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• Pablo Cesar Ruiz Guzman

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Pablo Cesar Ruiz Guzman. Código: 201320088604. Aeroelasticidad. Ingeniería Aeronáutica. Facultad de Ingeniería. 1. Determine the lift distribution and divergence speed of a fixed root rectangular wing of semi-span s, chord c and torsional rigidity GJ using modified strip theory such that 𝑦2

𝑎𝑤 (𝑦) = 𝑎𝑤 (1 − 𝑠2 )

𝑦 𝛿𝐿 = 𝑞𝐶𝑎𝑤 (𝜃0 + 𝜃𝑇 ) (1) 𝑠 Se reemplaza aw en la ecuación 1 𝑠 𝑦2 𝑦 𝐿 = ∫ 𝑞𝑎𝑤 (1 − 2 ) (𝜃0 + 𝜃𝑇 ) 𝑑𝑦 𝑠 𝑠 0 𝑠 2 𝑦 𝑦 𝑦3 𝐿 = 𝑞𝑎𝑤 ∫ (𝜃0 + 𝜃𝑇 − 2 𝜃0 − 3 𝜃𝑇 ) 𝑑𝑦 𝑠 𝑠 𝑠 0 2 3 𝑦 𝑦 𝑦4 𝑠 𝐿 = 𝑞𝑎𝑤 (𝜃0 𝑦 + 𝜃𝑇 − 2 𝜃0 − 3 𝜃𝑇 ) ⃒ 0 2𝑠 3𝑠 4𝑠 𝑠2 𝑠3 𝑠4 𝐿 = 𝑞𝑎𝑤 (𝜃0 𝑠 + 𝜃𝑇 − 2 𝜃0 − 3 𝜃𝑇 ) 2𝑠 3𝑠 4𝑠 𝑠 𝑠 𝑠 𝐿 = 𝑞𝑎𝑤 (𝑠𝜃0 + 𝜃𝑇 − 𝜃0 − 𝜃𝑇 ) 2 3 4 Se simplifican términos y se reduce la expresión de la mayor forma posible 2𝑠 𝑠 𝐿 = 𝑞𝑎𝑤 ( 𝜃0 + 𝜃𝑇 ) 3 4 𝑦 𝜃 = 𝜃𝑇 𝑠 1 𝑠 𝜕𝜃 2 𝑈 = ∫ 𝐺𝐽 ( ) 𝑑𝑦 2 0 𝜕𝑦 𝜕𝜃 𝜃𝑇 = 𝜕𝑦 𝑠 𝐺𝐽 𝑠 𝜃𝑇 2 𝑈 = ∫ ( ) 𝑑𝑦 2 0 𝑠 1

