<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/0d8b9f55-02e3-4bea-b2ac-5be7b063d710/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws3"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws3">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws3"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls2 ws3">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls2 ws3"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls2 ws3">SISTEMAS MECÂNICOS VIBRACIONAIS </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls2 ws3">COM MDOF \u2013 ANÁLISE MODAL </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls2 ws3">ANALÍTICA \u2013 PARTE 01: VIBRAÇÕES </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 ls2 ws3">LIVRES SEM AMORTECIMENTO<span class="fs3 fc2"> </span></div><div class="t m0 x1 h6 yb ff3 fs3 fc1 sc0 ls2 ws3"> Introdução: </div><div class="t m0 x1 h7 yc ff2 fs4 fc0 sc0 ls2 ws3">Uma <span class="_1 blank"> </span>vez <span class="_1 blank"> </span>obtidas <span class="_0 blank"> </span>as <span class="_1 blank"> </span>equações <span class="_2 blank"> </span>do <span class="_1 blank"> </span>movimento, <span class="_0 blank"> </span>nosso <span class="_1 blank"> </span>trabalho <span class="_1 blank"> </span>será <span class="_1 blank"> </span>resolver <span class="_1 blank"> </span>o <span class="_1 blank"> </span>conjunto </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 ls2 ws3">de equações diferenciais ordinárias (EDO\u2019s). </div><div class="t m0 x3 h7 ye ff2 fs4 fc0 sc0 ls2 ws3"> </div><div class="t m0 x4 h8 yf ff2 fs5 fc0 sc0 ls2 ws3">Figura 01: Exemplo de um<span class="_3 blank"></span> sistema com vários graus de lib<span class="_3 blank"></span>erdade </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 ls2 ws3">Importante <span class="_4 blank"> </span>ressaltar <span class="_4 blank"> </span>que <span class="_4 blank"> </span>o <span class="_4 blank"> </span>sistema <span class="_4 blank"> </span>representando <span class="_4 blank"> </span>pela <span class="_4 blank"> </span>figura <span class="_4 blank"> </span>01 <span class="_5 blank"> </span>corresponde <span class="_4 blank"> </span>a <span class="_4 blank"> </span>um </div><div class="t m0 x1 h7 y11 ff2 fs4 fc0 sc0 ls2 ws3">sistema <span class="_2 blank"> </span>acoplado <span class="_2 blank"> </span>de <span class="_6 blank"> </span>equações, <span class="_2 blank"> </span>o <span class="_2 blank"> </span>que <span class="_2 blank"> </span>pode <span class="_6 blank"> </span>dificultar <span class="_1 blank"> </span>determinadas <span class="_6 blank"> </span>análises, <span class="_2 blank"> </span>além <span class="_2 blank"> </span>de </div><div class="t m0 x1 h7 y12 ff2 fs4 fc0 sc0 ls2 ws3">não <span class="_6 blank"> </span>permitir <span class="_6 blank"> </span>uma <span class="_4 blank"> </span>generalização <span class="_6 blank"> </span>direta <span class="_6 blank"> </span>com <span class="_6 blank"> </span>sistemas <span class="_4 blank"> </span>com <span class="_6 blank"> </span>1 <span class="_6 blank"> </span>GDL. <span class="_6 blank"> </span> <span class="_4 blank"> </span>Nessas <span class="_6 blank"> </span>situações, </div><div class="t m0 x1 h7 y13 ff2 fs4 fc0 sc0 ls2 ws3">transformar <span class="_1 blank"> </span>o <span class="_1 blank"> </span>sistema <span class="_1 blank"> </span>para <span class="_1 blank"> </span>uma <span class="_1 blank"> </span>outra <span class="_1 blank"> </span>base <span class="_2 blank"> </span>de <span class="_1 blank"> </span>coordenadas <span class="_0 blank"> </span>pode <span class="_2 