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 analysisone 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 r c 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 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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