<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/b37187b0-b9e4-4a58-bbff-8c66e6797ca5/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws2">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws2"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls2 ws2">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls2 ws2"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls2 ws2">SISTEMAS MECÂNICOS VIBRACIONAIS </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls2 ws2">COM MDOF \u2013 MÉTODO DA EQUAÇÃO </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls2 ws2">DE LAGRANGE<span class="fs3 fc2"> </span></div><div class="t m0 x1 h6 ya ff3 fs3 fc1 sc0 ls2 ws2"> Introdução: </div><div class="t m0 x1 h7 yb ff2 fs4 fc0 sc0 ls2 ws2">Até o m<span class="_1 blank"></span>omento nós <span class="_1 blank"></span>só trabalhamos <span class="_1 blank"></span>com sistemas <span class="_1 blank"></span>com 1 <span class="_1 blank"></span>GDL <span class="_1 blank"></span>(Grau de liberdade). <span class="_1 blank"></span>Que </div><div class="t m0 x1 h7 yc ff2 fs4 fc0 sc0 ls2 ws2">são <span class="_2 blank"> </span>aqueles <span class="_0 blank"> </span>q<span class="_0 blank"> </span>ue <span class="_2 blank"> </span>necessitam <span class="_0 blank"> </span>de <span class="_2 blank"> </span>apenas <span class="_2 blank"> </span>uma <span class="_2 blank"> </span>coordenada <span class="_2 blank"> </span>para <span class="_2 blank"> </span>descrever <span class="_2 blank"> </span>o <span class="_2 blank"> </span>movimento </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 ls2 ws2">vibratório. </div><div class="t m0 x1 h8 ye ff2 fs4 fc0 sc0 ls2 ws2">Vimos que sistemas com múltiplos graus de<span class="_1 blank"></span> liberdade (MDOF<span class="fs5 ls0 v1">1</span>) podem ser modelados, </div><div class="t m0 x1 h7 yf ff2 fs4 fc0 sc0 ls2 ws2">por meio <span class="_1 blank"></span>d<span class="_1 blank"></span>e <span class="_1 blank"></span>associação <span class="_1 blank"></span>de massas<span class="_1 blank"></span>, r<span class="_1 blank"></span>igidezes e <span class="_1 blank"></span>amortecimen<span class="_1 blank"></span>tos, co<span class="_1 blank"></span>mo <span class="_1 blank"></span>u<span class="_0 blank"> </span>m si<span class="_1 blank"></span>stema com </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 ls2 ws2">1 <span class="_2 blank"> </span>GDL, <span class="_0 blank"> </span>no <span class="_2 blank"> </span>entanto, <span class="_2 blank"> </span>nem <span class="_2 blank"> </span>sempre <span class="_0 blank"> </span>teremos <span class="_2 blank"> </span>resultados <span class="_0 blank"> </span>confiáveis <span class="_2 blank"> </span>com <span class="_2 blank"> </span>esse <span class="_2 blank"> </span>modelo <span class="_0 blank"> </span>de <span class="_2 blank"> </span>1 </div><div class="t m0 x1 h7 y11 ff2 fs4 fc0 sc0 ls2 ws2">GDL. </div><div class="t m0 x1 h7 y12 ff2 fs4 fc0 sc0 ls2 ws2">Sendo assim, temos que trabalhar com os MDOF cujas característ<span class="_1 blank"></span>icas são: </div><div class="t m0 x3 h7 y13 ff4 fs4 fc0 sc0 ls2 ws0">\uf0b7<span class="ff5 ls1 ws2"> <span class="ff2 ls2">Várias Frequências Naturais, </span></span></div><div class="t m0 x3 h7 y14 ff4 fs4 fc0 sc0 ls2 ws0">\uf0b7<span class="ff5 ls1 ws2"> <span class="ff2 ls2">Vários Fatores de amortecimento. </span></span></div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 ls2 ws2">Para <span class="_3 blank"> </span>um <span class="_3 blank"> </span>sistema <span class="_3 blank"> </span>com <span class="_3 blank"> </span>vibrações <span class="_3 blank"> </span>livres, <span class="_3 blank"> </span>ele <span class="_3 blank"> </span>oscilará <span class="_3 blank"> </span>como <span class="_2 blank"> </span>uma <span class="_3 blank"> </span>combinação <span class="_3 blank"> </span>das <span class="_3 blank"> </span>várias </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 ls2 ws2">frequências naturais. </div><div class="t m0 x1 h7 y17 ff2 fs4 fc0 sc0 ls2 ws2">Outro <span class="_4 blank"> </span>ponto <span class="_4 blank"> </span>importante <span class="_4 blank"> </span>é <span class="_4 blank"> </span>o <span class="_4 blank"> </span>surgindo <span class="_4 blank"> </span>do <span class="_4 blank"> </span>\u201cModos <span class="_4 blank"> </span>de <span class="_4 blank"> </span>Vibrar\u201d, <span class="_4 blank"> </span>também <span class="_4 blank"> </span>chamados <span class="_4 blank"> </span>de </div><div class="t m0 x1 h7 y18 ff2 fs4 fc0 sc0 ls2 ws2">\u201cFormas <span class="_5 blank"> </span>Modais\u201d <span class="_5 blank"> </span>de <span class="_5 blank"> </span>uma <span class="_6 blank"> </span>estrutura. <span class="_5 blank"> </span>Sendo <span class="_6 blank"> </span>cada <span class="_5 blank"> </span>modo <span class="_5 blank"> </span>associado <span class="_6 blank"> </span>as <span class="_5 blank"> </span>respectivas </div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 ls2 ws2">frequências naturais e fator de amortecimento. </div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 ls2 ws2">Os <span class="_1 blank"></span>si<span class="_1 blank"></span>stemas <span class="_1 blank"></span>com <span class="_7 blank"></span>MDOF <span class="_7 blank"></span>são <span class="_1 blank"></span>mais <span class="_7 blank"></span>trabalhosos <span class="_7 blank"></span>para <span class="_1 blank"></span>se <span class="_7 blank"></span>obter <span class="_1 blank"></span>as <span class="_7 blank"></span>equações <span class="_7 blank"></span>do <span class="_1 blank"></span>movimento, </div><div class="t m0 x1 h7 y1b ff2 fs4 fc0 sc0 ls2 ws2">sendo o grau <span class="_1 blank"></span>de dificuldad<span class="_1 blank"></span>e em funç<span class="_1 blank"></span>ão do método <span class="_1 blank"></span>adotado. Até <span class="_1 blank"></span>então, temos <span class="_1 blank"></span>aplicado </div><div class="t m0 x1 h7 y1c ff2 fs4 fc0 sc0 ls2 ws2">o <span class="_0 blank"> </span>método <span class="_0 blank"> </span>do <span class="_0 blank"> </span>diagrama <span class="_0 blank"> </span>de <span class="_0 blank"> </span>corpo <span class="_0 blank"> </span>livre, <span