<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/6077f9b4-d4f4-4e7a-b8ed-9ab4b2092d23/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls5 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls5 ws2">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls5 ws2"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls5 ws2">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls5 ws2"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls5 ws2">VIBRAÇÃO LIVRE NÃO-AMORTECIDA<span class="ff2"> </span></div><div class="t m0 x1 h6 y8 ff2 fs0 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 h7 y9 ff3 fs3 fc1 sc0 ls5 ws2">Introdução: </div><div class="t m0 x1 h8 ya ff2 fs4 fc0 sc0 ls5 ws2">Vimos <span class="_1 blank"> </span>em <span class="_1 blank"> </span>aulas <span class="_1 blank"> </span>anteriores <span class="_1 blank"> </span>que <span class="_1 blank"> </span>a <span class="_1 blank"> </span><span class="fc1">vibração <span class="_1 blank"> </span>livre</span> <span class="_1 blank"> </span>é <span class="_1 blank"> </span>aquela <span class="_1 blank"> </span>que <span class="_1 blank"> </span>acontece <span class="_1 blank"> </span>sem <span class="_1 blank"> </span>a <span class="_1 blank"> </span>atuação </div><div class="t m0 x1 h8 yb ff2 fs4 fc0 sc0 ls5 ws2">de <span class="_2 blank"> </span>forças <span class="_2 blank"> </span>externas <span class="_2 blank"> </span>ao <span class="_2 blank"> </span>sistema. <span class="_2 blank"> </span>El<span class="_3 blank"></span>a <span class="_2 blank"> </span>dá <span class="_2 blank"> </span>início <span class="_2 blank"> </span>após <span class="_2 blank"> </span>uma <span class="_2 blank"> </span>perturbação <span class="_2 blank"> </span>inici<span class="_3 blank"></span>al <span class="_2 blank"> </span>que </div><div class="t m0 x1 h8 yc ff2 fs4 fc0 sc0 ls5 ws2">desaparece posteriormente. </div><div class="t m0 x1 h8 yd ff2 fs4 fc0 sc0 ls5 ws2">Também <span class="_4 blank"> </span>vimos <span class="_0 blank"> </span>que <span class="_4 blank"> </span>a <span class="_1 blank"> </span><span class="fc1">vibração <span class="_0 blank"> </span>não-amo<span class="_0 blank"> </span>rtecida</span> <span class="_0 blank"> </span>é <span class="_4 blank"> </span>aquela <span class="_4 blank"> </span>cuja <span class="_4 blank"> </span>energia <span class="_4 blank"> </span>de <span class="_4 blank"> </span>vibração <span class="_0 blank"> </span>não </div><div class="t m0 x1 h8 ye ff2 fs4 fc0 sc0 ls5 ws2">é dissipada e o sistema permanece vibrando continuamente. </div><div class="t m0 x1 h8 yf ff2 fs4 fc0 sc0 ls5 ws2">Também <span class="_5 blank"> </span>foi <span class="_5 blank"> </span>mostrado <span class="_5 blank"> </span>que <span class="_5 blank"> </span>sistemas <span class="_5 blank"> </span>reais <span class="_5 blank"> </span>podem <span class="_5 blank"> </span>ser <span class="_5 blank"> </span>m<span class="_3 blank"></span>odelados <span class="_5 blank"> </span>como <span class="_5 blank"> </span>um <span class="_5 blank"> </span>modelo </div><div class="t m0 x1 h8 y10 ff2 fs4 fc0 sc0 ls5 ws2">mais simples com 1 GDL (grau de liberdade), como mostrado na figura abai<span class="_3 blank"></span>xo: </div><div class="t m0 x3 h8 y11 ff2 fs4 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 h9 y12 ff2 fs5 fc0 sc0 ls5 ws2"> Sistema real <span class="_3 blank"></span> Modo de <span class="_3 blank"></span>Vibrar Modelo simples 1 GDL </div><div class="t m0 x1 h7 y13 ff3 fs3 fc1 sc0 ls5 ws2">Obtenção da Equação Diferencial Ordinária (EDO)<span class="_0 blank"> </span> </div><div class="t m0 x1 h8 y14 ff2 fs4 fc0 sc0 ls5 ws2">Um <span class="_2 blank"> </span>dos <span class="_2 blank"> </span>métodos <span class="_2 blank"> </span>mais <span class="_2 blank"> </span>comuns <span class="_2 blank"> </span>para <span class="_2 blank"> </span>a <span class="_2 blank"> </span>obtenção <span class="_2 blank"> </span>da <span class="_2 blank"> </span>EDO <span class="_2 blank"> </span>(Equação <span class="_2 blank"> </span>Difer<span class="_3 blank"></span>encial </div><div class="t m0 x1 h8 y15 ff2 fs4 fc0 sc0 ls5 ws2">Ordinária <span class="_4 blank"> </span>) <span class="_4 blank"> </span>é <span class="_4 blank"> </span>por <span class="_4 blank"> </span>meio <span class="_4 blank"> </span>da <span class="_1 blank"> </span>elaboração <span class="_4 blank"> </span>do <span class="_4 blank"> </span>DCL <span class="_4 blank"> </span>(Diagrama <span class="_4 blank"> </span>de <span class="_4 blank"> </span>Corpo <span class="_4 blank"> </span>Rígido) <span class="_4 blank"> </span>e<span class="_0 blank"> </span> <span class="_4 blank"> </span>ap<span class="_0 blank"> </span>licar <span class="_4 blank"> </span>as </div><div class="t m0 x1 h8 y16 ff2 fs4 fc0 sc0 ls5 ws2">equações de Newton-Euler, que são: </div><div class="t m0 x1 ha y17 ff4 fs4 fc0 sc0 ls0">\u2211<span class="ls5 ws0 v1">\ue728<span class="_6 blank"> </span>=<span class="_7 blank"> </span>\ue749.