6. CONCLUSÕES
6.1. Sugestão de trabalhos Futuros
A gPCE apresentou bons resultados e é uma alternativa aos métodos de Monte Carlo para aplicações em análise de incertezas de sistemas rotativas. A seguir são feitas algumas
sugestões de trabalhos futuros relevantes a este tema, os quais visam o prosseguimento da pesquisa:
• Estudo e formulação de métodos adaptativos da estimação dos coeficientes de expansão. Estes métodos calculam os coeficientes de expansão mais relevantes, já que termos de maior ordem acabam não tendo coeficientes elevados. Assim obtém-se uma aproximação com mesma precisão, porém com menos termos;
• Aplicação de grades esparsas na Inferência Bayesiana com Colocação estocástica;
• Aplicar expansão polinomial do caos generalizado em problemas de otimização estocástica.
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ANEXO A – MATRIZES DO ELEMNETO DE VIGA CILINDRICA
A partir das equações de energia cinética e energia potencial aplica-se a equação de Lagrange e obtém-se a equação de movimento do elemento de viga cilíndrica:
[𝑀]{𝑞̈} − ([𝐶] + 𝜔 ∙ [𝐺]){𝑞̇}([𝐾𝐵] − [𝐾𝐴] − [𝐾𝑇]){𝑞} = {𝐹} (A.1)
A seguir, são apresentadas as equações das matrizes.
Matriz de massa [𝑀] = [ 𝛾1 𝑠𝑖𝑚. 0 𝛾1 0 −𝛾2𝑙 𝛾3𝑙2 𝛾2𝑙 0 0 𝛾3𝑙2 𝛾4 0 0 𝛾5𝑙 𝛾1 0 𝛾4 −𝛾5𝑙 0 0 𝛾1 0 𝛾5𝑙 𝛾6𝑙2 0 0 𝛾 2𝑙 𝛾3𝑙2 −𝛾5𝑙 0 0 𝛾6𝑙2 −𝛾 2𝑙 0 0 𝛾3𝑙2] (A.2) Matriz giroscópica [𝐺] = [ 0 𝑎𝑛𝑡𝑖 − 𝑠𝑖𝑚. −𝛾7 0 𝛾8𝑙 0 0 0 𝛾8𝑙 −𝛾9𝑙2 0 0 −𝛾7 𝛾8𝑙 0 0 𝛾7 0 0 𝛾8𝑙 −𝛾7 0 𝛾8𝑙 0 0 −𝛾10𝑙2 𝛾8𝑙 0 0 0 𝛾8𝑙 𝛾10𝑙2 0 0 −𝛾8𝑙 −𝛾9𝑙 0 ] (A.3) Na qual:
𝛾1 = (156 + 3528Φ + 20160Φ2)𝛼𝑇+ 36𝛼𝑅 𝛾2 = (22 + 462Φ + 2520Φ2)𝛼𝑇+ (3 − 180Φ)𝛼𝑅 𝛾3 = (4 + 84Φ + 504Φ2)𝛼𝑇+ (4 + 60Φ + 1440Φ2)𝛼𝑅 𝛾4 = (54 + 1512Φ + 10080Φ2)𝛼 𝑇− 36𝛼𝑅 𝛾5 = (13 + 378Φ + 2520Φ2)𝛼 𝑇− (3 − 180Φ)𝛼𝑅 𝛾6 = −(3 + 84Φ + 504Φ2)𝛼𝑇− (1 + 60Φ − 720Φ2)𝛼𝑅 𝛾7 = 27𝛼𝑅 𝛾8 = 2(3 − 180Φ)𝛼𝑅 𝛾9= 2(4 + 60Φ + 1440Φ2)𝛼 𝑅 𝛾10 = 2(1 + 60Φ)𝛼𝑅 (A.4) e 𝛼𝑇 = 𝜌𝐴𝑙 420(1 + 12Φ)2 𝛼𝑅 = 𝜌𝐴𝑟 2 120𝑙(1 + 12Φ)2 (A.5)
Matriz de rigidez à flexão
[𝐾𝐵] = 𝐸𝐽𝑇 𝑙3(1 + 12Φ) [ 12 𝑠𝑖𝑚. 0 12 0 −6𝑙 𝛾11 6𝑙 0 0 𝛾11 −12 0 0 −6𝑙 12 0 −12 6 0 0 12 0 −6𝑙 𝛾12 0 0 6𝑙 𝛾11 6𝑙 0 0 𝛾12 −6𝑙 0 0 𝛾11] 𝛾11 = 4𝑙2(1 + 3Φ) 𝛾12 = 2𝑙2(1 − 6Φ) (A.6)
[𝐾𝐴] = 𝐹𝑥 30𝑙 [ 36 𝑠𝑖𝑚. 0 36 0 −3𝑙 4𝑙2 3𝑙 0 0 4𝑙2 −36 0 0 −3𝑙 36 0 −36 3𝑙 0 0 36 0 3𝑙 −𝑙2 0 0 3𝑙 4𝑙2 3𝑙 0 0 −𝑙2 −3𝑙 0 0 4𝑙2 ] (A.7)
