<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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 lsb wsb">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 lsb wsb"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 lsb wsb">SISTEMAS MECÂNICOS VIBRACIONAIS </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 lsb wsb">COM MDOF \u2013 ANÁLISE MODAL </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 lsb wsb">ANALÍTICA \u2013 PARTE 03: VIBRAÇÕES </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 lsb wsb">LIVRES COM AMORTECIMENTO </div><div class="t m0 x1 h5 yb ff3 fs2 fc0 sc0 lsb ws0">PROPORCIONAL<span class="fs3 fc2 wsb"> </span></div><div class="t m0 x1 h6 yc ff3 fs3 fc1 sc0 lsb wsb"> Introdução: </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 lsb wsb">Um <span class="_1 blank"> </span>sistema <span class="_1 blank"> </span>mecânico <span class="_1 blank"> </span>vibratório <span class="_1 blank"> </span>com <span class="_1 blank"> </span>vibração <span class="_1 blank"> </span>livre <span class="_1 blank"> </span>e <span class="_1 blank"> </span>com <span class="_1 blank"> </span>amortecime<span class="_2 blank"></span>nto <span class="_1 blank"> </span>pode <span class="_1 blank"> </span>ser </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 lsb wsb">descrito pela seguinte equação diferencial: </div><div class="t m0 x3 h8 yf ff4 fs4 fc0 sc0 lsb ws1">\ue879\ue89e</div><div class="t m0 x4 h8 y10 ff4 fs4 fc0 sc0 ls0">\u0308<span class="lsb ws2 v0">+ \ue86f\ue89e</span></div><div class="t m0 x5 h8 y10 ff4 fs4 fc0 sc0 ls1">\u0307<span class="lsb wsb v0">+ \ue877\ue89e<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\uead9 (\ue887)<span class="ff2"> </span></span></div><div class="t m0 x1 h9 y11 ff2 fs4 fc0 sc0 lsb wsb">Sendo <span class="_4 blank"> </span><span class="ff3 ls2">C</span> <span class="_4 blank"> </span>a <span class="_4 blank"> </span>matriz <span class="_4 blank"> </span>de <span class="_4 blank"> </span>amortecimento <span class="_4 blank"> </span>do <span class="_4 blank"> </span>tipo <span class="_4 blank"> </span>proporcional <span class="_4 blank"> </span>as <span class="_4 blank"> </span>matrizes <span class="_4 blank"> </span>de <span class="_4 blank"> </span>massa <span class="_1 blank"> </span><span class="ff3 ws3">M</span> <span class="_4 blank"> </span>e </div><div class="t m0 x1 h9 y12 ff2 fs4 fc0 sc0 lsb wsb">rigidez <span class="ff3 ws3">K.</span><span class="ls3"> </span><span class="ff4 v1">\ue86f<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue7d9\ue879 + \ue7da\ue877 (\ue888)</span><span class="v1"> </span></div><div class="t m0 x1 h7 y13 ff2 fs4 fc0 sc0 lsb wsb">Sendo <span class="_4 blank"> </span><span class="ff5 ws4">\uf061</span> <span class="_4 blank"> </span>e <span class="_4 blank"> </span><span class="ff5 ls2">\uf062</span> <span class="_1 blank"> </span>constantes <span class="_4 blank"> </span>determinadas <span class="_4 blank"> </span>a <span class="_4 blank"> </span>partir <span class="_4 blank"> </span>de <span class="_4 blank"> </span>métodos <span class="_4 blank"> </span>específicos <span class="_4 blank"> </span>de <span class="_4 blank"> </span>ajustes <span class="_4 blank"> </span>de<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 lsb wsb">modelos. </div><div class="t m0 x1 h6 y15 ff3 fs3 fc1 sc0 lsb wsb">Vibrações <span class="_5 blank"> </span>Livres <span class="_5 blank"> </span>em <span class="_5 blank"> </span>Sistemas <span class="_5 blank"> </span>com <span class="_5 blank"> </span>Amortecimento <span class="_5 blank"> </span>Proporcional,<span class="_0 blank"> </span> </div><div class="t m0 x1 h6 y16 ff3 fs3 fc1 sc0 lsb wsb">Subamortecido </div><div class="t m0 x1 h7 y17 ff2 fs4 fc0 sc0 lsb wsb">Aqui <span class="_2 blank"></span>o <span class="_6 blank"></span>problema <span class="_6 blank"></span>de <span class="_6 blank"></span>autovalor e <span class="_6 blank"></span>autovetor <span class="_6 blank"></span>associado <span class="_6 blank"></span>a <span class="_6 blank"></span>equação (a) <span class="_6 blank"></span>irá <span class="_6 blank"></span>envolver <span class="_6 blank"></span>soluções </div><div class="t m0 x1 h7 y18 ff2 fs4 fc0 sc0 lsb wsb">complexas. <span class="_0 blank"> </span>Assim, <span class="_0 blank"> </span>as <span class="_0 blank"> </span>raízes <span class="_0 blank"> </span>da <span class="_0 blank"> </span><span class="fc1">equação <span class="_0 blank"> </span>carac<span class="_0 blank"> </span>terística <span class="_0 blank"> </span></span>associada, <span class="_0 blank"> </span>irá <span class="_0 blank"> </span>envolver <span class="_0 blank"> </span>pares <span class="_0 blank"> </span>de </div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 lsb wsb">pólos <span class="_6 blank"></span>complexos <span class="_6 blank"></span>conjugados <span class="_7 blank"></span>para <span class="_7 blank"></span>cada <span class="_6 blank"></span>modo <span class="_6 blank"></span>de <span class="_6 blank"></span>vibrar <span class="_7 blank"></span>do <span class="_6 blank"></span>sistema, <span class="_6 blank"></span>considerando <span class="_7 blank"></span>o <span class="_6 blank"></span>caso </div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 lsb wsb">de um sistema subamortecido em todos os mod<span class="_2 blank"></span>os (0 < \u03be < 1). Ficando: </div><div class="t m0 x6 ha y1b ff4 fs4 fc0 sc0 ls4">\ue7e3<span class="fs5 