{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helve tica" 1 10 0 0 255 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "N ormal" -1 258 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 259 "" 0 "" {TEXT 256 25 "Moderne Physik mit Mapl e " }}{PARA 261 "" 0 "" {TEXT 257 9 "PDF-Buch " }{URLLINK 17 "Moderne \+ Physik mit Maple" 4 "http://mikomma.de/fh/modphys.pdf" "" }}{PARA 258 "" 0 "" {TEXT -1 19 "Update auf Maple 10" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 14 "Kapitel 3.1.1 " }}{PARA 258 " " 0 "" {TEXT -1 21 "Worksheet oszi_10.mws" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "International Thomson Publishing, Bonn 1995 filename: o szi.ms" }}{PARA 0 "" 0 "" {TEXT -1 102 "Autor: Komma \+ Datum: 7.8. 94" }}{PARA 0 "" 0 "" {TEXT -1 66 "Thema: Berechnug und Darstellung de r \334berlagerung von Schwingungen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 430 "Die Schwingung sgleichung kann als wichtiger Sonderfall der Newtonschen Bewegungsglei chung angesehen werden, oder -- mathematisch gesehen -- als gew\366hnl iche DG 2.Ordnung. Die Bestimmung der L\366sung wird deshalb in den en tsprechenden Abschnitten behandelt. An dieser Stelle geht es darum, wi e man die als gegeben vorausgesetzten L\366sungen darstellt und mit Ma ple handhabt. Wir wollen mit der \334berlagerung zweier Schwingungen b eginnen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x1:=a1*sin(omega*t-phi);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "x2:=a2*sin(Omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x:=x1+x2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "a1:=2: a2:=.4: phi:=Pi/2: Omega:=2: omega:=Omeg a/5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p1:=plot(x1,t,color =red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p2:=plot(x2,t,col or=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "p3:=plot(x,t) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(\{p1,p2,p3\}); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Animation zu verschiedenen Phasen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phi:='phi':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "animate(x,t=-10..10,phi=0..2*Pi-0.1,numpoints=200); \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Der Sonderfall der Schwebung mit a2=a1 und fast glei chen Frequenzen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "a1:='a1 ': a2:=a1: omega:='omega': Omega:='Omega': phi:='phi': x:=x1+x2; # m anchmal -a1*sin..." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "Mit Hilfe \+ des " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x:=a1*op(trigsubs(simplify(x /a1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Die Cosinus-Funktion kann als zeitabh \344ngige Amplitude aufgefa\337t werden, die Sinusfunktion beschreibt \+ dann die Phase:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ampl:=x/ op(3,x); # Numerierung nicht stabil!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "#ampl:=2*a1*cos(1/2*omega*t-1/2*phi-1/2*Omega*t);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "phase:=op(3,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "#phase:=sin(1/2*omega*t-1/2*phi+1/2 *Omega*t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Plot der beiden Fak toren und des Produktes:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "phi:=0: Omega:=5: omega:=.9*Omega: a1:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(\{ampl,phase,x\},t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Animat ion der Schwebungskurve:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "phi:='phi': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "animate(x, t=-10..10,phi=0..2*Pi-.1,numpoints=200,color=green);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "V on einem Oszilloskop kann z.B. eine Folge von Momentaufnahmen gemacht \+ werden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Omega:=5: om ega:=0.9*Omega:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t:='t': \+ ts:='ts': phi:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ania mpl:=subs(t=ts,ampl);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pl ot(\{seq(aniampl*phase,ts=1..8)\},t=-2..2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Animatio n der Momentaufnahmen (beim Oszilloskop m\374\337te man sich mit der T riggerung M\374he geben):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "ps:=seq(plot(aniampl*phase,t=-2..2),ts=seq(i*2*Pi/(Omega-omega)/10 ,i=0..19)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "display([ps ],insequence=true); # [ ] f\374r richtige Reihenfolge" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "Ber\374cksichtigung der D\344mpfung: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Die a nderen Gr\366\337en wieder unassignen (mit einer kleinen procedure, we il das \366fter vokommt):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "unas:=proc() unassign('omega','Omega','phi','k','a1'): end: unas() ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=exp(-k*t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xd:=x1*d;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a1:=2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "omega:=5: # 5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "k:=1/4: # 1/4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "phi:=0:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "xd;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Plot der ged\344mpften Schwingung mit Einh\374llender:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "plot(\{xd,a1*d,-a1*d\},t=0..20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Phasenportraits (v-x-Diagramme) zun\344chst ohne D\344mpfung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "v1:=diff(x1,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " plot([x1,v1,t=0..1],scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "phi:= Pi/3: plot([x1,v1,t=0..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "r\344umliche Darstellung d er Phasenkurve:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "spacecur ve([t,v1,x1],t=0..5,axes=boxed,numpoints=100);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Verbin dung der Kurvenpunkte mit der t-Achse" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "cylinderplot([z,t/2,t],t=0..20,z=0..5); # Verbindu ng der spacecurve (von oben) mit der Zeitachse (senkrecht)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 29 "Phasenportraits mit D\344mpfu ng:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "unas();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " vd:=diff(xd,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=1: o mega:=5: k:=1/4: phi:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot([xd,vd,t=0..5]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "#with(plots): \+ # geh\366rt endlich in den .ini-file od. zusammen mit restart!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "spacecurve([t,vd,xd],t=0..5, axes=boxed,numpoints=100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "k:=.1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cylinderplot([z*exp(-k*t),t/2,t],t= 0..20,z=0..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Lissajousfiguren (mit D\344mpfung, falls \+ erw\374nscht):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "#restart: w ith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "x:=ax*exp(-k x*t)*sin(ox*t+px);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "y:=ay *exp(-ky*t)*cos(oy*t+py);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ax:=1: kx:=0.2: ox:=1: p x:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "ay:=1: \+ ky:=0: oy:=4: py:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot([x,y,t=0..2*Pi/ox]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "In Bewegung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "px:='px':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "animate([x,y, t=0..2*Pi/ox],px=0..Pi,numpoints=100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "\"r\344umlich e Lissajous-Figuren\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s etoptions3d(axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " px:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "spacecurve([cos(o x*t),sin(ox*t),y],t=0..2*Pi,axes=boxed,numpoints=100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "In Bewegung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "i:='i ': py:=0: px:=i*Pi/20: ox:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "sl:=seq(spacecurve([cos(ox*t+px),sin(ox*t+px),y],t=0..2*Pi,axes= boxed,numpoints=100),i=1..20): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display3d([sl],insequence=true); " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 36 "Epizyklen in komplexer Schreibweise:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;w ith(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "o1:=r1*exp( I*Omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "o2:=r2*exp(I *omega*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "o:=o1+o2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "theta:=argument(o);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "r:=abs(o);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "r1:=2: Omeg a:=-1: # auch negative Kreisfrequenzen sind erlaubt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r2:=1: omeg a:=7:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Plot in Polarkoordinaten (mit Zeitmessung):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seto ptions(scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "start:=time();" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "pol arplot([evalc(r),evalc(theta),t=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "dauer:=time()-start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Parameterplot :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "y:=evalc(Im(o)); x:=ev alc(Re(o));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "start:=time( );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot([x,y,t=0..2*Pi]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dauer:=time()-start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni-tuebingen.de" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }