<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/7d559f82-dc99-4ff1-8bd8-f911b7edbd02/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls4 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls4 ws7">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls4 ws7"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls4 ws7">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls4 ws7"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls4 ws7">SISTEMAS MECÂNICOS VIBRACIONAIS </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls4 ws7">COM MDOF \u2013 ANÁLISE MODAL </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls4 ws7">ANALÍTICA \u2013 PARTE 02: EXERCÍCIO: </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 ls4 ws7">VIBRAÇÕES LIVRES SEM </div><div class="t m0 x1 h5 yb ff3 fs2 fc0 sc0 ls4 ws0">AMORTECIMENTO<span class="fs3 fc2 ws7"> </span></div><div class="t m0 x1 h6 yc ff3 fs3 fc1 sc0 ls4 ws7"> Introdução: </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 ls4 ws7">Na aula anterior nós desenvolvemos as equações para um MDOF Livre e sem </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 ls4 ws7">amortecimento e agora vamos resolver um exercício de fixa<span class="_1 blank"></span>ção: </div><div class="t m0 x1 h6 yf ff3 fs3 fc1 sc0 ls4 ws7">Exercício </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 ls4 ws7">Para o sistema da figura, calcule as frequências naturais e os modos de v<span class="_1 blank"></span>ibrar, </div><div class="t m0 x1 h7 y11 ff2 fs4 fc0 sc0 ls4 ws7">considerando que os coeficientes de amortecimento viscosos C<span class="fs5 ls0 v1">1</span> = C<span class="fs5 ls1 v1">2</span> = C<span class="fs5 ws1 v1">3</span> = 0, k<span class="fs5 ls0 v1">1</span> = k<span class="fs5 ls1 v1">2</span> = </div><div class="t m0 x1 h7 y12 ff2 fs4 fc0 sc0 ls4 ws2">k<span class="fs5 ws1 v1">3</span><span class="ws7"> = k e m<span class="fs5 ls0 v1">1</span> = m<span class="fs5 ls1 v1">2</span> = m. </span></div><div class="t m0 x3 h7 y13 ff2 fs4 fc0 sc0 ls4 ws7"> </div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 ls4 ws7">Figura 01: Exercício: Vibrações Livres sem amortecimento </div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 ls4 ws7">Este exercício é o mesmo que resolvemos na aula 20, quand<span class="_1 blank"></span>o usamos as equações de </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 ls4 ws7">Lagrange para obtenção da equação do movimento e as m<span class="_1 blank"></span>atrizes correspondentes. </div><div class="t m0 x1 h7 y17 ff2 fs4 fc0 sc0 ls4 ws7">Vamos, portanto, aproveitar os resultados: </div><div class="t m0 x1 h8 y18 ff2 fs4 fc0 sc0 ls4 ws7">Para a matriz <span class="ff3 ws3">M</span> temos: </div><div class="t m0 x4 h9 y19 ff4 fs4 fc0 sc0 ls4 ws4">\ue879 = \ued64<span class="v2">\ue749</span></div><div class="t m0 x5 ha y1a ff4 fs6 fc0 sc0 ls2">\ueb35<span class="fs4 ls4 v3">0</span></div><div class="t m0 x6 hb y1b ff4 fs4 fc0 sc0 ls4 ws5">0 \ue749<span class="fs6 