<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls1 ws7">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls1 ws7"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls1 ws7">MÉTODOS NUMÉRICOS PARA A </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls1 ws7">SOLUÇÃO DE EQUAÇÕES DE </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls1 ws0">MOVIMENTO<span class="fs3 fc2 ws7"> </span></div><div class="t m0 x1 h6 ya ff3 fs3 fc1 sc0 ls1 ws7"> Introdução: </div><div class="t m0 x1 h7 yb ff2 fs4 fc0 sc0 ls1 ws7">Na <span class="_1 blank"> </span>disciplina <span class="_1 blank"> </span><span class="fc1">cálculo <span class="_1 blank"> </span>numérico</span>, <span class="_1 blank"> </span>existente <span class="_1 blank"> </span>na <span class="_1 blank"> </span>maioria <span class="_1 blank"> </span>dos <span class="_1 blank"> </span>cursos <span class="_1 blank"> </span>de <span class="_1 blank"> </span>engenharia, </div><div class="t m0 x1 h7 yc ff2 fs4 fc0 sc0 ls1 ws7">estudamos <span class="_2 blank"> </span>uma <span class="_2 blank"> </span>série <span class="_2 blank"> </span>de <span class="_2 blank"> </span>técnicas <span class="_2 blank"> </span>matemáticas <span class="_2 blank"> </span>para <span class="_3 blank"> </span>aproximação <span class="_2 blank"> </span>de <span class="_2 blank"> </span>soluções <span class="_2 blank"> </span>de </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 ls1 ws7">equações. Muitos vezes, mal sabemos sua apli<span class="_4 blank"></span>cação e ficamos muita vezes perdidos. </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 ls1 ws7">Nessa <span class="_5 blank"> </span>aulas <span class="_5 blank"> </span>vamos <span class="_5 blank"> </span>apresentar <span class="_5 blank"> </span>uma <span class="_5 blank"> </span>das <span class="_5 blank"> </span>aplicações <span class="_5 blank"> </span>onde <span class="_5 blank"> </span>o <span class="_5 blank"> </span>cálculo <span class="_5 blank"> </span>numérico <span class="_5 blank"> </span>é <span class="_5 blank"> </span>uma </div><div class="t m0 x1 h7 yf ff2 fs4 fc0 sc0 ls1 ws7">ferramenta <span class="_6 blank"> </span>poderosa <span class="_6 blank"> </span>na <span class="_6 blank"> </span>obtenção <span class="_6 blank"> </span>de <span class="_6 blank"> </span>soluções <span class="_6 blank"> </span>para <span class="_7 blank"> </span>problemas <span class="_6 blank"> </span>que <span class="_6 blank"> </span>envolvem </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 ls1 ws7">equações diferenciais ordinárias não lineares. </div><div class="t m0 x1 h6 y11 ff3 fs3 fc1 sc0 ls1 ws7">Equações Diferenciais Ordinárias \u2013 Abordagem Numérica<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y12 ff2 fs4 fc0 sc0 ls1 ws7">Em <span class="_8 blank"> </span>problemas <span class="_8 blank"> </span>de <span class="_8 blank"> </span>engenharia, <span class="_8 blank"> </span>mais <span class="_8 blank"> </span>especificamente <span class="_8 blank"> </span>em <span class="_8 blank"> </span>dinâmica <span class="_8 blank"> </span>de <span class="_8 blank"> </span>sistemas </div><div class="t m0 x1 h7 y13 ff2 fs4 fc0 sc0 ls1 ws7">mecânicos, <span class="_9 blank"> </span>a <span class="_9 blank"> </span>aplicação <span class="_9 blank"> </span>da <span class="_9 blank"> </span>2ª <span class="_9 blank"> </span>lei <span class="_a blank"> </span>de <span class="_9 blank"> </span>newton <span class="_9 blank"> </span>gera <span class="_9 blank"> </span>sistemas <span class="_9 blank"> </span>de <span class="_9 blank"> </span>EDO\u2019s <span class="_a blank"> </span>que <span class="_9 blank"> </span>são, <span class="_9 blank"> </span>em <span class="_9 blank"> </span>sua </div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 ls1 ws7">maioria, <span class="_b blank"> </span>não-lineares, <span class="_b blank"> </span>cujas <span class="_b blank"> </span>soluções <span class="_b blank"> </span>analíticas <span class="_b blank"> </span>são <span class="_b blank"> </span>muito <span class="_b blank"> </span>complexas, <span class="_a blank"> </span>o<span class="_0 blank"> </span>u <span class="_b blank"> </span>não <span class="_b blank"> </span>possui </div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 ls1 ws7">soluções. </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 ls1 ws7">Nesses caso, quando <span class="_0 blank"> </span>a equação diferencial não <span class="_0 blank"> </span>pode se<span class="_0 blank"> </span>r integr<span class="_0 blank"> </span>ada <span class="_0 blank"> </span>de forma f<span class="_0 blank"> </span>echada e </div><div class="t m0 x1 h7 y17 ff2 fs4 fc0 sc0 ls1 ws7">pelos métodos convencionais, nós lançamos mão da abordagem numérica. </div><div class="t m0 x1 h7 y18 ff2 fs4 fc0 sc0 ls1 ws7">Abordagem <span class="_2 blank"> </span>numérica <span class="_2 blank"> </span>permite <span class="_2 blank"> </span>a <span class="_3 blank"> </span>obtenção <span class="_2 blank"> </span>de <span class="_2 blank"> </span>valores <span class="_2 blank"> </span>aproximados <span class="_3 blank"> </span>das <span class="_2 blank"> </span>funções <span class="_2 blank"> </span>a </div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 ls1 ws7">serem <span class="_1 blank"> </span>trabalhadas. <span class="_1 blank"> </span>É <span class="_1 blank"> </span>a <span class="_1 blank"> </span>base <span class="_1 blank"> </span>dos <span class="_1 blank"> </span>softwares <span class="_1 blank"> </span>de <span class="_1 blank"> </span>simulações <span class="_1 blank"> </span>e<span class="_0 blank"> </span>xistentes, <span class="_1 blank"> </span>cada <span class="_1 blank"> </span>um </div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 ls1 ws7">utilizando um ou mais métodos de abordagem numérica. </div><div class="t m0 x1 h7 y1b ff2 fs4 fc0 sc0 ls1 ws7">Equação diferencial é aquela que env<span class="_4 blank"></span>olve derivadas de uma função na sua expressão. </div><div class="t m0 x1 h7 y1c ff2 fs4 fc0 sc0 ls1 ws7">Temos <span class="_c blank"> </span>as <span class="_c blank"> </span>chamadas <span class="_c blank"> </span>Equação <span class="_c blank"> </span>Diferencial <span class="_c blank"> </span>Ordin<span class="_4 blank"></span>ária <span class="_c blank"> </span>(EDO) <span class="_c blank"> </span>que <span class="_c blank"> </span>possui <span class="_c blank"> </span>apenas <span class="_c blank"> </span>01 </div><div class="t m0 x1 h8 y1d ff2 fs4 fc0 sc0 ls1 ws7">variável independente, <span class="ff4">y(x), </span>como por exemplo: <span class="ff5 fs5 ws1 v1">\uebd7\uebec</span></div><div class="t m0 x3 h9 y1e ff5 fs5 fc0 sc0 ls1 ws2">\uebd7\uebeb <span class="fs4 ws3 v2">=<span class="_3 blank"> </span>\ue754<span class="_3 blank"> </span>+ \ue755<span class="ff2 ws7"> .<span class="_0 blank"> </span> </span></span></div><div class="t m0 x1 h7 y1f ff2 fs4 fc0 sc0 ls1 ws7">Se <span class="_9 blank"> </span>tivermos <span class="_9 blank"> </span>mais <span class="_9 blank"> </span>de <span class="_a blank"> </span>uma <span class="_9 blank"> </span>variável, <span class="_9 blank"> </span>temos <span class="_9 blank"> </span>uma <span class="_9 blank"> </span>Equação <span class="_a blank"> </span>Diferencial <span class="_9 blank"> </span>Parcial <span class="_9 blank"> </span>como <span class="_9 blank"> </span>por </div><div class="t m0 x1 h8 y20 ff2 fs4 fc0 sc0 ls1 ws7">exemplo <span class="ff5 fs5 ws1 v1">\uec21\ueb36\uec0f</span></div><div class="t m0 x4 ha y21 ff5 fs5 fc0 sc0 ls1 ws4">\uec21\uebeb\ueb36 <span class="fs4 ls0 v2">+</span><span class="ws1 v3">\uec21\ueb36\uec0f</span></div><div class="t m0 x5 h9 y21 ff5 fs5 fc0 sc0 ls1 ws5">\uec21\uebec\ueb36 <span class="fs4 ws6 v2">=<span class="_3 blank"> </span>\ue7d82\ue7e0 <span class="ff2 ws7"> = 0.<span class="_0 blank"> </span> </span></span></div><div class="t m0 x1 h7 y22 ff2 fs4 fc0 sc0 ls1 ws7">Nosso foco aqui será nas Equações Diferenciais Ordinári<span class="_4 blank"></span>as (EDO).<span class="_0 blank"> </span> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h6 y23 ff3 fs3 fc1 sc0 ls1 ws7">Pontos Importantes sobre as EDO\u2019s: </div><div class="t m0 x1 h7 y24 ff2 fs4 fc0 sc0 ls1 ws7">As <span class="_0 blank"> </span>equações <span class="_0 blank"> </span>diferenciais <span class="_0 blank"> </span>o<span class="_0 blank"> </span>rdinárias <span class="_0 blank"> </span>não <span class="_0 blank"> </span>possui <span class="_d blank"> </span>uma <span class="_0 blank"> </span>única <span class="_0 blank"> </span>solução, <span class="_0 blank"> </span>e <span class="_d blank"> </span>sim <span class="_0 blank"> </span>um <span class="_0 blank"> </span>conjunto, </div><div class="t m0 x1 h7 y25 ff2 fs4 fc0 sc0 ls1 ws7">ou família, de soluções. </div><div class="t m0 x1 h7 y26 ff2 fs4 fc0 sc0 ls1 ws7">Para <span class="_a blank"> </span>obtermos <span class="_b blank"> </span>uma <span class="_b blank"> </span>particularização <span class="_a blank"> </span>de <span class="_a blank"> </span>uma <span class="_b blank"> </span>EDO <span class="_b blank"> </span>temos <span class="_a blank"> </span>que <span class="_b blank"> </span>estabelecer <span class="_a blank"> </span>condições </div><div class="t m0 x1 h7 y27 ff2 fs4 fc0 sc0 ls1 ws7">suplementares, se <span class="_2 blank"> </span>estas forem <span class="_2 blank"> </span>estipuladas no <span class="_2 blank"> </span>mesmo ponto, <span class="_2 blank"> </span>temos as <span class="_2 blank"> </span>chamadas </div><div class="t m0 x1 h7 y28 ff2 fs4 fc0 sc0 ls1 ws7">condições Iniciais e uma PVI (problema com valores Inici<span class="_4 blank"></span>ais). </div><div class="t m0 x1 h7 y29 ff2 fs4 fc0 sc0 ls1 ws7">Por <span class="_0 blank"> </span>outro <span class="_0 blank"> </span>lado, <span class="_0 blank"> </span>se<span class="_0 blank"> </span> <span class="_0 blank"> </span>as <span class="_0 blank"> </span>condições <span class="_0 blank"> </span>estabel<span class="_0 blank"> </span>ecidas <span class="_0 blank"> </span>forem <span class="_d blank"> </span>em <span class="_0 blank"> </span>pontos <span class="_0 blank"> </span>diferentes <span class="_0 blank"> </span>temos <span class="_d blank"> </span>uma </div><div class="t m0 x1 h7 y2a ff2 fs4 fc0 sc0 ls1 ws7">condição de contorno e consequente uma PVC (problema com valores d<span class="_4 blank"></span>e contorno).<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y2b ff2 fs4 fc0 sc0 ls1 ws7">As equações diferenciais podem ser: </div><div class="t m0 x6 h7 y2c ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Lineares quando obedecem a uma função do tipo y = a.x </span></span></div><div class="t m0 x6 hb y2d ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Não Lineares para as demais funções, por exemplo<span class="fs6 ls3 v4">1</span> </span></span><span class="ls4">\uf06c</span><span class="ff2 ws7">\u201d(x) + </span>\uf06c<span class="ff2 ls4">\u2019<span class="fs6 ls5 v4">2</span><span class="ls1 ws7">(x) = 1 </span></span></div><div class="t m0 x1 h6 y2e ff3 fs3 fc1 sc0 ls1 ws7">Métodos Numéricos Aplicados as EDO\u2019s<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 ls1 ws7">Os métodos de integração numérica tem duas características prin<span class="_4 blank"></span>ciapais: </div><div class="t m0 x6 h7 y30 ff2 fs4 fc0 sc0 ls1 ws9">1.<span class="ff7 ls6 ws7"> </span><span class="ws7">Não <span class="_a blank"> </span>pretendem <span class="_a blank"> </span>satisfazer <span class="_b blank"> </span>as <span class="_a blank"> </span>equações <span class="_a blank"> </span>diferenciais <span class="_a blank"> </span>governantes <span class="_a blank"> </span>em <span class="_b blank"> </span>todos <span class="_a blank"> </span>os </span></div><div class="t m0 x7 h7 y31 ff2 fs4 fc0 sc0 ls1 ws7">tempo t, mas somente em intevalos de tempos discretos separados por \u2206t. </div><div class="t m0 x6 h7 y32 ff2 fs4 fc0 sc0 ls1 ws9">2.<span class="ff7 ls7 ws7"> </span><span class="ws7">Entende-se <span class="_2 blank"> </span>que <span class="_2 blank"> </span>há <span class="_3 blank"> </span>um <span class="_2 blank"> </span>tipo <span class="_2 blank"> </span>de <span class="_2 blank"> </span>variação <span class="_3 blank"> </span>adequada <span class="_2 blank"> </span>do <span class="_2 blank"> </span>deslocamento <span class="_2 blank"> </span>x, <span class="_2 blank"> </span>da </span></div><div class="t m0 x7 h7 y33 ff2 fs4 fc0 sc0 ls1 ws7">velocidade x\u2019 e da aceleração x\u201d; </div><div class="t m0 x1 h7 y34 ff2 fs4 fc0 sc0 ls1 ws7">A <span class="_9 blank"> </span>ideia <span class="_9 blank"> </span>básica <span class="_9 blank"> </span>de <span class="_9 blank"> </span>grande <span class="_d blank"> </span>parte <span class="_9 blank"> </span>dos <span class="_9 blank"> </span>métodos <span class="_9 blank"> </span>numéricos <span class="_9 blank"> </span>é <span class="_9 blank"> </span>ser <span class="_9 blank"> </span>capaz <span class="_9 blank"> </span>de <span class="_9 blank"> </span>construir <span class="_9 blank"> </span>uma </div><div class="t m0 x1 h7 y35 ff2 fs4 fc0 sc0 ls1 ws7">solução para uma equação do tipo x\u2019(t) = f(x,t) dada uma condição <span class="ff4 ws9">x(t<span class="fs6 wsa v5">o</span><span class="ls4 ws7">) = x</span><span class="fs6 wsa v5">o</span></span>. </div><div class="t m0 x1 h7 y36 ff2 fs4 fc0 sc0 ls1 ws7">Busca-se <span class="_9 blank"> </span>defini<span class="_4 blank"></span>r <span class="_9 blank"> </span>t<span class="fs6 ls3 v5">1</span>, <span class="_d blank"> </span>t<span class="fs6 ls5 v5">2</span>, <span class="_9 blank"> </span>\u2026,t<span class="fs6 wsa v5">n</span> <span class="_9 blank"> </span>e <span class="_d blank"> </span>c<span class="_0 blank"> </span>alcular <span class="_d blank"> </span>aproximaçõ<span class="_0 blank"> </span>es <span class="_9 blank"> </span>numéric<span class="_4 blank"></span>as <span class="_9 blank"> </span>para <span class="_9 blank"> </span><span class="ff4 ws9">x<span class="fs6 ws7 v5">i <span class="_d blank"> </span></span>(t<span class="fs6 ls8 v5">i</span><span class="ls9">)</span></span> <span class="_9 blank"> </span>baseado <span class="_9 blank"> </span>em </div><div class="t m0 x1 h7 y37 ff2 fs4 fc0 sc0 ls1 ws7">informações passadas, ou <span class="_0 blank"> </span>seja, obtidas <span class="_0 blank"> </span>no passo <span class="_0 blank"> </span>anterior, ou nas <span class="_0 blank"> </span>condições iniciais, <span class="_0 blank"> </span>no </div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 ls1 ws7">ínicio dos cálculos. </div><div class="t m0 x1 h7 y39 ff2 fs4 fc0 sc0 ls1 ws7">Existem vários métodos númericos, aplicáveis a EDO\u2019s, como por exemplo: </div><div class="t m0 x6 h7 y3a ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Taylor; </span></span></div><div class="t m0 x6 h7 y3b ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Diferenças Finitas; </span></span></div><div class="t m0 x6 h7 y3c ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Runge- Kutta; </span></span></div><div class="t m0 x6 h7 y3d ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Newmark; </span></span></div><div class="t m0 x6 h7 y3e ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Houbolt; </span></span></div><div class="t m0 x6 h7 y3f ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Método de Wilson; </span></span></div><div class="t m0 x6 h7 y40 ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> <span class="ff2 ls1">Entre outros </span></span></div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 ls1 ws7">Vários desses métodos estão presentes em softw<span class="_4 blank"></span>ares de matemática, como Matlab e </div><div class="t m0 x1 h7 y42 ff2 fs4 fc0 sc0 ls1 ws7">SciLab, de tal maneira, que basta você compreender como<span class="_4 blank"></span> usar esses métodos e </div><div class="t m0 x1 h7 y43 ff2 fs4 fc0 sc0 ls1 ws7">como programá-los em tais softwares. </div><div class="t m0 x1 h2 y44 ff1 fs0 fc0 sc0 ls1 ws7"> <span class="_0 blank"> </span> <span class="_e blank"></span> </div><div class="t m0 x1 hc y45 ff2 fs7 fc0 sc0 ls1 wsb">1<span class="fs8 ws7 v6"> A notação <span class="ff6 lsa">\uf06c</span>\u2019 e <span class="ff6 lsa">\uf06c</span>\u201d são forma<span class="_4 blank"></span>s de representar, <span class="_4 blank"></span>respectivamente, as derivad<span class="_4 blank"></span>as de primeira e <span class="_4 blank"></span>de segunda </span></div><div class="t m0 x1 hd y46 ff2 fs8 fc0 sc0 ls1 ws7">ordem da função <span class="ff6 lsa">\uf06c</span>. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h7 y47 ff2 fs4 fc0 sc0 ls1 ws7">Vamos abordar nessa aula, o Método de Taylor e o Método d<span class="_4 blank"></span>e Runge-Kutta, para mais </div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls1 ws7">detalhes sobre outros métodos, consulte livros especializados como, p<span class="_4 blank"></span>or exemplo, </div><div class="t m0 x1 h7 y49 ff2 fs4 fc0 sc0 ls1 ws7">Vibrações Mecânicas do Singiresu RAO. </div><div class="t m0 x1 h6 y4a ff3 fs3 fc1 sc0 ls1 ws7">Método de Série de Taylor (PSD) </div><div class="t m0 x1 h7 y4b ff2 fs4 fc0 sc0 ls1 ws7">Podemos utilizar a série de Taylor para a resolução de qualquer tipo de <span class="_4 blank"></span>EDO, no </div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 ls1 ws7">entanto, os resultados em termos de eficiência computacional<span class="_4 blank"></span> são limitados para </div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 ls1 ws7">EDO\u2019s de ordem baixa. </div><div class="t m0 x1 h7 y4e ff2 fs4 fc0 sc0 ls1 ws7">Aproxima-se a função x(t) em um ponto em torno de t = tn+1 por uma <span class="_4 blank"></span>série: </div><div class="t m0 x8 he y4f ff5 fs4 fc0 sc0 lsb">\ue754<span class="ls1 wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls1 wsd v9">\uebe1\ueb3e\ueb35 </span><span class="lsc v7">)</span><span class="ls1 wse">\u2248<span class="_3 blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsd v9">\uebe1</span><span class="lse v7">)</span><span class="wsf">+<span class="_2 blank"> </span>\ue754<span class="_f blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsf v9">\uebe1</span><span class="wsc v7">)</span><span class="ws10">.<span class="_5 blank"> </span>\u2206\ue750<span class="_2 blank"> </span>+<span class="_2 blank"> </span>\ue754<span class="_f blank"></span>\u0308 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsd v9">\uebe1</span><span class="ls10 v7">)</span><span class="wsc va">(</span><span class="ws11 vb">\u2206\ue750</span><span class="wsc va">)</span><span class="fs5 vc">\ueb36</span></span></span></span></div><div class="t m0 x9 hf y50 ff5 fs4 fc0 sc0 ls11">2<span class="ls1 ws3 vd">+ \ue754<span class="_10 blank"></span>\u20db</span></div><div class="t m0 xa h10 y51 ff5 fs4 fc0 sc0 ls1 wsc">(<span class="v8">\ue750</span><span class="fs5 lsf ve">\uebe1</span><span class="ls10">)</span><span class="vb">(</span><span class="ws11 vf">\u2206\ue750</span><span class="vb">)</span><span class="fs5 v3">\ueb37</span></div><div class="t m0 xb h11 y50 ff5 fs4 fc0 sc0 ls11">6<span class="ls1 ws3 vd">+ \u2026<span class="_5 blank"> </span>+ \ue754<span class="_d blank"> </span></span><span class="fs5 lsd v10">\uebe1</span><span class="ls1 wsc vd">(\ue750</span><span class="fs5 lsd v11">\uebe1</span><span class="ls12 vd">)</span><span class="ls1 wsc v12">(</span><span class="ls1 ws11 v13">\u2206\ue750 </span><span class="ls1 wsc v12">)<span class="fs5 v4">\uebe1</span></span></div><div class="t m0 xc h12 y50 ff5 fs4 fc0 sc0 ls1 ws12">\ue74a! <span class="ff2 ws7 vd"> </span></div><div