Chapter Finite Difference Method for Ordinary Differential Equations

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1 Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence mehod? The finie diffeence mehod is sed o solve odinay diffeenial eqaions ha have condiions imposed on he bonday ahe han a he iniial poin. These poblems ae called bonday-vale poblems. In his chape, we solve second-ode odinay diffeenial eqaions of he fom d y f (, y, y' ), a b, () d wih bonday condiions y( a) y a and y( b) yb () Many academics efe o bonday vale poblems as posiion-dependen and iniial vale poblems as ime-dependen. Tha is no necessaily he case as illsaed by he following eamples. The diffeenial eqaion ha govens he deflecion y of a simply sppoed beam nde nifomly disibed load (Fige ) is given by d y q( L ) () d EI whee locaion along he beam (in) E Yong s modls of elasiciy of he beam (psi) I second momen of aea (in ) q nifom loading inensiy (lb/in) L lengh of beam (in) The condiions imposed o solve he diffeenial eqaion ae y ( ) () y ( L) Clealy, hese ae bonday vales and hence he poblem is consideed a bonday-vale poblem. 8.7.

2 8.7. Chape 8.7 y q L Fige Simply sppoed beam wih nifom disibed load. Now conside he case of a canileveed beam wih a nifomly disibed load (Fige ). The diffeenial eqaion ha govens he deflecion y of he beam is given by d y q( L ) () d EI whee locaion along he beam (in) E Yong s modls of elasiciy of he beam (psi) I second momen of aea (in ) q nifom loading inensiy (lb/in) L lengh of beam (in) The condiions imposed o solve he diffeenial eqaion ae y ( ) (6) dy ( ) d Clealy, hese ae iniial vales and hence he poblem needs o be consideed as an iniial vale poblem. y q L Fige Canileveed beam wih a nifomly disibed load.

3 Finie Diffeence Mehod 8.7. Eample The deflecion y in a simply sppoed beam wih a nifom load q and a ensile aial load T is given by d y Ty q( L ) (E.) d EI EI whee locaion along he beam (in) T ension applied (lbs) E Yong s modls of elasiciy of he beam (psi) I second momen of aea (in ) q nifom loading inensiy (lb/in) L lengh of beam (in) y q T T L Fige Simply sppoed beam fo Eample. Given, T 7 lbs, q lbs/in, L 7 in, E Msi, and I in, a) Find he deflecion of he beam a ". Use a sep size of " and appoimae he deivaives by cenal divided diffeence appoimaion. b) Find he elaive e eo in he calclaion of y (). Solion a) Sbsiing he given vales, d y 7y () (7 ) 6 6 d ( )() ( )() d y 6 7 y 7. (7 ) (E.) d d y Appoimaing he deivaive a node i by he cenal divided diffeence d appoimaion,

4 8.7. Chape 8.7 i i i Fige Illsaion of finie diffeence nodes sing cenal divided diffeence mehod. d y y y y d ( ) We can ewie he eqaion as yi yi yi 6 7 y 7. (7 ) i i i ( ) Since, we have nodes as given in Fige i i i (E.) i i i i (E.) 7 Fige Finie diffeence mehod fom o 7 wih. The locaion of he nodes hen is 7 Wiing he eqaion a each node, we ge Node : Fom he simply sppoed bonday condiion a, we obain y (E.) Node : Rewiing eqaion (E.) fo node gives y y y 6 7 y 7. (7 ) () 7.6y.y.6y 7. ()(7 ).6y.y.6y 9.7 (E.6) Node : Rewiing eqaion (E.) fo node gives y y y 6 7 y 7. (7 ) () 7.6y.y.6y 7. ()(7 ).6y.y.6y 9.7 (E.7) Node : Fom he simply sppoed bonday condiion a 7, we obain y (E.8)

