<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/3fe67c8e-14d9-41ec-897a-65abd2b7427c/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsa"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsa">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsa"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 lsb wsa">Vibrações Mecânicas<span class="_0 blank"> </span> \u2013 Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 lsb wsa"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 lsb wsa">TIPOS DE AMORTECIMENTO: DE </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 lsb wsa">COULOMB, HISTERÉTICO E </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 lsb ws0">PROPORCIONAL<span class="fs3 fc2 wsa"> </span></div><div class="t m0 x1 h6 ya ff3 fs3 fc1 sc0 lsb wsa"> Introdução: </div><div class="t m0 x1 h7 yb ff2 fs4 fc0 sc0 lsb wsa">Até <span class="_0 blank"> </span>o <span class="_0 blank"> </span>presente <span class="_0 blank"> </span>momento, <span class="_0 blank"> </span>nós <span class="_0 blank"> </span>apenas <span class="_1 blank"> </span>falamos <span class="_0 blank"> </span>do <span class="_0 blank"> </span>amortecimento <span class="_0 blank"> </span>viscoso <span class="_0 blank"> </span>que <span class="_0 blank"> </span>é </div><div class="t m0 x1 h7 yc ff2 fs4 fc0 sc0 lsb wsa">um dos mais simples para ser modelado matematicamente. </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 lsb wsa">No <span class="_2 blank"> </span>entanto, <span class="_2 blank"> </span>existem <span class="_2 blank"> </span>outros <span class="_2 blank"> </span>tipos <span class="_2 blank"> </span>de <span class="_2 blank"> </span>amortecimento, <span class="_2 blank"> </span>que <span class="_2 blank"> </span>estão <span class="_2 blank"> </span>baseados <span class="_2 blank"> </span>na </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 lsb wsa">dissipação <span class="_2 blank"> </span>de <span class="_3 blank"> </span>energia <span class="_3 blank"> </span>por <span class="_2 blank"> </span>atrito <span class="_3 blank"> </span>seco, <span class="_2 blank"> </span>ou <span class="_3 blank"> </span>na <span class="_3 blank"> </span>dissipação <span class="_2 blank"> </span>de <span class="_3 blank"> </span>energia <span class="_2 blank"> </span>devido <span class="_3 blank"> </span>a </div><div class="t m0 x1 h7 yf ff2 fs4 fc0 sc0 lsb wsa">distorções <span class="_4 blank"></span>internas <span class="_4 blank"></span>aos <span class="_4 blank"></span>corpos <span class="_4 blank"></span>devido <span class="_4 blank"></span>a de<span class="_4 blank"></span>formações, ou <span class="_4 blank"></span>obtidos <span class="_4 blank"></span>empiricamente </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 lsb wsa">(testes experimentais), entre outros. </div><div class="t m0 x1 h6 y11 ff3 fs3 fc1 sc0 lsb wsa">Amortecimento de Coulomb </div><div class="t m0 x1 h8 y12 ff2 fs4 fc0 sc0 lsb wsa">O <span class="_2 blank"> </span>amortecimento <span class="_2 blank"> </span>de <span class="_2 blank"> </span>coulomb <span class="_2 blank"> </span>é <span class="_2 blank"> </span>aquele <span class="_2 blank"> </span>relacionado <span class="_2 blank"> </span>ao <span class="_5 blank"> </span><span class="ff3 fc2">atrito <span class="_2 blank"> </span>seco</span>, <span class="_2 blank"> </span>também </div><div class="t m0 x1 h7 y13 ff2 fs4 fc0 sc0 lsb wsa">conhecido <span class="_0 blank"> </span>como <span class="_0 blank"> </span>amortecimento <span class="_0 blank"> </span>constante <span class="_0 blank"> </span>pois <span class="_1 blank"> </span>não <span class="_0 blank"> </span>depende <span class="_0 blank"> </span>do <span class="_1 blank"> </span>deslocamento, </div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 lsb wsa">nem da velocidade. </div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 lsb wsa">Depende apenas da força normal e do coeficiente de atrito, <span class="ff4 ws1">\uf06d</span>. </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 lsb wsa">Para <span class="_6 blank"> </span>uma <span class="_6 blank"> </span>força <span class="_6 blank"> </span>com <span class="_6 blank"> </span>excitação <span class="_6 blank"> </span>harmônica <span class="_6 blank"> </span>podemos <span class="_6 blank"> </span>aproximá<span class="_0 blank"> </span>-lo <span class="_6 blank"> </span>ao </div><div class="t m0 x1 h7 y17 ff2 fs4 fc0 sc0 lsb wsa">amortecimento <span class="_7 blank"> </span>viscoso <span class="_7 blank"> </span>por <span class="_7 blank"> </span>meio <span class="_8 blank"> </span>de <span class="_7 blank"> </span>uma <span