<span class="_8 blank"> </span>\ue73d <span class="ff2 ws2"> <span class="ff5 ls1">\uf0e0</span> Relativo às forças externas </span></span></div><div class="t m0 x1 h8 y18 ff2 fs4 fc0 sc0 ls5 ws2">Onde: </div><div class="t m0 x4 hb y19 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> </span><span class="ls3">\uf0e5</span><span class="ff2 ws2">F <span class="ff5 ls1">\uf0e0</span> somatório das forças externas, no SI dado em N (Newton) </span></div><div class="t m0 x4 hb y1a ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">m <span class="ff5 ls1">\uf0e0</span> massa do sistema, no SI dado em kg (quilograma) </span></span></div><div class="t m0 x4 hc y1b ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">a <span class="ff5 ls1">\uf0e0</span> aceleração, no SI dado em m/s<span class="fs6 ls4 v2">2</span> (metro por segundo ao quadrado) </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/6077f9b4-d4f4-4e7a-b8ed-9ab4b2092d23/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls5 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls5 ws2">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls5 ws2"> </div></div><div class="t m0 x1 ha y1c ff4 fs4 fc0 sc0 ls0">\u2211<span class="ls5 v1">\ue72f</span></div><div class="t m0 x5 hd y1d ff4 fs7 fc0 sc0 ls6">\uebc0<span class="fs4 ls5 ws3 v3">=<span class="_7 blank"> </span>\ue72b .<span class="_5 blank"> </span>\ue7e0</span></div><div class="t m0 x6 ha y1e ff4 fs4 fc0 sc0 ls7">\u0308<span class="ff2 ls5 ws2 v4"> <span class="ff5 ls1">\uf0e0</span> Relativo ao Momento de Inércia de massa </span></div><div class="t m0 x1 h8 y1f ff2 fs4 fc0 sc0 ls5 ws2">Onde: </div><div class="t m0 x4 hb y20 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> </span><span class="ls3">\uf0e5</span><span class="ff2 ws4">M<span class="fs6 ls8 v5">G</span><span class="ws2"> <span class="ff5 ls1">\uf0e0</span> somatório de momentos no centro de gravid<span class="_3 blank"></span>ade G </span></span></div><div class="t m0 x4 hb y21 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">I <span class="_0 blank"> </span><span class="ff5 ls1">\uf0e0</span> <span class="_4 blank"> </span>momento <span class="_0 blank"> </span>de <span class="_0 blank"> </span>inércia <span class="_0 blank"> </span>de <span class="_4 blank"> </span>massa <span class="_0 blank"> </span>com <span class="_4 blank"> </span>relação <span class="_0 blank"> </span>ao <span class="_0 blank"> </span>eixo <span class="_0 blank"> </span>que <span class="_4 blank"> </span>passa <span class="_0 blank"> </span>no <span class="_0 blank"> </span>centro <span class="_4 blank"> </span>de </span></span></div><div class="t m0 x7 h8 y22 ff2 fs4 fc0 sc0 ls5 ws2">gravidade </div><div class="t m0 x4 ha y23 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> </span><span class="ff4">\ue7e0</span></div><div class="t m0 x8 ha y24 ff4 fs4 fc0 sc0 ls7">\u0308<span class="ff2 ls5 ws2 v4"> <span class="ff5 ls1">\uf0e0</span> aceleração angular </span></div><div class="t m0 x1 h8 y25 ff2 fs4 fc0 sc0 ls5 ws2">Suponhamos <span class="_5 blank"> </span>que a <span class="_5 blank"> </span>massa <span class="_8 blank"> </span>m <span class="_5 blank"> </span>foi colocada <span class="_5 blank"> </span>na <span class="_5 blank"> </span>posição </div><div class="t m0 x1 h8 y26 ff2 fs4 fc0 sc0 ls5 ws2">onde <span class="_9 blank"> </span>a <span class="_9 blank"> </span>figura <span class="_9 blank"> </span>está <span class="_9 blank"> </span>tracejada, <span class="_9 blank"> </span>fazendo <span class="_9 blank"> </span>o <span class="_9 blank"> </span>DCL, <span class="_9 blank"> </span>teremos </div><div class="t m0 x1 h8 y27 ff2 fs4 fc0 sc0 ls5 ws2">uma força de mola oposta ao movimento, ou seja, na direção oposta a di<span class="_3 blank"></span>reção positiva </div><div class="t m0 x1 h8 y28 ff2 fs4 fc0 sc0 ls5 ws2">de <span class="_5 blank"> </span>x. <span class="_5 blank"> </span>No sis<span class="_3 blank"></span>tema <span class="_5 blank"> </span>não <span class="_5 blank"> </span>há <span class="_5 blank"> </span>nada p<span class="_3 blank"></span>romovendo <span class="_5 blank"> </span>dissipação <span class="_5 blank"> </span>de <span class="_5 blank"> </span>energia, o<span class="_3 blank"></span>u <span class="_5 blank"> </span>sej<span class="_0 blank"> </span>a, <span class="_8 blank"> </span>não <span class="_5 blank"> </span>há </div><div class="t m0 x1 h8 y29 ff2 fs4 fc0 sc0 ls5 ws2">amortecedores. </div><div class="t m0 x1 ha y2a ff2 fs4 fc0 sc0 ls5 ws2">Aplicando a equação da lei de Newton, vem: <span class="ff4">\u2212\ue747\ue754<span class="_6 blank"> </span>=<span class="_7 blank"> </span>\ue749 \ue754<span class="_a blank"></span>\u0308<span class="_b blank"> </span><span class="ff2"> </span></span></div><div class="t m0 x1 ha y2b ff2 fs4 fc0 sc0 ls5 ws2">Colocando todos os mesmos do mesmo lado da equação: <span class="ff4">\ue893 \ue89e</span></div><div class="t m0 x9 ha y2c ff4 fs4 fc0 sc0 ls9">\u0308<span class="ls5 ws5 v1">+<span class="_c blank"> </span>\ue891\ue89e = \uead9<span class="ff3 ws2"> . <span class="ff2">Essa é a </span></span></span></div><div class="t m0 x1 h8 y2d ff2 fs4 fc0 sc0 ls5 ws2">EDO (Equação Diferencial Ordinária) do sistema. </div><div class="t m0 x1 h8 y2e ff2 fs4 fc0 sc0 ls5 ws2">Pensando <span class="_0 blank"> </span>agora <span class="_4 blank"> </span>em <span class="_0 blank"> </span>um <span class="_0 blank"> </span>sistema <span class="_4 blank"> </span>massa <span class="_4 blank"> </span>mola <span class="_0 blank"> </span>como <span class="_4 blank"> </span>movimento <span class="_0 blank"> </span>vertical, <span class="_4 blank"> </span>como </div><div class="t m0 x1 h8 y2f ff2 fs4 fc0 sc0 ls5 ws2">o indicado <span class="_0 blank"> </span>na figura <span class="_0 blank"> </span>ao <span class="_0 blank"> </span>lado. Nesse <span class="_0 blank"> </span>sistema, ao <span class="_0 blank"> </span>ser colocado <span class="_0 blank"> </span>o <span class="_0 blank"> </span>peso, a <span class="_0 blank"> </span>mola irá </div><div class="t m0 x1 h8 y30 ff2 fs4 fc0 sc0 ls5 ws2">se <span class="_4 blank"> </span>alongar <span class="_4 blank"> </span>até <span class="_4 blank"> </span>que <span class="_4 blank"> </span>a <span class="_4 blank"> </span>força <span class="_4 blank"> </span>peso <span class="_4 blank"> </span>se <span class="_4 blank"> </span>iguale <span class="_4 blank"> </span>com <span class="_4 blank"> </span>a <span class="_4 blank"> </span>força <span class="_4 blank"> </span>de <span class="_4 blank"> </span>mola, <span class="_4 blank"> </span>passando <span class="_4 blank"> </span>para </div><div class="t m0 x1 h8 y31 ff2 fs4 fc0 sc0 ls5 ws2">uma nova condição <span class="_5 blank"> </span>de equilíbrio estático. Essa defor<span class="_3 blank"></span>mação é chama<span class="_3 blank"></span>da de </div><div class="t m0 x1 hb y32 ff2 fs4 fc0 sc0 ls5 ws2">deformação <span class="_9 blank"> </span>estática <span class="_9 blank"> </span><span class="ff6 ws1">\uf064</span><span class="fs6 ws6 v5">ST</span>. <span class="_9 blank"> </span>Se <span class="_9 blank"> </span>considerarmos <span class="_9 blank"> </span>esse <span class="_9 blank"> </span>novo <span class="_9 blank"> </span>ponto <span class="_9 blank"> </span>de <span class="_9 blank"> </span>equilíbrio, <span class="_b blank"> </span>a </div><div class="t m0 x1 h8 y33 ff2 fs4 fc0 sc0 ls5 ws2">força <span class="_0 blank"> </span>peso não <span class="_0 blank"> </span>influenciará <span class="_0 blank"> </span>no mo<span class="_0 blank"> </span>vimento oscilatório. <span class="_0 blank"> </span>De <span class="_0 blank"> </span>tal <span class="_0 blank"> </span>sorte, <span class="_0 blank"> </span>que <span class="_0 blank"> </span>a ED<span class="_0 blank"> </span>O </div><div class="t m0 x1 ha y34 ff2 fs4 fc0 sc0 ls5 ws2">será idêntica ao sistema horizontal, ou seja, <span class="ff4">\ue893 \ue89e</span></div><div class="t m0 xa ha y35 ff4 fs4 fc0 sc0 ls9">\u0308<span class="ls5 ws7 v1">+ \ue891\ue89e<span class="_7 blank"> </span>=<span class="_7 blank"> </span>\uead9<span class="ff2 ws2"> </span></span></div><div class="t m0 x1 he y36 ff3 fs3 fc1 sc0 ls5 ws2">Solução da Equação Diferencial Ordinária (EDO): <span class="_0 blank"> </span><span class="ff4">\ue893 \ue89e</span></div><div class="t m0 xb he y37 ff4 fs3 fc1 sc0 lsa">\u0308<span class="ls5 ws8 v0">+ \ue891\ue89e<span class="_6 blank"> </span>=<span class="_6 blank"> </span>\uead9<span class="ff3 ws2"> </span></span></div><div class="t m0 x1 hf y38 ff2 fs4 fc0 sc0 ls5 ws2">Se dividirmos a equação acima por m temos: <span class="ff4 ws9">\ue754<span class="_a blank"></span>\u0308 + <span class="fs7 v6">\uebde</span></span></div><div class="t m0 xc h10 y39 ff4 fs7 fc0 sc0 lsb">\uebe0<span class="fs4 ls5 ws2 v7">\ue754<span class="_6 blank"> </span>=<span class="_7 blank"> </span>0 <span class="ff2">(a). </span></span></div><div class="t m0 x1 hb y3a ff2 fs4 fc0 sc0 ls5 ws2">Podemos definir frequência natural angular (<span class="ff6">\uf077</span></div><div class="t m0 xd h11 y3b ff2 fs6 fc0 sc0 ls5 ws6">n<span class="fs4 ws2 v8">) como: <span class="ff4">\ue7f1</span></span></div><div class="t m0 xe h12 y3c ff4 fs7 fc0 sc0 lsc">\uebe1<span class="fs4 ls5 ws2 v3">=<span class="_7 blank"> </span> <span class="lsd v9">\ueda7</span><span class="fs7 v6">\uebde</span></span></div><div class="t m0 xf h13 y3d ff4 fs7 fc0 sc0 lse">\uebe0<span class="ff2 fs4 ls5 ws2 v7"> </span></div><div class="t m0 x1 h8 y3e ff2 fs4 fc0 sc0 ls5 ws2">Onde: </div><div class="t m0 x4 ha y3f ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> </span><span class="ff4">\ue7f1</span></div><div class="t m0 x10 h14 y40 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="ff2 fs4 ls5 ws2 v3"> <span class="ff5 ls1">\uf0e0</span> frequência natural angular, no SI dado em rad/s (radianos por Segundo) </span></div><div class="t m0 x4 hb y41 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">K <span class="ff5 ls1">\uf0e0</span> rigidez ou coeficiente de mola, no SI d<span class="_3 blank"></span>ado em N/m (Newton por metro) </span></span></div><div class="t m0 x4 hb y42 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">m <span class="ff5 ls1">\uf0e0</span> massa do sistema, no SI dado em kg (quilograma) </span></span></div><div class="t m0 x1 hb y43 ff2 fs4 fc0 sc0 ls5 ws2">Substituindo <span class="ff6">\uf077</span></div><div class="t m0 x11 h11 y44 ff2 fs6 fc0 sc0 ls5 ws2">n <span class="fs4 ws4 v8">na</span><span class="ls3"> </span><span class="fs4 v8">equação (a), fica: <span class="ff4 wsa">\ue754<span class="_a blank"></span>\u0308<span class="_6 blank"> </span>+ \ue7f1</span></span></div><div class="t m0 x12 h14 y45 ff4 fs7 fc0 sc0 ls5">\uebe1</div><div class="t m0 x12 h15 y46 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 ws2 va">.