.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion" h re more compliced hn wh undeermined coefficien cn hndle. We will coninue hi dicuion on Fridy, wih few more ble enrie including "he del (impule) funcion". f wih f Ce M F f e d for M commen u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off f f d F G "convoluion" for invering produc of Lplce rnform The uni ep funcion wih jump = i defined o be, u =,. IThi funcion i lo clled he "Heviide" funcion, e.g. in Mple nd Wolfrm lph. In Wolfrm lph i' lo clled he "he" funcion. Oliver Heviide w n ccomplihed phyici in he 8'. The nme i no becue he grph i hevy on one ide. :-) hp://en.wikipedi.org/wiki/oliver_heviide > wih plo : plo Heviide, =.., color = green, ile = `grph of uni ep funcion` ; grph of uni ep funcion.6.2 2 2 Noice h echniclly he vericl line hould no be here - more precie picure would hve olid poin, nd hollow circle,, for he grph of u. In erm of Lplce rnform inegrl definiion i doen' cully mer wh we define u o be.
Then, ; i.e. u =, ; i.e. nd h grph h i horizonl rnlion by o he righ, of he originl grph, e.g. for = 2: Exercie ) Verify he ble enrie u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off
Exercie 5) Explin why he decripion bove led o he differenil equion iniil vlue problem for x x x =.2 co u x = x = 5b) Find x. Show h fer he pren op puhing, he child i ocilling wih n mpliude of excly meer (in our linerized model).
Picure for he wing: > plo plo. in, =.. Pi, color = blck : plo2 plo Pi in, = Pi..2 Pi, color = blck : plo plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo5 plo., =.. Pi, color = blck, lineyle = 2 : plo6 plo., =.. Pi, color = blck, lineyle = 2 : diply plo, plo2, plo, plo4, plo5, plo6, ile = `dvenure he winge` ; dvenure he winge 2 4 6 8 2 6 2 Alerne pproch vi Chper 5: ep ) olve for. ep 2) Then olve nd e x = y for. x x =.2 co x = x = y y = y = x y = x
f, wih f Ce M F f e d for M verified c f c 2 f 2 c F c 2 F 2 2 n, n 2 2 n! n ( e ( co k in k coh k inh k e co k e in k e f u f u f f f n, n 2 k 2 ( k 2 k 2 ( 2 k 2 ( k k 2 k 2 ( k 2 k 2 ( k 2 k 2 F e e F e F f 2 F f f n F n f... f n
f d F f 2 f n f, n f co k 2 k in k 2 k in k k co k e n e, n F F n F n F d 2 k 2 2 k 2 2 2 k 2 2 2 k 2 2 2 n! n f g d F G f wih period p e p p f e d Lplce rnform ble
Mh 225-4 Fri Apr 6.5, EP7.6 piecewie nd implule forcing. Announcemen: Wrm-up Exercie:
Lplce ble enrie for ody. f wih f Ce M F f e d for M commen u uni ep funcion e for urning componen on nd off =. f u e F more compliced on/off e uni impule/del "funcion" EP 7.6 impule funcion nd he operor. Conider force f cing on n objec for only on very hor ime inervl, for exmple when b hi bll. Thi impule p of he force i defined o be he inegrl p f d nd i meure he ne chnge in momenum of he objec ince by Newon' econd lw m v = f m v d = m v = f d = p Since he impule p only depend on he inegrl of f, nd ince he exc form of f i unlikely o be known in ny ce, he eie model i o replce f wih conn force hving he me ol impule, i.e. o e f = p d, where d, i he uni impule funcion given by = p.,, d, =,. Noice h d, d = d =. Here' grph of d 2,., for exmple:
2 4
Since he uni impule funcion i liner combinion of uni ep funcion, we could olve differenil equion wih impule funcion o-conruced. A fr Lplce rnform goe, i' even eier o ke he limi for he Lplce rnform d,, nd hi effecively model impule on very hor ime cle. d, = u u d, = e e = e e. In Lplce lnd we cn ue L'Hopil' rule (in he vrible ) o ke he limi : lim e e = e e lim = e. The reul in ime pce i no relly funcion bu we cll i he "del funcion" nywy, nd viulize i funcion h i zero everywhere excep =, nd h i i infinie = in uch wy h i inegrl over ny open inervl conining equl one. A explined in EP7.6, he del "funcion" cn be hough of in rigorou wy liner rnformion, no funcion. I cn lo be hough of he derivive of he uni ep funcion u, nd hi i conien wih he Lplce ble enrie for derivive of funcion. In ny ce, hi led o he very ueful Lplce rnform ble enry uni impule funcion e for impule forcing
Exercie ) Revii he wing from Wednedy' noe nd olve he IVP below for x. In hi ce he pren i providing n impule ech ime he child pe hrough equilibrium poiion fer compleing cycle. x x =.2 2 4 6 8 x = x =.
> > wih plo : plo plo. in, =.. Pi, color = blck : plo2 plo Pi in, = Pi..2 Pi, color = blck : plo plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo4 plo Pi, = Pi..2 Pi, color = blck, lineyle = 2 : plo5 plo., =.. Pi, color = blck, lineyle = 2 : plo6 plo., =.. Pi, color = blck, lineyle = 2 : diply plo, plo2, plo, plo4, plo5, plo6, ile = `Wednedy dvenure he winge` ; Wednedy dvenure he winge 2 4 6 8 2 4 6 8 2 > impule oluion: five equl impule o ge me finl mpliude of meer - Exercie : > f.2 Pi um Heviide k 2 Pi in k 2 Pi, k =..4 : > plo f, =..2 Pi, color = blck, ile = `lzy pren on Fridy` ; > Or, n impule = nd noher one =. > g.2 Pi 2 in Heviide Pi in Pi : > plo g, =..2 Pi, color = blck, ile = `very lzy pren` ; > lzy pren on Fridy 2 4 6 8 4 2 very lzy pren 2 4 6 8 4 2