Prediction of modal properties of circular disc with pre-stressed fields

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1 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ Pedcon of odal popees of ccula dsc h pe-sessed felds Mlan Naď 1* Rasslav Ďuš 1 bo Nánás 1 1 Slovak Unvesy of echnology n Baslava Faculy of Maeals Scence and echnology n nava Jána Bou nava Slovak Republc Absac. Many sucual eleens n echncal devces as ell as soe ools n anufacung equpen such as sa blade o have a geoec shape sla o a ccula dsc. In any cases hese ccula dscs us ee he equed dynac popees. One of he echnques o acheve he equed dynac popees of ccula dsc s based on nang pe-sessed felds n he dsc plane. In-plane esdual sesses ae ceaed by appopae echnologcal eaens n seleced dsc pa fo exaple by ollng of annulus leadng o plasc defoaon hch causes he change of volue dsbuon n hs dsc pa. he effecs of n-plane esdual sesses on odal popees of ccula dsc ae analysed. he naual fequences odfcaons dependng on poson and dh and also on he change n hckness of he oll-pesessed annulus ae nvesgaed n hs pape. Keyods: ccula dsc vbaon odal popees pe-sessed felds fne eleen ehod 1 Inoducon hee ae any suaons and applcaons n he acouscal and sucual pobles hee he vbang ccula dsc have o be odfed o acheve convenen dynacal popees [1]. hs s anly concened h he equeens fo he so-called unng of he odal popees of he vbang ccula dsc by eans of echnologcal eaens ha nduce he esdual sesses n he dsc plane. he naual fequences of ccula dsc claped on nne adus ae vayng hen he localed plasc defoaon caused fo exaple by ollensonng nduces esdual sesses. he naual fequences of ccula dsc claped on nne adus ae vayng hen he localed plasc defoaon caused fo exaple by ollensonng nduces esdual sesses. A sla effec esulng fo esdual sesses can also be acheved by phase ansfoaon dung echnologcal eaens. Dung he pocess of nang dsc n-plane sesses usng he oll-ensonng pocess a dsc s copessed hn a cean annula conac one beeen o opposng olles. he conac one of ccula dsc s plascally defoed and he esdual sesses ae occued n hole dsc plane. hen he effecs of esdual sesses nduced by oll-enson on odal popees naual fequences ode shapes can be analysed. he appopae condons fo coespondng echnologcal eaens can be pedced usng he naual fequency * Coespondng auho: lan.nad@suba.sk Revees: Mlan Sága Mlan Vaško he Auhos publshed by EDP Scences. hs s an open access acle dsbued unde he es of he Ceave Coons Abuon Lcense 4.0 hp://ceavecoons.og/lcenses/by/4.0/.

2 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ chaacescs hee he naual fequency values ae dependng on paaees causng nplane dsc sesses. he consdeed copuaonal pocedue fo he pleenaon of nplane dsc esdual sesses s based on dea hch s sla o he foaon of heoelasc sesses.he naual fequency chaacescs fo vaous ollng posons fo vaous ollng dephs and dhs of he annulus ae obaned by odal analyss usng Fne Eleen Mehod FEM. he ole of esdual sesses obaned by ollng can be assessed fo he change n naual fequences and odal shapes. Foulaon of he poble Ceang pe-sessed felds n a ccula dsk causes a change n s spaal popees.e. a change n he dsbuon of s ass and sffness paaees. As a consequence of hese changes he odfcaon of he dynac popees of a gven ccula dsc occus..1 heoecal appoach o odellng of a odfed dynac syse he geneal equaon of oon fo undaped syse hou exenal excaon s defned by Mu Ku 0 1 hee M s ass ax K s sffness ax of he syse u and u ae dsplaceens and acceleaons vecos especvely. Equaon 1 can be ansfoed [5] usng he ansfoaon equaons u Φq Φ MΦ I Φ KΦ Λ hee Φ [ 1... N ] s ax of odal vecos s specal ax.. Egenvalue poble of he syse 1 can be en n he fo Φ KΦ Φ MΦ q Λ I q 0 3 hee s naual angula fequency. When he syse descbed by equaon 1 s odfed [4] hen he odfcaon of he syse has o be ncopoaed hough ass and sffness changes of syse paaees. he ass and sffness popees of he syse ae odfed and equaon 1 becoes M M u K K u 0 4 he odfcaon aces M and K chaacese he ass and sffness odfcaons n he spaal odel. he paccal odfcaon s no caed ou on aces bu on physcal coponens o paaees of he sucue. Usng equaons he egenvalue poble of odfed syse 4 s [ Φ K K Φ Φ M M Φ ] q 0 5 hee s he naual angula fequency of odfed syse. Equaon 5 povdes he ne naual angula fequences and ne odal vecos of he syse afe sucual odfcaon. hen -h naual angula fequency s expessed K K. 6 M M

