Chape 8.7 Fne Dffeence Mehod fo Odnay Dffeenal Eqaons Afe eadng hs chape, yo shold be able o. Undesand wha he fne dffeence mehod s and how o se o solve poblems. Wha s he fne dffeence mehod? The fne dffeence mehod s sed o solve odnay dffeenal eqaons ha have condons mposed on he bonday ahe han a he nal pon. These poblems ae called bonday-vale poblems. In hs chape, we solve second-ode odnay dffeenal eqaons of he fom d y f ( x, y, y' ), a x b, () dx wh bonday condons y ( a) y a and y ( b) yb () Many academcs efe o bonday vale poblems as poson-dependen and nal vale poblems as me-dependen. Tha s no necessaly he case as llsaed by he followng examples. The dffeenal eqaon ha govens he deflecon y of a smply sppoed beam nde nfomly dsbed load (Fge ) s gven by d y qx( L x) () dx EI whee x locaon along he beam (n) E Yong s modls of elascy of he beam (ps) I second momen of aea (n ) q nfom loadng nensy (lb/n) L lengh of beam (n) The condons mposed o solve he dffeenal eqaon ae y ( x ) () y ( x L) Clealy, hese ae bonday vales and hence he poblem s consdeed a bonday-vale poblem. 8.7.
8.7. Chape 8.7 y q x L Fge Smply sppoed beam wh nfom dsbed load. Now consde he case of a canleveed beam wh a nfomly dsbed load (Fge ). The dffeenal eqaon ha govens he deflecon y of he beam s gven by d y q( L x) () dx EI whee x locaon along he beam (n) E Yong s modls of elascy of he beam (ps) I second momen of aea (n ) q nfom loadng nensy (lb/n) L lengh of beam (n) The condons mposed o solve he dffeenal eqaon ae y ( x ) (6) dy ( x ) dx Clealy, hese ae nal vales and hence he poblem needs o be consdeed as an nal vale poblem. y q x L Fge Canleveed beam wh a nfomly dsbed load.
Fne Dffeence Mehod 8.7. Example The deflecon y n a smply sppoed beam wh a nfom load q and a ensle axal load T s gven by d y Ty qx( L x) (E.) dx EI EI whee x locaon along he beam (n) T enson appled (lbs) E Yong s modls of elascy of he beam (ps) I second momen of aea (n ) q nfom loadng nensy (lb/n) L lengh of beam (n) y q T x T L Fge Smply sppoed beam fo Example. Gven, T 7 lbs, q lbs/n, L 7 n, E Ms, and I n, a) Fnd he deflecon of he beam a x ". Use a sep sze of x " and appoxmae he devaves by cenal dvded dffeence appoxmaon. b) Fnd he elave e eo n he calclaon of y (). Solon a) Sbsng he gven vales, d y 7y () x(7 x) 6 6 dx ( )() ( )() d y y 7. x(7 x) (E.) dx d y Appoxmang he devave a node by he cenal dvded dffeence dx appoxmaon,
8.7. Chape 8.7 Fge Illsaon of fne dffeence nodes sng cenal dvded dffeence mehod. d y y y y dx ( x) We can ewe he eqaon as y y y y 7. (7 ) x x ( x) Snce x, we have nodes as gven n Fge (E.) (E.) x x x x 7 Fge Fne dffeence mehod fom x o x 7 wh x. The locaon of he nodes hen s x x x x x x x x x x 7 Wng he eqaon a each node, we ge Node : Fom he smply sppoed bonday condon a x, we oban y (E.) Node : Rewng eqaon (E.) fo node gves y y y y 7. x (7 x ) ().6y.y.6y 7. ()(7 ).6y.y.6y 9.7 (E.6) Node : Rewng eqaon (E.) fo node gves y y y y 7. x(7 x ) ().6y.y.6y 7. ()(7 ).6y.y.6y 9.7 (E.7) Node : Fom he smply sppoed bonday condon a x 7, we oban y (E.8)
Fne Dffeence Mehod 8.7. Eqaons (E.-E.8) ae smlaneos eqaons wh nknowns and can be wen n max fom as y.6..6 y 9.7.6..6 y 9.7 y The above eqaons have a coeffcen max ha s dagonal (we can se Thomas algohm o solve he eqaons) and s also scly dagonally domnan (convegence s gaaneed f we se eave mehods sch as he Gass-Sedel mehod). Solvng he eqaons we ge, y y.8 y.8 y y ) y( x ) y.8" ( The exac solon of he odnay dffeenal eqaon s deved as follows. The homogeneos pa of he solon s gven by solvng he chaacesc eqaon m m ±. Theefoe,.x.x yh Ke K e The pacla pa of he solon s gven by y p Ax Bx C Sbsng he dffeenal eqaon (E.) gves d y p y 7. x(7 x) p dx d ( Ax Bx C) ( Ax Bx C) 7. x(7 x) dx A ( Ax Bx C) 7. x(7 x) Ax Bx (A C).6 x 7. x Eqang ems gves A. B.6 6 A C Solvng he above eqaon gves A.7 B 8. C.7
8.7.6 Chape 8.7 The pacla solon hen s.7x 8.x.7 y p The complee solon s hen gven by.x.x y.7x 8.x.7 Ke K e Applyng he followng bonday condons y ( x ) y ( x 7) we oban he followng sysem of eqaons K K.7.9K.8997K.7 These eqaons ae epesened n max fom by K.7.9.8997 K.7 A nmbe of dffeen nmecal mehods may be lzed o solve hs sysem of eqaons sch as he Gassan elmnaon. Usng any of hese mehods yelds K.77666 K.9777 Sbsng hese vales back no he eqaon gves.x.x y.7x 8.x.7.776666 e.9777 e Unlke ohe examples n hs chape and n he book, he above expesson fo he deflecon of he beam s dsplayed wh a lage nmbe of sgnfcan dgs. Ths s done o mnmze he ond-off eo becase he above expesson nvolves sbacon of lage nmbes ha ae close o each ohe. b) To calclae he elave e eo, we ms fs calclae he vale of he exac solon a y..() y ( ).7() 8.().7.776666 e.().9777 e y ( ). The e eo s gven by E Exac Vale Appoxmae Vale E. (.8) E. The elave e eo s gven by Te Eo % Te Vale. %. %
Fne Dffeence Mehod 8.7.7 Example Take he case of a pesse vessel ha s beng esed n he laboaoy o check s ably o whsand pesse. Fo a hck pesse vessel of nne ads a and oe ads b, he dffeenal eqaon fo he adal dsplacemen of a pon along he hckness s gven by d d d d (E.) The nne ads a and he oe ads b 8, and he maeal of he pesse vessel s ASTM A6 seel. The yeld sengh of hs ype of seel s 6 ks. Two san gages ha ae bonded angenally a he nne and he oe ads mease nomal angenal san as / a.776 / b.86 (E.a,b) a he maxmm needed pesse. Snce he adal dsplacemen and angenal san ae elaed smply by, (E.) hen.776.87' ' a b.86 8.769' ' The maxmm nomal sess n he pesse vessel s a he nne ads E d σ max ν ν a d a whee E Yong s modls of seel (E Ms) ν Posson s ao ( ν.) The faco of safey, FS s gven by Yeld sengh of seel σ max a and s gven by (E.7) FS (E.8) a) Dvde he adal hckness of he pesse vessel no 6 eqdsan nodes, and fnd he adal dsplacemen pofle b) Fnd he maxmm nomal sess and faco of safey as gven by eqaon (E.8) c) Fnd he exac vale of he maxmm nomal sess as gven by eqaon (E.8) f s gven ha he exac expesson fo adal dsplacemen s of he fom C C. Calclae he elave e eo.
8.7.8 Chape 8.7 Solon a b - n a - b Fge Nodes along he adal decon. a) The adal locaons fom a o b ae dvded no n eqally spaced segmens, and hence eslng n n nodes. Ths wll allow s o fnd he dependen vaable nmecally a hese nodes. A node along he adal hckness of he pesse vessel, d (E.9) d ( ) d (E.) d Sch sbsons wll conve he odnay dffeenal eqaon no a lnea eqaon (b wh moe han one nknown). By wng he eslng lnea eqaon a dffeen pons a whch he odnay dffeenal eqaon s vald, we ge smlaneos lnea eqaons ha can be solved by sng echnqes sch as Gassan elmnaon, he Gass-Sedel mehod, ec. Sbsng hese appoxmaons fom Eqaons (E.9) and (E.) n Eqaon (E.) (E.) ( ) ( ) ( ) ( ) (E.) Le s beak he hckness, b a, of he pesse vessel no n nodes, ha s a s node and b s node n. Tha means we have n nknowns. We can we he above eqaon fo nodes,..., n. Ths wll gve s n eqaons. A he edge nodes, and n, we se he bonday condons of