ls5 v2">\uebdc</span><span class="lsb ws5">= \u2212\ue7e6<span class="_6 blank"></span><span class="fs5 ls6 v2">\uebdc<span class="fs4 lsb ws6 v3">. \ue7f1</span><span class="ls7">\uebdc<span class="fs4 lsb ws6 v3">±<span class="_8 blank"> </span>\ue746<span class="_0 blank"></span>. \ue7f1</span><span class="lsb ws7">\uebe1\uebdc <span class="fs4 ws1 v3">\ueda7<span class="ws8 v0">1 \u2212 \ue7e6<span class="_6 blank"></span><span class="fs5 v4">\uebdc</span></span></span></span></span></span></span></div><div class="t m0 x7 hb y1c ff4 fs5 fc0 sc0 ls8">\ueb36<span class="fs4 lsb wsb v5"> \ue73f\ue74b\ue749 \ue745<span class="_9 blank"> </span>=<span class="_3 blank"> </span>1,2,<span class="_a blank"> </span>\u2026<span class="_a blank"> </span>,<span class="_a blank"> </span>\ue74a (\ue73f<span class="_b blank"> </span>)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y1d ff2 fs4 fc0 sc0 lsb wsb">Onde: </div><div class="t m0 x8 hc y1e ff5 fs4 fc0 sc0 lsb ws4">\uf0b7<span class="ff6 ls9 wsb"> <span class="ff2 lsb">n <span class="ff7 ls2">\uf0e0</span> Número de modos do sistema; </span></span></div><div class="t m0 x8 hc y1f ff5 fs4 fc0 sc0 lsb ws4">\uf0b7<span class="ff6 ls9 wsb"> </span><span class="ff2 ws9">\u03be<span class="fs6 lsa v6">i</span><span class="wsb"> <span class="ff7 ls2">\uf0e0</span> Fator de amortecimento modal associado ao i-ésimo modo de vibrar; </span></span></div><div class="t m0 x8 hc y20 ff5 fs4 fc0 sc0 lsb ws4">\uf0b7<span class="ff6 ls9 wsb"> </span>\uf077<span class="ff2 fs6 wsa v6">ni</span><span class="ff2 wsb"> <span class="ff7 ls2">\uf0e0</span> a i-ésima frequência natural. </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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h7 y21 ff2 fs4 fc0 sc0 lsb wsb">Para <span class="_c blank"> </span>o <span class="_c blank"> </span>caso <span class="_c blank"> </span>particular <span class="_c blank"> </span>de <span class="_c blank"> </span>amortecimento <span class="_c blank"> </span>do <span class="_c blank"> </span>tipo <span class="_c blank"> </span>proporcional, <span class="_c blank"> </span>os <span class="_c blank"> </span>fatores <span class="_c blank"> </span>de </div><div class="t m0 x1 h8 y22 ff2 fs4 fc0 sc0 lsb wsb">amortecimento modal <span class="ff4 ws1">\ue7e6<span class="_6 blank"></span><span class="fs5 ls6 v2">\uebdc<span class="ff8 fs4 lsb wsb v3"> <span class="ff2">podem ser aproximados pela equação: </span></span></span></span></div><div class="t m0 x9 hd y23 ff4 fs4 fc0 sc0 lsb ws1">\ue7e6<span class="_6 blank"></span><span class="fs5 lsc v2">\uebdc<span class="fs4 lsd v3">=<span class="lsb v7">1</span></span></span></div><div class="t m0 xa he y24 ff4 fs4 fc0 sc0 lse">2<span class="ls2 v8">\ued6c</span><span class="lsb ws1 v8">\ue7d9<span class="_0 blank"></span>\ue7f1</span><span class="fs5 lsb wsc v9">\uebe1\uebdc </span><span class="lsf v8">+</span><span class="lsb va">\ue7da</span></div><div class="t m0 xb hf y24 ff4 fs4 fc0 sc0 lsb ws1">\ue7f1<span class="fs5 wsd v2">\uebe1\uebdc </span><span class="ls10 v8">\ued70</span><span class="wsb v8"> <span class="_4 blank"> </span>(\ue740)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y25 ff2 fs4 fc0 sc0 lsb wsb">Para <span class="_8 blank"> </span>so<span class="_0 blank"> </span>lucionar <span class="_8 blank"> </span>o <span class="_d blank"> </span>problema <span class="_8 blank"> </span>de <span class="_d blank"> </span>autovalor <span class="_8 blank"> </span>e <span class="_d blank"> </span>autovetor<span class="_0 blank"> </span> <span class="_d blank"> </span>é <span class="_8 blank"> </span>interessante <span class="_d blank"> </span>reescrever <span class="_8 blank"> </span>a </div><div class="t m0 x1 h7 y26 ff2 fs4 fc0 sc0 lsb wsb">equação (a) de uma forma mais conveniente. </div><div class="t m0 x1 h7 y27 ff2 fs4 fc0 sc0 lsb wsb">A <span class="_4 blank"> </span>p<span class="_2 blank"></span>rincipal <span class="_4 blank"> </span>di<span class="_6 blank"></span>f<span class="_0 blank"> </span>erença <span class="_4 blank"> </span>a<span class="_6 blank"></span>gora <span class="_4 blank"> </span>é <span class="_b blank"> </span>que <span class="_4 blank"> </span>os <span class="_4 blank"> </span>autov<span class="_6 blank"></span>alores <span class="_4 blank"> </span>e <span class="_4 blank"> </span>os <span class="_b blank"> </span>autovetores <span class="_4 blank"> </span>sã<span class="_6 blank"></span>o <span class="_1 blank"> </span><span class="fc1 ws9">complexos</span>, <span class="_4 blank"> </span>ou </div><div class="t m0 x1 h7 y28 ff2 fs4 fc0 sc0 lsb wsb">seja, <span class="_4 blank"> </span>os <span class="_b blank"> </span>autovalores <span class="_4 blank"> </span>estão <span class="_b blank"> </span>relacionados <span class="_4 blank"> </span>diretamente <span class="_b blank"> </span>aos <span class="_4 blank"> </span>fatores <span class="_e blank"> </span>de <span class="_e blank"> </span>amortecimento <span class="_b blank"> </span>e </div><div class="t m0 x1 h7 y29 ff2 fs4 fc0 sc0 lsb wsb">frequência <span class="_7 blank"></span>natural <span class="_6 blank"></span>para <span class="_6 blank"></span>cada <span class="_6 blank"></span>modo <span class="_7 blank"></span>e <span class="_6 blank"></span>os <span class="_6 blank"></span>autovetores <span class="_7 blank"></span>aos <span class="_6 blank"></span>modos <span class="_6 blank"></span>de <span class="_7 blank"></span>vibrar <span