<span class="_5 blank"> </span>\ue754<span class="_6 blank"> </span>=<span class="_7 blank"> </span>0 <span class="ff2">(b) </span></span></div><div class="t m0 x1 h16 y47 ff2 fs4 fc0 sc0 ls5 ws2">Admitindo que a resposta da EDO é do tipo: <span class="ff4 ls11">\ue754<span class="ls5 wsb v9">(</span><span class="ls12">\ue750<span class="ls13 v9">)</span><span class="ls5 wsc">=<span class="_7 blank"> </span>\ue725 .<span class="_5 blank"> </span>\ue741</span></span></span></div><div class="t m1 x13 h17 y48 ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x14 h14 y48 ff4 fs7 fc0 sc0 ls14">\uebe7<span class="ff2 fs4 ls5 ws2 va"> onde C e <span class="ff6 ls1">\uf06c</span> são constantes a </span></div><div class="t m2 x1 h8 y49 ff2 fs4 fc0 sc0 ls5 ws2">serem determinadas. </div><div class="t m2 x1 h16 y4a ff2 fs4 fc0 sc0 ls5 ws2">Derivando até 2ª ordem, vem: <span class="ff4 wsd">\ue754<span class="_a blank"></span>\u0307 <span class="wsb v9">(</span><span class="ls12">\ue750<span class="ls13 v9">)</span></span><span class="wse">=<span class="_7 blank"> </span>\ue725<span class="_4 blank"> </span>. \ue748<span class="_0 blank"> </span>. \ue741</span></span></div><div class="t m1 x15 h17 y4b ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x16 h18 y4b ff4 fs7 fc0 sc0 ls15">\uebe7<span class="ff2 fs4 ls5 ws2 va"> e <span class="ff4 wsd">\ue754<span class="_a blank"></span>\u0308 <span class="wsb v9">(</span><span class="ls12">\ue750<span class="ls13 v9">)</span></span><span class="wsc">=<span class="_7 blank"> </span>\ue725 .</span></span></span></div><div class="t m1 x17 h19 y4c ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x18 h14 y4d ff4 fs7 fc0 sc0 ls16">\ueb36<span class="fs4 ls5 vb">\ue741</span></div><div class="t m1 x19 h17 y4b ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x1a h14 y4b ff4 fs7 fc0 sc0 ls15">\uebe7<span class="ff2 fs4 ls5 ws2 va"> </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/6077f9b4-d4f4-4e7a-b8ed-9ab4b2092d23/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls5 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls5 ws2">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls5 ws2"> </div></div><div class="t m0 x1 ha y4e ff2 fs4 fc0 sc0 ls5 ws2">Substituindo na equação (b), fica: <span class="ff4 wsc">\ue725 .</span></div><div class="t m1 x1b h19 y4e ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x1c h14 y4f ff4 fs7 fc0 sc0 ls16">\ueb36<span class="fs4 ls5 vb">\ue741</span></div><div class="t m1 x1d h17 y50 ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x1e h15 y50 ff4 fs7 fc0 sc0 ls17">\uebe7<span class="fs4 ls5 wsa va">+ \ue7f1</span></div><div class="t m2 x1f h14 y51 ff4 fs7 fc0 sc0 ls5">\uebe1</div><div class="t m2 x1f h15 y50 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 wsc va">.<span class="_5 blank"> </span>\ue725 \ue741</span></div><div class="t m1 x20 h17 y50 ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x21 h15 y50 ff4 fs7 fc0 sc0 ls18">\uebe7<span class="fs4 ls5 ws5 va">= 0<span class="ff2 ws2"> </span></span></div><div class="t m2 x1 ha y52 ff2 fs4 fc0 sc0 ls5 ws2">Simplificando, vem: <span class="ff4 wsc">\ue725 .<span class="_5 blank"> </span>\ue741</span></div><div class="t m1 x22 h17 y53 ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x23 h15 y53 ff4 fs7 fc0 sc0 ls15">\uebe7<span class="fs4 ls5 va">(</span></div><div class="t m1 x24 h19 y52 ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x25 h14 y54 ff4 fs7 fc0 sc0 ls19">\ueb36<span class="fs4 ls5 wsa vb">+ \ue7f1</span></div><div class="t m2 x26 h14 y55 ff4 fs7 fc0 sc0 ls5">\uebe1</div><div class="t m2 x26 h15 y53 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 wsf va">) = 0<span class="ff2 ws2"> </span></span></div><div class="t m2 x1 h8 y56 ff2 fs4 fc0 sc0 ls5 ws2">Se <span class="_0 blank"> </span>a <span class="_0 blank"> </span>constante <span class="_0 blank"> </span>C <span class="_0 blank"> </span>for <span class="_4 blank"> </span>zero <span class="_0 blank"> </span>teríamos <span class="_0 blank"> </span>uma <span class="_0 blank"> </span>solução <span class="_0 blank"> </span>trivial <span class="_0 blank"> </span>da <span class="_0 blank"> </span>equação <span class="_0 blank"> </span>anterior, <span class="_0 blank"> </span>então <span class="_0 blank"> </span>não </div><div class="t m2 x1 ha y57 ff2 fs4 fc0 sc0 ls5 ws2">vamos <span class="_7 blank"> </span>considerá-la. <span class="_7 blank"> </span>Sabendo <span class="_7 blank"> </span>também <span class="_7 blank"> </span>que <span class="_7 blank"> </span><span class="ff4">\ue741</span></div><div class="t m1 xa h17 y58 ff6 fs8 fc0 sc0 ls5">\uf06c</div><div class="t m2 x27 h14 y58 ff4 fs7 fc0 sc0 ls15">\uebe7<span class="ff2 fs4 ls5 ws2 va"> <span class="_7 blank"> </span>nunca <span class="_7 blank"> </span>é <span class="_7 blank"> </span>zero, <span class="_7 blank"> </span>então, <span class="_7 blank"> </span>para <span class="_7 blank"> </span>que <span class="_7 blank"> </span>a </span></div><div class="t m2 x1 ha y59 ff2 fs4 fc0 sc0 ls5 ws2">expressão <span class="_5 blank"> </span>acima <span class="_9 blank"> </span>seja <span class="_5 blank"> </span>verdadeira, <span class="_5 blank"> </span>temos <span class="_5 blank"> </span><span class="ff4">(</span></div><div class="t m1 x28 h19 y59 ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x15 h14 y5a ff4 fs7 fc0 sc0 ls1a">\ueb36<span class="fs4 ls5 wsa vb">+ \ue7f1</span></div><div class="t m2 x29 h14 y5b ff4 fs7 fc0 sc0 ls5">\uebe1</div><div class="t m2 x29 h15 y5c ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 ws5 va">) = 0<span class="ff2 ws2">. <span class="_5 blank"> </span> <span class="_5 blank"> </span> <span class="_5 blank"> </span> <span class="_5 blank"> </span>Isolando <span class="_5 blank"> </span><span class="ff6 ls1">\uf06c</span>, <span class="_9 blank"> </span>temo<span class="_0 blank"> </span>s <span class="_9 blank"> </span> </span></span></div><div class="t m1 x2a h19 y59 ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x2b h14 y5a ff4 fs7 fc0 sc0 ls1b">\ueb36<span class="fs4 ls5 vb">=</span></div><div class="t m2 x1 ha y5d ff4 fs4 fc0 sc0 ls5 ws2">\u2212 \ue7f1</div><div class="t m2 x5 h14 y5e ff4 fs7 fc0 sc0 lsf">\uebe1<span class="ff2 fs4 ls5 ws2 v3"> e finalmente chegamos a: </span></div><div class="t m1 x2c h19 y5d ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x2d ha y5d ff4 fs4 fc0 sc0 ls1c">=<span class="ls3 v0">\ueda5</span><span class="ls5 wsb">\u2212\ue7f1</span></div><div class="t m2 x2e h14 y5e ff4 fs7 fc0 sc0 ls5">\uebe1</div><div class="t m2 x2e h14 y5f ff4 fs7 fc0 sc0 ls5">\ueb36</div><div class="t m0 x28 h8 y5d ff2 fs4 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 h8 y60 ff2 fs4 fc0 sc0 ls5 ws2">Portanto, temos uma solução no conjunto dos números imagin<span class="_3 blank"></span>ários </div><div class="t m1 x2f h19 y60 ff6 fs9 fc0 sc0 ls5">\uf06c</div><div class="t m2 x30 hd y61 ff4 fs7 fc0 sc0 ls5 ws10">\ueb35,\ueb36 <span class="fs4 ws11 v3">=<span class="_7 blank"> </span>±\ue745 .<span class="_5 blank"> </span>\ue7f1</span></div><div class="t m2 x31 h14 y61 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="ff2 fs4 ls5 ws2 v3"> </span></div><div class="t m2 x1 h16 y62 ff2 fs4 fc0 sc0 ls5 ws2">Então a solução da EDO será: <span class="ff4 ls11">\ue754<span class="ls5 wsb v9">(</span><span class="ls12">\ue750<span class="ls13 v9">)</span><span class="ls5 ws5">= \ue725</span></span></span></div><div class="t m2 x32 hd y63 ff4 fs7 fc0 sc0 ls1d">\ueb35<span class="fs4 ls1e v3">\ue741</span><span class="ls5 ws12 v6">\uebdc.\uec20</span><span class="fsa ls1f vc">\uecd9</span><span class="fs4 ls5 wsa v3">+ \ue725</span></div><div class="t m2 x27 hd y63 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls1e v3">\ue741</span><span class="ls5 ws12 v6">\ueb3f\uebdc.\uec20</span><span class="fsa ls20 vc">\uecd9</span><span class="ff2 fs4 ls5 ws2 v3"> </span></div><div class="t m2 x1 h16 y64 ff2 fs4 fc0 sc0 ls5 ws2">Lembrando a relação de Euler <span class="ff8"> <span class="ff4 ls1e">\ue741<span class="fs7 ls5 ws13 vd">±\uebdc\uec0f </span><span class="ls5 ws5">= cos<span class="wsb v9">(</span><span class="ls21">\ue7e0<span class="ls22 v9">)</span></span><span class="ws14">±<span class="_c blank"> </span>\ue745 .<span class="_5 blank"> </span>\ue74f\ue741\ue74a (\ue7e0 )</span></span></span></span> temos: </div><div class="t m2 x33 h16 y65 ff4 fs4 fc0 sc0 ls3">x<span class="ls5 wsb v9">(</span>t<span class="ls13 v9">)</span><span class="ls5 ws5">= \ue725</span></div><div class="t m2 x34 h1a y66 ff4 fs7 fc0 sc0 ls10">\ueb35<span class="fs4 ls5 wsb v3">[cos<span class="v9">(</span>\ue7f1</span></div><div class="t m2 x35 h1a y66 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls12 v3">\ue750<span class="ls22 v9">)</span><span class="ls5 ws11">+<span class="_c blank"> </span>\ue745 .<span class="_5 blank"> </span>\ue74f\ue741\ue74a (\ue7f1</span></span></div><div class="t m2 x28 hd y66 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 wsa v3">\ue750)]<span class="_d blank"> </span>+ \ue725</span></div><div class="t m2 x36 h1a y66 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 wsb v3">[cos<span class="v9">(</span>\ue7f1</span></div><div class="t m2 x37 h1a y66 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls12 v3">\ue750<span class="ls22 v9">)</span><span class="ls5 ws15">\u2212<span class="_c blank"> </span>\ue745 .