blank"> </span>ser <span class="_1 blank"> </span>útil <span class="_0 blank"> </span>e <span class="_2 blank"> </span>ganha <span class="_1 blank"> </span>um </div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 ls2 ws3">destaque <span class="_5 blank"> </span>em <span class="_5 blank"> </span>dinâmica<span class="_0 blank"> </span> <span class="_5 blank"> </span>estrutural. <span class="_5 blank"> </span>Esse<span class="_0 blank"> </span> <span class="_5 blank"> </span>assunto <span class="_5 blank"> </span>é<span class="_0 blank"> </span>, <span class="_5 blank"> </span>na <span class="_5 blank"> </span>literatura, referenciad<span class="_3 blank"></span>o <span class="_5 blank"> </span>como </div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 ls2 ws3">análise <span class="_7 blank"></span>modal <span class="_7 blank"></span>analítica, <span class="_7 blank"></span>que <span class="_7 blank"></span>veremos <span class="_7 blank"></span>mais <span class="_7 blank"></span>alguns <span class="_3 blank"></span>detalhes <span class="_7 blank"></span>para <span class="_7 blank"></span>problemas <span class="_7 blank"></span>de <span class="_7 blank"></span>vibrações </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 ls2 ws3">livres sem amortecimento. </div><div class="t m0 x1 h6 y17 ff3 fs3 fc1 sc0 ls2 ws3">Vibrações Livres em Sistema Sem Amortecimento<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y18 ff2 fs4 fc0 sc0 ls2 ws3">Considerando o <span class="_8 blank"> </span>sistema indicado <span class="_8 blank"> </span>na <span class="_8 blank"> </span>figura 01 <span class="_8 blank"> </span>com coeficiente <span class="_8 blank"> </span>de amortecimento </div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 ls2 ws3">viscoso, <span class="_6 blank"> </span>C, <span class="_4 blank"> </span>igual <span class="_6 blank"> </span>a <span class="_6 blank"> </span>0 <span class="_4 blank"> </span>e <span class="_6 blank"> </span>forças <span class="_4 blank"> </span>de <span class="_6 blank"> </span>excitação, <span class="_6 blank"> </span>F <span class="_4 blank"> </span>igual <span class="_6 blank"> </span>a <span class="_6 blank"> </span>zero. <span class="_4 blank"> </span>A <span class="_6 blank"> </span>equação <span class="_6 blank"> </span>do <span class="_4 blank"> </span>movimento </div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 ls2 ws3">ficará: </div><div class="t m0 x5 h7 y1b ff4 fs4 fc0 sc0 ls2 ws3">\ue879.<span class="_5 blank"> </span>\ue754<span class="_7 blank"></span>\u0308<span class="_9 blank"> </span>+ \ue877.<span class="_5 blank"> </span>\ue754<span class="_9 blank"> </span>=<span class="_a blank"> </span>\uead9 (\ue887)<span class="ff2"> </span></div><div class="t m0 x1 h7 y1c ff2 fs4 fc0 sc0 ls2 ws3">Sendo a solução do tipo: </div><div class="t m0 x6 h9 y1d ff4 fs4 fc0 sc0 ls2 ws0">\ue754<span class="_a blank"> </span>=<span class="_a blank"> </span>\ue8b6\ue741 <span class="fs6 ws1 v1">\uebdd\uec20\uebe7 </span><span class="ws3"> (\ue73e<span class="_0 blank"> </span>)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y1e ff2 fs4 fc0 sc0 ls2 ws3">Onde: </div><div class="t m0 x7 h7 y1f ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff2 ls1">\u0424 é o <span class="fc1 ls2">vetor amplitudes <span class="fc0">que indicam as formas modais do problema. </span></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="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/0d8b9f55-02e3-4bea-b2ac-5be7b063d710/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws3"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws3">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws3"> </div></div><div class="t m0 x1 h7 y20 ff2 fs4 fc0 sc0 ls2 ws3">Derivando e substituindo em (a), vem: </div><div class="t m0 x8 h9 y21 ff4 fs4 fc0 sc0 ls2 ws4">\u2212\ue7f1<span class="fs6 ls3 v1">\ueb36</span><span class="ws5">\ue879. \ue8b6. \ue741<span class="_1 blank"> </span><span class="fs6 ws6 v1">\uebdd<span class="_0 blank"> </span>\uec20\uebe7 </span><span class="ws0">+<span class="_8 blank"> </span>\ue877.