class="_6 blank"></span>que <span class="_6 blank"></span>neste <span class="_6 blank"></span>caso </div><div class="t m0 x1 h7 y2a ff2 fs4 fc0 sc0 lsb wsb">por serem <span class="_8 blank"> </span>complexos devem <span class="_8 blank"> </span>ser <span class="_8 blank"> </span>descritos por <span class="_8 blank"> </span>uma <span class="_8 blank"> </span>amplitude e <span class="_8 blank"> </span>uma <span class="_8 blank"> </span>fase, o <span class="_8 blank"> </span>que </div><div class="t m0 x1 h7 y2b ff2 fs4 fc0 sc0 lsb wsb">significa <span class="_e blank"> </span>dizer <span class="_4 blank"> </span>que <span class="_4 blank"> </span>os <span class="_4 blank"> </span>modos <span class="_4 blank"> </span>de <span class="_4 blank"> </span>vibrar <span class="_e blank"> </span>apresentam <span class="_4 blank"> </span>uma <span class="_4 blank"> </span>fase <span class="_4 blank"> </span>na <span class="_4 blank"> </span>mesma <span class="_e blank"> </span>coordenada. </div><div class="t m0 x1 h7 y2c ff2 fs4 fc0 sc0 lsb wsb">Isto tudo é induzido pela presença de amortecimento no sistema. </div><div class="t m0 x1 h7 y2d ff2 fs4 fc0 sc0 lsb wsb">Importante ress<span class="_0 blank"> </span>altar que <span class="_0 blank"> </span>é <span class="_0 blank"> </span>muito <span class="_0 blank"> </span>comum <span class="_0 blank"> </span>se <span class="_0 blank"> </span>desconsiderar o <span class="_0 blank"> </span>efeito <span class="_0 blank"> </span>do <span class="_0 blank"> </span>amortecimento </div><div class="t m0 x1 h7 y2e ff2 fs4 fc0 sc0 lsb wsb">no <span class="_0 blank"> </span>cálculo <span class="_b blank"> </span>de <span class="_0 blank"> </span>mo<span class="_0 blank"> </span>dos <span class="_0 blank"> </span>de <span class="_b blank"> </span>vibrar <span class="_0 blank"> </span>e <span class="_b blank"> </span>frequências <span class="_0 blank"> </span>naturais, <span class="_b blank"> </span>caso <span class="_b blank"> </span>a <span class="_0 blank"> </span>estrutura <span class="_b blank"> </span>seja <span class="_0 blank"> </span>levemente </div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 lsb wsb">amortecida <span class="_b blank"> </span>e <span class="_e blank"> </span>o <span class="_e blank"> </span>f<span class="_0 blank"> </span>ator <span class="_b blank"> </span>de <span class="_e blank"> </span>amortecimento <span class="_e blank"> </span>possa <span class="_e blank"> </span>ser <span class="_e blank"> </span>aproxi<span class="_0 blank"> </span>mado <span class="_e blank"> </span>a <span class="_e blank"> </span>zero, <span class="_e blank"> </span>o <span class="_e blank"> </span>que <span class="_e blank"> </span>significa </div><div class="t m0 x1 h7 y30 ff2 fs4 fc0 sc0 lsb wsb">dizer que os pólos do sistema estão muitos próximos do eixo i<span class="_2 blank"></span>maginário. </div><div class="t m0 x1 h7 y31 ff2 fs4 fc0 sc0 lsb wsb">A <span class="_3 blank"> </span>seguir <span class="_3 blank"> </span>apresentamos <span class="_f blank"> </span>as <span class="_3 blank"> </span>duas <span class="_f blank"> </span>formas <span class="_3 blank"> </span>padrão <span class="_3 blank"> </span>muito <span class="_f blank"> </span>usadas <span class="_3 blank"> </span>para <span class="_3 blank"> </span>a <span class="_f blank"> </span>solução <span class="_3 blank"> </span>do </div><div class="t m0 x1 h7 y32 ff2 fs4 fc0 sc0 lsb wsb">problema de autovalor e autovetor de um sistema com am<span class="_6 blank"></span>ortecimento proporcional.<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y33 ff2 fs4 fc1 sc0 lsb wsb">Forma 1: Dobrando o número de equações e diminuindo a ordem: </div><div class="t m0 xc h8 y34 ff4 fs4 fc0 sc0 lsb">\ue879</div><div class="t m0 xd h8 y35 ff4 fs4 fc0 sc0 ls11">\uede9<span class="lsb ws2 v2">\ue755<span class="_7 blank"></span>\u0307<span class="_f blank"> </span>+ \ue877</span></div><div class="t m0 xe h8 y35 ff4 fs4 fc0 sc0 ls12">\uede9<span class="lsb wsb v2">\ue755<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\uead9 (\ue88a)<span class="ff2"> </span></span></div><div class="t m0 x1 h8 y36 ff2 fs4 fc0 sc0 lsb wsb">Sendo <span class="_6 blank"></span><span class="ff4">\ue879</span></div><div class="t m0 xf h8 y37 ff4 fs4 fc0 sc0 ls11">\uede9<span class="lsb wsb v2"> \ue88b \ue877</span></div><div class="t m0 x10 h8 y37 ff4 fs4 fc0 sc0 ls12">\uede9<span class="ff3 lsb wsb v2"> <span class="_6 blank"></span><span class="ff2">matrizes <span class="_6 blank"></span>simétricas <span class="_6 blank"></span>de <span class="_6 blank"></span>ordem <span class="_6 blank"></span>2n <span class="_6 blank"></span>x <span class="_6 blank"></span>2n <span class="_6 blank"></span>e <span class="ff3 ls2">y</span> <span class="_7 blank"></span>o <span class="_6 blank"></span>vetor <span class="_6 blank"></span>de <span class="_2 blank"></span>estados <span class="_6 blank"></span>definidos <span class="_6 blank"></span>por: </span></span></div><div class="t m0 xd h8 y38 ff4 fs4 fc0 sc0 lsb">\ue879</div><div class="t m0 x4 h10 y39 ff4 fs4 fc0 sc0 ls13">\uede9<span class="lsb wse v2">=<span class="_3 blank"> </span>\uf242 </span><span class="lsb wsf vb">\uead9 \ue879</span></div><div class="t m0 xe h11 y3a ff4 fs4 fc0 sc0 lsb ws10">\ue879<span class="_10 blank"> </span>\ue86f <span class="wsb vc">\uf243<span class="_e blank"> </span> <span class="_4 blank"> </span>(\ue741<span class="_0 blank"></span>)<span