11 Nonlinearities in finite element analysis
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- Rudolf Harmon
- 5 years ago
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1 Noleares e eleme aalyss e s sar wh some smlao eamples: Shell olg hp:// Sampg hp:// Sal shp mas hp:// Faseer os hp:// Hemmg hp://
2 Ra-54.3 Nmeral Mehos Srral Egeerg Coes. Moellg prples a boary ale problems egeerg sees. Eergy mehos a bas D e eleme mehos - bars/ros beams hea so seepage elerosas 3. Bas D a 3D e eleme mehos - hea so seepage 4. Nmeral mplemeao ehqes o e eleme mehos 5. Absra ormlao a aray o e eleme mehos 6. Fe eleme mehos or Eler Beroll beams 7. Fe eleme mehos or mosheo beams 8. Fe eleme mehos or Krhho oe plaes 9. Fe eleme mehos or Resser Ml plaes. Fe eleme mehos or D a 3D elasy. Era lere: Noleares e eleme aalyss. Era lere: Fe eleme mehos or me-epee problems Ra-54.3 / 6 / JN 65
3 Noleares e eleme aalyss Coes. Nolear ompaoal mehas. Geomeral oleares e eleme aalyss 3. Maeral oleares e eleme aalyss 4. earzao a solo algorhms earg oome A. Oerall ew a e ersag o oleares egeerg B. Ably o ormlae a sole smple moel problems o olear mehas Reerees J. N. Rey A Iroo o Nolear Fe Eleme Aalyss 4: hapers 3. 3 A.3; Boe & Woo Nolear Com Mehas or FE Aalyss: hapers Ra-54.3 / 5 / JN 66
4 . Ulzg or aog oleares Wha a be ahee by ersag olear pheomea mehas a e eleme mehos? Ra-54.3 / 4 / JN 67
5 . Ulzg or aog oleares Frs o all ersag o he are reqres ersag o oleares. Howeer olear pheomea are o oly a eresg researh el or geeral rosy. Namely beer or oae pros or seres a be esge by aog or eplog oleares: Ra-54.3 / 5 / JN 68
6 . Ulzg or aog oleares Frs o all ersag o he are reqres ersag o oleares. Howeer olear pheomea are o oly a eresg researh el or geeral rosy. Namely beer or oae pros or seres a be esge by aog or eplog oleares: ew maerals: omposes power meals power oree shape memory alloys aomaerals bomaerals large eormaos: blg olg orm-g oas a eraos: maser-slae oa l-srre erao speal loags: mpa loas ollower loas ml-physs a ople problems Ra-54.3 / 5 / JN 69
7 . Nolear ompaoal mehas wo ma sores o oleares are maeral oleary a geomeral oleary. Amog or ao o hese bas sores ere ypes o olear problems egeerg mehas a be grope as ollows: Ra-54.3 / 5 / JN 63
8 . Nolear ompaoal mehas wo ma sores o oleares are maeral oleary a geomeral oleary. Amog or ao o hese bas sores ere ypes o olear problems egeerg mehas a be grope as ollows: Geomeral oleary ables rames membraes plae shells large splaemes a/or roaos b small sras Fe eormaos meal ormg yre mehas large splaemes a/or roaos a large sras 3 Maeral oleary polymers seel oree sol olear/eoloal sra-sress ose respose Ra-54.3 / 5 / JN 63
9 . Nolear ompaoal mehas wo ma sores o oleares are maeral oleary a geomeral oleary. Amog or ao o hese bas sores ere ypes o olear problems egeerg mehas a be grope as ollows: Geomeral oleary ables rames membraes plae shells large splaemes a/or roaos b small sras Fe eormaos meal ormg yre mehas large splaemes a/or roaos a large sras 3 Maeral oleary polymers seel oree sol olear/eoloal sra-sress ose respose 4 Sably problems blg sap hrogh eg or shear bas meals geomeral a/or maeral sables 5 Nolear boary oos oa problems speal loags 6 Cople problems hermomehaal l-srre pezoelery Ra-5.35 / 5 / JN 63
10 . Nolear ompaoal mehas I boh lear a olear om mehas eormao a moo o a om s ee by he ollowg oeps: Kemas: eormaos Kes: eqaos o moos a sresses 3 hermoyams: eergy eqales 4 Cose behaor: e s. ema/hermoyam arables Ra-5.35 / 4 / JN 633
11 . Nolear ompaoal mehas I boh lear a olear om mehas eormao a moo o a om s ee by he ollowg oeps: Kemas: eormaos Kes: eqaos o moos a sresses 3 hermoyams: eergy eqales 4 Cose behaor: e s. ema/hermoyam arables Nmeral aalyss as e eleme mehos reerre as olear ompaoal mehas whe apple o olear om mehas s a way o smlae a aalyze omple sysems sbe o omple loags: A Nolear om mehas 4 B Nolear eergy mehos as e eleme ormlaos C Solo ehqes or reslg goerg olear algebra eqaos Remar. A esrbes a oos sysem B a C s sree appromaos. Ra-5.35 / 4 / JN 634