<span class="_5 blank"> </span>\ue74f\ue741\ue74a (\ue7f1</span></span></div><div class="t m2 x38 hd y66 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 ws16 v3">\ue750 )]<span class="ff2 ws2"> </span></span></div><div class="t m2 x1 h16 y67 ff2 fs4 fc0 sc0 ls5 ws2">Simplificando chegamos a: <span class="ff4 ls3">x<span class="ls5 wsb v9">(</span>t<span class="ls13 v9">)</span><span class="ls5 ws5">= (\ue725</span></span></div><div class="t m2 x39 hd y2a ff4 fs7 fc0 sc0 ls23">\ueb35<span class="fs4 ls5 wsa v3">+ \ue725</span></div><div class="t m2 x3a h1a y2a ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 wse v3">). cos<span class="wsb v9">(</span>\ue7f1</span></div><div class="t m2 x36 h1a y2a ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls12 v3">\ue750<span class="ls24 v9">)</span><span class="ls5 ws7">+ (\ue725</span></span></div><div class="t m2 x3b hd y2a ff4 fs7 fc0 sc0 ls23">\ueb35<span class="fs4 ls5 wsa v3">\u2212 \ue725</span></div><div class="t m2 x3c hd y2a ff4 fs7 fc0 sc0 ls10">\ueb36<span class="fs4 ls5 wse v3">). \ue74f\ue741\ue74a(\ue7f1</span></div><div class="t m2 x3d hd y2a ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 ws16 v3">\ue750 )]<span class="ff2 ws2"> </span></span></div><div class="t m2 x1 h8 y68 ff2 fs4 fc0 sc0 ls5 ws2">Chamando C<span class="fs6 ls25 v5">1</span> + C<span class="fs6 v5">2 </span>= B e C<span class="fs6 ls25 v5">1</span> \u2013 C<span class="fs6 ls4 v5">2</span> = A, podemos escrever: </div><div class="t m2 x23 h16 y69 ff4 fs4 fc0 sc0 ls5 wsb">\ue89e<span class="v9">(</span><span class="ls1">\ue89a<span class="ls13 v9">)</span></span><span class="wse">=<span class="_7 blank"> </span>\ue6ef. \ue899\ue88b\ue894</span><span class="v9">(</span>\ue8d3<span class="fs7 ls26 v4">\ue894</span><span class="ls1">\ue89a<span class="ls22 v9">)</span></span><span class="wse">+<span class="_c blank"> </span>\ue86e. \ue86f\ue895\ue899(\ue8d3<span class="fs7 ls26 v4">\ue894</span></span>\ue89a)]<span class="ff3 ws2"> </span></div><div class="t m2 x1 h8 y6a ff2 fs4 fc0 sc0 ls5 ws2">A solução final dessa equação irá depender das condições iniciai<span class="_3 blank"></span>s do problema. </div><div class="t m2 x1 h16 y6b ff2 fs4 fc0 sc0 ls5 ws2">Fazendo <span class="ff4 ls11">\ue754<span class="ls5 wsb v9">(</span><span class="ls1">0<span class="ls13 v9">)</span><span class="ls5">=<span class="_9 blank"> </span> <span class="_9 blank"> </span>\ue754<span class="fs7 ls27 v4">\uebe2</span></span><span class="ff2"> e </span><span class="ls5 wsd">\ue754<span class="_a blank"></span>\u0307 <span class="wsb v9">(</span><span class="ls1">0<span class="ls13 v9">)</span></span><span class="ws5">= \ue752</span></span></span></span></div><div class="t m2 x3e h14 y6c ff4 fs7 fc0 sc0 ls27">\uebe2<span class="ff2 fs4 ls5 ws2 v3"> , temos: </span></div><div class="t m2 x1 h16 y6d ff4 fs4 fc0 sc0 ls11">\ue754<span class="ls5 wsb v9">(</span><span class="ls1">0<span class="ls13 v9">)</span><span class="ls5 ws17">=<span class="_7 blank"> </span>\ue723. \ue74f\ue741\ue74a<span class="_0 blank"> </span><span class="wsb v9">(</span>\ue7f1</span></span></div><div class="t m2 x3f h1a y6e ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls1 v3">0<span class="ls22 v9">)</span><span class="ls5 ws18">+<span class="_c blank"> </span>\ue724 .<span class="_5 blank"> </span>\ue725\ue74b\ue74f (\ue7f1</span></span></div><div class="t m2 x40 hd y6e ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 wsb v3">0)]<span class="ff2 ws2"> = </span>\ue754</span><span class="ls27">\uebe2<span class="ff2 fs4 ls5 ws2 v3"> <span class="ff5 ls1">\uf0e0</span> <span class="ff3">B = <span class="ff4 wsb">\ue89e</span></span></span><span class="ls28">\ue895<span class="ff2 fs4 ls5 ws2 v3"> </span></span></span></div><div class="t m2 x1 h16 y6f ff4 fs4 fc0 sc0 ls5 wsd">\ue754<span class="_a blank"></span>\u0307 <span class="wsb v9">(</span><span class="ls1">0<span class="ls13 v9">)</span></span><span class="ws17">=<span class="_7 blank"> </span>\ue723. \ue7f1</span></div><div class="t m2 x33 h1a y70 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 wse v3">. \ue73f\ue74b\ue74f<span class="wsb v9">(</span>\ue7f1</span></div><div class="t m2 x34 h1a y70 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls1 v3">0<span class="ls22 v9">)</span><span class="ls5 ws19">\u2212<span class="_c blank"> </span>\ue724 .<span class="_5 blank"> </span>\ue7f1</span></span></div><div class="t m2 x3e hd y70 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 ws17 v3">. \ue74f\ue741\ue74a(\ue7f1</span></div><div class="t m2 x41 hd y70 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 ws5 v3">0)] = \ue752</span></div><div class="t m2 x42 hd y70 ff4 fs7 fc0 sc0 ls27">\uebe2<span class="ff2 fs4 ls5 ws2 v3"> <span class="ff5 ls1">\uf0e0</span> v<span class="fs6 ls29 v5">o</span>= A. <span class="ff4">\ue7f1</span></span></div><div class="t m2 x17 h12 y70 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="ff2 fs4 ls5 ws2 v3"> <span class="ff5 ls1">\uf0e0</span> <span class="ff4 wsb">\ue86d</span><span class="ff3"> = <span class="_b blank"> </span></span></span><span class="ls5 ws1a ve">\ue89c<span class="fsa vf">\ue895</span></span></div><div class="t m2 x43 h14 y71 ff4 fs7 fc0 sc0 ls3">\ue8d3<span class="fsa ls5 vf">\ue894</span></div><div