... (1) Thus: ... (2) Where f(x + y) is a function to be found, in here we assume symmetry and the moments are a function on x + y only.

Transcription

1 Ec d Numeicl Soluio fo ge eflecio of Elsic No-Pismic Ples B Fid A. Choue P.E. S.E. 7 Fid A. Choue ll ighs eseved Geel Soluio fo smme Cse I: Sig ih Eq. 7 d 8 9 d Eq 8 fom Theo of Ples d Shells Timosheo d Woios-Kiege e hve:.... Thus: f o... Whee f is fucio o e foud i hee e ssume smme d he momes e fucio o ol. As esul e hve d.. e m d susiue i Eq. e hve: Sucul Elecicl d Foudio Egiee FAC Ssems Ic h Ave. NW Sele WA

2 m f m m d m.. The soluio of Eq. fom efoe:.. C dm C dm m f C dm m f m. Ad.. C dm m f C dm m f m 6 Ad.... C dm m f m f m m f m f Fom Timosheo Eq8 e hve:

3 o q q m q m dmdm T m.. 8 Whee T is he ol mome o he le segme Susiuig Eq. 7 i Eq. 8 d egig e hve: f m f m o. f m dm C. f m. f m e:. f m dm C. f m dm C T m T m... 9 u.... Ad iege Eq. 9 ih esec o ields: u u u T m dm. Eq. is Quic equio i u d hs fou oos o e fo coodies d d he ohes e ossil imgi. Ad he soluio is foud umeicll susiuig c i Eq. 6 d 7. A leive efeed soluio fo Eq. c e foud eessig i &. Reiig Eq. 6 o & m o u u u & &

4 Susiue Eq. i Eq. ields: dm m T & & &. e θ & so i is defied evehee d susiue i Eq. eilds dm m T si θ θ.. Equio hs o sic oos i θ d he es e diffes π muliles. Thus & of Eq. hs ol o el oos fo coodies d. Geel Soluio fo i-smme Cse II: Chgig Eq. o fucio o hus g o... Whee g - is fucio o e foud i hee e ssume i-smme d he momes e fucio o - ol. As esul e hve d.. 6 e - d susiue i Eq. e hve:

5 g d.. 7 The soluio of Eq. 7 fom efoe:.. C d C d f C d g m. 8 Ad.. C d g C d g m 9 Ad.... C d g g g g.. Fom Timosheo Eq8 e hve:

6 o o q q q q q dd T.. Whee T- is he ol mome o he le segme Susiuig Eq. 9 i Eq. d egig e hve: g g o.g d C.g.g e:.g d C.g d C T T... v... Ad iege Eq. ih esec o ields: v v v T d. Eq. is Quic equio i v d hs fou oos o e fo coodies - d - d he ohes e ossil imgi. Ad he soluio is foud umeicll susiuig c i Eq. 8 9 d. A leive efeed soluio fo Eq. c e foud eessig i &. Reiig Eq. 9 o 6

7 & m o v v v & & Susiue Eq. i Eq. ields: & & T & d. 6 e & θ so i is defied evehee d susiue i Eq. eilds si T θ d θ.. 7 Equio 7 hs o sic oos i θ d he es e diffes π muliles. Thus & of Eq. 6 hs ol o el oos fo coodies - d -. Geel Soluio fo Cse III: Chgig Eq. o fucio o hus o h... 8 Whee h is fucio o e foud i hee e ssume he momes e fucio o ol. As esul e hve 7

8 8 d.. 9 e d susiue i Eq. e hve: h... Ad..

9 9 Fom Timosheo Eq8 e hve: dd q o q q o q q T.. Whee T is he ol mome o he le segme Susiuig Eq. i Eq. d egig e hve: T... O [ ] [ ] o T T && & && & &&

10 e θ θ θ cos so & && &. Susiue Eq. i Eq. ields; [ ] [ ] cos si cos cos si cos cos T θ θ θ θ θθ θ θ θθ & & &.. 6 d T si si si si θ θ θ θ θ. 7 Equio 7 hs o sic oos i θ d he es e diffes π muliles. Thus & of Eq. hs ol o el oos fo coodies / / d / /. Geel Soluio Cse IV: The folloig is he geel soluio fo veicl lods o he le ihou uclig i he Cesi coodies usig Tlo s d Fouie s eeseio. We s eiig he il diffeeil Eq. 8 Timosheo d Woios-Kiege oo usig Eq. 7 d Eq. 8 s follos: q. 8 Eq. 8 c e ie s: q 9