12 . Geomeral oleares e eleme aalyss arge sras geomerally olear eormaos mply les eg erre large sra measres or oms. Oe mesoal sra measres. e s oser a rss member o al legh a area A srehe o a al legh l a al area a. Egeerg sra or Cahy sra: E l : l l l l For ery small hages legh hs sra measre es o small sra qay. Ra-5.35 / 4 / JN 635
13 . Geomeral oleares e eleme aalyss arge sras geomerally olear eormaos mply les eg erre large sra measres or oms. Oe mesoal sra measres. e s oser a rss member o al legh a area A srehe o a al legh l a al area a. Egeerg sra or Cahy sra: E l : l l l l For ery small hages legh hs sra measre es o small sra qay. ogarhm sra or Hey sra or re sra: l l l l l l l Ra-5.35 / 4 / JN 636
14 . Geomeral oleares e eleme aalyss arge sras geomerally olear eormaos mply les eg erre large sra measres or oms. Oe mesoal sra measres. e s oser a rss member o al legh a area A srehe o a al legh l a al area a. Egeerg sra or Cahy sra: E l : l l l l For ery small hages legh hs sra measre es o small sra qay. ogarhm sra or Hey sra or re sra: l l l l l l Remar. Sras a a o be easly a ompaoally heaply geeralze o hree-mesoal oms. l Ra-5.35 / 4 / JN 637
15 638 3 Gree sra: 4 Almas sra: l l G ; ; l l A. Geomeral oleares e eleme aalyss Ra-5.35 / 4 / JN
16 639 3 Gree sra: 4 Almas sra: Remar. For ery small hages legh oe aga hese sra measres es o small sra qay: l l G ; ; l l A l l l l l l l l l l l l G. Geomeral oleares e eleme aalyss Ra-5.35 / 4 / JN
17 . Geomeral oleares e eleme aalyss Com sra measres. Small sra lear esor ompoes o he womesoal ase are o he orm y y yy y y y Ra-5.35 / 4 / JN 64
18 64 Com sra measres. Small sra lear esor ompoes o he womesoal ase are o he orm whle Gree sras ge large sra olear esor ompoes he orm y y y y y yy. Y X Y X X X E Y Y Y E X X X E y y y y y y yy y. Geomeral oleares e eleme aalyss Ra-5.35 / 4 / JN
19 . Geomeral oleares e eleme aalyss Aally loae geomerally olear elas bar problem. e s sar rom he geeral ormla o he prple o ral wor or eormable boes applable or boh lear a olear om heores: W V σ : ε V EAεε S S b V b V W e Ra-5.35 / 4 / JN 64
20 . Geomeral oleares e eleme aalyss Aally loae geomerally olear elas bar problem. e s sar rom he geeral ormla o he prple o ral wor or eormable boes applable or boh lear a olear om heores: W V EAεε Nε σ : ε V S b b S N V b V EAε W e Ra-5.35 / 4 / JN 643
21 . Geomeral oleares e eleme aalyss Aally loae geomerally olear elas bar problem. e s sar rom he geeral ormla o he prple o ral wor or eormable boes applable or boh lear a olear om heores: W se V EAεε Nε σ : ε V ε ' b ' b N ' ' S S a N V EAε b b V W e ε ' ' ' ' ' Ra-5.35 / 4 / JN 644
22 . Geomeral oleares e eleme aalyss Iegrag by pars ges he orm N '' N ' N ' b Ra-5.35 / 4 / JN 645
23 . Geomeral oleares e eleme aalyss Iegrag by pars ges he orm N '' N ' N ' a ally he ereal eqao a boary oos as N '' ; N EA N or 3 N or. ol EA ' ' : N l b N ol Ra-5.35 / 4 / JN 646
24 . Geomeral oleares e eleme aalyss Iegrag by pars ges he orm N '' a ally he ereal eqao a boary oos as N or 3 N or. whh a be ompare o he lear ase N ' N ' N '' ; N EA N' ; N EA N or 3 N or. l EA ' ' : N b Remar. Noleary s learly mple by he olear seo orer aal sra. ol EA' : N l l N ol Ra-5.35 / 4 / JN 647
25 . Geomeral oleares e eleme aalyss Fe eleme ormlao o a aally loae geomerally olear elas bar problem. As sal we sar rom he wea orm ere aboe N ' ' a : b EA' N EAε EA ' ' ' ' ' EA' b : l ' Ra-5.35 / 4 / JN 648
26 649 Fe eleme ormlao o a aally loae geomerally olear elas bar problem. As sal we sar rom he wea orm ere aboe a he hoose he e eleme ral a es os sasyg he esseal boary oos as sal: : ' ' ' ' : ' ' ' ' ' ' l b EA a EA EA EAε N b N. h h. Geomeral oleares e eleme aalyss Ra-5.35 / 4 / JN
