A equa\303\247\303\243o da onda amortecida von Dr. M. P\303\266hlmann Email: Poehlmann001@aol.com Nesta planilha determinamos a solu\303\247\303\243o da equa\303\247\303\243o da onda amortecida com as condi\303\247\303\265es iniciais e de contorno de acordo com o m\303\251todo FOURIER. O amortecimento das vibra\303\247\303\265es \303\251 descrito na equa\303\247\303\243o diferencial parcial pelo coeficiente de amortecimento \316\262. Devido ao amortecimento, as vibra\303\247\303\265es diminuem em amplitude com o tempo.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">1. Representa\303\247\303\243o da equa\303\247\303\243o de onda amortecida</Font></Text-field> A equa\303\247\303\243o de onda \303\251 uma equa\303\247\303\243o diferencial parcial. A equa\303\247\303\243o de onda unidimensional com o coeficiente de amortecimento \316\262 descreve o comportamento vibrat\303\263rio de uma haste el\303\241stica apoiada em ambos os lados, que oscila ao longo do tempo. Este comportamento de decad\303\252ncia ao longo do tempo corresponde ao comportamento de vibra\303\247\303\243o natural de uma haste el\303\241stica. A equa\303\247\303\243o de onda com o coeficiente de amortecimento \316\262 \303\251: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== 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 LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkidUc2IjYkSSJ4R0YsSSJ0R0YsRi9GLyIiIiooIiIjRjBJJWJldGFHRixGMEYoRjBGMComKUYzRjJGMEYqRjBGMComKUkiY0dGLEYyRjAtRiU2JC1GJTYkRipGLkYuRjA= A solu\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInRGJ0YvRjJGQUZBRkE= depende da localiza\303\247\303\243o xe do tempo de avan\303\247o t. O amortecimento das vibra\303\247\303\265es \303\251 descrito por um coeficiente positivo \316\262 na equa\303\247\303\243o (1.1). Se considerarmos o caso de uma haste el\303\241stica apoiada em ambos os lados, que \303\251 excitada para oscilar ap\303\263s ser defletida, as seguintes condi\303\247\303\265es iniciais e de contorno s\303\243o necess\303\241rias para a solu\303\247\303\243o: - condi\303\247\303\265es iniciais (tempo zero) 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 Ly1JInVHNiI2JEkieEdGJSIiIS1JJHBoaUdGJTYjRic= No momento zero a oscila\303\247\303\243o tem a deflex\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjItSSNtb0dGJDYtUSIuRidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlVGMg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiYtRiM2JS1JI21uR0YkNiRRIjJGJ0YyLyUrZXhlY3V0YWJsZUdGMUYyRjIvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYlLUYsNiVRInVGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0Y+RjJGMi1GNjYkLUYjNictRiw2JVEieEYnRk1GTy1JI21vR0YkNi1RIixGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRk4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2JFEiMEYnRjJGPkYyRjItRlk2LVEiPUYnRjJGZm4vRmluRjFGam5GXG9GXm9GYG9GYm8vRmVvUSwwLjI3Nzc3NzhlbUYnL0Zob0ZicC1GWTYtUSomdW1pbnVzMDtGJ0YyRmZuRmBwRmpuRlxvRl5vRmBvRmJvL0Zlb1EsMC4yMjIyMjIyZW1GJy9GaG9GaHAtRiw2JVEnJiM5NDY7RidGL0YyLUZZNi1RJyZzZG90O0YnRjJGZm5GYHBGam5GXG9GXm9GYG9GYm9GZG8vRmhvRmZvLUYsNiVRJyYjOTY2O0YnRi9GMi1GNjYkLUYjNiVGVUY+RjJGMi1GWTYtUSI7RidGMkZmbkZobkZqbkZcb0Zeb0Zgb0Zib0Zkb0ZjcC1GWTYtUSJ+RidGMkZmbkZgcEZqbkZcb0Zeb0Zgb0Zib0Zkb0ZgcS1GLDYlUSRBQjJGJ0ZNRk8tRlk2LVEqJmNvbG9uZXE7RidGMkZmbkZgcEZqbkZcb0Zeb0Zgb0Zib0ZhcEZjcC1GLDYlUSIlRidGTUZPLUZZNi1RIjpGJ0YyRmZuRmBwRmpuRlxvRl5vRmBvRmJvRmFwRmNwRj5GMg== Ly0tJkkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjNiNJInVHRio2JEkieEdGKiIiISwkKiZJJWJldGFHRioiIiItSSRwaGlHRio2I0YwRjUhIiI= A mudan\303\247a na oscila\303\247\303\243o no tempo zero \303\251 a mesma 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. - condi\303\247\303\265es de fronteira 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 Ly1JInVHNiI2JCIiIUkidEdGJUYn 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 Ly1JInVHNiI2JEkibEdGJUkidEdGJSIiIQ== A haste el\303\241stica tem o comprimento LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. No limite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJsRidGL0YyRjk= a haste \303\251 articulada e as vibra\303\247\303\265es nos pontos de apoio s\303\243o zero. 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 Estamos procurando uma solu\303\247\303\243o para a equa\303\247\303\243o da onda amortecida (1.1) que atenda \303\240s condi\303\247\303\265es iniciais e de contorno (1.2) a (1.5).
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">2. A solu\303\247\303\243o do problema</Font></Text-field> A equa\303\247\303\243o diferencial parcial (1.1) pode ser resolvida separando as vari\303\241veis \342\200\213\342\200\213e transformando-as em um sistema de equa\303\247\303\265es diferenciais ordin\303\241rias.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">2.1. Separa\303\247\303\243o das vari\303\241veis </Font></Text-field> Encontramos uma solu\303\247\303\243o para a equa\303\247\303\243o diferencial parcial (1.1) separando as vari\303\241veis \342\200\213\342\200\213(x e t).Est\303\241 na fun\303\247\303\243o de solu\303\247\303\243o buscada 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uma fun\303\247\303\243o puramente dependente da localiza\303\247\303\243o e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSJ+RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=uma fun\303\247\303\243o puramente dependente do tempo. N\303\263s formulamos a solu\303\247\303\243o para 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em forma : 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 Ly1JInVHNiI2JEkieEdGJUkidEdGJSomLUkiRkdGJTYjRiciIiItSSJUR0YlNiNGKEYt Se inserirmos a abordagem do produto (2.1.1) na equa\303\247\303\243o diferencial parcial (1.1), as vari\303\241veis \342\200\213\342\200\213podem ser separadas. Voc\303\252 obt\303\251m: 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 LywoKiYtSSJURzYiNiNJInRHRichIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYsNiRGJUYpRikiIiJGMSoqIiIjRjFGJUYqSSViZXRhR0YnRjFGL0YxRjEqJClGNEYzRjFGMSooLUkiRkdGJzYjSSJ4R0YnRiopSSJjR0YnRjNGMS1GLDYkLUYsNiRGOEY7RjtGMQ== Os lados direito e esquerdo da equa\303\247\303\243o (2.1.2) tornam-se um par\303\242metro de separa\303\247\303\243o negativo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUScmIzk1NTtGJy8lJ2l0YWxpY0dGNEYvLUYsNi1RIn5GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGUEYv igualado. Isso d\303\241 duas equa\303\247\303\265es diferenciais ordin\303\241rias, a saber - a equa\303\247\303\243o dependente da localiza\303\247\303\243o: 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 LyomKUkiY0c2IiIiIyIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRio2JC1JIkZHRiY2I0kieEdGJkYyRjJGKCwkKiZGL0YoSSdsYW1iZGFHRiZGKCEiIg== - a equa\303\247\303\243o dependente do tempo: 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 LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiVEc2IjYjSSJ0R0YsRi5GLiIiIiooIiIjRi9JJWJldGFHRixGL0YoRi9GLyomRipGLylGMkYxRi9GLywkKiZGKkYvSSdsYW1iZGFHRixGLyEiIg== A solu\303\247\303\243o da equa\303\247\303\243o dependente da posi\303\247\303\243o (2.1.3) leva ao chamado "problema de autovalor". Teoricamente, a equa\303\247\303\243o diferencial possui um n\303\272mero infinito de solu\303\247\303\265es. Na maioria dos casos, no entanto, os primeiros 10 a 20 valores pr\303\263prios s\303\243o suficientes para uma representa\303\247\303\243o e determina\303\247\303\243o suficientes da solu\303\247\303\243o.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">2.2. Solu\303\247\303\243o do problema do autovalor local</Font></Text-field> As condi\303\247\303\265es de contorno separadas da equa\303\247\303\243o (1.4) e (1.5) para resolver a equa\303\247\303\243o (2.1.3) lidas 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 NiQvLUkiRkc2IjYjIiIhRigvLUYlNiNJImxHRiZGKA== Precisamos da solu\303\247\303\243o geral da equa\303\247\303\243o diferencial dependente da posi\303\247\303\243o (2.1.3). Isto \303\251 : 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 Ly1JIkZHNiI2I0kieEdGJSwmKiZJJF9DMUdGJSIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKihJImNHRiUhIiIpSSdsYW1iZGFHRiUjRisiIiNGK0YnRitGK0YrKiZJJF9DMkdGJUYrLUkkY29zR0YuRjFGK0Yr Assim, definimos a fun\303\247\303\243o de posi\303\247\303\243o F: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKHVuYXBwbHlGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSRyaHNGJ0YvRjItRlM2JC1GIzYlLUYsNiVRI0c2RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjktRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGL0YyRltvRjlGOUZbb0Y5 Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJkkkX0MxR0YlIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqKEkiY0dGJSEiIilJJ2xhbWJkYUdGJSNGLCIiI0YsRiRGLEYsRiwqJkkkX0MyR0YlRiwtSSRjb3NHRi9GMkYsRixGJUYlRiU= A partir das condi\303\247\303\265es de contorno (2.2.1), obtemos o seguinte sistema de equa\303\247\303\265es com a fun\303\247\303\243o F: 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 PCQvSSRfQzJHNiIiIiEvLCYqJkkkX0MxR0YlIiIiLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqKEkiY0dGJSEiIilJJ2xhbWJkYUdGJSNGKyIiI0YrSSJsR0YlRitGK0YrKiZGJEYrLUkkY29zR0YuRjFGK0YrRiY= Para determinar os valores pr\303\263prios da solu\303\247\303\243o, configuramos a matriz das constantes _C1 e _C2. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEnbGluYWxnRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiUtRiw2JVEqZ2VubWF0cml4RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjY2JC1GIzYnLUYsNiVRI0c3RidGL0YyLUkjbW9HRiQ2LVEiLEYnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GNjYmLUYjNictRiw2JVEkX0MxRidGL0YyRk8tRiw2JVEkX0MyRidGL0YyRj1GQEZAL0ZDUSJ8ZnJGJy9GRlEifGhyRidGPUZARkAtRlA2LVEiO0YnRkBGU0ZVRldGWUZlbkZnbkZpbkZbby9GX29RLDAuMjc3Nzc3OGVtRictRiw2JVEjRzhGJ0YvRjItRlA2LVEqJmNvbG9uZXE7RidGQEZTL0ZWRj9GV0ZZRmVuRmduRmluL0Zcb0ZjcEZicC1GLDYlUSIlRidGL0YyLUZQNi1RIjpGJ0ZARlNGanBGV0ZZRmVuRmduRmluRltxRmJwRj1GQA== PTYiNiQ7IiIiIiIjRiVFXFtsJTYkRidGJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGIzYjKihJImNHRiMhIiIpSSdsYW1iZGFHRiMjRiZGJ0YmSSJsR0YjRiY2JEYmRidGJjYkRiZGJiIiITYkRidGJy1JJGNvc0dGLEYv Os valores pr\303\263prios \316\273 s\303\243o determinados a partir do determinante da matriz