11 e us eess ih ccue oimio usig Tlo esio olomil ove he ievls d fo ecgul oio of he le his eeseio hs o eis sice e e o ssumig les deflecig o ifii d fom hsics fo eve lod hee is uique deflecio o e gueed i he elsic elm givig cei ouded fucio ove h oio of he le. Thus he fucio c e eeseed Tlo esio olomil usig Tlo heoem ih uique coefficies fo eve lod so: Thus e: The g... [ g ].. h [ h ] g h. 6 No eess g h q d i Fouie s eeseio ove he ievl d fo ecgul oio of he le i Cesi coodies s follos: m m mπ π cos si. 7

12 g m h q Whee: g h q m m m m m m m m m g m mπ π cos si mπ π si cos mπ π hm si cos q m mπ π si si mπ π cos si d d mπ π g cos si d d mπ π si cos d d mπ π h si cos d d mπ π q si si d d Eve hough hese e o full eeseio of Fouie seies hese fucios ill give ec vlues i he ievls d. B susiuig Eq. 7 hough i Eq. 9 d diffeeiig he i is equied ields: m mπ g m mπ mπ π h π m m π mπ π h m mπ m π mπ π si si g m m qm mπ π si si.. 7 Equig ems esul i he folloig equio o e sisfied:

13 q h m h g g m m m m m m m m m π π π π π π.. 8 Equio 8 is o e equed em em fo eve m. d.. Thus he soluio c e chieved selecig se of coefficies fo. d. d fo ccele se of d m he iege Eq. hough 6 umeicll ih ccele ccuc d see if Eq. 8 is sisfied fo eve d m. If i is o sisfied he uded ih ccele covesio lgoihm such s Neo Rhso ehod ih he Jcoi mi. As e sid efoe hee is oe deflecio e lod hus he coefficie mus e uique hus hee is oe oo fo he soluio of Eq. 8. The iiil veco c e e leig g d h so h he esul gives he soluio fo smll deflecios s s see Timosheo Eq 8 his mes Eq. 8: q m m m m m π π π π.. 9 I is see h his ecomes mi ivesio fo if m d he mi o e iveed is m m iveed mi. All of he coefficies of he lef of Eq. o c e evlued ecl ihou umeicl iegios. The oosed geel soluio omises o e ec fo ccele ccuc fo he deflecio. I he el old deedig o he licio hee ls defied ccele olece se he egiee. To cool his olece i is es o eese Eq. s follos:. 6 Whe > his is doe i ode o hve moe covesio seies ih good eeseio of he deflecio fo d.

14 Boud Codiio: The oud codiio of he le he edges o iell c e chieved selecig covesio seies fo Eq. 6 h sisfies he oud codiios fo se of d. This ill ecome evide he folloig licio emles. Secio : A. Clmed Recgul Ple he oud edge: B isecio d Eq. 6 mus hve oo d o hve zeo deflecios. Also fo clmed le d so Eq. 6 mus hve ohe oo d o hve zeo sloes. This c lso e sid fo he oudies d Eq. 6 c e ie s:.. 6 Thus d c e ie s: Usig Eq. 6 d 6 e c fid he momes oud oig he sloes e zeo he oud such h / - ec. d he momes ecomes:

15 6.. 6 We c see fom Eq. 6 d 6 h he momes o he oudies e diffee fucios ih diffee coefficies d Eq. 6 is eeseio of clmed le. B. Clmed Recgul Ple he oud edge d smmeic lod: Fo smmeic lod e fis sle he deflecio o he cee of he le d loo he deflecio e hve:.. 66 Whee Tlo esio is doe o d o oi smme d he oos d ± ± sisfied he oud codiios. Whe slig c o coe of he le Eq. 66 ecomes:.. 67

16 6 Ad he momes ecomes: Ad e sho smme i he momes. No e see he momes / d /. Fis e fid d fom Eq. 67 s follos: 7 Susiue / d / i Eq. 7 ields:

17 7 7 Sice / d / he momes he cee ecomes: 7 7 Fo sque le he is he sme fo he coodie d he d e c ie d susiue i Eq. 67 ields:. 7 C. Clmed Recgul Ple ih Clmed ellise Buil i he cee See Fig. Fig. - Clmed Recgul Ple ih Clmed ellise Buil i he cee / / d c