27 . Geomeral oleares e eleme aalyss hs resls aer a ew seps ome here a eqao sysem K wh he ore eor ompable or = a he splaeme eor ow or = as beore or he lear ase b Ra-5.35 / 4 / JN 65
28 . Geomeral oleares e eleme aalyss hs resls aer a ew seps ome here a eqao sysem K wh he ore eor ompable or = a he splaeme eor ow or = as beore or he lear ase b whle he sess mar ompable or = ow epes o he ows: K K K Remar. he sess mar s osymmer whle s eres are sll oerae a arrow agoal ba ormg a ba mar e o loally sppore ral a es os. Ra-5.35 / 4 / JN 65
29 . Geomeral oleares e eleme aalyss hs resls aer a ew seps ome here a eqao sysem K wh he ore eor ompable or = a he splaeme eor ow or = as beore or he lear ase b whle he sess mar ompable or = ow epes o he ows: K K K Remar. he sess mar s osymmer whle s eres are sll oerae a arrow agoal ba ormg a ba mar e o loally sppore ral a es os. Remar. he sysem a o be sole by saar mar erse mehos whh s he mos sga eree bewee lear a olear FE mehos. Ra-5.35 / 4 / JN 65
30 . Geomeral oleares e eleme aalyss By ag a loser loo o he eral eergy beh he sess mar a be see as a eomposo o he symmer lear par a a ople o olear pars o ere orers: a EA' EA' ' EA ' ' ' 3 ' ' EA ' 3 ' Ra-5.35 / 4 / JN 653
31 . Geomeral oleares e eleme aalyss By ag a loser loo o he eral eergy beh he sess mar a be see as a eomposo o he symmer lear par a a ople o olear pars o ere orers: a K EA' ' EA K K l EA' K K ' ' ' 3 3 ' ol ' EA ' 3 ' Ra-5.35 / 4 / JN 654
32 . Geomeral oleares e eleme aalyss By ag a loser loo o he eral eergy beh he sess mar a be see as a eomposo o he symmer lear par a a ople o olear pars o ere orers: a K EA' ' EA K K l EA' K K ' ' ' 3 3 ' ol ' EA ' ' Remar. I D a 3D problems he olear sess par wll be mh more omplae ha he rre D moel problem alhogh he prples are he same. Remar. I he e we shall o be erese he sess mar he orm aboe raher ha s learzao so alle ageal sess mar. 3 Ra-5.35 / 4 / JN 655
33 .3 Maeral oleares e eleme aalyss Aally loae maerally olear elas bar problem. e s sar rom he geeral ormla o he prple o ral wor or eormable boes applable or boh lear a olear om heores: W V EAεε Nε σ : ε V S b b S N V b V E Aε E W l e ' A' : N l N ol Ra-5.35 / 4 / JN 656
34 .3 Maeral oleares e eleme aalyss Aally loae maerally olear elas bar problem. e s sar rom he geeral ormla o he prple o ral wor or eormable boes applable or boh lear a olear om heores: W se V EAεε Nε σ : ε V ε ' a N ' b b S S b N ε ' V ' b V E Aε E W l e ' A' : N l N ol Ra-5.35 / 4 / JN 657
35 .3 Maeral oleares e eleme aalyss Iegrag by pars ges he orm aalog o he lear ase N' N N b a ally he ereal eqao a boary oos N' ; N E Aε E N or l ' A' : N l N ol 3 N or. Ra-5.35 / 4 / JN 658
36 .3 Maeral oleares e eleme aalyss Iegrag by pars ges he orm aalog o he lear ase N' N N b a ally he ereal eqao a boary oos 3 N' N N ; or or N E Aε E. whh a be ompare o he lear ase whh oly he ormal ore s ere e o he sra-sress relao lear ol N N E A ε E E A ε Remar. Noleary s learly mple by he olear aal sress. l E l ' A' : N A' : N l l l ' A' : N l N ol N ol Ra-5.35 / 4 / JN 659
37 .3 Maeral oleares e eleme aalyss Fe eleme ormlao o a aally loae maerally olear elas bar problem. As sal we sar rom he wea orm ere aboe N ' a : b N E Aε EA' ' ' E l ' A' : N b : l l N ol Ra-5.35 / 4 / JN 66
38 66 Fe eleme ormlao o a aally loae maerally olear elas bar problem. As sal we sar rom he wea orm ere aboe a he hoose he e eleme ral a es os sasyg he esseal boary oos as sal: : ' ' ' : : ' ' ' l b EA a N N A E Aε E N b N ol l l h h.3 Maeral oleares e eleme aalyss Ra-5.35 / 4 / JN
39 .3 Maeral oleares e eleme aalyss hs resls aer a ew seps ome here a eqao sysem K wh he ore eor ompable or = a he splaeme eor ow or = as beore or he lear ase b whle he sess mar ompable or = ow epes o he ows: K K K Ra-5.35 / 4 / JN 66