class="_0 blank"> </span>no <span class="_0 blank"> </span>entanto, <span class="_0 blank"> </span>existem <span class="_0 blank"> </span>o <span class="_0 blank"> </span>método <span class="_2 blank"> </span>das <span class="_0 blank"> </span>equações <span class="_0 blank"> </span>de<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y1d ff2 fs4 fc0 sc0 ls2 ws2">Lagrange, métodos dos modos normais, também chamada de análise mo<span class="_0 blank"> </span>dal analítica. </div><div class="t m0 x1 h7 y1e ff2 fs4 fc0 sc0 ls2 ws2">Agora iremos abordar o método das equações de Lagran<span class="_1 blank"></span>ge. </div><div class="t m0 x1 h6 y1f ff3 fs3 fc1 sc0 ls2 ws2">Equações de Movimento </div><div class="t m0 x1 h7 y20 ff2 fs4 fc0 sc0 ls2 ws2">A figura a seguir representa um sistema com vários graus de li<span class="_1 blank"></span>berdade: </div><div class="t m0 x1 h2 y21 ff1 fs0 fc0 sc0 ls2 ws2"> <span class="_0 blank"> </span> <span class="_7 blank"></span> </div><div class="t m0 x1 h9 y22 ff2 fs6 fc0 sc0 ls2 ws1">1<span class="fs7 ws2 v2"> MDOF \u2013 Do inglês, Multiple <span class="_1 blank"></span>Degree of Freedom, ou<span class="_1 blank"></span> seja, múltiplos graus de <span class="_1 blank"></span>liberdade.<span class="fc3"> </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="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/b37187b0-b9e4-4a58-bbff-8c66e6797ca5/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws2">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws2"> </div></div><div class="t m0 x4 h7 y23 ff2 fs4 fc0 sc0 ls2 ws2"> </div><div class="t m0 x5 ha y24 ff2 fs7 fc0 sc0 ls2 ws2">Figura 01 \u2013 Exemplo de <span class="_1 blank"></span>um sistema com múltiplos g<span class="_1 blank"></span>raus de liberdade (M<span class="_1 blank"></span>DOF) </div><div class="t m0 x1 h7 y25 ff2 fs4 fc0 sc0 ls2 ws2">As equações do movimento serão descritas por: </div><div class="t m0 x6 hb y26 ff6 fs4 fc0 sc0 ls2 ws3">\ue879\ue754<span class="_7 blank"></span>\u0308<span class="_5 blank"> </span>+ \ue86f\ue754<span class="_7 blank"></span>\u0307<span class="_5 blank"> </span>+ \ue877.<span class="_8 blank"> </span>\ue754<span class="_5 blank"> </span>=<span class="_6 blank"> </span>\ue872<span class="ff2 ws2"> </span></div><div class="t m0 x1 h7 y27 ff2 fs4 fc0 sc0 ls2 ws2">Onde: </div><div class="t m0 x3 hc y28 ff4 fs4 fc0 sc0 ls2 ws0">\uf0b7<span class="ff5 ls1 ws2"> </span><span class="ff3 ws4">M<span class="ff2 ws2">, </span><span class="ls3">C<span class="ff2 ws2"> e </span></span>K<span class="ff2 ws2"> são matrizes de massa, rigidez e amortecimento do sistema; </span></span></div><div class="t m0 x3 hd y29 ff4 fs4 fc0 sc0 ls2 ws0">\uf0b7<span class="ff5 ls1 ws2"> <span class="ff2 ls2">x = [ x<span class="fs5 ls0 v3">1</span>, x<span class="fs5 ls4 v3">2</span>, x<span class="fs5 ws5 v3">3</span>, \u2026 , x<span class="fs5 ws5 v3">n</span><span class="ws6">]<span class="ff3 fs5 ls5 v1">T</span></span> <span class="ff7 ls3">\uf0e0</span> vetor deslocamento em cada coordenada </span></span></div><div class="t m0 x7 h7 y2a ff2 fs4 fc0 sc0 ls2 ws2">generalizada; </div><div class="t m0 x3 he y2b ff4 fs4 fc0 sc0 ls2 ws0">\uf0b7<span class="ff5 ls1 ws2"> <span class="ff2 ls2">F <span class="ff7 ls3">\uf0e0</span> Forças que excitam o sistema; </span></span></div><div class="t m0 x1 h7 y2c ff2 fs4 fc0 sc0 ls2 ws2">Essas matrizes, obteremos por meio das equações de Lagr<span class="_1 blank"></span>ange. </div><div class="t m0 x1 h6 y2d ff3 fs3 fc1 sc0 ls2 ws2">Equações <span class="_9 blank"> </span>da <span class="_9 blank"> </span>Energia <span class="_9 blank"> </span>Cinética, <span class="_9 blank"> </span>da <span class="_9 blank"> </span>Energia <span class="_9 blank"> </span>Potencial <span class="_9 blank"> </span>e <span class="_9 blank"> </span>da <span class="_a blank"> </span>Energia </div><div class="t m0 x1 h6 y2e ff3 fs3 fc1 sc0 ls2 ws2">Dissipativa </div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 ls2 ws2">Antes de <span class="_8 blank"> </span>aplicarmos as <span class="_8 blank"> </span>equações de <span class="_8 blank"> </span>lagrang<span class="_1 blank"></span>e, temos que <span class="_8 blank"> </span>obter <span class="_8 blank"> </span>as equa<span class="_1 blank"></span>ções para </div><div class="t m0 x1 hb y30 ff2 fs4 fc0 sc0 ls2 ws2">descrever <span class="_5 blank"> </span>a <span class="_5 blank"> </span><span class="fc1">energia <span class="_5 blank"> </span>cinética <span class="_5 blank"> </span>T<span class="_0 blank"> </span> <span class="_5 blank"> </span></span><span class="ff6 ws7">(\ue754</span></div><div class="t m0 x8 hf y31 ff6 fs8 fc0 sc0 ls6">\ueb35<span class="fs4 ls2 ws8 v4">, \ue754</span><span class="ls7">\ueb36<span class="fs4 ls2 ws8 v4">, \u2026 , \ue754<span class="_1 blank"></span><span class="fs8 ls8 v5">\uebe1<span class="fs4 ls2 ws9 v4">,<span class="_8 blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="wsa">,<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb36</span><span class="ws8">, \u2026 , \ue754<span class="_7 blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls8 v5">\uebe1</span><span class="ws7">)<span class="ff2 ws2">, <span class="_d blank"> </span>a <span class="_5 blank"> </span><span class="fc1">energia <span class="_5 blank"> </span>potencial <span class="_5 blank"> </span>V<span class="_0 blank"> </span></span> </span></span></span></span></span></span></span></span></div><div class="t m0 x1 hb y32 ff6 fs4 fc0 sc0 ls2 ws7">(\ue754</div><div class="t m0 x9 hf y33 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 wsb v4">, \ue754</span><span class="ls9">\ueb36<span class="fs4 ls2 wsb v4">, \u2026 , \ue754<span class="_1 blank"></span><span class="fs8 ls8 v5">\uebe1<span class="fs4 ls2 ws7 v4">)<span class="ff2 ls3 ws2"> e <span class="fc1 ls2">energia dissipativ<span class="_1 blank"></span>a<span class="ff3 lsa"> </span><span class="ff6">\ue727</span></span></span></span></span></span></span></div><div class="t m0 xa hf y33 ff6 fs8 fc1 sc0 lsb">\uebd7<span class="fs4 ls2 ws2 v4"> <span class="fc0 ws7">(\ue754</span></span></div><div class="t m0 xb hf y33 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws8 