18 8 B isecio d Eq. 6 mus hve oo d d he coou of he ellise o hve zeo deflecios. Also fo clmed le d d he coou of he ellise so Eq. 6 mus hve ohe oo d d he coou of he ellise o hve zeo sloes. This c lso e sid fo he oudies d Eq. 6 c e ie s Eq. 6 ih he ellise coou: d c d c d c 76 Oe c see Eq. 76 c e eded o give ohe eessio o Tlo olomil d he soluio c e oied.. Clmed Tigul Ple: Simil o h e hve doe efoe isecio d d he edge of igle Eq. 6 mus hve oo d d he edge of igle see Fig o hve zeo deflecios. Also fo clmed le d he edge of igle d he edge of igle he Eq. 6 mus hve ohe oo d d he edge of igle o hve zeo sloes d Eq. 6 c e ie s:. 77

19 Fig. - Clmed Tigul Ple ll sides Oe c see Eq. 77 c e eded o give ohe eessio o Tlo olomil d he soluio c e oied. E. To Side Clmed Recgul Ple ih Clmed ellise Buil he edge Simil o h e hve doe efoe isecio d d he edge of he ellise Eq. 6 mus hve oo d d he edge of he ellise see Fig o hve zeo deflecios. Also fo clmed le d he edge of he ellise d he edge of he ellise he Eq. 6 mus hve ohe oo d d he edge of he ellise o hve zeo sloes d Eq. 6 c e ie s:. 78 9

20 Fig. - Clmed Ple ll sides Oe c see Eq. 78 c e eded o give ohe eessio o Tlo olomil d he soluio c e oied. F. To Side Clmed Recgul Ple ih Clmed Polomil Cuve Buil he edge Simil o h e hve doe efoe isecio d d he edge of he cuve Eq. 6 mus hve oo d d he edge of he cuve see Fig o hve zeo deflecios. Also fo clmed le d he edge of he cuve d he edge of he cuve he Eq. 6 mus hve ohe oo d d he edge of he cuve o hve zeo sloes d Eq. 6 c e ie s: [ R ] Whee he cuve is eeseed s olomil: R.. 8

21 R Fig. - Clmed Ple ll sides Oe c see Eq. 79 d Eq. 8 c e eded o give ohe eessio o Tlo olomil d he soluio c e oied. G. iscussio o Simle s d Clmed Ples ih ge eflecio: Clmed les e sicll simle suoed le ih secific mome he edge h me he sloes he edge fl o zeo. If hee is lge deflecio o such le hee he le is uched o is suo d he le lile ovesized ove he suo he lge deflecio ill cuse he le o deflec eough he edges of he le o loose is suo he edge hee he ficio he suo due o he veicl lod is igoed ssumed smooh. This c he he he edig sesses i he le did o ech filue hich is he cse i mos hi les see Fig. fo ecgul le. Becuse he le is clmed his olem of loosig suo he edge is o liel o he i eli u i is moe liel fo uched simle suoed les hee he ficio suo due o he veicl lod is igoed ssumed smooh. This c he i ddiio o evese deflecio veicll i secio of he suo see Secio fo delig ih his olem Hoeve he ove soluio offes o solve vious clmed le olems h hs o ee solved efoe d if lge deflecio is

22 o issue he he iiil vlue of Eq. 9 is sufficie fo soluio hich equies ol oe mi ivesio. We give he equios fo loosig suos fo smmeic lodig o ecgul le ih edge codiio s follos: [ ] [ ] d d / / 8 Fig. loss of suo i le Secio : A. Recgul Siml Suoed Ple: B isecio d Eq. 6 mus hve oo d o hve zeo deflecios. This c lso e sid fo he oudies d Eq. 6 c e ie s:.. 8

23 Thus d c e ie s: Thus d c e ie s: Usig he equio fo fidig he momes he edges e ie: [ ] [ ] [ ] [ ] 87 Whe solvig he o equio fo / d / d e see he ol Eq. 87 is sisfied is he d 88 Simill he gume holds fo d so d 89