Matriz de rigidez à torção
[𝐾𝑇] = 𝑀𝑥 [ 0 0 1/𝑙 0 0 0 −1/𝑙 0 0 0 0 1/𝑙 0 0 0 −1/𝑙 1/𝑙 0 0 −1/2 −1/𝑙 0 0 1/2 0 1/𝑙 1/2 0 0 −1/𝑙 −1/2 0 0 0 −1/𝑙 0 0 0 1/𝑙 0 0 0 0 −1/𝑙 0 0 0 1/𝑙 −1/𝑙 0 0 −1/2 1/𝑙 0 0 1/2 0 −1/𝑙 1/2 0 0 1/𝑙 −1/2 0 ] A (A.7) Com Φ =12𝐸𝐽𝑇𝜒 𝐺𝐴𝑙2 (A.8) 𝜒 = 6(1 + 𝜐) (1 +𝑟𝑖 2 𝑟𝑒2) 2 (7 + 6𝜐) (1 +𝑟𝑖2 𝑟𝑒2) 2 + (20 + 12𝜐)𝑟𝑖 2 𝑟𝑒2 (A.9)
ANEXO B – MODELOS DE MANCAIS HIDRODINÂMICOS
Modelo linear
Um dos modelos adotados nesse trabalho foi desenvolvido por Machado e Cavalca (2009) e o programa para a determinação dos coeficientes por Tuckmantel (2010). Nesse modelo, os coeficientes dinâmicos são determinados a partir das forças de reação do fluido lubrificante. {𝐹𝑦 𝐹𝑧} = − ∫ ∫ 𝑝 ∙ { sen(𝜃) 𝑐𝑜𝑠(𝜃)} ∙ 𝑅 ∙ 𝑑𝜃𝑑𝑧 𝜃2 𝜃1 𝐿/2 −𝐿/2 (B.1)
sendo 𝑝 é a distribuição de pressão no fluido, 𝐿 é o comprimento axial do mancal, 𝑅 é o raio do mancal e 𝜃 é a coordenada angular no sentido anti-horário.
A equação de Reynolds modela a distribuição de pressão dentro da folga radial do mancal: 𝜕 𝜕𝑧(ℎ 3𝜕𝑝 𝜕𝑧) + 𝜕 𝜕𝑦(ℎ 3𝜕𝑝 𝜕𝑦) = 6𝜇𝑓𝑈 𝜕ℎ 𝜕𝑧+ 12𝜇 𝜕ℎ 𝜕𝑡 (B.2)
sendo 𝑦 e 𝑧 as coordenadas cartesianas, horizontal e vertical, respectivamente, 𝑡 é o tempo, 𝜇𝑓 é a viscosidade absoluta do fluido, 𝑈 é a velocidade periférica linear do mancal e ℎ é a espessura de filme, que é função da geometria do mancal.
ℎ = ℎ0+ 𝑧 ∙ cos(𝜃) + 𝑦 ∙ 𝑠𝑒𝑛(𝜃) (B.3)
sendo ℎ0 é a folga radial e 𝑦 e 𝑧 as coordenadas do centro do rotor.
As forças de reação são funções do deslocamento do centro do rotor, 𝑦 e 𝑧, e de suas velocidades instantâneas, 𝑦̇ e 𝑧̇. Realizando a expansão por série de Taylor de primeira ordem em torno do ponto de equilíbrio estático (𝑦0, 𝑧0), para pequenos deslocamentos do centro
𝐹𝑦 = 𝐹𝑦0 + 𝐾𝑦𝑧Δ𝑧 + 𝐾𝑦𝑦Δ𝑦 + 𝐶𝑦𝑧Δ𝑧̇ + 𝐶𝑦𝑦Δ𝑦̇
𝐹𝑧= 𝐹𝑧0+ 𝐾𝑧𝑧Δ𝑧 + 𝐾𝑧𝑦Δ𝑦 + 𝐶𝑧𝑧Δ𝑧̇ + 𝐶𝑧𝑦Δ𝑦̇ (B.4)
Desta forma, os coeficientes dinâmicos são as derivadas parciais:
𝐾𝑧𝑦= (𝜕𝐹𝑧 𝜕𝑦)0 𝐶𝑧𝑦 = (𝜕𝐹𝑧
𝜕𝑦̇)0
(B.5)
de forma similar, os outros coeficientes são calculados e adicionados nas matrizes de rigidez e amortecimento.
Modelo não linear
Um método baseado nos trabalhos de Capone (1986 e 1991) também será utilizado. Este utiliza um modelo não linear do mancal para calcular as forças hidrodinâmicas.