constante: 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 LywkLUkkc2luRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqKEkiY0dGKSEiIilJJ2xhbWJkYUdGKSMiIiIiIiNGMUkibEdGKUYxRi0iIiE= Resolvemos a equa\303\247\303\243o (2.2.6) \316\273 em e receba LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVExX0VudkFsbFNvbHV0aW9uc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVGMUYvRjItRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk= 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 PCMvSSdsYW1iZGFHNiIqKilJI1BpRyUqcHJvdGVjdGVkRyIiIyIiIilJJV9aMXxpckc2IiIiIyIiIilJImNHNiIiIiMiIiIpSSJsRzYiIiIjISIi 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 os autovalores do problema local: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYrLUYsNiVRJEcxMUYnRi9GMi1GNjYmLUYjNiUtSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGRUZFLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUkjbW9HRiQ2LVEiPUYnRkUvJSZmZW5jZUdGSS8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zeby1GLDYlUSJrRidGL0YyLUZRNi1RIixGJ0ZFRlQvRldGMUZYRlpGZm5GaG5Gam4vRl1vUSYwLjBlbUYnL0Zgb1EsMC4zMzMzMzMzZW1GJy1GLDYlUSRHMTBGJ0YvRjJGPUZHRkVGRS1GUTYtUSI7RidGRUZURmdvRlhGWkZmbkZobkZqbkZob0Zfby1GUTYtUSJ+RidGRUZURlZGWEZaRmZuRmhuRmpuRmhvL0Zgb0Zpby1GLDYlUSRHMTJGJ0YvRjItRlE2LVEqJmNvbG9uZXE7RidGRUZURlZGWEZaRmZuRmhuRmpuRlxvRl9vLUYsNiVRIiVGJ0YvRjItRlE2LVEiOkYnRkVGVEZWRlhGWkZmbkZobkZqbkZcb0Zfb0ZHRkU= L0knbGFtYmRhRzYiKiopSSNQaUclKnByb3RlY3RlZEciIiMiIiIpSSJrR0YkRilGKilJImNHRiRGKUYqKUkibEdGJEYpISIi Definimos a seguinte fun\303\247\303\243o dos valores pr\303\263prios LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2J1EjRVdGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjEvJSxtYXRodmFyaWFudEdRLGJvbGQtaXRhbGljRicvJStmb250d2VpZ2h0R1ElYm9sZEYnLUkobWZlbmNlZEdGJDYmLUYjNiYtRiw2J1Eia0YnRi9GMkY0RjdGLy9GNUY5RjdGL0ZCRjctSSNtb0dGJDYvUSJ+RidGL0ZCRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWEYvRkJGNw==: 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 Zio2I0kia0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiopSSNQaUclKnByb3RlY3RlZEciIiMiIiIpRiRGLUYuKUkiY0dGJUYtRi4pSSJsR0YlRi0hIiJGJUYlRiU= 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 Queremos agora determinar a autofun\303\247\303\243o do problema local. A solu\303\247\303\243o geral considerando a condi\303\247\303\243o de contorno 1 \303\251: 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 Ly1JIkZHNiI2I0kieEdGJSomSSRfQzFHRiUiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyoqSSNQaUdGLkYqSSJrR0YlRipJImxHRiUhIiJGJ0YqRio= A autofun\303\247\303\243o \303\251 agora determinada a partir da solu\303\247\303\243o (2.2.10) de acordo com a f\303\263rmula: 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 KihJJF9DMUc2IiIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKipJI1BpR0YpRiVJImtHRiRGJUkibEdGJCEiIkkieEdGJEYlRiUpLUkkSW50R0YoNiQqJilGIyIiI0YlKUYmRjhGJS9GMTsiIiFGLyNGJUY4RjA= Ao calcular a integral e simplificar o resultado, obtemos: 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 KigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IyoqSSNQaUdGJiIiIkkia0dGKEYsSSJsR0YoISIiSSJ4R0YoRixGLCkiIiMjRixGMkYsKUYuRjNGLw== Agora, a autofun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRUZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSJrRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJ4RidGL0YyRkFGQUZB a ser definida 