3 . Copuaonal odel of ccula dsc h pesessed felds he geneal shape of ccula dsc of oue adus 0 nne adus v and hckness h 0 Fg.1 s consdeed. he aeal of ccula dsc s soopc and hoogeneous. he nne adus v specfes a ccle hee he dsc s claped by flanges. o odfy he odal popees of a ccula dsc he case h pe-sessed annulus feld s consdeed. he geoecal shape of dsc n-plane pesessed felds ae defned by ddle adus and dh of each pesessed feld.e. 1 and b 1. Fg. 1. he ccula dsc h pesessed felds he fundaenal consdeaons and devaon of equaons of oon ae based on Kchhoff s plae heoy assupons. Usng Kchhoff s plae heoy he feld of dsplaceens n he cylndcal coodnaes can by en as u u v v 7 hee u v and ae dsplaceens of pon layng on neual plane of he ccula dsc n coodnae decons. Geneally he san-dsplaceen elaons n he cylndcal coodnaes fo pesessed ccula dsc can be en n he fo ~ 0 8 esp. v v u u v u ~ 9 hee ] [ ] [ ~ b b s a odfed eavsde funcon descbng he poson of he pe-sessed one x = 0 fo x < 0; x = 1 fo x > 0 3 MAEC Web of Confeences hps://do.og/ /aecconf/ MMS 017

4 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ ae he nal sans nseed n pe-sessed aea and ae he paal devaons. Geneally he sess-san elaons unde consdeaon of nal sesses and nal sans ae gven by ~ σ D ε 0 ε σ 10 hee and ae sess and san vecos and ae nal sess and nal san vecos D s elascy ax. Usng he fne eleen foulaon he equaon of oon fo a fee vbaon of n-plane sessed dsc s descbed by expesson Mu K K u 0 11 hee M s ass ax K s sffness ax K s sffness ax esulng fo sess dsbuon nduced by ollng u and u ae veco of nodal acceleaons and veco of nodal dsplaceens especvely. We noe ha he ass dsbuon of ccula dsc afe ollng s no changed bu he bendng sffness s consdeably changed. Equaon 10 can be ansfoed o odal coodnaes usng he ansfoaon equaons. Afe applyng he hese ansfoaons he equaon of oon 11 can be used o deenaon of he naual angula fequences and ode shapes of he ccula dsc h oll-ensonng nduced esdual sess dsbuon. We oban he follong egenvalue poble hee K M K K M 0 1 K s -h naual angula fequency s egenveco descbng -h odal shape of he ccula dsc. 3 Nuecal sulaon and esuls We consde a ccula dsc Fg. 1 of he oue adus 0 = 10 flange adus s v = 5 hckness h = 1.8. he dh of plascally defoed annulus s assued as 10. hs dh s seleced abaly and s consdeed as a epesenave value fo planned expeenal vefcaon of nvesgaed phenoenon. he npu daa used fo nuecal analyss of ccula dsc ae noduced n he able 1. able 1. Inpu daa Young odulus E [GPa] 10.0 Posson nube [-] 0.3 densy [kg -3 ] coeffcen of heal expanson [K -1 ] oue adus of dsc 0 [] 10.0 flange adus v [] 5.0 dsc hckness h 0 [] 1.8 dh of plasc-afeced one b 1 [] 10.0 deph of ollng of he plasc-afeced one [] {0.0; 1.0;.0; 3.0; 4.0} 4