class="ff2"> </span></span></div><div class="t m0 xd h8 y3b ff4 fs4 fc0 sc0 lsb">\ue877</div><div class="t m0 xd h10 y3c ff4 fs4 fc0 sc0 ls14">\uede9<span class="lsb wse v2">=<span class="_3 blank"> </span>\uf242 </span><span class="lsb ws11 vb">\uead9 \ue879</span></div><div class="t m0 x11 h11 y3d ff4 fs4 fc0 sc0 lsb ws12">\ue879<span class="_10 blank"> </span>\ue86f <span class="wsb vc">\uf243<span class="_e blank"> </span> <span class="_e blank"> </span>(\ue742<span class="_b blank"> </span>)<span class="ff2"> </span></span></div><div class="t m0 x12 h12 y3e ff4 fs4 fc0 sc0 lsb ws13">y = \uf244<span class="ws1 vd">\ue754<span class="_7 blank"></span>\u0307</span></div><div class="t m0 x13 h11 y3f ff4 fs4 fc0 sc0 ls15">\ue754<span class="lsb wsb vc">\uf245<span class="_e blank"> </span> <span class="_e blank"> </span>(\ue743<span class="_0 blank"></span>)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y40 ff2 fs4 fc0 sc0 lsb wsb">A solução da equação ficará: <span class="_11 blank"> </span><span class="ff4 ws14 ve">\ue755<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue8b8.<span class="_a blank"> </span>\ue741 <span class="fs5 ws15 vb">\ueb3f\uec12\uebe7 </span><span class="wsb"> (\u210e<span class="_0 blank"></span>)<span class="ff2"> </span></span></span></div><div class="t m0 x8 hc y41 ff6 fs4 fc0 sc0 lsb wsb">\u2022 <span class="_12 blank"> </span><span class="ff5 ls2">\uf06c</span><span class="ff2"> <span class="ff7 ls2">\uf0e0</span> os 2n autovalores; </span></div><div class="t m0 x8 hc y42 ff6 fs4 fc0 sc0 lsb wsb">\u2022 <span class="_12 blank"> </span><span class="ff5 sc1 ls16">\uf059</span><span class="ff2"> <span class="ff7 ls2">\uf0e0</span> matriz modal de ordem 2n x 2n; </span></div><div class="t m0 x1 h8 y43 ff2 fs4 fc0 sc0 lsb wsb">Assim <span class="_13 blank"> </span>como <span class="_13 blank"> </span>sem <span class="_13 blank"> </span>amortecimento <span class="_5 blank"> </span>a <span class="_13 blank"> </span>matriz <span class="_13 blank"> </span>modal<span class="_0 blank"> </span><span class="ff3"> <span class="_13 blank"> </span> <span class="_13 blank"> </span><span class="ff4 ls4">\ue8b8</span></span> <span class="_13 blank"> </span>satifaz <span class="_13 blank"> </span>a <span class="_13 blank"> </span>relação <span class="_13 blank"> </span>de </div><div class="t m0 x1 h7 y44 ff2 fs4 fc1 sc0 lsb ws9">ortogonalidade<span class="fc0 wsb">, assim: <span class="_14 blank"> </span><span class="ff4 ws1 ve">\ue8d2<span class="fs5 ls17 v2">\uebdc</span>\ue879</span></span></div><div class="t m0 x14 h8 y45 ff4 fs4 fc0 sc0 ls11">\uede9<span class="lsb ws1 v2">\ue8d2</span><span class="fs5 ls18 v5">\uebdd</span><span class="lsb wsb v2">=<span class="_3 blank"> </span>\uead9 ,<span class="_a blank"> </span> \ue73f\ue74b\ue749 \ue745<span class="_f blank"> </span>\u2260<span class="_3 blank"> </span>\ue746<span class="_0 blank"></span> (\ue745<span class="_0 blank"></span>)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y46 ff2 fs4 fc0 sc0 lsb wsb"> </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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h7 y21 ff2 fs4 fc1 sc0 lsb wsb">Forma 2: <span class="_a blank"> </span>Descrever a <span class="_a blank"> </span>equação do movimento <span class="_a blank"> </span>a partir <span class="_a blank"> </span>da realização <span class="_a blank"> </span>no espaço <span class="_a blank"> </span>de </div><div class="t m0 x1 h7 y22 ff2 fs4 fc1 sc0 lsb wsb">estados. </div><div class="t m0 x1 h8 y47 ff2 fs4 fc0 sc0 lsb wsb">Assim isolando o vetor de aceleração <span class="ff4 ws16">\ue754<span class="_7 blank"></span>\u0308 <span class="ff2 wsb"> dentro da equação (a), temos: </span></span></div><div class="t m0 x15 h8 y48 ff4 fs4 fc0 sc0 lsb ws5">\ue754<span class="_7 blank"></span>\u0308<span class="_15 blank"> </span>= \u2212\ue879<span class="fs5 ws17 vb">\ueb3f\ueb35 </span><span class="ws2">\ue877\ue754<span class="_d blank"> </span>\u2212 \ue879<span class="fs5 ws17 vb">\ueb3f\ueb35 </span><span class="wsb">\ue86f\ue754<span class="_7 blank"></span>\u0307<span class="_4 blank"> </span> (\ue746)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y49 ff2 fs4 fc0 sc0 lsb wsb">Definindo o vetor de estados como: </div><div class="t m0 x12 h13 y4a ff4 fs4 fc0 sc0 lsb ws18">z = \uf244<span class="vc">\ue754</span></div><div class="t m0 x13 h12 y4b ff4 fs4 fc0 sc0 lsb ws19">\ue754<span class="_7 blank"></span>\u0307 <span class="wsb vd">\uf245<span class="_e blank"> </span> <span class="_4 blank"> </span>(\ue747<span class="_0 blank"></span>)<span class="ff2"> </span></span></div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 lsb wsb">Pode-se então chegar à realização <span class="_2 blank"></span>no espaço de estados da equação de <span class="_6 blank"></span>movimento do </div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 lsb wsb">sistema para o caso de vibrações livres: </div><div class="t m0 x16 h8 y4e ff4 fs4 fc0 sc0 lsb">\ue8a0</div><div class="t m0 x17 h8 y4f ff4 fs4 fc0 sc0 ls19">\u0307<span class="lsb wsb v0">=<span class="_3 blank"> </span>\ue86d.<span class="_a blank"> </span>\ue8a0 (\ue749)<span class="ff2"> </span></span></div><div class="t m0 x1 h9 y50 ff2 fs4 fc0 sc0 lsb wsb">Sendo <span class="_16 blank"> </span><span class="ff3">A <span class="_16 blank"> </span></span>a <span class="_16 blank"> </span>matriz <span class="_16 blank"> </span>dinâmica <span class="_16 blank"> </span>do <span class="_16 blank"> </span>sistema <span class="_16 blank"> </span>função <span class="_16 blank"> </span>das <span class="_16 blank"> </span>matrizes <span class="_16 blank"> </span>de <span class="_16 blank"> </span>massa <span class="_15 blank"> </span><span class="ff3">M, <span class="_16 blank"> </span></span>de </div><div class="t m0 x1 h9 y51 ff2 fs4 fc0 sc0 lsb wsb">amortecimento proporcional<span class="ff3"> C </span>e de rigidez<span class="ff3"> K</span>, dada por: </div><div class="t m0 x18 h14 y52 ff4 fs4 fc0 sc0 lsd ws1a">\ue86d=\uf242 <span class="lsb ws1b vd">0 \ue72b</span></div><div class="t m0 xc h11 y53 ff4 fs4 fc0 sc0 lsb ws1">\u2212\ue879<span class="fs5 ws17 vb">\ueb3f\ueb35 </span><span class="ws1c">\ue877 \u2212\ue879<span class="fs5 ws17 vb">\ueb3f\ueb35 </span><span class="ls1a">\ue86f</span><span class="wsb vc">\uf243<span class="_e blank"> </span> <span class="_e blank"> </span>(\ue74a<span class="_0 blank"></span>)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h8 y54 ff2 fs4 fc0 sc0 lsb wsb">Sendo <span class="_0 blank"> </span><span class="ff3 ws3">I</span><span class="ls1b"> a<span class="_6 blank"></span> <span class="fc1 lsb">m<span class="_0 blank"> </span>atriz <span class="_0 blank"> </span>identidade <span class="_0 blank"> </span><span class="fc0">de <span class="_b blank"> </span>ordem n <span class="_b blank"> </span>x <span class="_0 blank"> </span>n. <span class="_0 blank"> </span>As <span class="_0 blank"> </span>fre<span class="_0 blank"> </span>quências <span class="_0 blank"> </span>naturais <span class="_b blank"> </span><span class="ff4 ws1">\ue7f1<span class="fs5 wsd v2">\uebe1\uebdc </span>,</span> <span class="_0 blank"> </span>os <span class="_0 blank"> </span>modos <span class="_0 blank"> </span>de </span></span></span></div><div class="t m0 x1 h8 y55 ff2 fs4 fc0 sc0 lsb wsb">vibrar <span class="_7 blank"></span><span class="ff4">\ue7d6</span></div><div class="t m0 x19 h15 y56 ff4 fs5 fc0 sc0 ls17">\uebdc<span class="ff2 fs4 lsb wsb v3"> <span class="_7 blank"></span>e <span class="_6 blank"></span>os <span class="_6 blank"></span>fatores <span class="_6 blank"></span>de <span class="_7 blank"></span>amortecimento <span class="_6 blank"></span><span class="ff4 ws1">\ue7e6<span class="_6 blank"></span><span class="fs5 ls17 v2">\uebdc<span class="fs4 ls16 wsb v3"> <span class="ff2 lsb"> <span class="_6 blank"></span>são <span class="_6 blank"></span>extraídos <span class="_6 blank"></span>diretamente <span class="_7 blank"></span>do <span class="_6 blank"></span>conhecimento </span></span></span></span></span></div><div class="t m0 x1 h9 y57 ff2 fs4 fc0 sc0 lsb wsb">da matriz dinâmica <span class="ff3 ws3">A</span> a partir da solução do problema de autovalor e autov<span class="_2 blank"></span>etor. </div><div class="t m0 x1 h7 y58 ff2 fs4 fc0 sc0 lsb wsb">Resolvendo, temos: <span class="_17 blank"> </span><span class="ff4 ws1 v1">det<span class="vf">(</span><span class="wsb">\ue86d \u2212 \ue7e3\ue875<span class="ls1c vf">)</span>=<span class="_3 blank"> </span>\uead9 (\ue895)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y59 ff2 fs4 fc0 sc0 lsb wsb">Que conduz ao seguinte resultado: </div><div class="t m0 x1a h8 y5a ff4 fs4 fc0 sc0 lsb wsb">\ue723\ue8d2<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue7e3\ue8d2 (\ue896)<span class="ff2"> </span></div><div class="t m0 x1 h7 y5b ff2 fs4 fc0 sc0 lsb wsb">Para ilustrar, vamos ao exercício. </div><div class="t m0 x1 h6 y5c ff3 fs3 fc1 sc0 lsb wsb">Exercício </div><div class="t m0 x1 h7 y5d ff2 fs4 fc0 sc0 lsb wsb">Considere o <span class="_6 blank"></span>sistema mecânico da fi<span class="_6 blank"></span>gura com m<span class="fs6 ls1d v6">1</span> = <span class="_6 blank"></span>m<span class="_0 blank"> </span><span class="fs6 ls1e v6">2</span> = <span class="_6 blank"></span>1 kg, c<span class="fs6 ls1d v6">1</span> = <span class="_6 blank"></span>c<span class="fs6 wsa v6">2</span> = c<span class="fs6 v6">3 </span></div><div class="t m0 x1b h7 y5d ff2 fs4 fc0 sc0 lsb wsb">= 20 (N.s/<span class="_6 blank"></span>m) e k<span class="fs6 ls1d v6">1</span> </div><div class="t m0 x1 h7 y5e ff2 fs4 fc0 sc0 lsb wsb">= k<span class="fs6 ls1e v6">2</span> <span class="_a blank"> </span>= k<span class="fs6 v6">3 </span>= 1500 <span class="_a blank"> </span>(N/m). Pede-se <span class="_a blank"> </span>o cálculo <span class="_a blank"> </span>das frequências<span class="_6 blank"></span> naturais, dos <span class="_a blank"> </span>fatores d<span class="_6 blank"></span>e<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y5f ff2 fs4 fc0 sc0 lsb wsb">amortecimento modal e dos modos de vibrar do sistema. </div><div class="t m0 x1c h7 y60 ff2 fs4 fc0 sc0 lsb wsb"> </div><div class="t m0 x1 h7 y61 ff2 fs4 fc0 sc0 lsb wsb">Figura 01: Exercício: Vibrações Livres com amortecimento proporcion<span class="_2 blank"></span>al </div><div class="t m0 x1 h7 y62 ff2 fs4 fc1 sc0 lsb wsb">Resolução: </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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h7 y21 ff2 fs4 fc0 sc0 lsb wsb">Este <span class="_0 blank"> </span>e<span class="_0 blank"> </span>xercício <span class="_0 blank"> </span>é <span class="_b blank"> </span>o <span class="_b blank"> </span>mesmo <span class="_0 blank"> </span>que <span class="_b blank"> </span>resolvemos <span class="_b blank"> </span>na <span class="_0 blank"> </span>aula <span class="_b blank"> </span>20, <span class="_b blank"> </span>quando <span class="_b blank"> </span>usamos <span class="_b blank"> </span>as <span class="_0 blank"> </span>e<span class="_0 blank"> </span>quações <span class="_0 blank"> </span>de </div><div class="t m0 x1 h7 y22 ff2 fs4 fc0 sc0 lsb wsb">Lagrange <span class="_1 blank"> </span>para <span class="_4 blank"> </span>obtenção <span class="_1 blank"> </span>da <span class="_1 blank"> </span>equação <span class="_1 blank"> </span>do <span class="_1 blank"> </span>movimento <span class="_4 blank"> </span>e<span class="_0 blank"> </span> <span class="_1 blank"> </span>as <span class="_4 blank"> </span>matrizes <span class="_1 blank"> </span>correspondentes, </div><div class="t m0 x1 h7 y63 ff2 fs4 fc0 sc0 lsb wsb">portanto, vamos, portanto, aproveitar os resultados: </div><div class="t m0 x3 h16 y64 ff4 fs4 fc0 sc0 lsb ws1d">\ue879 = \ued64<span class="ws1 vc">\ue749<span class="_2 blank"></span><span class="fs5 ls1f v2">\ueb35<span class="fs4 lsb v3">0</span></span></span></div><div class="t m0 x1d h17 y65 ff4 fs4 fc0 sc0 lsb ws1e">0 \ue749<span class="fs5 ls20 v2">\ueb36</span><span class="ws13 vd">\ued68 = \uf242</span><span class="ws1f v10">1 0</span></div><div class="t m0 x1e h11 y66 ff4 fs4 fc0 sc0 lsb ws1f">0 1<span class="ws1 vc">\uf243<span class="ff2 wsb"> </span></span></div><div class="t m0 x1f h16 y67 ff4 fs4 fc0 sc0 lsd ws20">\ue877=\ued64<span class="_18 blank"></span><span class="lsb ws1 vc">\ue747<span class="_6 blank"></span><span class="fs5 ls21 v2">\ueb35<span class="fs4 lsb ws2 v3">+ \ue747</span><span class="ls22">\ueb36<span class="fs4 lsb v3">\u2212\ue747</span><span class="lsb">\ueb36</span></span></span></span></div><div class="t m0 x20 h18 y68 ff4 fs4 fc0 sc0 lsb ws1">\u2212\ue747<span class="fs5 ls23 v2">\ueb36</span>\ue747<span class="fs5 ls24 v2">\ueb36</span><span class="ws2">+ \ue747<span class="fs5 ls8 v2">\ueb37</span><span class="ws13 v11">\ued68 = \uf242<span class="_16 blank"> </span></span><span class="ws21 v10">3000 \u22121500</span></span></div><div class="t m0 x21 h19 y69 ff4 fs4 fc0 sc0 lsb ws22">\u22121500<span class="_19 blank"> </span>3000 <span class="ws1 vc">\uf243<span class="ff2 wsb"> </span></span></div><div class="t m0 x22 h1a y6a ff4 fs4 fc0 sc0 lsb ws5">\ue725<span class="_9 blank"> </span>= \uf242<span class="v8">\ue73f</span></div><div class="t m0 x23 h15 y6b ff4 fs5 fc0 sc0 ls24">\ueb35<span class="fs4 lsb ws2 v3">+ \ue73f<span class="_6 blank"></span><span class="fs5 ls25 v2">\ueb36<span class="fs4 lsb ws1 v3">\u2212\ue73f</span><span class="lsb">\ueb36</span></span></span></div><div class="t m0 x14 h1b y6c ff4 fs4 fc0 sc0 lsb ws1">\u2212\ue73f<span class="_6 blank"></span><span class="fs5 ls26 v2">\ueb36<span class="fs4 lsb v3">\ue73f</span><span class="ls24">\ueb36<span class="fs4 lsb ws2 v3">+ \ue73f<span class="_6 blank"></span><span class="fs5 ls20 v2">\ueb37<span class="fs4 lsb ws5 v12">\uf243 = \uf242<span class="_15 blank"> </span><span class="ws23 vd">40 \u221220</span></span></span></span></span></span></div><div class="t m0 x24 h11 y6d ff4 fs4 fc0 sc0 lsb ws24">\u221220<span class="_19 blank"> </span>40 <span class="ws1 vc">\uf243<span class="ff2 wsb"> </span></span></div><div class="t m0 x1 h7 y6e ff6 fs0 fc0 sc0 ls27 wsb"> <span class="ff2 fs4 lsb"> </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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg5.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">5 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h7 y21 ff2 fs4 fc0 sc0 lsb wsb">Escrevendo as matrizes auxiliares: </div><div class="t m0 x25 h8 y6f ff4 fs4 fc0 sc0 lsb">\ue879</div><div class="t m0 x26 h10 y70 ff4 fs4 fc0 sc0 ls13">\uede9<span class="lsb wse v2">=<span class="_3 blank"> </span>\uf242 </span><span class="lsb wsf vb">\uead9 \ue879</span></div><div class="t m0 x3 h1c y71 ff4 fs4 fc0 sc0 lsb ws10">\ue879<span class="_10 blank"> </span>\ue86f <span class="ws5 vc">\uf243 = \uf24e</span><span class="ws1f v13">0 0</span></div><div class="t m0 x7 h1d y72 ff4 fs4 fc0 sc0 lsb ws1f">0 0<span class="_1a blank"> </span><span class="v14">1 0</span></div><div class="t m0 x27 h8 y72 ff4 fs4 fc0 sc0 lsb ws1f">0 1</div><div class="t m0 x7 h8 y73 ff4 fs4 fc0 sc0 lsb ws1f">1 0</div><div class="t m0 x7 h1d y74 ff4 fs4 fc0 sc0 lsb ws1f">0 1<span class="_19 blank"> </span><span class="ws25 v14">40 \u221220</span></div><div class="t m0 x28 h1e y74 ff4 fs4 fc0 sc0 lsb ws24">\u221220<span class="_19 blank"> </span>40 <span class="ls4 v15">\uf24f</span><span class="ff2 wsb v15"> </span></div><div class="t m0 x29 h8 y75 ff4 fs4 fc0 sc0 lsb">\ue877</div><div class="t m0 x2a h10 y76 ff4 fs4 fc0 sc0 ls14">\uede9<span class="lsb ws5 v2">= \uf242</span><span class="lsb ws26 vb">\u2212\ue879 \uead9</span></div><div class="t m0 x2b h1f y77 ff4 fs4 fc0 sc0 lsb ws27">\uead9 \ue877<span class="ws18 vc">\uf243 = \uf24e</span><span class="ws28 v13">\u22121 0</span></div><div class="t m0 x5 h1d y78 ff4 fs4 fc0 sc0 lsb ws29">0 \u22121<span class="_1b blank"> </span><span class="wsb v14">10 <span class="_1c blank"> </span> 0</span></div><div class="t m0 x2c h8 y78 ff4 fs4 fc0 sc0 lsb wsb">0 <span class="_1d blank"> </span> 0</div><div class="t m0 xe h8 y79 ff4 fs4 fc0 sc0 lsb wsb">0 <span class="_1c blank"> </span> 0</div><div class="t m0 xe h20 y7a ff4 fs4 fc0 sc0 lsb wsb">0 <span class="_1c blank"> </span> 0<span class="_1e blank"> </span><span class="ws2a v14">3000 \u22121500</span></div><div class="t m0 x2d h21 y7b