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 Zio2JEkia0c2IkkieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyoqSSNQaUdGLiIiIkYkRjNJImxHRiUhIiJGJkYzRjMpIiIjI0YzRjdGMylGNEY4RjVGJUYlRiU= Os autovalores e autofun\303\247\303\265es do problema local s\303\243o assim: 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 Ly1JI0VXRzYiNiNJImtHRiUqKilJI1BpRyUqcHJvdGVjdGVkRyIiIyIiIilGJ0YsRi0pSSJjR0YlRixGLSlJImxHRiVGLCEiIg== 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 Ly1JI0VGRzYiNiRJImtHRiVJInhHRiUqKC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKipJI1BpR0YtIiIiRidGMkkibEdGJSEiIkYoRjJGMikiIiMjRjJGNkYyKUYzRjdGNA== Com isso, o problema de autovalores da solu\303\247\303\243o local \303\251 calculado. Agora podemos definir a solu\303\247\303\243o geral para o problema geral da equa\303\247\303\243o de onda amortecida. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVExX0VudkFsbFNvbHV0aW9uc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVGPUYvRjItRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0vJStleGVjdXRhYmxlR0Y9Rjk=
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">2.3. Solu\303\247\303\243o geral da equa\303\247\303\243o de onda amortecida</Font></Text-field> A solu\303\247\303\243o geral para o problema da equa\303\247\303\243o da onda amortecida \303\251 apresentada na forma de uma expans\303\243o em s\303\251rie. Formulamos uma expans\303\243o em s\303\251rie a partir do produto da solu\303\247\303\243o temporal e espacial na forma: 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 Ly1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiotJkkiVEdGJTYjSSJrR0YlNiNGKCIiIi1JJHNpbkdGKzYjKipJI1BpR0YsRjZGNEY2SSJsR0YlISIiRidGNkY2KSIiIyNGNkY/RjYpRjxGQEY9L0Y0O0Y2SSlpbmZpbml0eUdGLA== 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\303\251 a solu\303\247\303\243o de tempo para o problema, que \303\251 determinada a seguir. A fun\303\247\303\243o seno descreve o comportamento local da solu\303\247\303\243o da equa\303\247\303\243o de onda.
<Text-field style="Heading 2" layout="Heading 2">2.4. Resolvendo o problema do tempo</Text-field> A solu\303\247\303\243o de tempo segue da equa\303\247\303\243o diferencial (2.1.4). Se inserirmos os valores pr\303\263prios de acordo com (2.2.9) na equa\303\247\303\243o diferencial (2.1.4), obtemos 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 LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLSZJIlRHNiI2I0kia0dGLTYjSSJ0R0YtRjFGMSIiIiooIiIjRjJJJWJldGFHRi1GMkYoRjJGMiomRipGMilGNUY0RjJGMiwkKixGKkYyKUkjUGlHRiZGNEYyKUYvRjRGMilJImNHRi1GNEYyKUkibEdGLUY0ISIiRkE= Para resolver esta equa\303\247\303\243o diferencial de tempo, precisamos das condi\303\247\303\265es iniciais no ponto no tempo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. O valor inicial 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resulta da seguinte integral levando em considera\303\247\303\243o a autofun\303\247\303\243o de acordo com (2.2.13): 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 Ly0mSSJURzYiNiNJImtHRiY2IyIiIS1JJGludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkKiotSSRwaGlHRiY2I0kieEdGJiIiIi1JJHNpbkdGLTYjKipJI1BpR0YuRjZGKEY2SSJsR0YmISIiRjVGNkY2KSIiIyNGNkY/RjYpRjxGQEY9L0Y1O0YqRjw= O segundo valor inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JC1JJW1zdWJHRiQ2JS1GLDYlUSJURicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictRiM2JS1GLDYlUSJrRidGQEZCRkBGQi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGMi1GNjYkLUYjNiQtSSNtbkdGJDYkRktGMkYyRjJGMg== obtemos com o integral: 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 Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMmSSJUR0YpNiNJImtHRik2IyIiIS1JJGludEdGJjYkLCQqLEklYmV0YUdGKSIiIi1JJHBoaUdGKTYjSSJ4R0YpRjctSSRzaW5HRiY2IyoqSSNQaUdGJ0Y3Ri5GN0kibEdGKSEiIkY7RjdGNykiIiMjRjdGREY3KUZBRkVGQkZCL0Y7O0YwRkE= Agora podemos resolver e obter a equa\303\247\303\243o diferencial ordin\303\241ria (2.4.1) com os valores iniciais (2.4.2) e (2.4.3) 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 Ly0mSSJURzYiNiNJImtHRiY2I0kidEdGJiosKSIiIyMiIiJGLUYvKUkibEdGJkYuISIiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiQqJi1JJHBoaUdGJjYjSSJ4R0YmRi8tSSRzaW5HRjU2IyoqSSNQaUdGNkYvRihGL0YxRjJGPUYvRi8vRj07IiIhRjFGLy1JJGV4cEdGNTYjLCQqJkklYmV0YUdGJkYvRipGL0YyRi8tSSRjb3NHRjU2IyosRjFGMkZCRi9GKEYvSSJjR0YmRi9GKkYvRi8= A solu\303\247\303\243o de tempo, portanto, muda na forma de uma fun\303\247\303\243o cosseno geral de t. O amortecimento das vibra\303\247\303\265es \303\251 descrito pela fun\303\247\303\243o exponencial negativa.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">2.5. Formula\303\247\303\243o da solu\303\247\303\243o geral para o problema de vibra\303\247\303\243o</Font></Text-field> Do produto da solu\303\247\303\243o espacial e temporal, obtemos a solu\303\247\303\243o geral para o problema da vibra\303\247\303\243o. Al\303\251m disso, o somat\303\263rio dos autovalores deve ser levado em considera\303\247\303\243o. Se inserirmos a solu\303\247\303\243o temporal (2.4.4) na solu\303\247\303\243o geral (2.3.1), obtemos a solu\303\247\303\243o geral do problema proposto: 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 Ly1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCQqLiIiIyIiIkkibEdGJSEiIi1JJGludEdGKzYkKiYtSSRwaGlHRiU2I0YnRjItSSRzaW5HRis2IyoqSSNQaUdGLEYySSJrR0YlRjJGM0Y0RidGMkYyL0YnOyIiIUYzRjItSSRleHBHRis2IywkKiZJJWJldGFHRiVGMkYoRjJGNEYyLUkkY29zR0YrNiMqLEYzRjRGQEYyRkFGMkkiY0dGJUYyRihGMkYyRjxGMkYyL0ZBO0YySSlpbmZpbml0eUdGLA== A equa\303\247\303\243o (2.5.1) descreve o comportamento vibrat\303\263rio da equa\303\247\303\243o da onda amortecida de uma haste el\303\241stica com suporte articulado em ambos os lados. Revis\303\243o da solu\303\247\303\243o: No momento zero (in\303\255cio das oscila\303\247\303\265es) a condi\303\247\303\243o inicial (1.2) deve ser atendida. Em t = 0, obt\303\251m-se (2.5.1): 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 Ly1JInVHNiI2JEkieEdGJSIiIS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCQqKiIiIyIiIkkibEdGJSEiIi1JJGludEdGKzYkKiYtSSRwaGlHRiU2I0YnRjItSSRzaW5HRis2IyoqSSNQaUdGLEYySSJrR0YlRjJGM0Y0RidGMkYyL0YnO0YoRjNGMkY8RjJGMi9GQTtGMkkpaW5maW5pdHlHRiw= Se a solu\303\247\303\243o estiver correta, a soma de acordo com a equa\303\247\303\243o (2.5.2) deve ser igual \303\240 deflex\303\243o inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== estar. A verifica\303\247\303\243o gr\303\241fica \303\251 realizada mais abaixo. JSFH