40 .3 Maeral oleares e eleme aalyss hs resls aer a ew seps ome here a eqao sysem K wh he ore eor ompable or = a he splaeme eor ow or = as beore or he lear ase b whle he sess mar ompable or = ow epes o he ows: K K K Remar. he sess mar s osymmer whle s eres are sll oerae a arrow agoal ba ormg a ba mar e o loally sppore ral a es os. Remar. he sysem aboe a o be sole by saar mar erse mehos whh s he mos sga eree bewee lear a olear FE mehos. Ra-5.35 / 4 / JN 663
41 .3 Maeral oleares e eleme aalyss By ag a loser loo o he eral eergy beh he sess mar a be see as a eomposo o he symmer lear par a a ople o olear pars o ere orers: a K K l EA' ' ' K EA' ' EA ' K ol ' Ra-5.35 / 4 / JN 664
42 .3 Maeral oleares e eleme aalyss By ag a loser loo o he eral eergy beh he sess mar a be see as a eomposo o he symmer lear par a a ople o olear pars o ere orers: a K K l EA' ' ' K EA' ' EA ' K ol ' Remar. Maeral oleares a be ase by olear elasy olear elas maeral behaor as elaso-plasy so-plasy e. as well. Remar. I parlar elas behaor meas ere loa-respose pah or loag a loag phases pah epeee whle or elas behaor hese pahs are eal b wh oppose reos. hs ms be ae o ao he solo algorhms. Ra-5.35 / 4 / JN 665
43 .4 earzao a solo algorhms Solg a olear eqao sysem a se o olear algebra eqaos as K a be aomplshe by sg a Newo Raphso erao whh oles learzaos o he eqlbrm eqaos reqrg reoal eraes. Frs he loa s ge wh a salar mlpler a a reeree loa as re. Ra-5.35 / 4 / JN 666
44 .4 earzao a solo algorhms Solg a olear eqao sysem a se o olear algebra eqaos as K a be aomplshe by sg a Newo Raphso erao whh oles learzaos o he eqlbrm eqaos reqrg reoal eraes. Frs he loa s ge wh a salar mlpler a a reeree loa as re. Seo he sysem s wre a geeral orm as g : r : K re wh r eog he eral ore eor epeg o he ow eor. re Ra-5.35 / 4 / JN 667
45 .4 earzao a solo algorhms Solg a olear eqao sysem a se o olear algebra eqaos as K a be aomplshe by sg a Newo Raphso erao whh oles learzaos o he eqlbrm eqaos reqrg reoal eraes. Frs he loa s ge wh a salar mlpler a a reeree loa as re. Seo he sysem s wre a geeral orm as g : r : K re wh r eog he eral ore eor epeg o he ow eor. g g g g. hr he eqao s learze a a eqalbrm po : Remar. Appromae eqaly s al he eghborhoo o po. re Ra-5.35 / 5 / JN 668
46 669 Dreoal eraes appearg he learze orm are smply where he age sess mar wll play a esseal role he Newo Raphso erao base o he learze sysem whh mples he Newo Raphso erao seps as ollows: : re g g K r re re re re g g r K r K.4 earzao a solo algorhms K Ra-5.35 / 4 / JN
47 67 Dreoal eraes appearg he learze orm are smply where he age sess mar wll play a esseal role he Newo Raphso erao base o he learze sysem whh mples he Newo Raphso erao seps as ollows: : re g g K r g g sop sop re sop re re re re re or he e reme Se or whh l Ierae wh r r K r K r K.4 earzao a solo algorhms K Ra-5.35 / 4 / JN
48 .4 earzao a solo algorhms Choose Ireme seps...: a. Choose he e loa reme b. Ierao seps...; Choose olerae : b. Se b. For ompe b.3 Compe b.4 Upae sh ha K r Sole Compe r b.5 Che he oergee es ssase: ; g ; oe wh ; a se K r re r re I I re se sop o rer ba o b.. sop a sar he e reme a a. Ra-5.35 / 4 / JN 67
49 .4 earzao a solo algorhms Remar. I so alle moe Newo Raphso meho he age sess mar s o pae or eah erae sep oly or eah remeal sep. Newo Raphso erao Moe Newo Raphso erao Ra-5.35 / 4 / JN 67
50 .4 earzao a solo algorhms Remar. I so alle ar legh mehos or Rs mehos a aoal osra s apple or eraos orer o pree sap-hrogh ear lm pos rom A o A a rom B o B he gre o he rgh. Newo Raphso erao Ar legh meho Remar. I erae solo algorhms apple o problems olg elas oleares he pah-epeee o he ose mar hae o be ae o ao sep b.4 o he Newo Raphso erao aboe: Aoher erao or some oher ehqe s apple or ompg sress remes rom sra remes orer o say o he yel srae a o ollow he low rle. Ra-5.35 / 4 / JN 673
