ME 501A Seminar in Engineering Analysis Page 1

Transcription

1 Phse-plne Anlsis of Ordinr November, 7 Phse-plne Anlsis of Ordinr Lrr Cretto Mechnicl Engineering 5A Seminr in Engineering Anlsis November, 7 Outline Mierm exm two weeks from tonight covering ODEs nd Lplce trnsforms Review lst clss Introduction to phse-plne nlsis Look t two simultneous ODEs d / nd d / plotted s vs. Look t different criticl points for different sstems of equtions Review Lplce Trnsforms Use trnsform tbles to trnsform terms in differentil eqution for (t) into n lgebric eqution for Y(s) Derivtive trnsforms give initil conditions on (t) nd its derivtives Mnipulte Y(s) eqution to sum of individul terms, Y(s) subterms, tht ou cn find in trnsform tbles Mnipultion m require use of method of prtil frctions Review Prtil Frctions Method to convert frction with severl fctors in denomintor into sum of individul fctors (in denomintor) Exmple is F(s) = /(s+)(s+b) Write /(s+)(s+b) = A/(s+) + B/(s+b) Multipl b (s+)(s+b) nd equte coefficients of like powers of s = A(s + b) + B(s + ) A + B = for s terms nd = ba + B for s terms 3 4 Review Prtil Frctions II A + B = for s terms nd = ba + B for s terms Solving for A nd B gives A = -B = /(b ) Result: /(s+)(s+b) = /[(b )(s + )] /[(b )(s + b)] So f(t) = [e -t e -bt ]/(b ) This ctull mtches tble entr Follow sme bsic process for more complex frctions Specil rules for repeted fctors nd complex fctors 5 Review Prtil Frction Rules Repeted frctions for repeted fctors An An A A n n n ( s ) ( s ) ( s ) ( s ) s Complex fctors (s + i)(s + + i) As B ( s i )( s i ) ( s ) Pure imginr fctor As B s ( s i )( s i ) s Rel squred fctors A B s s s 6 ME 5A Seminr in Engineering Anlsis Pge

2 Phse-plne Anlsis of Ordinr November, 7 Review Sstems of ODEs Appl Lplce trnsforms to sstems of equtions b trnsforming ll ODEs Trnsform ODE terms like k to Y k (s), d k / to sy k (s) k (), etc. Trnsform ll ODEs in sstem then use Gussin elimintion to get n eqution for onl one Y k (s) Get inverse trnsform from Y k (s) to k (t) Repet for ll ODEs 7 Review Group Exercise Solve 9= e -t with () = nd () = b Lplce trnsforms Trnsform differentil eqution: s Y(s) s() () 9Y(s)= /(s +) Substitute initil conditions nd solve result for Y(s) s Y(s) 9Y(s) = /(s +) (s 9)Y(s) = + /(s +) 8 Review Group Exercise II (s 9)Y(s) = + /(s +) 9 9 Use prtil frctions for lst term Set sums of like powers to zero 9 Review Group Exercise III s terms: s terms: 4 s terms: s eqution gives B = A Substituting B = A into s eqution gives A + A + C = or C = 3A A = /4 Substitute B = A nd C = 3A into s eqution to get = 3A 3(A) 9( 3A) Review Group Exercise IV From A = /4 nd B = A: B = /4 From A = /4 nd C = -3A: C = -3/ From trnsform tble sinh sinh Bsic Phse Plne Equtions Look t solution of sstem of two firstorder utonomous (no t dependence) homogenous equtions Cn be single second order eqution written s two first order equtions d c d k d v m m dv c v m d v k m ME 5A Seminr in Engineering Anlsis Pge

3 Phse-plne Anlsis of Ordinr November, 7 Phse Plne Anlsis Look t solutions of sstems of equtions, here use two equtions s n exmple Find certin points, clled criticl points, tht hve prticulr behvior depending on the eigenvlues of the ODE s This leds to discussion of stbilit; will solution tend to zero or increse without bound? 3 Wht is Stble Solution? Plot trjectories (plot one dependent vrible ginst the other) For ODE s d/ nd dv/, plot vs. v If t one time the trjector is within distnce of point P nd for ll future times it remins within distnce of P, the solution is stble The solution is unstble if it is not stble Wnt to find criteri for stble solutions 4 Undmped Vibrtions Exmple Eqution: d x/ + x = ( = k/m) Solution: x = (v /)sin t + x cos t As sstem of equtions dx/ = v nd dv/ = d x/ = x Define = x nd = v to get sstem of equtions s d / = nd d / = 5 Phse Plne Introduction Usul plot shows solutions for x = nd v = dx/ = s function of time t 6 Phse Plne Introduction II Phse plne plot shows s function of with t s prmeter Sme s previous plot is displcement nd is velocit Time differs long plot Phse Plne Introduction III t -.5 Initil point (t = ) repets periodicll ME 5A Seminr in Engineering Anlsis Pge 3

4 Phse-plne Anlsis of Ordinr November, 7 Generl Form Write s mtrix eqution d/ = A Generl form for two equtions nd solutions in terms of Ax = x eigenvlues nd eigenvectors is C d A t t t t C x e C x e () () x() e Cx() e t t Cx() e Cx() e 9 Stble Solution Criteri Look t the sstem of two equtions d/ = A Autonomous sstems (no t dependence) We will show tht stbilit depends on the trce of A = + = +, nd the determinnt nd the discriminnt, = (trce A) 4det A Review eigenvlues for x mtrix from September lecture Two-b-two Mtrix Eigenvlues Qudrtic eqution with two roots for eigenvlues ( )( ) Eigenvlue solutions ( ) ( ) 4( ) Det A ( ) Trce A Return to Previous Exmple For undmped vibrtions we hd This gives trjector slope s d d () Continue Undmped Vibrtion Hve seprble solution giving eqution for ellipse d d d d C,, v x,,, x v v, x 3 Wht Hppens if = =? Autonomous sstem of two equtions d d d d Trjector slope, d /d, depends on vlues of A nd m be indeterminte t = = = = is clled criticl point For multidimensionl sstems = 4 ME 5A Seminr in Engineering Anlsis Pge 4

5 Phse-plne Anlsis of Ordinr November, 7 Tpes of Criticl Points Criticl points re points on - plot tht re clssified depending on the trjector shpes t or ner these points Centers Improper Nodes Proper Nodes Sddle Points Spirl Points Our First Criticl Point The undmped.5 vibrtions.5 solution is n ellipse tht does.5 not go through = = This tpe of -.5 criticl point is - clled center Improper Node Unstble Proper Node Stble All trjectories, except two of them hve the sme limiting direction of the tngents The two exceptions will hve different direction RepetedEigenvlues_files/imge3.gif 7 x hse_portrit_unstble_proper_node.svg 4/4/Phse_Portrit_Stble_Proper_Node.svg 8 Unstble Sddle Point 4 Stble Sddle Point ME 5A Seminr in Engineering Anlsis Pge 5

6 Phse-plne Anlsis of Ordinr November, 7 Spirl Points Unstble Spirl Source Asmptoticll Unstble Asmptoticll Stble ME 5A Seminr in Engineering Anlsis Pge 6