Assim como o método linear, este começa com a solução da equação de Reynolds. Porém, o mancal é considerado curto. Desta forma, os diferenciais de pressão dependentes da direção circunferencial na Equação de Reynolds podem ser desconsiderados e a pressão resultante é dada pela equação:
𝑝 =1 2( 𝑙 𝑑) 2 [∙(𝑦 − 2𝑧̇) ∙ sen(𝜃) − (𝑧 − 2𝑦̇) ∙ cos(𝜃) (1 − 𝑦 ∙ cos(𝜃) − 𝑧 ∙ sen(𝜃))3 ] ∙ (4𝑥2− 1) (B.6)
sendo 𝑙 e 𝑑 o comprimento e o diâmetro do mancal, respectivamente, e 𝜃 a coordenada angular no mancal. Assim as forças hidrodinâmicas podem ser determinadas pela equação: 𝑓ℎ = { 𝑓ℎ𝑦 𝑓ℎ𝑧} = −𝜇𝜔 (𝑟 ∙ 𝑙)3 (𝑐 ∙ 𝑑)2 [(𝑦 − 2𝑧̇)2+ (𝑧 − 2𝑦̇)2]12 1 − 𝑦2− 𝑧2 ∙ {3𝑦 ∙ 𝑣 − 𝑔 ∙ sen(𝛼) − 2𝑏 ∙ cos (𝛼) 3𝑧 ∙ 𝑣 − 𝑔 ∙ cos(𝛼) − 2𝑏 ∙ sen (𝛼)} (B.7)
sendo 𝑣 =2 + (𝑧 ∙ cos(𝛼) − 𝑦 ∙ sen(𝛼)) ∙ 𝑔 1 − 𝑦2− 𝑧2 𝑏 =𝑦 ∙ cos(𝛼) + 𝑧 ∙ sen(𝛼) 1 − 𝑦2− 𝑧2 𝑔 = 𝜋 √1 − 𝑦2 − 𝑧2− 2 √1 − 𝑦2− 𝑧2tan −1(𝑧 ∙ cos(𝛼) − 𝑦 ∙ sen(𝛼) √1 − 𝑦2− 𝑧2 ) 𝛼 = tan−1(𝑧 + 2𝑦̇ 𝑦 − 2𝑧̇) − 𝜋 2sign ( 𝑧 + 2𝑦̇ 𝑦 − 2𝑧̇) − 𝜋 2sign(𝑧 + 2𝑦̇) (B.8)
As forças hidrodinâmicas calculadas pela são adicionadas no lado direito da equação de movimento, como uma força de excitação externa.
ANEXO C – OBTENÇÃO DE DISTRIBUIÇÕES CONHECIDAS
PELO PRINCÍPIO DA MÁXIMA ENTROPIA
Algumas distribuições conhecidas, como uniforme, exponencial, gaussiana etc., podem ser obtidas através do Princípio da Máxima Entropia. Algumas distribuições serão apresentadas no presente anexo, Sampaio e Lima (2012) apresentam mais outras distribuições.
Distribuição uniforme
Caso seja conhecido apenas os limites finitos [𝑎, 𝑏] da variável aleatório, a equação (4.10) será escrita como:
𝑝(𝜆) = 𝟙[𝑎,𝑏](𝜆) exp(−𝛾0) (C.1)
Como
𝜕2
𝜕𝑝(𝜆)2ℎ(𝑝, 𝛾0) > 0 (C.2)
A distribuição que maximiza a entropia de informação é a distribuição uniforme:
𝑝(𝜆) = 𝟙[𝑎,𝑏](𝜆) ∙ 1
𝑏 − 𝑎 (C.3)
Distribuição exponencial
Caso seja conhecida a média 𝐸[𝜆] = 𝜇 e seu suporte seja [0, ∞), tem-se os vínculos e função densidade de probabilidade:
∫ 𝑝(𝜆) ∞ 0 𝑑𝜆 = 1, ∫ 𝑝(𝜆) ∙ 𝜆 ∞ 0 𝑑𝜆 = 𝜇 (C.4) 𝑝(𝜆) = 𝟙[𝑎,𝑏](𝜆) exp(−𝛾0− 𝛾1∙ 𝜆) (C.5)
Calculando os multiplicadores de Lagrange 𝛾0 e 𝛾1, obtêm-se uma distribuição
idêntica à da distribuição exponencial:
𝑝(𝜆) = 𝟙[0,∞)(𝜆) ∙1
𝜇exp (− 𝜆
𝜇) (C.6)
Distribuição gaussiana
Caso a variável aleatória esteja no suporte (−∞, ∞) , com média 𝐸[𝜆] = 𝜇 e segundo momento 𝐸[𝜆2] = 𝛼2 tem-se os vínculos e função densidade de probabilidade: