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ASE 365 - Lecture 15

ASE 365 - Lecture 15 - 2DOF systems—recap F x x x = K C M...

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Unformatted text preview: 2DOF systems—recap F x x x = + + K C M : system prototype" " for EOMs Derived k 1 c1 m 1 c2 m 2 k 2 F2 F1 x2 x 1 (coupled) full some but diagonal some symmetric, Matrices EOMs d transforme ; s" coordinate " of set different Chose ops row elementary by back it got symmetry; matrix Lost matrix T with tion transforma coordinate ed Systematiz Another 2DOF system x(t ) θ(t ) F(t ) c a b L k1 k2 m, IC C FB D fS1 fS2 F C a b c ) ( ) ( 2 1 2 1 θ θ b x k a x k F f f F x m S S +--- =-- = b b x k a a x k Fc b f a f Fc I S S C ) ( ) ( 2 1 2 1 θ θ θ +-- + =- + = Measured relative to equilibrium (ignore weight): F b k a k x k k x m = +- + + + θ ) ( ) ( 2 1 2 1 Fc b k a k x b k a k I C = + + +- + θ θ ) ( ) ( 2 2 2 1 2 1 EOMs: Matrix form: = + +- +- + + Fc F x b k a k b k a k b k a k k k x I m C θ θ 2 2 2 1 2 1 2 1 2 1 Matrices are symmetric. Equations are elastically coupled, but inertially uncoupled. Coordinate transformation e1(t ) F(t ) c a b L k1 k2 m, IC C e2(t ) θ(t ) x(t ) = - = - + = ⇒ - = ⇒ + =- = 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 , e e T e e a b L e e a b a b x x b a e e b x e a x e θ θ θ θ = - + +- +- + + - Fc F e e a b L b k a k b k a k b k a k k k e e a b L I m C 2 1 2 2 2 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 Introduce into EOMs: = + +- + + + - Fc F e e b a b k a b a k b a k a b k L e e I I ma mb L C C 2 1 2 1 2 1 2 1 ) ( ) ( ) ( ) ( 1 1 = - + - Fc F e e b k a k k k e e I I ma mb L C C 2 1 2 1 2 1 2 1 1 . 1 1 1 - = a b L T T by multiply Now - = - - + - - Fc F a b L e e b k a k k k a b L e e I I ma mb a b L C C 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 +- = +-- + + +-- + ) ( ) ( 1 ) ( ) ( ) ( ) ( 1 1 2 1 2 1...
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ASE 365 - Lecture 15 - 2DOF systems—recap F x x x = K C M...

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