5 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ he analysed dsc s assued o be pefecly fxed n egon v. he oue edge of ccula dsc s fee. he change n dsk sffness afe ollng hch s epesened by he odfed sffness ax K us be deened fo he esdual sess dsbuon n he dsc plane. o deene he esdual sess dsbuon he ehod of heoelasc sess loadng s used []. he heoelasc expanson nduces a sess dsbuon hch s analogous o he sess dsbuon naed by ollng. he dependence beeen epeaue and deph of oll-ensonng s appoxaely descbed by equaon 13 h 0 hee s Posson nube s he coeffcen of heal expanson h 0 s dsc hckness and s deph of oll-ensonng. he aces M K and addonal ax K hch follos fo sess dsbuon asng fo ollng n hs odel analogy h heoelasc expanson s used ae calculaed auoacally by ANSYS. he calculaon pocesses fo deenaon of naual angula fequences and ode shapes ae ealsed by ANSYS. he dsbuon of adal and angenal esdual sesses nduced n plane of ccula dscs h one oll-enson annulus fo vaous paaees s shon n Fg.. Induced sesses [MPa] Radus of ccula dsc [] Fg.. Dsbuon of adal and angenal esdual sesses n dsc plane 1 = 50 ; = 1 ; b 1 = 10 In Fg. 3 he naual fequency cuves of he odal shapes 0/1 0/0 0/ 0/3 nodal ccles/nodal lnes fo dffeen deph of ollng = calculaed by FEM hen 1 vaes fo 003 o 011 ae shon. he naual fequences of ccula dsc befoe oll-ensonng ae aked by 1 = 0.0. he endency of cuves fo ode shapes 0/1 and 0/0 dffes fo cuves fo ode shapes 0/ and 0/3. he naual fequences of he ode shapes 0/ and 0/3 ncease h c unl he axu values nea c ae eached; hen hey decease. Conay o hs he naual fequences of he ode shapes 0/1 and 0/0 decease h 1 and fo each he nu; hen hey ncease. 5

6 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ a b c d Fg. 3. Dependency of naual fequency on cene adus of pe-sessed annulus 1 fo ndvdual and fo fs fou ode shapes 6

7 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ In Fg. 4 he naual fequency cuves of he ndvdual odal shapes 0/1 0/0 0/ 0/3 n dependency of deph of ollng ae shon. he naual fequences of he odal shapes 0/1 and 0/0 decease h 1 unl he nu values nea ae eached; hen hey ncease. Conay o hs he naual fequences of he odal shapes 0/ and 0/3 ncease h c and fo each he nu; hen hey ncease. he effec of deph of ollng on naual fequency s evden fo hese gaphs. a b c d Fg. 4. Dependency of naual fequency of he fs fou ode shapes on cene adus of pe-sessed annulus 1 and fo vaous Conclusons he heoecal foulaon and calculaon odel fo analyss of dynacal popees of ccula dsc h esdual sess dsbuon ae pesened. Fne eleen analyss fo esang he naual fequences as used. Fo cean ean adus of pe-sessed annulus 1 he naual fequences of ode shapes 0/ and 0/3 becoe salle han hose befoe 7

8 MAEC Web of Confeences MMS 017 hps://do.og/ /aecconf/ oll-ensonng and he equed effec leadng o goh of naual fequences caused by pe-sesses sae of ccula dsc canno be acheved. heefoe he appopae ollng poson s necessay o be deened fo naual fequency chaacescs calculaed fo vaous ean adus of pe-sessed annulus. hs ehod of sucual odfcaon can be used o solve he vaous desgn pobles of echancal syses [3] [6] and s vey effecve o odfcaon of he dynac popees of sla sucual eleens. he auhos sh o hank he fnancal suppo of he pojec VEGA 1/1010/16 and pojec IP MF 1603/017. Refeences 1. G. Iael J. Bukovecky L. Gulan Influence of Nonsandad Loads ono Lfe of Chosen Modules of Moble Wokng Machnes. Machne Desgn M. Sága R. Bedná M. Vaško Conbuon o odal and specal neval fne eleen analyss. Vbaon Pobles ICOVP 011 Spnge Poceedngs n Physcs M. Naď Modfcaon of Modal Chaacescs of Vbang Sucual Eleens. Scenfc Monogaphs Köhen F. Kuaan S. Yano Vbaon Analyss of a Ccula Dsc ensoned by Rollng Usng Fne Eleen Mehod. Achve of Appled Mechancs I. Maukevč L. Gulan G. Iael Moble Wokng Machnes - heoy and Consucons of Basc Modules. [n slovak] 1 s ed. SU Baslava M. Oečný Š. Segľa R. uňady Mehodology fo unng a Se-acve Dynac Vbaon Absobe on a Passve Suspended Sea. Appled Mechancs and Maeals