Chapter 9. Application of Partial Differential Equations in Mechanical Engineering Analysis

Transcription

1 Applied Egieeig Alsis - slides o clss echig* Chpe 9 Applicio o Pil Dieeil Eqios i Mechicl Egieeig Alsis * Bsed o he ook o Applied Egieeig Alsis i-r Hs plished Joh Wile & Sos 8 (ISBN ) (Chpe 9 pplicio o PDEs) i-r Hs

2 Chpe Leig Ojecives Le he phsicl meig o pil deivives o cios. Le h hee e diee ode o pil deivives desciig he e o chges o cios epeseig el phsicl qiies. Le he wo commol sed echiqe o solvig pil dieeil eqios () Iegl som mehods h iclde he Lplce som o phsicl polems coveig hl-spce d he Foie som mehod o polems h cove he eie spce; () he sepio o vile echiqe. Le he se o he sepio o vile echiqe o solve pil dieeil eqios elig o he codcio i solids d viio o solids i mlidimesiol ssems.

3 9. Iodcio A pil dieeil eqio is eqio h ivolves pil deivives. Like odi dieeil eqios Pil dieeil eqios o egieeig lsis e deived egiees sed o he phsicl lws s sipled i Chpe 7. Pil dieeil eqios c e cegoied s Bod-vle polems o Iiil-vle polems o Iiil-od vle polems : () he Bod-vle polems e he oes h he complee solio o he pil dieeil eqio is possile wih speciic od codiios. () he Iiil-vle polems e hose pil dieeil eqios o which he complee solio o he eqio is possile wih speciic iomio oe picl is (i.e. ime poi) Solios o mos hese polems eqie speciied oh od d iiil codiios. 3

4 9. Pil Deivives (p.85): A pil deivive epeses he e o chge o cio ivolvig moe h oe vile ( i miimm d 4 i mimm). M phsicl pheome eed o e deied moe h oe vile s i he ollowig isce: Emple o pil deivives: he mie empees somewhee i Clioi deped o whee d whee his empee is coed. heeoe he mgide o he empee eeds o e epessed i mhemicl om o () i which he viles d i he cio idice he locio which he empee is mesed d he vile idices he ime o he d o he moh o he e which he meseme is ke. he e o chge o he mgide o he empee i.e. he deivives o he cio () eeds o e del wih he chge o EACH o ll hese 4 viles ccoed wih his cio. I ohe wods we m hve ll ogehe 4 (o js oe) sch deivives o e cosideed i he lsis. Ech o hese 4 deivive is clled pil deivive o he cio () ecse ech deivive s we will epess mhemicll c ol epese p (o whole) o he deivive o his cio h ivolves mli-viles. hee e wo kids o idepede viles i pil deivives: () Spil viles epeseed () i ecgl coodie ssem o () i clidicl pol coodie ssem d () he empol vile epeseed ime. 4

5 9. Pil Deivives: - Co d Mhemicl epessios o pil deivives (p.86) We hve leed om Secio..5. (p.33) h he deivive o cio wih ol oe vile sch s () c e deied mhemicll i he ollowig epessio wih phsicl meig show i Fige 9..: d ( ) d im ( ) ( ) (.9) () ge lie: d ( ) d () Fige 9. Fo cios ivolvig wih moe h oe idepede vile e.g. d epessed i cio () we eed o epess he deivive o his cio wih BOH o he idepede viles d sepel s show elow: he pil deivive o cio () wih espec o ol m e epessed i simil w s we did wih cio () i Eqio (.9) o i he ollowig w: (9.) We oice h we eed he ohe idepede vile s cos i he ove epessio o he pil deivive o cio () wih espec o vile. Likewise he deivive o cio () wih espec o he ohe vile is epessed s: ( ) ( ) ( ) im (9.) 5

6 9. Pil Deivives: - Co d Mhemicl epessios o highe odes o pil deivives: Highe ode o pil deivives c e epessed i simil w s o odi cios sch s: ( ) ( ) ( ) (9.3) im ( ) ( ) d ( ) im (9.4) hee eiss ohe om o secod ode pil deivives wih coss dieeiios wih espec o is viles i he om: ( ) ( ) (9.5) 9.3 Solio Mehods o Pil Dieeil Eqios (PDEs) (p.87) hee e me ws o solve PDEs licll; Amog hese e: () sig iegl som mehods somig oe vile o pmeic domi e ohe i he eqios h ivolve pil deivives wih mli-viles. Foie som d Lplce som mehods e mog hese popl mehods. he ece ville meicl mehods sch s he iie eleme mehod s will pese i Chpe oes mch pcicl vles i solvig polems ivolvig eemel comple geome d pescied phsicl codiios. he le mehod ppes hvig eplced mch eo eqied i solvig PDEs sig clssicl mehods. Wih edil ville digil compes d odle commecil sowe sch d ANSYS code his mehod hs ee widel cceped ids. he clssicl solio mehods ppes less i demd i egieeig lsis s ime evolves. 6

7 9.3 Solio Mehods o Pil Dieeil Eqios-Co d 9.3. he sepio o viles mehod (p.87): he essece o his mehod is o sepe he idepede viles sch s d ivolved i he cios d pil deivives ppeed i he PDEs. We will illse he piciple o his solio echiqe wih cio F() i pil dieeil eqio. he pocess egis wih ssmpio o he oigil cio F() o e podc o hee cios ech ivolves ol oe o he hee idepede viles s epessed i Eqio (9.6) s show elow: F() = () () 3 () (9.6) whee () is cio o vile ol () is cio o vile ol d 3 () is cio o vile ol Eqio (9.6) hs eecivel seped he hee idepede viles i he oigil cio F() io he podc o hee sepe cios; ech cosiss o ol oe o he hee idepede viles. he 3 sepe cio d 3 i Eqio (9.6) will e oied solvig 3 idividl odi dieeil eqios ivolvig sepio coss. We m h se he mehods o solvig odi dieeil eqios leed i Chpes 7 d 8 o solve hese 3 odi dieeil eqios. he pil dieeil eqio h ivolve he cio F() d is pil deivives c hs e solved eqivle odi dieeil eqios vi he sepio elioship show i Eqio (9.6). I geel PDEs wih idepede viles c e seped io odi dieeil eqios wih (-) sepio coss. he me o eqied give codiios o complee solios o he seped odi dieeil eqios is eql o he odes o he seped odi dieeil eqios. 7

8 9.3 Solio Mehods o Pil Dieeil Eqios-Co d 9.3. Lplce som mehod o solio o pil dieeil eqios (p.88): We hve leed o se Lplce som mehod o solve odi dieeil eqios i Secio 6.6 i which he ol vile s ivolved wih he cio i he dieeil eqio () ms cove he hl spce o (o<< ). Solio o he dieeil eqio () is oied coveig his eqio io lgeic eqio Lplce somio wih he somed epessio F(s) i which s is he Lplce som pmee. he solio o he odi dieeil eqio () is oied iveig he F(s) i is eslig epessio. We hve lso se he Lplce som mehod o solve pil dieeil eqio i Emple 6.9 (p.94) e hvig leed how o som pil deivives i Secio Foie som mehod o solio o pil dieeil eqios (p.88): Foie som egieeig lsis eeds o sis he codiios h he viles h e o e somed Foie som shold cove he eie domi o (- ). Mhemicll i hs he om: i e d F (9.7) he ivese Foie som is: i F F e d (9.8) he ollowig le 9. peses ew sel oml o Foie soms o ew seleced cios. Fcios o Foie som () Ae Foie som F(ω) () (-) F(ω)e -iω () δ()* (3) ()* (iω) - (4) e (5) ()si (6) ()cos *δ() = Del cio o implsive cio d () is he i sep cio. Boh hese cios e deied i Secio.4. 8

9 9.3 Solio Mehods o Pil Dieeil Eqios-Co d Foie som mehod o solio o pil dieeil eqios:-co d Emple 9. Solve he ollowig pil dieeil eqio sig Foie som mehod. - < < (9.) whee he coeicie α is cos. he eqio sisies he ollowig speciied codiio: (9.) Solio We will som vile i he cio () i Eqio (9.) sig Foie som i Eqio (97): i e d * Appl he ove iegl o he le-hd-side o Eqio (9.) will ield: e i e d Eqio (9.) hs he om e he somio: d * * d d * om Eqio (9.) d i i * d e d o he igh-hd-side o Eq. (9.) Eqio () is is ode odi dieeil eqio ivolvig he cio *(ω) d he mehod o oiig he geel solio o his eqio is ville i Chpe 7. A his poi we eed o som he speciied codiio i Eqio (9.) he Foie som deied i Eqio () o he ollowig epessio: * i i e d e d g () () (c) 9

10 9.3 Solio Mehods o Pil Dieeil Eqios-Co d Foie som mehod o solio o pil dieeil eqios:-co d Emple 9.- Co d We will solve he is ode ODE i Eqio () wih he solio o *(ω) i Eqio () d oi: * e g he solio o he pil dieeil eqio i Eqio (9.) wih he speciied codiio i Eqio (9.) c hs e oied iveig he som *(ω) o () sig Eqio (9.8) he ollowig epessio: (e) i i * e d g e e d whee g(ω) is ville i Eqio (c) o e he Foie somed speciied codiio o () i Eqio (9.). (d)

11 9.4 Pil Dieeil Eqios o He Codcio i Solids (p.9) 9.4. He codcio i egieeig lsis We hve leed om Secio 7.5 (p.7) h empee viios i medi is idced he smissios. his viio o empee i medi (solids o lids) is clled empee ield. He se is ve impo ch o mechicl d eospce egieeig lses ecse m mchies d devices i oh hese egieeig disciplies e vlele o he. Accodig o sisics ove 6% o elecoics devices i he US Aioce iled o cios de o ecessive heig. Ecessive he low c lso esl i high empee ields i he scl medi which m esl i seios heml sesses i ddiio o sigiic deeioio o meil segh d pope chges s peseed i Secio 7.5. I his secio we will deive he pil dieeil eqios o he codcio i solids i oh ecgl d clidicl pol coodie ssems d solve hese eqios sig sepios o viles echiqe. Alhogh m o hese polems c lso e solved dvced meicl echiqes sch s iie dieece d iie eleme mehods he clssic solios s will e peseed i his chpe howeve will oe egiees wih solios whee i he solid sce which he meicl mehods co oe he sme. hese meicl mehods howeve e oe sed o siios h ivolve compliced geome lodig d od codiios.

12 9.4. Deivio o pil dieeil eqios o he codcio lsis He codcio eqio is sed o deemie he empee disiios idced he codcio i solids eihe he geeio he solids o he om eel soces. his eqio will e deived om he lw o cosevio o eeg i picl he is lw o hemodmics. Fige 9.3 o deive he mhemicl epessio o he cse: B eeig o Fige 9.3 solid wih volme is sjeced o he low i he om o he l q() om eel soces o smll eleme (i he smll ope cicle) i he ige. he he levig he eleme is q(+ ) wih desigig he spil viles o () i ecgl coodie ssem o (θ) i clidicl pol coodie ssem. Sice he is om o eeg we m se he lw o cosevio o eeg i he ollowig lock digms

13 9.4. Deivio o pil dieeil eqios o he codcio lsis Co d We m se he ollowig mhemicl epessios o epese he phsicl qiies i he solid show i Fige 9.3. Fom he lock digm o eeg cosevio d he ove mhemicl epeseios o phsicl qiies i he lock digm we m eslish he ollowig pil dieeil eqio o he empee viios i he eie solid o e: Fige 9.3 c k Q (9.3) whee k = heml codcivi o he solid meil Q()= he geeio he meil (sch s Ohm heig o Q=iR wih i eig he elecic ce i Ampee d R is he elecic esisce o he meil i Ohms. 3

14 9.4.3 He codcio eqio i ecgl coodie ssem he geel he codcio eqio i Eqio (9.3) will ke he ollowig om wih () = (): Q k k k c (9.4) i which k k d k e he heml codciviies o he solid log he - - d - coodies especivel He codcio eqio i clidicl pol coodie ssem: He codcio eqio i his coodie ssem is oied epdig Eqio (9.8) s ollows wih () = (θ): Q k k k k c (9.4) whee k k θ d k e heml codciviies o he meil log he - θ- d -coodie especivel. 4

15 9.4.5 Geel he codcio eqio (p.93): heml codciviies k k d k i Eqio (9.4) d k k θ d k i Eqio (9.4) e sed o he codcio lsis o solids wih hei hemophsicl popeies vig i diee diecios sch s o ie ilme composies. Fo mos egieeig lses sch viio o hemophsicl popeies do o eis. Coseqel geelie he codcio eqio m e epessed s ollows Q k (9.5) whee k = heml codcivi o he meil d Q() is he he geeed he meil pe i volme d ime. he smol α i Eqio (9.5) is heml disivi o he meil wih is vle eqls o: k c i is oe sed s mese o how s he c low codcio i solids Iiil codiios: Complee solio o he codcio eqio i Eqio (9.5) ivolves deemiig me o i coss ccodig o speciic iiil d od codiios. hese codiios e ecess o sle he el phsicl codiios io mhemicl epessios. Iiil codiios e eqied ol whe delig wih sie he se polems i which empee ield i solid chges wih elpsig ime. he commo iiil codiio i solid c (9.6) e epessed mhemicll s: whee he empee ield () is speciied cio o he spil coodies ol I m pcicl pplicios he iiil empee disiio () i Eqio (9.6) c e ssiged wih cos vle sch s oom empee o C o iom empee codiio i he solid. 5

16 9.4.6 Bod codiios: Speciic od codiios e eqied i oiig complee solios i he se lses sig he geel he codcio eqio i (9.5). Fo pes o od codiios e ville o his pposes.s will e peseed elow. ) Pescied sce empee s (): his pe o od codiio is sed o hve he empee he sce o he solid sce mesed eihe chig hemocoples o he sce sce o some o-coc mehods sch s ied heml imgig scig cme. he mhemicl epessio o his cse kes he om: s (9.7) s whee s is he coodies o he od sce whee empee e speciied o e s () ) Pescied he l od codiio q s (): M sces hve hei sces eposed o he soce o he sik i sch siios he is eig spplied o o emoved om he solids hogh is oside sce. he mhemicl slio o he he l o o om solid sce c e edil cied o sig he Foie lw o he codcio deied i Eqio (7.5). he mhemicl omlio o he he l coss solid od sce c e epessed s: qs s (9.7) k i s whee k is he heml codcivi o he solid meil. he smol is he dieeiio log i he owd-dw oml o he od sce S i. We m epess Eqio (9.7) o he odies h e impemele o he low o od h is hemll isled s: s (9.7c) 6

17 9.4.6 Bod codiios Co d: 3) Covecive od codiios: Fige 9.5 his pe o od codiio pplies whe he solid sce is eihe i coc wih lid o is smeged i lids s oe hppe i eli. Le s deive he mhemicl epessios o he od codiios eeig o he skech i Fige 9.5. We is ecogie h hee is phsicl ie h eds ee he low ewee he solid sce d is coced lid. his ie is oe ecogied s he od le h c e chceied ilm esisce h is eql o /h wih h eig he ilm coeicie s deied i Eqio (7.9) i Secio Phsicll i mes h he empee o he solid sce s he empee o he sodig lk lid. he ollowig wo () mhemicl epessios e deived o epese he ove phsicl pheomeo: Fom he c h o he is eig soed he iece o he solid d lid which leds o he ollowig Eqli: k He low i solid = He low i lid o i he om: s h k h k s s h s (9.7d) he ove eqio ivolves he lows i solids codcio d he lows i lids covecio. I is oe eeed o e he mied od codiios. his epessio o od codiio cll cold e sed o polems ivolvig pescied sce empees i Eqio (9.7) wih h We m lso pove h leig h = i Eqio (9.7d) will led o hemll isled od codiio wih q s = i Eqio (9.7). 7

18 Emple 9.3 (p.95) Show he ppopie od codiios o log hick wll pipe coiig ho sem low iside he pipe lk empee s wih he se coeicie h s. he oside wll o he pipe is i coc wih cold i empee o d wih he se coeicie h s illsed i Fige 9.6. Fige 9.6 Solio A commo logicl hpohesie mde i his pe o egieeig lsis is h he will low is pimil log he posiive dil diecio () i log pipe sch s i his emple ecse o he gee empee gdie coss he pipe wll h h log he legh. So he dil diecio is he picipl diecio o he low. Coseqel we will cco o wo od sces i his lsis i.e. he ie sce wih = d he oside sce =. Sice he se coeicies o oh he sem iside he pipe (h s ) d he he se coeicie o he i oside he pipe (h ) e give we m se Eqio (9.7d) o eslish he covecive od codiios oh sides o he pipe wll s ollows: () A ie od wih = : d hs k d k () A he oside od wih = : d h k d k i which k = heml codcivi o he pipe meil hs k h k s 8

19 Emple 9.4 (p.96) Fid he empee disiio i log hick wll pipe wih ie d oside dii d especivel sig he hee pes o od codiios i Eqios (9.7d). Codiios o eslishig he mhemicl epessios o hese od codiios wih ho sem iside he pipe d he cool sodig i oside he pipe e idiced i Fige 9.7. Solio Fige 9.7 We dop he sme picipl s descied i he ls emple h he shoe he low ph log he dil diecio o he pipe eles s o ssme he picipl empee viio i he pipe wll is wih he dis vile (). Coseqel we m ssme h he empee cio h we desie i his lsis is () ol. hs selec he elev ems i he PDE i (9.4) we will hve he elev dieeil eqio o he om: d d () d d Solio o he dieeil eqio i () m e oied eihe sig Eqio (8.6) o e-gig he ems h i he ollowig om o: d d () d d om which we ge he solio () iegig Eqio () wice wih espec o vile ledig o he om: c c (c) whee c d c e wo i coss 9

20 Emple 9.4-Co d We hve deived he geel solio o he empee coss he pipe wll o e: (c) c c We will deemie he wo i coss c d c sig he 3 diee ses o od codiios peseed i Secio s ollows: (A) Wih pescied od codiios i Eqio (9.7): Wih he give codiios o: o e he empee he ie sce wih () = d () = he oside sce o he pipe we will deemie he wo coss i Eqio (c) o e: c d c which leds o he ollowig complee solio: (d) (B) Wih pescied he l q coss he ie sce d he oside sce: d q ie sce: (e) d k oside sce: () q We m deemie he coss c d c i Eqio (c) o e: q c d c k k which leds o he comolee solio o Eqio () o e: q (g) k

21 Emple 9.4-Co d (c) Wih mied od codiios i Eqio (9.7d): he ppopie od codiios e: ie pipe sce: s s s k h k h d d (h) k h k h d d oside pipe sce: (j) Ssie (h) d (j) io Eqio (c) we will ge: h h kh kh h h c s s s s h k h h kh kh h h c s s s s d he empee disiio i he pipe wll () m e oied ssiig he coss c d c i he ove epessios io he solio i Eqio (c).