<span class="_4 blank"> </span>\ue8b6\ue741 <span class="fs6 ws7 v1">\uebdd<span class="_0 blank"> </span>\uec20\uebe7 </span><span class="ws3">=<span class="_a blank"> </span>\uead9 (\ue889)<span class="ff2"> </span></span></span></span></div><div class="t m0 x1 h7 y22 ff2 fs4 fc0 sc0 ls2 ws3">Colocando em evidência, fica: </div><div class="t m0 x9 h9 y23 ff4 fs4 fc0 sc0 ls4">\ue741<span class="fs6 ls2 ws1 v1">\uebdd\uec20\uebe7 </span><span class="ls2 ws8 v2">[</span><span class="ls2 ws4">\u2212\ue7f1<span class="fs6 ls3 v1">\ueb36</span><span class="ws9">\ue879 + \ue877<span class="ws8 v2">]</span><span class="ws3">.<span class="_b blank"> </span>\ue8b6<span class="_a blank"> </span>=<span class="_a blank"> </span>\uead9 (\ue88a)<span class="ff3"> </span></span></span></span></div><div class="t m0 x1 h9 y24 ff2 fs4 fc0 sc0 ls2 ws3">Como <span class="ff4 ls4">\ue741<span class="fs6 ls2 wsa v1">\uebdd\uec20\uebe7 </span></span> é sempre diferente de zero, a expressão somente será zero, se: </div><div class="t m0 xa ha y25 ff4 fs4 fc0 sc0 ls2 ws8">[<span class="ws4 v3">\ue877<span class="_8 blank"> </span>\u2212<span class="_c blank"> </span>\ue7f1</span><span class="fs6 ls3 v4">\ueb36</span><span class="ls5 v3">\ue879</span>]<span class="ws3 v3">.<span class="_b blank"> </span>\ue8b6<span class="_a blank"> </span>=<span class="_a blank"> </span>\uead9 (\ue88b)<span class="ff3"> </span></span></div><div class="t m0 x1 hb y26 ff2 fs4 fc0 sc0 ls2 ws3">A equação (e) representa um problema clássico <span class="ff3 fc1 ls1">de </span><span class="fc1">autovalor e autovetor</span>, que pode </div><div class="t m0 x1 h7 y27 ff2 fs4 fc0 sc0 ls2 ws3">ser descrito como: </div><div class="t m0 xb ha y28 ff4 fs4 fc0 sc0 ls2 ws8">[<span class="ls6 v3">\ue72f</span><span class="fs6 wsb v4">\ueb3f\ueb35 </span><span class="wsc v3">\ue877 \u2212 \ue7e3\ue875</span>]<span class="ws3 v3">.<span class="_5 blank"> </span>\ue8b6<span class="_a blank"> </span>=<span class="_a blank"> </span>\uead9 (\ue88c)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y29 ff2 fs4 fc0 sc0 ls2 ws3">Sendo: </div><div class="t m0 x7 hc y2a ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff2 ls2">I <span class="ff7 ls1">\uf0e0</span> a matriz identidade de ordem n x n; </span></span></div><div class="t m0 x7 hc y2b ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff2 ls2">n <span class="ff7 ls1">\uf0e0</span> o número de graus de liberdade; </span></span></div><div class="t m0 x7 h9 y2c ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> </span><span class="ff4 ws4">\ue7e3<span class="_a blank"> </span>=<span class="_a blank"> </span>\ue7f1<span class="fs6 ls3 v1">\ueb36</span><span class="ff7 ls1">\uf0e0</span><span class="ff2 ws3"> relacionados diretamente às frequências naturais do sistema. </span></span></div><div class="t m0 x1 h7 y2d ff2 fs4 fc0 sc0 ls2 ws3">Escrevendo o problema de autovalor e autovetor n<span class="_3 blank"></span>a forma padrão, fica: </div><div class="t m0 xc h9 y2e ff4 fs4 fc0 sc0 ls6">\ue72f<span class="fs6 ls2 wsb v1">\ueb3f\ueb35 </span><span class="ls2 ws3">\ue877.