class="_7 blank"> </span>razão <span class="_7 blank"> </span>de <span class="_8 blank"> </span>amortecimento <span class="_7 blank"> </span>equivalente, </div><div class="t m0 x1 h7 y18 ff2 fs4 fc0 sc0 ls0">\u03be<span class="fs5 lsb ws2 v1">eq</span><span class="lsb wsa">. <span class="_9 blank"> </span>Calculado <span class="_8 blank"> </span>por <span class="_9 blank"> </span>meio <span class="_9 blank"> </span>dos <span class="_9 blank"> </span>trabalhos <span class="_9 blank"> </span>realizados, <span class="_9 blank"> </span>considerando <span class="_9 blank"> </span>que <span class="_9 blank"> </span>W<span class="fs5 ws2 v1">viscoso</span> <span class="_9 blank"> </span>= </span></div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 ls1">W<span class="fs5 lsb ws2 v1">coulomb</span><span class="lsb wsa">. </span></div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 lsb wsa">Assim: <span class="_a blank"> </span><span class="ff5 ws3 v2">\ue7e6<span class="_4 blank"></span><span class="fs6 ws4 v3">\uebd8\uebe4 <span class="fs4 ls2 v4">=</span><span class="ws5 v5">\ueb36.\uec0d</span></span></span></div><div class="t m0 x3 h9 y1b ff5 fs6 fc0 sc0 lsb ws5">\uec17.\uebe5<span class="_0 blank"> </span>.\uebc6<span class="_0 blank"></span><span class="fs4 wsa v6"> , <span class="ff2">com <span class="ff5 ws6">\ue74e =<span class="_b blank"> </span></span></span><span class="fs6 v7">\uec20</span></span></div><div class="t m0 x4 h9 y1b ff5 fs6 fc0 sc0 lsb ws5">\uec20<span class="fs7 ls3 v8">\uecd9</span><span class="ff2 fs4 wsa v6"> e <span class="ff5 ws7">\ue7de<span class="_c blank"> </span>= </span></span><span class="ls4 v9">\uebbf</span><span class="fs7 va">\uecd1</span></div><div class="t m0 x5 ha y1b ff5 fs6 fc0 sc0 lsb">\uebbf</div><div class="t m0 x6 hb y1c ff5 fs7 fc0 sc0 ls5">\uecda<span class="ff2 fs4 lsb wsa vb"> </span></div><div class="t m0 x1 h7 y1d ff2 fs4 fc0 sc0 lsb wsa">Onde: </div><div class="t m0 x7 hc y1e ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> </span><span class="ff5 ws3">\ue7e6<span class="_4 blank"></span><span class="fs6 ws8 v3">\uebd8\uebe4 <span class="ff6 fs4 ls1 v4">\uf0e0<span class="ff2 lsb wsa"> Fator de amortecimento equivalen<span class="_4 blank"></span>te;<span class="_0 blank"> </span> </span></span></span></span></div><div class="t m0 x7 hc y1f ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> <span class="ff5 ls7">\ue7de</span><span class="ls0"> <span class="ff6 ls1">\uf0e0</span><span class="lsb"> Razão entre as amplitudes das forças; </span></span></span></div><div class="t m0 x7 h7 y20 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 wsa"> <span class="_d blank"> </span>r <span class="ff6 ls1">\uf0e0</span> Razão de frequências; </span></div><div class="t m0 x7 hc y21 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> </span><span class="ff5">\ue728</span></div><div class="t m0 x8 hd y22 ff5 fs6 fc0 sc0 ls8">\uebd9<span class="ff2 fs4 ls0 wsa v4"> <span class="ff6 ls1">\uf0e0<span class="ff2 lsb"> amplitude da força de atrito (Coulomb) definido por <span class="ff5">\ue728</span></span></span></span></div><div class="t m0 x9 hd y22 ff5 fs6 fc0 sc0 ls9">\uebd9<span class="fs4 lsb ws9 v4">=<span class="_e blank"> </span>\ue7e4<span class="_0 blank"></span>. \ue749<span class="_0 blank"></span>. \ue743<span class="_0 blank"></span><span class="ff2 wsa"> </span></span></div><div class="t m0 x7 hc y23 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> </span><span class="ff5">\ue728</span></div><div class="t m0 x8 ha y24 ff5 fs6 fc0 sc0 lsa">\uebe2<span class="ff6 fs4 ls1 v4">\uf0e0<span class="ff2 lsb wsa"> amplitude da força de excitação; </span></span></div><div class="t m0 x7 h7 y25 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 wsa"> <span class="_d blank"> </span>M <span class="ff6 ls1">\uf0e0</span> fator de ampliação; </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/3fe67c8e-14d9-41ec-897a-65abd2b7427c/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsa"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsa">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsa"> </div></div><div class="t m0 x7 hc y26 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> <span class="ff5 lsc">\ue7f1</span><span class="ls0"> <span class="ff6 ls1">\uf0e0</span><span class="lsb"> frequência angular de excitação; </span></span></span></div><div class="t m0 x7 hc y27 ff4 fs4 fc0 sc0 lsb ws1">\uf0b7<span class="ff2 ls6 wsa"> </span><span class="ff5 ws3">\ue7f1<span class="_4 blank"></span><span class="fs6 lsd v3">\uebe1<span class="ff6 fs4 ls1 v4">\uf0e0<span class="ff2 lsb wsa"> frequência angular natural; </span></span></span></span></div><div class="t m0 x1 h7 y28 ff2 fs4 fc0 sc0 lsb wsa">O fator de ampliação M para esse caso pode ser calculado por: </div><div class="t m0 xa he y29 ff5 fs4 fc0 sc0 lsb wsa">\ue72f<span class="_c blank"> </span>=<span class="_e blank"> </span> <span class="ls0 vc">\ueda9</span><span class="wsb vd">1 \u2212 \uf240</span><span class="ws3 ve">4\ue7de</span></div><div class="t m0 xb hf y2a ff5 fs4 fc0 sc0 lse">\ue7e8<span class="lsb ws3 v6">\uf241<span class="fs6 vf">\ueb36</span></span></div><div class="t m0 xc hc y2b ff5 fs4 fc0 sc0 lsb ws3">(<span class="wsb v1">1 \u2212 \ue74e<span class="_1 blank"> </span></span><span class="fs6 lsf v10">\ueb36</span>)<span class="fs6 ls10 v10">\ueb36</span><span class="ff2 wsa vb"> </span></div><div class="t m0 x1 hc y2c ff2 fs4 fc0 sc0 lsb wsa">Mas somente é válida se <span class="ff5 wsc">\ue7de<span class="_c blank"> </span>< <span class="fs6 v7">\uec17</span></span></div><div class="t m0 xd ha y2d ff5 fs6 fc0 sc0 ls11">\ueb38<span class="ff2 fs4 lsb wsa v6">. </span></div><div class="t m0 x1 h6 y2e ff3 fs3 fc1 sc0 lsb wsa">Amortecimento Histerético<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 lsb wsa">Esse <span class="_4 blank"></span>amortecimento <span class="_4 blank"></span>é <span class="_4 blank"></span>o<span class="_4 blank"></span>bservado <span class="_4 blank"></span>em <span class="_4 blank"></span>função <span class="_4 blank"></span>no <span class="_4 blank"></span>atrito <span class="_f blank"></span>interno <span class="_f blank"></span>dos <span class="_4 blank"></span>corpos <span class="_4 blank"></span>quando </div><div class="t m0 x1 h7 y30 ff2 fs4 fc0 sc0 lsb wsa">sujeitos <span class="_1 blank"> </span>a <span class="_1 blank"> </span>deformações <span class="_0 blank"> </span>elásticas. <span class="_1 blank"> </span>Evidências <span class="_1 blank"> </span>empíricas <span class="_1 blank"> </span>mostram <span class="_1 blank"> </span>que <span class="_1 blank"> </span>a <span class="_1 blank"> </span>energia </div><div class="t m0 x1 h7 y31 ff2 fs4 fc0 sc0 lsb wsa">dissipada <span class="_10 blank"> </span>é <span class="_10 blank"> </span>independente <span class="_10 blank"> </span>da <span class="_10 blank"> </span>frequência, <span class="_10 blank"> </span>mas <span class="_10 blank"> </span>proporcional <span class="_10 blank"> </span>ao <span class="_10 blank"> </span>quadrado <span class="_10 blank"> </span>da </div><div class="t m0 x1 h7 y32 ff2 fs4 fc0 sc0 lsb wsa">amplitude. </div><div class="t m0 x1 h7 y33 ff2 fs4 fc0 sc0 lsb wsa">A <span class="_1 blank"> </span>resposta <span class="_7 blank"> </span>livre <span class="_1 blank"> </span>de <span class="_7 blank"> </span>um <span class="_1 blank"> </span>sistema <span class="_1 blank"> </span>com <span class="_7 blank"> </span>esse <span class="_1 blank"> </span>tipo <span class="_7 blank"> </span>de <span class="_1 blank"> </span>amortecimento <span class="_7 blank"> </span>é <span class="_1 blank"> </span>similar <span class="_7 blank"> </span>a <span class="_1 blank"> </span>de </div><div class="t m0 x1 h7 y34 ff2 fs4 fc0 sc0 lsb wsa">um <span class="_1 blank"> </span>sistema <span class="_7 blank"> </span>com <span class="_1 blank"> </span>amortecimento <span class="_1 blank"> </span>viscoso. <span class="_7 blank"> </span>De <span class="_7 blank"> </span>tal <span class="_1 blank"> </span>forma <span class="_1 blank"> </span>que <span class="_7 blank"> </span>podemos <span class="_1 blank"> </span>definir <span class="_7 blank"> </span>um </div><div class="t m0 x1 h7 y35 ff2 fs4 fc1 sc0 lsb wsa">coeficiente <span class="_f blank"></span>de <span class="_4 blank"></span>amortecimento <span class="_f blank"></span>his<span class="_0 blank"> </span>terético <span class="_f blank"></span>adimensional, <span class="_4 blank"></span>h<span class="fc0">, <span class="_4 blank"></span>a <span class="_4 blank"></span>partir <span class="_f blank"></span>do <span class="_4 blank"></span>decremento </span></div><div class="t m0 x1 h10 y36 ff2 fs4 fc0 sc0 lsb wsa">logarítmico, <span class="ff4 ws1">\uf064</span>, definido por: <span class="_11 blank"> </span><span class="ff5 wsd v11">\u210e = <span class="v12">\ue7dc</span></span></div><div class="t m0 xe hc y37 ff5 fs4 fc0 sc0 ls12">\ue7e8<span class="ff2 lsb wsa vb"> </span></div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 lsb wsa">Assim temos uma razão de amortecimento viscoso equivalente: </div><div class="t m0 xf h11 y39 ff5 fs4 fc0 sc0 lsb ws3">\ue7e6<span class="_f blank"></span><span class="fs6 wse v3">\uebd8\uebe4 <span class="fs4 ls13 v4">=<span class="lsb v12">\ue7dc</span></span></span></div><div class="t m0 x10 hc y3a ff5 fs4 fc0 sc0 ls12">\ue7e8<span class="ff2 lsb wsa vb"> </span></div><div class="t m0 x1 h7 y3b ff2 fs4 fc0 sc0 lsb wsa">Que nos leva ao fator de Ampliação: </div><div class="t m0 x11 h12 y3c ff5 fs4 fc0 sc0 lsb ws7">\ue72f<span class="_c blank"> </span>= <span class="v5">\ue73a</span></div><div class="t m0 x12 hd y3d ff5 fs6 fc0 sc0 ls14">\uebe3<span class="fs4 lsb wsf v4">. \ue747</span></div><div class="t m0 x13 h13 y3e ff5 fs4 fc0 sc0 ls15">\ue728<span class="ls16 vb">=</span><span class="lsb v13">1</span></div><div class="t m0 x14 h14 y3f ff5 fs4 fc0 sc0 ls0">\ueda5<span class="lsb ws3 v14">(</span><span class="lsb wsb v0">1 \u2212 \ue74e<span class="_1 blank"> </span><span class="fs6 lsf v15">\ueb36</span><span class="ws3 v14">)</span><span class="fs6 ls17 v15">\ueb36</span><span class="ws10">+ \u210e<span class="fs6 ls11 v15">\ueb36</span><span class="ff2 wsa v16"> </span></span></span></div><div class="t m0 x1 h7 y40 ff2 fs4 fc0 sc0 lsb wsa">Caso o estudo esteja relacionado ao desbalanceamento de máquinas rotativas, </div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 lsb wsa">temos o fator de ampliação por desbalanceamento: </div><div class="t m0 x11 h12 y42 ff5 fs4 fc0 sc0 lsb ws11">\u039b = <span class="v5">\ue73a</span></div><div class="t m0 xa hd y43 ff5 fs6 fc0 sc0 ls18">\uebe3<span class="fs4 lsb ws9 v4">. \ue749</span></div><div class="t m0 x15 h13 y44 ff5 fs4 fc0 sc0 ls0">\ue749<span class="fs6 lsa v3">\uebe2</span><span class="ls19">\ue740<span class="ls16 vb">=</span><span class="lsb v13">1</span></span></div><div class="t m0 x14 h14 y45 ff5 fs4 fc0 sc0 ls0">\ueda5<span class="lsb ws3 v14">(</span><span class="lsb ws12 v0">1 \u2212 \ue74e<span class="_1 blank"> </span><span class="fs6 ls1a v15">\ueb36</span><span class="ws3 v14">)</span><span class="fs6 ls1b v15">\ueb36</span><span class="ws10">+ \u210e<span class="fs6 lsf v15">\ueb36</span><span class="ff2 wsa v16"> </span></span></span></div><div class="t m0 x1 h6 y46 ff3 fs3 fc1 sc0 lsb wsa">Amortecimento Proporcional<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y47 ff2 fs4 fc0 sc0 lsb wsa">Amortecimento <span class="_3 blank"> </span>proporcional <span class="_3 blank"> </span>é <span class="_2 blank"> </span>um <span class="_3 blank"> </span>tipo <span class="_3 blank"> </span>comum <span class="_3 blank"> </span>de <span class="_2 blank"> </span>amortecimento <span class="_9 blank"> </span>usado <span class="_2 blank"> </span>para </div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 lsb wsa">modelar <span class="_1 blank"> </span>sistemas <span class="_7 blank"> </span>na <span class="_1 blank"> </span>prática <span class="_7 blank"> </span>e <span class="_7 blank"> </span>de <span class="_1 blank"> </span>uma <span class="_7 blank"> </span>força <span class="_1 blank"> </span>empírica. <span class="_7 blank"> </span>A <span class="_1 blank"> </span>ideia <span class="_7 blank"> </span>é <span class="_7 blank"> </span>assumir <span class="_1 blank"> </span>que <span class="_7 blank"> </span>o </div><div class="t m0 x1 h7 y49 ff2 fs4 fc0 sc0 lsb wsa">amortecimento <span class="_0 blank"> </span>é <span class="_1 blank"> </span>proporcional <span class="_0 blank"> </span>aos <span class="_1 blank"> </span>parâmetros <span class="_1 blank"> </span>de <span class="_0 blank"> </span>r<span class="_0 blank"> </span>igidez <span class="_0 blank"> </span>equivalente <span class="_1 blank"> </span>da <span class="_0 blank"> </span>m<span class="_0 blank"> </span>assa </div><div class="t m0 x1 h7 y4a ff2 fs4 fc0 sc0 lsb wsa">do sistema, como: <span class="_12 blank"> </span><span class="ff5 ws9 v17">\ue889<span class="_e blank"> </span>=<span class="_e blank"> </span>\ue8bb. \ue893<span class="_5 blank"> </span>+<span class="_10 blank"> </span>\ue8bc. \ue891</span><span class="v17"> </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/3fe67c8e-14d9-41ec-897a-65abd2b7427c/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsa"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsa">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsa"> </div></div><div class="t m0 x1 h7 y26 ff2 fs4 fc0 sc0 lsb wsa">Sendo <span class="ff4 ws1">\uf061</span> e <span class="_4 blank"></span><span class="ff4 ls1">\uf062<span class="ff2 lsb"> duas constantes a s<span class="_4 blank"></span>erem determinadas, de um modo<span class="_4 blank"></span> geral, a partir </span></span></div><div class="t m0 x1 h7 y4b ff2 fs4 fc0 sc0 lsb wsa">de testes experimentais e usando técnicas de ajuste de<span class="_4 blank"></span> modelos e otimização.<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 lsb wsa">As <span class="_13 blank"> </span>características <span class="_13 blank"> </span>do <span class="_13 blank"> </span>sistema <span class="_13 blank"> </span>físico <span class="_13 blank"> </span>podem <span class="_13 blank"> </span>não <span class="_13 blank"> </span>ser <span class="_13 blank"> </span>a <span class="_13 blank"> </span>mesmas <span class="_13 blank"> </span>daquelas </div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 lsb wsa">modeladas pelo amortecimento proporcional. </div><div class="t m0 x1 h7 y4e ff2 fs4 fc0 sc0 lsb wsa">O seu uso é bastante frequente em softwares de elementos fini<span class="_4 blank"></span>tos.<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y4f ff2 fs4 fc0 sc0 lsb wsa">O fator de amortecimento para sistemas com amortecimento prop<span class="_4 blank"></span>orcional é </div><div class="t m0 x1 h7 y50 ff2 fs4 fc0 sc0 lsb wsa">escrito em função das constantes <span class="ff4 ws1">\uf061</span><span class="ls0"> e </span><span class="ff4 ws1">\uf062</span>, conforme: </div><div class="t m0 x15 h11 y51 ff5 fs4 fc0 sc0 lsb ws7">\ue8c8 = <span class="v12">\ueada</span></div><div class="t m0 x16 h13 y52 ff5 fs4 fc0 sc0 ls1c">\ueadb<span class="ls1 vb">\ued6c</span><span class="lsb ws9 vb">\ue8bb. \ue8d3</span><span class="fs6 ls1d v18">\ue894</span><span class="ls1e vb">+</span><span class="lsb v13">\ue8bc</span></div><div class="t m0 x17 h15 y52 ff5 fs4 fc0 sc0 lsb ws3">\ue8d3<span class="fs6 ls1f v3">\ue894</span><span class="ls1 vb">\ued70</span><span class="ff2 wsa vb"> </span></div><div class="t m0 x1 h7 y53 ff2 fs4 fc0 sc0 lsb wsa">A grande vantagem desse tipo de modelagem é a simpli<span class="_4 blank"></span>ficação em simulações </div><div class="t m0 x1 h7 y54 ff2 fs4 fc0 sc0 lsb wsa">que ele proporciona em sistemas com múltiplos graus de<span class="_4 blank"></span> liberdade, uma vez </div><div class="t m0 x1 h7 y55 ff2 fs4 fc0 sc0 lsb wsa">que o problema de auto-valor e auto-vetor são idênticos aos problemas com </div><div class="t m0 x1 h7 y56 ff2 fs4 fc0 sc0 lsb wsa">amortecimento viscoso. </div><div class="t m0 x1 h7 y57 ff2 fs4 fc0 sc0 lsb wsa">Para exemplificar o conteúdo apresentado, vamos a dois exemplos: </div><div class="t m0 x1 h6 y58 ff3 fs3 fc1 sc0 lsb wsa">Exercício Resolvido 01: Amortecimento de Coulomb<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y59 ff2 fs4 fc0 sc0 lsb wsa">Calcule a amplitude de vibração em regime permanente de um sistema mass<span class="_4 blank"></span>a<span class="_0 blank"> </span>-</div><div class="t m0 x1 h7 y5a ff2 fs4 fc0 sc0 lsb wsa">mola com amortecimento de coulomb, sabendo<span class="_4 blank"></span> que é a massa m=100 kg, a </div><div class="t m0 x1 h16 y5b ff2 fs4 fc0 sc0 lsb wsa">rigidez é 10<span class="fs5 v19">5 </span>N/m e <span class="ff4 ws1">\uf06d</span>=0,08 e a força de excitação F = 300.sen(40t). </div><div class="t m0 x1 h8 y5c ff2 fs4 fc1 sc0 lsb ws13">Resolução<span class="ff3 wsa">: </span></div><div class="t m0 x1 h7 y5d ff2 fs4 fc0 sc0 lsb wsa">Calculando a frequência angular natural: </div><div class="t m0 x18 h17 y5e ff5 fs4 fc0 sc0 lsb ws3">\ue7f1<span class="_f blank"></span><span class="fs6 ls20 v3">\uebe1<span class="fs4 ls21 v4">=<span class="ls22 v14">\ueda7</span></span><span class="lsb v5">\uebde</span></span></div><div class="t m0 x19 h18 y5f ff5 fs6 fc0 sc0 ls23">\uebe0<span class="fs1 fc3 ls24 wsa v6"> </span><span class="fs4 ls21 v6">=<span class="ls25 v1a">\ueda7</span></span><span class="lsb ws5 v1b">\ueb35\ueb34<span class="fs7 v4">\uec31</span></span></div><div class="t m0 x14 h19 y5f ff5 fs6 fc0 sc0 lsb ws5">\ueb35\ueb34\ueb34<span class="ff2 fs2 fc3 wsa v6"> <span class="fs4 fc0">= 31,6 (rad/s) </span></span></div><div class="t m0 x1 h7 y60 ff2 fs4 fc0 sc0 lsb wsa">Calculando a razão de frequência: </div><div class="t m0 x1a hc y61 ff5 fs4 fc0 sc0 lsb ws6">\ue74e =<span class="_b blank"> </span><span class="fs6 v7">\uec20</span></div><div class="t m0 x1b h18 y62 ff5 fs6 fc0 sc0 lsb ws5">\uec20<span class="fs7 ls26 v8">\uecd9</span><span class="fs1 fc3 ls24 wsa v6"> </span><span class="fs4 ls27 v6">=</span><span class="v1b">\ueb38\ueb34</span></div><div class="t m0 x1c h19 y62 ff5 fs6 fc0 sc0 lsb ws5">\ueb37\ueb35,\ueb3b<span class="ff2 fs2 fc3 wsa v6"> <span class="fs4 fc0">= 1,3 </span></span></div><div class="t m0 x1 h7 y63 ff2 fs4 fc0 sc0 lsb wsa">Calculando a razão de forças: </div><div class="t m0 x1d hc y64 ff5 fs4 fc0 sc0 lsb ws7">\ue7de<span class="_c blank"> </span>= <span class="fs6 ls4 vb">\uebbf</span><span class="fs7 v6">\uecd1</span></div><div class="t m0 x1e ha y65 ff5 fs6 fc0 sc0 lsb">\uebbf</div><div class="t m0 x1f h1a y66 ff5 fs7 fc0 sc0 ls28">\uecda<span class="fs2 fc3 ls29 wsa vb"> </span><span class="fs4 ls21 vb">=</span><span class="fs6 lsb ws5 v1c">\uec13.\uebe0<span class="_0 blank"></span>.\uebda</span></div><div class="t m0 x20 ha y65 ff5 fs6 fc0 sc0 lsb">\uebbf</div><div class="t m0 x1b h1b y66 ff5 fs7 fc0 sc0 ls2a">\uecda<span class="fs1 fc3 ls24 wsa vb"> </span><span class="fs4 ls21 vb">=</span><span class="fs6 lsb ws5 v1c">\ueb34,\ueb34\ueb3c.\ueb35\ueb34\ueb34.\ueb3d,\ueb3c</span></div><div class="t m0 x21 h19 y65 ff5 fs6 fc0 sc0 lsb ws14">\ueb37\ueb34\ueb34 <span class="ff2 fs2 fc3 wsa v6"> <span class="fs4 fc0">= 0,261 </span></span></div><div class="t m0 x1 h7 y67 ff2 fs4 fc0 sc0 lsb wsa">Calculando o fator de ampliação: </div><div class="t m0 x22 h1c y68 ff5 fs4 fc0 sc0 lsb ws7">\ue72f<span class="_c blank"> </span>= <span class="ws3 v15">\ueda8</span><span class="fs6 ws5 v12">\ueb35\ueb3f\uf240</span><span class="fs7 ws15 v9">\uec30\ued05</span></div><div class="t m0 x23 h1d y69 ff5 fs7 fc0 sc0 ls2b">\ued0f<span class="fs6 lsb ws5 v19">\uf241</span><span class="lsb va">\uec2e</span></div><div class="t m0 x24 h1e y6a ff5 fs6 fc0 sc0 ls2c">(<span class="lsb ws16 v1">\ueb35\ueb3f\uebe5 </span><span class="fs7 ls2d v4">\uec2e</span>)<span class="fs7 ls2d v4">\uec2e</span><span class="ff2 fs4 ls0 wsa v18"> <span class="ff5 ls21">=<span class="lsb ws3 v15">\ueda8</span></span></span><span class="lsb ws5 v1c">\ueb35\ueb3f\uf240</span><span class="fs7 lsb ws15 v1d">\uec30.\uec2c,\uec2e\uec32\uec2d</span></div><div class="t m0 x17 h1f y69 ff5 fs7 fc0 sc0 ls2e">\ued0f<span class="fs6 ls2f v19">\uf241</span><span class="lsb vd">\uec2e</span></div><div class="t m0 xe ha y6a ff5 fs6 fc0 sc0 ls2c">(<span class="lsb ws5 v1">\ueb35\ueb3f\ueb35,\ueb37</span><span class="fs7 ls30 v4">\uec2e</span>)<span class="fs7 ls31 v4">\uec2e</span><span class="ff2 fs4 lsb wsa v18"> = 1,37 </span></div><div class="t m0 x1 h7 y6b ff2 fs4 fc0 sc0 lsb wsa">Calculando a amplitude: </div><div class="t m0 x1 hc y6c ff2 fs4 fc0 sc0 lsb wsa">Temos, <span class="ff5 ws7">\ue72f<span class="_c blank"> </span>= <span class="fs6 vb">\uebd1</span></span></div><div class="t m0 x25 h20 y6d ff5 fs7 fc0 sc0 ls32">\uecdb<span class="fs6 lsb ws5 v1e">.\uebde</span></div><div class="t m0 x26 ha y6e ff5 fs6 fc0 sc0 lsb">\uebbf</div><div class="t m0 x27 hb y6f ff5 fs7 fc0 sc0 ls33">\uecda<span class="ff2 fs4 lsb wsa vb">, isolando X<span class="fs5 ls34 v1">p</span>, vem: </span></div><div class="t m0 x28 hc y70 ff5 fs4 fc0 sc0 lsb">\ue73a</div><div class="t m0 x29 hd y71 ff5 fs6 fc0 sc0 ls35">\uebe3<span class="fs4 ls21 v4">=</span><span class="lsb v5">\uebbf</span></div><div class="t m0 x2a h20 y72 ff5 fs7 fc0 sc0 ls36">\uecda<span class="fs6 lsb ws5 v1e">.\uebc6</span></div><div class="t m0 x18 h21 y73 ff5 fs6 fc0 sc0 ls37">\uebde<span class="fs2 fc3 ls38 wsa v6"> </span><span class="fs4 ls21 v6">=</span><span class="lsb ws5 v1b">\ueb37\ueb34\ueb34.\ueb35,\ueb37\ueb3b</span></div><div class="t m0 x12 h9 y74 ff5 fs6 fc0 sc0 lsb ws5">\ueb35\ueb34<span class="fs7 ls39 v4">\uec31</span><span class="ff2 fs4 ls0 wsa v6"> </span><span class="fs4 ws3 v6">=<span class="ff2 wsa"> 0,0411 (m)<span class="fs2 fc3"> </span>= 4,1 (mm) </span></span></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/3fe67c8e-14d9-41ec-897a-65abd2b7427c/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsb wsa"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsb wsa">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsb wsa"> </div></div><div class="t m0 x1 h7 y75 ff2 fs4 fc0 sc0 lsb wsa">Chegando assim, ao resultado desejado. </div><div class="t m0 x1 h6 y76 ff3 fs3 fc1 sc0 lsb wsa">Exercício Resolvido 02: Amortecimento Histerético<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y77 ff2 fs4 fc0 sc0 lsb wsa">Uma <span class="_1 blank"> </span>bomba <span class="_7 blank"> </span>com <span class="_1 blank"> </span>125 <span class="_7 blank"> </span>kg <span class="_1 blank"> </span>é <span class="_7 blank"> </span>instalada <span class="_1 blank"> </span>em <span class="_7 blank"> </span>cima <span class="_1 blank"> </span>de <span class="_1 blank"> </span>um <span class="_7 blank"> </span>suporte <span class="_1 blank"> </span>formado <span class="_7 blank"> </span>por <span class="_1 blank"> </span>uma </div><div class="t m0 x1 h16 y78 ff2 fs4 fc0 sc0 lsb wsa">viga <span class="_14 blank"> </span>engastada-livre <span class="_14 blank"> </span>de <span class="_14 blank"> </span>aço <span class="_14 blank"> </span>com <span class="_14 blank"> </span>E <span class="_15 blank"> </span>= <span class="_14 blank"> </span>210x10<span class="fs5 ls34 v19">9</span> <span class="_14 blank"> </span>(N/m<span class="_0 blank"> </span><span class="fs5 ls34 v19">2</span>), <span class="_14 blank"> </span>com <span class="_14 blank"> </span>0,8 <span class="_14 blank"> </span>(m) <span class="_14 blank"> </span>de </div><div class="t m0 x1 h16 y79 ff2 fs4 fc0 sc0 lsb wsa">comprimento e<span class="_4 blank"></span> perfil <span class="_10 blank"> </span>T como <span class="_10 blank"> </span>momento <span class="_10 blank"> </span>de inércia <span class="_10 blank"> </span>de área <span class="_10 blank"> </span>de 4,5.10<span class="fs5 ws2 v19">-6</span><span class="ws13">(m<span class="fs5 ls34 v19">4</span></span>). </div><div class="t m0 x1 h7 y7a ff2 fs4 fc0 sc0 lsb wsa">Quando <span class="_9 blank"> </span>um <span class="_9 blank"> </span>teste <span class="_3 blank"> </span>de <span class="_9 blank"> </span>vibrações <span class="_9 blank"> </span>livres <span class="_9 blank"> </span>é <span class="_9 blank"> </span>feito <span class="_9 blank"> </span>a <span class="_3 blank"> </span>razão <span class="_9 blank"> </span>de <span class="_9 blank"> </span>amplitudes <span class="_9 blank"> </span>em <span class="_3 blank"> </span>ciclos </div><div class="t m0 x1 h7 y7b ff2 fs4 fc0 sc0 lsb wsa">sucessivos é de <span class="_0 blank"> </span>2,5:1. Determine a resposta <span class="_0 blank"> </span>da máquina ao <span class="_0 blank"> </span>desbalanceamento </div><div class="t m0 x1 h7 y7c ff2 fs4 fc0 sc0 lsb wsa">0,25 <span class="_0 blank"> </span>kg.m <span class="_0 blank"> </span>quando <span class="_0 blank"> </span>a <span class="_0 blank"> </span>bomba <span class="_0 blank"> </span>opera <span class="_1 blank"> </span>a <span class="_0 blank"> </span>2.000 <span class="_0 blank"> </span>RPM <span class="_0 blank"> </span>e <span class="_0 blank"> </span>o <span class="_0 blank"> </span>amortecimento <span class="_0 blank"> </span>é <span class="_0 blank"> </span>assumido </div><div class="t m0 x1 h7 y7d ff2 fs4 fc0 sc0 lsb wsa">como sendo histerético. </div><div class="t m0 x1 h8 y7e ff2 fs4 fc1 sc0 lsb ws13">Resolução<span class="ff3 wsa">: </span></div><div class="t m0 x1 h7 y7f ff2 fs4 fc0 sc0 lsb wsa">Calculando a rigidez equivalente: </div><div class="t m0 x1 h7 y80 ff2 fs4 fc0 sc0 lsb wsa">Para uma viga, temos: </div><div class="t m0 x29 hc y81 ff5 fs4 fc0 sc0 lsb ws7">\ue747<span class="_c blank"> </span>= <span class="fs6 ws5 v7">\ueb37.\uebbe.\uebc2</span></div><div class="t m0 x2a h22 y82 ff5 fs6 fc0 sc0 ls3a">\uebc5<span class="fs7 ls3b v4">\uec2f</span><span class="fs2 fc3 ls38 wsa v6"> </span><span class="fs4 ls21 v6">=</span><span class="lsb ws5 v1b">\ueb37.\ueb36\ueb35\ueb34.\ueb35\ueb34</span><span class="fs7 ls30 v1f">\uec35</span><span class="lsb ws5 v1b">.\ueb38,\ueb39.\ueb35\ueb34</span><span class="fs7 lsb ws15 v1f">\uec37\uec32</span></div><div class="t m0 x2b h9 y82 ff5 fs6 fc0 sc0 lsb ws5">\ueb34,\ueb3c<span class="fs7 ls3c v4">\uec2f</span><span class="fs4 wsa v6"> <span class="ff3">= 5,27 .106(N/m)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y83 ff2 fs4 fc0 sc0 lsb wsa">Calculando a frequência angular natural: </div><div class="t m0 x2c h17 y84 ff5 fs4 fc0 sc0 lsb ws3">\ue7f1<span class="_f blank"></span><span class="fs6 ls20 v3">\uebe1<span class="fs4 ls21 v4">=<span class="ls22 v14">\ueda7</span></span><span class="lsb v5">\uebde</span></span></div><div class="t m0 x2d h18 y85 ff5 fs6 fc0 sc0 ls23">\uebe0<span class="fs1 fc3 ls24 wsa v6"> </span><span class="fs4 ls21 v6">=<span class="ls3d v1a">\ueda7</span></span><span class="lsb ws5 v1b">\ueb39,\ueb36\ueb3b.\ueb35\ueb34<span class="fs7 v4">\uec32</span></span></div><div class="t m0 x2e h19 y85 ff5 fs6 fc0 sc0 lsb ws17">\ueb35\ueb36\ueb39 <span class="ff2 fs2 fc3 wsa v6"> <span class="ff3 fs4 fc0">= 205,3 (rad/s)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y86 ff2 fs4 fc0 sc0 lsb wsa">Calculando a frequência de excitação: </div><div class="t m0 x2f hc y87 ff5 fs4 fc0 sc0 lsb ws18">\ue7f1<span class="_c blank"> </span>=<span class="_10 blank"> </span>2\ue7e8 <span class="fs6 ws5 v7">\uebcb\uebe2\uebe7\uebd4çã\uebe2</span></div><div class="t m0 x24 h9 y88 ff5 fs6 fc0 sc0 lsb ws19">\ueb3a\ueb34 <span class="ff2 fs4 ls0 wsa v6"> </span><span class="fs4 ws18 v6">=<span class="_e blank"> </span>2\ue7e8 </span><span class="ws5 v1b">\ueb36\ueb34\ueb34\ueb34</span></div><div class="t m0 x30 ha y88 ff5 fs6 fc0 sc0 lsb ws1a">\ueb3a\ueb34 <span class="ff2 fs4 ls0 wsa v6"> <span class="ff3 lsb">= 209,4 (rad/s)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y89 ff2 fs4 fc0 sc0 lsb wsa">Calculando a razão de frequência: </div><div class="t m0 x31 hc y8a ff5 fs4 fc0 sc0 lsb ws6">\ue74e =<span class="_b blank"> </span><span class="fs6 v7">\uec20</span></div><div class="t m0 x23 h18 y8b ff5 fs6 fc0 sc0 lsb ws5">\uec20<span class="fs7 ls26 v8">\uecd9</span><span class="fs1 fc3 ls24 wsa v6"> </span><span class="fs4 ls3e v6">=</span><span class="v1b">\ueb36\ueb34\ueb3d,\ueb38</span></div><div class="t m0 x32 h19 y8b ff5 fs6 fc0 sc0 lsb ws5">\ueb36\ueb34\ueb39,\ueb37<span class="ff2 fs2 fc3 wsa v6"> <span class="fs4 fc0">= 1,02 </span></span></div><div class="t m0 x1 h7 y8c ff2 fs4 fc0 sc0 lsb wsa">Calculando o decremento logarítmico: </div><div class="t m0 x33 hc y8d ff5 fs4 fc0 sc0 lsb ws1b">\ue7dc<span class="_c blank"> </span>=<span class="_e blank"> </span>\ue748\ue74a \uf240<span class="fs6 ls3f v7">\uebeb</span><span class="fs7 v20">\uec2d</span></div><div class="t m0 x34 h22 y8e ff5 fs6 fc0 sc0 ls3f">\uebeb<span class="fs7 ls36 v8">\uecda</span><span class="fs4 ls40 v6">\uf241<span class="fs2 fc3 ls29 wsa"> </span><span class="lsb ws1b">=<span class="_e blank"> </span>\ue748\ue74a \uf240</span></span><span class="lsb ws5 v1b">\ueb36,\ueb39</span></div><div class="t m0 x35 h9 y8e ff5 fs6 fc0 sc0 ls41">\ueb35<span class="fs4 lsb ws3 v6">\uf241<span class="ff2 fs2 fc3 wsa"> <span class="ff3 fs4 fc0">= 0,916<span class="ff2"> </span></span></span></span></div><div class="t m0 x1 h7 y8f ff2 fs4 fc0 sc0 lsb wsa">Calculando o coeficiente de amortecimento histerético: </div><div class="t m0 x2d hc y90 ff5 fs4 fc0 sc0 lsb ws1c">\u210e = <span class="fs6 v7">\uec0b</span></div><div class="t m0 x23 h23 y91 ff5 fs6 fc0 sc0 ls42">\uec17<span class="fs8 fc3 ls43 wsa v6"> </span><span class="fs4 ls21 v6">=</span><span class="lsb ws5 v1b">\ueb34,\ueb3d\ueb35\ueb3a</span></div><div class="t m0 x36 h19 y91 ff5 fs6 fc0 sc0 ls44">\uec17<span class="ff2 fs2 fc3 lsb wsa v6"> <span class="ff3 fs4 fc0">= 0,292<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y92 ff2 fs4 fc0 sc0 lsb wsa">Calculando a amplitude: </div><div class="t m0 x1 hc y93 ff2 fs4 fc0 sc0 lsb wsa">Temos que o fator de ampliação, <span class="ff5 ws7">\ue7c9<span class="_c blank"> </span>= <span class="fs6 vb">\uebd1</span></span></div><div class="t m0 xc h20 y94 ff5 fs7 fc0 sc0 ls32">\uecdb<span class="fs6 lsb ws5 v1e">.\uebe0</span></div><div class="t m0 x14 h9 y95 ff5 fs6 fc0 sc0 ls45">\uebe0<span class="fs7 ls36 v8">\uecda</span><span class="ls46">\uebd7<span class="fs4 ls47 v6">=</span><span class="lsb v1b">\ueb35</span></span></div><div class="t m0 x37 h24 y96 ff5 fs6 fc0 sc0 lsb ws5">\ueda5<span class="ls2c v0">(</span><span class="ws16 v1">\ueb35\ueb3f\uebe5 </span><span class="fs7 ls30 v4">\uec2e</span><span class="ls2c v0">)<span class="fs7 ls2d v4">\uec2e</span></span><span class="v1">\ueb3e\uebdb</span><span class="fs7 ls48 v4">\uec2e</span><span class="ff2 fs4 wsa vc">, isolando X<span class="fs5 ls34 v1">p</span>, vem: </span></div><div class="t m0 x38 hc y97 ff5 fs4 fc0 sc0 lsb">\ue73a</div><div class="t m0 x26 hd y98 ff5 fs6 fc0 sc0 ls35">\uebe3<span class="fs4 ls21 v4">=</span><span class="ls45 v5">\uebe0</span><span class="fs7 ls36 vb">\uecda</span><span class="lsb v5">\uebd7</span></div><div class="t m0 x29 h25 y99 ff5 fs6 fc0 sc0 ls49">\uebe0<span class="lsb v1b">\ueb35</span></div><div class="t m0 x2c h26 y9a ff5 fs6 fc0 sc0 lsb ws5">\ueda5<span class="ls2c v0">(</span><span class="ws16 v1">\ueb35\ueb3f\uebe5 </span><span class="fs7 ls30 v4">\uec2e</span><span class="ls2c v0">)<span class="fs7 ls2d v4">\uec2e</span></span><span class="v1">\ueb3e\uebdb</span><span class="fs7 ls48 v4">\uec2e</span><span class="ff2 fs4 ls0 wsa vc"> <span class="ff5 ls21">=</span></span><span class="v21">\ueb34,\ueb36\ueb39</span></div><div class="t m0 x39 h25 y99 ff5 fs6 fc0 sc0 lsb ws1d">\ueb35\ueb36\ueb39 <span class="v1b">\ueb35</span></div><div class="t m0 x14 h26 y9a ff5 fs6 fc0 sc0 lsb ws5">\ueda5<span class="ls2c v0">(</span><span class="wsa v1">\ueb35\ueb3f \ueb35,\ueb34\ueb36</span><span class="fs7 ls30 v4">\uec2e</span><span class="ls2c v0">)<span class="fs7 ls2d v4">\uec2e</span></span><span class="v1">\ueb3e\ueb34,\ueb36\ueb3d\ueb36</span><span class="fs7 ls2d v4">\uec2e</span><span class="ff2 fs4 wsa vc"> <span class="ff5 ws3">=<span class="_4 blank"></span><span class="ff3 wsa"> 7,06 (mm)<span class="ff2"> </span></span></span></span></div><div class="t m0 x1 h7 y9b ff2 fs4 fc0 sc0 lsb wsa">Chegando assim, ao resultado desejado. </div><div class="t m0 x1 h2 y9c ff1 fs0 fc0 sc0 lsb wsa"> </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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