5 Finie Diffeence Mehod 8.7. Eqaions (E.-E.8) ae simlaneos eqaions wih nknowns and can be wien in mai fom as y.6..6 y y 9.7 y The above eqaions have a coefficien mai ha is idiagonal (we can se Thomas algoihm o solve he eqaions) and is also sicly diagonally dominan (convegence is gaaneed if we se ieaive mehods sch as he Gass-Siedel mehod). Solving he eqaions we ge, y y.8 y.8 y y ) y( ) y.8" ( The eac solion of he odinay diffeenial eqaion is deived as follows. The homogeneos pa of he solion is given by solving he chaaceisic eqaion 6 m m. Theefoe,.. yh Ke K e The paicla pa of he solion is given by y p A B C Sbsiing he diffeenial eqaion (E.) gives d y p 6 7 y 7. (7 ) p d d 6 7 ( A B C) ( A B C) 7. (7 ) d 6 7 A ( A B C) 7. (7 ) A B (A C).6 7. Eqaing ems gives 6 7 A 7. 6 B.6 6 A C Solving he above eqaion gives A.7 B 8. C.7

6 8.7.6 Chape 8.7 The paicla solion hen is y p The complee solion is hen given by.. y Ke K e Applying he following bonday condiions y ( ) y ( 7) we obain he following sysem of eqaions K K.7.9K.8997K.7 These eqaions ae epesened in mai fom by K K.7 A nmbe of diffeen nmeical mehods may be ilized o solve his sysem of eqaions sch as he Gassian eliminaion. Using any of hese mehods yields K K.9777 Sbsiing hese vales back ino he eqaion gives.. y e.9777 e Unlike ohe eamples in his chape and in he book, he above epession fo he deflecion of he beam is displayed wih a lage nmbe of significan digis. This is done o minimize he ond-off eo becase he above epession involves sbacion of lage nmbes ha ae close o each ohe. b) To calclae he elaive e eo, we ms fis calclae he vale of he eac solion a y.. () y( ).7() 8.() e. ().9777 e y ( ). The e eo is given by E = Eac Vale Appoimae Vale E. (.8) E. The elaive e eo is given by Te Eo % Te Vale. %. %

7 Finie Diffeence Mehod Eample Take he case of a pesse vessel ha is being esed in he laboaoy o check is abiliy o wihsand pesse. Fo a hick pesse vessel of inne adis a and oe adis b, he diffeenial eqaion fo he adial displacemen of a poin along he hickness is given by d d (E.) d d The inne adis a and he oe adis b 8, and he maeial of he pesse vessel is ASTM A6 seel. The yield sengh of his ype of seel is 6 ksi. Two sain gages ha ae bonded angenially a he inne and he oe adis mease nomal angenial sain as.776 / a / b.86 (E.a,b) a he maimm needed pesse. Since he adial displacemen and angenial sain ae elaed simply by, (E.) hen ' ' a b ' ' The maimm nomal sess in he pesse vessel is a he inne adis E d ma a d a whee E Yong s modls of seel (E= Msi) Poisson s aio (.) The faco of safey, FS is given by Yield sengh of seel ma a and is given by (E.7) FS (E.8) a) Divide he adial hickness of he pesse vessel ino 6 eqidisan nodes, and find he adial displacemen pofile b) Find he maimm nomal sess and faco of safey as given by eqaion (E.8) c) Find he eac vale of he maimm nomal sess as given by eqaion (E.8) if i is given ha he eac epession fo adial displacemen is of he fom C C. Calclae he elaive e eo.

8 8.7.8 Chape 8.7 Solion a i b i- i+ n a i- i i+ b Fige Nodes along he adial diecion. a) The adial locaions fom a o b ae divided ino n eqally spaced segmens, and hence esling in n nodes. This will allow s o find he dependen vaiable nmeically a hese nodes. A node i along he adial hickness of he pesse vessel, d i i i (E.9) d d i i (E.) d Sch sbsiions will conve he odinay diffeenial eqaion ino a linea eqaion (b wih moe han one nknown). By wiing he esling linea eqaion a diffeen poins a which he odinay diffeenial eqaion is valid, we ge simlaneos linea eqaions ha can be solved by sing echniqes sch as Gassian eliminaion, he Gass-Siedel mehod, ec. Sbsiing hese appoimaions fom Eqaions (E.9) and (E.) in Eqaion (E.) i i i i i i (E.) i i i i i (E.) i i i Le s beak he hickness, b a, of he pesse vessel ino n nodes, ha is a is node i and b is node i n. Tha means we have n nknowns. We can wie he above eqaion fo nodes,..., n. This will give s n eqaions. A he edge nodes, i and i n, we se he bonday condiions of