Fne Dffeence Mehod 8.7.9 a n b Ths gves a oal of n eqaons. So we have n nknowns and n lnea eqaons. These can be solved by any of he nmecal mehods sed fo solvng smlaneos lnea eqaons. We have been asked o do he calclaons fo n, ha s a oal of 6 nodes. Ths gves b a n 8.6 " A node, a ",.87 " (E.) A node,.6.6" (E.).6.6 (.6)(.6) (.6).6 (.6)(.6).7778.88.7 (E.) A node,.6.6 6.".6.6 ( 6.)(.6) 6..6 ( 6.)(.6).7778.8.66 (E.6) A node, A node, A node, 6..6 6.8".6.6 ( 6.8)(.6) 6.8.6 ( 6.8)(.6).7778.8.9 (E.7) 6.8.6 7..6.6 ( 7.)(.6) ( 7.).6 ( 7.)(.6).7778.799. (E.8) 7..6 8.769 b (E.9) Wng Eqaon (E.) o (E.9) n max fom gves
8.7. Chape 8.7.7778.88.7778.7.8.7778.66.8.7778.9.799.87..769 The above eqaons ae a -dagonal sysem of eqaons and specal algohms sch as Thomas algohm can be sed o solve sch a sysem of eqaons..87.66..7.68.769 b) To fnd he maxmm sess, s gven by Eqaon (E.7) as E d σ max ν ν a d a 6 E ps ν..87 a d a d.66.87.6.767 The maxmm sess n he pesse vessel hen s 6.87 σmax.(. 767 )..7 ps So he faco of safey FS fom Eqaon (E.8) s 6 FS.6896.7 c) The dffeenal eqaon has an exac solon and s gven by he fom C C (E.) whee C and C ae fond by sng he bonday condons a a and b.
Fne Dffeence Mehod 8.7. C ( a) ( ).87 C() C ( b) ( 8).769 C(8) 8 gvng C.6 C.6 Ths.6.6 (E.) d.6.6 (E.) d E d σ max ν ν a d a.6 6.6( ).6..6..8 ps The e eo s E.8.7.689 The absole elave e eo s.8.7.8.7% Example The appoxmaon n Example d d s fs ode accae, ha s, he e eo s of O( ). The appoxmaon d (E.) d ( ) s second ode accae, ha s, he e eo s O( ) ) Mxng hese wo appoxmaons wll esl n he ode of accacy of O( ) and O( ) ), ha s O( ). So s bee o appoxmae
8.7. Chape 8.7 d (E.) d ( ) becase hs eqaon s second ode accae. Repea Example wh he moe accae appoxmaons. Solon a) Repeang he poblem wh hs appoxmaon, a node n he pesse vessel, d d ( ) d d Sbsng Eqaons (E.) and (E.) n Eqaon (E.) gves ( ) ( ) ( ) ( ) ( ) ( ) A node, a " (E.) (E.) (E.).87" (E.6) A node,.6.6" (.6)(.6) (.6) (.6) (.6).6 (.6)(.6).697.87.966 (E.7) A node,.6.6 6. ".6 A node, (E.8).86.9 ( 6.)(.6).6.6 6..6 ( 6.)(.6) 6..6 6. 8 ".6 A node, (E.9).77.9 ( 6.8)(.6).6.6 6.8.6 ( 6.8)(.6) 6.8.6 7. ".66 A node, (E.) 7..78.89 ( 7.)(.6).6.6 ( ).6 ( 7.)(.6) 7..6 8 " / b.769 " (E.) Wng Eqaons (E.6) h (E.) n max fom gves
Fne Dffeence Mehod 8.7..697.87.6.966.86.6.9.77.66.9.78.87.89.769 The above eqaons ae a -dagonal sysem of eqaons and specal algohms sch as Thomas algohm can be sed o solve sch eqaons..87".6".9 ".689".86 ".769 " d b) d a ( ).87.6.9 (.6).9 6.87 σ max.(.9 )..666 ps Theefoe, he faco of safey FS s 6 FS.666.7 c) The e eo n calclang he maxmm sess s E.8.666 8 ps The elave e eo n calclang he maxmm sess s 8.8.6% Table Compasons of adal dsplacemens fom wo mehods. exac s ode nd ode
8.7. Chape 8.7.87.87..87..6.6.66.6.6. 6....6.9.876 6.8.68.7.87.689.6 7..8.68.9.86 9.66 8.769.769..769. ORDINARY DIFFERENTIAL EQUATIONS Topc Fne Dffeence Mehods of Solvng Odnay Dffeenal Eqaons Smmay Texbook noes of Fne Dffeence Mehods of solvng odnay dffeenal eqaons Majo Geneal Engneeng Ahos Aa Kaw, Cong Ngyen, Lke Snyde Dae Jly 7, Web Se hp://nmecalmehods.eng.sf.ed