<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/98de2b23-5c6b-4aaf-92dd-b05cf194e2ae/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsf wsd">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsf wsd"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 lsf wsd">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 lsf wsd"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 lsf wsd">VIBRAÇÃO LIVRE AMORTECIDA \u2013 </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 lsf wsd">SUPERAMORTECIMENTO E </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 lsf wsd">AMORTECIMENTO CRÍTICO \u2013 </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 lsf wsd">DECREMENTO LOGARÍTMICO<span class="ff2"> </span></div><div class="t m0 x1 h6 yb ff3 fs3 fc1 sc0 lsf wsd">Introdução: </div><div class="t m0 x1 h7 yc ff2 fs4 fc0 sc0 lsf wsd">Vimos em aulas anteriores que um sistema com massa-mola-</div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 lsf wsd">amortecedor com vibração livre, representado na figura ao </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 lsf wsd">lado tem sua EDO (Equação Diferencial Ordinária) é do tipo: </div><div class="t m0 x3 h8 yf ff4 fs4 fc0 sc0 lsf wsd">\ue749 \ue754<span class="_1 blank"></span>\u0308<span class="_2 blank"> </span>+ \ue73f<span class="_0 blank"> </span>\ue754<span class="_1 blank"></span>\u0307<span class="_2 blank"> </span>+ \ue747\ue754<span class="_2 blank"> </span>=<span class="_3 blank"> </span>0<span class="ff2"> </span></div><div class="t m0 x1 h9 y10 ff2 fs4 fc0 sc0 lsf wsd">A solução da equação acima é <span class="ff4 ls0">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls2 v1">)</span><span class="lsf">=<span class="_3 blank"> </span> \ue741<span class="_4 blank"> </span><span class="fs5 ws1 v2">\ueb3f\uec15 \uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls4 v2">\uebe7</span><span class="ws2">. \uf242\ue726<span class="ls5 v4">1</span><span class="ls6">\ue741</span><span class="fs5 ws3 v2">\uebdc .\uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ws4 v2">.<span class="v0">\ueda5<span class="ws1 v0">\ueb35\ueb3f\uec15<span class="fs6 ls7 v3">\uec2e</span><span class="wsd"> \uebe7<span class="_3 blank"> </span></span></span></span></span><span class="ws5">+ \ue726<span class="ls5 v4">2</span><span class="ws6">\ue741\ue7f1 <span class="fs5 ws7 v2">\ueb3f\uebdc .\uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ws4 v2">\ueda5<span class="ws1 v0">\ueb35\ueb3f\uec15 <span class="fs6 ls7 v3">\uec2e</span><span class="wsd"> \uebe7<span class="_4 blank"> </span></span></span></span><span class="ws0">\uf243</span></span></span></span></span></span></span>, </div><div class="t m0 x1 h7 y11 ff2 fs4 fc0 sc0 lsf wsd">onde: </div><div class="t m0 x1 ha y12 ff1 fs4 fc0 sc0 lsf ws8">\u03be<span class="ff2 wsd"> <span class="ff5 ls5">\uf0e0</span> Fator de amortecimento </span></div><div class="t m0 x1 ha y13 ff6 fs4 fc0 sc0 lsf ws9">\uf077<span class="ff2 fs7 ls8 v5">n</span><span class="ff2 wsd"> <span class="ff5 ls5">\uf0e0</span> frequência angular natural </span></div><div class="t m0 x1 h7 y14 ff2 fs4 fc0 sc0 lsf wsd">Vimos também que quando, 0 < <span class="ff1 ws8">\u03be</span> < 1, temos uma vibração subamortecida, quando <span class="ff1 ws8">\u03be</span> = </div><div class="t m0 x1 h7 y15 ff2 fs4 fc0 sc0 lsf wsd">1 temos o amortecimento crítico e, por fim, <span class="ff1 ws8">\u03be</span> > 1, temos o amortecimento </div><div class="t m0 x1 h7 y16 ff2 fs4 fc0 sc0 lsf wsd">supercrítico. </div><div class="t m0 x1 hb y17 ff2 fs4 fc0 sc0 lsf wsd">O fator de amortecimento é definido por <span class="ff4 wsa">\ue7e6 =<span class="_5 blank"> </span><span class="fs5 v6">\uebd6</span></span></div><div class="t m0 x4 hc y18 ff4 fs5 fc0 sc0 lsf ws4">\uebd6<span class="fs6 ls9 v7">\uecce</span><span class="ff2 fs4 wsd v8"> , onde: </span></div><div class="t m0 x1 ha y19 ff2 fs4 fc0 sc0 ls5 wsd">c <span class="ff5">\uf0e0</span><span class="lsf"> coeficiente de amortecimento viscoso </span></div><div class="t m0 x1 ha y1a ff2 fs4 fc0 sc0 lsa">c<span class="fs7 lsf wsb v5">c</span><span class="lsf wsd"> <span class="ff5 ls5">\uf0e0</span> coeficiente de amortecimento críti<span class="_6 blank"></span>co </span></div><div class="t m0 x1 h7 y1b ff2 fs4 fc0 sc0 lsf wsd">O coeficiente de amortecimento críti<span class="_6 blank"></span>co c<span class="fs7 wsb v5">c</span> pode definido como c<span class="_0 blank"> </span><span class="fs7 wsb v5">c</span> = 2.m.<span class="ff6 ws9">\uf077</span><span class="fs7 wsb v5">n</span>, onde m é a </div><div class="t m0 x1 h7 y1c ff2 fs4 fc0 sc0 lsf wsd">massa do sistema. </div><div class="t m0 x1 h7 y1d ff2 fs4 fc0 sc0 lsf wsd">Também foi definido a frequência angular natural amortecida <span class="ff6">\uf077</span></div><div class="t m0 x5 hd y1e ff2 fs7 fc0 sc0 lsb">d<span class="fs4 lsf wsd v9"> que é definida por </span></div><div class="t m0 x1 h8 y1f ff4 fs4 fc0 sc0 lsf ws0">\ue7f1<span class="fs5 lsc va">\uebd7</span><span class="wsd">=<span class="_7 blank"> </span> <span class="_7 blank"> </span>\ue7f1</span></div><div class="t m0 x6 he y20 ff4 fs5 fc0 sc0 lsd">\uebe1<span class="fs4 lsa v3">\ueda5<span class="lsf wsc v0">1<span class="_8 blank"> </span>\u2212<span class="_8 blank"> </span>\ue7e6 </span></span><span class="lse v8">\ueb36</span><span class="ff2 fs4 lsf wsd v3"> </span></div><div class="t m0 x1 h7 y21 ff2 fs4 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h7 y22 ff2 fs4 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h7 y23 ff2 fs4 fc0 sc0 lsf wsd">Para um sistema com vibração subamortecida, chegamos à seguinte sol<span class="_6 blank"></span>ução: </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/98de2b23-5c6b-4aaf-92dd-b05cf194e2ae/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsf wsd">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsf wsd"> </div></div><div class="t m0 x1 hb y24 ff4 fs4 fc0 sc0 ls0">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls2 v1">)</span><span class="lsf wsd">=<span class="_3 blank"> </span> \ue741<span class="_4 blank"> </span><span class="fs5 ws1 v2">\ueb3f\uec15 \uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls4 v2">\uebe7</span><span class="ws2">. \uf242\ue754<span class="fs5 ls10 va">\uebe2</span><span class="wse">. \ue73f\ue74b\ue74f<span class="ws0 v1">(<span class="vb">\ue7f1</span></span><span class="fs5 ls11 va">\uebd7</span></span></span></span>\ue750<span class="ls12 v1">)</span><span class="ls13">+<span class="fs5 lsf v6">\uebe9</span></span></span></div><div class="t m0 x7 hf y25 ff4 fs6 fc0 sc0 ls14">\uecda<span class="fs5 lsf wsf vc">\ueb3e\uec15 .\uec20</span><span class="ls3">\uecd9<span class="fs5 lsf ws4 vc">.\uebeb</span><span class="lsf">\uecda</span></span></div><div class="t m0 x8 h10 y26 ff4 fs5 fc0 sc0 lsf ws4">\uec20<span class="fs6 ls3 v7">\uecd9</span>\uec20<span class="fs6 ls15 v7">\ueccf</span><span class="fs4 ws2 v8">. \ue74f\ue741\ue74a<span class="ws0 v1">(<span class="vb">\ue7f1</span></span></span><span class="ls11 vd">\uebd7</span><span class="fs4 ls1 v8">\ue750</span><span class="fs4 ws0 ve">)<span class="vb">\uf243<span class="ff2 wsd">, que também pode ser escrita </span></span></span></div><div class="t m0 x1 h9 y27 ff2 fs4 fc0 sc0 lsf wsd">em termos de amplitude C e ângulo de fase <span class="ff7 ws10">\u03d5</span>, a saber: <span class="ff4 ls16">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls17 v1">)</span><span class="lsf ws11">=<span class="_3 blank"> </span>\ue725 .<span class="_7 blank"> </span>\ue741<span class="_4 blank"> </span><span class="fs5 ws1 v2">\ueb3f\uec15 \uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls4 v2">\uebe7</span><span class="ws0">.</span></span></span></span> <span class="ff4 ws0">\ue74f\ue741\ue74a<span class="_0 blank"> </span><span class="v1">(</span>\ue7f1<span class="fs5 ls11 va">\uebd7</span><span class="ws12">\ue750 +</span></span></div><div class="t m0 x1 h11 y28 ff4 fs4 fc0 sc0 lsa">\u2205<span class="lsf ws0 v1">)</span><span class="ff2 lsf wsd"> sendo <span class="ff4">\ue725<span class="_2 blank"> </span>=<span class="_3 blank"> </span> <span class="_9 blank"> </span><span class="fs5 ls18 v6">\ueda5<span class="ls19 v1">(</span></span><span class="fs5 v6">\uebe9</span></span></span></div><div class="t m0 x9 h12 y29 ff4 fs6 fc0 sc0 ls1a">\uecda<span class="fs5 lsf wsf vc">\ueb3e\uec15 \uec20</span><span class="ls3">\uecd9<span class="fs5 ls1b vc">\uebeb</span><span class="ls14">\uecda<span class="fs5 lsf ws4 vf">)</span><span class="ls7 v2">\uec2e</span><span class="fs5 lsf ws4 vc">\ueb3e<span class="ls19 v1">(</span>\uebeb</span>\uecda<span class="fs5 lsf ws4 vc">.\uec20</span><span class="ls1c">\ueccf<span class="fs5 ls19 vf">)</span><span class="lsf v2">\uec2e</span></span></span></span></div><div class="t m0 xa h13 y2a ff4 fs5 fc0 sc0 lsf ws4">\uec20<span class="fs6 ls1d v7">\ueccf</span><span class="ff2 fs4 wsd v8"> e <span class="ff4 ws13">\u2205<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue73d\ue74e\ue73f\ue750\ue743 \uf240<span class="_a blank"> </span></span></span><span class="ls1b v10">\uebeb</span><span class="fs6 ls14 v11">\uecda</span><span class="v10">.\uec20</span><span class="fs6 v11">\ueccf</span></div><div class="t m0 xb h14 y2a ff4 fs5 fc0 sc0 lsf">\uebe9</div><div class="t m0 xc h15 y2b ff4 fs6 fc0 sc0 ls14">\uecda<span class="fs5 lsf wsf vc">\ueb3e\uec15 .\uec20</span><span class="ls3">\uecd9<span class="fs5 lsf ws4 vc">.\uebeb</span></span>\uecda<span class="fs4 lsf ws0 v12">\uf241<span class="ff2 wsd"> onde x<span class="fs7 ls1e v5">o</span> é a posição </span></span></div><div class="t m0 x1 h7 y2c ff2 fs4 fc0 sc0 lsf wsd">inicial e v<span class="fs7 ls1e v5">o</span> a velocidade inicial. </div><div class="t m0 x1 h7 y2d ff2 fs4 fc0 sc0 lsf wsd">Quando plotamos um gráfico da resposta temos: </div><div class="t m0 xd h7 y2e ff2 fs4 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h8 y2f ff2 fs4 fc0 sc0 lsf wsd">Sendo a curva tracejada vermelha definida por: <span class="ff4 fc2 ls6">\ue741<span class="fs5 lsf ws1 v2">\ueb3f\uec15 \uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls4 v2">\uebe7</span></span> </div><div class="t m0 x1 h7 y30 ff2 fs4 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h6 y31 ff3 fs3 fc1 sc0 lsf wsd">Vibração Livre com Super Amortecimento ou <span class="_0 blank"> </span>Amortecimento </div><div class="t m0 x1 h6 y32 ff3 fs3 fc1 sc0 lsf wsd">Supercrítico (\u03be > 1) </div><div class="t m0 x1 h8 y33 ff2 fs4 fc0 sc0 lsf wsd">Para <span class="ff4 ls1f">\ue7e6</span> > 1 implica que as raízes da equação <span class="ff4 ws14">\u2212\ue7e6 \ue7f1</span></div><div class="t m0 xe he y34 ff4 fs5 fc0 sc0 ls20">\uebe1<span class="fs4 lsf ws15 v3">± \ue7f1</span></div><div class="t m0 xf he y34 ff4 fs5 fc0 sc0 ls21">\uebe1<span class="fs4 lsa v3">\ueda5<span class="ls22 v0">\ue7e6</span></span><span class="ls23 v8">\ueb36</span><span class="fs4 lsf ws5 v3">\u2212 1<span class="ff2 wsd"> sejam reais e </span></span></div><div class="t m0 x1 h7 y35 ff2 fs4 fc0 sc0 lsf wsd">distintas. </div><div class="t m0 x1 h7 y36 ff2 fs4 fc0 sc0 lsf wsd">Dessa forma, podemos escrever a solução da como: </div><div class="t m0 x1 h16 y37 ff4 fs4 fc0 sc0 ls0">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls2 v1">)</span><span class="lsf ws0">=<span class="ff2 wsd"> </span><span class="ws16">A.<span class="_9 blank"> </span>\ue741 <span class="fs5 ws1 v8">\uf240\ueb3f\uec15 \ueb3e<span class="ws4 v0">\ueda5<span class="ls24 v0">\uec15<span class="fs6 ls7 v3">\uec2e</span><span class="lsf">\ueb3f\ueb35\uf241\uec20<span class="fs6 ls25 v7">\uecd9</span><span class="ls26">\uebe7</span></span></span></span></span><span class="ws17">+<span class="_8 blank"> </span>\ue724.<span class="_9 blank"> </span>\ue741<span class="_4 blank"> </span><span class="fs5 wsd v8">\uf240\ueb3f\uec15 \u2013<span class="ws4 v0">\ueda5<span class="ls24 v0">\uec15<span class="fs6 ls27 v3">\uec2e</span><span class="lsf">\ueb3f\ueb35\uf241\uec20<span class="fs6 ls25 v7">\uecd9</span><span class="ls28">\uebe7</span></span></span></span></span><span class="ff2 wsd"> </span></span></span></span></span></div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 lsf wsd"> Onde A e B são constantes reais definidas a partir das condições iniciais de posição </div><div class="t m0 x1 h7 y39 ff2 fs4 fc0 sc0 lsf wsd">inicial x<span class="fs7 ls1e v5">o</span> e velocidade inicial v<span class="fs7 ls1e v5">o</span>. Resolvendo e após algumas manipulações algébricas </div><div class="t m0 x1 h7 y3a ff2 fs4 fc0 sc0 lsf wsd">chegamos: </div><div class="t m0 x1 h17 y3b ff4 fs4 fc0 sc0 lsf ws18">A = <span class="fs5 v13">\uebe9</span></div><div class="t m0 x10 h12 y3c ff4 fs6 fc0 sc0 ls14">\uecda<span class="fs5 lsf wsf vc">\ueb3e\uf240\uec15 \ueb3e<span class="ws4 v0">\ueda5<span class="ls24 v0">\uec15</span></span></span><span class="ls7 v2">\uec2e</span><span class="fs5 lsf ws4 vc">\ueb3f\ueb35\uf241\uec20</span><span class="ls3">\uecd9<span class="fs5 lsf ws4 vc">.\uebeb</span><span class="lsf">\uecda</span></span></div><div class="t m0 x11 h18 y3d ff4 fs5 fc0 sc0 lsf ws4">\ueb36\uec20<span class="fs6 ls3 v7">\uecd9</span><span class="v0">\ueda5</span><span class="ls24">\uec15<span class="fs6 ls27 v3">\uec2e</span></span><span class="ws19">\ueb3f\ueb35 <span class="ff2 fs4 wsd ve"> e <span class="ff4 ws1a">B<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\u2212 </span></span><span class="v14">\uebe9</span></span></div><div class="t m0 x12 h12 y3c ff4 fs6 fc0 sc0 ls14">\uecda<span class="fs5 lsf wsd vc">\ueb3e\uf240\uec15 \ueb3f<span class="ws4 v0">\ueda5<span class="ls24 v0">\uec15</span></span></span><span class="ls7 v2">\uec2e</span><span class="fs5 lsf ws4 vc">\ueb3f\ueb35\uf241\uec20</span><span class="ls25">\uecd9<span class="fs5 lsf ws4 vc">.\uebeb</span><span class="lsf">\uecda</span></span></div><div class="t m0 x13 h19 y3d ff4 fs5 fc0 sc0 lsf ws4">\ueb36\uec20<span class="fs6 ls3 v7">\uecd9</span><span class="v0">\ueda5</span><span class="ls24">\uec15<span class="fs6 ls7 v3">\uec2e</span></span><span class="ws1b">\ueb3f\ueb35 <span class="ff2 fs4 wsd ve"> </span></span></div><div class="t m0 x1 h7 y3e ff2 fs4 fc0 sc0 lsf wsd">Nesse sistema, a sua resposta é sem oscilação e </div><div class="t m0 x1 h7 y3f ff2 fs4 fc0 sc0 lsf wsd">quando perturbado, retorna de forma exponen<span class="_6 blank"></span>cial a </div><div class="t m0 x1 h7 y40 ff2 fs4 fc0 sc0 lsf wsd">sua posição de equilíbrio como podemos visualizar na </div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 lsf wsd">figura ao lado. </div><div class="t m0 x1 h7 y42 ff2 fs4 fc0 sc0 lsf wsd"> </div><div class="c x14 y43 w3 h1a"><div class="t m0 x0 h8 y44 ff4 fs4 fc2 sc0 lsf wsd"> </div></div><div class="c x15 y43 w4 h1a"><div class="t m0 x0 h8 y44 ff4 fs4 fc2 sc0 lsf">\ue741</div></div><div class="c x16 y43 w5 h1a"><div class="t m0 x0 h14 y45 ff4 fs5 fc2 sc0 lsf">\ueb3f</div></div><div class="c x17 y43 w6 h1a"><div class="t m0 x0 h14 y45 ff4 fs5 fc2 sc0 lsf">\uec15</div></div><div class="c x18 y43 w7 h1a"><div class="t m0 x0 h14 y45 ff4 fs5 fc2 sc0 lsf">\uec20</div></div><div class="c x19 y43 w8 h1a"><div class="t m0 x0 h1b y46 ff4 fs6 fc2 sc0 lsf">\uecd9</div></div><div class="c x1a y43 w9 h1a"><div class="t m0 x0 h14 y45 ff4 fs5 fc2 sc0 lsf">\uebe7</div></div><div class="c x1b y43 wa h1c"><div class="t m0 x0 h1d y44 ff8 fs4 fc2 sc0 lsf wsd"> </div></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/98de2b23-5c6b-4aaf-92dd-b05cf194e2ae/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsf wsd">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsf wsd"> </div></div><div class="t m0 x1 h6 y47 ff3 fs3 fc1 sc0 lsf wsd">Vibração Livre com Amortecimento Crítico (<span class="_0 blank"> </span>\u03be = 1) </div><div class="t m0 x1 h8 y48 ff2 fs4 fc0 sc0 lsf wsd">Para termos ksi igual a um implica que a raízes da equação <span class="ff4 ws14">\u2212\ue7e6 \ue7f1</span></div><div class="t m0 x1c he y49 ff4 fs5 fc0 sc0 ls20">\uebe1<span class="fs4 lsf ws5 v3">± \ue7f1</span></div><div class="t m0 x1d he y49 ff4 fs5 fc0 sc0 lsd">\uebe1<span class="fs4 lsa v3">\ueda5<span class="ls22 v0">\ue7e6</span></span><span class="ls23 v8">\ueb36</span><span class="fs4 lsf ws5 v3">\u2212 1<span class="ff2 wsd"> sejam </span></span></div><div class="t m0 x1 h8 y4a ff2 fs4 fc0 sc0 lsf wsd">raízes reais e iguais, pois com ksi igual a 1, o resultado da <span class="ff4 lsa v0">\ueda5<span class="ls22 v0">\ue7e6<span class="fs5 ls23 vd">\ueb36</span><span class="lsf ws5">\u2212 1</span></span></span> é igual a zero. </div><div class="t m0 x1 h9 y4b ff2 fs4 fc0 sc0 lsf wsd">Assim a solução será <span class="ff4 ls16">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls17 v1">)</span><span class="lsf ws0">=</span></span></span> <span class="ff4 ls6">\ue741<span class="fs5 lsf ws4 v2">\ueb3f\uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls29 v2">\uebe7</span><span class="lsf ws0 v1">[(</span><span class="lsf">\ue752</span></span></div><div class="t m0 x13 he y4c ff4 fs5 fc0 sc0 ls2a">\uebe2<span class="fs4 lsf ws5 v3">+ \ue7f1</span></div><div class="t m0 x1e h1e y4c ff4 fs5 fc0 sc0 lsd">\uebe1<span class="fs4 lsf ws0 v3">\ue754</span><span class="ls2b">\uebe2<span class="fs4 lsf ws0 v15">)<span class="ws5 vb">\ue750 + \ue754</span></span><span class="ls10">\uebe2<span class="fs4 lsf ws0 v15">]<span class="ff2 wsd vb"> </span></span></span></span></div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 lsf wsd">Este sistema não oscila, quando perturbado retorna para o ponto de equilí<span class="_6 blank"></span>brio no </div><div class="t m0 x1 h7 y4e ff2 fs4 fc0 sc0 lsf wsd">tempo mais rápido. </div><div class="t m0 x1 h7 y4f ff2 fs4 fc0 sc0 lsf wsd">Sua aplicação prática é em sistemas de amorteciment<span class="_6 blank"></span>o de portas e também, em </div><div class="t m0 x1 h7 y50 ff2 fs4 fc0 sc0 lsf wsd">sistemas de recolhimento de armas de fogo. </div><div class="t m0 x1 h7 y51 ff2 fs4 fc0 sc0 lsf wsd">Fazendo uma simulação para algumas condições iniciais, temos o indicado n<span class="_6 blank"></span>a figura: <span class="_0 blank"> </span> </div><div class="t m0 x1 h1f y52 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y53 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y54 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y55 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y56 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y57 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y58 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y59 ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y5a ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y5b ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h1f y5c ff2 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x1 h6 y5d ff3 fs3 fc1 sc0 lsf wsd">Comparação entre Movimentos com Tipos Diferentes de </div><div class="t m0 x1 h6 y5e ff3 fs3 fc1 sc0 lsf ws1c">Amortecimento<span class="fs4 wsd"> </span></div><div class="t m0 x1 h7 y5f ff2 fs4 fc0 sc0 lsf wsd">Para entendermos melhor o efeito do tipo de amorteciment<span class="_6 blank"></span>o, observe o <span class="_0 blank"> </span>gráfico </div><div class="t m0 x1 h7 y60 ff2 fs4 fc0 sc0 lsf wsd">abaixo: </div><div class="t m0 x1f h7 y61 ff2 fs4 fc0 sc0 lsf wsd"> </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/98de2b23-5c6b-4aaf-92dd-b05cf194e2ae/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsf wsd">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsf wsd"> </div></div><div class="t m0 x1 h7 y62 ff2 fs4 fc0 sc0 lsf wsd">A curva em vermelho representa um sistema com vibração<span class="_6 blank"></span> livre sem amortecimento, </div><div class="t m0 x1 h7 y63 ff2 fs4 fc0 sc0 lsf wsd">veja que ela oscila continuamente, pois não tem nenhum elemento di<span class="_6 blank"></span>ssipando </div><div class="t m0 x1 h7 y64 ff2 fs4 fc0 sc0 lsf wsd">energia. </div><div class="t m0 x1 h7 y65 ff2 fs4 fc0 sc0 lsf wsd">A curva em azul, trata-se de um sistema com vibração livre com subamortecimento, </div><div class="t m0 x1 h7 y66 ff2 fs4 fc0 sc0 lsf wsd">note que a amplitude da vibração vai diminuindo com o tempo. </div><div class="t m0 x1 h7 y67 ff2 fs4 fc0 sc0 lsf wsd">Já a curva em verde, temos um sistema com vibração livre superamortecida, v<span class="_6 blank"></span>eja que </div><div class="t m0 x1 h7 y68 ff2 fs4 fc0 sc0 lsf wsd">o sistema no oscila, retorna de forma exponencial ao ponto de equil<span class="_6 blank"></span>íbrio. </div><div class="t m0 x1 h7 y69 ff2 fs4 fc0 sc0 lsf wsd">E por fim, a curva em azul marinho, que representa um sistema com vibração livre </div><div class="t m0 x1 h7 y6a ff2 fs4 fc0 sc0 lsf wsd">com amortecimento crítico, note que é o sistema que mais rapid<span class="_6 blank"></span>amente retorna a </div><div class="t m0 x1 h7 y6b ff2 fs4 fc0 sc0 lsf wsd">posição de equilíbrio. </div><div class="t m0 x1 h6 y6c ff3 fs3 fc1 sc0 lsf wsd">Decremento Logarítmico (<span class="ff6 sc1 ws1d">\uf064</span>) </div><div class="t m0 x1 h7 y6d ff2 fs4 fc0 sc0 lsf wsd">Em análise de sistemas existentes, muitas vezes não temos as suas informações de </div><div class="t m0 x1 h7 y6e ff2 fs4 fc0 sc0 lsf wsd">rigidez e amortecimento. Existem alguns métodos que perm<span class="_6 blank"></span>item estimar esses </div><div class="t m0 x1 h7 y6f ff2 fs4 fc0 sc0 lsf wsd">valores, para sistemas com vibração livre, podemos utilizar o método d<span class="_6 blank"></span>o decremento </div><div class="t m0 x1 h7 y70 ff2 fs4 fc0 sc0 lsf wsd">logarítmico. </div><div class="t m0 x1 h7 y71 ff2 fs4 fc0 sc0 lsf wsd">Por definição, o decremento logarítmico, <span class="ff6 sc2 ws9">\uf064</span>, é o logaritmo natural da razão entre duas </div><div class="t m0 x1 hb y72 ff2 fs4 fc0 sc0 lsf wsd">amplitudes, <span class="ff4 ws1e">\ue7dc<span class="_2 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240<span class="_b blank"> </span><span class="fs5 ws1f v6">\uebeb(\uebe7 )</span></span></div><div class="t m0 x20 h20 y73 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="ff9 lsf wsd"> </span><span class="lsf ws1f">(\uebe7\ueb3e\uebe7 <span class="fs6 ls1c v7">\ueccf</span><span class="ls19">)</span><span class="fs4 ws0 v8">\uf241<span class="ff2 wsd">, onde <span class="ff3 ls5">t<span class="fs7 lsf ws20 v5">d</span></span> é o período para sistema subamortecidos. </span></span></span></div><div class="t m0 x1 hb y74 ff2 fs4 fc0 sc0 lsf wsd">Podemos generalizar a equação acima como: <span class="ff4 ws21">\ue7dc<span class="_2 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240<span class="fs5 ls1b v6">\uebeb</span><span class="fs6 v16">\uecda</span></span></div><div class="t m0 x21 h13 y75 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 ls2c v7">\uec2d</span><span class="fs4 lsf ws21 v8">\uf241<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240</span><span class="v10">\uebeb</span><span class="fs6 lsf v11">\uec2d</span></div><div class="t m0 x22 h13 y75 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 ls7 v7">\uec2e</span><span class="fs4 lsf ws1e v8">\uf241<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240</span><span class="v10">\uebeb</span><span class="fs6 lsf ws22 v11">\uecd9\uec37\uec2e</span></div><div class="t m0 x23 h20 y75 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 lsf ws23 v7">\uecd9\uec37\uec2d </span><span class="fs4 lsf ws0 v8">\uf241<span class="ff2 wsd"> ou </span></span></div><div class="t m0 x1 hb y76 ff4 fs4 fc0 sc0 lsf wsd">\ue7dc<span class="_2 blank"> </span>=<span class="_3 blank"> </span> <span class="_c blank"> </span><span class="fs5 v6">\ueb35</span></div><div class="t m0 x10 h13 y77 ff4 fs5 fc0 sc0 ls2d">\uebe1<span class="fs4 lsf ws24 v8">ln<span class="_9 blank"> </span>\uf240</span><span class="ls1b v10">\uebeb</span><span class="fs6 lsf v11">\uecda</span></div><div class="t m0 x24 h20 y77 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 ls3 v7">\uecd9</span><span class="fs4 lsf ws0 v8">\uf241<span class="ff2 wsd">, onde x é uma amplitude. </span></span></div><div class="t m0 x1 h9 y78 ff2 fs4 fc0 sc0 lsf wsd">Lembrando que a resposta do sistema é do tipo <span class="ff4 ls0">\ue754<span class="lsf ws0 v1">(</span><span class="ls1">\ue750<span class="ls2 v1">)</span><span class="lsf">=<span class="_3 blank"> </span>\ue73a \ue741<span class="_4 blank"> </span><span class="fs5 ws1 v2">\ueb3f\uec15 \uec20</span><span class="fs6 ls3 v3">\uecd9</span><span class="fs5 ls29 v2">\uebe7</span><span class="ws0">\ue74f\ue741\ue74a(\ue7f1<span class="fs5 ls11 va">\uebd7</span><span class="ws5">\ue750<span class="_d blank"> </span>+ \u2205)</span></span></span></span></span> </div><div class="t m0 x1 hb y79 ff2 fs4 fc0 sc0 lsf wsd">podemos escrever: <span class="ff4 ws21">\ue7dc<span class="_2 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240<span class="fs5 ls1b v6">\uebeb</span><span class="fs6 v16">\uecda</span></span></div><div class="t m0 x25 h21 y7a ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 ls2c v7">\uec2d</span><span class="fs4 lsf ws21 v8">\uf241<span class="_d blank"> </span>=<span class="_e blank"> </span>\ue748\ue74a \uf240</span><span class="lsf wsd v10">\uebd1 \uebd8<span class="_4 blank"> </span></span><span class="fs6 lsf ws22 v17">\uec37\ued0d\ued18<span class="ls25 v18">\uecd9</span><span class="ls2e">\uecdf</span></span><span class="lsf ws4 v10">\uebe6\uebd8\uebe1(\uec20</span><span class="fs6 lsf wsd v11">\ueccf <span class="_0 blank"> </span></span><span class="ls2f v10">\uebe7</span><span class="fs6 ls1a v11">\uecda</span><span class="lsf ws4 v10">\ueb3e\u2205)<span class="ff2 wsd"> </span></span></div><div class="t m0 x26 h13 y7a ff4 fs5 fc0 sc0 lsf wsd">\uebd1 \uebd8<span class="_4 blank"> </span><span class="fs6 ws22 v3">\uec37\ued0d\ued18<span class="ls25 v18">\uecd9</span><span class="ls2e">\uecdf</span></span><span class="ws4">\uebe6\uebd8\uebe1(\uec20<span class="fs6 ls30 v7">\ueccf</span></span> \uebe7<span class="fs6 ls7 v7">\uec2d</span><span class="ws4">\ueb3e\u2205)<span class="_6 blank"></span><span class="ff2 wsd"> <span class="ff4 fs4 ws0 v8">\uf241</span><span class="fs4 v8"> onde <span class="ff4 ws0">\ue750<span class="fs5 lsc va">\uebd7</span><span class="wsd">=<span class="_d blank"> </span> <span class="fs5 ws4 v6">\ueb36\uec17</span></span></span></span></span></span></div><div class="t m0 x27 hc y7a ff4 fs5 fc0 sc0 lsf ws4">\uec20<span class="fs6 ls1c v7">\ueccf</span><span class="ff2 fs4 wsd v8">. </span></div><div class="t m0 x1 hb y7b ff2 fs4 fc0 sc0 lsf wsd">Resolvendo a equação, chegamos em: <span class="ff4 ws25">\ue7dc<span class="_e blank"> </span>= </span><span class="ff9 ls31"> </span><span class="ff4 fs5 ws4 v6">\ueb36\uec17\uec15</span></div><div class="t m0 x28 h22 y7c ff4 fs5 fc0 sc0 lsf ws4">\ueda5<span class="ws1 v0">\ueb35\ueb3f\uec15 <span class="fs6 ls7 v3">\uec2e</span><span class="ff2 fs4 wsd ve">, ou seja, conhecendo dois picos </span></span></div><div class="t m0 x1 h7 y7d ff2 fs4 fc0 sc0 lsf wsd">subsequentes podemos determinar o fator de amortecimen<span class="_6 blank"></span>to <span class="ff1 ws8">\u03be</span>; </div><div class="t m0 x1 h6 y7e ff3 fs3 fc1 sc0 lsf wsd">Exemplo (<span class="ff6 sc1 ws1d">\uf064</span>) </div><div class="t m0 x1 h7 y7f ff2 fs4 fc0 sc0 lsf wsd">Considere um sistema massa-mola-amortecedor cuja resposta está demonstrado no </div><div class="t m0 x1 h7 y80 ff2 fs4 fc0 sc0 lsf wsd">gráfico abaixo. Estime os coeficientes </div><div class="t m0 x1 h7 y81 ff2 fs4 fc0 sc0 lsf wsd">equivalentes de rigidez e amortecimento </div><div class="t m0 x1 h7 y82 ff2 fs4 fc0 sc0 lsf wsd">viscoso desse sistema, se a massa m = 20 kg e o </div><div class="t m0 x1 h7 y83 ff2 fs4 fc0 sc0 lsf wsd">deslocamento inicial x<span class="fs7 ls1e v5">o</span> = 0,01 (m). </div><div class="t m0 x1 h7 y84 ff2 fs4 fc0 sc0 lsf wsd">Resolução: </div><div class="t m0 x1 h7 y85 ff2 fs4 fc0 sc0 lsf wsd">Calculando o Decremento Logarítmico (<span class="ff6 ws9">\uf064</span>): </div><div class="c x29 y86 wa h23"><div class="t m0 x0 h1d y87 ff8 fs4 fc2 sc0 lsf wsd"> </div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/98de2b23-5c6b-4aaf-92dd-b05cf194e2ae/bg5.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 lsf wsd">5 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 lsf wsd"> </div></div><div class="t m0 x1 hb y24 ff6 fs4 fc0 sc0 lsf ws9">\uf064<span class="ff2 wsd"> = ? <span class="ff5 ls5">\uf0e0</span><span class="ls32"> </span><span class="ff4 ws1e">\ue7dc<span class="_e blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240<span class="fs5 ls1b v6">\uebeb</span><span class="fs6 v16">\uecda</span></span></span></div><div class="t m0 x2a h13 y26 ff4 fs5 fc0 sc0 ls1b">\uebeb<span class="fs6 ls2c v7">\uec2d</span><span class="fs4 lsf ws21 v8">\uf241<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue748\ue74a \uf240 </span><span class="lsf ws4 v10">\ueb34,\ueb34\ueb35</span></div><div class="t m0 x1b h20 y26 ff4 fs5 fc0 sc0 lsf ws4">\ueb34,\ueb34\ueb34\ueb39<span class="fs4 ws26 v8">\uf241 =<span class="ff2 wsd"> 0,693 </span></span></div><div class="t m0 x1 h7 y88 ff2 fs4 fc0 sc0 lsf wsd">Calculando o fator de amortecimento (<span class="ff1 ws8">\u03be</span>): </div><div class="t m0 x1 hb y89 ff1 fs4 fc0 sc0 lsf ws8">\u03be<span class="ff2 wsd"> =? <span class="ff5 ls5">\uf0e0</span> <span class="ff4">\ue7e6<span class="_2 blank"> </span>=<span class="_3 blank"> </span> <span class="_f blank"> </span><span class="fs5 v6">\uec0b</span></span></span></div><div class="t m0 x2b h24 y8a ff4 fs5 fc0 sc0 lsf ws4">\u221a<span class="ws27 v0">\ueb38\uec17<span class="fs6 ls7 v3">\uec2e</span><span class="ws28">\ueb3f\uec0b <span class="fs6 ls33 v3">\uec2e</span><span class="ff2 fs4 wsd ve"> <span class="ff4">=<span class="_3 blank"> </span> <span class="_10 blank"> </span></span></span></span></span><span class="v10">\ueb34,\ueb3a\ueb3d\ueb37</span></div><div class="t m0 x2c h10 y8b ff4 fs5 fc0 sc0 lsf ws4">\ueda5<span class="ws27 vb">\ueb38\uec17</span><span class="fs6 ls7 vf">\uec2e</span><span class="vb">\ueb3f\ueb34,\ueb3a\ueb3d\ueb37</span><span class="fs6 ls34 vf">\uec2e</span><span class="fs4 wsd ve"> <span class="_7 blank"> </span>=<span class="ff2"> 0,11 </span></span></div><div class="t m0 x1 h7 y8c ff2 fs4 fc0 sc0 lsf wsd">Calculando a frequência angular natural amortecida (<span class="ff6">\uf077</span></div><div class="t m0 x2d hd y8d ff2 fs7 fc0 sc0 lsb">d<span class="fs4 lsf wsd v9">): </span></div><div class="t m0 x1 hb y8e ff6 fs4 fc0 sc0 lsf ws9">\uf077<span class="ff2 fs7 lsb v5">d</span><span class="ff2 wsd"> =? <span class="ff5 ls5">\uf0e0</span> <span class="ff4 ws0">\ue7f1<span class="fs5 lsc va">\uebd7</span><span class="wsd">=<span class="_d blank"> </span> <span class="_9 blank"> </span><span class="fs5 ws4 v6">\ueb36\uec17</span></span></span></span></div><div class="t m0 x2e h13 y8f ff4 fs5 fc0 sc0 ls2f">\uebe7<span class="fs6 ls35 v7">\ueccf</span><span class="ff2 fs4 lsf wsd v8"> <span class="ff4">=<span class="_3 blank"> </span> <span class="_5 blank"> </span></span></span><span class="lsf ws4 v10">\ueb36\uec17</span></div><div class="t m0 x16 h20 y8f ff4 fs5 fc0 sc0 lsf ws4">\ueb34,\ueb34\ueb3a<span class="ff2 fs4 wsd v8"> <span class="ff4 ws0">=<span class="ff3 wsd"> 104,7 (rad/s)</span></span> </span></div><div class="t m0 x1 h7 y90 ff2 fs4 fc0 sc0 lsf wsd">Calculando a frequência angula natural (<span class="ff6">\uf077</span></div><div class="t m0 x2f hd y91 ff2 fs7 fc0 sc0 lsf wsb">n<span class="fs4 wsd v9">): </span></div><div class="t m0 x1 h8 y92 ff6 fs4 fc0 sc0 lsf ws9">\uf077<span class="ff2 fs7 ls8 v5">n</span><span class="ff2 wsd"> =? <span class="ff5 ls5">\uf0e0</span> <span class="ff4 ws0">\ue7f1<span class="fs5 lsc va">\uebd7</span><span class="ws25">= \ue7f1</span></span></span></div><div class="t m0 x2a he y93 ff4 fs5 fc0 sc0 lsd">\uebe1<span class="fs4 lsa v3">\ueda5<span class="lsf wsc v0">1<span class="_8 blank"> </span>\u2212<span class="_8 blank"> </span>\ue7e6 </span></span><span class="lse v8">\ueb36</span><span class="ff2 fs4 lsf wsd v3"> <span class="ff5 ls5">\uf0e0</span> <span class="ff4">\ue7f1</span></span></div><div class="t m0 x30 h25 y93 ff4 fs5 fc0 sc0 ls36">\uebe1<span class="fs4 ls37 v3">=</span><span class="lsf ws4 v19">\uec20<span class="fs6 v7">\ueccf</span></span></div><div class="t m0 x31 h26 y94 ff4 fs5 fc0 sc0 lsf ws4">\ueda5<span class="ws1 v0">\ueb35\ueb3f\uec15 <span class="fs6 ls7 v3">\uec2e</span><span class="ff2 ls38 wsd"> </span><span class="fs4 ls39 ve">=</span></span><span class="v1a">\ueb35\ueb34\ueb38,\ueb3b</span></div><div class="t m0 x32 h14 y94 ff4 fs5 fc0 sc0 lsf ws4">\ueda5<span class="vb">\ueb35\ueb3f\ueb34,\ueb35\ueb35</span></div><div class="t m0 x33 h27 y95 ff4 fs6 fc0 sc0 ls27">\uec2e<span class="ff2 fs5 ls3a wsd va"> </span><span class="fs4 lsf wsd v2"> <span class="_7 blank"> </span>=<span class="ff3"> 105,3 (rad/s)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y96 ff2 fs4 fc0 sc0 lsf wsd">Calculando a rigidez (k): </div><div class="t m0 x1 h8 y97 ff2 fs4 fc0 sc0 lsf wsd">k = ? <span class="ff5 ls5">\uf0e0</span> <span class="ff4">\ue7f1</span></div><div class="t m0 x34 h25 y98 ff4 fs5 fc0 sc0 ls36">\uebe1<span class="fs4 ls3b v3">=<span class="ls3c v1">\ueda7</span></span><span class="lsf v19">\uebde</span></div><div class="t m0 x35 h20 y99 ff4 fs5 fc0 sc0 ls3d">\uebe0<span class="ff2 fs4 lsf wsd v8"> <span class="ff5 ls5">\uf0e0</span> <span class="ff4 wse">k<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue749. \ue7f1</span></span></div><div class="t m0 x36 he y98 ff4 fs5 fc0 sc0 lsd">\uebe1<span class="lse v6">\ueb36</span><span class="ff2 fs4 lsf wsd v3"> <span class="ff4">=<span class="_3 blank"> </span>20.105,3 </span></span><span class="ls3e v6">\ueb36</span><span class="fs4 ls3f v3">=<span class="ff3 lsf wsd"> 2,22 x 10<span class="fs7 ws20 v1b">5</span> (N/m)<span class="ff2"> </span></span></span></div><div class="t m0 x1 h7 y9a ff2 fs4 fc0 sc0 lsf wsd">Calculando o fator de amortecimento viscoso (c): </div><div class="t m0 x1 h8 y9b ff2 fs4 fc0 sc0 lsf wsd">c = ? <span class="ff5 ls5">\uf0e0</span> <span class="ff4 wse">c<span class="_3 blank"> </span>=<span class="_3 blank"> </span>\ue749<span class="_0 blank"></span>. \ue7f1</span></div><div class="t m0 x2a he y9c ff4 fs5 fc0 sc0 lsd">\uebe1<span class="ff2 fs4 lsf wsd v3">. <span class="ff1 ws8">\u03be<span class="ff4 wsd"> =</span></span> 2 x 20 x 105,3 x 0,11 = <span class="ff3">4,63 x 10<span class="fs7 ws20 v1b">2</span> (N.s/m)</span> </span></div><div class="t m0 x1 h2 y9d ff1 fs0 fc0 sc0 lsf wsd"> </div><div class="c x37 y9e wa h28"><div class="t m0 x0 h1d y9f ff8 fs4 fc2 sc0 lsf wsd"> </div></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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