24 We c see h is ue led susiuio i Eq. 8 d Eq. 86 h: d Whe usig Eq. 8 e fid { } 9 Thus se i he olomil of Eq. 8 d he codiio is sisfied O se.. 9 Simill fo d e hve he codiio:. 9 Thus he selecig hese coefficies of Eq. 9 d 9 d susiuig i Eq. 8 he oud codiio e sisfied. B. Simle Suoed Recgul Ple he oud edge d smmeic lod: Fo smmeic lod e fis sle he deflecio o he cee of he le d loo he deflecio e hve:.. 9

25 Whee Tlo esio is doe o d o oi smme d he oos d ± ± sisfied he oud codiios. Whe slig c o coe of he le Eq. 9 ecomes:.. 9 Ad d ecomes: Ad d ecomes:

26 6 As efoe e equie d. Fom Eq. 98 d Eq. 99 e mus hve: - d -. Thus he selecig hese coefficies d susiuig i Eq. 9 he oud codiio e sisfied. No e see he momes / d /. Fis e fid d fom Eq. 98 d Eq. 99 d usig Eq. ields: 6 6. Sice / d / he momes he cee ecomes: Fo sque le he is he sme fo he coodie d he d e c ie d susiuig i Eq. 9 ields:. ohe oud codiios c e sisfied ih fidig he oe Tlo seies esio. Oe moe codiios ill e ddessed fo he cse of ecgul le ih oe fee edge d he ohes e clmed edge.

27 7 Secio : A. Thee Sided fied ecgul Ple d oe fee edge: Coside Fig 6. Fig. 6 Thee sides clmed oe side fee The fucio z is he fil deflecio of he fee edge d c e eessed ih ccue oimio Tlo olomil s follos: i i i A z Ad s efoe he deflecio c e ie s.. We efoce he oud codiio of he momes: z

28 z z [ ] [ ] [ ] [ ] 6 Ad he legh of he le s: z d. 7 Which mes he soluio comliced u he soluio c e efoced cei coodies ig se of o i s fo emle dividig o icemes /T d i i/t i.. he ech Eq 6 d 7 is se of i equios o dd moe equios o Eq. 8 d he lgoihm of fidig lso fids A i. Oce A i is foud Eq. 6 d 7 c e used o veif he ccuc of ohe ois eside i. No-Pismic Ples If e hve se of sque les ih diffee hicess elded o ched ogehe o me oe ig le hee he eimee c hve igul les if eeded he e eed o mch he sloe d deflecio oud ech le. This c e doe mchig se of ois i d i fo he fucio z z he eimee simil o he ls emle. Eve hough his ss o loo lie fiie eleme i is much ee eeseio o iclude lge deflecio d moe ccue. Geel Soluio usig Poi ods d omes: Befoe givig he geel soluio ih oi lods d mome he oi of licio e ill evie he ove soluio usig oe dimesiol em siml suoed ih o lge deflecio. e us use se of Tlo olomil o oime he lsis fo siml suoed em ih lodig see Fig. 7 8

29 9 FIG 7. Simle S Bem Eessig he lod q i Fouie Seies ields si q π. 8 No e ic Tlo seies h sisfies he oud codiio. Thus he seies mus hve oo d d d & & & & We ill sho he ehvio of he soluio fo h 6 h 7 h 8 h d 9 h ode olomil d come he esuls. We fid he folloig equio sisfies he oud codiio: h ode: EI 9 6 h ode: EI 6 q

30 7 h ode: EI h ode: EI h ode: EI Whee 7 6 d e coss o e foud. No folloig he oosed soluio e eess he equio fo he sloe i Fouie seies s: u du u d cos cos hee cos π π π &. Whe iegig Eq. 9 d usig Eq. he diffeeiig hee ime o ge he essue d equig o Eq. 8 e oi se lie ssem of equios i i. Whe iveig he mi fo ech olomil equio e oi he folloig soluios:

31 h ode: / /.. 6 h ode / / / 6 7 h ode -.7E E / / / / 7 8 h ode E / / / / /. 8 9 h ode.7e E E / / / / K / / 9

32 No e ivesige sevel lodig codiio: - Eessig he lodig of Fig. 8 i Fouie Seies ields q c FIG 8. Pil od o Simle S Bem π c cos π q q si.. π Thus: π c cos..... π Susiuig i Eq d Eq. 9 d ghig he esuls fo c.7 ields Fig A d Fig B; Tle shos he esuls i comig ih he ec soluio. - Eessig he lodig of Fig. 9 i Fouie Seies o oi he essue fo oi lod ields π c πε si si π q q si π i

33 q c ε/ c ε/ c FIG 9. od o Simle S Bem o e coveed o Poi od e he oi lod P ε q hich is he lod ude q d susiue fo q i Eq. Thus q P i π ε si π c π si si πε.... e ε go o zeo d e ge he essue eeseig oi lod s follos: q Thus: P i π c π si si.. P π c si Susiuig i Eq d Eq. 9 d ghig he esuls fo c. ields Fig A d Fig B; Tle shos he esuls i comig ih he ec soluio.