𝐺𝐽𝜃𝑇2 𝑠 ∫ 𝑑𝑦 2𝑠 2 0 𝐺𝐽𝜃𝑇2 𝑈= 𝑠 2𝑠 2 𝐺𝐽 𝑈 = 𝜃𝑇2 2𝑠 𝑈=

𝑠

𝜕𝑤 = ∫ 𝑑𝐿𝑒𝑐 𝛿𝜃𝑇 0 𝑠

𝑦2 𝑦 𝑦 ) (𝜃0 + 𝜃𝑇 ) 𝑑𝑦 𝑒𝐶 𝛿𝜃𝑇 2 𝑠 𝑠 𝑠 0 𝑠 2 3 𝑦 𝑦 𝑦 𝑦 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ∫ (𝜃0 + 𝜃𝑇 − 2 𝜃0 − 3 𝜃𝑇 ) 𝑑𝑦 𝛿𝜃𝑇 𝑠 𝑠 𝑠 𝑠 0 𝑠 2 3 4 𝑦 𝑦 𝑦 𝑦 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ∫ ( 𝜃0 + 2 𝜃𝑇 − 3 𝜃0 − 4 𝜃𝑇 ) 𝑑𝑦 𝛿𝜃𝑇 𝑠 𝑠 𝑠 𝑠 0 2 3 4 𝑦 𝑦 𝑦 𝑦5 𝑠 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ( 𝜃0 + 2 𝜃𝑇 − 3 𝜃0 − 4 𝜃𝑇 ) ⃒ 𝛿𝜃𝑇 0 2𝑠 3𝑠 4𝑠 5𝑠 2 3 4 𝑠 𝑠 𝑠 𝑠5 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ( 𝜃0 + 2 𝜃𝑇 − 3 𝜃0 − 4 𝜃𝑇 ) 𝛿𝜃𝑇 2𝑠 3𝑠 4𝑠 5𝑠 𝑠 𝑠 𝑠 𝑠 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ( 𝜃0 + 𝜃𝑇 − 𝜃0 − 𝜃𝑇 ) 𝛿𝜃𝑇 2 3 4 5 𝑠 2𝑠 𝛿𝑤 = 𝑞𝑒𝐶 2 𝑎𝑤 ( 𝜃0 + 𝜃𝑇 ) 𝛿𝜃𝑇 4 15 Aplicando Lagrange a la energía potencial (U): 𝜕𝑈 𝜕(𝛿𝑤) = 𝜕𝜃𝑇 𝜕(𝛿𝜃𝑇 ) 𝜕𝑈 𝐺𝐽 𝐺𝐽 𝐺𝐽 = 𝜃𝑇2 = 2 𝜃𝑇 = 𝜃𝑇 𝜕𝜃𝑇 2𝑠 2𝑠 𝑠 𝐺𝐽 𝑠 2𝑠 𝜃𝑇 = 𝑞𝑒𝐶 2 𝑎𝑤 𝜃0 + 𝑞𝑒𝐶 2 𝑎𝑤 𝜃𝑇 𝑠 4 15 𝑑𝑒𝑠𝑝𝑒𝑗𝑎𝑚𝑜𝑠 𝜃𝑇 𝐺𝐽 2𝑠 𝑠 𝜃𝑇 − 𝑞𝑒𝐶 2 𝑎𝑤 𝜃𝑇 = 𝑞𝑒𝐶 2 𝑎𝑤 𝜃0 𝑠 15 4 𝐺𝐽 2𝑠 𝑠 2 2 𝜃𝑇 ( − 𝑞𝑒𝐶 𝑎𝑤 ) = 𝑞𝑒𝐶 𝑎𝑤 𝜃0 𝑠 15 4 2 𝑠 𝑞𝑒𝐶 𝑎𝑤 𝑠𝜃0 𝑞𝑒𝐶 2 𝑎𝑤 4 𝜃0 15𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 4 𝜃𝑇 = = = 𝜃 𝐺𝐽 15𝐺𝐽 − 2𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 60𝐺𝐽 − 8𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 0 2 𝑎 2𝑠 − 𝑞𝑒𝐶 𝑤 𝑠 15 15𝑠 15𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 60𝐺𝐽 𝜃𝑇 = 𝜃 8𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 0 1− 60𝐺𝐽 𝑒𝑐 2 𝑎𝑤 𝑠 2 15𝐺𝐽 𝑅= = 𝑞𝑑𝑖𝑣 = 2 15𝐺𝐽 𝑒𝑐 𝑎𝑤 𝑠 2 𝛿𝑤 = ∫ 𝑞𝐶𝑎𝑤 (1 −

15𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 15𝐺𝐽 = 1 − 2𝑞𝑒𝐶 2 𝑎𝑤 𝑠 2 15𝐺𝐽 Velocidad de divergencia 𝑉𝑑𝑖𝑣 = √

30𝐺𝐽 𝜌𝑒𝑐 2 𝑠 2 𝑎𝑤

2. Determine the lift distribution and divergence speed of a fixed root tapered rectangular wing of semi-span s and chord 𝑐 = 𝑐0 (1 −

𝑦2 ) 𝑠2

and torsional 𝐺𝐽 = 𝐺𝐽0 (1 −

𝑦2 ) 𝑠2

. use trip

theory. 𝑦2 ) 𝑠2 𝑦2 𝐺𝐽 = 𝐺𝐽0 (1 − 2 ) 𝑠 𝑦 𝛿𝐿 = 𝑞𝐶𝑎𝑤 (𝜃0 + 𝜃𝑇 ) (1) 𝑠 Se remplaza 𝐶 Y 𝜃 en la ecuación 1. 𝐶 = 𝐶0 (1 −

