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aS]xlYeUKas\\EVipp@=mdEjWat\\\\yGIV[iyE`S[DxkIxeweyxfIq]tFfCQl_>pKyexGy_?m:hitaxK?\\N`kbIbhYt]wrHhiwixQYcQp_GasFajpnvcIwCgnoPqnGw\\qwhqcBYZLP`Yvdhf\\gPZsxuNxpQVugIsfOlkNx`ag@fmJ`c@ogWqdLPdTGm?@orqheFdcA_q@]D?t:FyiHofAqDvlQ@apOsp_hb`rSisSPx^ph_?m\\YlIajKgpbGt\\nuNFrxQ^RhlJ>ijhZk?lMXl=oh[ikUv`lWhmGoFpZHIoMwnHns;@_W^ar?p=O[n`mNV^mp];nq_YZLwlWaw`Ga;f\\K@[rqsb?eSolPGbjQ[oObCvg[_iLin]hufqxBPuOYoQGfonmyviXisGi_wfaWVotf`of`uVaUP[`gtexdINpjV_eVsmvoOQoOW[<GtAyrAob;igP>fqVqSisSWpMnbfqvdpbaFkwN[bnkWVaPpiIWcEWxoixO?_HOtrYf>^_u`njhufqfqPbThwkopUXcpFkAp_qajn?wJIjW>yPoa>`onQwEA\\Hh[jigqVqSe_WfWQsZ?bvywyuyEMrIOGD?TXOgfsg[CR@IEIkdLOE^EgPSiSishm@V[G`ZXFj`Ahg@_@>\\aFyAQvJXla@sR_^IQo_Wj`GuGqgpWp_XnWQsXWjnX^Rftyy^_GpwFleGiTOZRgtnn\\gAq]hufqfqVetNug`coNxWPhoabSXj=AkKPpromVawfvkgIlnXmWaw`WcAWqOhm@qn[F[?Oum@gC>oOo\\xO\\I?jHavBfnWvhnXmWesSWyyb:avhYb>]xIgH:;B:;RLEdMCde?DR?6>\"\{\}-%+ANNOTATIONG6'-%)BOUNDS_XG6#,$-%&FloatG6#%)infinityG!""-%)BOUNDS_YGF'-%-BOUNDS_WIDTHGF'-%.BOUNDS_HEIGHTGF'-%)CHILDRENG6"Determine as amplitudes das respostas de estado estacion\303\241rio do sistema de vibra\303\247\303\243o mck viajando a uma velocidade constante v na dire\303\247\303\243o horizontal, para valores 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and 1.1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiYtRiw2JVEoJm9tZWdhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiw2JVEibkYnL0Y2USV0cnVlRicvRjlRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y4(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1GLzYmUSJuRicvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJ0Y/RjVGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJ0Y1Rjc=\303\251 a frequ\303\252ncia natural n\303\243o amortecida do sistema onedof simples). Ignore o tamanho da roda pequena e sua massa.
Use os seguintes valores m = 100 kg, k = 10000 N/m, c = 100 Ns/m, a = 1.75 m, and b = 0.05 m.
O perfil da curva \303\251 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2JkYrLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGK0YxRjRGNw==Derivando as equa\303\247\303\265es governantesrestart:Typesetting:-Settings(typesetdot = true):Desenhando um diagrama de corpo livre da massa m, podemos identificar a equa\303\247\303\243o governante do sistema oscilante, igualando a for\303\247a resultante no corpo \303\240 massa vezes a acelera\303\247\303\243o.
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a carta y(t) \303\251 usado para denotar o deslocamento vertical do corpo, a seguinte equa\303\247\303\243o pode ser derivada do diagrama mostrado acima.Eq:=m*diff(y(t),t$2) + c*diff(y(t),t) + k * y(t) = k*y__b(t) + c*diff(y__b(t),t)O lado direito da equa\303\247\303\243o acima pode ser identificado como a fun\303\247\303\243o de for\303\247a aplicada \303\240 equa\303\247\303\243o do oscilador amortecido de segunda ordem (lado esquerdo).
A express\303\243o da fun\303\247\303\243o de for\303\247a \303\251 escrita em termos do perfil da 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Definindo o perfil da estrada para a excita\303\247\303\243o de baseProfileEq:= y__b(t) = b*sin(2*Pi*x__b(t)/a)Para velocidade constante, a quantidade 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o que torna a equa\303\247\303\243o do perfilProfileEq:= y__b(t) = b*sin((2*Pi*v/a)*t)Se n\303\263s definirmos 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, ent\303\243o esta equa\303\247\303\243o pode ser escrita como
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2I1EhRidGRi8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUrZXhlY3V0YWJsZUdGNEYvProfileEq:= algsubs(2*Pi*v/a = omega,ProfileEq);A fun\303\247\303\243o de for\303\247a torna-se ent\303\243osubs([ProfileEq, diff(ProfileEq,t)], rhs(Eq))Que pode ainda ser combinado 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Onde 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e 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
Esta simplifica\303\247\303\243o transforma a equa\303\247\303\243o governante emEq:=m*diff(y(t),t$2) + c*diff(y(t),t) + k * y(t) =k*A*sin(omega*t+phi)Dividindo ambos os lados por m e introduzindo a defini\303\247\303\243o de frequ\303\252ncia natural n\303\243o amortecida 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e o fator de amortecimento 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n\303\263s reescrevemos a equa\303\247\303\243o comoEq:=diff(y(t),t$2) + 2*zeta*omega__n*diff(y(t),t) + omega__n^2* y(t) =k*A*sin(omega*t+phi)Podemos supor que a solu\303\247\303\243o ser\303\241 da forma 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TrialSolution:= y(t) = B*sin(omega*t) + C*cos(omega*t);Substituir a solu\303\247\303\243o de teste na equa\303\247\303\243o governante nos d\303\241Eq1:=expand(subs(TrialSolution, lhs(Eq)-rhs(Eq)=0));Eq2:=collect(Eq1,[sin(omega*t),cos(omega*t)]);Coletando termos para 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\303\263s obtemos as duas seguintes equa\303\247\303\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EqA:=-2*C*omega*zeta*omega__n - A*cos(phi)*k - B*omega^2 + B*omega__n^2 = 0:EqB:=2*B*omega*zeta*omega__n - A*sin(phi)*k - C*omega^2 + omega__n^2*C = 0:collect(EqA,{B,C});collect(EqB,{B,C});Pela solu\303\247\303\243o destas duas equa\303\247\303\265es, n\303\263s podemos conseguir as express\303\265es para B e C.soln:=solve({EqA,EqB},{B,C}):simplify(soln[1])simplify(soln[2])Lembrando que a resposta est\303\241 na forma 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podemos identificar prontamente que a resposta de estado estacion\303\241rio do sistema pode ser expressa em termos da seguinte fun\303\247\303\243o harm\303\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Onde 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:=factor(simplify(subs(soln,sqrt(B^2+C^2))))arctan(simplify(subs(soln,C/B)))Use estas rela\303\247\303\265es para calcular as amplitudes para diferentes valores de omega como requerido na quest\303\243o.