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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 12 "S\372rl\363d\363 harm" }
{TEXT 256 0 "" }{TEXT -1 40 "onikus oszcill\341tor mozg\341sa 1 dimenz
i\363ban" }{MPLTEXT 0 21 0 "" }{TEXT -1 0 "" }}{PARA 19 "" 0 "" {TEXT 
-1 25 "Bartha Ferenc, SZTE, 2003" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 18 "A mozg\341segyenlet: " }{MPLTEXT 1 0 1 " " }{XPPEDIT 19 
1 "me := m*diff(v(t),t) = -a*x(t)-b*v(t)/abs(v(t))+F,v(t) = diff(x(t),
t):" "6#>%#meG6$/*&%\"mG\"\"\"-%%diffG6$-%\"vG6#%\"tGF0F),(*&%\"aGF)-%
\"xG6#F0F)!\"\"*(%\"bGF)-F.6#F0F)-%$absG6#-F.6#F0F7F7%\"FGF)/-F.6#F0-F
+6$-F56#F0F0" }}{PARA 11 "" 0 "" {TEXT -1 149 "Az F er\365 a tapad\341
si s\372rl\363d\341st jelenti. \332gy \351rtend\365, hogy F=0 amikor v
(t) nem nulla, a fordul\363pontokban (v=0) pedig |F|<c, ahol c valamil
yen \341lland\363. " }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{MPLTEXT 0 21 
12 "Param\351terek:" }{MPLTEXT 1 0 36 " m:=1:a:=3:b:=0.5:c:=1:x0:=3:v0
:=2:\n" }{MPLTEXT 0 21 24 "Kezd\365felt\351telek t=0-ban:" }{MPLTEXT 
1 0 23 " ini:=x(0)=x0,v(0)=v0:\n" }{MPLTEXT 0 21 16 "Szimul\341ci\363s
 id\365:" }{TEXT -1 1 " " }{MPLTEXT 1 0 9 "tmax:=18:" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 0 21 71 "El\365sz\366r a MAPLE megold\341s\341t \+
\341ll\355tjuk el\365 (tapad\341si s\372rl\363d\341s n\351lk\374l: F=0
)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F:=0:" }{TEXT -1 0 "" }
{MPLTEXT 1 0 47 "sol:=dsolve([me,ini],\{x(t),v(t)\},type=numeric):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "prx:=odeplot(sol,[t,x(t)],0..tmax,
title=`x(t)_MAPLE(rkf45)`):display(prx);\nprv:=odeplot(sol,[t,v(t)],0.
.tmax,title=`v(t)_MAPLE(rkf45)`):display(prv);" }}}{EXCHG {PARA 0 "" 
0 "" {MPLTEXT 0 21 399 "Most pedig ugyanez tapad\341ssal, saj\341t (Eu
ler-f\351le) megold\341ssal....\nFigyelj\374k meg (az al\341bb k\366ve
tkez\365 h:=..... sor m\363dos\355t\341s\341val), hogy \na) h=0.05 (du
rva) beoszt\341ssal a rezg\351s nemhogy nem csillapodik, hanem gerjed!
\nb) h=0.005 beoszt\341sn\341l a megold\341s l\341tsz\363lag eg\351sz \+
j\363, de elv\351tj\374k a tapad\341st! A hiba vizsg\341lat\341b\363l \+
(utols\363 k\351t \341bra) l\341that\363, hogy a kvalitat\355ve j\363n
ak l\341tsz\363 megold\341s nagyon pontatlan.  " }}{PARA 0 "" 0 "" 
{TEXT -1 44 "Legyen most a (j\363nak min\363s\355thet\365) l\351p\351s
k\366z: " }{MPLTEXT 1 0 9 "h:=0.001:" }{MPLTEXT 0 21 14 "\nEzzel durv
\341n " }{MPLTEXT 1 0 19 "imx:=floor(tmax/h);" }{MPLTEXT 0 21 42 "\npo
ntra osztottuk a [0,tmax] intervallumot" }{TEXT -1 0 "" }}{PARA 0 "" 
0 "" {MPLTEXT 0 21 121 "Nemigen sz\374ks\351ges minden oszt\363pontban
 elt\341rolni az \341br\341zol\341shoz a megfelel\365 f\374ggv\351ny
\351rt\351keket. Mondjuk el\351g lesz maximum " }{TEXT -1 0 "" }
{MPLTEXT 1 0 9 "npt:=100:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 48 "az \366
sszesb\365l, ami azt jelenti, hogy el\351g minden " }{MPLTEXT 1 0 28 "
step:=max(1,floor(imx/npt));" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 48 "step
-edik pontban meg\365rizni x(t) \351s v(t) \351rt\351k\351t" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 23 "Ime a rekurzi\363s ciklus:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 520 "t:=0:vi:=v0:xi:=x0:\nj:=0:\nfor i
 from 0 to imx do\nif ((i mod step)=0) then  #mentj\374k a kisz\341mol
t pontokat az \341br\341zol\341shoz\n j:=j+1:\n ex[j]:=xi:ev[j]:=vi:et
[j]:=t:\n rx[j]:=subs(sol(t),x('t')):rv[j]:=subs(sol(t),v('t'))\nend i
f:\nf1:=vi:f2:=-a/m*xi-b/m*signum(vi):\nif(vi*(vi+h*f2)>0 or a*abs(xi)
>c) then  #nem kell l\351pni, ha tapad \n  xi:=xi+h*f1:vi:=vi+h*f2:\ne
nd if:\nt:=t+h:\nod:\npex:=plot(\{[et[s],ex[s]] $s=1..j\},style=point,
title=`x(t)`):display(pex);\npev:=plot(\{[et[s],ev[s]] $s=1..j\},style
=point,title=`v(t)`):display(pev);" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 18 "\326sszehasonl\355t\341sok:" }{MPLTEXT 1 0 168 "\npd
x:=plot(\{[et[s],ex[s]-rx[s]] $s=1..j\},style=point,title=`Err_x(t)`):
display(pdx);\npdv:=plot(\{[et[s],ev[s]-rv[s]] $s=1..j\},style=point,t
itle=`Err_v(t)`):display(pdv);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 1 0" 147 }{VIEWOPTS 1 1 0 1 1 
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