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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 48 "MAT1154 \+
- Equa\347\365es Diferenciais e de Diferen\347as " }}{PARA 257 "" 0 "
" {TEXT -1 65 "Equa\347\365es  diferenciais e de diferen\347as lineare
s de segunda ordem." }{TEXT 257 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "
Reson\342ncia." }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 
"" {TEXT -1 17 "George Svetlichny" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "restart: with(DEtools):with(plots):" }}}{EXCHG {PARA 
258 "" 0 "" {TEXT -1 10 "Reson\342ncia" }{TEXT 258 172 " occore quando
 a amplitude da solu\347\343o de uma e.d.o. linear com coeficientes co
nstantes maximisa-se em torno de uma determinada frequencia de um term
o n\343o homog\352neo senoidal" }{TEXT -1 2 ". " }{TEXT 259 118 "Consd
ere uma massa amarrada a uma mola e deslizando com atrito numa supef
\355cie. Considere primeiro a equa\347\343o homog\352nea." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eq:=(D@@2)(y)(t)+D(y)(t)+(17/4)*y(t
)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(eq,y(t));" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Note que a frequ\352ncia dos fat
ores trigonometricos \351 " }{TEXT 260 1 "2" }{TEXT -1 78 ". Acrescent
amos agora um termo n\343o homog\352neo senoidal com frequ\352ncia gen
erica " }{TEXT 261 1 "f" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "eq2:=(D@@2)(y)(t)+D(y)(t)+(17/4)*y(t)=sin(f*t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol:=subs(dsolve(eq2,y(t)),y
(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Vamos pegar a solu\347
\343o com a contribui\347\343o da equa\347\343o homog\352nea nula, ist
o \351 " }{TEXT 262 9 "_C1=_C2=0" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "fun:=subs([_C1=0,_C2=0],sol);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "Vamos agora plotar estas fun\347\365es pa
ra os valores de " }{TEXT 263 1 "f" }{TEXT -1 83 " igual a 1, 3/2, 2, \+
5/2 e 3, usando as cores preto amarelo, vermelo, azul, e verde." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "fun1:=subs(f=1,fun): fun2:=
subs(f=3/2,fun): fun3:=subs(f=2,fun): fun4:=subs(f=5/2,fun): fun5:=sub
s(f=3,fun): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot([fun1,fun2,fun
3,fun4,fun5],t=0..20,y=-1..1,color=[black, yellow, red,blue,green]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Note que a onda vermelha \351 \+
a mais alta. Vamos agora pegar a amplitude da onda, que para uma expre
ss\343o da forma" }{TEXT 265 1 " " }}{PARA 0 "" 0 "" {TEXT 273 5 "y(t)
=" }{TEXT -1 1 " " }{TEXT 264 21 "a cos(ft) + b sin(ft)" }{TEXT -1 57 
" \351 raiz quadrada de a^2+b^2, o que \351 a raiz quadrada de (" }
{TEXT 266 0 "" }{TEXT -1 18 "y'(0)/f)^2+y(0)^2:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 64 "a:=simplify(subs(t=0,fun)):b:=simplify(subs(t=
0,diff(fun,t))/f):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "amp:=simplify
(sqrt(a^2+b^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(am
p, f=0..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Note o m\341ximo \+
em torno de " }{TEXT 267 4 "f=2." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
43 "Vamos agora fazer a mesma coisa sem atrito:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "eq3:=(D@@2)(y)(t)+4*y(t)=sin(f*t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol:=subs(dsolve(eq3,y(t)),y(t));" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Esta express\343o n\343o faz se
ntido para " }{TEXT 268 6 "f = 2 " }{TEXT -1 65 "portanto vamos resolv
er a equa\347\343o para este valore separadamente." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 47 "f:=2: sol2:=subs(dsolve(eq3,y(t)),y(t));f:=
'f':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "De novo vamos plotar as \+
solu\347\365es para os dois constantantes de integra\347\343o igual a \+
zero e para os mesmos valores de " }{TEXT 269 1 "f" }{TEXT -1 65 " de \+
antes. Primeiro colocamos a zero as constantes de integra\347\343o:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "fun:=subs([_C1=0,_C2=0],so
l);func:=subs([_C1=0,_C2=0],sol2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "fun1:=subs(f=1,fun): fun2:=subs(f=3/2,fun): fun3:=fu
nc: fun4:=subs(f=3/2,fun): fun5:=subs(f=3,fun): " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 87 "plot([fun1,fun2,fun3,fun4,fun5],t=0..20,y=-5..5,color
=[black, yellow, red,blue,green]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 106 "Note que agora a amplitude da onda vermela cresce sem limite. \+
Plotamos a amplitude das ondas em fun\347\343o de " }{TEXT 270 1 "f" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:=simplif
y(subs(t=0,fun)):b:=simplify(subs(t=0,diff(fun,t))/f):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "amp:=simplify(sqrt(a^2+b^2));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(amp,f=0..4,y=0..10);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "A resson\342ncia pode ser muito d
estrutiva. Soldados n\343o marcham sobre pontes em sincronia. Um exemp
lo classico de engenharia pessimamente feita \351 dado pela destrui
\347\343o da ponte de Tacoma Narrows nos Estados Unidos que ficou tota
lmente estra\347alhada ap\363s ter entrado em reson\342ncia com a for
\347a do vento. Veja detalhes em: " }{URLLINK 17 "Destrui\347\343o da \+
ponte Tacoma Narrows" 4 "http://www.lib.washington.edu/specialcoll/tnb
/" "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "N\343o h\341 reson\342nci
a para uma e.d.o. cuja equa\347\343o caracter\355stica tem raizes reai
s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eq3:=(D@@2)(y)(t)-(1/
4)*y(t)=sin(f*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sol:=s
ubs(dsolve(eq3,y(t)),y(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "fun:=subs([_C1=0,_C2=0],sol);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
36 "Plotamos a amplitude como fun\347\343o de " }{TEXT 274 1 "f" }
{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:=simplif
y(subs(t=0,fun)):b:=simplify(subs(t=0,diff(fun,t))/f):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "amp:=simplify(sqrt(a^2+b^2));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "plot(amp,f=0..10);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 65 "Existe o fen\364meno de reson\342ncia em equa\347\365es d
e diferen\347as tambem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "
eqd:=y(n+2)-y(n+1)+(25/36)*y(n)=sin(f*n);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "fun:=simplify(rsolve(\{eqd,y(0)=0,y(1)=0\},y(n)));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "seq1:=[[n,subs(f=2/4,f
un)]$n=0..50]: seq2:=[[n,subs(f=3/4,fun)]$n=0..50]: seq3:=[[n,subs(f=4
/4,fun)]$n=0..50]: seq4:=[[n,subs(f=5/4,fun)]$n=0..50]: seq5:=[[n,subs
(f=6/4,fun)]$n=0..50]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([seq
1,seq2,seq3,seq4,seq5],x=0..50,color=[blue,black,red,green,yellow]);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "A amplitude da onda vermelha, \+
" }{TEXT 275 5 "f = 1" }{TEXT -1 41 ", \351 a mais alta. H\341 uma res
on\342ncia entre " }{TEXT 271 5 "f=3/4" }{TEXT -1 3 " e " }{TEXT 272 
5 "f=5/4" }{TEXT -1 61 " mas ach\341-la com exatid\343o n\343o seria u
ma tarefa muito simples." }}}}{MARK "37 1 0" 77 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }