal en he hllw ylne een values he me paamee Cpygh Ses OPN CCSS
D S Masha e al / Naual Sene 7-79 77 empeaue 8 4 8 -S = = = 4 7 7 87 aal paamee Fgue 9 Vaan mensnless empeaue hugh he aal en he hllw ylne een values he elaxan me 7 xal sess σ 4 4 -S - 7 7 87 aal paamee Fgue Vaan mensnless axal sess hugh he aal en he hllw ylne een values he me paamee aal sess σ - - -S - -7-9 7 7 87 aal paamee xal sess σ 4 7 7 7 7 - = = = -S 7 7 87 aal paamee Fgue Vaan mensnless aal sess hugh he aal en he hllw ylne een values he me paamee Fgue Vaan mensnless axal sess hugh he aal en he hllw ylne een values he elaxan me 7 aal sess σ - - -S - -7 = = = -9 7 7 87 aal paamee Cumeenal sess σ 4 - - -S - 7 7 87 aal paamee Fgue Vaan mensnless aal sess hugh he aal en he hllw ylne een values he elaxan me 7 Fgue 4 Vaan mensnless umeenal sess hugh he aal en he hllw ylne een values he me paamee Cpygh Ses OPN CCSS
78 D S Masha e al / Naual Sene 7-79 Cumeenal sess σ 9 8 = = = -S - 7 7 87 aal paamee Fgue Vaan mensnless umeenal sess hugh he aal en he hllw ylne een values he elaxan me 7 Shea sess σ - -S - 7 7 87 aal paamee Fgue Vaan mensnless shea sess hugh he aal en he hllw ylne een values he me paamee Shea sess σ 8 4 = = = -4 -S -8 7 7 87 aal paamee Fgue 7 Vaan mensnless shea sess hugh he aal en he hllw ylne een values he elaxan me 7 mullayee hllw ylne een values he elaxan me when he mensnless me 7 CONCSIONS he nlusn he abve esuls may be gven as: Fgue llusaes ha he mensnless empeaue s slghly hange an he eenes beween C-D -S an G- ae vey small ny he uple hey C-D may gve esuls wh small elave e mpae wh hse gven by an Shulman s -S an Geen an nsay s G- hees Hweve he esuls -S an G- ae muh lse eah he see Fgue he pls esuls gven by an Shulman s hey shw ha he ee he mensnless me s slghly lea n he s laye bu n he sen an h layes he ee s n elae hs happene mensnless aal an axal splaemens see Fgues 4-7 an axal umeenal an shea sesses see Fgues -7 Hweve mensnless empeaue an aal sess he ees mensnless me s vey lea n he s laye an sa eease wh he nease aal en n he sen an h laye see Fgues 8-4 he ee he elaxan me an Shulman s hey n all physal waves splaemens empeaue an sesses s lea n he s laye bu s less nseable n he sen an h layes hs eveale ha he ee he elaxan me has n ee when he mensnless aus s neasng CKNOWDGMN hs pape s ully suppe by he Deanshp Sen eseah a Kng bul nvesy Gan N 8/48 FNCS [] M 9 hemelasy an evesble hem-ynams Junal pple Physs 7 4- [] HW an Shulman Y 97 geneale ynamal hey hemelasy Jnal Mehans an Physs Sls 99-9 [] Dhalwal S an Shee HH 98 Geneale hemelasy ansp mea Quaely pple Mahemas 8-8 [4] Ignaak J 979 nqueness n geneale hemelasy Junal hemal Sesses 7-7 [] Shee HH an Dhalwal 98 unqueness heem an a vaanal pnple geneale hemelasy Junal hemal Sesses - Cpygh Ses OPN CCSS
D S Masha e al / Naual Sene 7-79 79 [] Geen an nsay K 97 hemelasy Junal lasy -7 [7] Jane KC an ee Y 999 hemelasy mullayee ylne Junal hemal Sesses 7-74 [8] Kanl 97 Invesgan sess analyss n mpun ylnes une hgh pessue an empeaue MS hess CI Helwan [9] Shee HH an nwa MN 988 pblem n geneale hemelasy an nnely lng annula ylne Inenanal Junal ngneeng Sene - [] Yan YC an Chen CK 98 hemelas ansen espnse an nnely lng annula ylne mpse w een maeals Inenanal Junal ngneeng Sene 4 9-8 [] ee -Y Geneale uple ansen espnse -D mullayee hllw ylne Inenanal Cm- munans n Hea an Mass anse - [] Chen CK Hung CI an ee Y ansen hemal sesses analyss mullayee hllw ylne a Mehana 7-88 [] Chen CK Hung CI an ee Y hemelas ansen espnse mullayee hllw ylne wh nal neae pessue Junal hemal Sesses 4 987- [4] llam MN lsba K an buelegal hemal sesses n a hamn el an nne by wh a ula ylnal hle whu enegy sspan Junal hemal Sesses 7-7 [] enku M Masha DS an lsba K 9 enng analyss unnally gae plaes n he nex een hees hemelasy Mahemaal Pblems n ngneeng 9 le ID 9 - Cpygh Ses OPN CCSS