ls3 v4">\ueb36</span><span class="ws6 v5">\ued68<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h8 y1c ff2 fs4 fc0 sc0 ls4 ws7">Para a matriz <span class="ff3 ws3">K</span>, 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="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/7d559f82-dc99-4ff1-8bd8-f911b7edbd02/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls4 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls4 ws7">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls4 ws7"> </div></div><div class="t m0 x7 h9 y1d ff4 fs4 fc0 sc0 ls5 ws8">\ue877=\ued64<span class="_2 blank"></span><span class="ls4 v2">\ue747</span></div><div class="t m0 x8 ha y1e ff4 fs6 fc0 sc0 ls6">\ueb35<span class="fs4 ls4 ws9 v3">+ \ue747</span><span class="ls7">\ueb36<span class="fs4 ls4 ws6 v3">\u2212\ue747</span><span class="ls4">\ueb36</span></span></div><div class="t m0 x9 hc y1f ff4 fs4 fc0 sc0 ls4 ws6">\u2212\ue747<span class="fs6 ls8 v4">\ueb36</span>\ue747<span class="fs6 ls9 v4">\ueb36</span><span class="wsa">+ \ue747<span class="fs6 lsa v4">\ueb37</span></span><span class="v5">\ued68<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y20 ff2 fs4 fc0 sc0 ls4 ws7">Que particularizando para o presente exercício, fica: </div><div class="t m0 xa hd y21 ff4 fs4 fc0 sc0 ls4 wsb">\u2192 \ue877 = \uf242<span class="_3 blank"> </span><span class="wsc v6">2\ue747 \u2212\ue747</span></div><div class="t m0 x6 he y22 ff4 fs4 fc0 sc0 ls4 wsd">\u2212\ue747<span class="_4 blank"> </span>2\ue747 <span class="ws6 v2">\uf243<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y23 ff2 fs4 fc0 sc0 ls4 ws7">Para se calcular as frequências naturais e os modos de vibrar deste sistema deve-se </div><div class="t m0 x1 h7 y24 ff2 fs4 fc0 sc0 ls4 ws7">resolver o problema de autovalor (frequências na<span class="_1 blank"></span>turais) e autovetor (modos de vibrar) </div><div class="t m0 x1 h8 y25 ff2 fs4 fc0 sc0 ls4 ws7">associados as matrizes <span class="ff3 ws3">M</span><span class="lsb"> e </span><span class="ff3 ws3">K</span>. Assim: </div><div class="t m0 x1 h8 y26 ff2 fs4 fc0 sc0 ls4 ws7">Det (<span class="ff3">K </span>- <span class="ff5 lsb">\uf06c</span><span class="ff3 ws3">M</span>) = 0 </div><div class="t m0 x1 hf y27 ff2 fs4 fc0 sc0 ls4 ws7">Sendo <span class="ff4 wsb">\ue7e3 = \ue7f1<span class="_0 blank"> </span><span class="fs6 ls3 v7">\ueb36</span></span> e efetuando os cálculos correspondentes, temos: </div><div class="t m0 xb hd y28 ff4 fs4 fc0 sc0 ls4 ws7">\u2192<span class="_5 blank"> </span>\ue740\ue741\ue750 <span class="_6 blank"> </span>\uf240\uf242<span class="ws9 v6">2\ue747<span class="_7 blank"> </span>\u2212 \ue7e3\ue749<span class="_8 blank"> </span>\u2212\ue747</span></div><div class="t m0 xc he y29 ff4 fs4 fc0 sc0 ls4 wsa">\u2212\ue747<span class="_8 blank"> </span>2\ue747<span class="_7 blank"> </span>\u2212 \ue7e3\ue749<span class="_0 blank"> </span><span class="wse v2">\uf243\uf241 = 0<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y2a ff2 fs4 fc0 sc0 ls4 ws7">O que nos conduz a seguinte <span class="fc1">equação característica</span>: </div><div class="t m0 xd hf y2b ff4 fs4 fc0 sc0 ls4 ws6">(<span class="wsa v8">2\ue747<span class="_7 blank"> </span>\u2212 \ue7e3\ue749</span>)<span class="fs6 lsc v9">\ueb36</span><span class="wsf v8">\u2212<span class="_7 blank"> </span>\ue747 </span><span class="fs6 lsd v9">\ueb36</span><span class="wsb v8">= 0<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y2c ff2 fs4 fc0 sc0 ls4 ws7"> Expandindo este termo, chegamos a: </div><div class="t m0 xe h10 y2d ff4 fs4 fc0 sc0 ls4 ws6">\ue7e3<span class="fs6 lsc v7">\ueb36</span><span class="wsa">\u2212 4<span class="_5 blank"> </span><span class="va">\ue747</span></span></div><div class="t m0 x9 h11 y2e ff4 fs4 fc0 sc0 lse">\ue749<span class="ls4 ws9 v2">\ue7e3<span class="_7 blank"> </span>+ 3<span class="_5 blank"> </span></span><span class="lsf vb">\ue747</span><span class="fs6 ls4 vc">\ueb36</span></div><div class="t m0 xf h12 y2e ff4 fs4 fc0 sc0 ls10">\ue749<span class="fs6 ls11 vd">\ueb36</span><span class="ls4 wsb v2">= 0<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 hf y2f ff2 fs4 fc0 sc0 ls4 ws7">Resolvendo a equação encontramos os valores de <span class="ff4">\ue7e3</span></div><div class="t m0 x10 ha y30 ff4 fs6 fc0 sc0 ls4 ws10">\ueb35,\ueb36 <span class="fs4 ws6 v3">.<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 hf y31 ff2 fs4 fc0 sc0 ls4 ws7">Lembrando que <span class="ff4">\ue7e3</span></div><div class="t m0 x11 ha y32 ff4 fs6 fc0 sc0 ls4 ws11">\ueb35,\ueb36 <span class="fs4 ws7 v3">=<span class="_5 blank"> </span> \ue7f1</span><span class="ws7 v8">\uebe1 \ueb35,\ueb36</span></div><div class="t m0 x12 h13 y33 ff4 fs6 fc0 sc0 ls12">\ueb36<span class="ff2 fs4 ls4 ws7 ve">, encontramos <span class="ff4 ws6">\ue7f1</span></span><span class="ls4 ws7 vf">\uebe1 \ueb35,\ueb36<span class="_0 blank"> </span><span class="ff2 fs4 v3">: </span></span></div><div class="t m0 x1 h14 y34 ff4 fs4 fc0 sc0 ls4 ws6">\ue7f1<span class="fs6 ws11 v4">\uebe1\ueb35 </span><span class="ls5">=<span class="ls13 v10">\ueda7</span></span><span class="fs6 v11">\uebde</span></div><div class="t m0 x13 h15 y35 ff4 fs6 fc0 sc0 ls14">\uebe0<span class="ff2 fs7 fc3 ls4 ws7 v6"> </span><span class="ff6 fs4 lsb v6">\uf0e0<span class="ff2 ls4 ws7"> 1ª Frequência Natural </span></span></div><div class="t m0 x1 h14 y36 ff4 fs4 fc0 sc0 ls4 ws6">\ue7f1<span class="fs6 ws11 v4">\uebe1\ueb36 </span><span class="ls5">=</span><span class="v10">\ueda7</span><span class="fs6 ws12 v11">\ueb37\uebde</span></div><div class="t m0 x14 h15 y37 ff4 fs6 fc0 sc0 ls15">\uebe0<span class="ff2 fs7 fc3 ls4 ws7 v6"> </span><span class="ff6 fs4 lsb v6">\uf0e0<span class="ff2 ls4 ws7"> 2ª Frequência Natural </span></span></div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 ls4 ws7">Agora precisamos calcular os autovetores dos sistemas, lembrand<span class="_1 blank"></span>o que cada </div><div class="t m0 x1 h7 y39 ff2 fs4 fc0 sc0 ls4 ws7">frequência natural está associada a um modo de vibrar. </div><div class="t m0 x1 h7 y3a ff2 fs4 fc1 sc0 ls4 ws7">1º Modo de Vibrar (1ª Frequência Natural): </div><div class="t m0 x1 h14 y3b ff2 fs4 fc0 sc0 ls4 ws7">Substituindo <span class="ff4 ws6">\ue7f1<span class="fs6 ws11 v4">\uebe1\ueb35 </span><span class="ls5">=<span class="ls13 v10">\ueda7</span></span><span class="fs6 v11">\uebde</span></span></div><div class="t m0 x15 h16 y3c ff4 fs6 fc0 sc0 ls14">\uebe0<span class="ff2 fs4 ls4 ws7 v6"> em </span><span class="fs4 ls4 ws6 v6">[</span><span class="fs4 ls16 ws13 v6">\ue877\u2212\ue7f1<span class="_9 blank"></span><span class="fs6 lsa v7">\ueb36<span class="fs4 ls17 ve">\ue879<span class="ls4 ws6 v10">]<span class="ws14 v8">. \ue8b6<span class="_5 blank"> </span>=<span class="_7 blank"> </span>\uead9<span class="ff2 ws7">, temos: <span class="fc3"> </span></span></span></span></span></span></span></div><div class="t m0 x16 hd y3d ff4 fs4 fc0 sc0 ls18">\uf242<span class="ls4 ws15 v6">\ue747 \u2212\ue747</span></div><div class="t m0 x17 h17 y3e ff4 fs4 fc0 sc0 ls4 ws16">\u2212\ue747<span class="_a blank"> </span>\ue747 <span class="ws17 v2">\uf243 \ued5c</span><span class="v12">\ue7d4</span></div><div class="t m0 x18 h18 y3f ff4 fs6 fc0 sc0 ls4 ws12">\ueb35\ueb35</div><div class="t m0 x19 hf y40 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x18 h19 y41 ff4 fs6 fc0 sc0 ls4 ws18">\ueb36\ueb35 <span class="fs4 ls5 ws8 va">\ued60=0<span class="_2 blank"></span><span class="ff2 ls4 ws7"> </span></span></div><div class="t m0 x1 h7 y42 ff2 fs4 fc0 sc0 ls4 ws7">Sendo o 1º modo de vibrar definido por: </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/7d559f82-dc99-4ff1-8bd8-f911b7edbd02/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls4 ws7"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls4 ws7">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls4 ws7"> </div></div><div class="t m0 x1a hf y43 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x4 h1a y44 ff4 fs6 fc0 sc0 ls19">\ueb35<span class="fs4 ls5 v3">=<span class="ls4 ws6 v10">[<span class="v0">\ue7d4</span></span></span></div><div class="t m0 x1b ha y45 ff4 fs6 fc0 sc0 ls4 ws19">\ueb35\ueb35 <span class="fs4 v3">\ue7d4</span></div><div class="t m0 x1c h1b y45 ff4 fs6 fc0 sc0 ls4 ws1a">\ueb36\ueb35 <span class="fs4 ws6 v3">]</span><span class="ls1a v6">\uebcd</span><span class="ff2 fs4 ws7 v13"> </span></div><div class="t m0 x1 hf y46 ff2 fs4 fc0 sc0 ls4 ws7">Sendo <span class="ff4">\ue7d4</span></div><div class="t m0 x13 ha y47 ff4 fs6 fc0 sc0 ls4 ws18">\ueb35\ueb35 <span class="ff2 fs4 lsb ws7 v3"> e </span><span class="fs4 v3">\ue7d4</span></div><div class="t m0 x1d h18 y47 ff4 fs6 fc0 sc0 ls4 ws1a">\ueb36\ueb35 <span class="ff2 fs4 ws7 v3">valores das amplitudes nas coordenadas generalizadas 1 e 2 </span></div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls4 ws7">respectivamente. Resolvendo o sistema linear acima, temos: </div><div class="t m0 x1e hf y49 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x1f h18 y4a ff4 fs6 fc0 sc0 ls4 ws12">\ueb35\ueb35</div><div class="t m0 x1e hf y4b ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x1f h1c y4c ff4 fs6 fc0 sc0 ls4 ws1b">\ueb36\ueb35 <span class="fs4 wsb v14">= 1<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 ls4 ws7">Portanto os autovetores não são únicos, e podemos definir por exempl<span class="_1 blank"></span>o: </div><div class="t m0 x9 hf y4e ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x20 h1d y4f ff4 fs6 fc0 sc0 ls19">\ueb35<span class="fs4 ls4 wsb v3">= \uf244<span class="v6">1</span></span></div><div class="t m0 x21 h1e y50 ff4 fs4 fc0 sc0 lsb">1<span class="ls4 ws6 v2">\uf245<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 hf y51 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x22 ha y52 ff4 fs6 fc0 sc0 ls4 ws1a">\ueb35\ueb35 <span class="ff2 fs4 lsb ws7 v3"> e </span><span class="fs4 v3">\ue7d4</span></div><div class="t m0 x23 h18 y52 ff4 fs6 fc0 sc0 ls4 ws1a">\ueb36\ueb35 <span class="ff2 fs4 ws7 v3">tem mesmo sinal, isso significa que as massas irão oscilar em fase e como a </span></div><div class="t m0 x1 h7 y53 ff2 fs4 fc0 sc0 ls4 ws7">razão é 1, na mesma intensidade. </div><div class="t m0 x1 h7 y54 ff2 fs4 fc1 sc0 ls4 ws7">2º Modo de Vibrar (2ª Frequência Natural): </div><div class="t m0 x1 h14 y55 ff2 fs4 fc0 sc0 ls4 ws7">Substituindo <span class="ff4 ws6">\ue7f1<span class="fs6 ws11 v4">\uebe1\ueb35 </span><span class="ls5">=</span><span class="v10">\ueda7</span><span class="fs6 ws12 v11">\ueb37\uebde</span></span></div><div class="t m0 x24 h16 y56 ff4 fs6 fc0 sc0 ls15">\uebe0<span class="ff2 fs4 ls4 ws7 v6"> em </span><span class="fs4 ls4 ws6 v6">[</span><span class="fs4 ls16 ws13 v6">\ue877\u2212\ue7f1<span class="_9 blank"></span><span class="fs6 lsa v7">\ueb36<span class="fs4 ls17 ve">\ue879<span class="ls4 ws6 v10">]<span class="ws14 v8">. \ue8b6<span class="_5 blank"> </span>=<span class="_7 blank"> </span>\uead9<span class="ff2 ws7">, temos:<span class="fc3"> </span></span></span></span></span></span></span></div><div class="t m0 x16 hd y57 ff4 fs4 fc0 sc0 ls4 ws6">\uf242<span class="ws1c v6">\u2212\ue747 \u2212\ue747</span></div><div class="t m0 x17 h17 y58 ff4 fs4 fc0 sc0 ls4 ws1d">\u2212\ue747<span class="_b blank"> </span>\u2212\ue747 <span class="ws17 v2">\uf243 \ued5c</span><span class="v12">\ue7d4</span></div><div class="t m0 x18 h18 y59 ff4 fs6 fc0 sc0 ls4 ws12">\ueb36\ueb35</div><div class="t m0 x19 hf y5a ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x18 h19 y5b ff4 fs6 fc0 sc0 ls4 ws18">\ueb36\ueb36 <span class="fs4 ls5 ws8 va">\ued60=0<span class="_2 blank"></span><span class="ff2 ls4 ws7"> </span></span></div><div class="t m0 x1 h7 y5c ff2 fs4 fc0 sc0 ls4 ws7"> Sendo o 2º modo de vibrar definido por: </div><div class="t m0 x1a hf y5d ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x4 h1a y5e ff4 fs6 fc0 sc0 lsd">\ueb36<span class="fs4 ls5 v3">=<span class="ls4 ws6 v10">[<span class="v0">\ue7d4</span></span></span></div><div class="t m0 x1b ha y5f ff4 fs6 fc0 sc0 ls4 ws19">\ueb36\ueb35 <span class="fs4 v3">\ue7d4</span></div><div class="t m0 x1c h1b y5f ff4 fs6 fc0 sc0 ls4 ws1e">\ueb36\ueb36 <span class="fs4 ws6 v3">]</span><span class="ls1b v6">\uebcd</span><span class="ff2 fs4 ws7 v13"> </span></div><div class="t m0 x1 hf y60 ff2 fs4 fc0 sc0 ls4 ws7">Sendo <span class="ff4">\ue7d4</span></div><div class="t m0 x13 ha y61 ff4 fs6 fc0 sc0 ls4 ws18">\ueb36\ueb35 <span class="ff2 fs4 lsb ws7 v3"> e </span><span class="fs4 v3">\ue7d4</span></div><div class="t m0 x1d h18 y61 ff4 fs6 fc0 sc0 ls4 ws1a">\ueb36\ueb36 <span class="ff2 fs4 ws7 v3">valores das amplitudes nas coordenadas generalizadas 1 e 2 </span></div><div class="t m0 x1 h7 y62 ff2 fs4 fc0 sc0 ls4 ws7">respectivamente e resolvendo o sistema linear acima, temos: </div><div class="t m0 x8 hf y63 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x25 h18 y64 ff4 fs6 fc0 sc0 ls4 ws12">\ueb36\ueb35</div><div class="t m0 x8 hf y65 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x25 h1c y66 ff4 fs6 fc0 sc0 ls4 ws1f">\ueb36\ueb36 <span class="fs4 wsb v14">= \u22121<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 h7 y67 ff2 fs4 fc0 sc0 ls4 ws7">Novamente, os autovetores não são únicos e podemos defin<span class="_1 blank"></span>ir, por exemplo: </div><div class="t m0 x26 hf y68 ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x27 h1d y69 ff4 fs6 fc0 sc0 ls19">\ueb36<span class="fs4 ls4 wsb v3">= \uf244<span class="_c blank"> </span><span class="v6">1</span></span></div><div class="t m0 x19 h1e y6a ff4 fs4 fc0 sc0 ls4 ws6">\u22121<span class="v2">\uf245<span class="ff2 ws7"> </span></span></div><div class="t m0 x1 hf y6b ff4 fs4 fc0 sc0 ls4">\ue7d4</div><div class="t m0 x22 ha y6c ff4 fs6 fc0 sc0 ls4 ws1a">\ueb35\ueb35 <span class="ff2 fs4 lsb ws7 v3"> e </span><span class="fs4 v3">\ue7d4</span></div><div class="t m0 x23 h18 y6c ff4 fs6 fc0 sc0 ls4 ws1a">\ueb36\ueb35 <span class="ff2 fs4 ws7 v3">tem sinais opostos, isso significa que as massas irão oscilar em fase oposta, </span></div><div class="t m0 x1 h7 y6d ff2 fs4 fc0 sc0 ls4 ws7">ou seja, quando a massa m<span class="fs5 ls0 v1">1</span> for para direita, a massa m<span class="fs5 ls1 v1">2</span> vai para a esquerda, quando a </div><div class="t m0 x1 h7 y6e ff2 fs4 fc0 sc0 ls4 ws7">massa m<span class="fs5 ls0 v1">1</span> vai para esquerda, a massa m<span class="fs5 ls1 v1">2</span> vai para a direita, e como a razão é 1, será na </div><div class="t m0 x1 h7 y6f ff2 fs4 fc0 sc0 ls4 ws7">mesma intensidade. </div><div class="t m0 x1 h7 y70 ff2 fs4 fc0 sc0 ls4 ws7">E assim, nós finalizamos nosso exercício. </div><div class="t m0 x1 h7 y71 ff2 fs4 fc0 sc0 ls4 ws7"> </div><div class="t m0 x1 h2 y72 ff1 fs0 fc0 sc0 ls4 ws7"> </div><div class="t m0 x1 h2 y73 ff1 fs0 fc0 sc0 ls4 ws7"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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