34 - Eessig he lodig of Fig. i Fouie Seies o oi he essue fo mome ields πε cos π c π q q cos si 6 π i q c ε/ c ε/ c - q FIG. od o Simle S Bem o e coveed o ome e he mome.ε ε q.ε ε q qε hich is he mome fo lodig i Fig. d susiue fo q i Eq. 6 Thus q i πε cos π c π π cos si πε e ε go o zeo d e ge he essue eeseig ome s follos:

35 q Thus: i π c π π cos si... 8 π c π cos Susiuig i Eq d Eq. 9 d ghig he esuls fo c. ields Fig A d Fig B; Tle shos he esuls i comig ih he ec soluio. All of he esuls shos Tlo oimio follos Fouie oimio d he eo imoves ih ddig highe em o Tlo olomil. Fo he oi lod e c see h ude he essue dv d c so he essue oches ifii ude he lod. This hs he ecuse e oimed he cul she hich is discoiuous fucio coiuous fucio. As i he cul she cuve h oi he she is eve defied ude he lod sice i hs o vlues d he quesio ecomes hich vlue c e use. I cice he she is ls e s m V c V c fo oie iceme hee c e foud esig fo ulime vlues oud c. Thus he soluio fo he she is coec fo ll vlues ece ude he lod d c e e s m V c V c of he oime Tlo olomil. If e hve lod h hs ee oimed ih m oi lods he i is es o deemie h she vlue o use ude he lod sed o desig cice d eese Tlo olomil oimio s discoiuous fucio ih oi vlues ude he lod. If e diffeeie he she o oi he essue he he essue ude he lod c o e deemied usig Tlo olomil s coiuous fucio d should e e V c V c P. The eso he cul she digm hs discoiui ecuse i is ellig us i el life hee o such hig s oi lod d i eli i is some id of essue ih some smll ε oud he lod s i Fig 9.

36 Simill fom he mome digm e c see h ude he she d d c so he she oches ifii ude he lod. This hs he ecuse e oimed he cul mome fucio hich is discoiuous fucio coiuous fucio. As i he cul mome digm he oi of licio he mome is eve defied ude he mome sice i hs o vlues d he quesio ecomes hich vlue c e use. I desig cice he o mome is ls e s d - o c d c ih hei coesodig sig fo oie iceme hee c e foud esig fo ulime vlues oud c. Thus he soluio fo he mome is coec fo ll vlues ece ude he oi of licio d c e e s c d c of he oime Tlo olomil. Thus i is es o eese Tlo olomil oimio s discoiuous fucio ih oi vlues ude he mome. If e diffeeie he mome o oi he she he he she ude he lod c o e deemied usig Tlo olomil s V. coiuous fucio d should e e [ c c ] Fill if e diffeeie he she o oi he essue he he essue ude he lod c o e deemied usig Tlo olomil s coiuous fucio d should e e zeo. This c lso e see he usig slighed sled lie ised of veicl lie he oi of licio c i he mome digm he diffeeiig ice o ge he essue esulig i zeo essue. The eso he cul mome digm hs discoiui ecuse i is sig i el life hee o such hig s mome oi of licio d i eli i is some id of essue ih some smll ε oud he lod s i Fig. Fo emle if e o u mome usig iio of moo he i eli he iio of he moo could eve hve zeo dius d he dius c ol e s smll s ε d sfeig he lod c ol e ossile ioducig some id of is-smmeic essue he oi of licio fom -ε c ε. 6

37 Fifh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh Eo.%.%.%.%.9.6% -.6%.86% -.6%.8.% -.6% 9.7% -.66%.7.997% -.6%.7% -.%.6.98% -.8% 8.87% -.7%. -.6% -.% -8.67% -.778%. -.%.6% -.9% -.%. -.96%.86% -86.7% -.667%. -6.7%.% -.% -7.98%. -.%.69% -.78% -.9% Sih Ode Polomil Aoimio c/ eo m eo m i gh Eo i gh eo.%.%.%.%.9.7%.7% 8.79%.9%.8.86%.%.%.%.7 -.6% -.% -8.8%.% % -.6% -.86% -.%. -.6% -.% -8.67% -.778%..7% -.%.6% -.7%..8%.7%.86% -.68%..69%.% 9.8% -.8%. -.66%.8% 8.87% -.6% Seveh Ode Polomil Aoimio c/ eo m eo m I gh eo i gh eo.%.%.%.%.9.9%.7%.68%.9%.8.9%.% -.89%.9%.7 -.% -.9% -.6%.%.6 -.7% -.% -6.% -.%..6% -.6%.6% -.%..%.66%.9% -.8%..66%.76%.% -.76%. -.7% -.% -.96% -.%. -.8% -.8% -.6% -.67% Eighh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh eo.%.%.%.%.9.9% -.%.6% -.% %.% -7.96%.%.7 -.7%.7% -.6%.%.6.% -.6%.8%.9%..6% -.6%.6% -.%. -.89%.% -6.9% -.8%. -.9%.% % -.7%..% -.%.7% -.%. -.9% -.89% 6.6% -.67% Nieh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh Eo.%.%.%.%.9.% -.%.679% -.%.8 -.6%.% -6.79%.%.7 -.%.%.68%.89%.6.% -.6%.9%.6%. -.6% -.8% -.967%.%. -.6%.% -.8% -.86%..9%.%.8% -.%..% -.% 7.6% -.976%. -.96%.% -8.8% -.89% TABE TAYOR APPROXIATION FOR A RECTANGUAR PRESSURE 7

38 Fih Ode Polom il Aoim io Fifh Ode Polom il Aoim io q c ul Fouie Tlo.6. ome c ul ome Tlo / / Sih Ode Polom il Aoim io Sih Ode Polom il Aoim io q c ul Fouie Tlo.6.. ome c ul ome Tlo / / Seveh Ode Polom il Aoim io Seveh Ode Polom il Aoim io q c ul Fouie Tlo.6.. ome c ul ome Tlo - -. / / FIGURE A TAYOR APPROXIATION FOR A RECTANGUAR PRESSURE c/.7 8

39 Eighh Ode Polom il Aoim io Eighh Ode Polom il Aoim io q c ul Fouie Tlo.6. ome c ul ome Tlo / / Nieh Ode Polom il Aoim io Nieh Ode Polom il Aoim io q c ul Fouie Tlo.6.. ome c ul ome Tlo / / FIGURE B TAYOR APPROXIATION FOR A RECTANGUAR PRESSURE c/.7 9

40 Fifh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh eo ~. -6.%.9% 9.9% -8.%.9-8.%.9% -.% -.9%.8 -.%.% -9.% -.8%.7.%.% 7.% -.98%.6.8% -.% 9.7% -9.%..8% -.8%.69% -.6%..8% -.% 9.7% -9.%..%.% 7.% -.98%. -.%.% -9.% -.8%. -8.%.9% -.% -.9% Seveh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh eo ~. -6.9% -.% -6.9% -.%.9 -.% -.% -9.% -.87%.8.78%.%.6% -.%.7.%.% 8.6% -9.9%.6 -.% -.7% -.% -9.79%. -.6% -.% -7.8% -.%. -.% -.7% -.% -9.79%..%.% 8.6% -9.9%..78%.%.6% -.%. -.% -.% -9.% -.87% Sih Ode Polomil Aoimio c/ eo m eo m i gh eo I gh eo ~. -6.8%.% 99.% -9.96%.9 -.7%.% 7.% -7.8%.8.%.9% 7.8% -.7%.7.7% -.%.78% -.9%.6 -.% -.% -69.% -.6%. -.6% -.% -7.8% -.%. -.% -.% -69.% -.6%..7% -.%.78% -.9%..%.9% 7.8% -.7%. -.7%.% 7.% -7.8% Eighh Ode Polomil Aoimio c/ eo m eo m i gh eo I gh eo ~. -.8% -.%.96% -.%.9.8% -.%.9% -6.%.8.%.8% -7.% -8.9% %.6% -6.6% -7.8%.6.% -.% 7.6% -9.%..6% -.% 6.% -7.8%..% -.% 7.6% -9.%. -.77%.6% -6.6% -7.8%..%.8% -7.% -8.9%..8% -.%.9% -6.% Nieh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh eo ~. -.9%.% -9.99% -.7%.9.88% -.% 7.% -6.67%.8.7% -.% 8.8% -6.%.7 -.8%.6% -9.7% -7.8%.6 -.7%.% -7.% -6.86%..6% -.% 6.% -7.8%. -.7%.% -7.% -6.86%. -.8%.6% -9.7% -7.8%..7% -.% 8.8% -6.%..88% -.% 7.% -6.67% TABE TAYOR APPROXIATION FOR POINT OA