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">3. C\303\241lculo e anima\303\247\303\243o do comportamento de vibra\303\247\303\243o</Font></Text-field> Em um exemplo, queremos calcular as vibra\303\247\303\265es de uma haste el\303\241stica de acordo com a solu\303\247\303\243o (2.5.1).Para fazer isso, damos a seguinte deflex\303\243o inicial da haste (fun\303\247\303\243o inicial LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg==) : 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 Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKCMiIiIiIiVGLEYkRiwsJkkibEdGJUYsRiQhIiJGLEYsRiVGJUYl Com esta fun\303\247\303\243o inicial, obtemos a seguinte solu\303\247\303\243o da equa\303\247\303\243o da onda amortecida de (2.5.1) para o caso mostrado: 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 Ly1JInVHNiI2JEkieEdGJUkidEdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkLCQqMiMiIiIiIiNGMilJImxHRiVGM0YyLChGMyEiIiooSSNQaUdGLEYySSJrR0YlRjItSSRzaW5HRis2IyomRjlGMkY6RjJGMkYyKiZGM0YyLUkkY29zR0YrRj1GMkYyRjIpRjoiIiRGNylGOUZDRjctSSRleHBHRis2IywkKiZJJWJldGFHRiVGMkYoRjJGN0YyLUZBNiMqLEY1RjdGOUYyRjpGMkkiY0dGJUYyRihGMkYyLUY8NiMqKkY5RjJGOkYyRjVGN0YnRjJGMkY3L0Y6O0YySSlpbmZpbml0eUdGLA== Controle das condi\303\247\303\265es de contorno: Na equipe LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJsRidGL0YyRjk= as vibra\303\247\303\265es devem ser zero (suporte articulado da haste). 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 Ly1JInVHNiI2JCIiIUkidEdGJUYn 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 Ly1JInVHNiI2JEkibEdGJUkidEdGJSIiIQ== As condi\303\247\303\265es de limite foram atendidas! Agora queremos representar a solu\303\247\303\243o de acordo com a equa\303\247\303\243o (3.2). Para o c\303\241lculo, definimos a seguinte fun\303\247\303\243o da solu\303\247\303\243o (3.2) com um par\303\242metro de s\303\251rie geral N: 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 Zio2JUkieEc2IkkidEdGJUkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJCoyIyIiIiIiI0Y0KUkibEdGJUY1RjQsKEY1ISIiKihJI1BpR0YuRjRJImtHRiVGNC1JJHNpbkdGLTYjKiZGO0Y0RjxGNEY0RjQqJkY1RjQtSSRjb3NHRi1GP0Y0RjRGNClGPCIiJEY5KUY7RkVGOS1JJGV4cEdGLTYjLCQqJkklYmV0YUdGJUY0RiZGNEY5RjQtRkM2IyosRjdGOUY7RjRGPEY0SSJjR0YlRjRGJkY0RjQtRj42IyoqRjtGNEY8RjRGN0Y5RiRGNEY0RjkvRjw7RjRGJ0YlRiVGJQ== Tamb\303\251m definimos os seguintes par\303\242metros para o c\303\241lculo: 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 IiIj IyIiIiIiJg== IyIiIiIjUw== Agora podemos calcular e exibir a solu\303\247\303\243o da equa\303\247\303\243o da onda amortecida em uma anima\303\247\303\243o: 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De acordo com a condi\303\247\303\243o inicial 1, a deflex\303\243o inicial da haste deve ser igual \303\240 fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5NjY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInhGJy9GMFEldHJ1ZUYnL0YzUSdpdGFsaWNGJ0YyRjJGMg== estar. Verifica\303\247\303\243o ! : 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As curvas para a deflex\303\243o inicial s\303\243o congruentes. 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As curvas azuis descrevem o amortecimento das vibra\303\247\303\265es no centro da barra. As vibra\303\247\303\265es da haste (mostradas em vermelho) s\303\263 se movem dentro da curva de amortecimento azul. Pode ser visto claramente que a amplitude das oscila\303\247\303\265es diminui com o tempo e as oscila\303\247\303\265es ap\303\263s longos tempos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJnNyYXJyO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUY7Ni1RKCZpbmZpbjtGJ0Y+RkBGQ0ZFRkdGSUZLRk1GT0ZSRj5GPkY+ chega a uma paralisa\303\247\303\243o. JSFH