<span class="_5 blank"> </span>\ue8b6<span class="_a blank"> </span>=<span class="_a blank"> </span>\ue7e3.<span class="_5 blank"> </span>\ue8b6 <span class="ws8 v2">(<span class="v3">\ue88d</span></span><span class="fs6 ls7 v5">\ueada</span><span class="ws8 v2">)</span> \ue895\ue89b \ue86d.<span class="_b blank"> </span> \ue8b6<span class="_a blank"> </span>=<span class="_d blank"> </span>\ue7e3.<span class="_b blank"> </span>\ue8b6 (\ue88d<span class="fs6 ls7 v5">\ueadb</span><span class="ws8">)</span><span class="ff2"> </span></span></div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 ls2 ws3">Sendo: </div><div class="t m0 x7 h9 y30 ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff4 ls8 wsd">\ue86d=\ue72f<span class="_e blank"></span><span class="fs6 ls2 wsb v1">\ueb3f\ueb35 <span class="fs4 ls9 v6">\ue877<span class="ff2 ls2 ws3"> </span></span></span></span></span></div><div class="t m0 x7 h9 y31 ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> </span><span class="ff4 ws4">\ue7e3<span class="_a blank"> </span>=<span class="_d blank"> </span>\ue7f1 <span class="fs6 ls3 v1">\ueb36</span><span class="ff7 ls1">\uf0e0</span><span class="ff2 ws3"> Autovalores relacionados as frequências naturais dos sistemas; </span></span></div><div class="t m0 x7 hc y32 ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff4 ls1">\ue8b6<span class="ff2 ls2"> </span><span class="ff7">\uf0e0<span class="ff2 ls2"> Autovetores que representam os modos de vibrar dos sistemas; </span></span></span></span></div><div class="t m0 x1 hb y33 ff2 fs4 fc0 sc0 ls2 ws3">Mas afinal, o que são <span class="ff3 fc1 wse">\u201c</span><span class="fc1">Modos de Vibrar<span class="ff3 wse">\u201d</span></span>? </div><div class="t m0 x1 h7 y34 ff2 fs4 fc0 sc0 ls2 ws3">Também são chamados de formas de vibrar. Modo de vibrar nada mai<span class="_3 blank"></span>s é do que uma </div><div class="t m0 x1 h7 y35 ff2 fs4 fc0 sc0 ls2 ws3">razão de amplitudes. </div><div class="t m0 x1 h7 y36 ff2 fs4 fc0 sc0 ls2 ws3">Há <span class="_0 blank"> </span>vários <span class="_0 blank"> </span>métodos <span class="_0 blank"> </span>para <span class="_0 blank"> </span>solução <span class="_1 blank"> </span>do <span class="_0 blank"> </span>problema <span class="_0 blank"> </span>de <span class="_0 blank"> </span>autovalores <span class="_0 blank"> </span>e <span class="_1 blank"> </span>autovetores, <span class="_0 blank"> </span>como <span class="_0 blank"> </span>por </div><div class="t m0 x1 h7 y37 ff2 fs4 fc0 sc0 ls2 ws3">exemplo, <span class="_0 blank"> </span>o método <span class="_1 blank"> </span>de <span class="_0 blank"> </span><span class="fc1 wsf">Choleski</span>, <span class="_0 blank"> </span>com <span class="_0 blank"> </span>algorítmo <span class="_0 blank"> </span>pronto <span class="_0 blank"> </span>no <span class="_0 blank"> </span>Matlab <span class="_0 blank"> </span>ou <span class="_0 blank"> </span>Scilab, <span class="_0 blank"> </span>e <span class="_0 blank"> </span>os <span class="_0 blank"> </span>mais </div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 ls2 ws3">tradicionais, por meio do cálculo do <span class="fc1">determinante </span>da matriz. </div><div class="t m0 x1 h7 y39 ff2 fs4 fc0 sc0 ls2 ws3">Assim, temos: <span class="ff4"> </span> </div><div class="t m0 xd h9 y3a ff4 fs4 fc0 sc0 ls2 ws8">det<span class="v2">(</span><span class="ls6">\ue72f</span><span class="fs6 wsb v1">\ueb3f\ueb35 </span><span class="ws10">\ue877 \u2212 \ue7e3\ue875<span class="lsa v2">)</span><span class="ws3">=<span class="_d blank"> </span>\uead9 <span class="ff2 ls1">ou </span> \ue740\ue741\ue750<span class="_0 blank"> </span></span></span><span class="v2">[</span><span class="ws4">\ue877<span class="_8 blank"> </span>\u2212<span class="_c blank"> </span>\ue7f1 <span class="fs6 ls3 v1">\ueb36</span><span class="ls5">\ue879<span class="lsb v2">]</span></span><span class="ws11">= \uead9<span class="ff3 ws3"> (h)<span class="ff2"> </span></span></span></span></div><div class="t m0 x1 h9 y3b ff2 fs4 fc0 sc0 ls2 ws3">No <span class="_5 blank"> </span>caso, <span class="_5 blank"> </span>o <span class="_5 blank"> </span>problema <span class="_5 blank"> </span>de <span class="_5 blank"> </span>autovalor <span class="_5 blank"> </span>leva <span class="_5 blank"> </span>a uma <span class="_4 blank"> </span>equação <span class="_5 blank"> </span>algébrica <span class="_5 blank"> </span>em <span class="ff4 lsc">\ue7f1<span class="fs6 ls3 v1">\ueb36</span></span>. <span class="_5 blank"> </span>Como <span class="_5 blank"> </span>o<span class="_0 blank"> </span>s </div><div class="t m0 x1 hb y3c ff2 fs4 fc0 sc0 ls2 ws3">coeficientes <span class="ff3 wse">M</span> <span class="_c blank"> </span>e <span class="_8 blank"> </span><span class="ff3 wse">K</span> são, normalmente, re<span class="_0 blank"> </span>ais e simétricos <span class="_c blank"> </span>teremo<span class="_0 blank"> </span>s <span class="_8 blank"> </span><span class="ff3 wse">n</span> raízes reais e </div><div class="t m0 x1 hb y3d ff2 fs4 fc0 sc0 ls2 ws3">consequentemente <span class="ff3 wse">n</span> frequências naturais. </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/0d8b9f55-02e3-4bea-b2ac-5be7b063d710/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws3"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws3">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws3"> </div></div><div class="t m0 x1 hb y20 ff2 fs4 fc0 sc0 ls2 ws3">Se o <span class="_0 blank"> </span>sistema for <span class="_0 blank"> </span>estável implica <span class="_0 blank"> </span>em <span class="_0 blank"> </span><span class="ff3 wse">K</span> <span class="_0 blank"> </span>positiva e <span class="_0 blank"> </span>com <span class="_0 blank"> </span>as raízes <span class="_0 blank"> </span>positivas. Já <span class="_0 blank"> </span>um sistema<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y3e ff2 fs4 fc0 sc0 ls2 ws3">não restringido <span class="_c blank"> </span>aprese<span class="_0 blank"> </span>ntará modos <span class="_c blank"> </span>de <span class="_8 blank"> </span>corpo rígido <span class="_c blank"> </span>corre<span class="_0 blank"> </span>spondendo a <span class="_c blank"> </span>fre<span class="_0 blank"> </span>quências </div><div class="t m0 x1 h7 y3f ff2 fs4 fc0 sc0 ls2 ws3">naturais nulas. </div><div class="t m0 x1 h7 y40 ff2 fs4 fc0 sc0 ls2 ws3">Importante <span class="_8 blank"> </span>reparar <span class="_8 blank"> </span>que <span class="_8 blank"> </span>os <span class="_8 blank"> </span>modos <span class="_8 blank"> </span>de <span class="_8 blank"> </span>vibrar <span class="_8 blank"> </span>representam <span class="_8 blank"> </span>uma <span class="_8 blank"> </span>base <span class="_8 blank"> </span>ortogonal <span class="_d blank"> </span>no </div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 ls2 ws3">espaço. Assim, a matriz modal <span class="ff4 ls1">\ue8b6</span> apresenta as seguintes propriedades para <span class="ff4 ws12">\ue745<span class="_9 blank"> </span>\u2260<span class="_d blank"> </span>\ue746 :</span> </div><div class="t m0 xe hd y42 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls2 v7">\uebdc</span></div><div class="t m0 xf he y43 ff4 fs6 fc0 sc0 lsd">\uebcd<span class="fs4 ls2 ws8 v8">\ue879\ue8b6</span></div><div class="t m0 x10 hf y44 ff4 fs6 fc0 sc0 lse">\uebdd<span class="fs4 ls2 ws3 v9">=<span class="_d blank"> </span>\uead9 (\ue88f</span><span class="ls7">\ueada<span class="fs4 lsf v9">)<span class="ff2 ls2 ws3"> </span></span></span></div><div class="t m0 xe hd y45 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls2 v7">\uebdc</span></div><div class="t m0 xf he y46 ff4 fs6 fc0 sc0 lsd">\uebcd<span class="fs4 ls2 ws8 v8">\ue877\ue8b6</span></div><div class="t m0 x11 hf y47 ff4 fs6 fc0 sc0 ls10">\uebdd<span class="fs4 ls2 ws3 v9">=<span class="_d blank"> </span>\uead9 (\ue88f</span><span class="ls11">\ueadb<span class="fs4 ls2 ws8 v9">)<span class="ff2 ws3"> </span></span></span></div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls2 ws3">Sendo: </div><div class="t m0 x12 hc y49 ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff4 ls1">\ue8b6<span class="fs6 ls12 v5">\ue88f</span><span class="ff7">\uf0e0<span class="ff2 ls2"> o i-ésimo modo associado com a i-ésima frequência natural <span class="ff4 ws8">\ue7f1<span class="fs6 ws13 v5">\uebe1\uebdc </span></span>; </span></span></span></span></div><div class="t m0 x12 hd y4a ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> </span><span class="ff4">\ue8b6</span></div><div class="t m0 x13 h10 y4b ff4 fs6 fc0 sc0 ls13">\uebdd<span class="ff7 fs4 ls1 v9">\uf0e0<span class="ff2 ls2 ws3"> o j-ésimo modo associado com a j-ésima frequência natural <span class="ff4 ws8">\ue7f1</span></span></span><span class="ls2 ws14">\uebe1\uebdd <span class="ff2 fs4 ws3 v9">; </span></span></div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 ls2 ws3">Assim: </div><div class="t m0 x14 hd y4d ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls2 v7">\uebdc</span></div><div class="t m0 x15 he y4e ff4 fs6 fc0 sc0 lsd">\uebcd<span class="fs4 ls2 ws8 v8">\ue879\ue8b6</span><span class="ls14 va">\uebdc</span><span class="fs4 ls2 ws3 v8">=<span class="_d blank"> </span>1 (\ue746</span></div><div class="t m0 x16 hf y4f ff4 fs6 fc0 sc0 ls3">\ueb35<span class="fs4 ls2 ws8 v9">)<span class="ff2 ws3"> </span></span></div><div class="t m0 x17 hd y50 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls2 v7">\uebdc</span></div><div class="t m0 x18 he y51 ff4 fs6 fc0 sc0 lsd">\uebcd<span class="fs4 ls2 ws8 v8">\ue877\ue8b6</span><span class="ls14 va">\uebdc</span><span class="fs4 ls2 ws11 v8">= \ue7f1</span><span class="ls2 vb">\uebdc</span></div><div class="t m0 x19 he y51 ff4 fs6 fc0 sc0 ls15">\ueb36<span class="fs4 ls2 ws3 v8"> (\ue746</span></div><div class="t m0 x1a hf y52 ff4 fs6 fc0 sc0 ls15">\ueb36<span class="fs4 ls2 ws8 v9">)<span class="ff2 ws3"> </span></span></div><div class="t m0 x1 h7 y53 ff2 fs4 fc0 sc0 ls2 ws3">Nesses casos, os <span class="_0 blank"> </span>modos <span class="ff4 ls1">\ue8b6</span> <span class="_0 blank"> </span>são normalizados <span class="_0 blank"> </span>em relação a <span class="_0 blank"> </span>matriz massa, o <span class="_0 blank"> </span>que implica </div><div class="t m0 x1 h7 y54 ff2 fs4 fc0 sc0 ls2 ws3">que a matriz modal é ortonormal. </div><div class="t m0 x1 h7 y55 ff2 fs4 fc0 sc0 ls2 ws3">A <span class="_5 blank"> </span>matriz <span class="_5 blank"> </span>modal <span class="_5 blank"> </span><span class="ff4 ls1">\ue8b6</span> contém <span class="_4 blank"> </span>as <span class="_b blank"> </span>formas <span class="_5 blank"> </span>de <span class="_5 blank"> </span>vibrar <span class="ff4">\ue8b6</span></div><div class="t m0 x1b hf y56 ff4 fs6 fc0 sc0 ls3">\ueb35<span