v4">, \ue754</span>\ueb36<span class="fs4 ls2 ws8 v4">, \u2026 , \ue754<span class="_1 blank"></span><span class="fs8 ls8 v5">\uebe1<span class="fs4 ls2 ws9 v4">,<span class="_c blank"> </span>\ue754<span class="_7 blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="wsa">,<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb36</span><span class="ws8">, \u2026 , \ue754<span class="_7 blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls8 v5">\uebe1</span><span class="ws7">)<span class="ff2 ws2">. </span></span></span></span></span></span></span></div><div class="t m0 x1 h10 y34 ff2 fs4 fc0 sc0 ls2 ws2">Energia Cinética T: <span class="_e blank"> </span><span class="ff6 wsc">\ue736<span class="_5 blank"> </span>= <span class="fs8 v6">\ueb35</span></span></div><div class="t m0 xc h11 y35 ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 lsd v7">\ue749</span><span class="lse v8">\uebdc</span><span class="fs4 ls2 wsd v7">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v9">\uebdc</span></span></div><div class="t m0 xd h12 y36 ff6 fs8 fc0 sc0 lsf">\ueb36<span class="fs4 ls10 va">+</span><span class="ls2 v4">\ueb35</span></div><div class="t m0 xe h11 y35 ff6 fs8 fc0 sc0 ls11">\ueb36<span class="fs4 ls2 v7">\ue72b</span></div><div class="t m0 xf hf y37 ff6 fs8 fc0 sc0 ls12">\uebe0<span class="fs4 ls2 v4">\ue7e0</span></div><div class="t m0 x10 hb y38 ff6 fs4 fc0 sc0 ls13">\u0307<span class="fs8 ls2 vb">\uebdc</span></div><div class="t m0 x11 h13 y36 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="ff2 fs4 ls2 ws2 va"> </span></div><div class="t m0 x1 h10 y39 ff2 fs4 fc0 sc0 ls2 ws2">Energia Potencial V: <span class="_f blank"> </span><span class="ff6 wsc">\ue738<span class="_5 blank"> </span>= <span class="fs8 v6">\ueb35</span></span></div><div class="t m0 xc h11 y3a ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 ws7 v7">\ue747</span><span class="lse v8">\uebdc</span><span class="fs4 ls2 ws7 v7">\ue754</span><span class="ls14 v8">\uebdc</span><span class="ff2 fs4 ls2 ws2 v7"> </span></div><div class="t m0 x1 hb y3b ff2 fs4 fc0 sc0 ls2 ws2">Energia dissipativa*: <span class="_10 blank"> </span><span class="ff6">\ue727</span></div><div class="t m0 x12 h14 y3c ff6 fs8 fc0 sc0 ls15">\uebd7<span class="fs4 ls16 v4">=</span><span class="ls2 vc">\ueb35</span></div><div class="t m0 x13 h11 y3d ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue73f</span></div><div class="t m0 x14 hf y3c ff6 fs8 fc0 sc0 lse">\uebdc<span class="fs4 ls2 wsd v4">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v9">\uebdc</span></span></div><div class="t m0 xd h13 y3e ff6 fs8 fc0 sc0 ls7">\ueb36<span class="ff2 fs4 ls2 ws2 va"> </span></div><div class="t m0 x1 h7 y3f ff2 fs4 fc0 sc0 ls2 ws2">Observação: <span class="_a blank"> </span>A <span class="_9 blank"> </span>equação <span class="_9 blank"> </span>apresentada <span class="_a blank"> </span>para <span class="_9 blank"> </span>energia <span class="_a blank"> </span>dissipativa <span class="_9 blank"> </span>vale <span class="_9 blank"> </span>apenas <span class="_9 blank"> </span>para </div><div class="t m0 x1 h7 y40 ff2 fs4 fc0 sc0 ls2 ws2">amortecimento v<span class="_0 blank"> </span>iscoso, <span class="_11 blank"> </span>para <span class="_11 blank"> </span>outros <span class="_11 blank"> </span>tipos <span class="_11 blank"> </span>de <span class="_11 blank"> </span>amortecimento <span class="_11 blank"> </span>é <span class="_11 blank"> </span>necessário <span class="_11 blank"> </span>fazer <span class="_11 blank"> </span>a </div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 ls2 ws2">equivalência, já estudadas. </div><div class="t m0 x1 h6 y42 ff3 fs3 fc1 sc0 ls2 ws2">Equações de Lagrange </div><div class="t m0 x1 h7 y43 ff2 fs4 fc0 sc0 ls2 ws2">Definindo o Lagrangiano L: </div><div class="t m0 x15 hb y44 ff6 fs4 fc0 sc0 ls2 ws9">\ue72e<span class="_5 blank"> </span>=<span class="_6 blank"> </span>\ue736<span class="_11 blank"> </span>\u2212<span class="_12 blank"> </span>\ue738 <span class="ff2 ws2"> </span></div><div class="t m0 x1 h7 y45 ff2 fs4 fc0 sc0 ls2 ws2">A equação de Lagrange pode ser obtida pela expressão ger<span class="_1 blank"></span>al: </div><div class="t m0 x16 hb y46 ff6 fs4 fc0 sc0 ls2">\ue740</div><div class="t m0 x17 h15 y47 ff6 fs4 fc0 sc0 ls2 wse">\ue740\ue750 <span class="ls17 vd">\ued6c</span><span class="ws7 ve">\ue7f2\ue72e</span></div><div class="t m0 x18 h16 y48 ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_3 blank"> </span><span class="fs8 lse v5">\uebdc</span><span class="ls18 vd">\ued70</span><span class="ls19 vd">\u2212</span><span class="ws7 ve">\ue7f2\ue72e</span></div><div class="t m0 x19 h17 y48 ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls1a v5">\uebdc</span><span class="wsc vd">= \ue733</span></div><div class="t m0 x13 hf y49 ff6 fs8 fc0 sc0 lse">\uebdc<span class="fs4 ls2 ws2 v4"> ,<span class="_13 blank"> </span>\ue745<span class="_5 blank"> </span>=<span class="_6 blank"> </span>1,2,<span class="_c blank"> </span>\u2026<span class="_c blank"> </span>,<span class="_c blank"> </span>\ue74a<span class="_0 blank"> </span><span class="ff2"> </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="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/b37187b0-b9e4-4a58-bbff-8c66e6797ca5/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws2">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws2"> </div></div><div class="t m0 x1 h7 y4a ff2 fs4 fc0 sc0 ls2 ws2">Englobando o termo dissipativo e expandindo, fica: </div><div class="t m0 x1a hb y4b ff6 fs4 fc0 sc0 ls2">\ue740</div><div class="t m0 x1b h16 y4c ff6 fs4 fc0 sc0 ls2 wse">\ue740\ue750 <span class="ls1b vd">\ued6c</span><span class="ws7 ve">\ue7f2\ue736</span></div><div class="t m0 x1c h16 y4c ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_3 blank"> </span><span class="fs8 lse v5">\uebdc</span><span class="ls18 vd">\ued70</span><span class="ls1c vd">\u2212</span><span class="ws7 ve">\ue7f2\ue736</span></div><div class="t m0 x1d h16 y4c ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls1d v5">\uebdc</span><span class="ls1e vd">+</span><span class="ws7 ve">\ue7f2\ue738</span></div><div class="t m0 x1e h16 y4c ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="fs8 ls1f v5">\uebdc</span><span class="ls20 vd">+</span><span class="ve">\ue7f2\ue727</span></div><div class="t m0 x1f h13 y4d ff6 fs8 fc0 sc0 ls2">\uebd7</div><div class="t m0 x20 h17 y4c ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls21 v5">\uebdc</span><span class="wsc vd">= \ue733</span></div><div class="t m0 x21 hf y4e ff6 fs8 fc0 sc0 lse">\uebdc<span class="fs4 ls2 ws2 v4"> ,<span class="_13 blank"> </span>\ue745<span class="_5 blank"> </span>=<span class="_6 blank"> </span>1,2,<span class="_c blank"> </span>\u2026<span class="_c blank"> </span>,<span class="_c blank"> </span>\ue74a<span class="_0 blank"> </span><span class="ff2"> </span></span></div><div class="t m0 x1 hc y4f ff2 fs4 fc0 sc0 ls2 ws2">Sendo Q<span class="fs5 ls22 v3">i</span> a força externa aplicada na <span class="ff3 ls3">i</span> coordenada do sistema. </div><div class="t m0 x1 h7 y50 ff2 fs4 fc0 sc0 ls2 ws2">A <span class="_4 blank"> </span>partir <span class="_3 blank"> </span>dessa <span class="_4 blank"> </span>última <span class="_4 blank"> </span>equação <span class="_3 blank"> </span>é <span class="_4 blank"> </span>possível <span class="_4 blank"> </span>descrever <span class="_3 blank"> </span>a <span class="_4 blank"> </span>equação <span class="_4 blank"> </span>do <span class="_4 blank"> </span>movimento <span class="_3 blank"> </span>de <span class="_4 blank"> </span>um </div><div class="t m0 x1 h7 y51 ff2 fs4 fc0 sc0 ls2 ws2">sistema <span class="_4 blank"> </span>MDOF, <span class="_4 blank"> </span>sem <span class="_4 blank"> </span>precisar <span class="_4 blank"> </span>realizar <span class="_4 blank"> </span>um <span class="_4 blank"> </span>diagrama <span class="_4 blank"> </span>de <span class="_4 blank"> </span>corpo <span class="_4 blank"> </span>livre <span class="_4 blank"> </span>da <span class="_4 blank"> </span>cada <span class="_4 blank"> </span>termo <span class="_4 blank"> </span>do </div><div class="t m0 x1 h7 y52 ff2 fs4 fc0 sc0 ls2 ws2">sistema. Mostraremos passo a passo, no exemplo a seguir. </div><div class="t m0 x1 h6 y53 ff3 fs3 fc1 sc0 ls2 ws2">Exemplo de aplicação: Equações de Lagrange </div><div class="t m0 x1 h7 y54 ff2 fs4 fc0 sc0 ls2 ws2">Obtenha <span class="_1 blank"></span>a equação <span class="_1 blank"></span>do <span class="_1 blank"></span>movimento <span class="_1 blank"></span>para <span class="_1 blank"></span>o <span class="_1 blank"></span>sistema <span class="_1 blank"></span>da figura <span class="_7 blank"></span>abaixo usando <span class="_1 blank"></span>as equações </div><div class="t m0 x1 h7 y55 ff2 fs4 fc0 sc0 ls2 ws2">de Lagrange, assumindo que a força F<span class="fs5 ls0 v3">1</span>(t) atua na massa m<span class="fs5 ls0 v3">1</span>. </div><div class="t m0 x22 h7 y56 ff2 fs4 fc0 sc0 ls2 ws2"> </div><div class="t m0 x1 h7 y57 ff2 fs4 fc0 sc0 ls2 ws2">Resolução: </div><div class="t m0 x1 h7 y58 ff2 fs4 fc1 sc0 ls2 ws2">Passo 1: <span class="fc0">Calcular os termos das energias cinética T, potencial V e di<span class="_1 blank"></span>ssipativa:<span class="_0 blank"> </span> </span></div><div class="t m0 x1 h10 y59 ff2 fs4 fc0 sc0 ls2 ws2">Cinética: <span class="_14 blank"> </span><span class="ff6 ls23">\ue736<span class="ls2 ws7 vf">(</span><span class="ls2 ws9">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="wsa">,<span class="_c blank"> </span>\ue754<span class="_7 blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb36</span><span class="ls24 vf">)</span><span class="ls16">=</span><span class="fs8 v6">\ueb35</span></span></span></span></div><div class="t m0 x1f h11 y5a ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue749</span></div><div class="t m0 x23 hf y5b ff6 fs8 fc0 sc0 ls9">\ueb35<span class="fs4 ls2 ws9 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v5">\ueb35</span></span></div><div class="t m0 x24 h18 y5c ff6 fs8 fc0 sc0 lsf">\ueb36<span class="fs4 ls20 va">+</span><span class="ls2 v4">\ueb35</span></div><div class="t m0 x25 h11 y5a ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 lsd v7">\ue749</span><span class="ls9 v8">\ueb36</span><span class="fs4 ls2 wsa v7">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v9">\ueb36</span></span></div><div class="t m0 x26 h13 y5c ff6 fs8 fc0 sc0 ls9">\ueb36<span class="ff2 fs4 ls2 ws2 va"> </span></div><div class="t m0 x1 h19 y5d ff2 fs4 fc0 sc0 ls2 ws2">Potencial: <span class="_15 blank"> </span><span class="ff6 ls25">\ue738<span class="ls2 ws7 vf">(</span><span class="ls2">\ue754</span></span></div><div class="t m0 x27 h14 y5e ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws8 v4">, \ue754</span>\ueb36<span class="fs4 ls24 v10">)<span class="ls16 v11">=</span></span><span class="ls2 vc">\ueb35</span></div><div class="t m0 x28 h11 y5f ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue747</span></div><div class="t m0 x29 hf y5e ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws8 v4">. \ue754</span></div><div class="t m0 x21 h13 y60 ff6 fs8 fc0 sc0 ls2">\ueb35</div><div class="t m0 x2a h18 y61 ff6 fs8 fc0 sc0 lsf">\ueb36<span class="fs4 ls20 va">+</span><span class="ls2 v4">\ueb35</span></div><div class="t m0 x2b h1a y5f ff6 fs8 fc0 sc0 ls11">\ueb36<span class="fs4 ls2 ws7 v7">\ue747</span><span class="ls9 v8">\ueb36</span><span class="fs4 ls2 ws2 v7">.<span class="_c blank"> </span> <span class="ws7 vf">(</span>\ue754</span></div><div class="t m0 x26 h14 y5e ff6 fs8 fc0 sc0 lsf">\ueb35<span class="fs4 ls2 ws3 v4">\u2212 \ue754</span><span class="ls7">\ueb36<span class="fs4 ls2 ws7 v10">)</span></span><span class="v6">\ueb36</span><span class="fs4 ls20 v4">+</span><span class="ls2 vc">\ueb35</span></div><div class="t m0 x22 h11 y5f ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 ws7 v7">\ue747</span><span class="ls9 v8">\ueb37</span><span class="fs4 ls2 wsb v7">. \ue754</span><span class="ls2 v8">\ueb36</span></div><div class="t m0 x2c h13 y61 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="ff2 fs4 ls2 ws2 va"> </span></div><div class="t m0 x1 hb y62 ff2 fs4 fc0 sc0 ls2 ws2">Energia Dissipativa: <span class="_16 blank"> </span><span class="ff6">\ue727</span></div><div class="t m0 x2d h14 y63 ff6 fs8 fc0 sc0 lsb">\uebd7<span class="fs4 ls2 ws7 v10">(<span class="ws9 v11">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="wsa">,<span class="_c blank"> </span>\ue754<span class="_7 blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb36</span><span class="ls24 vf">)</span><span class="ls16">=</span><span class="fs8 v6">\ueb35</span></span></span></span></div><div class="t m0 x2e h11 y64 ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue73f</span></div><div class="t m0 x23 hf y63 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws11 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v5">\ueb35</span></span></div><div class="t m0 x24 h18 y65 ff6 fs8 fc0 sc0 ls26">\ueb36<span class="fs4 ls20 va">+</span><span class="ls2 v4">\ueb35</span></div><div class="t m0 x25 h11 y64 ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue73f</span></div><div class="t m0 x2f h14 y63 ff6 fs8 fc0 sc0 ls7">\ueb36<span class="fs4 ls2 ws2 v4">.