<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/f80e4ebf-9792-403e-87fc-06408416f1b4/bg1.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">1 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h4 y5 ff2 fs1 fc1 sc0 ls18 wsb">Vibrações Mecânicas \u2013 <span class="_0 blank"> </span>Resumo </div><div class="t m0 x1 h4 y6 ff2 fs1 fc1 sc0 ls18 wsb"> </div><div class="t m0 x1 h5 y7 ff3 fs2 fc0 sc0 ls18 wsb">VIBRAÇÕES FORÇADAS EM SISTEMAS </div><div class="t m0 x1 h5 y8 ff3 fs2 fc0 sc0 ls18 wsb">COM 1 GDL \u2013 ESTIMATIVA </div><div class="t m0 x1 h5 y9 ff3 fs2 fc0 sc0 ls18 wsb">EXPERIMENTAL DE IRF\u2019S, FRF\u2019S E <span class="_1 blank"></span>DO </div><div class="t m0 x1 h5 ya ff3 fs2 fc0 sc0 ls18 wsb">FATOR DE AMORTECIMENTO, \u03be \u2013 </div><div class="t m0 x1 h5 yb ff3 fs2 fc0 sc0 ls18 wsb">ANÁLISE ESPECTRAL<span class="fs3 fc2"> </span></div><div class="t m0 x1 h6 yc ff3 fs3 fc1 sc0 ls18 wsb"> Introdução: </div><div class="t m0 x1 h7 yd ff2 fs4 fc0 sc0 ls18 wsb">Vimos na <span class="_0 blank"> </span>aula 12 <span class="_0 blank"> </span>que por <span class="_0 blank"> </span>meio <span class="_0 blank"> </span>de uma <span class="_0 blank"> </span>célula de <span class="_0 blank"> </span>carga podemos me<span class="_0 blank"> </span>dir o sinal <span class="_0 blank"> </span>de <span class="_0 blank"> </span>força </div><div class="t m0 x1 h7 ye ff2 fs4 fc0 sc0 ls18 wsb">de <span class="_0 blank"> </span>excitação <span class="_0 blank"> </span>aplicada <span class="_2 blank"> </span>a <span class="_0 blank"> </span>um <span class="_2 blank"> </span>sistema <span class="_0 blank"> </span>mecânico <span class="_0 blank"> </span>vibracional, <span class="_2 blank"> </span>coletando<span class="_0 blank"> </span>-se <span class="_0 blank"> </span>qualquer <span class="_2 blank"> </span>sinal </div><div class="t m0 x1 h7 yf ff2 fs4 fc0 sc0 ls18 wsb">de resposta, que pode ser de aceleração, de velocidade ou de deslocamento, podemos </div><div class="t m0 x1 h7 y10 ff2 fs4 fc0 sc0 ls18 wsb">obter uma <span class="fc1">FRF (função de resposta em frequência).</span> </div><div class="t m0 x1 h7 y11 ff2 fs4 fc0 sc0 ls18 wsb"> Um dos métodos <span class="_0 blank"> </span>utilizados é aplicar a <span class="_0 blank"> </span>transformada de Fourier nos <span class="_0 blank"> </span>sinais de saída x(t) </div><div class="t m0 x1 h7 y12 ff2 fs4 fc0 sc0 ls18 wsb">e F(t) que são definidos no domínio contínuo, ficando: </div><div class="t m0 x1 h8 y13 ff4 fs4 fc0 sc0 ls0">\ue73a<span class="ls18 ws0 v1">(</span><span class="ls1">\ue7f1<span class="ls2 v1">)</span><span class="ls18 wsb">=<span class="_3 blank"> </span> <span class="_4 blank"> </span><span class="ls3 v2">\u222b</span><span class="ls4">\ue741</span><span class="fs5 ws1 v3">\ueb3f\uebdd\uec20\uebe7 </span><span class="ls5">\ue754</span><span class="ws0 v1">(</span></span></span>\ue750<span class="ls18 ws0 v1">)</span><span class="ls18">\ue740\ue750</span></div><div class="t m0 x3 h9 y14 ff4 fs5 fc0 sc0 ls18 ws2">\ueb3e\uebb6</div><div class="t m0 x4 ha y15 ff4 fs5 fc0 sc0 ls6">\ueb34<span class="ff2 fs4 ls18 wsb v4"> </span></div><div class="t m0 x1 h8 y16 ff2 fs4 fc0 sc0 ls18 wsb"> <span class="ff4 ws0">F<span class="v1">(</span><span class="ls1">\ue7f1<span class="ls2 v1">)</span></span><span class="wsb">=<span class="_3 blank"> </span> <span class="_4 blank"> </span><span class="ls7 v2">\u222b</span><span class="ls4">\ue741</span><span class="fs5 ws3 v3">\ueb3f\uebdd\uec20\uebe7 </span><span class="ls8">\ue728</span></span><span class="v1">(</span><span class="ls0">\ue750</span><span class="v1">)</span>\ue740\ue750</span></div><div class="t m0 x3 h9 y17 ff4 fs5 fc0 sc0 ls18 ws2">\ueb3e\uebb6</div><div class="t m0 x4 ha y18 ff4 fs5 fc0 sc0 ls9">\ueb34<span class="ff2 fs4 ls18 wsb v4"> </span></div><div class="t m0 x1 h7 y19 ff2 fs4 fc0 sc0 ls18 wsb">No <span class="_5 blank"> </span>entanto, <span class="_5 blank"> </span>na <span class="_5 blank"> </span>prática, <span class="_5 blank"> </span>o <span class="_5 blank"> </span>uso <span class="_5 blank"> </span>da <span class="_6 blank"> </span>transformada <span class="_6 blank"> </span>c<span class="_0 blank"> </span>ontínua <span class="_6 blank"> </span>de <span class="_5 blank"> </span>Fourier <span class="_5 blank"> </span>não <span class="_5 blank"> </span>é <span class="_5 blank"> </span>muito </div><div class="t m0 x1 h7 y1a ff2 fs4 fc0 sc0 ls18 wsb">aplicável, <span class="_7 blank"> </span>tendo <span class="_7 blank"> </span>em <span class="_7 blank"> </span>vista <span class="_7 blank"> </span>que <span class="_7 blank"> </span>os <span class="_7 blank"> </span>sinais <span class="_7 blank"> </span>não <span class="_8 blank"> </span>são <span class="_7 blank"> </span>contínuos <span class="_7 blank"> </span>e <span class="_7 blank"> </span>são <span class="_7 blank"> </span>coletados <span class="_7 blank"> </span>em </div><div class="t m0 x1 h7 y1b ff2 fs4 fc0 sc0 ls18 wsb">intervalos de tempo. </div><div class="t m0 x1 h7 y1c ff2 fs4 fc0 sc0 ls18 wsb">Nesse <span class="_9 blank"> </span>caso, <span class="_9 blank"> </span>o <span class="_9 blank"> </span>mais <span class="_2 blank"> </span>indicado <span class="_9 blank"> </span>é <span class="_9 blank"> </span>o <span class="_9 blank"> </span>uso <span class="_9 blank"> </span>da <span class="_9 blank"> </span>transformada <span class="_2 blank"> </span>discreta <span class="_9 blank"> </span>de <span class="_9 blank"> </span>Fourier <span class="_9 blank"> </span>nos <span class="_9 blank"> </span>vetores<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y1d ff2 fs4 fc0 sc0 ls18 wsb">discretizados <span class="ff5">x[n] </span>e <span class="ff5 ws4">F[n]</span>. </div><div class="t m0 x5 h8 y1e ff4 fs4 fc0 sc0 ls0">\ue73a<span class="ls18 ws0 v1">(<span class="v2">\ue7f1</span></span><span class="fs5 lsa v5">\uebde</span><span class="lsb v1">)</span><span class="ls18 ws5">=<span class="_7 blank"> </span>\uedcd<span class="_a blank"> </span>\ue754 <span class="ws0 v1">[</span><span class="lsc">\ue74a</span><span class="v1">]</span></span></div><div class="t m0 x6 h9 y1f ff4 fs5 fc0 sc0 ls18">\uebc7</div><div class="t m0 x7 hb y20 ff4 fs5 fc0 sc0 ls18 ws6">\uebe1\ueb40\ueb34<span class="_b blank"> </span><span class="fs4 ws7 v6">.<span class="_4 blank"> </span>\ue741 </span><span class="ws8 v7">\ueb3f\uebdd \uec20</span><span class="fs6 lsd v8">\uecd6</span><span class="lse v7">\uebe1</span><span class="fs4 wsb v6"> <span class="ff2"> </span></span></div><div class="t m0 x8 h8 y21 ff4 fs4 fc0 sc0 lsf">\ue728<span class="ls18 ws0 v1">(<span class="v2">\ue7f1</span></span><span class="fs5 ls10 v5">\uebde</span><span class="ls2 v1">)</span><span class="ls18 ws9">=<span class="_7 blank"> </span>\uedcd<span class="_a blank"> </span>\ue728 [\ue74a]</span></div><div class="t m0 x9 h9 y22 ff4 fs5 fc0 sc0 ls18">\uebc7</div><div class="t m0 x6 hb y23 ff4 fs5 fc0 sc0 ls18 ws6">\uebe1\ueb40\ueb34<span class="_c blank"> </span><span class="fs4 ws7 v6">.<span class="_4 blank"> </span>\ue741 </span><span class="wsa v7">\ueb3f\uebdd\uec20</span><span class="fs6 lsd v8">\uecd6</span><span class="ls11 v7">\uebe1</span><span class="ff2 fs4 wsb v6"> </span></div><div class="t m0 x1 h7 y24 ff2 fs4 fc0 sc0 ls18 wsb">Sendo <span class="_0 blank"> </span><span class="ff6">\uf077</span></div><div class="t m0 xa hc y25 ff2 fs7 fc0 sc0 ls12">k<span class="fs4 ls18 wsb v9"> <span class="_0 blank"> </span>o <span class="_2 blank"> </span>valor <span class="_0 blank"> </span>discreto <span class="_2 blank"> </span>da <span class="_0 blank"> </span>frequência <span class="_0 blank"> </span>em <span class="_2 blank"> </span>uma <span class="_0 blank"> </span>posição <span class="_2 blank"> </span>k <span class="_0 blank"> </span>dado <span class="_2 blank"> </span>por <span class="_2 blank"> </span><span class="ff4 ws0">\ue7f1<span class="fs5 ls13 v5">\uebde</span><span class="ls14">=<span class="ff5 ls15 wsb"> </span></span><span class="fs5 ws2 va">\ueb36.\uec17</span></span></span></div><div class="t m0 xb hd y26 ff4 fs5 fc0 sc0 ls16">\uebc7<span class="fs4 ls17 vb">\ue747<span class="ff2 ls18 wsb"> <span class="_0 blank"> </span>e <span class="_0 blank"> </span>N <span class="_2 blank"> </span>o </span></span></div><div class="t m0 x1 h7 y27 ff2 fs4 fc0 sc0 ls18 wsb">número <span class="_6 blank"> </span>de <span class="_6 blank"> </span>amostras <span class="_6 blank"> </span>calculadas. <span class="_6 blank"> </span>Vale <span class="_6 blank"> </span>ressaltar <span class="_6 blank"> </span>que <span class="_6 blank"> </span>pela <span class="_6 blank"> </span>natureza <span class="_5 blank"> </span>do <span class="_6 blank"> </span>processo <span class="_6 blank"> </span>de </div><div class="t m0 x1 h7 y28 ff2 fs4 fc0 sc0 ls18 wsb">amostragem <span class="_d blank"> </span>o <span class="_9 blank"> </span>sinal <span class="_d blank"> </span>no <span class="_e blank"> </span>domínio <span class="_9 blank"> </span>da <span class="_d blank"> </span>frequência <span class="_d blank"> </span>será <span class="_d blank"> </span>periodizado, <span class="_d blank"> </span>consequentemente, </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/f80e4ebf-9792-403e-87fc-06408416f1b4/bg2.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">2 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h7 y29 ff2 fs4 fc0 sc0 ls18 wsb">se os sinais têm <span class="_4 blank"> </span>N amostras temporais, somente <span class="_4 blank"> </span>N/2 amostras serão usadas para </div><div class="t m0 x1 h7 y2a ff2 fs4 fc0 sc0 ls18 wsb">descrevê-los em termos de frequência. </div><div class="t m0 x1 h8 y2b ff2 fs4 fc0 sc0 ls18 wsb">Podemos então, obter a FRF a partir da razão de <span class="ff4 ls0">\ue73a<span class="ls18 ws0 v1">(<span class="v2">\ue7f1</span></span><span class="fs5 lsa v5">\uebde</span><span class="ls18 ws0 v1">)</span></span> e <span class="ff4 lsf">\ue728<span class="ls18 ws0 v1">(<span class="v2">\ue7f1</span></span><span class="fs5 ls10 v5">\uebde</span><span class="ls18 ws0 v1">)</span></span>: </div><div class="t m0 xc he y2c ff4 fs4 fc0 sc0 ls19">\ue72a<span class="ls18 ws0 v1">(</span><span class="ls1">\ue7f1<span class="ls2 v1">)</span><span class="ls18 wsb">=<span class="_3 blank"> </span> <span class="_4 blank"> </span><span class="wsc vc">\ue73a (\ue7f1</span><span class="fs5 ls10 vd">\uebde</span><span class="vc">)</span></span></span></div><div class="t m0 xd hf y2d ff4 fs4 fc0 sc0 ls18 ws9">\ue728 (\ue7f1<span class="fs5 ls10 v5">\uebde</span><span class="ls1a">)</span><span class="ff2 wsb ve"> </span></div><div class="t m0 x1 h7 y2e ff2 fs4 fc0 sc0 ls18 wsb">Esse <span class="_2 blank"> </span>método <span class="_9 blank"> </span>é <span class="_2 blank"> </span>conhecido <span class="_2 blank"> </span>como <span class="_d blank"> </span><span class="fc1">Método <span class="_2 blank"> </span>da <span class="_9 blank"> </span>Varredura <span class="_2 blank"> </span>em <span class="_2 blank"> </span>Frequência<span class="_0 blank"> </span><span class="ff3 ls1b">.</span></span> <span class="_2 blank"> </span>Embora <span class="_9 blank"> </span>seja <span class="_2 blank"> </span>o </div><div class="t m0 x1 h7 y2f ff2 fs4 fc0 sc0 ls18 wsb">métodos <span class="_0 blank"> </span>mais <span class="_0 blank"> </span>simples, <span class="_0 blank"> </span>também <span class="_0 blank"> </span>não <span class="_0 blank"> </span>fornece <span class="_0 blank"> </span>bons <span class="_0 blank"> </span>resultados, <span class="_0 blank"> </span>tendo <span class="_0 blank"> </span>em <span class="_0 blank"> </span>vista <span class="_0 blank"> </span>que <span class="_0 blank"> </span>os <span class="_0 blank"> </span>a </div><div class="t m0 x1 h7 y30 ff2 fs4 fc0 sc0 ls18 wsb">razão <span class="_e blank"> </span>entre <span class="_e blank"> </span>ruídos <span class="_4 blank"> </span>nos <span class="_d blank"> </span>sinais <span class="_e blank"> </span>de <span class="_4 blank"> </span>en<span class="_1 blank"></span>trada <span class="_e blank"> </span>e <span class="_f blank"> </span>saída <span class="_e blank"> </span>pode <span class="_e blank"> </span>se<span class="_0 blank"> </span>r <span class="_e blank"> </span>amplificado <span class="_e blank"> </span>pela <span class="_f blank"> </span>equação </div><div class="t m0 x1 h7 y31 ff2 fs4 fc0 sc0 ls18 ws4">H(<span class="ff6 wsd">\uf077</span><span class="wsb">). </span></div><div class="t m0 x1 h7 y32 ff2 fs4 fc0 sc0 ls18 wsb">Por <span class="_f blank"> </span>esse <span class="_f blank"> </span>motivo, <span class="_f blank"> </span>é <span class="_f blank"> </span>mais <span class="_f blank"> </span>frequente <span class="_f blank"> </span>fazer <span class="_4 blank"> </span>a <span class="_f blank"> </span>estimativa, <span class="_e blank"> </span>f<span class="_0 blank"> </span>azendo <span class="_f blank"> </span>uso <span class="_f blank"> </span>de <span class="_f blank"> </span>c<span class="_2 blank"> </span>onceitos <span class="_e blank"> </span>de<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y33 ff2 fs4 fc0 sc0 ls18 wsb">processamento de <span class="_0 blank"> </span>sinais <span class="_0 blank"> </span>aleatórios, <span class="_0 blank"> </span>lançando m<span class="_0 blank"> </span>ão de <span class="_0 blank"> </span>conceitos <span class="_0 blank"> </span>básicos <span class="_0 blank"> </span>de <span class="_0 blank"> </span>estatística. </div><div class="t m0 x1 h7 y34 ff2 fs4 fc0 sc0 ls18 wsb">Este recu<span class="_0 blank"> </span>rso é <span class="_0 blank"> </span>conhecido <span class="_0 blank"> </span>como <span class="_0 blank"> </span><span class="fc1">Análise <span class="_0 blank"> </span>Espectral</span>, qu<span class="_0 blank"> </span>e passamos <span class="_0 blank"> </span>a <span class="_0 blank"> </span>nos aprofundarmos </div><div class="t m0 x1 h7 y35 ff2 fs4 fc0 sc0 ls18 wsb">na sequência. </div><div class="t m0 x1 h6 y36 ff3 fs3 fc1 sc0 ls18 wsb">Análise Espectral </div><div class="t m0 x1 h7 y37 ff2 fs4 fc0 sc0 ls18 wsb">O objetivo da <span class="_0 blank"> </span>análise espectral é <span class="_0 blank"> </span>descrever <span class="_0 blank"> </span>a distribuição sobre <span class="_0 blank"> </span>frequência da potência </div><div class="t m0 x1 h7 y38 ff2 fs4 fc0 sc0 ls18 wsb">contida em um sinal, com base em um conjunto finit<span class="_1 blank"></span>o de amostras. </div><div class="t m0 x1 h7 y39 ff2 fs4 fc0 sc0 ls18 wsb">Seu <span class="_6 blank"> </span>uso <span class="_6 blank"> </span>é <span class="_5 blank"> </span>muito <span class="_6 blank"> </span>útil <span class="_6 blank"> </span>em <span class="_5 blank"> </span>análise <span class="_6 blank"> </span>modal, <span class="_6 blank"> </span>vibro<span class="_0 blank"> </span>-acústica, <span class="_6 blank"> </span>identificação <span class="_5 blank"> </span>de <span class="_6 blank"> </span>sistemas, </div><div class="t m0 x1 h7 y3a ff2 fs4 fc0 sc0 ls18 wsb">telecomunicações, processamento de imagens, ent<span class="_1 blank"></span>re outros. </div><div class="t m0 x1 h7 y3b ff2 fs4 fc0 sc0 ls18 wsb">Para ilustrar, <span class="_4 blank"> </span>apresentamos a fi<span class="_1 blank"></span>gura abaixo, <span class="_4 blank"> </span>que representa <span class="_4 blank"> </span>a análise de <span class="_4 blank"> </span>vibrações </div><div class="t m0 x1 h7 y3c ff2 fs4 fc0 sc0 ls18 wsb">aplicadas <span class="_9 blank"> </span>na <span class="_9 blank"> </span><span class="fc1">Manutenção <span class="_d blank"> </span>Preditiva</span>. <span class="_9 blank"> </span>Consiste <span class="_9 blank"> </span>num <span class="_d blank"> </span>gráfico <span class="_9 blank"> </span>Amplitude <span class="_9 blank"> </span>por <span class="_d blank"> </span>Frequência. </div><div class="t m0 x1 h7 y3d ff2 fs4 fc0 sc0 ls18 wsb">Note que temos <span class="_0 blank"> </span>alguns picos preponderantes nesse <span class="_0 blank"> </span>diagrama, cada um corresponde a </div><div class="t m0 x1 h7 y3e ff2 fs4 fc0 sc0 ls18 wsb">elementos <span class="_10 blank"> </span>pertencentes <span class="_10 blank"> </span>ao <span class="_10 blank"> </span>equipamento, <span class="_10 blank"> </span>como <span class="_10 blank"> </span>o <span class="_10 blank"> </span>acoplamento <span class="_10 blank"> </span>(pico <span class="_10 blank"> </span>mais <span class="_10 blank"> </span>a </div><div class="t m0 x1 h7 y3f ff2 fs4 fc0 sc0 ls18 wsb">esquerda) e ao número de dentes da engrenagem grande (<span class="_1 blank"></span>pico mais a direita). </div><div class="t m0 xe h7 y40 ff2 fs4 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y41 ff2 fs4 fc0 sc0 ls18 wsb">Figura 1 \u2013 Exemplo do Uso do Espectro de Frequência </div><div class="t m0 x1 h10 y42 ff2 fs8 fc0 sc0 ls18 wsb">Fonte: <span class="_b blank"> </span><span class="fc3 wse">https://produto.mercadolivre.com<span class="_1 blank"></span>.br/MLB-851379065-vendo-dom<span class="_1 blank"></span>inio-analise-de-vibraco-</span></div><div class="t m0 x1 h10 y43 ff2 fs8 fc3 sc0 ls18 wse">manutenco-preditiva-tpm-_<span class="_1 blank"></span>JM<span class="fc0 wsb"> </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/f80e4ebf-9792-403e-87fc-06408416f1b4/bg3.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">3 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h7 y29 ff2 fs4 fc0 sc0 ls18 wsb">Assumindo <span class="_8 blank"> </span>que <span class="_11 blank"> </span>os <span class="_8 blank"> </span>sinais <span class="_8 blank"> </span>(entrada <span class="_11 blank"> </span>e <span class="_8 blank"> </span>saída) <span class="_8 blank"> </span>de <span class="_11 blank"> </span>um <span class="_8 blank"> </span>sistema <span class="_11 blank"> </span>linear <span class="_8 blank"> </span>qualquer <span class="_11 blank"> </span>são </div><div class="t m0 x1 h11 y2a ff2 fs4 fc0 sc0 ls18 ws4">aleatórios<span class="fs7 ls1c vf">1</span><span class="wsb">, <span class="_12 blank"> </span>não-periódicos<span class="fs7 ls1d vf">2</span> <span class="_12 blank"> </span>e <span class="_12 blank"> </span>não <span class="_12 blank"> </span>transientes<span class="fs7 wsf vf">3</span> <span class="_12 blank"> </span>o <span class="_12 blank"> </span>que <span class="_12 blank"> </span>não <span class="_12 blank"> </span>nos <span class="_12 blank"> </span>permite <span class="_12 blank"> </span>utilizar </span></div><div class="t m0 x1 h7 y44 ff2 fs4 fc0 sc0 ls18 wsb">diretamente as ferramentas de <span class="_6 blank"> </span>análise de Fourier, verificados até agora, <span class="_6 blank"> </span>portanto </div><div class="t m0 x1 h7 y45 ff2 fs4 fc0 sc0 ls18 wsb">teremos <span class="_f blank"> </span>que <span class="_4 blank"> </span>lançar <span class="_4 blank"> </span>mão <span class="_f blank"> </span>de <span class="_4 blank"> </span>outros <span class="_4 blank"> </span>recursos <span class="_f blank"> </span>que <span class="_4 blank"> </span>só <span class="_4 blank"> </span>será <span class="_f blank"> </span>possível <span class="_4 blank"> </span>se <span class="_f blank"> </span>apresentarmos </div><div class="t m0 x1 h7 y46 ff2 fs4 fc0 sc0 ls18 wsb">antes alguns conceitos e definições adicionais. </div><div class="t m0 x1 h6 y47 ff3 fs3 fc1 sc0 ls18 wsb">Processo Estocástico: </div><div class="t m0 x1 h7 y48 ff2 fs4 fc0 sc0 ls18 wsb">Dentro <span class="_3 blank"> </span>da <span class="_3 blank"> </span>teoria <span class="_7 blank"> </span>das <span class="_3 blank"> </span>probabilidades, <span class="_3 blank"> </span>um <span class="_3 blank"> </span>processo <span class="_3 blank"> </span>estocástico <span class="_3 blank"> </span>é <span class="_7 blank"> </span>uma <span class="_3 blank"> </span>família <span class="_3 blank"> </span>de </div><div class="t m0 x1 h7 y49 ff2 fs4 fc0 sc0 ls18 wsb">variáveis aleatórias <span class="_0 blank"> </span>representando <span class="_0 blank"> </span>a evolução <span class="_0 blank"> </span>de <span class="_0 blank"> </span>um sistema <span class="_0 blank"> </span>de <span class="_0 blank"> </span>valores <span class="_0 blank"> </span>com o <span class="_0 blank"> </span>tempo. </div><div class="t m0 x1 h7 y4a ff2 fs4 fc0 sc0 ls18 wsb">É <span class="_5 blank"> </span>a <span class="_5 blank"> </span>contraparte <span class="_5 blank"> </span>probabilística <span class="_5 blank"> </span>de <span class="_5 blank"> </span>um <span class="_5 blank"> </span>processo<span class="_0 blank"> </span> <span class="_5 blank"> </span>determinístico. <span class="_5 blank"> </span>Como <span class="_5 blank"> </span>ilustrado <span class="_5 blank"> </span>na<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y4b ff2 fs4 fc0 sc0 ls18 wsb">figura abaixo. </div><div class="t m0 x1 h7 y4c ff2 fs4 fc0 sc0 ls18 wsb">Graficamente <span class="_0 blank"> </span>podemos <span class="_0 blank"> </span>expressar <span class="_0 blank"> </span>por <span class="_2 blank"> </span>um <span class="_0 blank"> </span>conjunto <span class="_0 blank"> </span>de <span class="_2 blank"> </span>testes <span class="_0 blank"> </span>com <span class="_0 blank"> </span>amostras <span class="_2 blank"> </span>aleatórias </div><div class="t m0 x1 h7 y4d ff2 fs4 fc0 sc0 ls18 ws4">x<span class="fs7 ls12 v10">k</span><span class="wsb">[n] <span class="_0 blank"> </span>com <span class="_2 blank"> </span>k <span class="_2 blank"> </span>= <span class="_0 blank"> </span>1, <span class="_2 blank"> </span>2, <span class="_2 blank"> </span>\u2026, <span class="_0 blank"> </span>K <span class="_2 blank"> </span>realizações <span class="_0 blank"> </span>e <span class="_2 blank"> </span>n <span class="_0 blank"> </span>=<span class="_0 blank"> </span> <span class="_0 blank"> </span>1, <span class="_2 blank"> </span>2, <span class="_2 blank"> </span>\u2026, <span class="_0 blank"> </span>N <span class="_2 blank"> </span>pontos <span class="_0 blank"> </span>cada, <span class="_2 blank"> </span>de <span class="_2 blank"> </span>tal <span class="_0 blank"> </span>forma <span class="_2 blank"> </span>que <span class="_0 blank"> </span>só <span class="_2 blank"> </span>é </span></div><div class="t m0 x1 h7 y4e ff2 fs4 fc0 sc0 ls18 wsb">possível fazer analises das características médias deste processo. </div><div class="t m0 x1 h7 y4f ff2 fs4 fc0 sc0 ls18 wsb"> </div><div class="t m0 xf h7 y50 ff2 fs4 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y51 ff2 fs4 fc0 sc0 ls18 wsb">Figura 2 \u2013 Ilustração de um Processo Estocástico </div><div class="t m0 x1 h10 y52 ff2 fs8 fc0 sc0 ls18 wsb">Fonte: <span class="fc3 ws10">https://pt.wikipedia.o<span class="_1 blank"></span>rg/wiki/Processo_estoc%C3<span class="_1 blank"></span>%A1stico<span class="fc0 wsb"> </span></span></div><div class="t m0 x1 h6 y53 ff3 fs3 fc1 sc0 ls18 wsb">Momentos Estatísticos: </div><div class="t m0 x1 h7 y54 ff2 fs4 fc0 sc0 ls18 wsb">São <span class="_4 blank"> </span>métricas <span class="_f blank"> </span>utilizadas <span class="_4 blank"> </span>para <span class="_4 blank"> </span>descrever <span class="_f blank"> </span>as <span class="_4 blank"> </span>características <span class="_4 blank"> </span>de <span class="_4 blank"> </span>processos <span class="_f blank"> </span>estocásticos. </div><div class="t m0 x1 h7 y55 ff2 fs4 fc0 sc0 ls18 wsb">Por <span class="_d blank"> </span>exemplo, <span class="_e blank"> </span>o <span class="_e blank"> </span>valor <span class="_e blank"> </span>médio <span class="_d blank"> </span>de <span class="_e blank"> </span>um <span class="_e blank"> </span>sinal <span class="_d blank"> </span>x[<span class="_0 blank"> </span>n] <span class="_d blank"> </span>é<span class="_0 blank"> </span> <span class="_d blank"> </span>chamado <span class="_e blank"> </span>de <span class="_e blank"> </span>momento <span class="_d blank"> </span>de <span class="_e blank"> </span>1ª <span class="_e blank"> </span>ordem, </div><div class="t m0 x1 h7 y56 ff2 fs4 fc0 sc0 ls18 wsb">que pode ser calculado por: </div><div class="t m0 x10 h12 y57 ff4 fs0 fc0 sc0 ls1e">\ue749<span class="ls18 ws11 v1">(</span><span class="ls1f">\ue747<span class="ls20 v1">)</span><span class="ls18 ws12">= lim</span></span></div><div class="t m0 x9 h13 y58 ff4 fs9 fc0 sc0 ls18 ws13">\uebde\u2192<span class="ff5">\u221e</span></div><div class="t m0 x11 h14 y59 ff4 fs0 fc0 sc0 ls18">1</div><div class="t m0 x11 h15 y5a ff4 fs0 fc0 sc0 ls21">\ue747<span class="ls18 ws14 vb">\uedcd \ue754</span><span class="fs9 ls22 vf">\uebde</span><span class="ls18 ws11 vb">[\ue74a]</span></div><div class="t m0 x12 h13 y5b ff4 fs9 fc0 sc0 ls18">\uebc7</div><div class="t m0 x13 h13 y5c ff4 fs9 fc0 sc0 ls18 ws13">\uebde\ueb40\ueb35</div><div class="t m0 x14 h16 y57 ff2 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y5d ff2 fs4 fc0 sc0 ls18 wsb">Existem <span class="_6 blank"> </span>vár<span class="_0 blank"> </span>ios <span class="_6 blank"> </span>tipos <span class="_5 blank"> </span>de <span class="_5 blank"> </span>momentos <span class="_5 blank"> </span>estatí<span class="_0 blank"> </span>sticos, <span class="_5 blank"> </span>mas <span class="_5 blank"> </span>iremos <span class="_5 blank"> </span>citar <span class="_6 blank"> </span>apenas <span class="_5 blank"> </span>os <span class="_5 blank"> </span>mais </div><div class="t m0 x1 h7 y5e ff2 fs4 fc0 sc0 ls18 wsb">importantes: </div><div class="t m0 x1 h2 y5f ff1 fs0 fc0 sc0 ls18 wsb"> <span class="_0 blank"> </span> <span class="_13 blank"></span> </div><div class="t m0 x1 h17 y60 ff7 fs8 fc0 sc0 ls18 wsb">1 Quando pensam<span class="_1 blank"></span>os em um sistema aleat<span class="_1 blank"></span>ório significa que seus estado<span class="_1 blank"></span>s futuros não podem<span class="_1 blank"></span> ser previstos. </div><div class="t m0 x1 h17 y61 ff7 fs8 fc0 sc0 ls18 wsb">2 Não-periódicos significa <span class="_1 blank"></span>que não se repetem<span class="_1 blank"></span> após um período de<span class="_1 blank"></span> tempo. </div><div class="t m0 x1 h18 y62 ff7 fs8 fc0 sc0 ls18 wsb">3 Transiente significa que<span class="_1 blank"></span> acontece num inter<span class="_1 blank"></span>valo de tempo m<span class="_1 blank"></span>uito curto.<span class="ff1 fs9 fc4 v11"> </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/f80e4ebf-9792-403e-87fc-06408416f1b4/bg4.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">4 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h7 y29 ff2 fs4 fc1 sc0 ls18 wsb">Funções de Autocorrelação (FAC) R<span class="fs7 wsf v10">xx</span>(n,m): </div><div class="t m0 x15 h12 y63 ff4 fs0 fc0 sc0 ls18 ws11">\ue734<span class="fs9 ws15 v12">\uebeb\uebeb </span><span class="v1">(</span><span class="ws16">\ue74a,<span class="_e blank"> </span>\ue749 <span class="ls23 v1">)</span><span class="ws12">= lim</span></span></div><div class="t m0 x16 h13 y64 ff4 fs9 fc0 sc0 ls18 ws13">\uebde\u2192<span class="ff5">\u221e</span></div><div class="t m0 x17 h14 y65 ff4 fs0 fc0 sc0 ls18">1</div><div class="t m0 x17 h19 y66 ff4 fs0 fc0 sc0 ls24">\ue72d<span class="ls18 ws14 vb">\uedcd \ue754</span><span class="fs9 ls22 vf">\uebde</span><span class="ls25 vd">[</span><span class="ls26 vb">\ue74a</span><span class="ls25 vd">]</span><span class="ls18 ws17 vb">. \ue754</span><span class="fs9 ls22 vf">\uebde</span><span class="ls25 vd">[</span><span class="ls18 ws16 vb">\ue74a<span class="_4 blank"> </span>+<span class="_4 blank"> </span>\ue749 <span class="v1">]</span></span></div><div class="t m0 x18 h13 y67 ff4 fs9 fc0 sc0 ls18">\uebc7</div><div class="t m0 xe h13 y68 ff4 fs9 fc0 sc0 ls18 ws18">\uebde\ueb40\ueb35</div><div class="t m0 x19 h16 y63 ff2 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y69 ff2 fs4 fc1 sc0 ls18 wsb">Funções de Correlações Cruzadas (FCC) R<span class="fs7 wsf v10">Fx</span> (n,m): </div><div class="t m0 x15 h12 y6a ff4 fs0 fc0 sc0 ls18 ws11">\ue734<span class="fs9 ws19 v12">\uebbf\uebeb </span><span class="v1">(</span><span class="ws16">\ue74a,<span class="_e blank"> </span>\ue749 <span class="ls20 v1">)</span><span class="ws1a">= lim</span></span></div><div class="t m0 x16 h13 y6b ff4 fs9 fc0 sc0 ls18 ws13">\uebde\u2192<span class="ff5">\u221e</span></div><div class="t m0 x17 h14 y6c ff4 fs0 fc0 sc0 ls18">1</div><div class="t m0 x17 h15 y6d ff4 fs0 fc0 sc0 ls27">\ue72d<span class="ls18 ws14 vb">\uedcd \ue728</span></div><div class="t m0 x1a h1a y6e ff4 fs9 fc0 sc0 ls28">\uebde<span class="fs0 ls25 v13">[<span class="ls26 v2">\ue74a</span><span class="ls18 ws11">]<span class="ws17 v2">. \ue754</span></span></span><span class="ls22">\uebde<span class="fs0 ls25 v13">[<span class="ls18 ws16 v2">\ue74a<span class="_4 blank"> </span>+<span class="_4 blank"> </span>\ue749 <span class="v1">]</span></span></span></span></div><div class="t m0 x18 h13 y6f ff4 fs9 fc0 sc0 ls18">\uebc7</div><div class="t m0 xe h13 y70 ff4 fs9 fc0 sc0 ls18 ws13">\uebde\ueb40\ueb35</div><div class="t m0 x19 h16 y6a ff3 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y71 ff2 fs4 fc0 sc0 ls18 wsb">Onde m <span class="_6 blank"> </span>é <span class="_6 blank"> </span>o número <span class="_6 blank"> </span>de atrasos <span class="_6 blank"> </span>temporais. <span class="_6 blank"> </span>Importante salientar <span class="_6 blank"> </span>que <span class="_6 blank"> </span>a função <span class="_6 blank"> </span>de </div><div class="t m0 x1 h7 y72 ff2 fs4 fc0 sc0 ls18 wsb">autocorrelação <span class="_9 blank"> </span><span class="fc1 ws4">FAC<span class="_0 blank"> </span></span> <span class="_9 blank"> </span>é <span class="_d blank"> </span>a <span class="_9 blank"> </span>média <span class="_d blank"> </span>do <span class="_9 blank"> </span>produto <span class="_d blank"> </span>x<span class="fs7 ls12 v10">k</span> <span class="_d blank"> </span>[n] <span class="_d blank"> </span>e <span class="_9 blank"> </span>x<span class="fs7 ls12 v10">k</span> <span class="_d blank"> </span>[n+m] <span class="_9 blank"> </span>e<span class="_0 blank"> </span> <span class="_9 blank"> </span>a <span class="_d blank"> </span>função <span class="_9 blank"> </span>de <span class="_d blank"> </span>correlação </div><div class="t m0 x1 h7 y73 ff2 fs4 fc0 sc0 ls18 wsb">cruzada <span class="fc1 ws4">FCC<span class="ff3 wsb"> </span></span>é a média do produto entre duas sequências diferent<span class="_1 blank"></span>es F<span class="_0 blank"> </span><span class="fs7 ls12 v10">k</span>[n] e x<span class="fs7 ls12 v10">k</span>[n+m] </div><div class="t m0 x1 h6 y74 ff3 fs3 fc1 sc0 ls18 wsb">Processo Estacionário: </div><div class="t m0 x1 h7 y75 ff2 fs4 fc0 sc0 ls18 wsb">São aqueles cujas <span class="_0 blank"> </span>propriedades estatísticas não <span class="_0 blank"> </span>variam com <span class="_0 blank"> </span>o tempo. Se <span class="_0 blank"> </span>dividir o <span class="_0 blank"> </span>sinal </div><div class="t m0 x1 h7 y76 ff2 fs4 fc0 sc0 ls18 wsb">em <span class="_5 blank"> </span>várias <span class="_3 blank"> </span>partes <span class="_3 blank"> </span>e <span class="_5 blank"> </span>se <span class="_3 blank"> </span>calcular <span class="_5 blank"> </span>a <span class="_3 blank"> </span>distribuição <span class="_5 blank"> </span>de <span class="_3 blank"> </span>probabilidade <span class="_5 blank"> </span>co<span class="_0 blank"> </span>nstata<span class="_0 blank"> </span>-se <span class="_3 blank"> </span>que <span class="_5 blank"> </span>a </div><div class="t m0 x1 h7 y77 ff2 fs4 fc0 sc0 ls18 wsb">distribuição estatística é a mesma. Como ilustrado na figura abaixo. </div><div class="t m0 x1 h7 y78 ff2 fs4 fc0 sc0 ls18 wsb">Figura <span class="_2 blank"> </span>3 <span class="_2 blank"> </span>\u2013 <span class="_9 blank"> </span>Ilustração <span class="_2 blank"> </span>de <span class="_2 blank"> </span>um <span class="_2 blank"> </span>Pro<span class="_0 blank"> </span>cesso <span class="_2 blank"> </span>Estacionário <span class="_2 blank"> </span>(Esquerda) <span class="_2 blank"> </span>e <span class="_2 blank"> </span>Distribuição <span class="_9 blank"> </span>de <span class="_2 blank"> </span>partes </div><div class="t m0 x1 h7 y79 ff2 fs4 fc0 sc0 ls18 wsb">(direita) </div><div class="t m0 x1 h7 y7a ff2 fs4 fc0 sc0 ls18 wsb">Fonte: Notas de Aulas de Vibrações Mecânicas do Prof. Dr. Samuel da Si<span class="_1 blank"></span>lva </div><div class="t m0 x1 h6 y7b ff3 fs3 fc1 sc0 ls18 wsb">Processo Ergódico: </div><div class="t m0 x1 h7 y7c ff2 fs4 fc0 sc0 ls18 wsb">São <span class="_0 blank"> </span>aq<span class="_0 blank"> </span>ueles <span class="_0 blank"> </span>cujas <span class="_2 blank"> </span>propriedades <span class="_2 blank"> </span>médias <span class="_0 blank"> </span>calculadas <span class="_2 blank"> </span>no <span class="_2 blank"> </span>tempo <span class="_2 blank"> </span>para <span class="_0 blank"> </span>qualquer <span class="_9 blank"> </span>realização </div><div class="t m0 x1 h7 y7d ff2 fs4 fc0 sc0 ls18 wsb">são iguais as propriedades calculadas a partir das médias do conjunto. </div><div class="t m0 x1 h7 y7e ff2 fs4 fc0 sc0 ls18 wsb">Assim, <span class="_2 blank"> </span>as <span class="_0 blank"> </span>funções <span class="_2 blank"> </span>de <span class="_2 blank"> </span>autocorrelação <span class="_2 blank"> </span><span class="fc1 ws4">FAC</span><span class="ff3 fc2 ls29"> </span>e <span class="_2 blank"> </span>as <span class="_2 blank"> </span>funções <span class="_2 blank"> </span>de <span class="_2 blank"> </span>correlações <span class="_2 blank"> </span>cruzada<span class="ff3 fc2 ls29"> </span><span class="fc1 ws4">FCC</span> <span class="_2 blank"> </span>de </div><div class="t m0 x1 h7 y7f ff2 fs4 fc0 sc0 ls18 wsb">processos estacionários e ergódicos se tornam dependentes apenas do atraso m. </div><div class="t m0 x1 h7 y80 ff2 fs4 fc0 sc0 ls18 wsb">Assim R<span class="fs7 wsf v10">xx</span> (m,n) = R<span class="fs7 wsf v10">xx</span> (m) e R<span class="fs7 wsf v10">Fx</span> (m,n) = R<span class="fs7 wsf v10">Fx</span> (m). </div><div class="t m0 x1 h7 y81 ff2 fs4 fc0 sc0 ls18 wsb">Existem vári<span class="_1 blank"></span>os métodos <span class="_4 blank"> </span>temporais para <span class="_4 blank"> </span>se es<span class="_1 blank"></span>timar as <span class="_4 blank"> </span>correlações <span class="_4 blank"> </span>temporais p<span class="_1 blank"></span>ara </div><div class="t m0 x1 h7 y82 ff2 fs4 fc0 sc0 ls18 wsb">estimar as correlações, sendo o método de Levinson<span class="_0 blank"> </span>-Durbin, um dos mais conhecidos, </div><div class="t m0 x1 h7 y83 ff2 fs4 fc0 sc0 ls18 wsb">tendo algoritmos prontos, em software como Mathlab ou Scilab. </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/f80e4ebf-9792-403e-87fc-06408416f1b4/bg5.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">5 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h6 y84 ff3 fs3 fc1 sc0 ls18 wsb">Espectro de Potências </div><div class="t m0 x1 h7 y85 ff2 fs4 fc0 sc0 ls18 wsb">Uma <span class="_3 blank"> </span>vez <span class="_5 blank"> </span>apresentadas <span class="_3 blank"> </span>as <span class="_3 blank"> </span>definições <span class="_5 blank"> </span>básicas, <span class="_3 blank"> </span>é <span class="_3 blank"> </span>possível <span class="_5 blank"> </span>descrever <span class="_3 blank"> </span>o <span class="_3 blank"> </span>espectro <span class="_5 blank"> </span>de </div><div class="t m0 x1 h7 y86 ff2 fs4 fc0 sc0 ls18 wsb">potências <span class="_2 blank"> </span>de <span class="_9 blank"> </span>um <span class="_2 blank"> </span>sinal <span class="_2 blank"> </span>aleatório <span class="_2 blank"> </span>e <span class="_9 blank"> </span>discreto <span class="_2 blank"> </span>x[n], <span class="_2 blank"> </span>descrito <span class="_9 blank"> </span>por <span class="_2 blank"> </span>um <span class="_2 blank"> </span>processo <span class="_2 blank"> </span>estocástico,<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 y87 ff2 fs4 fc0 sc0 ls18 wsb">estacionário e ergódico. </div><div class="t m0 x1 h7 y88 ff2 fs4 fc0 sc0 ls18 wsb">Por <span class="_e blank"> </span>meio <span class="_e blank"> </span>da <span class="_e blank"> </span>transformada <span class="_e blank"> </span>de <span class="_e blank"> </span>Fourier <span class="_e blank"> </span>da <span class="_e blank"> </span>função <span class="_f blank"> </span>de <span class="_e blank"> </span>autocorrelação <span class="_e blank"> </span><span class="fc1">FAC <span class="_e blank"> </span>Rxx <span class="_e blank"> </span></span>(m) <span class="_e blank"> </span>em </div><div class="t m0 x1 h7 y89 ff2 fs4 fc0 sc0 ls18 wsb">função de frequência, <span class="ff6 wsd">\uf077</span>, vem: </div><div class="t m0 x1b h14 y8a ff4 fs0 fc0 sc0 ls18">\ue735</div><div class="t m0 x1c h1a y8b ff4 fs9 fc0 sc0 ls18 ws1b">\uebeb\uebeb <span class="fs0 ws11 v13">(<span class="ls2a v2">\ue7f1</span><span class="ls2b">)<span class="ls18 ws1c v2">=<span class="_14 blank"> </span>\uedcd \ue734</span></span></span><span class="ws1d">\uebeb\uebeb <span class="fs0 ls25 v13">[<span class="ls2c v2">\ue749</span>]<span class="ls18 ws1e v2">.<span class="_d blank"> </span>\ue741 </span></span><span class="ws1f v4">\ueb3f\uebdd .\uec20 .\uebe0</span></span></div><div class="t m0 x9 h1b y8c ff5 fs9 fc0 sc0 ls18">\u221e</div><div class="t m0 x1d h13 y8d ff4 fs9 fc0 sc0 ls2d ws20">\uebe0\ueb40\ueb3f<span class="ff5 ls18">\u221e</span></div><div class="t m0 x1e h16 y8a ff2 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h1c y8e ff2 fs4 fc0 sc0 ls18 wsb">Sendo <span class="ff4 ws21">\ue7f1<span class="_3 blank"> </span>=<span class="_3 blank"> </span>2<span class="_0 blank"> </span>. \ue7e8<span class="_3 blank"> </span><span class="fs5 va">\uebd9</span></span></div><div class="t m0 x1f h9 y8f ff4 fs5 fc0 sc0 ls18">\uebbf</div><div class="t m0 x20 h1d y90 ff4 fs6 fc0 sc0 ls2e">\uecde<span class="ff2 fs4 ls18 wsb ve"> , temos: </span></div><div class="t m0 x1 h7 y91 ff2 fs4 fc0 sc0 ls18 wsb"> </div><div class="t m0 x21 h1e y92 ff4 fs4 fc0 sc0 ls18">\ue735</div><div class="t m0 x22 h1f y93 ff4 fs5 fc0 sc0 ls18 ws22">\uebeb\uebeb <span class="fs4 ws23 v13">(\ue742 )<span class="_3 blank"> </span>=<span class="_15 blank"> </span>\uedcd<span class="_16 blank"> </span>\ue734</span>\uebeb\uebeb <span class="fs4 ws0 v14">[<span class="ls2f v2">\ue749</span>]<span class="ws7 v2">.<span class="_f blank"> </span>\ue741 </span></span><span class="ws24 v15">\ueb3f\ueb36.\uec17 .\uebdd .\uebd9</span></div><div class="t m0 x23 h9 y94 ff4 fs5 fc0 sc0 ls18">\uebbf</div><div class="t m0 x1e h20 y95 ff4 fs6 fc0 sc0 ls30">\uecde<span class="fs5 ls18 ws2 vb">.\uebe0</span></div><div class="t m0 x16 h9 y96 ff4 fs5 fc0 sc0 ls18">\uebb6</div><div class="t m0 x24 h9 y97 ff4 fs5 fc0 sc0 ls18 ws2">\uebe0\ueb40\ueb3f\uebb6</div><div class="t m0 x25 h7 y92 ff2 fs4 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 y98 ff2 fs4 fc0 sc0 ls18 wsb">Onde F<span class="fs7 ls31 v10">s</span> é taxa de amostragem e f é o vetor de frequências, a<span class="_1 blank"></span>mbos em Hz (Hertz) </div><div class="t m0 x1 h16 y99 ff2 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h6 y9a ff3 fs3 fc1 sc0 ls18 wsb">Densidade Espectral (PSD) </div><div class="t m0 x1 h7 y9b ff2 fs4 fc0 sc0 ls18 wsb">A <span class="_7 blank"> </span>part<span class="_1 blank"></span>ir <span class="_7 blank"> </span>do <span class="_3 blank"> </span>espectro <span class="_7 blank"> </span>de <span class="_3 blank"> </span>potências <span class="_7 blank"> </span>é <span class="_3 blank"> </span>possível <span class="_7 blank"> </span>escrever <span class="_3 blank"> </span>a <span class="_7 blank"> </span>densidade <span class="_3 blank"> </span>espectral <span class="_7 blank"> </span>de </div><div class="t m0 x1 h7 y9c ff2 fs4 fc0 sc0 ls18 wsb">potência (PSD) P<span class="fs7 wsf v10">xx</span>(f) do sinal <span class="ff5 ws4">x[n].</span> </div><div class="t m0 x26 h1e y9d ff4 fs4 fc0 sc0 ls18">\ue732</div><div class="t m0 x27 h21 y9e ff4 fs5 fc0 sc0 ls18 ws22">\uebeb\uebeb <span class="fs4 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="lsb">)<span class="ls18 wsb v2">=<span class="_3 blank"> </span> <span class="_4 blank"> </span><span class="vc">\ue735</span></span></span></span></div><div class="t m0 x28 h22 y9f ff4 fs5 fc0 sc0 ls18 ws22">\uebeb\uebeb <span class="fs4 ws23 v13">(\ue742 )</span></div><div class="t m0 x29 h1e ya0 ff4 fs4 fc0 sc0 ls18">\ue728</div><div class="t m0 x2a h23 ya1 ff4 fs5 fc0 sc0 ls33">\uebe6<span class="ff2 fs4 ls18 wsb v16"> </span></div><div class="t m0 x1 h7 ya2 ff2 fs4 fc0 sc0 ls18 wsb">Em <span class="_9 blank"> </span>que <span class="_9 blank"> </span>PSD <span class="_2 blank"> </span>é <span class="_9 blank"> </span>a <span class="_d blank"> </span>potência <span class="_9 blank"> </span>contida <span class="_2 blank"> </span>num <span class="_9 blank"> </span>sinal <span class="_9 blank"> </span>numa <span class="_9 blank"> </span>banda <span class="_9 blank"> </span>de <span class="_9 blank"> </span>frequência <span class="_2 blank"> </span>infinitesimal, </div><div class="t m0 x1 h7 ya3 ff2 fs4 fc0 sc0 ls18 wsb">donde surge a <span class="_0 blank"> </span>definição de densidade. <span class="_0 blank"> </span>Sua unidade é <span class="_0 blank"> </span>potência do sinal <span class="_0 blank"> </span>por unidade <span class="_0 blank"> </span>de </div><div class="t m0 x1 h7 ya4 ff2 fs4 fc0 sc0 ls18 wsb">frequência. </div><div class="t m0 x1 h7 ya5 ff2 fs4 fc0 sc0 ls18 wsb">Na <span class="_10 blank"> </span>prática, <span class="_10 blank"> </span>obt<span class="_1 blank"></span>er <span class="_10 blank"> </span>a <span class="_10 blank"> </span>PS<span class="_1 blank"></span>D <span class="_10 blank"> </span>a <span class="_10 blank"> </span>p<span class="_1 blank"></span>artir <span class="_10 blank"> </span>de <span class="_10 blank"> </span>uma <span class="_17 blank"> </span>FAC <span class="_17 blank"> </span>não <span class="_10 blank"> </span>é <span class="_10 blank"> </span>comum, <span class="_17 blank"> </span>usa<span class="_0 blank"> </span>-se <span class="_10 blank"> </span>mais </div><div class="t m0 x1 h7 ya6 ff2 fs4 fc0 sc0 ls18 wsb">frequentemente <span class="_18 blank"> </span>métodos <span class="_18 blank"> </span>não-paramétricos <span class="_18 blank"> </span>como<span class="_0 blank"> </span> <span class="_18 blank"> </span>o <span class="_18 blank"> </span>Periodograma, <span class="_18 blank"> </span>Welch, </div><div class="t m0 x1 h7 ya7 ff2 fs4 fc0 sc0 ls18 wsb">Correlgorama, <span class="_14 blank"> </span>entre <span class="_14 blank"> </span>outros, <span class="_14 blank"> </span>métodos <span class="_14 blank"> </span>paramétricos<span class="_1 blank"></span> <span class="_14 blank"> </span>como <span class="_14 blank"> </span>os <span class="_14 blank"> </span>Modelos <span class="_14 blank"> </span>auto<span class="_0 blank"> </span>-</div><div class="t m0 x1 h7 ya8 ff2 fs4 fc0 sc0 ls18 wsb">regressivos, Equações de Yule-Walker, entre outros, ou métodos de sub<span class="_1 blank"></span>espaço. </div><div class="t m0 x1 h7 ya9 ff2 fs4 fc0 sc0 ls18 wsb">O <span class="_11 blank"> </span>estimador <span class="_19 blank"> </span>espectral <span class="_11 blank"> </span>não-paramétrico <span class="_19 blank"> </span>mais <span class="_11 blank"> </span>simples <span class="_19 blank"> </span>e <span class="_11 blank"> </span>utilizado <span class="_19 blank"> </span>é <span class="_19 blank"> </span>o <span class="_11 blank"> </span>chamado </div><div class="t m0 x1 h7 yaa ff2 fs4 fc1 sc0 ls18 ws4">Periodograma<span class="fc0 wsb">, pensando em uma função de autocorrel<span class="_1 blank"></span>ação, temos: </span></div><div class="t m0 x10 h1e yab ff4 fs4 fc0 sc0 ls18">\ue732</div><div class="t m0 x26 h1e yac ff4 fs4 fc0 sc0 ls18">\uede0</div><div class="t m0 xc h24 yad ff4 fs5 fc0 sc0 ls18 ws22">\uebeb\uebeb <span class="fs4 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="lsb">)<span class="ls18 wsb v2">=<span class="_3 blank"> </span> <span class="_4 blank"> </span></span></span><span class="vc">|<span class="ls0 v2">\ue73a</span>(<span class="ls32 v2">\ue742</span>)|</span></span></div><div class="t m0 x28 h1e yae ff4 fs4 fc0 sc0 ls18">\ue728</div><div class="t m0 x2b h22 yaf ff4 fs5 fc0 sc0 ls34">\uebe6<span class="fs4 ls18 v13">\ue72e</span></div><div class="t m0 x2c h9 yb0 ff4 fs5 fc0 sc0 ls35">\ueb36<span class="ff2 fs4 ls18 wsb v17"> </span></div><div class="t m0 x1 h8 yb1 ff2 fs4 fc0 sc0 ls18 wsb">Sendo <span class="ff4 ls0">\ue73a<span class="ls18 ws0 v1">(</span><span class="ls32">\ue742<span class="ls18 ws0 v1">)</span></span></span> é a transformada discreta de Fourier do sinal aleatório x[n] com L pontos </div><div class="t m0 x1 h7 yb2 ff2 fs4 fc0 sc0 ls18 wsb">Se pensarmos numa de PSD cruzada entre dois sinais x[n] e y[n], temos: </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f80e4ebf-9792-403e-87fc-06408416f1b4/bg6.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">6 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1c h1e yb3 ff4 fs4 fc0 sc0 ls18">\ue732</div><div class="t m0 x2d h1e yb4 ff4 fs4 fc0 sc0 ls18">\uede0</div><div class="t m0 x2e h24 yb5 ff4 fs5 fc0 sc0 ls18 ws25">\uebeb\uebec <span class="fs4 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="lsb">)<span class="ls18 wsb v2">=<span class="_3 blank"> </span> <span class="_4 blank"> </span></span><span class="ls36 v18">\ue73a</span></span><span class="vc">(<span class="ls32 v2">\ue742</span>)<span class="ws26 v2">.<span class="_4 blank"> </span>\ue73b </span></span></span><span class="ls37 v19">\u2217</span><span class="fs4 ws0 v1a">(<span class="ls32 v2">\ue742</span>)</span></div><div class="t m0 x2f h1e yb6 ff4 fs4 fc0 sc0 ls18">\ue728</div><div class="t m0 x30 h23 yb7 ff4 fs5 fc0 sc0 ls34">\uebe6<span class="fs4 ls38 v13">\ue72e<span class="ff2 ls18 wsb ve"> </span></span></div><div class="t m0 x1 h1e yb8 ff2 fs4 fc0 sc0 ls18 wsb">Onde <span class="ff4 ls39">\ue73b<span class="fs5 ls37 v3">\u2217</span><span class="ls18 ws23">(\ue742)</span></span><span class="ff5"> </span>é o <span class="_4 blank"> </span>complexo co<span class="_1 blank"></span>njugado <span class="_4 blank"> </span>da transformada <span class="_4 blank"> </span>discreta de <span class="_4 blank"> </span>Fourier <span class="_4 blank"> </span>do sinal </div><div class="t m0 x1 h7 yb9 ff5 fs4 fc0 sc0 ls18 ws4">y[n]<span class="ff2 wsb">. </span></div><div class="t m0 x1 h7 yba ff2 fs4 fc0 sc0 ls18 wsb">Devido a <span class="_2 blank"> </span>problemas relacionados <span class="_0 blank"> </span>à <span class="_0 blank"> </span>resolução, <span class="_0 blank"> </span>polarização <span class="_0 blank"> </span>e <span class="_0 blank"> </span>variância, a <span class="_2 blank"> </span>relação acima </div><div class="t m0 x1 h7 ybb ff2 fs4 fc0 sc0 ls18 wsb">costuma <span class="_0 blank"> </span>produzir <span class="_2 blank"> </span>estimativas <span class="_0 blank"> </span>mais <span class="_2 blank"> </span>pobres, <span class="_2 blank"> </span>no <span class="_0 blank"> </span>entanto, <span class="_2 blank"> </span>pod<span class="_0 blank"> </span>em <span class="_2 blank"> </span>ser <span class="_0 blank"> </span>melhoradas <span class="_2 blank"> </span>com <span class="_0 blank"> </span>o<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 ybc ff2 fs4 fc0 sc0 ls18 wsb">uso <span class="_2 blank"> </span>de <span class="_2 blank"> </span>janelas, <span class="_9 blank"> </span>que <span class="_2 blank"> </span>dá <span class="_2 blank"> </span>origem <span class="_2 blank"> </span>ao <span class="_9 blank"> </span><span class="fc1">Periodograma <span class="_2 blank"> </span>Ponderado</span> <span class="_9 blank"> </span>e/ou <span class="_2 blank"> </span>utilização <span class="_2 blank"> </span>de <span class="_2 blank"> </span>divisão </div><div class="t m0 x1 h7 ybd ff2 fs4 fc0 sc0 ls18 wsb">de <span class="_2 blank"> </span>segmentos <span class="_9 blank"> </span>o <span class="_2 blank"> </span>que <span class="_9 blank"> </span>dá <span class="_2 blank"> </span>origem <span class="_9 blank"> </span>ao <span class="_9 blank"> </span><span class="fc1">Periodograma <span class="_2 blank"> </span>de <span class="_9 blank"> </span>Welch</span>. <span class="_9 blank"> </span>Mais <span class="_2 blank"> </span>detalhes <span class="_9 blank"> </span>consulte <span class="_2 blank"> </span>o </div><div class="t m0 x1 h7 ybe ff2 fs4 fc0 sc0 ls18 wsb">Livro <span class="fc1">Introdution to Spectral Anlysis</span> de P. Stoica e R. Moses da Editora Prentice Hall. </div><div class="t m0 x1 h7 ybf ff2 fs4 fc0 sc0 ls18 wsb">Uma <span class="_2 blank"> </span>outra <span class="_2 blank"> </span>aplicação <span class="_9 blank"> </span>comum <span class="_2 blank"> </span>de <span class="_2 blank"> </span>PSD <span class="_9 blank"> </span>é <span class="_2 blank"> </span>estimar <span class="_2 blank"> </span>de <span class="_9 blank"> </span>forma <span class="_2 blank"> </span>não <span class="_9 blank"> </span>paramétrica <span class="_2 blank"> </span>funções <span class="_2 blank"> </span>de </div><div class="t m0 x1 h7 yc0 ff2 fs4 fc0 sc0 ls18 wsb">transferência <span class="_f blank"> </span>de <span class="_4 blank"> </span>sistemas <span class="_4 blank"> </span>lineares <span class="_4 blank"> </span>e <span class="_f blank"> </span>invariantes <span class="_4 blank"> </span>com <span class="_4 blank"> </span>o <span class="_4 blank"> </span>tempo <span class="_f blank"> </span>a <span class="_4 blank"> </span>partir <span class="_4 blank"> </span>de <span class="_4 blank"> </span>dados <span class="_f blank"> </span>de<span class="_0 blank"> </span> </div><div class="t m0 x1 h7 yc1 ff2 fs4 fc0 sc0 ls18 wsb">entrada <span class="_9 blank"> </span>e <span class="_9 blank"> </span>saída <span class="_9 blank"> </span>obtidos <span class="_9 blank"> </span>de <span class="_9 blank"> </span>testes <span class="_9 blank"> </span>experimentais. <span class="_9 blank"> </span>Em <span class="_9 blank"> </span>outras <span class="_9 blank"> </span>palavras, <span class="_9 blank"> </span>conhecendo <span class="_9 blank"> </span>os </div><div class="t m0 x1 h7 yc2 ff2 fs4 fc0 sc0 ls18 wsb">sinais de excitação <span class="ff5">x[n] = F[n]</span> e de resposta <span class="ff5 ws4">y[n]</span> qual o sistema <span class="ff5 ws4">h[n]</span>? </div><div class="t m0 x1 h7 yc3 ff2 fs4 fc0 sc0 ls18 wsb">Pode-se <span class="_d blank"> </span>mostrar <span class="_e blank"> </span>que <span class="_e blank"> </span>a <span class="_d blank"> </span>função <span class="_e blank"> </span>de <span class="_e blank"> </span>correlação <span class="_d blank"> </span>cruzada <span class="_e blank"> </span>(FCC) <span class="_f blank"> </span><span class="ff5 ws4">R<span class="fs7 wsf v10">Fx</span><span class="wsb">[ <span class="_e blank"> </span>i <span class="_d blank"> </span>]</span></span> <span class="_e blank"> </span>entre <span class="_e blank"> </span>a <span class="_e blank"> </span>excitação </div><div class="t m0 x1 h7 yc4 ff5 fs4 fc0 sc0 ls18 ws4">F[n]<span class="ff2 wsb"> e <span class="_6 blank"> </span>a resposta </span>x[n]<span class="ff2 wsb"> <span class="_6 blank"> </span>é igual a <span class="_6 blank"> </span>convolução discreta entre a <span class="_6 blank"> </span>função de resposta a<span class="_0 blank"> </span> </span></div><div class="t m0 x1 h7 yc5 ff2 fs4 fc0 sc0 ls18 wsb">frequência (IRF) discreta <span class="_0 blank"> </span><span class="ff5 ws4">h[n]</span> <span class="_0 blank"> </span>e <span class="_0 blank"> </span>a função <span class="_0 blank"> </span>de autocorrelação (FAC) <span class="_0 blank"> </span>R<span class="_0 blank"> </span><span class="fs7 wsf v10">FF</span>[ i <span class="_0 blank"> </span>], c<span class="_0 blank"> </span>uja relação é </div><div class="t m0 x1 h7 yc6 ff2 fs4 fc0 sc0 ls18 wsb">conhecida como equação de Wiener-Hopf. </div><div class="t m0 x31 h25 yc7 ff4 fs0 fc0 sc0 ls18 ws11">\ue734<span class="fs9 ws27 v12">\uebbf\uebeb </span><span class="ws28">[\ue745]<span class="_6 blank"> </span>=<span class="_a blank"> </span>\uedcd<span class="_e blank"> </span>\u210e <span class="ls25 v1">[</span><span class="ls3a">\ue746<span class="ls25 v1">]</span></span><span class="ws17">. \ue734<span class="fs9 ws29 v12">\uebbf\uebbf </span><span class="ls25 v1">[</span><span class="ws2a">\ue745<span class="_4 blank"> </span>\u2212<span class="_4 blank"> </span>\ue746 <span class="v1">]</span></span></span></span></div><div class="t m0 x9 h1b yc8 ff5 fs9 fc0 sc0 ls18">\u221e</div><div class="t m0 x6 h13 yc9 ff4 fs9 fc0 sc0 ls18 ws2b">\uebdd \ueb40\ueb34</div><div class="t m0 x32 h16 yc7 ff2 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x1 h7 yca ff2 fs4 fc0 sc0 ls18 wsb">Dessa <span class="_d blank"> </span>forma, <span class="_d blank"> </span>por <span class="_e blank"> </span>meio <span class="_d blank"> </span>da <span class="_d blank"> </span>estimativa <span class="_e blank"> </span>das <span class="_d blank"> </span>FAC <span class="_e blank"> </span>e <span class="_d blank"> </span>FCC <span class="_d blank"> </span>pode<span class="_0 blank"> </span>-se <span class="_e blank"> </span>encontrar <span class="_d blank"> </span>a <span class="_d blank"> </span>função <span class="_e blank"> </span>de </div><div class="t m0 x1 h7 ycb ff2 fs4 fc0 sc0 ls18 wsb">resposta <span class="_6 blank"> </span>a <span class="_6 blank"> </span>frequência <span class="_6 blank"> </span>(IRF) <span class="_6 blank"> </span>discreta <span class="_6 blank"> </span><span class="ff5 ws4">h<span class="_0 blank"> </span>[n]</span>. <span class="_6 blank"> </span>Tal <span class="_6 blank"> </span>método <span class="_6 blank"> </span>também <span class="_6 blank"> </span>é <span class="_6 blank"> </span>conhecido <span class="_6 blank"> </span>como </div><div class="t m0 x1 h7 ycc ff2 fs4 fc1 sc0 ls18 wsb">Método das Correlações<span class="fc0">. </span></div><div class="t m0 x1 h7 ycd ff2 fs4 fc0 sc0 ls18 wsb">Também podemos fazer esta estimativa em termos espectrais, uti<span class="_1 blank"></span>lizando a </div><div class="t m0 x1 h7 yce ff2 fs4 fc0 sc0 ls18 wsb">Densidade Espectral de potência (PSD) e a PSD Cruzada entre os sinais <span class="ff5 ws4">F[n]</span><span class="ls3b"> e </span><span class="ff5 ws4">x[n]</span>. </div><div class="t m0 x1 h7 ycf ff2 fs4 fc0 sc0 ls18 wsb">Apresentamos a seguir, os estimadores espectrais clássicos mais comuns: </div><div class="t m0 x33 h1e yd0 ff4 fs4 fc0 sc0 ls18">\ue72a</div><div class="t m0 x34 h26 yd1 ff4 fs5 fc0 sc0 ls35">\ueb35<span class="fs4 ls18 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="lsb">)<span class="ls14 v2">=<span class="ff5 ls3c wsb"> </span></span></span></span><span class="ls31 v1b">\uebc9</span><span class="fs6 ls18 ws2c v1c">\uecb7\uece3 </span><span class="ls18 ws2 v1b">(\uebd9)</span></div><div class="t m0 x35 h27 yd2 ff4 fs5 fc0 sc0 ls31">\uebc9<span class="fs6 ls18 ws2d v12">\uecb7\uecb7 </span><span class="ls3d ws2e">(\uebd9)</span><span class="ff2 fs4 ls18 wsb vb"> <span class="ff8 ls3b">\uf0e0</span> Quando o ruído afeta mais os sinais de resposta </span></div><div class="t m0 x36 h1e yd3 ff4 fs4 fc0 sc0 ls18">\ue72a</div><div class="t m0 x37 h26 yd4 ff4 fs5 fc0 sc0 ls3e">\ueb36<span class="fs4 ls18 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="ls2">)<span class="ls14 v2">=<span class="ff5 ls3f wsb"> </span></span></span></span><span class="ls18 v1b">\uebc9</span></div><div class="t m0 x38 h28 yd5 ff4 fs6 fc0 sc0 ls18 ws2f">\uece3\uece3 <span class="fs5 ws2 v1d">(\uebd9)</span></div><div class="t m0 x39 h27 yd6 ff4 fs5 fc0 sc0 ls31">\uebc9<span class="fs6 ls18 ws30 v12">\uece3\uecb7 </span><span class="ls18 ws2">(\uebd9)<span class="ff2 fs4 wsb vb"> <span class="ff8 ls3b">\uf0e0</span> Quando o ruído afeta mais os sinais de entrada </span></span></div><div class="t m0 x3a h1e yd7 ff4 fs4 fc0 sc0 ls18">\ue72a</div><div class="t m0 x21 h29 yd8 ff4 fs5 fc0 sc0 ls40">\uebe9<span class="fs4 ls18 ws0 v14">(<span class="ls32 v2">\ue742</span><span class="lsb">)<span class="ls14 v2">=</span><span class="ls41 v2">\ueda5</span><span class="ls18 v2">\ue72a</span></span></span></div><div class="t m0 x3b h29 yd8 ff4 fs5 fc0 sc0 ls3e">\ueb35<span class="fs4 ls18 ws0 v14">(<span class="ls32 v2">\ue742</span>)<span class="v2">\ue72a</span></span></div><div class="t m0 x3c h29 yd8 ff4 fs5 fc0 sc0 ls3e">\ueb36<span class="fs4 ls18 ws0 v14">(<span class="ls32 v2">\ue742</span>)<span class="ls0 v2">\ue736</span><span class="ff2 wsb v2"> <span class="ff8 ls3b">\uf0e0</span> O mais genérico </span></span></div><div class="t m0 x1 h7 yd9 ff2 fs4 fc0 sc0 ls18 wsb">É possível verificar se a estimativa da função de resposta em frequência (<span class="_1 blank"></span>FRF) foi bem </div><div class="t m0 x1 h7 yda ff2 fs4 fc0 sc0 ls18 wsb">feita através da função de coerência: </div><div class="t m0 x8 h1e ydb ff4 fs4 fc0 sc0 ls18">\ue725</div><div class="t m0 x1b h24 ydc ff4 fs5 fc0 sc0 ls18 ws31">\uebbf\uebeb <span class="fs4 ws0 v14">(<span class="ls1 v2">\ue7f1</span><span class="ls2">)<span class="ls14 v2">=<span class="ff5 ls42 wsb"> </span></span></span><span class="vc">|<span class="v2">\ue735</span></span></span></div><div class="t m0 x12 h29 ydd ff4 fs5 fc0 sc0 ls18 ws31">\uebbf\uebeb <span class="fs4 ws0 v13">(\ue7f1)<span class="ls43 v1">|</span></span><span class="vd">\ueb36</span></div><div class="t m0 x17 h1e yde ff4 fs4 fc0 sc0 ls18">\ue735</div><div class="t m0 x3d h29 ydf ff4 fs5 fc0 sc0 ls18 ws32">\uebbf\uebbf <span class="fs4 ws0 v14">(<span class="ls1 v2">\ue7f1</span>)<span class="ws33 v2">. \ue735</span></span></div><div class="t m0 x3e h23 ydf ff4 fs5 fc0 sc0 ls18 ws22">\uebeb\uebeb <span class="fs4 ws0 v13">(\ue7f1)<span class="ff2 wsb ve"> </span></span></div><div class="t m0 x1 h7 ye0 ff2 fs4 fc0 sc0 ls18 wsb">Os valores de C<span class="fs7 wsf v10">FX</span><span class="ls44">(</span><span class="ff6 wsd">\uf077</span>) ficam entre 0 e 1, sendo que quanto mais próxi<span class="_1 blank"></span>mo de 1, melhor </div><div class="t m0 x1 h7 ye1 ff2 fs4 fc0 sc0 ls18 wsb">foi feita a estimativa. </div><div class="t m0 x1 h7 ye2 ff2 fs4 fc0 sc0 ls18 wsb"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x0 y0 w1 h1" alt="" src="https://files.passeidireto.com/f80e4ebf-9792-403e-87fc-06408416f1b4/bg7.png"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls18 wsb"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls18 wsb">7 </div><div class="c x1 y3 w2 h3"><div class="t m0 x0 h2 y4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="t m0 x1 h6 y84 ff3 fs3 fc1 sc0 ls18 wsb">Determinação Experimental do Fator de Amortecimento, <span class="_0 blank"> </span>\u03be </div><div class="t m0 x1 h7 y85 ff2 fs4 fc0 sc0 ls18 wsb">O fator de amortecimento pode s<span class="_1 blank"></span>er estimado com o uso da eq<span class="_1 blank"></span>uação do fator de </div><div class="t m0 x1 h2a ye3 ff2 fs4 fc0 sc0 ls18 wsb">ampliação <span class="ff4 ls45">\ue72f<span class="ls18 ws0 v1">(</span><span class="ls18 ws34">\ue74e,<span class="_a blank"> </span>\ue7e6 <span class="ls2 v1">)</span><span class="ls14">=</span></span><span class="ff5 ls46"> </span><span class="fs5 ls18 v1c">\uebd1</span></span></div><div class="t m0 x35 h28 ye4 ff4 fs6 fc0 sc0 ls47">\uecdb<span class="fs5 ls18 ws2 v1d">.\uebde</span></div><div class="t m0 x35 h2b ye5 ff4 fs5 fc0 sc0 ls48">\uebbf<span class="fs4 ls49 vb">=</span><span class="ls18 v1e">\ueb35</span></div><div class="t m0 x3f h2c ye6 ff4 fs5 fc0 sc0 ls18 ws2">\ueda5<span class="ls3d v0">(</span><span class="ws35 v0">\ueb35\ueb3f\uebe5 <span class="fs6 ls4a v1f">\uec2e</span><span class="ls3d v1">)</span><span class="fs6 ls4b v1f">\uec2e</span><span class="ws2">\ueb3e<span class="ls3d v1">(</span><span class="ws36">\ueb36.\uec15.\uebe5 <span class="ls3d v1">)</span><span class="fs6 ls4b v1f">\uec2e</span><span class="ff2 fs4 wsb vd"> visto na aula 09. </span></span></span></span></div><div class="t m0 x1 h7 ye7 ff2 fs4 fc0 sc0 ls18 wsb">Também <span class="_9 blank"> </span>é <span class="_2 blank"> </span>possível <span class="_9 blank"> </span>pelo <span class="_9 blank"> </span>o <span class="_9 blank"> </span>Decremento <span class="_9 blank"> </span>Logarítmico <span class="_9 blank"> </span>visto <span class="_9 blank"> </span>na <span class="_9 blank"> </span>aula <span class="_2 blank"> </span>07, <span class="_9 blank"> </span>mas <span class="_9 blank"> </span>para <span class="_9 blank"> </span>isso <span class="_9 blank"> </span>é </div><div class="t m0 x1 h7 ye8 ff2 fs4 fc0 sc0 ls18 wsb">necessário ter um martelo de impacto para apli<span class="_1 blank"></span>car um impulso ao sistema. </div><div class="t m0 x1 h7 ye9 ff2 fs4 fc0 sc0 ls18 wsb">Outra forma <span class="_0 blank"> </span>é extraindo <span class="_0 blank"> </span>experimentalmente FRF ou <span class="_0 blank"> </span>a IRF, <span class="_0 blank"> </span>pelos métodos <span class="_0 blank"> </span>vistos nessa </div><div class="t m0 x1 h7 yea ff2 fs4 fc0 sc0 ls18 wsb">aula. </div><div class="t m0 x1 h7 yeb ff2 fs4 fc0 sc0 ls18 wsb">O <span class="_5 blank"> </span>método <span class="_5 blank"> </span>mais <span class="_5 blank"> </span>Usado <span class="_5 blank"> </span>é <span class="_5 blank"> </span>Método <span class="_5 blank"> </span>\u201c<span class="_0 blank"> </span><span class="ff5">Quadrature <span class="_5 blank"> </span>peak <span class="_5 blank"> </span>picking\u201d<span class="_0 blank"> </span></span> <span class="_5 blank"> </span>válido <span class="_5 blank"> </span>para <span class="_5 blank"> </span>sistemas </div><div class="t m0 x1 h7 yec ff2 fs4 fc0 sc0 ls18 wsb">levemente <span class="_f blank"> </span>amortecidos <span class="_f blank"> </span>e <span class="_4 blank"> </span>consiste <span class="_f blank"> </span>em <span class="_f blank"> </span>me<span class="_0 blank"> </span>dir <span class="_f blank"> </span>duas <span class="_f blank"> </span>frequências <span class="_f blank"> </span>próximas<span class="_0 blank"> </span> <span class="_f blank"> </span>do <span class="_f blank"> </span>pico <span class="_4 blank"> </span>de </div><div class="t m0 x1 h1e yed ff2 fs4 fc0 sc0 ls18 wsb">ressonância. <span class="_2 blank"> </span>Amplitude <span class="_9 blank"> </span><span class="ff4">\ue7f1</span></div><div class="t m0 x21 h22 yee ff4 fs5 fc0 sc0 ls35">\ueb35<span class="ff5 fs4 ls18 wsb v13"> <span class="ff4 ws0">\ue741</span> <span class="ff4 ws0">\ue7f1</span></span>\ueb36<span class="ff2 fs4 ls18 wsb v13"> <span class="_2 blank"> </span>da <span class="_9 blank"> </span>FRF <span class="_2 blank"> </span>são <span class="_9 blank"> </span>0,707 <span class="_2 blank"> </span>(-3 <span class="_9 blank"> </span>dB) <span class="_2 blank"> </span>que <span class="_9 blank"> </span>é <span class="_2 blank"> </span>conhecido <span class="_9 blank"> </span>com <span class="_2 blank"> </span>ponto </span></div><div class="t m0 x1 h7 yef ff2 fs4 fc0 sc0 ls18 wsb">de meia potência. </div><div class="t m0 x1 h7 yf0 ff2 fs4 fc0 sc0 ls18 wsb">Assim, o fator de amortecimento pode ser estimado: </div><div class="t m0 x40 h2d yf1 ff4 fs0 fc0 sc0 ls18 ws37">\ue7e6<span class="_6 blank"> </span>= <span class="vd">1</span></div><div class="t m0 x41 h2e yf2 ff4 fs0 fc0 sc0 ls4c">2<span class="ls18 ws11 vb">\ued6c</span><span class="ls4d v1e">\ue7f1</span><span class="fs9 ls4e v20">\ueb36</span><span class="ls18 ws38 v1e">\u2212 \ue7f1<span class="fs9 v12">\ueb35</span></span></div><div class="t m0 x42 h14 yf2 ff4 fs0 fc0 sc0 ls18">\ue7f1</div><div class="t m0 x43 h2f yf3 ff4 fs9 fc0 sc0 ls4f">\uebe1<span class="fs0 ls18 ws11 v1c">\ued70<span class="ff2 wsb"> </span></span></div><div class="t m0 x1 h2 yf4 ff1 fs0 fc0 sc0 ls18 wsb"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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