𝑠

𝐿 = ∫ 𝑞 (𝐶0 (1 − 0 𝑠

𝑦2 𝑦 )) 𝑎𝑤 (𝜃0 + 𝜃𝑇 ) 𝑑𝑦 2 𝑠 𝑠

𝑦2 𝑦 ) 𝑎𝑤 (𝜃0 + 𝜃𝑇 ) 𝑑𝑦 2 𝑠 𝑠 0 𝑠 𝑦2 𝑦 𝐿 = 𝑞𝑎𝑤 ∫ (𝐶0 − 𝐶0 2 ) (𝜃0 + 𝜃𝑇 ) 𝑑𝑦 𝑠 𝑠 0 𝑠 𝑦 𝑦2 𝑦3 𝐿 = 𝑞𝑎𝑤 ∫ (𝐶0 𝜃0 + 𝐶0 𝜃𝑇 − 𝐶0 2 𝜃0 − 𝐶0 3 𝜃𝑇 ) 𝑑𝑦 𝑠 𝑠 𝑠 0 2 3 𝑦 𝑦 𝑦4 𝑠 𝐿 = 𝑞𝑎𝑤 (𝐶0 𝜃0 𝑦 + 𝐶0 𝜃𝑇 − 𝐶0 2 𝜃0 − 𝐶0 3 𝜃𝑇 ) ⃒ 0 2𝑠 3𝑠 4𝑠 2 3 𝑠 𝑠 𝑠4 𝐿 = 𝑞𝑎𝑤 (𝐶0 𝜃0 𝑠 + 𝐶0 𝜃𝑇 − 𝐶0 2 𝜃0 − 𝐶0 3 𝜃𝑇 ) 2𝑠 3𝑠 4𝑠 𝑠 𝑠 𝑠 𝐿 = 𝑞𝑎𝑤 (𝐶0 𝜃0 𝑠 + 𝐶0 𝜃𝑇 − 𝐶0 𝜃0 − 𝐶0 𝜃𝑇 ) 2 3 4 Se simplifican los términos 2𝑠 𝑠 𝐿 = 𝑞𝑎𝑤 𝐶0 ( 𝜃0 + 𝜃𝑇 ) 3 4 Para la segunda parte se hace lo mismo que la primera donde remplazamos 𝐺𝐽 en la ecuación 1 𝑠 𝛿𝜃 2 𝑈 = ∫ 𝐺𝐽 ( ) 𝛿𝑦 2 0 𝛿𝑦 𝜕𝜃 𝜃𝑇 = 𝜕𝑦 𝑠 1 𝑠 𝑦 2 𝜃𝑇 2 𝑈 = ∫ 𝐺𝐽𝑂 (1 − 2 ) ( ) 𝑑𝑦 2 0 𝑠 𝑠 𝐿 = ∫ 𝑞 (𝐶0 − 𝐶0

𝑠 2 1 𝜃𝑇 𝑦 2 𝜃𝑇2 𝑈 = 𝐺𝐽0 ∫ 2 − 4 𝑑𝑦 2 𝑠 0 𝑠 2 1 𝜃𝑇 𝑦 3 𝜃𝑇2 𝑈 = 𝐺𝐽𝑜 ( 2 𝑦 − ) 2 𝑠 3𝑠 4 1 𝜃𝑇2 𝑠 3 𝜃𝑇2 1 𝜃𝑇2 𝜃𝑇2 𝑈 = 𝐺𝐽0 ( 2 𝑠 − = 𝐺𝐽 ) ( − ) 2 𝑠 3𝑠 4 2 0 𝑠 3𝑠 2 𝐺𝐽0 2𝜃𝑇 𝐺𝐽0 2 𝑈= 𝜃 ( )= 2 3𝑠 3𝑠 𝑇 𝑠

𝛿𝑤 = ∫ 𝑑𝐿𝑒𝑐𝛿𝜃𝑇 0

Se toman los valores de las ecuaciones integrales y se remplazan en la principal 𝑠 𝑦2 𝑦 𝑦2 𝑦 𝛿𝑤 = ∫ (𝑞 (𝐶0 − 2 𝐶0 ) 𝑎𝑤 (𝜃0 + 𝜃𝑇 ) 𝑑𝑦) 𝑒 (𝐶0 − 2 𝐶0 ) ( 𝛿𝜃𝑇 ) 𝑠 𝑠 𝑠 𝑠 0 2