<span class="_c blank"> </span> <span class="ws7 vf">(</span><span class="ws11">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb35</span><span class="wsd">\u2212<span class="_11 blank"> </span>\ue754<span class="_7 blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb36</span><span class="ws7 vf">)</span><span class="fs8 ls26 v12">\ueb36</span><span class="ls20">+</span><span class="fs8 v6">\ueb35</span></span></span></span></div><div class="t m0 x30 h11 y64 ff6 fs8 fc0 sc0 lsc">\ueb36<span class="fs4 ls2 v7">\ue73f</span></div><div class="t m0 x31 hf y63 ff6 fs8 fc0 sc0 ls7">\ueb37<span class="fs4 ls2 wsa v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 v9">\ueb36</span></span></div><div class="t m0 x32 h13 y65 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="ff2 fs4 ls2 ws2 va"> </span></div><div class="t m0 x1 h7 y66 ff2 fs4 fc1 sc0 ls2 ws2">Passo 2: <span class="fc0">Aplicar Lagrange: </span></div><div class="t m0 x1 h7 y67 ff2 fs4 fc0 sc0 ls2 ws2">Coordenada x<span class="fs5 ls0 v3">1</span>: </div><div class="t m0 x1 h7 y68 ff2 fs4 fc0 sc0 ls2 ws2">Para a coordenada x<span class="fs5 ls0 v3">1</span> faremos passo a passo. </div><div class="t m0 x1 h7 y69 ff2 fs4 fc0 sc0 ls2 ws2">Lembrando que Q<span class="fs5 ls22 v3">i</span> está relacionada com a força externa que no nosso exemplo está </div><div class="t m0 x1 h7 y6a ff2 fs4 fc0 sc0 ls2 ws2">aplicada a massa m<span class="fs5 ls0 v3">1</span>, portanto Q<span class="fs5 ls0 v3">1</span> = F<span class="fs5 ls0 v3">1</span>. </div><div class="t m0 x1 h7 y6b ff2 fs4 fc0 sc0 ls2 ws2">A equação de Lagrange para x<span class="fs5 ls0 v3">1</span>, fica: </div><div class="t m0 x33 hb y6c ff6 fs4 fc0 sc0 ls2">\ue740</div><div class="t m0 x34 h16 y6d ff6 fs4 fc0 sc0 ls2 wse">\ue740\ue750 <span class="ls27 vd">\ued6c</span><span class="ws7 ve">\ue7f2\ue736</span></div><div class="t m0 x35 h16 y6d ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="ls18 vd">\ued70</span><span class="ls28 vd">\u2212</span><span class="ws7 ve">\ue7f2\ue736</span></div><div class="t m0 x15 hb y6d ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754</div><div class="t m0 x36 h1b y6e ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2a v13">+<span class="ls2 ws7 v14">\ue7f2\ue738</span></span></div><div class="t m0 x37 hb y6d ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754</div><div class="t m0 x38 h1b y6e ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls20 v13">+<span class="ls2 wsf v14">\ue7f2 \ue727</span></span></div><div class="t m0 xf h13 y6f ff6 fs8 fc0 sc0 ls2">\uebd7</div><div class="t m0 x39 h17 y6d ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls2b v5">\ueb35</span><span class="wsc vd">= \ue728</span></div><div class="t m0 x3a h1c y70 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="ff2 fs4 ls2 ws2 v4"> </span></div><div class="t m0 x1 h7 y71 ff2 fs4 fc0 sc0 ls2 ws2">Calculando os termos das derivadas: </div><div class="t m0 x1 h10 y72 ff2 fs4 fc0 sc0 ls2c ws2"> <span class="ff6 fs8 ls2 ws12 v6">\uec21\uebcd</span></div><div class="t m0 x3b h13 y73 ff6 fs8 fc0 sc0 ls2 ws13">\uec21\uebeb</div><div class="t m0 x3c h1d y74 ff6 fs8 fc0 sc0 ls2d">\u0307<span class="fs9 ls2e v15">\uec2d</span><span class="ff2 fs4 ls2 ws2 v7">: <span class="_17 blank"> </span>Esse termo é a derivada parcial de <span class="ff3 fc2 ws4">T</span> com relação a <span class="ff6 fc2 ws11">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb35</span><span class="ff2 fc0 ws2">, que fica: </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/b37187b0-b9e4-4a58-bbff-8c66e6797ca5/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws2">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws2"> </div></div><div class="t m0 xa hb y75 ff6 fs4 fc0 sc0 ls2 ws7">\ue7f2\ue736</div><div class="t m0 x3d h17 y76 ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754<span class="_b blank"></span>\u0307<span class="_2 blank"> </span><span class="fs8 ls2f v5">\ueb35</span><span class="wsc vd">= \ue749</span></div><div class="t m0 x3e hf y77 ff6 fs8 fc0 sc0 ls9">\ueb35<span class="fs4 ls2 ws9 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="ff2 ws2"> </span></span></div><div class="t m0 x3b h13 y78 ff6 fs8 fc0 sc0 ls2">\uebd7</div><div class="t m0 x1 h1e y79 ff6 fs8 fc0 sc0 ls2 ws14">\uebd7\uebe7 <span class="fs4 ls2c v7">\uf240</span><span class="ws12 v16">\uec21\uebcd</span></div><div class="t m0 x3 h13 y79 ff6 fs8 fc0 sc0 ls2 ws13">\uec21\uebeb</div><div class="t m0 x3f h1f y7a ff6 fs8 fc0 sc0 ls2d">\u0307<span class="fs9 ls30 v15">\uec2d</span><span class="fs4 ls2 ws7 v7">\uf241:<span class="ff2 ws2"> <span class="_18 blank"> </span>Esse termo é a derivada total de <span class="_c blank"> </span></span></span><span class="fc1 ls2 ws12 v16">\uec21\uebcd</span></div><div class="t m0 x28 h13 y79 ff6 fs8 fc1 sc0 ls2 ws15">\uec21 \uebeb</div><div class="t m0 x40 h20 y7a ff6 fs8 fc1 sc0 ls2d">\u0307<span class="fs9 ls31 v15">\uec2d</span><span class="ff2 fs4 fc0 ls2 ws2 v7"> , calculado na linha anterior, em </span></div><div class="t m0 x1 h7 y7b ff2 fs4 fc0 sc0 ls2 ws2">relação ao tempo, <span class="fc1 ws6">t</span>. Embora não esteja explicito, o termo relacionado ao tempo é </div><div class="t m0 x1 h10 y7c ff6 fs4 fc1 sc0 ls2 ws11">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb35</span><span class="ff2 fc0 ws2">que é o mesmo que<span class="fc1"> </span></span><span class="fs8 ws12 v6">\uebd7\uebeb</span></div><div class="t m0 x41 h21 y7d ff6 fs8 fc1 sc0 ls2 ws16">\uebd7\uebe7 <span class="ff3 fs4 fc2 ws4 v7">,<span class="ff2 fc0 ws2"> que resulta: </span></span></div><div class="t m0 x1e hb y7e ff6 fs4 fc0 sc0 ls2">\ue740</div><div class="t m0 x42 h16 y7f ff6 fs4 fc0 sc0 ls2 wse">\ue740\ue750 <span class="ls27 vd">\ued6c</span><span class="ws7 ve">\ue7f2\ue736</span></div><div class="t m0 x15 h17 y7f ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="ls32 vd">\ued70</span><span class="wsc vd">= \ue749</span></div><div class="t m0 x43 hf y80 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws9 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0308 <span class="fs8 ls7 v5">\ueb35</span><span class="ff2 ws2"> </span></span></div><div class="t m0 x3b h13 y81 ff6 fs8 fc0 sc0 ls2 ws12">\uec21\uebcd</div><div class="t m0 x1 h22 y82 ff6 fs8 fc0 sc0 ls2 ws13">\uec21\uebeb<span class="fs9 ls2e v15">\uec2d</span><span class="ff2 fs4 ws2 v7">: <span class="_19 blank"> </span>Esse termo é a derivada parcial de <span class="fc1">T </span>em relação a <span class="fc1 ws6">x<span class="fs5 ls0 v3">1</span></span>. Que fica: </span></div><div class="t m0 x44 hb y83 ff6 fs4 fc0 sc0 ls2 ws7">\ue7f2\ue736</div><div class="t m0 x45 hb y84 ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754</div><div class="t m0 x46 h23 y85 ff6 fs8 fc0 sc0 ls33">\ueb35<span class="fs4 ls2 wsc v13">= 0<span class="ff2 ws2"> </span></span></div><div class="t m0 x3b h13 y86 ff6 fs8 fc0 sc0 ls2 ws12">\uec21\uebcf</div><div class="t m0 x1 h22 y87 ff6 fs8 fc0 sc0 ls2 ws13">\uec21\uebeb<span class="fs9 ls2e v15">\uec2d</span><span class="ff2 fs4 ws2 v7">: <span class="_19 blank"> </span>Esse termo é a derivada parcial de <span class="fc1">V </span>em relação a <span class="fc1 ws6">x<span class="fs5 ls0 v3">1</span></span>. Que fica: </span></div><div class="t m0 x47 hb y88 ff6 fs4 fc0 sc0 ls2 ws7">\ue7f2\ue738</div><div class="t m0 x48 hb y89 ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue754</div><div class="t m0 x49 h23 y8a ff6 fs8 fc0 sc0 ls33">\ueb35<span class="fs4 ls2 wsc v13">= \ue747</span></div><div class="t m0 xa hf y8b ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws8 v4">. \ue754</span></div><div class="t m0 x36 hf y8b ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2 ws3 v4">+ \ue747</span><span class="ls7">\ueb36<span class="fs4 ls2 ws8 v4">. \ue754</span></span></div><div class="t m0 x4a hf y8b ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2 ws17 v4">\u2212 \ue747</span><span class="ls7">\ueb36<span class="fs4 ls2 ws7 v4">\ue754</span>\ueb36<span class="ff2 fs4 ls2 ws2 v4"> </span></span></div><div class="t m0 x1 h13 y8c ff6 fs8 fc0 sc0 ls2 ws13">\uec21\uebbe<span class="fs9 v15">\ueccf</span></div><div class="t m0 x1 h13 y8d ff6 fs8 fc0 sc0 ls2 ws15">\uec21 \uebeb</div><div class="t m0 x9 h24 y8e ff6 fs8 fc0 sc0 ls2d">\u0307<span class="fs9 ls34 v15">\uec2d</span><span class="ff2 fs4 ls2 ws2 v7">: <span class="_1a blank"> </span>Esse termo é a derivada parcial de <span class="fc1">Ed </span>em relação a <span class="ff6 fc2 ws9">\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls7 v5">\ueb35</span><span class="ff2 fc0 ws2">. Que fica: </span></span></span></div><div class="t m0 x48 hb y8f ff6 fs4 fc0 sc0 ls2 wsf">\ue7f2 \ue727</div><div class="t m0 x49 h13 y90 ff6 fs8 fc0 sc0 ls2">\uebd7</div><div class="t m0 x48 h17 y91 ff6 fs4 fc0 sc0 ls2 ws10">\ue7f2 \ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls2b v5">\ueb35</span><span class="wsc vd">= \ue73f</span></div><div class="t m0 xa hf y92 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws11 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb35</span><span class="ws3">+ \ue73f</span></span></div><div class="t m0 x2e hf y92 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="fs4 ls2 ws9 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb35</span><span class="ws3">\u2212 \ue73f</span></span></div><div class="t m0 x4b hf y92 ff6 fs8 fc0 sc0 ls7">\ueb36<span class="fs4 ls2 wsa v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 ls9 v5">\ueb36</span><span class="ff2 ws2"> </span></span></div><div class="t m0 x1 h7 y93 ff2 fs4 fc0 sc0 ls2 ws2">Substituindo na equação de Lagrange inicial<span class="_1 blank"></span>, colocando os termos em ordem </div><div class="t m0 x1 h7 y94 ff2 fs4 fc0 sc0 ls2 ws2">crescente e agrupando os termos semelhantes, fica: </div><div class="t m0 x4c hb y95 ff6 fs4 fc0 sc0 ls2">\ue754</div><div class="t m0 x4d hf y96 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="fs4 ls2 ws2 v4">: m</span></div><div class="t m0 x4e hf y96 ff6 fs8 fc0 sc0 ls6">\ueb35<span class="fs4 ls2 ws18 v4">.<span class="_c blank"> </span>x<span class="_1b blank"></span>\u0308 <span class="fs8 lsf v5">\ueb35</span><span class="ws3">+ (\ue73f</span></span></div><div class="t m0 x4f hf y96 ff6 fs8 fc0 sc0 lsf">\ueb35<span class="fs4 ls2 ws3 v4">+ \ue73f</span></div><div class="t m0 x8 hf y96 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="fs4 ls2 ws9 v4">).<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb35</span><span class="ws3">\u2212 \ue73f</span></span></div><div class="t m0 x2e hf y96 ff6 fs8 fc0 sc0 ls7">\ueb36<span class="fs4 ls2 wsd v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb36</span><span class="ws3">+ (\ue747</span></span></div><div class="t m0 x25 hf y96 ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2 ws17 v4">+ \ue747</span><span class="ls7">\ueb36<span class="fs4 ls2 ws8 v4">). \ue754</span></span></div><div class="t m0 x50 hf y96 ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2 ws3 v4">\u2212 \ue747</span><span class="ls7">\ueb36<span class="fs4 ls2 ws7 v4">\ue754</span><span class="ls2f">\ueb36<span class="fs4 ls2 wsc v4">= \ue728</span></span></span></div><div class="t m0 x51 h1c y96 ff6 fs8 fc0 sc0 ls7">\ueb35<span class="ff2 fs4 ls2 ws2 v4"> </span></div><div class="t m0 x1 h7 y97 ff2 fs4 fc0 sc0 ls2 ws2">Para a coordenada x<span class="fs5 ls4 v3">2</span>, aplicamos a mesma técnica, ficando: </div><div class="t m0 x52 hb y98 ff6 fs4 fc0 sc0 ls2 ws7">\ue754<span class="fs8 ls7 v5">\ueb36</span><span class="ws2">: m<span class="fs8 ls7 v5">\ueb36</span><span class="ws19">.