51 QUESIONS? ANSWERS ECURE BREAK!
52 Fe eleme mehos or me-epee problems e s sar wh some smlao eamples: Cg proess hp:// Bar braos l hp://
53 Ra-54.3 Nmeral Mehos Srral Egeerg Coes. Moellg prples a boary ale problems egeerg sees. Eergy mehos a bas D e eleme mehos - bars/ros beams hea so seepage elerosas 3. Bas D a 3D e eleme mehos - hea so seepage 4. Nmeral mplemeao ehqes o e eleme mehos 5. Absra ormlao a aray o e eleme mehos 6. Fe eleme mehos or Eler Beroll beams 7. Fe eleme mehos or mosheo beams 8. Fe eleme mehos or Krhho oe plaes 9. Fe eleme mehos or Resser Ml plaes. Fe eleme mehos or D a 3D elasy. Era lere: Noleares e eleme aalyss. Era lere: Fe eleme mehos or me-epee problems Ra-54.3 / 6 / JN 678
54 Fe eleme mehos or me-epee problems Coes. Parabol problems. Hyperbol problems 3. me egrao algorhms earg oome A. Ably o ere a apply e eleme mehos or me-epee problems B. Ably o lze a mpleme meral me egrao shemes or FEM Reerees ere oes: hapers 7 8 e boo: hapers A7.I 8 9 Ra-54.3 / 4 / JN 679
55 . All lse ompaoal moellg? I whh o saos oe has o ae o ao me-epee pheomea FE aalyss? Ra-54.3 / 4 / JN 68
56 . Parabol problems D moel problem Hea so oo problem D: e s oser oe-mesoal hea so. Forer law bls a ose relao bewee he hea l q a he emperare hrogh he hermal oy as q or D / 3D : q Ra-54.3 / 4 / JN 68
57 68 Hea so oo problem D: e s oser oe-mesoal hea so. Forer law bls a ose relao bewee he hea l q a he emperare hrogh he hermal oy as he rs law o hermoyams or he prple o oserao o eergy who he saoary sae assmpo mples he ollowg so eqao esrbg he problem or he hea sore wh he hea apay a mass esy ρ: q q. Parabol problems D moel problem Ra-54.3 / 4 / JN
58 683 Hea so oo problem D: e s oser oe-mesoal hea so. Forer law bls a ose relao bewee he hea l q a he emperare hrogh he hermal oy as he rs law o hermoyams or he prple o oserao o eergy who he saoary sae assmpo mples he ollowg so eqao esrbg he problem or he hea sore wh he hea apay a mass esy ρ: Remar. Cosolao o sols ollows he same eqao by replag he emperare by he pore pressre a he oees aorgly. q q. Parabol problems D moel problem Ra-54.3 / 4 / JN
59 . Parabol problems D moel problem Eler eqaos srog rom o he al-boary ale problem are he ollowg: a ' ' b q 3 q Ra-54.3 / 4 / JN 684
60 . Parabol problems D moel problem Eler eqaos srog rom o he al-boary ale problem are he ollowg: 3 emperare ow o o a q a hermal oy hea apay mass esy ge maeral aa hea spply ge loag aa - oma ge geomeral aa emperare o he boary ge esseal Drhle boary aa hea l o he boary ge aral Nema boary aa emperare o he boary ge esseal Drhle boary aa al emperare ge al aa ' ' b q q - oma ge aa. Ra-54.3 / 4 / JN 685
61 686. Mlply he ereal eqao by a smooh es o spee laer epeg o oly: ' ' ' '. Parabol problems wea orm Ra-54.3 / 4 / JN
62 687. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: ' ' ' ' ' '. Parabol problems wea orm Ra-54.3 / 4 / JN
63 688. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: ' ' ' ' ' ' ' ' ' '. Parabol problems wea orm Ra-54.3 / 4 / JN
64 689. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: 4. Ulze he aral boary oo b: ' ' ' ' ' ' ' ' ' ' ' q q. Parabol problems wea orm Ra-54.3 / 4 / JN
65 69. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: 4. Ulze he aral boary oo b: 5. Se a zero esseal boary oo a or he es o: ' ' ' ' ' ' ' ' ' ' ' q q. Parabol problems wea orm Ra-54.3 / 4 / JN
66 . Parabol problems wea orm ' ' q Ra-54.3 / 4 / JN 69
67 69 6. Mlply he al oo 3 by he es o a maeral osas a egrae oer he oma:. Parabol problems wea orm ' ' q Ra-54.3 / 4 / JN
68 Mlply he al oo 3 by he es o a maeral osas a egrae oer he oma: Wea orm. F sh ha sases a or all sasyg.. Parabol problems wea orm q b ' ' a ' ' q Ra-54.3 / 4 / JN
69 . Parabol problems wea orm Remar. he solo a he es o respeely hae o sasy he esseal boary oos a he reglary oos o he orm H ; orer o be emaally amssble. H Ra-54.3 / 4 / JN 694
70 695 Remar. he solo a he es o respeely hae o sasy he esseal boary oos a he reglary oos o he orm orer o be emaally amssble. Remar. me-epeey s sll prese he ormlao:. Parabol problems wea orm ; H H Ra-54.3 / 4 / JN b ' ' a q
71 . Parabol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e e e Ra-54.3 / 4 / JN 696
72 697. De he solo eral oma o sberals elemes wh oes a he eleme sze :. Choose a ral o or he e eleme appromao as a separae sm e e e e h h. Parabol problems e elemes Ra-54.3 / 4 / JN
73 . Parabol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e. Choose a ral o or he e eleme appromao as a separae sm h wh sable loal bass os e o some polyomal orer ow lear e Ra-54.3 / 4 / JN 698
74 . Parabol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e. Choose a ral o or he e eleme appromao as a separae sm h o some polyomal orer ow lear wh sable loal bass os e e he egrees o reeom h epe ow o me. Ra-54.3 / 4 / JN 699
75 . Parabol problems e elemes Esre ha he ral o sases he esseal boary oos: h Ra-54.3 / 4 / JN 7
76 7 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: h. Parabol problems e elemes Ra-54.3 / 4 / JN
77 7 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: 4. Iser he os ral a es o he wea orm: h ' ' a q. Parabol problems e elemes Ra-54.3 / 4 / JN
78 73 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: 4. Iser he os ral a es o he wea orm:... ' ' ' ' a q h. Parabol problems e elemes Ra-54.3 / 4 / JN
79 74 q ' '. Parabol problems e elemes Ra-54.3 / 4 / JN
80 75 q ' '. Parabol problems e elemes b Ra-54.3 / 4 / JN
81 76 Wh he mass mar sess mar ompable or = ore eor ompable or = a he splaeme eor ow or = q K K M M ' ' M K M q ' '. Parabol problems e elemes b Ra-54.3 / 4 / JN
82 . Parabol problems e elemes hs resls a smple eqao sysem M K Ra-54.3 / 4 / JN 77
83 . Parabol problems e elemes hs resls a smple eqao sysem M K Remar. he apay or mass mar M s symmer a pose-ee. Ra-54.3 / 4 / JN 78
84 . Parabol problems e elemes hs resls a smple eqao sysem M K Remar. he apay or mass mar M s symmer a pose-ee. Remar. hs s a semsree ormlao se me-epeey s sll oosly prese he sysem hrogh he me epee eors a. Ra-54.3 / 4 / JN 79
85 . Parabol problems e elemes hs resls a smple eqao sysem M K Remar. he apay or mass mar M s symmer a pose-ee. Remar. hs s a semsree ormlao se me-epeey s sll oosly prese he sysem hrogh he me epee eors a. Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o rs orer wh he ge al ale 3 o he orgal problem seg. Ra-54.3 / 4 / JN 7
86 . Parabol problems e elemes hs resls a smple eqao sysem M K Remar. he apay or mass mar M s symmer a pose-ee. Remar. hs s a semsree ormlao se me-epeey s sll oosly prese he sysem hrogh he me epee eors a. Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o rs orer wh he ge al ale 3 o he orgal problem seg. Remar. I a olear ase he sysem eqao wol hae he orm or more geerally M K M g. Ra-54.3 / 4 / JN 7
87 . Parabol problems e elemes Brea eerse Sole he ree osllaos o a mass-sprg-amper m sysem moelle by a lear seo orer orary ereal eqao wh osa oees he orm m e H: Use he ral o or erg he harasers eqao a sgsh he ases II ^ < 4m wo omple ogae roos erampg III ^ = 4m a real ople roo ral ampg I ^ > 4m wo s real roos oerampg.. Ra-54.3 / 4 / JN 7
88 . Hyperbol problems e s reall ha he sa eqlbrm or ore balae or a boy rom he Eler s laws or momem prples eqale geeralzaos o Newo s laws: Prple o lear momem a be wre he orm b S S R wh spaal rre ograo boy loa b = b srae rao = mass esy ρ = ρ a spaal eloy el =. 3 ollows Ra-54.3 / 4 / JN 73
89 . Hyperbol problems e s reall ha he sa eqlbrm or ore balae or a boy rom he Eler s laws or momem prples eqale geeralzaos o Newo s laws: Prple o lear momem a be wre he orm b S S wh spaal rre ograo boy loa b = b srae rao = mass esy ρ = ρ a spaal eloy el =. Cahy s law a Gass ergee heorem mply he orm rom whh oe obas he eqao o moo: b b S S S σ b σ b. σ S R 3 ollows Ra-54.3 / 4 / JN 74
90 . Hyperbol problems D moel problem Elasoyams o a aally loae ro: Eler eqaos are ow o he orm a b A E A ' N N ' 3a 3b b Ra-54.3 / 4 / JN 75
91 . Hyperbol problems D moel problem Elasoyams o a aally loae ro: Eler eqaos are ow o he orm a b A N N 3a A ross-seoal area; legh; E A ' ' b 3b aal splaeme ow o o a E Yog's mols mass esy ge maeral aa b N al splaeme al eloy ge al aa. - oma ge geomeral aa aal boy loa ge loag aa aal e po splaeme ge esseal/geomer boary aa aal e po ore ge aral/ore boary aa Ra-54.3 / 4 / JN 76
92 77. Mlply he ereal eqao by a smooh es o spee laer epeg o oly: ' ' ' ' b A E A b A E A. Hyperbol problems wea orm Ra-54.3 / 4 / JN
93 78. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: ' ' ' ' b A E A b A E A b EA A ' '. Hyperbol problems wea orm Ra-54.3 / 4 / JN
94 79. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' '. Hyperbol problems wea orm Ra-54.3 / 4 / JN
95 7. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: 4. Ulze he aral boary oo b: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' ' ' N N EA. Hyperbol problems wea orm Ra-54.3 / 4 / JN
96 7. Mlply he ereal eqao by a smooh es o spee laer epeg o oly:. Iegrae oer he -oma eral: 3. Iegrae by pars he le ha se: 4. Ulze he aral boary oo b: 5. Se a zero esseal boary oo a or he es o: ' ' ' ' b A E A b A E A b EA A ' ' b EA A EA EA ' ' ' ' ' N N EA. Hyperbol problems wea orm Ra-54.3 / 4 / JN
97 . Hyperbol problems wea orm A EA ' ' b N Ra-54.3 / 4 / JN 7
98 73 6. Mlply he al oos 3a a 3b by he es o a maeral osas a egrae oer he oma:. Hyperbol problems wea orm ' ' N b EA A Ra-54.3 / 4 / JN A A A A
99 74 6. Mlply he al oos 3a a 3b by he es o a maeral osas a egrae oer he oma: Wea orm. F sh ha sases a. Hyperbol problems wea orm ' ' N b EA A. sasyg b ' ' a A A A A N b EA A A A A A Ra-54.3 / 4 / JN
100 . Hyperbol problems wea orm Remar. he solo a he es o respeely hae o sasy he esseal boary oos a he reglary oos he orm H ; orer o be emaally amssble. H Ra-54.3 / 4 / JN 75
101 76 Remar. he solo a he es o respeely hae o sasy he esseal boary oos a he reglary oos he orm orer o be emaally amssble. Remar. me-epeey s sll prese he ormlao:. Hyperbol problems wea orm ; H H Ra-54.3 / 4 / JN. sasyg b ' ' a A A A A N b A E A
102 . Hyperbol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e e e Ra-54.3 / 4 / JN 77
103 78. De he solo eral oma o sberals elemes wh oes a he eleme sze :. Choose a ral o or he e eleme appromao as a separae sm e e e e Ra-54.3 / 4 / JN h h. Hyperbol problems e elemes
104 . Hyperbol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e. Choose a ral o or he e eleme appromao as a separae sm h wh sable loal bass os e o some polyomal orer ow lear e Ra-54.3 / 4 / JN 79
105 . Hyperbol problems e elemes. De he solo eral oma o sberals elemes wh oes a he eleme sze : h e e. Choose a ral o or he e eleme appromao as a separae sm h o some polyomal orer ow lear wh sable loal bass os e e he egrees o reeom h epe ow o me. Ra-54.3 / 4 / JN 73
106 . Hyperbol problems e elemes Esre ha he ral o sases he esseal boary oos: h Ra-54.3 / 4 / JN 73
107 73 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: h. Hyperbol problems e elemes Ra-54.3 / 4 / JN
108 733 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: 4. Iser he os ral a es o he wea orm: h ' ' a N b EA A. Hyperbol problems e elemes Ra-54.3 / 4 / JN
109 734 Esre ha he ral o sases he esseal boary oos: 3. Choose a es o o a smlar orm Galer meho wh he orrespog oo: 4. Iser he os ral a es o he wea orm:... ' ' ' ' a EA A N b EA A h. Hyperbol problems e elemes Ra-54.3 / 4 / JN
110 735 N b EA A ' '. Hyperbol problems e elemes Ra-54.3 / 4 / JN
111 736 N b EA A ' '. Hyperbol problems e elemes A A A A b Ra-54.3 / 4 / JN
112 737 N b EA A ' '. Hyperbol problems e elemes A A A A A A A A b Ra-54.3 / 4 / JN
113 738 N b EA A ' '. Hyperbol problems e elemes A A A A A A A A b Ra-54.3 / 4 / JN Wh he mass mar sess mar ompable or = ore eor ompable or = a he splaeme eor ow or =
114 739 hs resls a smple eqao sysem K M. Hyperbol problems e elemes Ra-54.3 / 4 / JN A A N b EA K K A M M : : : : : ' ' : : : : M M K M
115 74 hs resls a smple eqao sysem Remar. he mass mar M s symmer a pose-ee. K M. Hyperbol problems e elemes Ra-54.3 / 4 / JN A A N b EA K K A M M : : : : : ' ' : : : : M M K M
116 74 hs resls a smple eqao sysem Remar. he mass mar M s symmer a pose-ee. Remar. hs s a semsree ormlao se me-epeey s sll oosly prese he sysem hrogh he me epee eors a. K M. Hyperbol problems e elemes Ra-54.3 / 4 / JN A A N b EA K K A M M : : : : : ' ' : : : : M M K M
117 . Hyperbol problems e elemes Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o seo orer wh he ge al ales 3 o he orgal problem seg. Ra-54.3 / 4 / JN 74