𝑠

𝛿𝑤 = ∫ (𝑞 (𝐶0 − 0 𝑠

𝑦2 𝑌 𝑦2 𝐶 𝑎 𝛿𝜃 + 𝜃 𝛿𝜃 ) 𝑑𝑦) 𝑒 ) (𝜃 𝑊 0 𝑠2 0 𝑆 𝑡 𝑠2 𝑡 𝑡

𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒 ∫ (𝐶0 2 𝜃0 0

𝑌 𝑦2 𝑦3 𝑦4 𝑦5 + 𝐶0 2 2 𝜃𝑡 − 2 3 𝐶0 2 𝜃0 − 2 4 𝐶0 2 𝜃𝑡 + 5 𝐶0 2 𝜃0 𝑆 𝑠 𝑠 𝑠 𝑠

𝑦6 2 𝐶 𝜃 ) 𝑑𝑦 𝑠6 0 𝑡 𝑠 𝑌 𝑦2 𝑦3 𝑦4 𝑦5 𝑦6 𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒𝐶0 2 ∫ ( 𝜃0 + 2 𝜃𝑡 − 2 3 𝜃0 − 2 4 𝜃𝑡 + 5 𝜃0 + 6 𝜃𝑡 ) 𝑑𝑦 𝑠 𝑠 𝑠 𝑠 𝑠 0 𝑆 2 3 4 5 6 𝑦 𝑦 𝑦 𝑦 𝑦 𝑦7 𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒𝐶0 2 ( 2 𝜃0 + 2 𝜃𝑡 − 2 3 𝜃0 − 2 4 𝜃𝑡 + 5 𝜃0 + 6 𝜃𝑡 ) 𝑠 3𝑠 4𝑠 5𝑠 6𝑠 7𝑠 2 3 4 5 6 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠7 𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒𝐶0 2 ( 𝜃0 + 2 𝜃𝑡 − 2 3 𝜃0 − 2 4 𝜃𝑡 + 5 𝜃0 + 6 𝜃𝑡 ) 2𝑠 3𝑠 4𝑠 5𝑠 6𝑠 7𝑠 2 3 4 5 6 𝑠 𝑠 𝑠 𝑠 𝑠7 2 𝑠 𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒𝐶0 ( 𝜃0 + 2 𝜃𝑡 − 2 3 𝜃0 − 2 4 𝜃𝑡 + 5 𝜃0 + 6 𝜃𝑡 ) 2𝑠 3𝑠 4𝑠 5𝑠 6𝑠 7𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝛿𝑤 = 𝑞𝑎𝑤 𝛿𝜃𝑡 𝑒𝐶0 2 ( 𝜃0 + 𝜃𝑡 − 2 𝜃0 − 2 𝜃𝑡 + 𝜃0 + 𝜃𝑡 ) 2 3 4 5 6 7 𝑠 8𝑠 𝛿𝑊 = 𝑞𝑎𝑤 𝑒𝐶0 2 ( 𝜃0 + 𝜃 ) 𝛿𝜃𝑡 6 105 𝑡 Se aplica lagrange en la ecuación. 1 𝜕 (3𝑠 𝐺𝐽0 𝜃𝑡 2 ) 𝑑𝑈 1 2 = = 𝐺𝐽0 2𝜃𝑡 = 𝐺𝐽 𝜃 𝑑𝜃𝑡 𝑑𝜃𝑡 3𝑠 3𝑠 0 𝑡 𝑑(𝛿𝑊) 𝑠 8𝑠 = 𝑞𝑎𝑤 𝑒𝐶0 2 ( 𝜃0 + 𝜃) 𝑑(𝛿𝜃𝑡 ) 6 105 𝑡 2 𝑠 8𝑠 𝐺𝐽0 𝜃𝑡 = 𝑞𝑎𝑤 𝑒𝐶0 2 ( 𝜃0 + 𝜃) 3𝑠 6 105 𝑡 2 𝑠 8𝑠 𝐺𝐽0 𝜃𝑡 = 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃0 + 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃 3𝑠 6 105 𝑡 Se despeja 𝜃𝑡 de la ecuación y se simplifica 2 8𝑠 𝑠 𝐺𝐽0 𝜃𝑡 − 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃𝑡 = 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃0 3𝑠 105 6 +

2 8𝑠 𝑠 𝜃𝑡 ( 𝐺𝐽0 − 𝑞𝑎𝑤 𝑒𝐶0 2 ) = 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃0 3𝑠 105 6 2 2 210𝐺𝐽0 − 24 𝑞𝑎𝑤 𝑒𝐶0 𝑠 𝑠 𝜃𝑡 ( ) = 𝑞𝑎𝑤 𝑒𝐶0 2 𝜃0 315𝑠 6 𝑠 𝑞𝑎𝑤 𝑒𝐶0 2 6 𝜃0