class="t m0 x44 h1b y6f ff3 fs4 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 hf y72 ff2 fs4 fc0 sc0 ls5 ws2">E portanto, obtendo a resposta da EDO<span class="ff3">: <span class="ff4 ls2a">\ue89e<span class="ls5 wsb v9">(</span><span class="ls1">\ue89a<span class="ls13 v9">)</span><span class="ls2b">=<span class="fs7 ls5 ws1a v6">\ue89c<span class="fsa vf">\ue895</span></span></span></span></span></span></div><div class="t m0 x29 h1c y73 ff4 fs7 fc0 sc0 ls3">\ue8d3<span class="fsa ls2c vf">\ue894</span><span class="fs4 ls5 wse v7">. \ue899\ue88b\ue894<span class="wsb v9">(<span class="v1">\ue8d3</span></span></span><span class="ls26 v10">\ue894</span><span class="fs4 ls1 v7">\ue89a<span class="ls22 v9">)</span><span class="ls5 wse">+<span class="_c blank"> </span>\ue89e\ue895. \ue86f\ue895\ue899(\ue8d3</span></span><span class="ls26 v10">\ue894</span><span class="fs4 ls5 wsb v7">\ue89a)]<span class="ff3 ws2"> </span></span></div><div class="t m0 x1 h8 y74 ff2 fs4 fc0 sc0 ls5 ws2">Também podemos escrever a equação anterior na forma de uma amplit<span class="_3 blank"></span>ude e um </div><div class="t m0 x1 h16 y75 ff2 fs4 fc0 sc0 ls5 ws2">ângulo de fase, ficando: <span class="ff4 ls11">\ue754<span class="ls5 wsb v9">(</span><span class="ls12">\ue750<span class="ls13 v9">)</span><span class="ls5 wse">=<span class="_7 blank"> </span>\ue723. \ue74f\ue741\ue74a<span class="_0 blank"> </span><span class="wsb v9">(</span>\ue7f1</span></span></span></div><div class="t m0 x45 h1a y76 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 ws2 v3">\ue750<span class="_d blank"> </span>+ \u2205<span class="wsb v9">)</span><span class="ff2"> ou <span class="ff4 ls2d">\ue754<span class="ls5 wsb v9">(</span><span class="ls12">\ue750<span class="ls2e v9">)</span><span class="ls5 ws17">=<span class="_7 blank"> </span>\ue723. \ue73f\ue74b\ue74f<span class="_0 blank"> </span><span class="wsb v9">(</span>\ue7f1</span></span></span></span></span></div><div class="t m0 x46 h1a y76 ff4 fs7 fc0 sc0 lsf">\uebe1<span class="fs4 ls5 wsa v3">\ue750<span class="_d blank"> </span>\u2212 <span class="ff8 ws2"> <span class="ff4 ls3">\u2205<span class="ls5 wsb v9">)</span></span><span class="ff2"> onde: </span></span></span></div><div class="t m0 x4 hf y77 ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> <span class="ff2 ls5">A = <span class="ff4 wsb v8">\ueda7</span><span class="ff4 ls2f">\uf240<span class="fs7 ls5 v6">\uebe9</span></span></span></span></div><div class="t m0 x47 h1d y78 ff4 fsa fc0 sc0 ls5">\uecda</div><div class="t m0 x48 h1e y79 ff4 fs7 fc0 sc0 ls5 ws1a">\uec20<span class="fsa ls30 vf">\uecd9</span><span class="fs4 wsb v7">\uf241</span><span class="ls1a v11">\ueb36</span><span class="fs4 wsa v7">+ \ue754</span><span class="v10">\uebe2</span></div><div class="t m0 x49 h14 y7a ff4 fs7 fc0 sc0 ls10">\ueb36<span class="ff2 fs4 ls5 ws2 v12"> <span class="ff5 ls1">\uf0e0</span>A é a amplitude de deslocamento, no S<span class="_3 blank"></span>I dado em m </span></div><div class="t m0 x7 h8 y7b ff2 fs4 fc0 sc0 ls5 ws2">(metro); </div><div class="t m0 x4 hf y7c ff6 fs4 fc0 sc0 ls5 ws1">\uf0b7<span class="ff7 ls2 ws2"> </span><span class="ff4 ws1b">\u2205<span class="_7 blank"> </span>=<span class="_7 blank"> </span>\ue73d\ue74e\ue73f\ue750\ue743 \uf240<span class="fs7 ws1a v6">\uec20</span><span class="fsa ls30 vc">\uecd9</span><span class="fs7 ls31 v6">\uebeb</span><span class="fsa vc">\uecda</span></span></div><div class="t m0 x4a h14 y7d ff4 fs7 fc0 sc0 ls5">\uebe9</div><div class="t m0 x4b h1f y7e ff4 fsa fc0 sc0 ls32">\uecda<span class="fs4 ls5 wsb v13">\uf241<span class="ff2 ws2"> <span class="ff5 ls1">\uf0e0</span> \u03a6 é o ângulo de Fase, no SI dado em rad (radian<span class="_3 blank"></span>os) </span></span></div><div class="t m0 x1 h8 y7f ff2 fs4 fc0 sc0 ls5 ws2">Representando graficamente temos: </div><div class="t m0 x4c h8 y80 ff2 fs4 fc0 sc0 ls5 ws2"> <span class="_e blank"> </span> <span class="_f blank"> </span> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="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/6077f9b4-d4f4-4e7a-b8ed-9ab4b2092d23/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls5 ws2"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls5 ws2">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls5 ws2"> </div></div><div class="t m0 x1 h8 y81 ff2 fs4 fc0 sc0 ls5 ws2">No <span class="_4 blank"> </span>gráfico <span class="_0 blank"> </span>da <span class="_4 blank"> </span>esquerda <span class="_4 blank"> </span>temos <span class="_4 blank"> </span>a <span class="_4 blank"> </span>projeção <span class="_4 blank"> </span>do <span class="_4 blank"> </span>deslocamento <span class="_0 blank"> </span>x <span class="_4 blank"> </span>em <span class="_4 blank"> </span>relação<span class="_4 blank"> </span> <span class="_4 blank"> </span>a <span class="_0 blank"> </span>fre<span class="_0 blank"> </span>quência </div><div class="t m0 x1 h8 y82 ff2 fs4 fc0 sc0 ls5 ws2">natural <span class="_4 blank"> </span>angular, <span class="_4 blank"> </span><span class="ff1 ls1">\u03d5</span> <span