Um sistema de eixo consiste em um eixo de a\303\247o sem massa (circular) e tr\303\252s rotores, conforme mostrado abaixo. Estamos preocupados com o comportamento vibracional de tor\303\247\303\243o do sistema do rotor. Portanto, apenas a energia de deforma\303\247\303\243o torcional e as energias cin\303\251ticas de rota\303\247\303\243o dos tr\303\252s discos em torno do eixo do eixo s\303\243o consideradas. A flex\303\243o lateral \303\251 ignorada neste problema. Escolha um conjunto adequado de coordenadas e estabele\303\247a as equa\303\247\303\265es. 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movimento por vibra\303\247\303\243o de tor\303\247\303\243o do sistema DOF m\303\272ltiplo. Encontre uma das frequ\303\252ncias naturais e sua forma de modo correspondente (vetor). Os valores dos par\303\242metros s\303\243o d = 48 mm, L1 = L2 = L3 = L4 = 100 mm, G = 70 GPa, J1 = J2 = J3 = 0.01 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 Conjunto adequado de coordenadas: Vamos com tr\303\252s \303\242ngulos 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i = 1, 2,e 3 para as tr\303\252s in\303\251rcias. Energia cin\303\251tica do sistema restart: Digits:=2: T:= 1/2*J*diff(theta__1(t),t)^2+ 1/2*J*diff(theta__2(t),t)^2+ 1/2*J*diff(theta__3(t),t)^2

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Existem quatro elementos compat\303\255veis com propriedades geom\303\251tricas e materiais id\303\252nticas. Podemos presumir com seguran\303\247a que suas rigidezes torcionais tamb\303\251m s\303\243o id\303\252nticas. Vamos assumir k \303\251 a rigidez torcional. A rigidez torcional pode ser calculada a partir das propriedades fornecidas usando a express\303\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 params:=[L=.1, J = 0.01,G = 70e10, d = 0.048]: params := [op(params),subs(params,k = G * (Pi*d^4/32)/L)]: map(lprint,params):

L = .1J = .1e-1 G = .70e12 d = .48e-1 k = .38e7

Em termos disso, podemos escrever a energia potencial armazenada nos elementos compat\303\255veis V1:=k/2*theta__1(t)^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNWMUdGKCwkKigjIiIiIiIjRjFJImtHRihGMSktSSl0aGV0YV9fMUdGKDYjSSJ0R0YoRjJGMUYxNyNGLg==

V2:=k/2*(theta__2(t) - theta__1(t))^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNWMkdGKCwkKigjIiIiIiIjRjFJImtHRihGMSksJi1JKXRoZXRhX18yR0YoNiNJInRHRihGMS1JKXRoZXRhX18xR0YoRjghIiJGMkYxRjE3I0Yu

V3:=k/2*(theta__3(t)-theta__2(t))^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNWM0dGKCwkKigjIiIiIiIjRjFJImtHRihGMSksJi1JKXRoZXRhX18zR0YoNiNJInRHRihGMS1JKXRoZXRhX18yR0YoRjghIiJGMkYxRjE3I0Yu

V4:=k/2*(theta__3(t))^2

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNWNEdGKCwkKigjIiIiIiIjRjFJImtHRihGMSktSSl0aGV0YV9fM0dGKDYjSSJ0R0YoRjJGMUYxNyNGLg==

Formando a Lagrangiana L:=T-(V1+V2+V3+V4): eq1:=-simplify(VariationalCalculus:-EulerLagrange(L,t,theta__1(t))[1]) = 0: eq2:=-simplify(VariationalCalculus:-EulerLagrange(L,t,theta__2(t))[1]) = 0: eq3:=-simplify(VariationalCalculus:-EulerLagrange(L,t,theta__3(t))[1]) = 0: expand(eq1)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCgqJkkiSkdGKCIiIi1JJWRpZmZHRiY2JC1JKXRoZXRhX18xR0YoNiNJInRHRigtSSIkR0YmNiRGNyIiI0YwRjAqKEY7RjBJImtHRihGMEY0RjBGMComRj1GMC1JKXRoZXRhX18yR0YoRjZGMCEiIiIiITcjLywoKiZGL0YwLUYyNiQtRjI2JEY0RjdGN0YwRjBGPEYwRj5GQUZC

expand(eq2)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCoqJkkiSkdGKCIiIi1JJWRpZmZHRiY2JC1JKXRoZXRhX18yR0YoNiNJInRHRigtSSIkR0YmNiRGNyIiI0YwRjAqJkkia0dGKEYwLUkpdGhldGFfXzFHRihGNkYwISIiKihGO0YwRj1GMEY0RjBGMComRj1GMC1JKXRoZXRhX18zR0YoRjZGMEZAIiIhNyMvLCoqJkYvRjAtRjI2JC1GMjYkRjRGN0Y3RjBGMEY8RkBGQUYwRkJGQEZF

expand(eq3)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCgqJkkiSkdGKCIiIi1JJWRpZmZHRiY2JC1JKXRoZXRhX18zR0YoNiNJInRHRigtSSIkR0YmNiRGNyIiI0YwRjAqJkkia0dGKEYwLUkpdGhldGFfXzJHRihGNkYwISIiKihGO0YwRj1GMEY0RjBGMCIiITcjLywoKiZGL0YwLUYyNiQtRjI2JEY0RjdGN0YwRjBGPEZARkFGMEZC

Definindo o vetor de estado qVec:=Vector([theta__1(t),theta__2(t),theta__3(t)])

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

A matriz de massa (in\303\251rcia) xM:=Matrix(3,3,[J,0,0,0,J,0,0,0,J])

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

A matriz de rigidez xK,f:=LinearAlgebra:-GenerateMatrix(subs({diff(theta__1(t), t, t) = 0,diff(theta__2(t), t, t) = 0,diff(theta__3(t), t, t) = 0},[eq1,eq2,eq3]),convert(qVec,list)): xK

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

Equa\303\247\303\265es de movimento para vibra\303\247\303\243o livre 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 eq:= M*diff(q(t),t,t) + K*q(t) = 0

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNlcUdGKC8sJiomSSJNR0YoIiIiLUklZGlmZkdGJjYkLUkicUdGKDYjSSJ0R0YoLUkiJEdGJjYkRjkiIiNGMkYyKiZJIktHRihGMkY2RjJGMiIiITcjLywmKiZGMUYyLUY0NiQtRjQ2JEY2RjlGOUYyRjJGPkYyRkA=

Supondo uma solu\303\247\303\243o geral 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Substituindo nas equa\303\247\303\265es gerais do movimento, obtemos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRictSShtZmVuY2VkR0YkNiUtRiM2Ki1JI21vR0YkNi5RKiZ1bWludXMwO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRkwvJSpzZXBhcmF0b3JHRkwvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGam4tSSVtc3VwR0YkNiUtRiw2JlEnJiM5Njk7RicvJSdpdGFsaWNHRkxGSkZNLUYjNiYtSSNtbkdGJDYlUSIyRidGSkZNL0Zkb1EldHJ1ZUYnRkovRk5RJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLDYmUSJNRidGW3BGSkZdcC1GRzYuUSIrRidGSkZNRlBGUkZURlZGWEZaRmZuRmhuRltvLUYsNiZRIktGJ0ZbcEZKRl1wLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGSkZNRkpGTS1GRzYuUScmc2RvdDtGJ0ZKRk1GUEZSRlRGVkZYRlpGZm4vRmluRjcvRlxvRjctRiw2JlEiQUYnRltwRkpGXXBGXnEtRl5vNiUtRkc2LlEvJkV4cG9uZW50aWFsRTtGJ0ZKRk1GUEZSRlRGVkZYRlpGZm5GYXEvRlxvUSwwLjExMTExMTFlbUYnLUYjNiwtRiw2JlEiSUYnRltwRkpGXXBGXnFGYG9GXnEtRiw2JlEidEYnRltwRkpGXXBGW3AvJStmb3JlZ3JvdW5kR1EsWzEyOCwwLDEyOF1GJ0ZKLyUscGxhY2Vob2xkZXJHRlxwRl1wRl9wLUZHNi5RIj1GJ0ZKRk1GUEZSRlRGVkZYRlpGZm4vRmluUSwwLjI3Nzc3NzhlbUYnL0Zcb0Zecy1GaG82JUZhcEZKRk0tRkc2LlEifkYnRkpGTUZQRlJGVEZWRlhGWkZmbkZhcUZicUZic0ZbcUZKRk0=Onde A \303\251 o vetor desconhecido e \317\211 \303\251 a frequ\303\252ncia da vibra\303\247\303\243o Agora, para que isso se mantenha para sempre, devemos ter 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Para este sistema ter solu\303\247\303\265es n\303\243o triviais a matriz (-MLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStleGVjdXRhYmxlR0Y0RjU=+ K) deve ser singular. Ent\303\243o a condi\303\247\303\243o aqui \303\251 CharEq:=simplify(LinearAlgebra:-Determinant(-omega^2 * xM + xK))=0

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdDaGFyRXFHRigvLCoqJilJIkpHRigiIiQiIiIpSSZvbWVnYUdGKCIiJ0Y0ISIiKipGN0Y0KUYyIiIjRjRJImtHRihGNClGNiIiJUY0RjQqKiIjNUY0RjJGNClGPEY7RjQpRjZGO0Y0RjgqJkY+RjQpRjxGM0Y0RjQiIiE3I0Yu

eq2:=factor(subs({J1 = J, J2 = J, J3 = J},CharEq))

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvLCQqJiwmKiZJIkpHRigiIiIpSSZvbWVnYUdGKCIiI0Y0RjQqJkY3RjRJImtHRihGNCEiIkY0LCgqJilGM0Y3RjQpRjYiIiVGNEY0KipGP0Y0RjNGNEY5RjRGNUY0RjoqJkY3RjQpRjlGN0Y0RjRGNEY6IiIhNyNGLg==

A partir do qual podemos resolver os valores de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJ0YyRjQ= Existem m\303\272ltiplas frequ\303\252ncias pr\303\263prias, mas somos instru\303\255dos a escolher apenas uma. Portanto, escolhemos aquele que \303\251 mais f\303\241cil de resolver. Escolher o primeiro fator e igual\303\241-lo a zero. omega1:=solve(J*omega^2 - 2*k = 0, omega)

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdvbWVnYTFHRig2JCooSSJKR0YoISIiKSIiIyMiIiJGM0Y1KSomRjBGNUkia0dGKEY1RjRGNSwkRi9GMTckRi9GOQ==

Observe que essa frequ\303\252ncia n\303\243o \303\251 necessariamente a frequ\303\252ncia fundamental. \303\251 apenas uma frequ\303\252ncia que satisfaz a equa\303\247\303\243o das caracter\303\255sticas. Substituindo este valor no relacionamento (-MLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJStleGVjdXRhYmxlR0Y0RjU= + K) A = 0 e resolvendo para o vetor A Amatrix:=subs(params, -omega1[1]^2*xM + xK): solve(LinearAlgebra:-GenerateEquations(Amatrix,[A1,A2,A3]))

PCUvSSNBMUc2IiwkKiYkIiIiIiIhRilJI0EzR0YlRikhIiIvSSNBMkdGJSRGKkYqL0YrRis=

O vetor Eigen para a frequ\303\252ncia escolhida \303\251 ent\303\243o 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