ASE 365 - Lecture 22

# ASE 365 - Lecture 22 - Analytical dynamics In contrast to...

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Unformatted text preview: Analytical dynamics In contrast to Newtonian ( F =m a ): – Work/energy based – Scalar, rather than vector – Bypasses constraint forces unless needed – System rather than individual components – Streamlines analysis—one EOM per DOF – Independent of coordinates chosen – More abstract, less “physical” Generalized coordinates Planar dumbbell example: m 1 m 2 L x y motion). (planar freedom of degrees 3 only but : s coordinate 4 j i r j i r 2 2 2 1 1 1 , y x y x + = + = satisfied. lly automatica is : Constraint : s" coordinate d generalize " as , in , choose Or, 2 2 1 2 2 1 2 2 1 1 2 2 1 2 1 ) ( ) ( ) sin (cos , ) sin (cos L y y x x m m L m m m L m y x y x c c c c c c c =- +- + + + = + +- = + = j i r r j i r r j i r θ θ θ θ θ r c θ r 1 r 2 (x2,y2 ) (x1,y1 ) assumed. is motion D- 3 and s constraint kinematic of number the is particles, of number the is where : freedom of degrees of Number c N c N n- = 3 ) , , ( ), , , ( ), , , ( ), , , ( ), , , ( ), , , ( , ) , , ( ), , , ( ), , , ( 2 1 2 1 2 1 2 1 2 2 2 1 2 2 2 1 2 2 2 1 1 1 2 1 1 1 2 1 1 1 n N N n N N n N N n n n n n n q q q z z q q q y y q q q x x q q q z z q q q y y q q q x x q q q z z q q q y y q q q x x = = = = = = = = = : tion) transforma e (coordinat s coordinate d generalize of terms in s coordinate physical Express Principle of virtual work variations to related ) ( antaneous neous/inst contempora arbitrary s) constraint kinematic with t (consisten admissible imagined zero) terms order- higher (so mal infinitesi : ... , , , nts displaceme virtual define First, • = • • • • → • 1 1 1 t z z y x N δ δ δ δ δ s. derivative als, differenti like handled be Can force. (reaction) constraint force, applied where : particle th on force Resultant = = = + = i i i i i N i i f F f F R , , 2 , 1 , N i W i i i i i i , , 2 , 1 , = = ⋅ ≡ = r R r R...
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ASE 365 - Lecture 22 - Analytical dynamics In contrast to...

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