class="_4 blank"> </span>representa <span class="_4 blank"> </span>uma <span class="_4 blank"> </span>deflexão <span class="_4 blank"> </span>entre <span class="_4 blank"> </span>o <span class="_4 blank"> </span>início <span class="_4 blank"> </span>do <span class="_4 blank"> </span>movimento <span class="_4 blank"> </span>oscilatório <span class="_4 blank"> </span>e </div><div class="t m0 x1 h8 y83 ff2 fs4 fc0 sc0 ls5 ws2">o <span class="_b blank"> </span>pico <span class="_b blank"> </span>mais <span class="_1 blank"> </span>próximo. <span class="_b blank"> </span>Já <span class="_b blank"> </span>no <span class="_b blank"> </span>gráfico <span class="_b blank"> </span>da <span class="_b blank"> </span>direita <span class="_b blank"> </span>temos <span class="_b blank"> </span>a <span class="_b blank"> </span>projeção <span class="_b blank"> </span>do <span class="_b blank"> </span>deslocamento <span class="_1 blank"> </span>x <span class="_b blank"> </span>a </div><div class="t m0 x1 h8 y84 ff2 fs4 fc0 sc0 ls5 ws2">cada instante. </div><div class="t m0 x1 h7 y85 ff3 fs3 fc1 sc0 ls5 ws2">Exemplo: Vibração Livre Não-Amortecida em um<span class="_0 blank"> </span> sistema com 1 GDL<span class="_0 blank"> </span> </div><div class="t m0 x1 h8 y86 ff2 fs4 fc0 sc0 ls5 ws2">Seja o sistema mecânico abaixo, calcular a frequência natural amortecida, a respost<span class="_3 blank"></span>a </div><div class="t m0 x1 h8 y87 ff2 fs4 fc0 sc0 ls5 ws2">de vibração e a amplitude máxima. </div><div class="t m0 x1 h8 y88 ff2 fs4 fc0 sc0 ls5 ws2">Dados: m = 12 (kg) | k = 1.200 (N/m) | x<span class="fs6 ls29 v5">o</span> = 0,02 (m) | v<span class="fs6 ls29 v5">o</span> = 0 </div><div class="t m0 x1 ha y89 ff2 fs4 fc0 sc0 ls5 ws2">Frequência Natural: <span class="ff4">\ue7f1</span></div><div class="t m0 x4d h12 y8a ff4 fs7 fc0 sc0 ls33">\uebe1<span class="fs4 ls5 ws2 v3">=<span class="_7 blank"> </span> <span class="lsd v9">\ueda7</span><span class="fs7 v6">\uebde</span></span></div><div class="t m0 x25 h20 y8b ff4 fs7 fc0 sc0 ls34">\uebe0<span class="ff2 fs4 ls5 ws2 v7"> <span class="ff4 ls1c">=<span class="ff8 ls5"> <span class="ff4 wsb v0">\ueda7</span></span></span></span><span class="ls5 ws1a v14">\ueb35\ueb36\ueb34\ueb34</span></div><div class="t m0 x4e h10 y8b ff4 fs7 fc0 sc0 ls5 ws1c">\ueb35\ueb36 <span class="fs4 ls2a v7">=<span class="ff8 ls5 ws2"> 10 rad/s <span class="ff4 ws1d">\u2192 \ue88c</span></span></span><span class="ls35 v10">\ue894</span><span class="fs4 wsb v7">=<span class="ff3 ws2"> 1,59 (Hz)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h8 y8c ff2 fs4 fc0 sc0 ls5 ws2">Eq. de Movimento (sem força externa): </div><div class="t m0 x10 hf y8d ff4 fs4 fc0 sc0 ls5 wsb">\ue89e<span class="v9">(</span><span class="ls1">\ue89a<span class="ls2e v9">)</span><span class="ls2b">=</span></span><span class="fs7 ws1a v6">\ue89c</span><span class="fsa vc">\ue895</span></div><div class="t m0 x3f h21 y8e ff4 fs7 fc0 sc0 ls3">\ue8d3<span class="fsa ls2c vf">\ue894</span><span class="fs4 ls5 wse v7">. \ue899\ue88b\ue894<span class="wsb v9">(<span class="v1">\ue8d3</span></span></span><span class="ls26 v10">\ue894</span><span class="fs4 ls1 v7">\ue89a<span class="ls22 v9">)</span><span class="ls5 wse">+<span class="_c blank"> </span>\ue89e\ue895. \ue86f\ue895\ue899(\ue8d3</span></span><span class="ls26 v10">\ue894</span><span class="fs4 ls5 wsb v7">\ue89a)]<span class="ff3 fsb fc2 ws2"> </span><span class="ff5 sc1 ls36">\uf0e0</span><span class="ff3 ws2"> <span class="ff4 ws1e">\ue720(\ue71c)<span class="_7 blank"> </span>=<span class="_7 blank"> </span>\uead9, \uead9\ueadb. \ue86f\ue895\ue899(\ueada\uead9\ue89a)]</span> </span></span></div><div class="t m0 x1 hf y8f ff2 fs4 fc0 sc0 ls5 ws2">Amplitude Máxima: A = <span class="ff4 wsb v8">\ueda7</span><span class="ff4 ls2f">\uf240<span class="fs7 ls5 v6">\uebe9</span></span></div><div class="t m0 x4f h1d y90 ff4 fsa fc0 sc0 ls5">\uecda</div><div class="t m0 x50 h1e y91 ff4 fs7 fc0 sc0 ls5 ws1a">\uec20<span class="fsa ls30 vf">\uecd9</span><span class="fs4 wsb v7">\uf241</span><span class="ls1a v11">\ueb36</span><span class="fs4 wsa v7">+ \ue754</span><span class="v10">\uebe2</span></div><div class="t m0 x51 h22 y92 ff4 fs7 fc0 sc0 ls10">\ueb36<span class="ff2 fsb fc2 ls5 ws2 v12"> <span class="fs4 fc0 ls1">= </span></span><span class="fs4 ls5 wsb vf">\ueda7<span class="ls37 vf">\uf240</span></span><span class="ls5 v10">\ueb34</span></div><div class="t m0 x28 h1e y91 ff4 fs7 fc0 sc0 ls5 ws1a">\ueb35\ueb34<span class="fs4 wsb v7">\uf241</span><span class="ls19 v11">\ueb36</span><span class="fs4 wsa v7">+ 0,02</span><span class="ls38 ve">\ueb36</span><span class="ff3 fsb fc2 ws2 v7"> <span class="fs4 fc0">= 0,02 (m) </span></span></div><div class="t m0 x1 h8 y93 ff2 fs4 fc0 sc0 ls5 ws2">Traçando o gráfico desse sistema fica: </div><div class="t m0 x18 h8 y94 ff2 fs4 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 h8 y95 ff2 fs4 fc0 sc0 ls5 ws2"> </div><div class="t m0 x1 h2 y96 ff1 fs0 fc0 sc0 ls5 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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