<span class="_c blank"> </span>x<span class="_1b blank"></span>\u0308 <span class="fs8 lsf v5">\ueb36</span><span class="ws3">\u2212 \ue73f</span></span></span></div><div class="t m0 x53 hf y99 ff6 fs8 fc0 sc0 ls9">\ueb36<span class="fs4 ls2 ws11 v4">.<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb35</span><span class="ws3">+ (\ue73f</span></span></div><div class="t m0 x54 hf y99 ff6 fs8 fc0 sc0 lsf">\ueb36<span class="fs4 ls2 ws3 v4">+ \ue73f</span></div><div class="t m0 x55 hf y99 ff6 fs8 fc0 sc0 ls9">\ueb37<span class="fs4 ls2 wsa v4">).<span class="_c blank"> </span>\ue754<span class="_b blank"></span>\u0307 <span class="fs8 lsf v5">\ueb36</span><span class="ws17">+ (\ue747<span class="fs8 lsf v5">\ueb36</span><span class="ws3">+ \ue747<span class="fs8 ls7 v5">\ueb37</span><span class="ws8">). \ue754<span class="fs8 lsf v5">\ueb36</span></span>\u2212 \ue747<span class="fs8 ls7 v5">\ueb36</span>\ue754</span></span></span></div><div class="t m0 x56 hf y99 ff6 fs8 fc0 sc0 ls33">\ueb35<span class="fs4 ls2 wsc v4">= 0<span class="ff2 ws2"> </span></span></div><div class="t m0 x1 hb y9a ff2 fs4 fc1 sc0 ls2 ws2">Passo 3<span class="fc0">: Montar as matrizes e vetores: <span class="ff6 ws3">\ue879\ue754<span class="_b blank"></span>\u0308<span class="_d blank"> </span>+ \ue86f\ue754<span class="_7 blank"></span>\u0307<span class="_d blank"> </span>+ \ue877.<span class="_c blank"> </span>\ue754<span class="_5 blank"> </span>=<span class="_6 blank"> </span>\ue872<span class="ff2 ws2"> </span></span></span></div><div class="t m0 x1 h7 y9b ff2 fs4 fc0 sc0 ls2 ws2">Com base <span class="_11 blank"> </span>nas equações de <span class="_11 blank"> </span>Lagrange obtidas <span class="_11 blank"> </span>anteriormente, podemos <span class="_11 blank"> </span>montar as </div><div class="t m0 x1 h7 y9c ff2 fs4 fc0 sc0 ls2 ws2">matrizes e <span class="_11 blank"> </span>vetores. <span class="_11 blank"> </span>Na <span class="_11 blank"> </span>primeira linha <span class="_11 blank"> </span>estarão <span class="_11 blank"> </span>os <span class="_12 blank"> </span>componentes correspondentes <span class="_11 blank"> </span>a </div><div class="t m0 x1 h7 y9d ff2 fs4 fc0 sc0 ls2 ws2">coordenada <span class="_2 blank"> </span>x<span class="_0 blank"> </span><span class="fs5 ls0 v3">1</span>, <span class="_3 blank"> </span>na <span class="_2 blank"> </span>linha <span class="_3 blank"> </span>dois, <span class="_3 blank"> </span>os <span class="_3 blank"> </span>componentes <span class="_2 blank"> </span>relacionados <span class="_3 blank"> </span>a <span class="_3 blank"> </span>coordenada <span class="_2 blank"> </span>x<span class="_0 blank"> </span><span class="fs5 ls4 v3">2</span>, <span class="_3 blank"> </span>e <span class="_3 blank"> </span>assim </div><div class="t m0 x1 h7 y9e ff2 fs4 fc0 sc0 ls2 ws2">por diante, ficando: </div><div class="t m0 x1 h7 y9f ff2 fs4 fc1 sc0 ls2 ws2">Vetor Deslocamento: <span class="fc0">Aqui relacionamos todas as coordenadas generalizadas em uma </span></div><div class="t m0 x1 hc ya0 ff2 fs4 fc0 sc0 ls2 ws2">matriz <span class="_2 blank"> </span>de <span class="_2 blank"> </span>uma <span class="_2 blank"> </span>c<span class="_0 blank"> </span>oluna <span class="_2 blank"> </span>por<span class="ff3 fc2"> <span class="_2 blank"> </span>i</span> <span class="_3 blank"> </span>linhas, <span class="_2 blank"> </span>correspondentes <span class="_2 blank"> </span>ao <span class="_2 blank"> </span>número <span class="_3 blank"> </span>de <span class="_2 blank"> </span>graus <span class="_2 blank"> </span>de <span class="_2 blank"> </span>liberdade </div><div class="t m0 x1 h7 ya1 ff2 fs4 fc0 sc0 ls2 ws2">que o sistema possui, ficando: </div><div class="t m0 x12 h25 ya2 ff6 fs4 fc0 sc0 ls2 wsc">\ue754<span class="_5 blank"> </span>= \uf244<span class="v14">\ue754</span></div><div class="t m0 x3e h13 ya3 ff6 fs8 fc0 sc0 ls2">\ueb35</div><div class="t m0 x28 h26 ya4 ff6 fs4 fc0 sc0 ls2 ws7">\ue754<span class="fs8 ls7 v5">\ueb36</span><span class="ls3 v17">\uf245</span><span class="ff2 ws2 v17"> </span></div><div class="t m0 x1 hc ya5 ff2 fs4 fc1 sc0 ls2 ws2">Vetor Força F<span class="ff3 ls35">:</span> <span class="fc0">O mesmo que para o vetor deslocamento. </span></div><div class="t m0 x12 h27 ya6 ff6 fs4 fc0 sc0 ls2 wsc">\ue728<span class="_5 blank"> </span>= \uf244<span class="v18">\ue728</span></div><div class="t m0 x3e h13 ya7 ff6 fs8 fc0 sc0 ls2">\ueb35</div><div class="t m0 x55 h28 ya8 ff6 fs4 fc0 sc0 ls36">0<span class="ls2 ws7 v19">\uf245<span class="ff2 ws2"> </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="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/b37187b0-b9e4-4a58-bbff-8c66e6797ca5/bg5.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws2">5 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws2"> </div></div><div class="t m0 x1 hb y4a ff2 fs4 fc0 sc0 ls2 ws2">Note que não temos força na equação de <span class="ff6 ws7">\ue754<span class="fs8 ls7 v5">\ueb36</span></span>, daí o termo zero <span class="_1 blank"></span>na segunda linha. </div><div class="t m0 x1 hb ya9 ff2 fs4 fc1 sc0 ls2 ws2">Matriz <span class="_3 blank"> </span>Massa <span class="_4 blank"> </span>M<span class="ff3 ls35">:</span><span class="ls37"> </span><span class="fc0">A <span class="_3 blank"> </span>massa <span class="_4 blank"> </span>está <span class="_3 blank"> </span>relacionada <span class="_3 blank"> </span>ao <span class="_4 blank"> </span>termo <span class="_4 blank"> </span><span class="ff6 fc2 ws18">\ue754<span class="_b blank"></span>\u0308 <span class="ff2 fc0 ws2">. <span class="_3 blank"> </span>Na <span class="_4 blank"> </span>primeira <span class="_3 blank"> </span>coluna <span class="_4 blank"> </span>ficará <span class="_3 blank"> </span>os </span></span></span></div><div class="t m0 x1 hb yaa ff2 fs4 fc0 sc0 ls2 ws2">itens <span class="_3 blank"> </span>relacionados <span class="_2 blank"> </span>ao <span class="_3 blank"> </span>termo <span class="_3 blank"> </span><span class="ff6 ws18">x<span class="_1b blank"></span>\u0308 <span class="fs8 ls9 v5">\ueb35</span><span class="ff2 ws2">, <span class="_3 blank"> </span>na <span class="_3 blank"> </span>coluna <span class="_3 blank"> </span>dois, <span class="_3 blank"> </span>os <span class="_2 blank"> </span>termos <span class="_3 blank"> </span>relacionados <span class="_3 blank"> </span>ao <span class="_3 blank"> </span>item <span class="_3 blank"> </span></span><span class="ws19">x<span class="_1b blank"></span>\u0308 <span class="fs8 ls9 v5">\ueb36</span><span class="ff2 ws2">, <span class="_3 blank"> </span>e </span></span></span></div><div class="t m0 x1 h7 yab ff2 fs4 fc0 sc0 ls2 ws2">assim por diante, ficando: <span class="_1c blank"> </span> </div><div class="t m0 x57 h29 yac ff6 fs4 fc0 sc0 ls2 wsc">\ue72f<span class="_5 blank"> </span>= \ued64<span class="v19">\ue749</span></div><div class="t m0 x58 hf yad ff6 fs8 fc0 sc0 ls38">\ueb35<span class="fs4 ls2 v4">0</span></div><div class="t m0 x59 h2a yae ff6 fs4 fc0 sc0 ls2 ws1a">0 \ue749<span class="fs8 ls9 v5">\ueb36</span><span class="ls3 v18">\ued68</span><span class="ff2 ws2 v18"> </span></div><div class="t m0 x1 h7 yaf ff2 fs4 fc0 sc0 ls2 ws2">Note <span class="_4 blank"> </span>que <span class="_4 blank"> </span>na <span class="_4 blank"> </span>equação <span class="_4 blank"> </span>da <span class="_4 blank"> </span>coordenada <span class="_3 blank"> </span>x<span class="_0 blank"> </span><span class="fs5 ls0 v3">1</span>, <span class="_4 blank"> </span>não <span class="_4 blank"> </span>temos <span class="_4 blank"> </span>nenhum <span class="_4 blank"> </span>componen<span class="_0 blank"> </span>te <span class="_4 blank"> </span>de <span class="_4 blank"> </span>massa </div><div class="t m0 x1 hb yb0 ff2 fs4 fc0 sc0 ls2 ws2">relacionado ao termo <span class="ff6 ws19">x<span class="_1b blank"></span>\u0308 <span class="fs8 ls9 v5">\ueb36</span><span class="ff2 ws2">, daí o valor zero na segunda coluna da linha 1. </span></span></div><div class="t m0 x1 hb yb1 ff2 fs4 fc1 sc0 ls2 ws2">Matriz Amortecimento Viscoso C<span class="ff3 fc2 ls35">:</span><span class="fc0"> O amortecimento está relacionado ao termo <span class="ff6 fc2 ws18">\ue754<span class="_b blank"></span>\u0307 <span class="ff3 ls35">.</span><span class="ff2 fc0 ws2"> </span></span></span></div><div class="t m0 x1 h7 yb2 ff2 fs4 fc0 sc0 ls2 ws2">Observando os mesmos critérios para as outras propriedades, fica: <span class="_1d blank"> </span> </div><div class="t m0 x49 h2b yb3 ff6 fs4 fc0 sc0 ls2 wsc">\ue725<span class="_d blank"> </span>= \uf242<span class="vd">\ue73f</span></div><div class="t m0 x5a hf yb4 ff6 fs8 fc0 sc0 lsf">\ueb35<span class="fs4 ls2 ws3 v4">+ \ue73f</span></div><div class="t m0 x13 hf yb4 ff6 fs8 fc0 sc0 ls39">\ueb36<span class="fs4 ls2 ws7 v4">\u2212\ue73f</span></div><div class="t m0 x39 h13 yb4 ff6 fs8 fc0 sc0 ls2">\ueb36</div><div class="t m0 x12 hb yb5 ff6 fs4 fc0 sc0 ls2 ws7">\u2212\ue73f</div><div class="t m0 x46 hf yb6 ff6 fs8 fc0 sc0 ls3a">\ueb36<span class="fs4 ls2 v4">\ue73f</span></div><div class="t m0 xd hf yb6 ff6 fs8 fc0 sc0 lsf">\ueb36<span class="fs4 ls2 ws3 v4">+ \ue73f</span></div><div class="t m0 x5b h2c yb6 ff6 fs8 fc0 sc0 ls9">\ueb37<span class="fs4 ls2 ws7 v1a">\uf243<span class="ff2 ws2"> </span></span></div><div class="t m0 x1 h7 yb7 ff2 fs4 fc1 sc0 ls2 ws2">Matriz Rigidez K: <span class="fc0">A rigidez está relacionada ao termo </span><span class="ws6">x</span><span class="fc0">, repetindo os mesmos passos </span></div><div class="t m0 x1 h7 yb8 ff2 fs4 fc0 sc0 ls2 ws2">para os itens anteriores, fica: </div><div class="t m0 x1 h29 yb9 ff2 fs4 fc0 sc0 ls3b ws2"> <span class="ff6 ls2 wsc">\ue747<span class="_5 blank"> </span>= \ued64<span class="v19">\ue747</span></span></div><div class="t m0 x5c hf yba ff6 fs8 fc0 sc0 ls29">\ueb35<span class="fs4 ls2 ws3 v4">+ \ue747</span><span class="ls3c">\ueb36<span class="fs4 ls2 ws7 v4">\u2212\ue747</span><span class="ls2">\ueb36</span></span></div><div class="t m0 x5d h2d ybb ff6 fs4 fc0 sc0 ls2 ws7">\u2212\ue747<span class="fs8 ls3d v5">\ueb36</span>\ue747<span class="fs8 lsf v5">\ueb36</span><span class="ws3">+ \ue747<span class="fs8 ls7 v5">\ueb37</span><span class="ls3 v18">\ued68</span><span class="ff2 ws2 v18"> </span></span></div><div class="t m0 x1 h7 ybc ff2 fs4 fc0 sc0 ls2 ws2">Uma vez montadas as matrizes e vetores, podemos fazer os cálculos para o sistema. </div><div class="t m0 x1 h7 ybd ff2 fs4 fc0 sc0 ls2 ws2">Como, normalmente, existem uma <span class="_11 blank"> </span>quantidade de cálculos muito grande, é <span class="_11 blank"> </span>muito </div><div class="t m0 x1 h7 ybe ff2 fs4 fc0 sc0 ls2 ws2">frequente <span class="_3 blank"> </span>o <span class="_4 blank"> </span>uso <span class="_4 blank"> </span>de <span class="_3 blank"> </span>softwares <span class="_4 blank"> </span>para <span class="_3 blank"> </span>auxiliar <span class="_4 blank"> </span>nessa <span class="_3 blank"> </span>tarefa, <span class="_4 blank"> </span>inclu<span class="_0 blank"> </span>indo <span class="_3 blank"> </span>o <span class="_4 blank"> </span>emprego <span class="_4 blank"> </span>de <span class="_3 blank"> </span>um </div><div class="t m0 x1 h7 ybf ff2 fs4 fc0 sc0 ls2 ws2">método <span class="_1e blank"> </span>numérico <span class="_4 blank"> </span>para <span class="_1e blank"> </span>a <span class="_1e blank"> </span>solução <span class="_1e blank"> </span>de <span class="_1e blank"> </span>equações <span class="_4 blank"> </span>diferencias <span class="_1e blank"> </span>mais <span class="_1e blank"> </span>complexas. <span class="_4 blank"> </span>Um <span class="_1e blank"> </span>dos </div><div class="t m0 x1 h7 yc0 ff2 fs4 fc0 sc0 ls2 ws2">métodos mais <span class="_0 blank"> </span>empregado em <span class="_0 blank"> </span>vibrações é <span class="_0 blank"> </span>o <span class="_0 blank"> </span>Método de<span class="_0 blank"> </span> <span class="_0 blank"> </span>Newmark que fo<span class="_0 blank"> </span>i apresentado </div><div class="t m0 x1 h7 yc1 ff2 fs4 fc0 sc0 ls2 ws2">na aula 14. </div><div class="t m0 x1 h7 yc2 ff2 fs4 fc0 sc0 ls2 ws2"> </div><div class="t m0 x1 h7 yc3 ff2 fs4 fc0 sc0 ls2 ws2"> </div><div class="t m0 x1 h7 yc4 ff2 fs4 fc0 sc0 ls2 ws2"> </div><div class="t m0 x1 h2 yc5 ff1 fs0 fc0 sc0 ls2 ws2"> </div><div class="t m0 x1 h2 yc6 ff1 fs0 fc0 sc0 ls2 ws2"> </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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