class="ff2 fs4 ls2 ws3 v9">quando <span class="_5 blank"> </span>o <span class="_5 blank"> </span>sistema <span class="_5 blank"> </span>é exci<span class="_3 blank"></span>tado <span class="_5 blank"> </span>na </span></div><div class="t m0 x1 h7 y57 ff2 fs4 fc0 sc0 ls2 ws3">primeira <span class="_f blank"> </span>frequência <span class="_f blank"> </span>natural <span class="_f blank"> </span><span class="ff4 ws8">\ue8d3<span class="fs6 ws15 v5">\ue894\ueada </span></span>, <span class="_f blank"> </span><span class="ff4 ls1">\ue8b6<span class="fs6 ls15 v5">\ueb36</span></span>quando <span class="_f blank"> </span>o <span class="_10 blank"> </span>sistema <span class="_f blank"> </span>é <span class="_f blank"> </span>excitado <span class="_f blank"> </span>na <span class="_f blank"> </span>segunda </div><div class="t m0 x1 h7 y58 ff2 fs4 fc0 sc0 ls2 ws3">frequência natural <span class="ff4 ws8">\ue8d3<span class="fs6 ws15 v5">\ue894\ueadb </span></span> e assim por diante, dessa forma, a matriz é dada por: </div><div class="t m0 x1c h11 y59 ff4 fs4 fc0 sc0 ls2 ws11">\ueab4 = <span class="ws8 v2">[</span>\ueab4</div><div class="t m0 x1d h12 y5a ff4 fs6 fc0 sc0 ls7">\ueada<span class="fs4 ls2 ws3 v9"> \ueab4</span></div><div class="t m0 x19 h12 y5a ff4 fs6 fc0 sc0 ls16">\ueadb<span class="fs4 ls2 ws3 v9"> <span class="_5 blank"> </span>\u2026<span class="_5 blank"> </span>\ueab4</span></div><div class="t m0 x1e h13 y5a ff4 fs6 fc0 sc0 ls17">\ue716<span class="fs4 ls2 ws8 vc">]<span class="ws3 v3"> (\ue747<span class="_0 blank"> </span>)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y5b ff2 fs4 fc0 sc0 ls2 ws3">Uma <span class="_5 blank"> </span>vez ca<span class="_3 blank"></span>lculados <span class="_5 blank"> </span>os <span class="_5 blank"> </span>modos de <span class="_5 blank"> </span>vibrar <span class="_5 blank"> </span><span class="ff4 ls1">\ueab4</span> e <span class="_5 blank"> </span>as <span class="_5 blank"> </span>frequências <span class="_5 blank"> </span>naturais <span class="ff4 ws8">\ue7f1<span class="fs6 ws13 v5">\uebe1\uebdc </span></span> <span class="_5 blank"> </span>podemos </div><div class="t m0 x1 h7 y5c ff2 fs4 fc0 sc0 ls2 ws3">substituir <span class="_3 blank"></span>na equa<span class="_3 blank"></span>ção (b)<span class="_3 blank"></span> <span class="_3 blank"></span>para a <span class="_7 blank"></span>solução da <span class="_7 blank"></span>resposta de <span class="_7 blank"></span>vibração do <span class="_3 blank"></span>sistema, <span class="_7 blank"></span>desde que, </div><div class="t m0 x1 h7 y5d ff2 fs4 fc0 sc0 ls2 ws3">se conheça as condições iniciais. </div><div class="t m0 x1 h7 y5e ff2 fs4 fc0 sc0 ls2 ws3">Um <span class="_f blank"> </span>sistem<span class="_3 blank"></span>a <span class="_f blank"> </span>MDOF <span class="_11 blank"> </span>com <span class="_f blank"> </span>coorden<span class="_3 blank"></span>adas <span class="_11 blank"> </span>f<span class="_0 blank"> </span>ísicas <span class="_11 blank"> </span>também <span class="_f blank"> </span>pode <span class="_11 blank"> </span>ser <span class="_f blank"> </span>convertido <span class="_11 blank"> </span>em </div><div class="t m0 x1 h7 y5f ff2 fs4 fc0 sc0 ls2 ws3">coordenadas modais: </div><div class="t m0 x1f h7 y60 ff4 fs4 fc0 sc0 ls2 ws3">\ue754<span class="_a blank"> </span>=<span class="_a blank"> </span>\ue8b6\ue74d<span class="_0 blank"> </span> (\ue748<span class="_1 blank"> </span>)<span class="ff2"> </span></div><div class="t m0 x1 h7 y61 ff2 fs4 fc0 sc0 ls2 ws3">Onde: </div><div class="t m0 x20 hc y62 ff5 fs4 fc0 sc0 ls2 ws2">\uf0b7<span class="ff6 ls0 ws3"> <span class="ff2 ls1">q <span class="ff7">\uf0e0</span><span class="ls2"> vetor deslocamento em coordenadas modais; </span></span></span></div><div class="t m0 x1 h9 y63 ff2 fs4 fc0 sc0 ls2 ws3">Substituindo na eq. do movimento inicial e já pré-multiplicando por <span class="ff4 ls18">\ue7d4<span class="fs6 ls19 v1">\uebcd</span></span>: </div><div class="t m0 x21 h9 y64 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls1a v1">\ue880</span><span class="ls2 ws8">\ue879\ue8b6\ue897</span></div><div class="t m0 xf h14 y65 ff4 fs4 fc0 sc0 ls1b">\u0308<span class="ls2 ws10 v3">+ \ue8b6</span><span class="fs6 ls1a v4">\ue880</span><span class="ls2 ws3 v3">\ue877\ue8b6\ue897<span class="_d blank"> </span>=<span class="_a blank"> </span>\uead9 (\ue749<span class="_0 blank"> </span>)<span class="ff2"> </span></span></div><div class="t m0 x1 hb y66 ff2 fs4 fc0 sc0 ls2 ws3"> Assumindo que <span class="ff4 ls1">\ue8b6</span> é normalizada em relação a <span class="ff3 wse">M</span> e ortonormal, fica: </div><div class="t m0 xe h9 y67 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls1a v1">\ue880</span><span class="ls2 ws3">\ue879\ue8b6<span class="_d blank"> </span>=<span class="_a blank"> </span>\ue875 (\ue74a</span></div><div class="t m0 x22 hf y68 ff4 fs6 fc0 sc0 ls3">\ueb35<span class="fs4 ls2 ws8 v9">)<span class="ff2 ws3"> </span></span></div><div class="t m0 x23 h9 y69 ff4 fs4 fc0 sc0 ls1">\ue8b6<span class="fs6 ls1a v1">\ue880</span><span class="ls2 ws3">\ue877\ue8b6<span class="_d blank"> </span>=<span class="_a blank"> </span>\ue8b9 (\ue74a<span class="fs6 ls15 v5">\ueb36</span><span class="ws8">)</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="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/0d8b9f55-02e3-4bea-b2ac-5be7b063d710/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls2 ws3"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls2 ws3">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls2 ws3"> </div></div><div class="t m0 x1 h7 y6a ff2 fs4 fc0 sc0 ls2 ws3">Sendo: <span class="ff4 ws16">\ue8b9<span class="_d blank"> </span>=<span class="_a blank"> </span>\ue740\ue745\ue73d\ue743 (\ue7f1</span></div><div class="t m0 x24 he y6b ff4 fs6 fc0 sc0 ls2">\ueb35</div><div class="t m0 x24 he y6c ff4 fs6 fc0 sc0 ls15">\ueb36<span class="fs4 ls2 ws3 v8">,<span class="_5 blank"> </span> \ue7f1<span class="fs6 v7">\ueb36</span></span></div><div class="t m0 x25 he y6c ff4 fs6 fc0 sc0 ls3">\ueb36<span class="fs4 ls2 ws5 v8">, \u2026 , \ue7f1</span></div><div class="t m0 x18 he y6d ff4 fs6 fc0 sc0 ls2">\uebe1</div><div class="t m0 x15 he y6e ff4 fs6 fc0 sc0 ls3">\ueb36<span class="fs4 ls2 ws8 v6">)<span class="ff2 ws3"> </span></span></div><div class="t m0 x1 h7 y6f ff2 fs4 fc0 sc0 ls2 ws3">Aplicando a equação (m), chegamos a equação para o sistema MD<span class="_3 blank"></span>OF livre e sem </div><div class="t m0 x1 h7 y70 ff2 fs4 fc0 sc0 ls2 ws3">amortecimento escreta em uma base modal, dada por: </div><div class="t m0 x26 hd y71 ff4 fs4 fc0 sc0 ls2">\ue897</div><div class="t m0 x27 hd y72 ff4 fs4 fc0 sc0 ls1b">\u0308<span class="ls2 wsc v3">+ \ue8b9\ue897<span class="_d blank"> </span>=<span class="_a blank"> </span>\uead9<span class="ff2 ws3"> </span></span></div><div class="t m0 x1 h7 y73 ff2 fs4 fc0 sc0 ls2 ws3">Na próxima aula, veremos um exemplo para ilustrar melhor o conteúdo. </div><div class="t m0 x1 h2 y74 ff1 fs0 fc0 sc0 ls2 ws3"> </div><div class="t m0 x1 h2 y75 ff1 fs0 fc0 sc0 ls2 ws3"> </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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