class="t m0 x1 h13 y52 ff2 fs4 fc0 sc0 ls1 ws7">Onde <span class="ff5 ws13">\u2206\ue750<span class="_c blank"> </span>= \ue750<span class="fs5 ws14 v9">\uebe1\ueb3e\ueb35 </span><span class="ws3">\u2212 \ue750<span class="fs5 lsf v9">\uebe1</span></span></span>é o chamado passo de integração. </div><div class="t m0 x1 h13 y53 ff2 fs4 fc0 sc0 ls1 ws7">Quanto <span class="_5 blank"> </span>menor <span class="_5 blank"> </span>for <span class="ff5 ws11">\u2206\ue750</span><span class="ls10"> <span class="ff8 ls4">\uf0e0</span></span> menor <span class="_b blank"> </span>se<span class="_0 blank"> </span>rá <span class="_5 blank"> </span>o <span class="_5 blank"> </span>erro obt<span class="_4 blank"></span>ido <span class="_5 blank"> </span>ao <span class="_5 blank"> </span>final, ou <span class="_b blank"> </span>seja, mais <span class="_b blank"> </span>próximo </div><div class="t m0 x1 h7 y54 ff2 fs4 fc0 sc0 ls1 ws7">estaremos do resultado real. </div><div class="t m0 x1 h7 y55 ff2 fs4 fc0 sc0 ls1 ws7">Um caso <span class="_5 blank"> </span>particular para <span class="_5 blank"> </span>a série <span class="_5 blank"> </span>de taylor <span class="_5 blank"> </span>é a aproximção <span class="_5 blank"> </span>de 1ª <span class="_5 blank"> </span>ordem, conhecido </div><div class="t m0 x1 h7 y56 ff2 fs4 fc0 sc0 ls1 ws7">método de Euler: </div><div class="t m0 xd h14 y57 ff5 fs4 fc0 sc0 lsb">\ue754<span class="ls1 wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls1 wsd v9">\uebe1\ueb3e\ueb35 </span><span class="lsc v7">)</span><span class="ls1 wse">\u2248<span class="_3 blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsd v9">\uebe1</span><span class="lse v7">)</span><span class="wsf">+<span class="_2 blank"> </span>\ue754<span class="_f blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsf v9">\uebe1</span><span class="wsc v7">)</span><span class="ws11">.<span class="_5 blank"> </span>\u2206\ue750<span class="ff2 ws7"> </span></span></span></span></div><div class="t m0 x1 h6 y58 ff3 fs3 fc1 sc0 ls1 ws7">Procedimento: </div><div class="t m0 x1 h7 y59 ff2 fs4 fc0 sc0 ls1 ws7">O <span class="_3 blank"> </span>procedimento <span class="_2 blank"> </span>a <span class="_3 blank"> </span>ser <span class="_3 blank"> </span>adotado <span class="_2 blank"> </span>é <span class="_3 blank"> </span>conhecer <span class="_3 blank"> </span>as <span class="_2 blank"> </span>condições <span class="_3 blank"> </span>iniciais <span class="_3 blank"> </span>no <span class="_3 blank"> </span>instante <span class="_3 blank"> </span><span class="ff4 ws9">t<span class="fs6 wsa v5">o</span></span> <span class="_3 blank"> </span>e </div><div class="t m0 x1 h7 y5a ff2 fs4 fc0 sc0 ls1 ws7">prosseguir <span class="_b blank"> </span>na <span class="_b blank"> </span>aproximação <span class="_b blank"> </span>em <span class="_b blank"> </span>instantes <span class="_b blank"> </span>t<span class="_0 blank"> </span><span class="fs6 ls3 v5">1</span> <span class="_b blank"> </span>= <span class="_b blank"> </span>t<span class="fs6 ls13 v5">o</span> <span class="_b blank"> </span>+ <span class="_b blank"> </span>\u2206t <span class="_b blank"> </span><span class="ff8 ls4">\uf0e0</span> <span class="_5 blank"> </span>t<span class="fs6 ls14 v5">N</span> <span class="_b blank"> </span>= <span class="_b blank"> </span>t<span class="fs6 ls13 v5">o</span> <span class="_b blank"> </span>+ <span class="_b blank"> </span>N.\u2206t, <span class="_b blank"> </span>onde <span class="_b blank"> </span>N <span class="_b blank"> </span> <span class="_b blank"> </span>é<span class="_0 blank"> </span> <span class="_b blank"> </span>o </div><div class="t m0 x1 h7 y5b ff2 fs4 fc0 sc0 ls1 ws7">número de amostras a avaliar. </div><div class="t m0 x1 h14 y5c ff2 fs4 fc0 sc0 ls1 ws7">Para uma EDO do tipo: <span class="ff5 wsf">\ue754<span class="_f blank"></span>\u0308 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="lse v7">)</span></span><span class="ws10">+<span class="_2 blank"> </span>\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="lse v7">)</span></span><span class="ws15">+<span class="_2 blank"> </span>\ue754<span class="wsc v7">(</span><span class="ls15">\ue750<span class="ls16 v7">)</span></span><span class="ws13">= 0<span class="ff2 ws7">, com condições iniciais </span><span class="wsf">\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="wsc v7">)</span><span class="wse">,<span class="_11 blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="wsc v7">)</span><span class="ff2 ws7"> </span></span></span></span></span></span></span></div><div class="t m0 x1 h7 y5d ff2 fs4 fc0 sc0 ls1 ws7">conhecidas, tem-se: </div><div class="t m0 xe h14 y5e ff5 fs4 fc0 sc0 ls1 wsf">\ue754<span class="_f blank"></span>\u0308 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="ls16 v7">)</span>=<span class="_3 blank"> </span>\u2212\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="lse v7">)</span><span class="ws15">\u2212<span class="_2 blank"> </span>\ue754<span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="wsc v7">)</span><span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y5f ff2 fs4 fc0 sc0 ls1 ws7">Para um instante t<span class="fs6 ls3 v5">1</span> = t<span class="fs6 v5">o </span>+ \u2206t, as aproximações serão: </div><div class="t m0 xf h14 y60 ff5 fs4 fc0 sc0 ls18">\ue754<span class="ls1 wsc v7">(</span><span class="ls1">\ue750</span></div><div class="t m0 x10 h15 y61 ff5 fs5 fc0 sc0 ls19">\ueb35<span class="fs4 ls16 v14">)<span class="ls1 ws15 v8">=<span class="_3 blank"> </span>\ue754 </span><span class="ls1 wsc">(<span class="v8">\ue750</span></span></span><span class="ls1a">\uebe2<span class="fs4 ls1 ws11 v15">+<span class="_2 blank"> </span>\u2206\ue750<span class="ls16 v7">)</span><span class="ws15">\u2248<span class="_c blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750<span class="_4 blank"></span><span class="fs5 ls17 v9">\uebe2<span class="fs4 lse v14">)<span class="ls1 ws10 v8">+<span class="_2 blank"> </span>\ue754<span class="_f blank"></span>\u0307 (\ue750<span class="fs5 ls17 v9">\uebe2</span><span class="ws16">)\u2206\ue750<span class="ff2 ws7"> </span></span></span></span></span></span></span></span></span></span></div><div class="t m0 x11 h14 y62 ff5 fs4 fc0 sc0 ls1 ws10">\ue754<span class="_f blank"></span>\u0307 <span class="wsc v7">(</span>\ue750</div><div class="t m0 x12 h15 y63 ff5 fs5 fc0 sc0 ls19">\ueb35<span class="fs4 ls16 v14">)<span class="ls1 wsf v8">=<span class="_3 blank"> </span>\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750<span class="fs5 ls1a