ff4 fs4 fc0 sc0 lsb ws22">\u22121500<span class="_19 blank"> </span>3000 <span class="ws1 v15">\uf24f<span class="ff2 wsb"> </span></span></div><div class="t m0 x2e h12 y7c ff4 fs4 fc0 sc0 lsb ws13">y = \uf244<span class="ws1 vd">\ue754<span class="_7 blank"></span>\u0307</span></div><div class="t m0 x2f h22 y7d ff4 fs4 fc0 sc0 ls28">\ue754<span class="lsb ws5 vc">\uf245 = \ued5e</span><span class="lsb ws1 v16">\ue754<span class="_7 blank"></span>\u0307<span class="_b blank"> </span><span class="fs5 v2">\ueb35</span></span></div><div class="t m0 x1c h8 y7e ff4 fs4 fc0 sc0 lsb ws2b">\ue754<span class="_7 blank"></span>\u0307 <span class="fs5 v2">\ueb36</span></div><div class="t m0 x1c h8 y7f ff4 fs4 fc0 sc0 lsb ws1">\ue754<span class="_6 blank"></span><span class="fs5 v2">\ueb35</span></div><div class="t m0 x1c h23 y80 ff4 fs4 fc0 sc0 lsb ws1">\ue754<span class="fs5 ls8 v2">\ueb36</span><span class="v17">\ued62<span class="ff2 wsb"> </span></span></div><div class="t m0 x1 h7 y81 ff2 fs4 fc0 sc0 lsb wsb">O problema de autovalor e autovetor é então solucionado por: </div><div class="t m0 x30 h8 y82 ff4 fs4 fc0 sc0 lsb ws1">\ue740\ue741\ue750<span class="_0 blank"></span>\ued6b\ue879</div><div class="t m0 x31 h8 y83 ff4 fs4 fc0 sc0 ls11">\uede9<span class="fs5 lsb ws17 v18">\ueb3f\ueb35 </span><span class="lsb v2">\ue877</span></div><div class="t m0 x32 h24 y83 ff4 fs4 fc0 sc0 ls29">\uede9<span class="lsb ws2c v2">\u2212<span class="_8 blank"> </span>\ue7e3\ue875\ued6f<span class="_d blank"> </span>=<span class="_3 blank"> </span>\uead9<span class="_f blank"> </span>\u2192<span class="_3 blank"> </span>\ue88a\ue88b\ue89a </span><span class="lsb ws1 v2">\uf24c</span><span class="lsb ws2d va">\ueadd\uead9 \u2212 \ue8c5<span class="_1f blank"> </span>\u2212\ueadb\uead9</span></div><div class="t m0 x33 h1d y84 ff4 fs4 fc0 sc0 lsb ws2">\u2212\ueadb\uead9<span class="_1f blank"> </span>\ueadd\uead9 \u2212 \ue8c5<span class="_19 blank"> </span><span class="ls2 ws2a v14">\ueadc\uead9\uead9\uead9 \u2212\ueada\ueade\uead9\uead9</span></div><div class="t m0 x34 h8 y84 ff4 fs4 fc0 sc0 ls2 ws2a">\u2212\ueada\ueade\uead9\uead9 \ueadc\uead9\uead9\uead9</div><div class="t m0 x11 h8 y85 ff4 fs4 fc0 sc0 lsb wsb">\u2212\ueada <span class="_14 blank"> </span> \uead9</div><div class="t m0 x2f h20 y86 ff4 fs4 fc0 sc0 lsb wsb">\uead9 <span class="_20 blank"> </span> \u2212 \ueada<span class="_21 blank"> </span><span class="v14">\u2212\ue8c5 <span class="_19 blank"> </span> \uead9 </span></div><div class="t m0 x35 h25 y87 ff4 fs4 fc0 sc0 lsb wsb">\uead9 <span class="_20 blank"> </span> \u2212 \ue8c5 <span class="_22 blank"> </span><span class="ls2a v19">\uf24d</span><span class="ws5 v19">= \uead9</span><span class="ff2 v19"> </span></div><div class="t m0 x1 h7 y88 ff2 fs4 fc0 sc0 lsb wsb">Resolvendo, temos: <span class="_23 blank"> </span><span class="ff4 ws1 v1">\ue7e3<span class="_6 blank"></span><span class="fs5 ls2b v2">\ueb35<span class="fs4 lsb ws2e v3">=<span class="_3 blank"> </span>\u2212<span class="_0 blank"></span>10 + 37,4\ue746<span class="ff2 wsb"> </span></span></span></span></div><div class="t m0 x36 h8 y89 ff4 fs4 fc0 sc0 ls4">\ue7e3<span class="fs5 ls2c v2">\ueb36</span><span class="lsb ws2e">=<span class="_3 blank"> </span>\u221210 \u2212 37,4\ue746<span class="ff2 wsb"> </span></span></div><div class="t m0 x36 h8 y8a ff4 fs4 fc0 sc0 ls4">\ue7e3<span class="fs5 ls2b v2">\ueb37</span><span class="lsb wsb">=<span class="_3 blank"> </span>\u221230 + 60\ue746 <span class="ff2"> </span></span></div><div class="t m0 x36 h8 y8b ff4 fs4 fc0 sc0 ls4">\ue7e3<span class="fs5 ls2b v2">\ueb38</span><span class="lsb wsb">=<span class="_3 blank"> </span>\u221230 \u2212 60\ue746 <span class="ff2"> </span></span></div><div class="t m0 x1 h7 y8c ff2 fs4 fc0 sc0 lsb wsb">Temos que: </div><div class="t m0 x37 h8 y8d ff4 fs4 fc0 sc0 lsb wsb"> \ue7e3<span class="fs5 ls5 v2">\uebdc</span><span class="ws5">= \u2212\ue7e6</span></div><div class="t m0 x38 h26 y8e ff4 fs5 fc0 sc0 ls8">\ueb35<span class="fs4 lsb ws1 v3">\ue7f1</span><span class="lsb ws2f">\uebe1\ueb35 <span class="fs4 ws2 v3">± \ue746\ue7f1</span><span class="wsd">\uebe1\uebdc <span class="fs4 ws1 v3">\ueda7<span class="ws8 v0">1 \u2212 \ue7e6<span class="_6 blank"></span><span class="fs5 v4">\uebdc</span></span></span></span></span></div><div class="t m0 x39 h27 y8f ff4 fs5 fc0 sc0 ls20">\ueb36<span class="ff2 fs4 lsb wsb v5"> </span></div><div class="t m0 x1 h7 y90 ff2 fs4 fc0 sc0 lsb wsb">Portanto podemos calcular os parâmetros modais: </div><div class="t m0 x1 h8 y91 ff4 fs4 fc0 sc0 lsb ws1">\ue8d3<span class="fs5 ws30 v2">\ue894\ueada </span><span class="wsb">=<span class="_3 blank"> </span>38,7 <span class="ff2">(rad/s) <span class="_24 blank"> </span></span></span><span class="v1">\ue8d3</span><span class="fs5 ws31 v1a">\ue894\ueadb </span><span class="lsd v1">=</span><span class="ff2 wsb v1">67,1 (rad/s) </span></div><div class="t m0 x1d h8 y92 ff4 fs4 fc0 sc0 lsb ws1">\ue8c8<span class="fs5 ls2d v2">\ueada</span><span class="ws5">= 0,258<span class="ff2 wsb"> </span></span></div><div class="t m0 x1d h8 y93 ff4 fs4 fc0 sc0 lsb ws1">\ue8c8<span class="fs5 ls2d v2">\ueada</span><span class="ws5">= 0,447<span class="ff2 wsb"> </span></span></div><div class="t m0 x1 h7 y94 ff2 fs4 fc0 sc0 lsb wsb">Os autovalores <span class="ff5 sc1 ls16">\uf059</span> podem ser calculados a partir: </div><div class="t m0 x3a h8 y95 ff4 fs4 fc0 sc0 lsb ws18">A\uead0 = \u03bb\uead0<span class="ff2 wsb"> </span></div><div class="t m0 x1 h7 y96 ff2 fs4 fc0 sc0 lsb wsb">Fazendo os cálculos, obtemos: <span class="_16 blank"> </span><span class="ff4 ws5 v1">\uead0 = <span class="ws1 vf">[<span class="v1b">\uead0<span class="fs5 ls20 v2">\ueb35</span><span class="wsb"> \uead0<span class="fs5 ls20 v2">\ueb36</span> \uead0<span class="fs5 ls20 v2">\ueb37</span> \uead0<span class="fs5 ls20 v2">\ueb38</span></span></span>]</span></span><span class="v1"> </span></div><div class="t m0 x1 h7 y97 ff2 fs4 fc0 sc0 lsb wsb">Resolvendo, temos: </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/a08ead07-d0f8-4d91-b3cd-201b7b8e1fb3/bg6.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsb">6 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsb"> </div></div><div class="t m0 x1 h28 y98 ff4 fs4 fc0 sc0 lsb ws1">\ue7f0<span class="_6 blank"></span><span class="fs5 ls2c v2">\ueb35<span class="fs4 lsd v3">=<span class="ls2e v0">\ued66<span class="lsb v1c">0,707</span></span></span></span></div><div class="t m0 x3b h8 y99 ff4 fs4 fc0 sc0 lsb ws1">0,707</div><div class="t m0 x3c h8 y9a ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 \u2212 0,017\ue746</div><div class="t m0 x3c h29 y9b ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 \u2212 0,017\ue746<span class="_0 blank"></span><span class="ws1 v1d">\ued6a</span><span class="ff2 wsb v1d"> <span class="ff4"> \ue7f0<span class="fs5 ls2b v2">\ueb36</span><span class="lsd">=<span class="ls2e v0">\ued66</span></span><span class="ws1 v1c">0,707</span></span></span></div><div class="t m0 x3d h8 y99 ff4 fs4 fc0 sc0 lsb ws1">0,707</div><div class="t m0 x38 h8 y9a ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 + 0,017\ue746</div><div class="t m0 x38 h2a y9b ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 + 0,017\ue746<span class="_0 blank"></span><span class="ws1 v1d">\ued6a</span><span class="ff2 wsb v1d"> </span></div><div class="t m0 x3e h28 y9c ff4 fs4 fc0 sc0 ls2">\ue7f0<span class="fs5 ls2b v2">\ueb37</span><span class="lsd">=<span class="ls2e v0">\ued66<span class="lsb ws1 v1c">0,707</span></span></span></div><div class="t m0 x18 h8 y9d ff4 fs4 fc0 sc0 lsb ws1">\u22120,707</div><div class="t m0 x3f h8 y9e ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 \u2212 0,009\ue746</div><div class="t m0 x40 h29 y9f ff4 fs4 fc0 sc0 lsb ws2d">0,0047 + 0,009\ue746<span class="_c blank"> </span><span class="ls2f v1d">\ued6a</span><span class="wsb v1d"> <span class="_e blank"> </span>\ue7f0<span class="_6 blank"></span><span class="fs5 ls2c v2">\ueb38<span class="fs4 lsd v3">=<span class="ls2e v0">\ued66<span class="lsb ws1 v1c">0,707</span></span></span></span></span></div><div class="t m0 x41 h8 y9d ff4 fs4 fc0 sc0 lsb ws1">\u22120,707</div><div class="t m0 x42 h8 y9e ff4 fs4 fc0 sc0 lsb ws2d">\u22120,0047 + 0,009\ue746</div><div class="t m0 x2c h2a y9f ff4 fs4 fc0 sc0 lsb ws2d">0,0047 \u2212 0,009\ue746<span class="_c blank"> </span><span class="ws1 v1d">\ued6a</span><span class="ff2 wsb v1d"> </span></div><div class="t m0 x1 h9 ya0 ff3 fs4 fc2 sc0 lsb wsb">Pontos Importantes: </div><div class="t m0 x8 h8 ya1 ff2 fs4 fc0 sc0 lsb ws9">a)<span class="ff6 ls30 wsb"> </span><span class="wsb">A <span class="_4 blank"> </span>matriz <span class="_4 blank"> </span>m<span class="_0 blank"> </span>odal <span class="_4 blank"> </span><span class="ff4 ws1">\uead0</span> <span class="_1 blank"> </span>é <span class="_4 blank"> </span>matriz <span class="_4 blank"> </span>complexa <span class="_1 blank"> </span>o <span class="_4 blank"> </span>que <span class="_1 blank"> </span>significa <span class="_e blank"> </span>que <span class="_1 blank"> </span>os <span class="_4 blank"> </span>modos <span class="_1 blank"> </span>de <span class="_4 blank"> </span>vibrar </span></div><div class="t m0 xf h7 ya2 ff2 fs4 fc0 sc0 lsb wsb">possuem módulo e fase. </div><div class="t m0 x8 h8 ya3 ff2 fs4 fc0 sc0 lsb ws9">b)<span class="ff6 ls31 wsb"> </span><span class="wsb">A <span class="_a blank"> </span>matriz modal <span class="_a blank"> </span><span class="ff4 ws1">\uead0</span> apresent<span class="_6 blank"></span>a razão <span class="_a blank"> </span>tanto em <span class="_a blank"> </span>amplitudes <span class="_a blank"> </span>de deslocam<span class="_2 blank"></span>entos </span></div><div class="t m0 xf h7 ya4 ff2 fs4 fc0 sc0 lsb wsb">quanto <span class="_6 blank"></span>em <span class="_6 blank"></span>amplitudes <span class="_6 blank"></span>de <span class="_6 blank"></span>velocidade, <span class="_6 blank"></span>daí <span class="_7 blank"></span>su<span class="_0 blank"> </span>a <span class="_6 blank"></span>ordem <span class="_6 blank"></span>ser <span class="_6 blank"></span>2n <span class="_6 blank"></span>x <span class="_6 blank"></span>2n <span class="_6 blank"></span>e <span class="_6 blank"></span>não <span class="_6 blank"></span>n <span class="_6 blank"></span>x n <span class="_7 blank"></span>como<span class="_0 blank"> </span> </div><div class="t m0 xf h9 ya5 ff2 fs4 fc0 sc0 lsb wsb">no caso da matriz <span class="ff3 ws3">\u0424.</span> </div><div class="t m0 x1 h7 ya6 ff2 fs4 fc0 sc0 lsb wsb">E assim, nós finalizamos nosso exercício. </div><div class="t m0 x1 h7 ya7 ff2 fs4 fc0 sc0 lsb wsb"> </div><div class="t m0 x1 h2 ya8 ff1 fs0 fc0 sc0 lsb wsb"> </div><div class="t m0 x1 h2 ya9 ff1 fs0 fc0 sc0 lsb wsb"> </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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