9 Finie Diffeence Mehod a n b This gives a oal of n eqaions. So we have n nknowns and n linea eqaions. These can be solved by any of he nmeical mehods sed fo solving simlaneos linea eqaions. We have been asked o do he calclaions fo n, ha is a oal of 6 nodes. This gives b a n 8.6 " A node i, a ",.87" (E.) A node i,.6.6" (E.) (E.) A node, A node, A node, A node,.6.6 i 6." (E.6) 6..6 i 6.8" (E.7) i (E.8) i (E.9) b Wiing Eqaion (E.) o (E.9) in mai fom gives

10 8.7. Chape =..769 The above eqaions ae a i-diagonal sysem of eqaions and special algoihms sch as Thomas algoihm can be sed o solve sch a sysem of eqaions b) To find he maimm sess, i is given by Eqaion (E.7) as E d ma a d a 6 E psi..87 a d a d The maimm sess in he pesse vessel hen is 6.87 ma psi So he faco of safey FS fom Eqaion (E.8) is 6 FS c) The diffeenial eqaion has an eac solion and is given by he fom C C (E.) whee C and C ae fond by sing he bonday condiions a a and b.

11 Finie Diffeence Mehod 8.7. C ( a) ( ).87 C() C ( b) ( 8).769 C(8) 8 giving C.6 C. 6 Ths.6.6 (E.) d.6.6 (E.) d E d ma a d a psi The e eo is E The absole elaive e eo is % Eample The appoimaion in Eample d i i d is fis ode accae, ha is, he e eo is of O( ). The appoimaion d i i i (E.) d is second ode accae, ha is, he e eo is O Miing hese wo appoimaions will esl in he ode of accacy of O and O, ha is O. So i is bee o appoimae

12 8.7. Chape 8.7 d d i i (E.) becase his eqaion is second ode accae. Repea Eample wih he moe accae appoimaions. Solion a) Repeaing he poblem wih his appoimaion, a node i in he pesse vessel, d i i i d ( ) d i i d Sbsiing Eqaions (E.) and (E.) in Eqaion (E.) gives i i i i i i i i i i i i i A node i, a " i (E.) (E.) (E.).87" (E.6) A node i,.6.6" A node, (E.7) i ".6 A node, (E.8) i 6. 8".6 A node, (E.9) i 7. ".66 A node, (E.) i " / b.769 " (E.) Wiing Eqaions (E.6) h (E.) in mai fom gives

13 Finie Diffeence Mehod = The above eqaions ae a i-diagonal sysem of eqaions and special algoihms sch as Thomas algoihm can be sed o solve sch eqaions..87".6".9 ".689".86 ".769 " d b) d a (.6) ma psi Theefoe, he faco of safey FS is 6 FS c) The e eo in calclaing he maimm sess is E psi The elaive e eo in calclaing he maimm sess is 8.8.6% Table Compaisons of adial displacemens fom wo mehods. eac s ode nd ode

14 8.7. Chape ORDINARY DIFFERENTIAL EQUATIONS Topic Finie Diffeence Mehods of Solving Odinay Diffeenial Eqaions Smmay Tebook noes of Finie Diffeence Mehods of solving odinay diffeenial eqaions Majo Geneal Engineeing Ahos Aa Kaw, Cong Ngyen, Lke Snyde Dae Decembe, 9 Web Sie hp://nmeicalmehods.eng.sf.ed

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations 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