Equa\303\247\303\265es Diferenciais: ModelosLaborat\303\263rio de Instrumenta\303\247\303\243o Alg\303\251brica - L.I.A.www.lia.ifsc.usp.brEsmerindo Bernardes (sousa@ifsc.usp.br)DFCM-IFSC-USP08/2009 - 05/2014 - 05/2015O Maple possui pacotes contendo v\303\241rias rotinas destinadas \303\240 resolu\303\247\303\243o de v\303\241rios tipos de equa\303\247\303\265es diferenciais. Em nossos casos de estudos, procuraremos resolver nossas equa\303\247\303\265es diferenciais de duas formas: de forma anal\303\255tica, onde a solu\303\247\303\243o pode ser escrita em termos de fun\303\247\303\265es conhecidas, e de forma num\303\251rica, onde n\303\243o sabemos escrever a solu\303\247\303\243o em termos de fun\303\247\303\265es conhecidas. Iremos usar a rotina dsolve para buscarmos por solu\303\247\303\265es anal\303\255ticas e a rotina dsolve/numeric para determinarmos uma solu\303\247\303\243o numericamente. Entre v\303\241rios m\303\251todos num\303\251ricos escolheremos um conhecido por Runge-Kutta de ordem 7-8 (dsolve/numeric/dverk78).Rotinas auxiliares:1) Rotina para fazer a anima\303\247\303\243o de um p\303\252ndulo simples.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==: tempo inicialLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg==: tempo finalX: nome da fun\303\247\303\243o que ir\303\241 calcular a equa\303\247\303\243o de movimento\303\232ltimo argumento (opcional): [c1,c2,c3,c4]AnimaPendulo := proc(t1,t2,X) local c1,c2,c3,c4,massa,linha,prego,todos; option `Copyright LIA 2004`; description "Simula um p\303\252ndulo simples"; if nargs=4 then c1 := args[4,1]; #raio do disco usado para rep. a massa c2 := color=args[4,2]; #cor da massa c3 := color=args[4,3]; #cor da haste do p\303\252ndulo c4 := frames=args[4,4]; #num. de quadros na anima\303\247\303\243o else c1 := NULL; c2 := NULL; c3 := NULL; c4 := NULL; end if; prego := plottools[disk]([0,0], c1/4, color=black); massa := (t) -> plottools[disk]([sin(X(t)),-cos(X(t))], c1, c2); linha := (t) -> plottools[line]([0,0],[sin(X(t)),-cos(X(t))], thickness=2, c3); todos := (t) -> plots[display]([massa(t), linha(t), prego]); return(plots[animate](todos,[t],t=t1..t2, c4,axes=none,scaling=constrained))end:#AnimaPendulo(T_inicial,T_final,Theta2, [0.1,blue,red,100]);2) Caso anal\303\255tico: Coloca um disco de raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElcmFpb0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em movimento de acordo com a equa\303\247\303\243o hor\303\241ria 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:AnimaQuedaLivre := proc(Z,par,raio) local t,Tq,Massa,Curva; Tq := [fsolve(Z(t,op(par)),t,0..1000)]; Massa := t -> plots[display](plottools[disk]([0,Z(t,op(par))], raio, color=red)); Curva := plot(0,-raio..raio,0..par[2]); return(plots[animate](Massa,[t],t=0..Tq[1], background=Curva, frames=100, scaling=constrained))end proc:3) Caso num\303\251rico: Coloca um disco de raio LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElcmFpb0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= em movimento de acordo com a equa\303\247\303\243o hor\303\241ria LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiWkYnRi9GMi1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEidEYnRi9GMkY+Rj5GPkYrRj4=:AnimaQuedaLivre2 := proc(Z,Par,Tq,raio) local t,Massa,Curva; Massa := t -> plots[display](plottools[disk]([0,Z(t)], raio, color=red)); Curva := plot(0,-raio..raio,0..Par[2]); return(plots[animate](Massa,[t],t=0..Tq, background=Curva, frames=100, scaling=constrained))end proc:Solu\303\247\303\243o anal\303\255tica:Oscilador harm\303\264nico:Equa\303\247\303\243o diferencial + condi\303\247\303\265es iniciais: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omega := 'omega';edo := {diff(x(t),t$2) + (omega[1]^2)*x(t) = 0, x(0)=2, D(x)(0)=0};Solu\303\247\303\243o:sols := dsolve(edo,{x(t)});X := unapply(rhs(sols),t);Velocidade:Vx := unapply(diff(X(t),t),t);Energia cin\303\251tica:T := unapply((m/2)*(Vx(t)^2),t);Energia potencial:V := unapply((k/2)*(X(t)^2),t);Casos espec\303\255ficos:m := 1; k := (2*Pi)^2;omega := [sqrt(k/m)];An\303\241lise:plot(X(t),t=0..2);plot([T(t),V(t),T(t)+V(t)],t=0..2, color=[black,blue,red]);Solu\303\247\303\243o num\303\251rica:Oscilador harm\303\264nico ideal:Equa\303\247\303\243o diferencial + condi\303\247\303\265es iniciais: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Casos espec\303\255ficos:m := 1; k := (2*Pi)^2;omega := [sqrt(k/m)];EDO:edo := {diff(x(t),t$2) + (omega[1]^2)*x(t) = 0, x(0)=2, D(x)(0)=0};Solu\303\247\303\243o:sol := dsolve(edo, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria:Xn := rhs(sol[2]);Velocidade:Vxn := rhs(sol[3]);Energia cin\303\251tica:T := unapply((m/2)*(Vxn(t)^2),t);Energia potencial:V := unapply((k/2)*(Xn(t)^2),t);An\303\241lise:plot(Xn(t),t=0..2);plot([T(t),V(t),T(t)+V(t)],t=0..2, color=[black,blue,red]);Oscilador harm\303\264nico amortecido:Equa\303\247\303\243o diferencial + condi\303\247\303\265es iniciais: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 : freq\303\274\303\252ncia 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 : amortecimentoCasos espec\303\255ficos:m := 1; k := (2*Pi)^2; b := Pi;omega := [sqrt(k/m),b/m];EDO:edo := {diff(x(t),t$2) + omega[2]*diff(x(t),t) + (omega[1]^2)*x(t) = 0, x(0)=2, D(x)(0)=0};Solu\303\247\303\243o:sol := dsolve(edo, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria:Xn := rhs(sol[2]);Velocidade:Vxn := rhs(sol[3]);Energia cin\303\251tica:T := unapply((m/2)*(Vxn(t)^2),t);Energia potencial:V := unapply((k/2)*(Xn(t)^2),t);An\303\241lise:plot(Xn(t),t=0..4);plot([T(t),V(t),T(t)+V(t)],t=0..2, color=[black,blue,red]);P\303\252ndulo simples: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Sem dissipa\303\247\303\243oEqua\303\247\303\243o hor\303\241ria:Equa\303\247\303\243o diferencial resultante da segunda lei de Newton aplicada a um p\303\252ndulo simples sem amortecimento:EDO := diff(theta(t),t$2) + omega^2*sin(theta(t))=0; 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 nosso objetivo \303\251 determinar como o \303\242ngulo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmdGhldGFGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= depende do tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= de forma a satisfazer esta equa\303\247\303\243o diferencial em qualquer instante de tempo. Sabe-se que, dado duas condi\303\247\303\265es iniciais, geralmente a posi\303\247\303\243o (angular) inicial, 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, e a velocidade (angular) inicial, 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, esta equa\303\247\303\243o diferencial tem uma \303\272nica solu\303\247\303\243o. Embora esta solu\303\247\303\243o possa ser encontrada utilizando-se t\303\251cnicas anal\303\255ticas, \303\251 conveniente resolv\303\252-la numericamente. Para utilizarmos um m\303\251todo num\303\251rico, devemos dar valores num\303\251ricos para todos os par\303\242metros e condi\303\247\303\265es iniciais da equa\303\247\303\243o diferencial a ser resolvida. Na equa\303\247\303\243o diferencial acima, nos temos um par\303\242metro, a frequ\303\252ncia (angular) natural LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSVtc3ViR0YkNiUtRiw2JlEnJiM5Njk7RicvRjBRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y+L0YzUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JVEiMEYnRj9GQUY/RkEvJS9zdWJzY3JpcHRzaGlmdEdGSEY/RkFGK0Y/RkE=, e duas condi\303\247\303\265es iniciais. Assim, \303\251 conveniente atribuirmos valores a estas tr\303\252s informa\303\247\303\265es na forma de uma lista, a qual denominaremos de lista de par\303\242metros: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNidGKy1GIzYnLUYsNiZRIlBGJ0YvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGMi1JI21vR0YkNi5RIj1GJ0Y8L0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlUtSShtZmVuY2VkR0YkNictRiM2Ki1JJW1zdWJHRiQ2JS1GLDYmUScmIzk2OTtGJy9GMEY+RjxGQy1GIzYlLUkjbW5HRiQ2JVEiMEYnRjxGQ0Y8RkMvJS9zdWJzY3JpcHRzaGlmdEdGY28tRkA2LlEiLEYnRjxGQ0ZFL0ZIRjFGSUZLRk1GT0ZRL0ZUUSYwLjBlbUYnL0ZXUSwwLjMzMzMzMzNlbUYnLUYjNictRiw2JlEnJiM5NTI7RidGXW9GPEZDLUZANi1RMCZBcHBseUZ1bmN0aW9uO0YnRkNGRUZHRklGS0ZNRk9GUUZqby9GV0ZbcC1GWTYlRl5vRjxGQ0Y8RkNGZm8tRiM2KEYrLUYjNictRiw2JlEiREYnRl1vRjxGQ0ZjcC1GWTYlLUYjNiVGYHBGPEZDRjxGQ0Y8RkNGY3BGZ3BGPEZDRitGPEZDRjxGQy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y8RkNGK0Y8RkNGK0Y8RkM=, onde 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 \303\251 a forma de informar ao Maple o valor da primeira derivada de 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 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEidEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GK0Y+. Tamb\303\251m \303\251 conveniente construirmos uma rotina (Monta_EDO) para preparar a equa\303\247\303\243o diferencial a ser resolvida numericamente.Condi\303\247\303\265es iniciais:CIs := theta(0)=o, D(theta)(0)=w;Rotina para escrever a equa\303\247\303\243o diferencial com os valores de seus par\303\242metros e das consid\303\247\303\265es iniciais: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2Jy1GIzYqLUYsNiZRKCZvbWVnYTtGJy9GMEY0RjJGPC1GOTYuUSIsRidGMkY8Rj4vRkFGMUZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRictRiM2Jy1GLDYmUSgmdGhldGE7RidGWUYyRjwtRjk2LVEwJkFwcGx5RnVuY3Rpb247RidGPEY+RkBGQkZERkZGSEZKRmhuL0ZQRmluLUZSNiUtRiM2JS1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8RjJGPEYyRjxGWi1GIzYoLUYsNiVRIUYnRi9GNS1GIzYnLUYsNiZRIkRGJ0ZZRjJGPEZhby1GUjYlLUYjNiVGXm9GMkY8RjJGPEYyRjxGYW9GZW9GMkY8Rl9wRjJGPEYyRjwvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGMkY8Monta_EDO := unapply({EDO,CIs}, omega,o,w);Algumas equa\303\247\303\265es diferenciais particulares (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):1) \303\202ngulo muito pequeno e velocidade inicial nula:Par1 := 2*Pi,Pi/20,0;edo1 := Monta_EDO(Par1);2) \303\202ngulo grande (horizontal) e velocidade inicial nula:Par2 := 2*Pi,Pi/2,0;edo2 := Monta_EDO(Par2);3) \303\202ngulo nulo (vertical) e velocidade inicial grande:`Velocidade inicial` = evalf(4*Pi);Par3 := 2*Pi,0,12.57;edo3 := Monta_EDO(Par3);Solu\303\247\303\265es num\303\251ricas..Um m\303\251todo num\303\251rico muito eficaz neste caso \303\251 conhecido como Runge-Kutta. H\303\241 v\303\241rias vers\303\265es deste m\303\251todo implementadas no Maple. Em ordem crescente de precis\303\243o, elas s\303\243o rkf45, gear e dverk78. Use-as com muito cuidado. A vari\303\241vel Digits controla a precis\303\243o destas rotinas. Quanto maior o valor da vari\303\241vel Digits, maior a precis\303\243o em todos os c\303\241lculos num\303\251ricos e mais lento os c\303\241lculos. A rotina dsolve aceita duas op\303\247\303\265es, abserr=Float(1,-8) e relerr=Float(1,-8), as quais tamb\303\251m podem ser usadas para controlar a precis\303\243o das solu\303\247\303\265es num\303\251ricas.Digits := 10;eh1 := dsolve(edo1, numeric, method=dverk78, output=listprocedure);eh2 := dsolve(edo2, numeric, method=dverk78, output=listprocedure);eh3 := dsolve(edo3, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria (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):Theta1 := rhs(eh1[2]);Theta2 := rhs(eh2[2]);Theta3 := rhs(eh3[2]);Velocidade (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):Omega1 := rhs(eh1[3]);Omega2 := rhs(eh2[3]);Omega3 := rhs(eh3[3]);Acelera\303\247\303\243o (via EDO):Alpha1 := t -> -Par1[1]^2*sin(Theta1(t));Alpha2 := t -> -Par2[1]^2*sin(Theta2(t));Alpha3 := t -> -Par3[1]^2*sin(Theta3(t));Gr\303\241ficos da equa\303\247\303\265es hor\303\241rias: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Simula\303\247\303\243o:AnimaPendulo(0,4,Theta3, [0.1,blue,red,100]);Energia mec\303\242nica:Tendo estudado as caracter\303\255sticas principais de qualquer p\303\252ndulo ideal, devemos agora considerar um caso concreto. Isto significa escolhermos valores para o comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= do p\303\252ndulo e para a massa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (todos os valores em mks). Para explorarmos tamb\303\251m a conserva\303\247\303\243o da energia mec\303\242nica, vamos iniciar o movimento do p\303\252ndulo a partir da posi\303\247\303\243o de equil\303\255brio est\303\241vel (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSVtc3ViR0YkNiUtRiw2JVEmdGhldGFGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjBGJ0Y/Rj8vJS9zdWJzY3JpcHRzaGlmdEdGRi1JI21vR0YkNi1RIj1GJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZ25GQ0Y/RitGPw==) e com uma velocidade angular inicial previamente escolhida. Tamb\303\251m \303\251 conveniente adotar o zero de energia potencial nesta posi\303\247\303\243o de equil\303\255brio. Assim a energia mec\303\242nica (conservada) ser\303\241 igual \303\240 energia cin\303\251tica dada no in\303\255cio do movimento.Uma rotina para calcular a energia cin\303\251tica (em Joules) em qualquer instante de tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (em segundos):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, 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, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: nome da rotina (fornecida pela solu\303\247\303\243o num\303\251rica) usada para calcular o valor de 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSW9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: momento de in\303\251rcia (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)Ec := (t,V,Io) -> Io*(V(t)^2)/2;Uma rotina para calcular a energia potencial (em Joules) em qualquer instante de tempo: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, 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, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: nome da rotina (fornecida pela solu\303\247\303\243o num\303\251rica) usada para calcular o valor de 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: freq\303\274\303\252ncia angular natural (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) Ep := (t,X,Io,w0) -> Io*(w0^2)*(1-cos(X(t)));Uma rotina para calcular o per\303\255odo usando somente a energia mec\303\242nica (v\303\241lida somente para oscila\303\247\303\265es):To = (sqrt(2)/'omega[0]')*Int(1/sqrt('Epsilon'+cos(theta)),theta=theta[1]..theta[2]);int(1/sqrt(a+cos(tt)),tt);\303\211 melhor resolver esta integral numericamente, com uma precis\303\243o determinada por N d\303\255gitos:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdzBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: freq\303\274\303\252ncia angular naturalLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: energia mec\303\242nicaLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUFJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: pontos de retornoTo := (w0,E,PR,N) -> evalf(sqrt(2)/w0)*evalf[N](Int(1/sqrt(E+cos(t)),t=PR[1]..PR[2]));Uma rotina para calcular o per\303\255odo usando somente a energia mec\303\242nica inicial (v\303\241lida somente para rota\303\247\303\265es):Tr = (sqrt(2)/'2*omega[0]')*Int(1/sqrt('Epsilon'+cos(theta)),theta=0..2*Pi);\303\211 melhor resolver esta integral numericamente, com uma precis\303\243o determinada por N d\303\255gitos:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdzBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: freq\303\274\303\252ncia angular naturalLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: energia mec\303\242nicaTr := (w0,E,N) -> evalf(sqrt(2)/2/w0)*evalf[N](Int(1/sqrt(E+cos(t)),t=0..2*Pi));EDO:EDO := diff(theta(t),t$2) + omega[0]^2*sin(theta(t))=0; omega[0]='sqrt(g/l)';Condi\303\247\303\265es iniciais:CIs := theta(0)=o, D(theta)(0)=w;Rotina para escrever a equa\303\247\303\243o diferencial com os valores de seus par\303\242metros e das consid\303\247\303\265es iniciais: Monta_EDO := unapply({EDO,CIs}, omega[0],o,w);Escolha seus valores preferidosComprimento do p\303\252ndulo (em metros):l := 1;Massa (em kilograma):m := 1;Acelera\303\247\303\243o da gravidade (m/s/s):g := 981/100;Freq\303\274\303\252ncia angular natural (rad/s):w0 := sqrt(g/l);Momento de in\303\251rcia (kg.m.m):Io := m*(l^2);Unidade de energia (em Joules):e := Io*(w0^2);Escolha um valor para a energia mec\303\242nica (em Joules):E := (5/2)*e;E := (2001/1000)*e;E := 1999999*e/1000000;Velocidade angular inicial correspondente (considerando 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):Vai := sqrt(2*E/Io);Vo = evalf(Vai);Uma constante muito \303\272til (adimensional):Epsilon := E/e-1;Gr\303\241fico da energia potencial:plot([E,e*(1-cos(q))], q=-Pi..Pi, axes=boxed, color=[red,blue], legend=["Energia mec\303\242nica","Energia potencial"], labelfont=[SYMBOL]);Pontos de retorno (somente para oscila\303\247\303\265es):PR := [-solve(cos(t)=-Epsilon), solve(cos(t)=-Epsilon)];"Em graus" = evalf(PR*180/Pi);Per\303\255odo (somente para oscila\303\247\303\265es):p := To(w0,Epsilon,PR,15);if op(2,p)=infinity then p:=20 end if;Per\303\255odo (somente para rota\303\247\303\265es):p := Tr(w0,Epsilon,15);Somente para pequenas amplitudes: compare a freq\303\274\303\252ncia calculada pelo per\303\255odo (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) com a freq\303\274\303\252ncia natural LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmb21lZ2FGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIwRidGNUY1LyUvc3Vic2NyaXB0c2hpZnRHRj1GNQ==: 2*Pi/'T' - 'omega[0]' = evalf(2*Pi/p - w0);Par\303\242metros (previamente escolhidos):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 := w0,0,Vai;Solu\303\247\303\243o num\303\251rica da equa\303\247\303\243o diferencial do p\303\252ndulo sem dissipa\303\247\303\243o:edo := Monta_EDO(Par);Digits := 12;eh := dsolve(edo, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria (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)Theta := rhs(eh[2]);Velocidade angular (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):Omega := rhs(eh[3]);Acelera\303\247\303\243o (via EDO):Alpha := t -> -Par[1]^2*sin(Theta(t));Gr\303\241fico da equa\303\247\303\243o hor\303\241ria:plot(Theta(t),t=0..p, labels=["t","theta"]);Todo mundo junto:plot([Theta(t),Omega(t),Alpha(t)], t=0..2*p, color=[blue,red,black], legend=["Posi\303\247\303\243o","Velocidade","Acelera\303\247\303\243o"]);Espa\303\247o de fase:plots[spacecurve]([Theta(t),Omega(t),Alpha(t)],t=0..p, labels=["q","w","a"], labelfont=[SYMBOL], title="Espa\303\247o de fase 3D", axes=boxed, shading=zhue, numpoints=400);Simula\303\247\303\243o:AnimaPendulo(0,p,Theta, [0.1,blue,red,200]);Situa\303\247\303\243o energ\303\251tica:EC := t -> Ec(t,Omega,Io);EP := t -> Ep(t,Theta,Io,w0);plot([EC(t),EP(t),EC(t)+EP(t)], t=0..p, color=[red,blue,black], legend=["Cin\303\251tica", "Potencial", "Mec\303\242nica"], axes=boxed);`Energia mec\303\242nica inicial e Cin\303\251tica + Potencial` = [evalf(E), EC(p/2)+EP(p/2)];Com dissipa\303\247\303\243oEqua\303\247\303\243o hor\303\241ria:Equa\303\247\303\243o diferencial resultante da segunda lei de Newton aplicada a um p\303\252ndulo simples com amortecimento (dissipa\303\247\303\243o, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, F e v s\303\243o vetores):EDO := diff(theta(t),t$2) + omega[0]^2*sin(theta(t)) + omega[1]*diff(theta(t),t$1)=0; omega[0]='sqrt(g/l)'; omega[1]='b/m/l';O nosso objetivo \303\251 determinar como o \303\242ngulo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmdGhldGFGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= depende do tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= de forma a satisfazer esta equa\303\247\303\243o diferencial em qualquer instante de tempo. Sabe-se que, dado duas condi\303\247\303\265es iniciais, geralmente a posi\303\247\303\243o (angular) inicial, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEmdGhldGFGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnRjxGPEY8RjxGK0Y8, e a velocidade (angular) inicial, 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, esta equa\303\247\303\243o diferencial tem uma \303\272nica solu\303\247\303\243o. Embora esta solu\303\247\303\243o possa ser encontrada utilizando-se t\303\251cnicas anal\303\255ticas, \303\251 conveniente resolv\303\252-la numericamente. Para utilizarmos um m\303\251todo num\303\251rico, devemos dar valores num\303\251ricos para todos os par\303\242metros e condi\303\247\303\265es iniciais da equa\303\247\303\243o diferencial a ser resolvida. Na equa\303\247\303\243o diferencial acima, nos temos dois par\303\242metros, a freq\303\274\303\252ncia (angular) natural LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmb21lZ2FGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIwRidGNUY1LyUvc3Vic2NyaXB0c2hpZnRHRj1GNQ== e a "freq\303\274\303\252ncia" de amortecimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmb21lZ2FGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIxRidGNUY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ==, e duas condi\303\247\303\265es iniciais. Assim, \303\251 conveniente atribuirmos valores a estas tr\303\252s informa\303\247\303\265es na forma de uma lista, a qual denominaremos de lista de par\303\242metros: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEiUEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JKG1mZW5jZWRHRiQ2Ji1GIzYrLUklbXN1YkdGJDYlLUYsNiVRJm9tZWdhRicvRjBGQkY+LUYjNiQtSSNtbkdGJDYkUSIwRidGPkY+LyUvc3Vic2NyaXB0c2hpZnRHRl9vLUY7Ni1RIixGJ0Y+RkAvRkRGMUZFRkdGSUZLRk0vRlBRJjAuMGVtRicvRlNRLDAuMzMzMzMzM2VtRictRlo2JUZmbi1GIzYkLUZdbzYkUSIxRidGPkY+RmBvRmJvLUYjNiYtRiw2JVEmdGhldGFGJ0ZpbkY+LUY7Ni1RMCZBcHBseUZ1bmN0aW9uO0YnRj5GQEZDRkVGR0ZJRktGTUZmby9GU0Znby1GVTYkRmpuRj5GPkZiby1GIzYnRistRiM2Ji1GLDYlUSJERidGaW5GPkZmcC1GVTYkLUYjNiRGY3BGPkY+Rj5GZnBGanBGPkYrRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+RitGPg==, onde 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 \303\251 forma de dizer ao Maple o valor da primeira derivada de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEmdGhldGFGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y8LyUmZmVuY2VHRjsvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUi1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjJGPEY8RjxGK0Y8 em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRiw2JVEidEYnRi9GMi1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjBGJ0Y+Rj5GK0Y+. Tamb\303\251m \303\251 conveniente construirmos uma rotina (EDO) para montar a equa\303\247\303\243o diferencial a ser resolvida numericamente. Rotina para escrever a equa\303\247\303\243o diferencial com os valores de seus par\303\242metros e das condi\303\247\303\265es iniciais: 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Condi\303\247\303\265es iniciais:CIs := theta(0)=o, D(theta)(0)=w;Rotina para escrever a equa\303\247\303\243o diferencial com os valores de seus par\303\242metros e das consid\303\247\303\265es iniciais: Monta_EDO := unapply({EDO,CIs}, omega[0],omega[1],o,w);Algumas equa\303\247\303\265es diferenciais particulares:1) \303\202ngulo muito pequeno e velocidade inicial nula:Par1 := 2*Pi,1/2,Pi/20,0;edo1 := Monta_EDO(Par1);2) \303\202ngulo grande (horizontal) e velocidade inicial nula:Par2 := 2*Pi,1/2,Pi/2,0;edo2 := Monta_EDO(Par2);3) \303\202ngulo nulo (vertical) e velocidade inicial grande:`Velocidade inicial` = evalf(4*Pi);Par3 := 2*Pi,1/2,0,12.57;edo3 := Monta_EDO(Par3);Um m\303\251todo num\303\251rico muito eficaz neste caso \303\251 conhecido como Runge-Kutta. H\303\241 v\303\241rias vers\303\265es deste m\303\251todo implementadas no Maple. Em ordem crescente de precis\303\243o, elas s\303\243o rkf45, gear e dverk78. Use-as com muito cuidado:Digits := 12;eh1 := dsolve(edo1, numeric, method=dverk78, output=listprocedure);eh2 := dsolve(edo2, numeric, method=dverk78, output=listprocedure);eh3 := dsolve(edo3, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria (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):Theta1 := rhs(eh1[2]);Theta2 := rhs(eh2[2]);Theta3 := rhs(eh3[2]);Velocidade angular (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):Omega1 := rhs(eh1[3]);Omega2 := rhs(eh2[3]);Omega3 := rhs(eh3[3]);Acelera\303\247\303\243o (via EDO):Alpha1 := t -> -Par1[1]^2*sin(Theta1(t));Alpha2 := t -> -Par2[1]^2*sin(Theta2(t));Alpha3 := t -> -Par3[1]^2*sin(Theta3(t));Gr\303\241fico da equa\303\247\303\243o hor\303\241ria:plot(Theta1(t),t=0..12, labels=["t","q"], labelfont=[SYMBOL], numpoints=400, title="Equa\303\247\303\243o hor\303\241ria");Simula\303\247\303\243o:T_inicial := 0;T_final := 20;AnimaPendulo(T_inicial,T_final,Theta2, [0.1,blue,red,100]);Energia mec\303\242nica:Tendo estudado as caracter\303\255sticas principais de qualquer p\303\252ndulo ideal, devemos agora considerar um caso concreto. Isto significa escolhermos valores para o comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= do p\303\252ndulo e para a massa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (todos os valores em mks). Para explorarmos tamb\303\251m a conserva\303\247\303\243o da energia mec\303\242nica, vamos iniciar o movimento do p\303\252ndulo a partir da posi\303\247\303\243o de equil\303\255brio est\303\241vel (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSVtc3ViR0YkNiUtRiw2JVEmdGhldGFGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRictRiM2JC1JI21uR0YkNiRRIjBGJ0Y/Rj8vJS9zdWJzY3JpcHRzaGlmdEdGRi1JI21vR0YkNi1RIj1GJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZ25GQ0Y/RitGPw==) e com uma velocidade angular inicial previamente escolhida. Tamb\303\251m \303\251 conveniente adotar o zero de energia potencial nesta posi\303\247\303\243o de equil\303\255brio. Assim a energia mec\303\242nica ser\303\241 igual \303\240 energia cin\303\251tica dada no in\303\255cio do movimento.Uma rotina para calcular a energia cin\303\251tica (J) em qualquer instante de tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: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, 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, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: nome da rotina (fornecida pela solu\303\247\303\243o num\303\251rica) usada para calcular o valor de 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjSW9GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: momento de in\303\251rcia (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)Ec := (t,V,Io) -> Io*(V(t)^2)/2;Uma rotina para calcular a energia potencial (J) em qualquer instante de tempo: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, 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, 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: nome da rotina (fornecida pela solu\303\247\303\243o num\303\251rica) usada para calcular o valor de 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: freq\303\274\303\252ncia angular natural (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) Ep := (t,X,Io,w0) -> Io*(w0^2)*(1-cos(X(t)));Uma rotina para calcular o per\303\255odo usando somente a energia mec\303\242nica inicial (n\303\243o vale para rota\303\247\303\265es, somente para oscila\303\247\303\265es):To = (sqrt(2)/'omega[0]')*Int(1/sqrt('Epsilon'+cos(theta)),theta=theta[1]..theta[2]);\303\211 melhor resolver esta integral numericamente, com uma precis\303\243o determinada por N d\303\255gitos:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdzBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: freq\303\274\303\252ncia angular naturalLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: energia mec\303\242nicaLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUFJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: pontos de retornoTo := (w0,E,PR,N) -> evalf(sqrt(2)/w0)*evalf[N](Int(1/sqrt(E+cos(t)),t=PR[1]..PR[2]));Uma rotina para calcular o per\303\255odo usando somente a energia mec\303\242nica inicial (n\303\243o vale para oscila\303\247\303\265es, somente para rota\303\247\303\265es):Tr = (sqrt(2)/'2*omega[0]')*Int(1/sqrt('Epsilon'+cos(theta)),theta=0..2*Pi);\303\211 melhor resolver esta integral numericamente, com uma precis\303\243o determinada por N d\303\255gitos:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjdzBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn: freq\303\274\303\252ncia angular naturalLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: energia mec\303\242nicaTr := (w0,E,N) -> evalf(sqrt(2)/2/w0)*evalf[N](Int(1/sqrt(E+cos(t)),t=0..2*Pi));Escolha seus valores preferidosComprimento do p\303\252ndulo (m):l := 1;Massa (kg):m := 1;Acelera\303\247\303\243o da gravidade (m/s/s):g := 981/100;Freq\303\274\303\252ncia angular natural (rad/s):w0 := sqrt(g/l);Momento de in\303\251rcia (kg.m.m):Io := m*(l^2);Unidade de energia (J):e := Io*(w0^2);Escolha um valor para a energia mec\303\242nica (J):E := 5*e/2;#E := 1999999*e/1000000;Velocidade angular inicial (rad/s) correspondente (considerando 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):Vai := sqrt(2*E/Io);`Velocidade angular inicial` = evalf(Vai);Uma constante muito \303\272til (adimensional):Epsilon := E/e-1;Gr\303\241fico da energia potencial:plot([E,e*(1-cos(q))], q=-Pi..Pi, axes=boxed, color=[red,blue], legend=["E","V"], labelfont=[SYMBOL]);Pontos de retorno (somente para oscila\303\247\303\265es):PR := [-solve(cos(t)=-Epsilon), solve(cos(t)=-Epsilon)];"Graus" = evalf(PR*180/Pi);Per\303\255odo (somente para oscila\303\247\303\265es, se estiver rodando, calcule o tempo de uma volta completa):p := To(w0,Epsilon,PR,15);Per\303\255odo (somente para rota\303\247\303\265es):p := Tr(w0,Epsilon,15);Somente para pequenas amplitudes: compare a freq\303\274\303\252ncia calculada pelo per\303\255odo (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) com a freq\303\274\303\252ncia natural LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEmb21lZ2FGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSIwRidGNUY1LyUvc3Vic2NyaXB0c2hpZnRHRj1GNQ==: 2*Pi/'T' - 'omega[0]' = evalf(2*Pi/p - w0);EDO:EDO := diff(theta(t),t$2) + omega[0]^2*sin(theta(t)) + omega[1]*diff(theta(t),t$1)=0; omega[0]='sqrt(g/l)'; omega[1]='b/m/l';Condi\303\247\303\265es iniciais:CIs := theta(0)=o, D(theta)(0)=w;Rotina para escrever a equa\303\247\303\243o diferencial com os valores de seus par\303\242metros e das consid\303\247\303\265es iniciais: Monta_EDO := unapply({EDO,CIs}, omega[0],omega[1],o,w);Solu\303\247\303\243o num\303\251rica da equa\303\247\303\243o diferencial do p\303\252ndulo com dissipa\303\247\303\243o (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):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 := w0,1/2,0,Vai;edo := Monta_EDO(Par);Digits := 10;eh := dsolve(edo, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria (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):Theta := rhs(eh[2]);Velocidade angular (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):Omega := rhs(eh[3]);Acelera\303\247\303\243o angular (via EDO):Alpha := t -> -Par[1]^2*sin(Theta(t)) - Par[2]*Omega(t);Gr\303\241fico da equa\303\247\303\243o hor\303\241ria:plot(Theta(t),t=0..10*p, labels=["t","theta"]);Espa\303\247o de fase:plots[spacecurve]([Theta(t),Omega(t),Alpha(t)],t=0..10*p, labels=["q","w","a"], labelfont=[SYMBOL], title="Espa\303\247o de fase 3D", axes=boxed, shading=zhue, numpoints=500);Simula\303\247\303\243o:AnimaPendulo(0,10*p,Theta, [0.1,blue,red,100]);Situa\303\247\303\243o energ\303\251tica:EC := t -> Ec(t,Omega,Io);EP := t -> Ep(t,Theta,Io,w0);EM := t -> EC(t)+EP(t);plot([EC(t),EP(t),EM(t)], t=0..4*p, color=[red,blue,magenta], legend=["Cin\303\251tica", "Potencial", "Mec\303\242nica"], axes=boxed);Pot\303\252ncia calculada diretamente da varia\303\247\303\243o de energia mec\303\242nica:Pot1 := (t,dt) -> (EM(t+dt)-EM(t))/dt;Veja o comportamento da primeira derivada em fun\303\247\303\243o do valor de dt:plot(Pot1(1,1/i),i=1..100);Pot\303\252ncia calculada diretamente da for\303\247a dissipativa: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, 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, 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, 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Pot2 := -Io*Par[2]*l*(Omega(t)^2);plot([Pot1(t,1/100),Pot2(t)],t=0..4*p, color=[red,blue], numpoints=200, legend=["Num\303\251rica", "Exata"]);Oscilador harm\303\264nico amortecido 2:Equa\303\247\303\243o diferencial + condi\303\247\303\265es iniciais: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 : freq\303\274\303\252ncia 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 : amortecimentoCasos espec\303\255ficos:m := 1; k := (2*Pi)^2; b := Pi;omega := 40;LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMqJi1JJXNxcnRHNiRGJEkoX3N5c2xpYkc2IjYjSSZvbWVnYUdGKyIiIi1JIi9HRik2IywkSSNQaUdGJCIiI0YuEDO:edo := {diff(x(t),t$2) + omega*x(t) + (omega/2)*x(t)^2 + (omega/4)*x(t)^3 = 0, x(0)=-10/100, D(x)(0)=0};Solu\303\247\303\243o:sol := dsolve(edo, numeric, method=dverk78, output=listprocedure);Equa\303\247\303\243o hor\303\241ria:Xn := rhs(sol[2]);Velocidade:Vxn := rhs(sol[3]);Energia cin\303\251tica:T := unapply((m/2)*(Vxn(t)^2),t);Energia potencial:V := unapply((k/2)*(Xn(t)^2),t);An\303\241lise:plot(Xn(t),t=0..10);plot([T(t),V(t),T(t)+V(t)],t=0..2, color=[black,blue,red]);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