22 9.5 Solio o Pil Dieeil Eqios o sie He Codcio Alsis (p.98) he pil dieeil eqio peseed elow d lso i i Eqio (9.5) is vlid o he geel cse o he codcio i solids icldes sie cses i which he idced empee ield () vies wih ime. Q (9.5) k whee = he posiio veco d = ime. Q()= he he geeio he meil i i volme d ime k α = heml codcivi d heml disivi o he meil especivel wih k o e mese o how well meil c codc he d he le α is mese o how s he meil c codc he. he posiio veco m e i ecgl coodies: () o i clidicl pol coodie ssem (θ). he complei i sie he codcio lsis is h o ol we eed o speci he posiio () whee he empee o he solid is ccoed o we will lso eed o speci he ime which he empee o he solid occs. We hs eed o speci oh heod d iiil codiios sch s descied i Secio o complee solio o he empee iled i he solid. I his secio we will demose how he sepio o viles echiqe descied i Secio 9.3 will e sed o solve his pe o polems i oh ecgl d clidicl pol coodie ssems.

23 9.5. sie he codcio lsis i ecgl coodie ssem (p.98) he cse h we will pese hee ivolves lge l sl mde o meil wih heml codcivi k. he sl hs hickess L s illsed i Fige 9.8. I hs iiil empee disiio h c e descied speciied cio o () d he empees o oh is ces e miied empee ime >. We eed o deemie he empee viio i he sl wih ime i.e. () i he ige e he empee o oh ces o he sl e miied. he phsicl siio o his emple is h he l sl hs iiil empee viio hogh is hickess is cio (.) = () give empee disiio. Boh is sces e miied cos empee ime > + o >. Oe m imgie h he empee i he sl will coiosl vig wih ime il he empee i he eie sl eches iom empee. he ppose o o sseqe lsis howeve is o id he sie empee () i he sl eoe i eches he lime iom empee o. We m lso ecogie c h he geome o lge l sl is good ppoimio o he siio o cicl clide wih lge dimee wih lge io o D/d i which D is he omil dimee o he hollow clide d d is he hickess o he wll o he hollow clides. he solio oied om his lsis o l sl m hs e sed o lge hollow clides sch s pesse vessels o lge dimees sch s o cle eco vessels i cle powe pls. 3

24 9.5. sie he codcio lsis i ecgl coodie ssem Co d (p.99) he goveig dieeil eqio o he oemeioed phsicl siio m e dedced om he codcio eqios i Eqios (9.4) d (9.5) wih he heml codcivi o he sl meil k = k = k =k o eig isoopic meil. he em Q() i Eqio (9.4) d Q() i Eqio (9.5) e deleed ecse he sl does o geee he isel. Coseqel he eqio h mches he he pese phsicl siio ecomes: Wih he iiil codiio (IC): d he ollowig od codiios (BC): L L (9.8) (9.9) (9.9) (9.9c) We m solve he pil dieeil eqio i Eqio (9.8) sig Lplce som mehod descied i Secio 6.5. (p. 8) o 9.3 (p.87) somig he vile o pmeic domi o se he sepio o viles echiqe s descied i Secio Howeve we m cicmve o eo i he solio o Eqio (9.8) sig he sepio o viles mehod wih coveig he o-homogeeos BCs i Eqio (9.9c) o homogeeos BCs he ollowig ssiio o () o (): () = () (9.) 4

25 9.5. sie he codcio lsis i ecgl coodie ssem Co d he ove elio i Eqio (9.) will esl i he evised PDEs i Eqio (9.8) io he ollowig om: (9.) wih he evised iiil codiio: d he coveed od codiios: o L L () L We e ow ed o solve he eqio i (9.) d he ssocie iiil d od codiios i Eqios (c) sig he sepio o viles mehod s peseed elow: We will poceed leig: () = X()τ() (9.) Ssiig he elioship i Eqio (9.) io Eqio (9.) will led o he ollowig epessios: leds o: () (c) d his eqli c ow e epessed i odi deivives. i which he pil deivives o eihe sides e oied. 5

26 9.5. sie he codcio lsis i ecgl coodie ssem Co d he epessio h we js deived s show elow c e epessed i sligh diee om e e-gig he ems o ohe eqli: d X d X d d he ove epessio shows ve ieesig iqe ee: he LHS o he ove epessio ivolves he vile ol = he RHS o he sme epessio ivolves he vile ol he ONLY codiio sch eqli c eis is o hve oh sides o he epessio o eql CONSAN!! (we m pove h he cos ms e NEGAIVE cos). Coseqel we m hve he ollowig vlid eqli: (9.3) whee β is he sepio cos d i c e eihe posiive o egive cos. Eqio (9.3) esls i he ollowig sepe odi dieeil eqios: d X X (9.4) d d X d X d d d (9.5) d 6

27 9.5. sie he codcio lsis i ecgl coodie ssem Co d d X X (9.4) d d X d X d d d (9.5) d he solio X() d τ() i especive Eqios (9.4) d (9.5) eqies he speciic codiios o oh hese eqios. Eqio (9.) is sed i cojcio wih hose give iiil d od codiios i Eqios (c) will ge s he ollowig eqied eqivle codiios: X()= d X(L) = (e e) o Eqio (9.4). Solio X() i Eqio (9.4) is edil od om Secio 8. wih he om: X() = A cosβ + B siβ () he i cos A i Eqio () c e deemied Eqio (e) o e eo which leves Bsiβ=. he se o he give codiio i Eqio (e) leds o BsiβL= which leds o eihe B= o siβl=; Sice B (o void o-ivil solio o X()=) he ol choice o s is o le siβl = (9.6) We will qickl elie h hee e mliple vles o he sepio cos β h sis Eqio (9.6). hese e: β = π wih = 3.Aleivel we m epess he sepio cos β i he ollowig om: ( ) (9.7) L Coseqel he cio X() 3 X B si B si B3si... L L L (9.8) i Eqio (9.4) kes he om: B si L ( 3...) 7

28 9.5. sie he codcio lsis i ecgl coodie ssem Co d (9.5) d d Solio o his is ode dieeil eqio is: C e (9.9) whee C wih = 3 e mli-vled iegio coss coespodig o he mlivled β i he solio. CBe si e si (9.3) L L he mli-vled cos coeicies =C B i Eqio (9.3) m e deemied he ls ville iiil codiio i Eqio () i which (o) = ()-. Coseqel we hve: si (9.3) L whee () d e he give iiil empee disiio i he sl d he cocig lk lid empee especivel. 8

29 9.5. sie he codcio lsis i ecgl coodie ssem Co d Deemiio o he mli-vled cos coeicies i Eqio (9.3) o P.3: We will se he ohogoli pope o iegls o igoome cios o he ove sk. he wo pplicle popeies e peseed elow: p m i m si si d (9.3) p p p / i m Followig seps e ke i deemiig he coeicie wih = 3. i Eqio (9.3): Sep : Mlipl oh side o Eqio (9.6) wih cio si L si L si L si L si si L L Sep : Iege oh sides o Eqio (g) wih iegio limis o (L): L si L L L d si si d si d Sep 3: Mke se o he ohogoli o he hmoios cios like sie d cosie wih he L elioships i Eqio (9.3): L si d ledig o: L L L si d L L L L (9.33) We hs hve he solio o Eqio (9.) o e: d e si si L L he solio o () i Eqio (9.8) o he empee disiio i he sl c hs e oied he elioship epessed i Eqio (9.) o ke he om: L d e (9.34) si si L L L I will o e hd o s o evisge h ( ) i Eqio (9.34) solio i eli. L L (g) (h) 9

30 9.5. sie he codcio lsis i clidicl pol coodie ssem (p.33) hee e m mechicl egieeig eqipme hvig geome h c e ee deied clidicl pol coodies (θ) sch s illsed i he ige o he igh: Clides pipes wheels disks ec. ll i o his kid o geome sch sshow i Fige I is desile o kow how o hdle he codcio i solids o hese geome. Fige 9.9 We will pese he cse o solvig he codcio polem sig he sepio vile echiqe i solid clide wih dis s show i Fige 9.9. he clide is iiill wih give empee disiio o (). I is smeged i lid wih lk lid empee. ime + +. he siio i el pplicio is like hvig ho od solid clide iiill wih empee viio om ho cee coolig dow owds is cicmeece sce descied cio (). I is clssicl cse o qechig opeio i mel omig opeio. he sodig cocig liqid coole empee is vigoosl gied so h he he se coeicie h o he lid he coc sce m e eed s i Eqio (9.7d) o p 95 ledig o he od empee o he solid clide o e s sed i he polem. he empee ield i he solid clide m e epeseed he cio () i which = dil coodie d is he ime io he he codcio i he solid. 3

31 9.5. sie he codcio lsis i clidicl pol coodie ssem Co d (p.33) he pplicle PDE o he ce pplicio m e dedced om Eqio (9.4) doppig he secod d ohe ems i he igh-hd-side o h eqio eslig i: (9.35) k whee c is he heml disivi o he clide meil wih ρ d c eig he mss desi d speciic he o he clide meil especivel. () We will hve he give iiil codiio: d od codiios: he ohe ieplici od codiio o solid clides o disks is h he empee he cee o he clide o disk ms e iie vle ll imes. Covesel his implici od codiio o he ce cse me o e: o iie vle () wih he PDE i (9.35) d he iiil d od codiios speciied i Eqios () () d () s speciied ove we m poceed o solve o he sie empee disiio () i Eqio (9.35) sig he sepio o viles echiqe simil o wh we did i he poceedig Secio () Agi o he sme eso s i he pevios cse we will is cove he o-homogeeos od codiio i Eqio () o he om o homogeeos codiio leig: (.) = () - (c) Accodigl Eqio (9.35) d he oigil iiil d od codiios will hve he oms: wih d o o (9.36) (d) (e) 3

32 We hs hve he PDE: wih codiios: o o (9.36) (d) (e) (9.37) Upo ssiig he ove elio i Eqio (9.37) io Eqio (9.36) will esl i he ollowig Epessios:: R R R o R R R () Eqio () oes he legiimc o coveig he pil deivives o R() d τ() o odi deivives s show elow: d d R dr (g) d R d d We oice h he LHS o Eqio (g) ivolves vile ol whees he RHS o he sme epessio ivolve he ohe vile ol. he ol w h sch eqli c eis is o oh sides i Eqio (g) o e eql o sme egive sepio cos β. We hs hve he ollowig elioship: d d R d R d dr d (9.38) 3

33 9.5. sie he codcio lsis i clidicl pol coodie ssem Co d Solio o pil dieeil eqio: wih codiios: o o d d R d R d dr d d R d d d dr d (9.36) We c hs spli Eqio (9.38) io he ollowig wo sepe odi dieeil eqios: R (d) (e) he solio o Eqio (9.39) is ideicl o Eqio (9.9) i he om: ce whee he cos coeicies c wih = 3.. is mlivled iegio coss. (9.39) (9.4) We oice h Eqio (9.4) is specil cse o he Bessel eqio i Eqio (.7) o p.56 wih ode =. Coseqel he solio o Eqio (9.4) c e epessed he Bessel cios give i Eqio (.8) o he sme pge wih = i he ollowig om: R() = A J (β) + B Y (β) (9.4) whee he cos coeicies A d B will e deemied he od codiios sipled i Eqios (d) d (e). (h) 33

34 9.5. sie he codcio lsis i clidicl pol coodie ssem Co d Solio o pil dieeil eqio: wih codiios: o o d R dr R d d R() = A J (β) + B Y (β) (9.36) We hve solve he dieeil eqio i Eqio (9.4) o e he epessio give i Eqio (9.4): (d) (e) (9.4) (9.4) whee A d B e wo i coss o e deemied he wo od codiios pplicle i his cse e: R() o () d hs (). Fo he codiio R() we will hve om Eqio (h) i he om: R()=AJ ()+BY o () we elie h J () =. om Fige.45 (p.56) Y o () - s idiced i he sme ige. he le idices h R() heeoe () - ( oded empee he cee o he solid clide which is oviosl o elisic solio. he ol w h we m void his elisic siio is o le he cos B =. Coseqel we hve he solio i Eqio (h) o ke he om: R() = A J (β) (j) he od codiio i Eqio (e) will led o he epessio: R() =A J (β) = which eqies eihe: A = o J (β) =. Sice he coeicie B i Eqio (h) is led se o e eo () o le A= will me he cio R()= cceple ivil solio o he empee (). We e hs le wih he ol opio o hve: J (β) = (9.4) Eqio (9.4) oes he vles o he sepio cos β i Eqio (9.38) ecse J () = is eqio h hs mliple oos (see Fige.45() o p.56 like si(βl) = i Eqio (9.6) o p.3. he oos o he eqio J (β) = i Eqio (9.4) m e od eihe om he Fige.45() o p.56 o om mh hdooks. 34

35 9.5. sie he codcio lsis i clidicl pol coodie ssem Co d Solio o pil dieeil eqio: (9.36) wih codiios: o (d) o (e) R() = A J (β) e c Sice oh A d C e coss d he le C is mlivled coss wih = 3. we m epess he complee solio () i he om: J e (9.44) whee he mli-vled cos m e deemied he codiios i Eqios (d) d (e). We hs hve he ollowig epessio e ppl he iiil codiio i Eqio (d): J J J J (9.45) whee ()- i Eqio (d) e give codiios wih he PDE i Eqio (9.35) d i Eqio (945) m e deemied ollowig simil pocede s olied i Secio 9.5. sig he ohogoli popeies o igoomeic cios i Eqio (9.3) o p.3. Howeve we will se he Foie-Bessel elio i deemiig he coeicies i Eqio (9.45) i he pese cse. 35

36 9.5. sie he codcio lsis i clidicl pol coodie ssem Co d Solio o pil dieeil eqio: (9.36) wih codiios: o (d) o (e) he Foie-Bessel elio hs he om (p.37): m m m i d J i d J J o diee gmes i he Bessel cios i he iegl o sme gmes i he Bessel cios i he iegl J J J J J J J J J We will mlipl oh sides o Eqio (9.45) he ollowig seies o Bessel cios:... 3 J J J s show i he ollowig epessio: d he epsio o oh sides o he ove epessio will esl i: J J J J J (k) Iegig oh sides o Eqio (k) wih espec o vile will esl i: d J J J J d J o d J J J J d J (l) he Foie-Bessel elio eles s o elimie he d p o he Bessel cios d esl i: d J d J (m) d J J We m hs oi he mli-vled coeicie o e: (9.46) 36

37 9.5. sie he codcio lsis i clidicl pol coodie ssem Ed Solio o pil dieeil eqio: wih codiios: o o (9.36) (d) (e) he solio o () i Eqio (9.36) hs hs he om: e J (9.44) whee he coeicies e oied o he iegl i Eqio (9.46): J J d We m oi he sie empee disiio i he clide () he elio deived om Eqio (c) s: () = i +(). We hs hve he solio o he empee disiio i he clide o e: e J (9.47) whee he mli-vled coeicies e comped om Eqio (9.46) (9.46) We oice he ppeces o Bessel cios i he solio o his polem. I is oml o see sch ppeces o Bessel cios i solid geome ivolvig cicl geome sch s clides disks d eve solids o spheicl geome. 37

38 9.6 Solio o Pil Dieeil Eqios o Sed-Se He Codcio Alsis Oe we e eqied o id he empee disiios i solid mchie sces wih sle he low pes which mkes he empee disiios i he solids idepede o ime viio i.e. he sed se he codcio. Followig e emples o he low i he mchies i sed-se codiios: Je egie-gs ie l he echge: es wih is IC chip wih he spede: 38

39 9.6 Pil Dieeil Eqios o Sed-Se He Codcio Alsis (p.38) Mhemicl epeseio o mli-dimesiol he codcio i solids is ville sig he pil dieeil eqio wiho he em eled o ime vile. he PDE i Eqio (9.5) o p.93 is edced o he ollowig om Q (9.48) k whee he posiio veco epeses () i ecgl coodie ssem o (θ) i clidicl pol coodie ssem. Eqio (9.48) is he edced o he Lplce eqios i he ollowig om i o he is geeed he solid: (9.49) We will demose he solio o PDEs o sed-se he codcios i mli-dimesiol solid sce compoes sig sepio viles echiqe i oh ecgl d clidicl pol coodie ssems i he sseqe peseios. 39

40 9.6. Sed-Se He Codcio Alsis i Recgl Coodie Ssem (p.38) We will demose he se o he Lplce eqio i Eqio (9.49) o he empee disiio i sqe ple wih empee i is hee edges miied cos empees o C d he ohe edge o C s illsed i Fige 9.. Solid ple sce compoes e commo i he he spedes i iel comsio egies l he echges d he spedes o micochips s illsed i he ls slide. I he pese cse he lows om he he soce he op ce wih highe empee owds he he siks i he ohe edges lowe empees. hee is o he lows coss he hickess o he ple wih ssmpio Fige 9. h oh he pl ces o he ple is hemll isled. he idced empee ield hs coves ove he ple e o he ple d he empee disiio i he ple o he ple is epeseed he cio (). We hve he pplicle PDE epessed i Eqio (9.5) d speciied od codiios i Eqios () () (3) d (4). wih d (9.5) () () (3) (4) 4

41 9.6. Sed-Se He Codcio Alsis i Recgl Coodie Ssem Co d Solio o Pil Dieeil Eqio sig Sepio o Viles Mehod (p.39): Bod codiios: d (9.5) hee e wo viles d i he solio o empee cio () we will hs le: () = X() Y() () i which cio X() ivolves ol vile d cio Y() ivolves vile ol. Ssie Eqio () io Eqio (9.5) d e e-gig ems ields he ollowig epessio: d X d Y X d Y d We will se he sme gme s we did i Secios 9.5. d 9.5. h he ol w he ove eqli c eis is o hve oh sides o he eqli o e eql o egive sepio cos. () () (3) (4) We will hs hve he ollowig eqli: X d X d Y d Y d (c) 4

42 9.6. Sed-Se He Codcio Alsis i Recgl Coodie Ssem Co d Solio o PDE i Eqio (9.5) sig Sepio o Viles Mehod-Co d: Eqio (c) leds o he spli o he PDE i Eqio (9.5) io wo odi dieeil eqios (ODEs) e he sepio o he viles d s show elow: he oigil PDE i Eqio (9.5): d X d X X() = X() = d Y Y d Y() = Boh Eqios (d) d (e) e homogeeos d ode dieeil eqios wih he solios mehods ville i Secio 8.. We will show he solios o hese wo eqios i he ollowig oms: Solio o Eqio (d): X Acos Bsi Solio o Eqio (e): Y() = C coshβ + D sihβ (k) We m oi he epessio o he mli-vled sepio cos β o e he solio o he chceisic eqio o si(β) = d he cos coeicie A = po ssiig he od codiios i Eqios () d () io Eqio (g). We hve hs oied he mli-vled sepio coss β = (π)/l wih L= d = 3. om he oos o he eqio si(β) =. We c hs epess he cio s: (j) X() = B siβ i which B wih = 3. e he mli-vled cos coeicies o e deemied le. (d) () () (e) (3) (g) 4

43 9.6. Sed-Se He Codcio Alsis i Recgl Coodie Ssem Co d Solio o Pil Dieeil Eqio (9.5) sig Sepio o Viles Mehod-Co d: he oigil pil dieeil Eqio (9.5): d X d X X() = X() = d Y Y d Y() = (d) () () (e) (3) Ne we will solve Eqio (e) wih solio (p.3): Y() = C coshβ + D sihβ (k) he od codiio i Eqio (3) wold mke he cos coeicie C =. Coseqel we will hve he cio Y() o ke he om: Y() = D sihβ wih =3 (m) We hve oi he solio () o Eqio (9.5) e ssiig he epessios o X() i Eqio (j) d Y() i Eqio (m) io Eqio () d esl i: X Y BD si sih si sih () 43

44 9.6. Sed-Se He Codcio Alsis i Recgl Coodie Ssem Co d Solio o Pil Dieeil Eqio (9.5) sig Sepio o Viles Mehod-Co d: d X Y (9.5) () () (3) (4) BD si sih si sih he kow coeicies i Eqio (m) m e deemied sig he emiig od codiio i Eqio (4) h () = which leds o: si sih o sih si (p) B ollowig he sme pocede i i sig he ohogoli o igoomeic cios i Secio 9.5. o p.98 We will deemie he coss i Eqio (p) o e: Ledig o he solio: cos sih wih 3... () (q) cos si sih (9.5) sih 44

45 9.6. Sed-Se He Codcio Alsis i Clidicl Pol Coodie Ssem (p.3) We will eploe how he sepio o viles echiqe m e sed i sed-se he codcio lsis i clidicl pol coodie ssem his cse illsio. he cse we hve hee ivolves solid clide wih dis d legh L wih empee he cicmeece d he oom ed miied o C d he empee he op sce is sjeced o empee disiio h is speciied cio F() s show i Fige 9.. We elie he phsicl siio i which Fige 9. he lows om he op ed o he clide i ohhe dil d logidil diecio. We m hs desige he empee i he clide () i clidicl pol coodie ssem. he goveig PDE o () i sed-se he codcio s descied ove m e oied selecig he igh ems i Eqio (9.4) i clidicl pol coodie ssem i he ollowig om: (9.5) wih speciied od codiios: d () = () () () 45

46 9.6. Sed-Se He Codcio Alsis i Clidicl Pol Coodie Ssem Co d Solio o he Pil Dieeil Eqio sig Sepio o Vile echiqe: (9.5) wih speciied od codiios: () = () ( )) () = Followig he sl pocedes i sepio o vile echiqe (p.3) we le: () = R() Z() whee he cios R() d Z() ivolve ol oe vile d especivel. Upo ssiig he ove epessio i Eqio () io Eqio (9.5) d e e-gig he ems we will ge he ollowig eqli: R R Z (c) R R Z he ol w h he ove eqli c ei is hvig oh sides o e eql o cos: We hs hve: R d R d R dr d Z d Z d We hve hs spli he PDE i Eqio (9.5) io wo sepe ODEs s ollows: d R dr R d d d Z Z d Sisig he codiios: R() = R() Z() = (3) () (d) (e) () (g) (g) (g3) 46

47 9.6. Sed-Se He Codcio Alsis i Clidicl Pol Coodie Ssem Co d Solio o he Pil Dieeil Eqio sig Sepio o Vile echiqe-co d: he solio o he ODE i Eqio (e) ivolves Bessel cios s i he cse i Secio 9.5. d R A J BY i Eqio (9.4) o p.35 o ke he om: : he codiio speciied codiio i Eqio (g) esls i hvig he cos coeicie i he ove epessio o e: B = ecse he secod em i he ove solio i he ove epessio co e llowed i he epessio ecse Y () - which is o elisic. Hece we hve: J (β ) = (j) he sepio cos β is oied om Eqio (j) d hee e mliple oos o h eqio wih: β = β β β 3...β wih = 3.. he solio o ohe ODE i Eqio () is: Z() = C cosh(β) + D sih(β) Ssiio o he codiio Z() = i Eqio (3) io Eqio (k) will led o he cos C =. We will hs hve: Z() = D sih (β). Howeve sice Z() ivolve he mli-vled β We m epess Z() i he om: Z() = D sih(β ) (m) We c hs epess he solio () i Eqio (9.5) i he om o: () = [A J (β )][D sih(β )] wih = 3.. o i moe compc om: J sih Whee e mli-vled cos i he he ove eqio h m e oied sig he Foie-Bessel elio s epessed o P. 37 eslig i he ollowig om: F L sih J L J L L d wih (k) (9.53) () 47

48 9.7 Pil Dieeil Eqios o svese Viio o Cle Sces (p.34) svese viio o sigs (eqivle o log leile cle sces i eli) e sed commol sed i sces sch s powe smissio lies g wies sspesio idges. hese sces leile i e e vlele o eso viios which m esl i devsios i plic se d pope losses o o socie. Log powe smissio lies Rdio owe sppoed g wies he wold mos Golde Ge Sspesio Bidge A cle sspesio Bidge he vege o collpsig: 48

49 9.7. Deivio o pil dieeil eqio o ee viio o cle sces We egi o deivio o mh models o he viio lsis o sigs (eqivle o log leile cles) wih iiil sgged shpe h c e descied cio () s illsed i Fige 9.5. Followig ideliios (o hpoheses) wee mde i he deivio o mhemicl modes o ee viio lsis o cle sces: Fige 9.5 A Log Cle Iiill i Sicll Eqiliim Se () he cle is s leile s sig. I mes h he cle hs o segh o esis edig. Hece we will eclde he edig mome d she oces i o sseqe deivios. () hee eiss esio i he sig i is ee-hg sic se s show i Fige 9.5. his esio is so lge h he weigh o he mss o he cle is egleced i he lsis. (3) Eve smll segme o he cle log is legh i.e. he segme wih legh moves i he veicl diecio ol dig viio. (4) he veicl moveme o he cle log he legh is smll so he slope o he delecio cve o he cle is smll. (5) he mss o he cle log he legh is cos i.e. he cle is mde o sme meil log is eie legh. 49

50 9.7. Deivio o pil dieeil eqio o ee viio o cle sces Co d A sligh iseos lel moveme o he cle i Fige 9.5 ime = will esl i lell vie p-d-dow i he - ple s show i Fige 9.6() () Iseos shpe ime () Foces o segme (Deil A) Fige 9.6 Shpe o viig Cle Fige 9.7 Fee-od Foce digm o viig cle Le he mss pe i legh o he sig e desiged m. he ol mss o sig i icemel legh i Fige 9.6 () d 9.7 will hs e (m). he codiio o dmic eqiliim ime s illsed i Fige 9.7 ccodig o Newo s secod lw peseed i he eqio o moio hs he ollowig elioship: 5

51 9.7. Deivio o pil dieeil eqio o ee viio o cle sces Co d We meioed i he ls slide h he PDF h we will se o model o ee lel viio lsis o he cle M e deived om Newo s Secod lw o dmics: We he elie h he mss o he segme o he cle i Fige 9.7 m e epessed o e: M= m i which m= mss o he cle pe i legh d he cceleio is eql o: whee ()= iseos delecio (mgide) o he Fig. 9.7 Fee-od oce digm viig cle. We ssme his dmic oce cs he mss cee s show i Fige 9.7. We m deive he ollowig epessio o he dmic oce eqiliim o smll secio o he cle ime : We m delee he em: Psi(α+ α) i he ove epessio ecse oh P d α e smll. We hs hve he ollowig o o he deivio: si ( ) B sice d si we will hve he ollowig epessio e ssiios o he ove elioships i he dmic oce eqiliim eqio: ( ) P ( ) m 5

52 9.7. Deivio o pil dieeil eqio o ee viio o cle sces Co d ) ( ) ( m P I we divide eve em i he ls epessio we will oi he ollowig epessio: ) ( ) ( m P B imposig he codiio h he cio o he lel delecio () o he viig cle vies (chges) is mgides coiosl log he cle legh i he -coodie i.e. d he iceme o () i.e. is smll eogh o e egleced (i.e. ) he ove epessio m e epessed i he ollowig om: ) ( ) ( ) ( im o ) ( m P We hs hve he PDE o he ee viio lsis o log leile cle i he om o: ) ( ) ( (9.54) whee m P wih P = esio i he sig wih i o Newo (N) d m = mss o he cle pe i legh i kg/m. he i o he cos i Eqio (9.54) is hs m/s. 5 o

53 9.7. Solio o PDE o ee viio lsis o cle sces (p.38) We will demose he pplicio o Eqio (9.54) o he ee viio lsis o log cle sce illsed i Fige 9.8. he cle iiill hs he shpe i he doed cve i Fige 9.8 h c e descied cio (). Lel viio o he cle wih iseos mgides () is idced o he cle smll iseos disce wih sligh veicl psh o he cle dowwd h podces he iseos shpe o he cle s show i he sloid cve i he sme ige ime. he ee viio o he cle wih he lel mplides () is ssied he mss o he cle meil d is ihei elsici o he cle. O lsis is o solve () o he phsicl siio descied ove. We will se Eqio (9.54) o solve o he () he sepio o viles echiqe s we did i Secio 9.6 o he codcio lsis. We will hs hve he ollowig mhemicl model o he solio: ( ) ( ) he PDE: (9.54) he iiil codiios: he ed (od) codiios: L L Fige 9.8 (9.55) (9.55) (9.56) (9.56) 53

54 9.7. Solio o pil dieeil eqio o ee viio lsis o cle sces Co d ) ( ) ( he pil dieeil eqio: (9.54) he iiil codiios: he ed (od) codiios: L L (9.55) (9.55) (9.56) (9.56) Solio o Pil Dieeil Eqio (9.54) Sepio o Viles Mehod (p.39): We will eed o sepe hese wo viles d om he cio () i Eqio (9.54) leig: () = X() () (9.57) he elio i Eq. (9.57) leds o: X X X ' ( X X X ' d X " X " d Ssiig he ove epessios io Eqio (9.54) will led o: ) ( ) ( ) ( ) ( d X d X d d LHS = = RHS = cos (-β ) ) ( ) ( ) ( ) ( d X d X d d (9.58) We hs hve: 54

55 9.7. Solio o pil dieeil eqio o ee viio lsis o cle sces Co d We will hs ge wo odi dieeil eqios om (9.58): d ( ) ( ) d d X ( ) X ( ) d d ( ) d ( ) d X ( ) X ( ) d (9.59) (9.6) Ae pplig he sme sepio o viles s illsed i Eq. (9.57) o he speciied codiios i Eqios (9.55) d (9.56) we ge he wo ses o ODEs wih speciic codiios i he ollowig epessios: d ( ) d X ( ) ( ) X ( ) d d () = () (9.59) (9.6) (9.6) X() = (9.6) d ( ) (9.6) X(L) = (9.6) d Boh Eqios (9.59) d (9.6) e lie d ode ODEs wih hei solios o e i he ollowig oms: () = A Si(β) + B Cos(β) (9.63) X() = C Si(β) + D Cos(β) (9.64) 55

56 9.7. Solio o pil dieeil eqio o ee viio lsis o cle sces Co d he lel mplide o viio cle () i Fige 9.8 o he solio o Eqio (9.54) c hs e epessed ssiig he epessios i Eqios (9.63) d (9.64) i Eqio (9.57) o give: () = A Si(β) + B Cos(β) X() = C Si(β) + D Cos(β) () = [A Si(β) + B Cos(β)][ C Si(β) + D Cos(β)] whee A B C d D e i coss eed o e deemied om he give iiil d od codiios give i Eqs. (9.6) d (9.6) Deemiio o i coss: Le s s wih he solio: X() = C Si(β) + D Cos(β) i Eq. (9.64): Fom Eq. (9.6): X() = : C Si (β*) + D Cos (β*) = which mes h D = X() = C Si(β) Now om Eq. (9.6): X(L) = : X(L) = = C Si(βL) A his poi we hve he choices o leig C = o Si (βl) = om he ove elioship. A cel look hese choices will coclde h C (wh?) we hs hve: Si (βl) = he ove epessio is scedel eqio wih iiie me o oos o he solios wih βl= π π 3π 4π 5π π i which is iege me. We m hs oi he vles o he sepio cos β o e: ( ) (9.66) L 56

57 9.7. Solio o pil dieeil eqio o ee viio lsis o cle sces Co d Now i we ssie he solio o X() i Eq. (9.64) wih D= d β = π/l wih = 3.. io he solio o () epessed i he ollowig om: () = [A Si(β) + B Cos(β)][ C Si(β) + D Cos(β)] We will ge: ( ) ASi BCos CSi ( = 3..) L L L B comiig coss A B d C i he ove epessio we hve he ieim solio o () o e: ( ) Si Cos Si ( = 3..) L L L We e ow ed o se he wo iiil codiios i Eqs (9.55.) d (9.55) o deemie coss d i he ove epessio: ( ) Le s is look he codiio i Eq. (9.55): ( ) Cos Si Si L L L L B sice Si (wh?) L = ( ) Cos Si L L hs he ol emiig coss o e deemied e: i he ove epessio. 57

58 9.7. Solio o pil dieeil eqio o ee viio lsis o cle sces Co d Deemiio o cos coeicies i he ollowig epessio (p.3): ( ) Cos Si L L he ls emiig codiio o (o) = () i Eqio (9.55) will e sed o his ppose i which () is he give iiil shpe o he sig. hs leig () = () we will hve: wih L L Si hee e me o ws o deemie he coeicies i he ove epessio. Wh we will do is o ollow he ohogoli o igoomeic cios i Secio 9.5. (p.3) o deemie he coeicie i he ollowig w: L ( ) Si d (9.68) L L he complee solio o he mplide o lel viig sig () ecomes: L ( ) ( Si ) d Cos Si L L L L (9.69) 58

59 9.7.3 Covegece o Seies Solios (p.3) Solio o pil dieeil eqios he sepio o viles echiqe sch s peseed i Secios 9.5 o 9.7 iclde smmios o iiie me o ems ssocied wih he iiie me o oos o scedel eqio (o chceisic eqios s meioed i Chpe 4.he solio i Eqio (9.69) o he PDE i Eqio (9.54) is lso i he om o iiie seies. Nmeicl solios o hese eqios c e oied smmig p he solios wih ech ssiged vle o h is wih = 3.o ve lge iege me. I oml cicmsces hese iiie seies solios shold covege il pidl so oe eeds ol o sm p ppoimel doe ems wih he me p o o esol cce solios o he polems.. Howeve he eec o he covegeces o iiie seies sch s he oe i Eqio (9.69) o he ccc o he licl esls emis coce o egiees i hei lses. We will demose he covegece o seies solio eled o Eqio (9.69) o he viio o log cle simil o he siio depiced i Fige 9.8 wih L = m d he cos coeicie = m/s. We ssme h he iiil shpe o he cle c e descied cio.5 si L he mgide o he mplide o viig cle = 5 m = secod is om Eqio (9.69) is o he om: 5 cos6 4 si 4 si d o i he om wih meicl vles o = 3.: (5) =

60 9.7.3 Covegece o Seies Solios Co d We sed he MicoSo Ecel sowe o compe he meicl vles o (5) wih = 3 6 wih he comped esls show i he ollowig le: E- 3.8 E-.8 E E-3 3. E E E-3.4 E E-3-4. E-3 d wih moe ems wih ddiiol vles o (p = 3) i Fige 9.9: 5.77 E-4 4. E Fige 9.9 Covegece o iiie seies solio o Eqio (9.69) = 5 = We oseved om his picl cse o meicl solios o he iiie seies solio o Eqio (969) h iclsio o he is ems i he seies (i.e. = 3..) wold oe esol cce solio o (5) ecse o he coios dimiishig o he eecs o he vles o (5) wih he iclsio o ems wih ddiiol ems wih -vles s illsed i his ige. 6

61 9.7.4 Modes o Viio o Cle Sces (p.33) L Viio e = + X X = = : () Iiil shpe Iseos d ime : () We hve js deived he solio o he AMPLIUDES o viig cles () o e: L ( ) ( Si ) d Cos Si L L L L (9.69) We elie om he ove epessio h he solio cosiss o INFINIE me o ems wih = = = 3 Wh i mes is h ech em loe i he iiie seies i Eqio (9.69) is VALID solio. Hece: () wih oe em wih = ol is oe possile solio d () wih = ol is ohe possile solio d so o d so oh. Coseqel ecse he solio () lso epeses he INSANANEOUS SHAPE o he viig sig hee cold e m POSSIBLE iseos shpe o he viig sig depedig o wh he ems i Eq. (9.69) e sed. Pedicig he possile oms (o INSAANEOUS SHAPES) o viig sig is clled MODAL ANALYSIS 6

62 9.7.4 Modes o Viio o Cle Sces Co d he Fis hee Modes o Viig Cles: We will se he solio i Eq (9.69) o deive he is hee modes o viig sig. Mode wih = i Eq. (9.69): ( ) Cos Si L L (9.69) he SHAPE o he Mode viig sig c e illsed ccodig o Eq (9.7) s: Fige 9. We oseve h he mimm mplides o viio occ he mid-sp o he sig As illsed i Fige 9.. he coespodig eqec o viio is oied om he coeicie i he gme o he cosie cio wih ime i Eqio (9.69) i.e.: / L (9.7) whee P = esio i Newo o pods d m = mss desi o sig/i legh i kg/m 3 o slgs/i. L L P m 6

63 9.7.4 Modes o Viio o Cle Sces Co d Mode wih = i Eq. (9.69): we will hve he mplide o he viig cle o e: () Cos Si L L (9.7) Possile shpe o he cle i Mode viio: Feqec o Mode viio: / L P (9.7) L L m Mode 3 wih = 3 i Eq. (9.69): Possile shpe o he Cle i Mode 3 viio: 3 3 () Cos Si L L 3 3 (9.73) 3 / L 3 3 P Feqec o Mode 3 viio: 3 (9.74) L L m 63

64 Phsicl Impoce o Modl Alsis i Viio o Cle Sces Modl lsis povides egiees wih ciicl iomio o whee he possile mimm mplides m eis whe he sig vies d he coespodig eqec o occece. Ideiicio o locios o mimm mplide llows egiees o pedic possile locios o scl ile d hs he vlele locio o sig (log cle) sces. O cose he mliple me o l eqecies sch s idiced i he Eqios (9.7) (9.7) d (9.74) o he cle i Fige 9.8 wih Mode me =3 e he idicos o wh he eqecies o he pplied iemie lods shold e voided o his kid o sces i ode o void he devsig eso viio o he sce. Modl lsis o cle sces sch s illsed i Figes is hs ciicll impo p o he lsis. 64

65 Emple 9.6 A meicl cse illsio o viio lsis o cle sce (p.35). A leile cle m log is ied oh eds wih esio o 5 N i he ee-hg se (see he ige i he igh. he cle hs dimee o cm d wih mss desi ρ =.7 g/cm 3. I he cle egis o vie iseos smll disce om is iiil shpe h c e descied he cio () =.5(-/). Deemie he ollowig: ) he pplicle dieeil eqio o he mplides o viio o he cle epeseed () i mees i which is he ime io he viio wih i o secod (s) ) he mhemicl epessios o he pplicle iiil d ed codiios c) he solio o () o he dieeil eqio i mees d) he solio o mplide o he viig cle i Mode i.e. () wih he mgide d locio o he mimm delecio o he cle i his mode o viio. e) he meicl vles o he eqecies o he is d secod mode o viio ) he phsicl sigiicce o hese mode shpes. 65

66 Emple 9.6 A meicl cse illsio o viio lsis o cle sce-co d. Solio (p.36): We elie he ollowig speciic codiios: he legh o he cle L = m wih dimee d = cm =. m he cle is mde o lmim wih mss desi ρ =.7 g/cm 3 he cle is sjeced o esio P = 5 N d wih iiil sg descied he cio ():.5 ) he pplicle dieeil eqio o he mplides o viio is Eqio (9.54) ( ) ( ) (9.54) wih he cos coeicie i he ove eqio deemied he ollowig epessio: P i which P = esio i he cle = 5 N d m = mss pe i legh which m eeds o e comped wih give codiios. he mss pe i legh o he cle is m = M/L whee M = ol mss o he cle wih M = ρv wih V eig he volme o he cle. d 4 3 We will ge he volme o he cle e comped he epessio V L 7.85 m 4 We will hs hve he ol mss o he cle M = ρv = (.7 3 )( ) =. kg ledig o he mss pe i legh o he cle o e. kg/m. he cos coeicie ccodig o he epessio i Eqio (9.54) is: P m s m. / he pplicle PDE i eqio (9.54) hs kes he om: () 66

67 Emple 9.6 A meicl cse illsio o viio lsis o cle sce-co d. Solio Co d (p.36) ) he mhemicl epessios o give iiil d ed codiios: he iiil codiios:.5 he ed codiios: L ( ) L c) he solio o () o Eqio () sisig he give codiios i Eqios ( ) d Eqios (c d c) will e oied s ollows: he solio o Eqio () is simil o h o Eqio (9.69) wih = m/s i he ollowig epessio: si d cos si o 5.5 si dcos5.5si.34 () () (c) (c) (d) o 3. cos5.5 si.34 3 (e) 67

68 Emple 9.6 A meicl cse illsio o viio lsis o cle sce-co d. Solio Co d (p.3) d) he mplide o he viig cle i Mode i.e. () wih he mgide d locio o he mimm delecio o he cle i his mode o viio: he eqied solio is oied leig = i Eqio (e) s:. cos5.5 si cos5.5 si () he mimm mplide occs he mid-sp o he cle = 5 m d he ime whe cos5.5 =.. We hs hve he mimm mplide m =.376 m o.376 cm = 5 m d ime 5.5 = π o ime = π/5.5 =. s. e) he meicl vles o he eqecies o he is d secod mode o viio: We m se Eqios (9.7) d (9.7) o compe he meicl vles o he eqecies o he is d secod mode o viio s ollows: P 5. H L m. 43 P 5 o Mode d 4. H L m. 86 o Mode ) he phsicl sigiicce o hese mode shpes o he desig egiee Egiees will se he ocomes o he ove modl lsis o dvise he ses o his cle sce o possiili o devsig eso viio o he cle sce shold he eqec o pplied cclic oce sch s wid oce coicides o he l eqecies comped i P (e) i he solios. he ses will lso e mde we o he locios whee mimm mplides o viio m occ s he mode shpes idice i he modl lsis. he shold void plcig delice chmes o hese locios o he cle sce o void poeil dmges de o ecessive viio hese locios. 68

69 9.8 Pil Dieeil Eqios o svese Viio o Memes (p.38) Solids o ple geome sch s hi ples e commo ppece i mchies d sces. hi ples (o hi diphgms) c e s smll s pied elecic cici ods wih micomees i sie o s lge s loos i ildig sces. Like leile cles hi leile ples e omll leile d e vlele o svese viio. I some cses hese ples m pe de o eso viios eslig i sigiic loss o pope d eve hm lives. his secio will deive ppopie PDEs h llow egiees o ssess he mplides i ee viio o hi ples h e leile eogh o e simled o hi memes. Egiees m se his mhemicl model o hei modl lsis o he se desig o hese pes o mchie compoes d sces. 69

70 9.8. Deivio o pil dieeil eqio o ple viios (p.38) We will deive he mhemicl model o he svese viio o hi ples wih he ollowig ideliios d hpoheses: ) he deivio o mhemicl epessios is sed o he lel (veicl) displceme o solids o ple geome h e leile d oe o esisce o edig. I eli he sce is he descipio o memes i he sseqe lsis. )hi ples wih sppoed lge ple es h e siciel leile i lel deomios. Fige 9.3 svese viio o hi ple 3) Beig leile hee is o she sess i he deomed hi ples. 4) he hi ple is iiill l wih is edges ied. hee is iiil sg epeseed cio () ssied i-ple esio P pe i legh o he ple i ll diecios. he esio P is lge eogh o jsi eglecig he weigh o he ple. 5) Fige 9.3 deies he ple i he () ple wih lel displceme () he mplide o viio o he ple he locios deied he - coodies d ime. 6) Eve p o he ple vies i he diecio pepedicl o he ple sce o he ple i.e. i he -coodie s illsed i Fige 9.4 he slopes o he deomed sce o he ple ll edges e smll. 7) he mss pe i e o he ple desiged he smol (m) is iom hogho he ple. We oice h he solio o he mplides o viig meme (o hi ple) () ow ivolves 3 idepede viles: d. We m well imge h i wold e mch moe compliced lsis polem h he cses h we hve coveed so i his ook. 7

71 9.8. Deivio o pil dieeil eqio o ple viios co d (p.39) Fige 9.4 A ee-od digm o oces i eleme o viig meme ime Fige 9.4 is ee-od digm which shows ll oces cig o smll deomed eleme o he ple dig lel viio. he siio sisies dmic eqiliim codiio wih he smmio o ll oces pese ime e eql o eo. Mhemicll we m epess his codiio i he om: F he idced dmic oce F Newo s secod lw pls mjo ole i he omlio o he ove eqiliim o oces. Mhemicll his oce m e epessed s: F m Fom Fige 9.4 we hve he ollowig dmic eqiliim codiios: Psi P si P si P si m (9.75) whee m = mss pe i e o he ple meil. Ideliio No. 6 idices h oh gles α d β e smll ledig o he ollowig ppoime elioships: si si si si 7

72 9.8. Deivio o pil dieeil eqio o ple viios co d Ssiig he ove 4 ppoime elioships io Eqio (9.75) will esl i he ollowig epessio: m P P he ollowig epessio is oied dividig he ove epessio : m P Give h he lel deomio o he ple coiosl vig wih he locios o he ple deied he - d -coodie we shold hve he ollowig elioships show i he e slide. 7

73 im d im 9.8. Deivio o pil dieeil eqio o ple viios co d he eqiliim eqio i (9.75) hs hs he ollowig om wih d o coios viio o he mplide o viio o he ple i oh - d -coodies wih: m P o i he om o: (9.76) whee he cos i Eqio (9.76) hs he simil om s i Eqio (9.54) wih diee meig: m P (9.77) whee P is he esio pe i legh wih i N/m d m is he mss pe i e kg/m. he cos hs hs i o m/s. 73

74 9.8. Solio o Pil Dieeil Eqio o hi Ple Viio (p.33) We will se Eqio (9.76) o compe he mgides o svese viig hi ple sch s compe mose pd idced sligh iseos disce i he -diecio i Fige 9.5. We will hve he ollowig PDE d he give ppopie iiil d od codiios o he solio o he mgides o he viig ple give ime i.e. () i Eqio (9.76): Fige 9.5 Pl view o leile hi ple degoig svese viio. A) he od codiios: ( c c (9.76) B) he iiil codiios: (c) g (c) he cio g() i Eqio (c) is ohe give cio h descies he veloci o he ple coss he ple o he ple he icepio o he viio. () () () () 74

75 9.8. Solio o Pil Dieeil Eqio o hi Ple Viio Co d A) he od codiios: () ( c () c B) he iiil codiios: (c) g (9.76) () () (c) We will se he sepio o viles echiqes o solve he ove eqios wih he speciied od d iiil codiios. his echiqe eqies he solio () o Eqio (9.76) o e he podc o 3 sepe cios ech cois ol oe o he 3 idepede viles s: Z() = X()Y()() (9.78) Ssiig he epessio i Eqio (9.78) io Eqio (976) will led o he ollowig epessio: LHS = X d X d Y d Y d he eqli o oh sides i he ove epess is possile i oh sides eql o cos he piciple o mhemics. We hs hve he ollowig vlid epessio ised: d d d X d Y d LHS = X d Y d d whee λ i Eqio (d) is he is sepio cos i his lsis = RHS = RHS Eqio (d) esls i he ollowig odi dieeil eqios (ODE): he is odi dieeil d X X eqio o cio X(): d d ohe eqli leds o d Y d he d ODE: Y d d (d) (e) () 75

76 9.8. Solio o Pil Dieeil Eqio o hi Ple Viio Co d A) he od codiios: ( c c B) he iiil codiios: g (9.76) () () () () (c) (c) Fo he sme eso; he vlidi o he eqli i Eqio () eqies h oh sides o he eqli o e sme cos s show elow: d d d Y d Y LHS = = RHS whee μ is he secod sepio cos i epessio (g). (g) We m deive ohe wo odi dieeil eqios om he epessio i Eqio (g): Y d Y d d d (h) (j) We hve hs sepe he viles i PDE i (9.76) oo 3 ODEs sig he sepio o viles echiqe i Eqio (9.78) s show elow: X d X d Y d Y d d d (h) (j) (e) (9.76) 76

77 9.8. Solio o Pil Dieeil Eqio o hi Ple Viio Co d (9.76) d X d d Y d X Y d (j) d A) he od codiios: () ( c () c B) he iiil codiios: (c) g (e) (h) () () (c) We oice h ll he 3 odi dieeil eqios (ODE) i Eqios (e) (h) d (j) e d ode lie ODEs. Solios o hese eqios e ville i Secio 8. (p.43) s show elow: X() = c cosλ + c siλ Y() = c 3 cosμ + c 4 siμ c5 cos c6 si We m ollow he simil pocedes peseed i Secio 9.5. (p.98) 9.6. (p.38) d 9.7. (p.38) i deemiig he coss c c c 3 d c 4 i Eqios (k)d (k) d we m deemied he wo sepio coss λ d μ o e: λ=mπ/ wih m = 3. d μ=π/c wih = 3. especivel s well s c =c 3 = sig he codiios i Eqios() d (). Coseqel we will hve: X Y m si m c si c4 si wih c c c si wih m (k) (k) (K3) (m) (m) 77

78 9.8. Solio o Pil Dieeil Eqio o hi Ple Viio Co d X d X d Y d Y d d d (h) (j) (e) (9.76)... 3 si si m wih m c c X m si si 4 4 wih c c c Y (m) (m) he cos c 5 d c 6 ivolved wih he solio i Eqio (k3) m e deemied wih he iiil codiios speciied i Eqios (c) d (c) o ():i he ollowig w: c m c c m c c c m c Y X m si cos si si () c m m m Now i we le: (p) We m epess he solio o PDE i Eqio (9.76) i he ollowig om: B A c m m m m m m si cos si si (9.79) whee he mli-vled cos coeicies A m d B m e deemied he wo emiig iiil codiios i Eqios (c) d (c) wih he oms: dd c m c A c m si si 4 dd c m g c B c m m si si 4 (9.8) (9.8) wih m = 3. 78

79 9.8.3 Nmeicl solio o he Pil Dieeil Eqio o hi ple viio (p.334) Nmeicl solio o he mplides o svese viio o leile ples give i Eqio (9.79) wih coeicies i Eqios (9.8) d (9.8) is mch moe edios d compliced h oe wold imgie. Howeve meicl solio c oe egiees mch eeded pecepio o he l eqecies o he ple sces mch moe so h wh we m oseve om he licl solios h we m oi om he oemeioed mh epessios i he oemeioed eqios. Wh we will pese i his secio is he meicl solio o Eqio (9.79) o he shpes o hi leile ple ( compe mose pd) illsed i Fige 9.5 o is is hee modes i viio. Dimesios o his ple is show i lowe ige i he igh wih he edges = d c=5 d hickess o.85. he pd is mde o sheic e so i is leile. he pd hs ied edges wih iiil sggig h c e descied cio () = (-)(5-) wih i-ple esio P =.5 l /i Viio o he pd idced sligh iseos disce lel o he pd om sic eqiliim codiio (i.e. eo veloci) wih which g() =.i Eqio (c). 79

80 9.8.3 Nmeicl solio o he Pil Dieeil Eqio o hi ple viio-co d We will se Eqio (9.76) o solve o he mgides o his hi ple: wih he ollowig od codiios: () ( () (9.76) c d he iiil codiios: = (-)(5-) (c) g = he cos coeicie i he RHS o Eqio (9.76) c e comped o e: Pg.5 l / i3. / s.55 l m / i i c 353.5i/ s he eqec ω m eqied o compe he peiods is epessed i Eqio (p) i Secio 9.8. wih eigevles λ m = mπ/ d μ = π/5 wih m = 3..s show i he sme Secio. he mode shpes o his ple om ee lel viio lsis is comped he ollowig epessio: m Amsi si Am cosm 5 whee A (9.8) m 6 c 3 3 m 6 m () () (c) (d) (9.8) We elie h m = 3. d = d c=5 i oh Eqios (9.8) d (9.8). 8

81 9.8.3 Nmeicl solio o he Pil Dieeil Eqio o hi ple viio-co d Gphicl solios o he is hee (3) modes o ee viio o he hi ple wih m= = d 3: Modl lsis o ple viio is ve impo egieeig lsis h eles o he se desig o his pe o sces ecse m sch sces e epeced o svive cclic lod pplicios. Sch siio is vlele o scl iles i eso viio shold he eqec o he pplied cclic lods coicides wih l eqecies o he ple od i he modl lsis. Solios o hese l eqecies o ples o give geome d meil popeies eqies he solio o he shpe o he deomed ples vios modes d i will lso povide egiees wih possile shpes o he ple de ech o hese modes o viio. I-deph descipios o eso viio d modl lsis o sces wee peseed i oh Secios 8.7. d 8.9. Nl eqecies o he ple illsed i Fige 9.5 eqies s o compe he mplides o he ple () i Eqio (9.8) give elow wih speciic codiios s peseed i Eqios ( c d c): m Amsi si Am cosm (9.8) 5 m 6c Whee he coeicies: Am (9.8) m Ad he l eqecies: m Wih: m = 3. d = d c=5 i he ove epessios. m Redes e emided h o ple viio lsis Mode viio is oied wih m== Mode viio wih m== d Mode 3 viio wih m==3 e sed i he ove omlios. m c (9.83) 8

82 9.8.3 Nmeicl solio o he Pil Dieeil Eqio o hi ple viio-co d Nl eqecies o he is hee (3) modes o ee viio o he hi ple wih m= = d 3: We hve oied he epessio o he mplide o ple () i ee-viio lsis show i Eqio (9.8) om which we m ge he l eqecies o he ple om he coeicie i he gme o he cosie cio: cos(ω m ) i Eqio (9.8). Hece he l eqecies o he ple e: ω m wih m=3.. d = 3.. m Amsi si Am cosm (9.8) 5 m 6c whee Am (9.8) m d Pg m m m.5 l / i3. / s.55 l m / i c i We m compe he l eqecies o he is 3 modes o e: Mode wih m==: 353.5i/ s (p) (d) Mode wih m==: Mode 3 wih m==3: 8

83 9.8.3 Nmeicl solio o he Pil Dieeil Eqio o hi ple viio-co d Shpes o he viig hi ple vios modes: whee d Pg A m.5 l / i3. / s 6.55 l m c m / i i m m m m m c Amsi si A 5 cos m 353.5i/ s m (9.8) (9.8) (p) (d) Modl shpes o hi ples eqie meicl solios o () o Eqio (9.8) wih m==3 which is ve edios jo. I will lso e ge del o loios eos o oi gphicl epeseios o hese shpes. Coseqel we will se commecill ville MLAB sowe (vesio R5) ville he ho s hos ivesi o peom hese compios d pese he comped modl shpes o he ples i gphs o he solios. A oveview o his sowe will e descied i Secio.5. o Chpe (p.376) wih ips/op iles o his licl polem peseed i Cse i Appedi 4 (P.473). Gphicl displs o he is wo modl shpes o he hi ple wih ime = /8 d /4 secods o Mode (m==) d Mode wih m== =/8 d ¼ secods will e show i he e wo slides. 83

84 Modl ONE shpes o he hi ple : () = () =/8 secod d (c) =/4 secods () A = pek.6 () A =/8 secod pek.7 (c) A = /4 secods Pek.4 84

85 Modl WO shpes o he hi ple : (c) =/8 secod d (d) =/4 secods (c) A =/8 secod peks.5 (d) A =/4 secod peks.3 hese modl shpes povide egiees wih possile shpe chges o he ple i viios. he illsed shpes lso idice whee he pek mplides o viio o he leile ple wold occ om which he desig egiee shold ke pecio o o plcig delice chmes hese locios o void possile dmges de o ecessive deomio o he ple sce. he comped l eqecies wih: =78.94 d/s =57.88 d/s d 3 = 36.8 d/s will emid he poeil ses o his ple sce o void sch eqecies whe pplig iemie lods i ode o void he devsig eso viio o his hi ple. 85