118 . Hyperbol problems e elemes Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o seo orer wh he ge al ales 3 o he orgal problem seg. Remar. I a olear ase he sysem wol hae he orm M K or more geerally M g. Ra-54.3 / 4 / JN 743
119 . Hyperbol problems e elemes Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o seo orer wh he ge al ales 3 o he orgal problem seg. Remar. I a olear ase he sysem wol hae he orm M K or more geerally M g. Remar. I ase o ampe braos sos ampg mar C s le he sysem as M C K. Oe so alle Raylegh ampg mar ampg pheomea wh some osas α a β. C M K s se o esrbe he Ra-54.3 / 4 / JN 744
120 . Hyperbol problems e elemes Remar. he sreze eqao sysem s a ople sysem o orary ereal eqaos o seo orer wh he ge al ales 3 o he orgal problem seg. Remar. I a olear ase he sysem wol hae he orm M K or more geerally M g. Remar. I ase o ampe braos sos ampg mar C s le he sysem as M C K. Oe so alle Raylegh ampg mar ampg pheomea wh some osas α a β. M K s se o esrbe he Remar. I may ases a agoal lmpe mass mar s se o appromae he orgal osse mass mar orer o smply he eqao solo phase. C Ra-54.3 / 4 / JN 745
121 .3 me egrao algorhms Geeralze rapezoal amly o me egrao mehos or parabol problems. For he al ale problem problem M K perhaps he mos well ow a ommoly se solo algorhms are members o he geeralze rapezoal amly ossg o me seps ollowg he eqaos M K Ra-54.3 / 4 / JN 746
122 .3 me egrao algorhms Geeralze rapezoal amly o me egrao mehos or parabol problems. For he al ale problem problem M K perhaps he mos well ow a ommoly se solo algorhms are members o he geeralze rapezoal amly ossg o me seps ollowg he eqaos M K Remar. Dere ales or parameer α ge ere shemes: α = s alle orwar erees or orwar Eler; α = ½ rapezoal rle mpo rle or Cra Nholso; α = bawar erees or bawar Eler. Remar. Dere mplemeaos es or mehos o hs ype. Ra-54.3 / 4 / JN 747
123 .3 me egrao algorhms Geeralze rapezoal me egrao algorhm a be wre as ollows: a. b. Sole M or he al ale K me seps... or he me reme : ~ b. Compe preor b. Sole rom eqao ~ M K K ~ b.3 Compe : a rer o b.. Ra-54.3 / 4 / JN 748
124 .3 me egrao algorhms Geeralze rapezoal me egrao algorhm a be wre as ollows: a. b. Sole M or he al K ale me seps... or he me reme : ~ b. Compe preor b. Sole rom eqao ~ M K K ~ b.3 Compe a rer o b.. Remar. Dere ales or parameer α ge ere meho ypes: or α = he mehos s alle epl oherwse mpl whle mpl-epl mehos par o K s reae mplly a par eplly. Remar. Wh a agoal M sep b. s sraghorwar or a epl meho. : Ra-54.3 / 4 / JN 749
125 75 Newmar amly o me egrao mehos or hyperbol a parabolhyperbol problems. For he al ale problem problem perhaps he mos wely se solo algorhms are members o he Newmar amly ossg o me seps ollowg he eqaos le s eoe.3 me egrao algorhms K C M a a a a K M Ma Ra-54.3 / 4 / JN a : :
126 .3 me egrao algorhms Newmar amly o me egrao mehos or hyperbol a parabolhyperbol problems. For he al ale problem problem M C K perhaps he mos wely se solo algorhms are members o he Newmar amly ossg o me seps ollowg he eqaos le s eoe Ma M K a a a : a : Remar. Dere ales or parameer β a γ ge shemes wh ere sably ooally sable ooally sable a aray haraserss. a Ra-54.3 / 4 / JN 75
127 75 Newmar me egrao algorhm a be wre as ollows:.3 me egrao algorhms a rer o b.. ~ a ~ Compe b.3 ~ ~ rom Sole b. ~ ~ Compe preors b. : or me reme me seps b. : a ales or he al Sole a.... a a K C a K C M a a a K C Ma a Ra-54.3 / 4 / JN
128 .3 me egrao algorhms Remar. I so alle mlsep mehos arables or he e me sep are ompe by ag o ao o oly he preos sep a b some o he oher preeeg seps as well. a a Ra-54.3 / 4 / JN 753
129 Frher lerare or e eleme mehos mehas a srral egeerg Johso C.: Nmeral Solo o Paral Dereal Eqaos by he Fe Eleme Meho Bahe K.-J. & Chapelle D.: he Fe Eleme Aalyss o Shells: Fameals Zeewz O. C. & aylor R..: he Fe Eleme Meho: Is Bass a Fameals Zeewz O. C. & aylor R..: he Fe Eleme Meho or Sol a Srral Mehas Carle P. G.: he Fe Eleme Meho or Ellp Problems Asworh M. & Oe J..: A poseror Error Esmao Fe Eleme Aalyss New MS leel orses ompaoal mehas a Aalo Uersy Egeerg Compaos II / 6 7 Fe Eleme Mehos Cl Egeerg V / 6 7
130 HANK YOU FOR FOOWING HE COURSE PEASE GIVE FEEDBACK! HE END
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