卡門方程是一個模擬平板變形的四階橢圓型非線性偏微分方程組:[1] Von Karman equation U Maple plot Von Karman equation w Maple plot Δ Δ ( u ) = a ( ( w x y ) 2 − w x x w y y ) {\displaystyle \Delta \Delta (u)=a((w_{xy})^{2}-w_{xx}w_{yy})} Δ Δ ( w ) = b ( u y y w x x + u x x w y y − 2 u x y w x y ) + c {\displaystyle \Delta \Delta (w)=b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy})+c} 其中 Δ = ∂ ∂ x 2 + ∂ ∂ y 2 {\displaystyle \Delta ={\frac {\partial }{\partial x^{2}}}+{\frac {\partial }{\partial y^{2}}}} Remove ads通解 卡門方程有下列解析解[2] u := ( 1 / 2 ∗ ( A [ 3 ] ∗ x 3 + A [ 2 ] ∗ x 2 + A [ 1 ] ∗ x + A [ 0 ] ) ) ∗ y 2 + y − ( 1 / 10 ) ∗ x 5 ∗ A [ 3 ] + x 3 + x 2 + x {\displaystyle u:=(1/2*(A[3]*x^{3}+A[2]*x^{2}+A[1]*x+A[0]))*y^{2}+y-(1/10)*x^{5}*A[3]+x^{3}+x^{2}+x} w := ∫ ( ( x − t ) ∗ f ( t ) , t = 0.. x ) + x {\displaystyle w:=\int ((x-t)*f(t),t=0..x)+x} 其中 f ″ ( x ) = b ∗ ( A [ 3 ] ∗ x 3 + A [ 2 ] ∗ x 2 + A [ 1 ] ∗ x + A [ 0 ] ) ∗ f ( x ) + c {\displaystyle f''(x)=b*(A[3]*x^{3}+A[2]*x^{2}+A[1]*x+A[0])*f(x)+c} Remove ads特解 當 A [ 2 ] = A [ 3 ] = 0 {\displaystyle A[2]=A[3]=0} 時 f ( x ) = A i r y A i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) {\displaystyle f(x)=AiryAi(-1.3200061217959123977*x+2.0087049679503014748)} ∗ C 2 + A i r y B i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) {\displaystyle *_{C}2+AiryBi(-1.3200061217959123977*x+2.0087049679503014748)} ∗ C 1 − 2.2727167324939371067 ∗ P i ∗ ( − ( I n t ( A i r y B i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) , x ) ) ∗ A i r y A i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) + ( I n t ( A i r y A i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) , x ) ) ∗ A i r y B i ( − 1.3200061217959123977 ∗ x + 2.0087049679503014748 ) ) {\displaystyle *_{C}1-2.2727167324939371067*Pi*(-(Int(AiryBi(-1.3200061217959123977*x+2.0087049679503014748),x))*AiryAi(-1.3200061217959123977*x+2.0087049679503014748)+(Int(AiryAi(-1.3200061217959123977*x+2.0087049679503014748),x))*AiryBi(-1.3200061217959123977*x+2.0087049679503014748))} 因此 u = ( 1 / 2 ∗ ( − 2.3 ∗ x + 3.5 ) ) ∗ y 2 + y + x 3 + x 2 + x {\displaystyle u=(1/2*(-2.3*x+3.5))*y^{2}+y+x^{3}+x^{2}+x} v = ∫ ( ( x − t ) ∗ ( A i r y A i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) + A i r y B i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) − 2.2727167324939371067 ∗ P i ∗ ( − ( I n t ( A i r y B i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) , t ) ) ∗ A i r y A i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) + ( I n t ( A i r y A i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) , t ) ) ∗ A i r y B i ( − 1.3200061217959123977 ∗ t + 2.0087049679503014748 ) ) ) , t ) + x {\displaystyle v=\int ((x-t)*(AiryAi(-1.3200061217959123977*t+2.0087049679503014748)+AiryBi(-1.3200061217959123977*t+2.0087049679503014748)-2.2727167324939371067*Pi*(-(Int(AiryBi(-1.3200061217959123977*t+2.0087049679503014748),t))*AiryAi(-1.3200061217959123977*t+2.0087049679503014748)+(Int(AiryAi(-1.3200061217959123977*t+2.0087049679503014748),t))*AiryBi(-1.3200061217959123977*t+2.0087049679503014748))),t)+x} Remove ads參考文獻Loading content...Loading related searches...Wikiwand - on Seamless Wikipedia browsing. On steroids.Remove ads