𝜃𝑡 = ( 𝜃𝑡 =

210𝐺𝐽0 − 24 𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2 ) 315𝑠 315𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2

1260𝐺𝐽0 − 144𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2

𝜃0

Dividimos entre 1260𝐺𝐽0 arriba y abajo 315𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2 315𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2 1260𝐺𝐽0 4𝑥315 𝐺𝐽0 𝜃𝑡 = 𝜃0 2 2 𝜃0 = 144𝑞𝑎𝑤 𝑒𝐶0 𝑠 36𝑞𝑎𝑤 𝑒𝐶0 2 𝑠 2 1− 1− 1260𝐺𝐽0 315𝐺𝐽0 2 2 𝑎𝑤 𝑒𝐶0 𝑠 1 1 315𝐺𝐽0 𝑅= 𝑅= 𝑞𝑑𝑖𝑣 = = 315𝐺𝐽0 𝑞𝑑𝑖𝑣 𝑅 𝑎𝑤 𝑒𝐶0 2 𝑠 2 315𝑞 4𝑞𝑑𝑖𝑣 𝜃𝑡 = 𝜃 36𝑞 0 1−𝑞 𝑑𝑖𝑣 𝑞𝑑𝑖𝑣 =

315𝐺𝐽0 𝑎𝑤 𝑒𝐶0

2 2; 𝑠

1 2 315𝐺𝐽0 𝜌𝑣 𝑑𝑖𝑣 = 2 𝑎𝑤 𝑒𝐶0 2 𝑠 2

Velocidad de divergencia 360𝐺𝐽0 𝑣𝑑𝑖𝑣 = √ 𝜌𝑎𝑤 𝑒𝐶0 2 𝑠 2 3. Determine the divergence speed and lift distribution for a rectangular fixed root wing of semi-span s, chord c, bending rigidity EI and torsional rigidity GJ for wing sweep ᴧ. Use strip theory. dL = qaw cdy [(θ0 + θ)Cos Λ + Sin Λ] s

L = qaw cdy ∫0 Cos Λθ0 + θ0 KSin Λ + θCos Λ + θSin Λ s y y L = qaw cdy ∫ Cos Λθ0 + θ0 KSin Λ + θ0 Cos Λ + θT Sin Λ s s 0

S

y2 y2 θT Cos Λ + θT KSin Λ)| 2s 2s 0 s s L = qaw c (θ0 Cos ΛS + θ0 SKSin Λ + θT Cos Λ + θT KSin Λ) 2 2 L = qaw c (θ0 Cos Λy + θ0 yKSin Λ +

s y CSin Λ δw = − ∫ qaw cdy[(θ0 + θ)Cos Λ + KSin Λ] ( + ) δK Cos Λ 4 0 s CCos Λ + ∫ qaw Cdy [(θ0 + θ)Cos Λ + KSin Λ ] δθ 4 0 s y CSin Λ [(θ δw = qaw C + ) dy 0 + θ)Cos Λ + KSin Λ]δK ∫ ( 4 0 Cos Λ s CCos Λ + qaw C[(θ0 + θ)Cos Λ + KSin Λ]δθ ∫ dy 4 0 S

δw = −qaw C[(θ0 + θ)Cos Λ + KSin Λ]δK (

y2 CySin Λ + )| 2Cos Λ 4 0

CyCos Λ S + qaw C[(θ0 + θ)Cos Λ + KSin Λ]δK ( )| 4 0 S2 CSSin Λ δw = −qaw C((θ0 + θ)Cos Λ + KSin Λ)δK ( + ) 2Cos Λ 4 CSCos Λ + qaw C[(θ0 + θ)Cos Λ + KSin Λ]δθ ( ) 4 2 −S CSSin Λ CSCos Λ δw = qaw C[(θ0 + θ)Cos Λ + KSin Λ] [( − δθ] ) δK + 2Cos Λ 4 4 1 1 U = K k K 2 + K θ θ2 2 2 S2 CSSin Λ K k K = −qaw C[(θ0 + θ)Cos Λ + KSin Λ] ( + ) 2Cos Λ 4 𝜃0 𝑆 2 𝜃𝑆 2 𝐾𝑆 2 𝑆𝑖𝑛 Λ 𝐶𝑆𝜃0 𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ 𝐶𝑆𝜃𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ 𝐶𝑆𝐾 𝑆𝑖𝑛2 Λ 𝐾𝑘 𝐾 = −𝑞𝑎𝑤 𝐶 ( + + + + + ) 2 2 2𝐶𝑜𝑠 Λ 4 4 4 𝑆 2 𝑆𝑖𝑛 Λ 𝐶𝑆 𝑆𝑖𝑛2 Λ 𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ + ) + 𝑞𝑎𝑤 𝐶𝜃 ( + ) 2𝐶𝑜𝑠 Λ 4 2 4 𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ + 𝑞𝑎𝑤 𝐶𝜃0 ( + )=0 2 4

𝐾𝑘 𝐾 + 𝑞𝑎𝑤 𝐶𝐾 (

𝑆 2 𝑆𝑖𝑛 Λ 𝐶𝑆 𝑆𝑖𝑛2 Λ 𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ 𝐾 [𝐾𝑘 + 𝑞𝑎𝑤 𝐶 ( + )] + 𝜃 [𝑞𝑎𝑤 𝐶 ( + )] 2𝐶𝑜𝑠 Λ 4 2 4 𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ = −𝜃0 [𝑞𝑎𝑤 𝐶 ( + )] 2 4 K θ θ = qaw

C 2 SCos Λ [(θ0 + θ)Cos Λ + KSin Λ] 4

𝐶𝑆𝜃0 𝐶𝑜𝑠 2 Λ 𝐶𝑆𝜃𝐶𝑜𝑠 2 Λ 𝐶𝑆𝐾𝑆𝑖𝑛 Λ𝐶𝑜𝑠 Λ 𝐾𝜃 𝜃 = 𝑞𝑎𝑤 𝐶 [ + + ] 4 4 4 𝐶𝑆𝐶𝑜𝑠 2 Λ 𝐶𝑆𝑆𝑖𝑛 Λ𝐶𝑜𝑠 Λ 𝐶𝑆𝐶𝑜𝑠 2 Λ 𝐾𝜃 𝜃 − 𝑞𝑎𝑤 𝐶𝜃 [ ] − 𝑞𝑎𝑤 𝐶𝐾 [ ]=0 ] − 𝑞𝑎𝑤 𝐶𝜃0 [ 4 4 4 𝐶𝑆𝐶𝑜𝑠 2 Λ 𝑞𝑎𝑤 𝐶 2 𝑆𝑆𝑖𝑛 Λ𝐶𝑜𝑠 Λ 𝑞𝑎𝑤 𝐶 2 𝑆𝐶𝑜𝑠 2 Λ 𝜃 (𝐾𝜃 − 𝑞𝑎𝑤 𝐶 [ ]) − 𝐾 [ ] = 𝜃0 [ ] 4 4 4

𝑆 2 𝑆𝑖𝑛 Λ 𝐶𝑆 𝑆𝑖𝑛2 Λ 𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ + ) 𝑞𝑎𝑤 𝐶 ( + ) 2𝐶𝑜𝑠 Λ 4 2 4 𝐾 [ ] 2 2 𝜃 𝑞𝑎𝑤 𝐶 𝑆𝑆𝑖𝑛 Λ𝐶𝑜𝑠 Λ 𝐶𝑆𝐶𝑜𝑠 Λ − 𝐾𝜃 − 𝑞𝑎𝑤 𝐶 [ ] 4 4 [ ] 2 𝑆 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ −𝑞𝑎𝑤 𝐶 ( + ) 2 4 = 𝜃0 𝑞𝑎𝑤 𝐶 2 𝑆𝐶𝑜𝑠 2 Λ [ ] 4 𝑆 2 𝑆𝑖𝑛 Λ 𝐶𝑆 𝑆𝑖𝑛2 Λ 𝐶𝑆𝐶𝑜𝑠 2 Λ [𝐾𝑘 + 𝑞𝑎𝑤 𝐶 ( + ])] ) (𝐾𝜃 − 𝑞𝑎𝑤 𝐶 [ 2𝐶𝑜𝑠 Λ 4 4 𝐾𝑘 + 𝑞𝑎𝑤 𝐶 (

− [𝑞𝑎𝑤 𝐶 (

𝑆 2 𝐶𝑆𝐶𝑜𝑠 Λ 𝑆𝑖𝑛 Λ −𝑞𝑎𝑤 𝐶 2 𝑆𝑆𝑖𝑛 Λ𝐶𝑜𝑠 Λ + ]] = 0 )[ 2 4 4