v9">\uebe2</span><span class="ws11">+<span class="_12 blank"> </span>\u2206\ue750 <span class="ls16 v7">)</span></span></span></span>\u2248<span class="_3 blank"> </span>\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls17 v9">\uebe2</span><span class="lse v7">)</span><span class="ws10">+<span class="_12 blank"> </span>\ue754<span class="_f blank"></span>\u0308 <span class="ff2 ws7"> </span><span class="wsc">(\ue750<span class="fs5 ls17 v9">\uebe2</span><span class="ws16">)\u2206\ue750 <span class="ff2 ws7"> </span></span></span></span></span></span></div><div class="t m0 x1 h13 y64 ff2 fs4 fc0 sc0 ls1 ws7">E portanto a função <span class="ff5 ws10">\ue754<span class="_e blank"></span>\u0308 <span class="ff2 ws7"> </span><span class="wsc">(\ue750<span class="_4 blank"></span><span class="fs5 ls17 v9">\uebe2<span class="fs4 ls1 v15">)<span class="ff2 ws7"> será aproximada para: </span></span></span></span></span></div><div class="t m0 xe h14 y65 ff5 fs4 fc0 sc0 ls1 ws10">\ue754<span class="_f blank"></span>\u0308 <span class="ff2 ws7"> </span><span class="wsc v7">(</span>\ue750</div><div class="t m0 x13 h15 y66 ff5 fs5 fc0 sc0 ls19">\ueb35<span class="fs4 lsc v14">)<span class="ls1 ws10 v8">\u2248<span class="_3 blank"> </span>\u2212\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(</span>\ue750</span></span></div><div class="t m0 x14 h15 y66 ff5 fs5 fc0 sc0 ls19">\ueb35<span class="fs4 ls1b v14">)<span class="ls1 ws15 v8">\u2212<span class="_12 blank"> </span>\ue754 </span><span class="ls1 wsc">(<span class="v8">\ue750</span></span></span></div><div class="t m0 x15 h15 y66 ff5 fs5 fc0 sc0 ls19">\ueb35<span class="fs4 ls1 wsc v14">)<span class="ff2 ws7 v8"> </span></span></div><div class="t m0 x1 h7 y67 ff2 fs4 fc0 sc0 ls1 ws7">E assim por diante, até chegar a t<span class="fs6 ls14 v5">N</span>. </div><div class="t m0 x1 h7 y68 ff2 fs4 fc0 sc0 ls1 ws7">A <span class="_5 blank"> </span>principal <span class="_5 blank"> </span>desvantagem <span class="_b blank"> </span>dess<span class="_0 blank"> </span>e <span class="_b blank"> </span>método <span class="_5 blank"> </span>é <span class="_5 blank"> </span>a <span class="_5 blank"> </span>necessidade <span class="_5 blank"> </span>de <span class="_5 blank"> </span>se <span class="_5 blank"> </span>verificar <span class="_5 blank"> </span>valores <span class="_b blank"> </span>das </div><div class="t m0 x1 h7 y69 ff2 fs4 fc0 sc0 ls1 ws7">derivadas de ordem mais alta da função x(t) a ser aproxima<span class="_4 blank"></span>da. </div><div class="t m0 x1 h6 y6a ff3 fs3 fc1 sc0 ls1 ws7">Método de Runge-Kutta: </div><div class="t m0 x1 h14 y6b ff2 fs4 fc0 sc0 ls1 ws7">Este <span class="_5 blank"> </span>método <span class="_5 blank"> </span>visa <span class="_5 blank"> </span>aproveitar <span class="_5 blank"> </span>as <span class="_5 blank"> </span>qualidades <span class="_5 blank"> </span>da <span class="_5 blank"> </span>série <span class="_5 blank"> </span>de <span class="_5 blank"> </span>Taylor <span class="_5 blank"> </span>para <span class="_5 blank"> </span>aproximar <span class="ff5 ls18">\ue754<span class="ls1 wsc v7">(</span><span class="ls15">\ue750<span class="ls1 wsc v7">)</span></span></span>, </div><div class="t m0 x1 h14 y6c ff2 fs4 fc0 sc0 ls1 ws7">eliminando a necessidade de cálculo das derivadas de <span class="ff5 lsb">\ue754<span class="ls1 wsc v7">(</span><span class="ls15">\ue750<span class="ls1 wsc v7">)</span></span></span>. </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h14 y47 ff2 fs4 fc0 sc0 ls1 ws7">Em <span class="_5 blank"> </span>compensação é <span class="_5 blank"> </span>necessário <span class="_5 blank"> </span>calcular <span class="ff5 wsf">\ue754<span class="_f blank"></span>\u0307 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="ls16 v7">)</span></span><span class="ws16">=<span class="_c blank"> </span>\ue742 <span class="wsc v7">(</span><span class="ws11">\ue754 ,<span class="_5 blank"> </span>\ue750 <span class="wsc v7">)</span><span class="ff2 ws7"> <span class="_5 blank"> </span>em vários <span class="_5 blank"> </span>pontos <span class="_5 blank"> </span>e também </span></span></span></span></div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls1 ws7">temos <span class="_b blank"> </span>um <span class="_b blank"> </span>métodos<span class="_0 blank"> </span> <span class="_b blank"> </span>de <span class="_b blank"> </span>Runge-Kutta <span class="_b blank"> </span>para <span class="_b blank"> </span>cad<span class="_0 blank"> </span>a <span class="_b blank"> </span>ordem <span class="_b blank"> </span>considerado, <span class="_b blank"> </span>quanto <span class="_b blank"> </span>maior <span class="_5 blank"> </span>a </div><div class="t m0 x1 h7 y49 ff2 fs4 fc0 sc0 ls1 ws7">ordem, menor <span class="_5 blank"> </span>será o erro, mas e<span class="_4 blank"></span>m compensação, <span class="_5 blank"> </span>maior o recurso computacio<span class="_4 blank"></span>nal </div><div class="t m0 x1 h7 y6d ff2 fs4 fc0 sc0 ls1 ws7">necessário. </div><div class="t m0 x1 h6 y6e ff3 fs3 fc1 sc0 ls1 ws7">Runge-Kutta de 4ª Ordem: </div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 ls1 ws7">É um dos métodos de runge-kutta mais populares, que produz bons resultados com </div><div class="t m0 x1 h7 y6f ff2 fs4 fc0 sc0 ls1 ws7">um recurso computacional razoável. </div><div class="t m0 x1 h7 y70 ff2 fs4 fc0 sc0 ls1 ws7">É definido pela expressão: </div><div class="t m0 x16 h10 y71 ff5 fs4 fc0 sc0 lsb">\ue754<span class="ls1 wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 ls1c v9">\uebe1</span><span class="ls1 ws3">+ 1<span class="ls16 v7">)</span><span class="wse">\u2248<span class="_3 blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750</span></span><span class="fs5 lsd v9">\uebe1</span><span class="ls1b v7">)</span><span class="ls1d">+</span><span class="vb">1</span></span></span></div><div class="t m0 x17 h16 y72 ff5 fs4 fc0 sc0 ls1e">6<span class="ls1 wsc vf">(<span class="v8">\ue747</span></span></div><div class="t m0 x9 h15 y73 ff5 fs5 fc0 sc0 ls1f">\ueb35<span class="fs4 ls1 ws3 v15">+ 2\ue747</span>\ueb36<span class="fs4 ls1 ws3 v15">+ 2\ue747</span><span class="ls20">\ueb37<span class="fs4 ls1 ws3 v15">+ \ue747</span><span class="ls19">\ueb38<span class="fs4 ls1 wsc v14">)<span class="ws11 v8">.<span class="_5 blank"> </span>\u2206\ue750 <span class="ff2 ws7"> </span></span></span></span></span></div><div class="t m0 x1 h7 y74 ff2 fs4 fc0 sc0 ls1 ws7">Onde k<span class="fs6 ls3 v5">1</span>, k<span class="fs6 ls5 v5">2</span>, k<span class="fs6 wsa v5">3</span> e k<span class="fs6 ls21 v5">4</span> são constantes a serem calculadas pelas equações: </div><div class="t m0 x18 h13 y75 ff5 fs4 fc0 sc0 ls1">\ue747</div><div class="t m0 x19 h15 y76 ff5 fs5 fc0 sc0 ls22">\ueb35<span class="fs4 ls1 ws11 v15">=<span class="_3 blank"> </span>\u2206\ue750 .<span class="_5 blank"> </span>\ue742<span class="_d blank"> </span>\ued6b\ue750</span><span class="lsd">\uebe1<span class="fs4 ls1 ws15 v15">,<span class="_5 blank"> </span>\ue754 <span class="wsc v7">(<span class="v8">\ue750<span class="_4 blank"></span><span class="fs5 lsf v9">\uebe1<span class="fs4 ls1 v14">)<span class="ls4 v8">\ued6f<span class="ff2 ls1 ws7"> </span></span></span></span></span></span></span></span></div><div class="t m0 x12 h10 y77 ff5 fs4 fc0 sc0 ls1 wsc">\ue747<span class="fs5 ls23 v9">\ueb36</span><span class="ws11">=<span class="_3 blank"> </span>\u2206\ue750 .<span class="_13 blank"> </span>\ue742<span class="_13 blank"> </span><span class="ls4 v0">\ued6c</span></span>\ue750<span class="fs5 ls1c v9">\uebe1</span><span class="ls1d">+</span><span class="vb">\u2206\ue750</span></div><div class="t m0 x1a h17 y78 ff5 fs4 fc0 sc0 ls24">2<span class="ls1 ws15 vd">,<span class="_5 blank"> </span>\ue754 </span><span class="ls1 wsc vf">(<span class="v8">\ue750</span></span><span class="fs5 lsf v11">\uebe1</span><span class="ls1b vf">)</span><span class="ls0 vd">+</span><span class="ls1 v13">\ue747</span></div><div class="t m0 x1b h18 y79 ff5 fs5 fc0 sc0 ls1">\ueb35</div><div class="t m0 x1c h19 y78 ff5 fs4 fc0 sc0 ls25">2<span class="ls1 wsc vd">\ued70</span><span class="ff2 ls1 ws7 vd"> </span></div><div class="t m0 x12 h10 y7a ff5 fs4 fc0 sc0 ls1 wsc">\ue747<span class="fs5 ls23 v9">\ueb37</span><span class="ws11">=<span class="_3 blank"> </span>\u2206\ue750 .<span class="_5 blank"> </span>\ue742<span class="_12 blank"> </span><span class="ls4 v0">\ued6c</span></span>\ue750<span class="fs5 ls1c v9">\uebe1</span><span class="ls0">+</span><span class="vb">\u2206\ue750</span></div><div class="t m0 x1a h17 y7b ff5 fs4 fc0 sc0 ls24">2<span class="ls1 wse vd">,<span class="_5 blank"> </span>\ue754 </span><span class="ls1 wsc vf">(<span class="v8">\ue750</span></span><span class="fs5 lsd v11">\uebe1</span><span class="ls1b vf">)</span><span class="ls1d vd">+</span><span class="ls1 wsc v13">\ue747<span class="fs5 v9">\ueb36</span></span></div><div class="t m0 x1c h19 y7b ff5 fs4 fc0 sc0 ls26">2<span class="ls4 vd">\ued70</span><span class="ff2 ls1 ws7 vd"> </span></div><div class="t m0 x1d h14 y7c ff5 fs4 fc0 sc0 ls1 wsc">\ue747<span class="fs5 ls23 v9">\ueb38</span><span class="ws11">=<span class="_3 blank"> </span>\u2206\ue750 .<span class="_5 blank"> </span>\ue742<span class="_d blank"> </span>(\ue750<span class="fs5 ls1c v9">\uebe1</span>+<span class="_12 blank"> </span>\u2206\ue750,<span class="_13 blank"> </span>\ue754 </span><span class="v7">(</span>\ue750<span class="fs5 lsd v9">\uebe1</span><span class="lse v7">)</span><span class="ws3">+ \ue747<span class="fs5 ls19 v9">\ueb37</span></span>)<span class="ff2 ws7"> </span></div><div class="t m0 x1 h6 y7d ff3 fs3 fc1 sc0 ls1 ws7">Método de Newmark: </div><div class="t m0 x1 h7 y7e ff2 fs4 fc0 sc0 ls1 ws7">Um dos métodos mais populares para resolução <span class="_4 blank"></span>de EDO\u2019s de 2ª ordem. </div><div class="t m0 x1 h7 y7f ff2 fs4 fc0 sc0 ls1 ws7">É <span class="_0 blank"> </span>baseado <span class="_d blank"> </span>na <span class="_d blank"> </span>premissa <span class="_0 blank"> </span>de <span class="_d blank"> </span>que <span class="_d blank"> </span>a <span class="_d blank"> </span>aceleração <span class="_0 blank"> </span>varia <span class="_0 blank"> </span>linearmente <span class="_d blank"> </span>entre <span class="_0 blank"> </span>d<span class="_0 blank"> </span>ois <span class="_0 blank"> </span>intervalos <span class="_d blank"> </span>de </div><div class="t m0 x1 h7 y80 ff2 fs4 fc0 sc0 ls1 ws7">tempo. </div><div class="t m0 x1 h7 y81 ff2 fs4 fc0 sc0 ls1 ws7">Softwares <span class="_2 blank"> </span>como <span class="_3 blank"> </span>o <span class="_2 blank"> </span>MatLab <span class="_2 blank"> </span>e <span class="_3 blank"> </span>SciLab <span class="_2 blank"> </span>já <span class="_3 blank"> </span>possuem <span class="_2 blank"> </span>recursos <span class="_3 blank"> </span>para <span class="_2 blank"> </span>a <span class="_3 blank"> </span>sua <span class="_2 blank"> </span>aplicação <span class="_3 blank"> </span>e </div><div class="t m0 x1 h7 y82 ff2 fs4 fc0 sc0 ls1 ws7">também é a base do software Ansys®, usado para análise de elementos fini<span class="_4 blank"></span>tos.<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y83 ff2 fs4 fc0 sc0 ls1 ws7">Considerando a equação de movimento do sistema descrita pelas matrizes de <span class="_0 blank"> </span>massa e </div><div class="t m0 x1 h7 y84 ff2 fs4 fc0 sc0 ls1 ws7">rigidez e com o amortecimento sendo do tipo proporcional a massa e/ou rigi<span class="_4 blank"></span>dez </div><div class="t m0 x1e h13 y85 ff5 fs4 fc0 sc0 ls1 ws17">\ue72f<span class="_0 blank"> </span>\ue754<span class="_e blank"></span>\u0308<span class="_c blank"> </span>+ \ue725<span class="_d blank"> </span>\ue754<span class="_e blank"></span>\u0307<span class="_1 blank"> </span>+ \ue72d\ue754<span class="_c blank"> </span>=<span class="_c blank"> </span>\ue728<span class="_0 blank"> </span><span class="ff2 ws7"> </span></div><div class="t m0 x1 h13 y86 ff2 fs4 fc0 sc0 ls1 ws7">Onde: <span class="_d blank"> </span><span class="ff5 ws15">\ue754 ,<span class="_14 blank"></span>\u0308<span class="_6 blank"> </span>\ue754<span class="_f blank"></span>\u0307<span class="_a blank"> </span><span class="ff2 ws7"> </span><span class="wsc">\ue741<span class="ff2 ws7"> </span><span class="lsb">\ue754</span><span class="ff2 ws7"> <span class="_9 blank"> </span>os <span class="_d blank"> </span>vetores <span class="_9 blank"> </span>aceleração, <span class="_d blank"> </span>velocidade <span class="_d blank"> </span>e <span class="_9 blank"> </span>deslocamento <span class="_d blank"> </span>o <span class="_9 blank"> </span>esquema <span class="_d blank"> </span>geral <span class="_9 blank"> </span>do </span></span></span></div><div class="t m0 x1 h7 y87 ff2 fs4 fc0 sc0 ls1 ws7">Método de Newmark fica: </div><div class="t m0 x1f h14 y88 ff5 fs4 fc0 sc0 ls1 ws10">\ue754<span class="_f blank"></span>\u0307 <span class="wsc v7">(</span><span class="ws11">\ue750<span class="_3 blank"> </span>+<span class="_12 blank"> </span>\u2206\ue750 <span class="ls16 v7">)</span><span class="ls27">=</span><span class="ff2 ws7"> </span></span>\ue754<span class="_e blank"></span>\u0307 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="ls1b v7">)</span></span><span class="ws11">+<span class="_12 blank"> </span>\u2206\ue750 <span class="wsc v7">[(</span><span class="ws18">1 \u2212 <span class="ff6 ws8">\uf061</span><span class="wsc v7">)</span><span class="wsf">.<span class="_13 blank"> </span>\ue754<span class="_f blank"></span>\u0308 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="lse v7">)</span></span><span class="ws16">+<span class="_2 blank"> </span>\ue7d9 <span class="wsc v7">(</span><span class="ws11">\ue750<span class="_2 blank"> </span>+<span class="_2 blank"> </span>\u2206\ue750 <span class="wsc v7">)]</span><span class="ff2 ws7"> </span></span></span></span></span></span></div><div class="t m0 x20 h10 y89 ff5 fs4 fc0 sc0 ls18">\ue754<span class="ls1 wsc v7">(</span><span class="ls1 ws11">\ue750<span class="_2 blank"> </span>+<span class="_12 blank"> </span>\u2206\ue750 <span class="lsc v7">)</span><span class="wse">=<span class="_c blank"> </span>\ue754 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="ls1b v7">)</span></span><span class="ws9">+<span class="_12 blank"> </span>\u2206\ue750 \ue754<span class="_e blank"></span>\u0307<span class="_9 blank"> </span><span class="wsc v7">(</span><span class="ls15">\ue750<span class="lse v7">)</span></span><span class="ws3">+ \ued64<span class="ls4 v0">\ued6c</span><span class="vb">1</span></span></span></span></span></div><div class="t m0 x9 h16 y8a ff5 fs4 fc0 sc0 ls28">2<span class="ls1 ws19 vd">\u2212<span class="_12 blank"> </span>\ue7da </span><span class="ls29 vd">\ued70</span><span class="ls1 ws10 vd">\ue754<span class="_e blank"></span>\u0308 <span class="wsc v7">(</span><span class="ls15">\ue750<span class="ls1b v7">)</span><span class="ls1 ws9">+<span class="_12 blank"> </span>\ue7da\ue754 (\ue750<span class="_3 blank"> </span>+<span class="_12 blank"> </span>\u2206\ue750 )\ued68<span class="_5 blank"> </span>\u2206 \ue750<span class="_d blank"> </span><span class="fs5 ls2a v16">\ueb36</span><span class="ff2 ws7"> </span></span></span></span></div><div class="t m0 x1 h7 y8b ff2 fs4 fc0 sc0 ls1 ws7">Onde: </div><div class="t m0 x6 h7 y8c ff6 fs4 fc0 sc0 ls1 ws8">\uf0b7<span class="ff7 ls2 ws7"> </span>\uf061<span class="ff2 ls4 ws7"> e <span class="ff6">\uf062</span><span class="ls1"> </span><span class="ff8">\uf0e0</span><span class="ls1"> </span></span>\uf061<span class="ff2 ws7"> e </span><span class="ls4">\uf062</span><span class="ff2 ws7"> <span class="ff8 ls4">\uf0e0</span> </span></div><div class="t m0 x1 h7 y8d ff2 fs4 fc0 sc0 ls1 ws7">Os <span class="_0 blank"> </span>parâmetros <span class="_d blank"> </span><span class="ff6 fs0 ls2b">\uf061</span> <span class="_0 blank"> </span>e <span class="_d blank"> </span><span class="ff6 fs0 ws1a">\uf062</span> <span class="_0 blank"> </span>são <span class="_d blank"> </span>chamados <span class="_0 blank"> </span>de <span class="_d blank"> </span>\u201cparâmetros <span class="_0 blank"> </span>de <span class="_d blank"> </span>Newmark\u201d <span class="_0 blank"> </span> <span class="_d blank"> </span>e <span class="_d blank"> </span>indicam <span class="_0 blank"> </span>quanto <span class="_d blank"> </span>a </div><div class="t m0 x1 h7 y8e ff2 fs4 fc0 sc0 ls1 ws7">aceleração <span class="_d blank"> </span>ao <span class="_d blank"> </span>final <span class="_d blank"> </span>do <span class="_9 blank"> </span>int<span class="_4 blank"></span>ervalo <span class="_d blank"> </span>entra <span class="_d blank"> </span>nas <span class="_9 blank"> </span>equações <span class="_0 blank"> </span>de <span class="_9 blank"> </span>velocid<span class="_4 blank"></span>ade <span class="_d blank"> </span>e <span class="_d blank"> </span>deslocamento <span class="_d blank"> </span>ao </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg5.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">5 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h7 y47 ff2 fs4 fc0 sc0 ls1 ws7">final <span class="_6 blank"> </span>do <span class="_7 blank"> </span>intervalo <span class="_6 blank"> </span>\u2206<span class="_0 blank"> </span>t <span class="_6 blank"> </span>e <span class="_7 blank"> </span>também <span class="_6 blank"> </span>estão <span class="_7 blank"> </span>relacionados <span class="_6 blank"> </span>a <span class="_7 blank"> </span>precisão <span class="_6 blank"> </span>e <span class="_7 blank"> </span>estabilidade </div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls1 ws7">dessejados. </div><div class="t m0 x1 h7 y8f ff2 fs4 fc0 sc0 ls1 ws7">Por exemplo: </div><div class="t m0 x1 h8 y90 ff2 fs4 fc0 sc0 ls4 ws7">Se <span class="ff6 ls1 ws8">\uf061<span class="ff5 ws7"> </span><span class="ls4">\uf03d</span></span><span class="ff5 ls12"> <span class="fs5 ls1 v1">\ueb35</span></span></div><div class="t m0 x7 ha y91 ff5 fs5 fc0 sc0 ls2c">\ueb36<span class="fs4 ls1 ws7 v2"> \ue741<span class="ff2"> <span class="ff6 ls2d">\uf062<span class="ff5 ls27">=</span></span></span><span class="fs5 v1">\ueb35</span></span></div><div class="t m0 x21 h1a y91 ff5 fs5 fc0 sc0 ls1 ws1">\ueb3a<span class="ff2 fs4 ws7 v2"> teremos os cálculos baseados no método da aceleração lin<span class="_4 blank"></span>ear<span class="_0 blank"> </span> </span></div><div class="t m0 x1 h8 y92 ff2 fs4 fc0 sc0 ls4 ws7">Se <span class="ff6 ls1 ws8">\uf061<span class="ff5 ws7"> </span><span class="ls4">\uf03d</span></span><span class="ff5 ls12"> <span class="fs5 ls1 v1">\ueb35</span></span></div><div class="t m0 x7 ha y93 ff5 fs5 fc0 sc0 ls2c">\ueb36<span class="ff2 fs4 ls1 ws7 v2"> <span class="ff5 wsc">\ue741</span> <span class="ff6 ls2d">\uf062<span class="ff5 ls27">=</span></span></span><span class="ls1 v3">\ueb35</span></div><div class="t m0 x5 h1a y93 ff5 fs5 fc0 sc0 ls1 ws1">\ueb38<span class="ff2 fs4 ws7 v2"> teremos os cálculos baseados no método da aceleração constant<span class="_4 blank"></span>e.<span class="_0 blank"> </span> </span></div><div class="t m0 x1 h7 y94 ff2 fs4 fc0 sc0 ls1 ws7">Para finalizar vamos fazer um exemplo: </div><div class="t m0 x1 h6 y95 ff3 fs3 fc1 sc0 ls1 ws7">Exemplo: Método de Runge-Kutta<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y96 ff2 fs4 fc0 sc0 ls1 ws7">Determine a solução <span class="_0 blank"> </span>da seguinte equação <span class="_0 blank"> </span>usando <span class="_0 blank"> </span>o método de <span class="_0 blank"> </span>Runge-kutta de <span class="_0 blank"> </span>quarta </div><div class="t m0 x1 h7 y97 ff2 fs4 fc0 sc0 ls1 ws7">ordem. </div><div class="t m0 x1 h13 y98 ff5 fs4 fc0 sc0 ls1 ws1b">\ue754<span class="_f blank"></span>\u0307<span class="_8 blank"> </span>=<span class="_c blank"> </span>\ue754<span class="_2 blank"> </span>\u2212<span class="_2 blank"> </span>1,5.<span class="_5 blank"> </span>\ue741 <span class="fs5 ws1c v16">\ueb3f\ueb34,\ueb39\uebe7 </span><span class="ff2 ws7"> | Com x<span class="fs6 ls13 v5">o</span> = 1; </span></div><div class="t m0 x1 h7 y99 ff2 fs4 fc0 sc0 ls1 ws7">Resolução: </div><div class="t m0 x1 h7 y9a ff2 fs4 fc0 sc0 ls1 ws7">Como <span class="_1 blank"> </span>a <span class="_1 blank"> </span>quantidade <span class="_c blank"> </span>de <span class="_1 blank"> </span>cálculos <span class="_1 blank"> </span>envolvidos <span class="_c blank"> </span>é<span class="_0 blank"> </span> <span class="_1 blank"> </span>muito <span class="_c blank"> </span>grande, <span class="_1 blank"> </span>vamos <span class="_1 blank"> </span>montar <span class="_1 blank"> </span>um </div><div class="t m0 x1 h7 y9b ff2 fs4 fc0 sc0 ls1 ws7">algorítimo no Matlab. </div><div class="t m0 x6 h7 y9c ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">UU(1) </span></div><div class="t m0 x6 h7 y9d ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">XX(1) =1.0 </span></div><div class="t m0 x6 h7 y9e ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">NEQ=1 </span></div><div class="t m0 x6 h7 y9f ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Passo=40 </span></div><div class="t m0 x6 h7 ya0 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">DeltaT=0.1 </span></div><div class="t m0 x6 h7 ya1 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">R = 0.0 </span></div><div class="t m0 x6 h7 ya2 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Write (58,10) </span></div><div class="t m0 x6 h7 ya3 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Format (//, 3X, 5H I , 10H , tempo(I), 7X, 5H X(1),/) </span></div><div class="t m0 x6 h7 ya4 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Do 40 l=1, Passo </span></div><div class="t m0 x6 h7 ya5 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Call RK4 (T, DeltaT, NEQ, XX, F, YI, YJ, YK, YL, UU) </span></div><div class="t m0 x6 h7 ya6 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Tempo(I)=T </span></div><div class="t m0 x6 h7 ya7 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Do 20 J=1, NEQ </span></div><div class="t m0 x6 h7 ya8 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">X(I,J)=XX(J) </span></div><div class="t m0 x6 h7 ya9 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Write (58,30) I, tempo(I), (X(I,J), J=1, NEQ) </span></div><div class="t m0 x6 h7 yaa ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Format (2x, 15, F10.4, 2X, E15.8, 2X, E15.8) </span></div><div class="t m0 x6 h7 yab ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Continue </span></div><div class="t m0 x6 h7 yac ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Stop </span></div><div class="t m0 x6 h7 yad ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">End </span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/e98ac46f-3c8a-457f-ba3d-a8d8ae0a56cf/bg6.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls1 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws7">6 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="t m0 x1 h7 y47 ff2 fs4 fc0 sc0 ls1 ws7">Agora a subrotina: </div><div class="t m0 x6 h7 yae ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Subroutine FUN (X, F, N, T) </span></div><div class="t m0 x6 h7 yaf ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">Dimension X(N), F(N) </span></div><div class="t m0 x6 h7 yb0 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">F(I)=X(1)-1,5*EXP(-0,5*T) </span></div><div class="t m0 x1 h7 yb1 ff2 fs4 fc0 sc0 ls1 ws7">Fazendo a simulação, chegamos aos resultados, vamos apresentar os 8 primeiros: </div><div class="t m0 x1 h7 yb2 ff2 fs4 fc0 sc0 ls1 ws7">Resultados: </div><div class="t m0 x6 h7 yb3 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">I <span class="_16 blank"> </span>| <span class="_17 blank"> </span>Tempo <span class="_18 blank"></span> | <span class="_19 blank"> </span> X(1) </span></div><div class="t m0 x6 h7 yb4 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">1 <span class="_1a blank"> </span>| <span class="_17 blank"> </span>0,1000 | <span class="_19 blank"> </span>0,95122939 E+00 </span></div><div class="t m0 x6 h7 yb5 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">2 <span class="_1b blank"> </span>| <span class="_17 blank"> </span>0,2000 <span class="_f blank"></span> | <span class="_19 blank"> </span>0,90483737 E+00 </span></div><div class="t m0 x6 h7 yb6 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">3 <span class="_1a blank"> </span>| <span class="_17 blank"> </span>0,3000 <span class="_4 blank"></span> | <span class="_19 blank"> </span>0,86070788 E+00 </span></div><div class="t m0 x6 h7 yb7 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">4 <span class="_1b blank"> </span>| <span class="_17 blank"> </span>0,4000 <span class="_f blank"></span> | <span class="_19 blank"> </span>0,81873059 E+00 </span></div><div class="t m0 x6 h7 yb8 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">5 <span class="_1c blank"> </span>| <span class="_17 blank"> </span>0,5000 <span class="_e blank"></span> | <span class="_19 blank"> </span>0,77880061 E+00 </span></div><div class="t m0 x6 h7 yb9 ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">6 <span class="_1b blank"> </span>| <span class="_17 blank"> </span>0,6000 <span class="_f blank"></span> | <span class="_19 blank"> </span>0,740818<span class="_4 blank"></span>02 E+00 </span></div><div class="t m0 x6 h7 yba ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">7 <span class="_1a blank"> </span>| <span class="_17 blank"> </span>0,<span class="_0 blank"> </span>7000 | <span class="_19 blank"> </span>0,70468783 E+00 </span></div><div class="t m0 x6 h7 ybb ff7 fs4 fc0 sc0 ls1 ws7">\u2022 <span class="_15 blank"> </span><span class="ff2">8 <span class="_1b blank"> </span>| <span class="_17 blank"> </span>0,8000 | <span class="_1d blank"> </span>0,67031974 E+00 </span></div><div class="t m0 x1 h7 ybc ff2 fs4 fc0 sc0 ls1 ws7"> </div><div class="t m0 x1 h7 ybd ff2 fs4 fc0 sc0 ls1 ws7"> </div><div class="t m0 x1 h7 ybe ff2 fs4 fc0 sc0 ls1 ws7"> </div><div class="t m0 x1 h7 ybf ff2 fs4 fc0 sc0 ls1 ws7"> </div><div class="t m0 x1 h2 yc0 ff1 fs0 fc0 sc0 ls1 ws7"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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