41 Fifh Ode Polomil Aoimio Fifh Ode Polom il Aoim io Vc ul Fouie Tlo ome cul ome Tlo / / Sih Ode Polom il Aoim io Sih Ode Polom il Aoim io q c ul Fouie Tlo ome cul ome Tlo / / Seveh Ode Polom il Aoim io Seveh Ode Polom il Aoim io V c ul Fouie Tlo.. ome c ul ome Tlo / / FIGURE A TAYOR APPROXIATION FOR A POINT OA c/.

42 Eighh Ode Polom il Aoim io Eighh Ode Polom il Aoim io V c ul Fouie Tlo ome c ul ome Tlo / / Nieh Ode Polom il Aoim io Nieh Ode Polom il Aoim io V c ul Fouie Tlo. ome c ul ome Tlo / / FIGURE B TAYOR APPROXIATION FOR A POINT OA c/.

43 Fifh Ode Polomil Aoimio c/ eo m eo m I gh eo i gh Eo.%.9% 8.868% -8.%.9.%.66%.8% -7.8%.8.% -.99%.86% -.887%.7.9% -.88% 7.9% -.87%.6.6% -.998% 8.6% -.7%..7%.99%.98% -.%..6% -.998% 8.6% -6.79%..9% -.88% 7.9% -.87%..% -.99%.86% -.887%..%.66%.8% -9.9% Seveh Ode Polomil Aoimio c/ eo m eo m i gh Eo i gh Eo.8% -.6% 66.87% -.%.9.69%.6%.7% -.7%.8.66%.76% 9.77% -.7%.7.79%.89% 6.68% -8.87%.6.86%.%.6% -7.69%..8%.89% 9.9% -.7%..86%.%.6% %..79%.89% 6.68%.%..66%.76% 9.77% -.7%..69%.6%.7% -.7% Sih Ode Polomil Aoimio c/ eo m eo m i gh eo i gh Eo.%.% 7.% -9.96%.9.8%.8%.% -.79%.8.8%.% 7.% -.7%.7.%.9%.877% %.6.7% -.% 7.97% -8.9%..7%.99%.98% -.%..7% -.% 7.97% -.%..%.9%.877% -.8%..8%.% 7.% -.7%..8%.8%.% -.79% Eighh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh Eo.8% -.9% 7.7% -.97%.9.%.89%.7% -.98%.8.%.%.% %.7.6%.%.% -.%.6.%.79% 9.8% -.96%..8%.89% 9.9% -.7%..%.79% 9.8% -.88%..6%.%.% -.9%..%.%.% %..%.89%.7% -.98% Nieh Ode Polomil Aoimio c/ eo m eo m i gh eo i gh eo.%.% 8.9% -.66%.9.6% -.7%.68% -8.7%.8.%.9%.8% -.%.7.%.7% 6.7% -6.87%.6.6%.669% 9.9% -.%..%.76%.% -.%..6%.669% 9.9%.87%..%.7% 6.7%.8%..%.9%.8% -.%..6% -.7%.68% -8.7% TABE TAYOR APPROXIATION FOR A OENT

44 Fifh Ode Polom il Aoim io Fifh Ode Polom il Aoim io c ul Fouie Tlo ome cul ome Tlo / / Sih Ode Polom il Aoim io Sih Ode Polom il Aoim io q c ul Fouie Tlo ome c ul ome Tlo / / Seveh Ode Polom il Aoim io Seveh Ode Polom il Aoim io c ul Fouie Tlo ome c ul ome Tlo / -.6 / FIGURE A TAYOR APPROXIATION FOR A OENT c/.

45 Eighh Ode Polom il Aoim io Eighh Ode Polom il Aoim io c ul Fouie Tlo ome c ul ome Tlo / / Nieh Ode Polom il Aoim io Nieh Ode Polom il Aoim io c ul Fouie Tlo ome c ul ome Tlo / / FIGURE B TAYOR APPROXIATION FOR A OENT c/.

46 Fom his eecise d emles e o h o eec fom ou le soluio. B usig Timosheo Eq. he oi lod o le c e ioduced usig he folloig essue: q m m si mπ si π. Whee m P mπ ξ πη si si. Bse o he lsis i oe dimesio e coclude h he she he oi ude he lod c e e s ± [ m Q ξ η Q ξ η Q ξ η Q ξ η ] d ± m Q ξ η Q ξ η Q ξ η Q ξ η.. fo oie icemes hee c e foud esig fo ulime vlues oud he oi ξ η. If e hve lod h hs ee oimed ih m oi lods he i is es o deemie h she vlue o use ude he lod sed o desig d eese Tlo olomil oimio s discoiuous fucio ih oi vlues ude he lod. If e diffeeie he she o oi he essue he he essue ude he lod c o e deemied usig Tlo olomil s coiuous fucio d should e e P Q ξ η Q ξ η o P Q ξ η Q ξ η. Fo he mome essue equio i he diecio use Eq. d Eq. d le hee e o oi lod i oosie diecio he coodie ξ ε η d ξ ε η he Eq. ecomes 6

47 m P π η mπ ξ ε mπ ξ ε 8P mπε mπξ πη si si si si cos si No le he mome due o he oi lod e lied he oi ξ η s εp. B susiuig i Eq. e hve: m mπε si mπξ πη mπ cos si mπε. e ε go o zeo he e he essue fo mome i he diecio usig Eq. ih he folloig: m mπξ πη mπ cos si. 6 Reeig he ove lsis fo he essue fo mome i he diecio usig Eq. ih he folloig: m mπ ξ πη π si cos... 7 Ad he fou momes suoudig he oi ude he oi of licio c e e s ξ η ξ η ξ η ξ η d ξ η ξ η ξ η ξ η.. 8 fo oie icemes hee c e foud esig fo ulime vlues oud he oi ξ η. Thus i is es o deemie he mome vlues o ude he oi of licio sed o desig cice d eese Tlo olomil oimio s discoiuous 7

48 8 fucio ih oi vlues ude he lod. If e diffeeie he mome o oi he she he he she ude he lod c o e deemied usig Tlo olomil s coiuous fucio d should e e [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] m m d m m η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ Q Q. 9 Comme o selecig d : A cicl selecio of d e ecommeded Pofesso Joh So sig he oi lod i cocee sl c e see s coe ogig i he hicess of he sl. Thus solue smlles iceme of d is he hicess of he le shed i cicle. This lso ecomes esicio he sudividig essue fucio io oi lods fo lge deflecio lsis d i ecomes codiio of usig he soluio i elm of elsici. ods h eed fie icemes he ice he hicess of he sl co e oimed io oi lods d should e ddessed diffeel he lge deflecio is he issue. If lge deflecio is o of coce he Eq. 9 is he es leive d hs ee he mehods used i sdd cice fo ges. Fil lsis: As e c see lod c oimed oi lods fo moe cosevive soluio. Hoeve he iceme of divisios hs limiig vlue s discussed ove. The ieesig is he Cesi soluio is i is sufficie soluio d ohe coodies sfomios e o ecess ovided he lod is coied i he oud codiio. Fo emle fo siml suoed cicul le he oud codiio c e sisfied s log s he lod is coied i he cicle.

49 Fill fo lge deflecio ih oi lod P i mome i he diecio i d mome i he diecio i he oi of licio ξi ηi mes he coodie ξ i ηi ecomes ohe coodie i i i he fil lge deflecio of he le. Thus e se of equios is equies so hee is o chge i legh o he oigil oi ξ hus fo ecgul le e hve: i ηi ξ i η i i i d [ ] i d [ ] i Wih hese ddiiol equios he soluio c e foud ecl fo oi lods momes. The soluio is simil o fidig he coefficies fo lge deflecio of ems. Thus e s ih iiil vlue fo he Tlo olomil coefficies lus he coefficies i i d he fee oudies d ude umeicll d he soluio ecomes ec. 9