set2 - 1 2. Some analytical models of nonlinear physical...

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Unformatted text preview: 1 2. Some analytical models of nonlinear physical systems (a) Discrete Dynamical Systems: 1. Inverted Double Pendulum : Cons ider the double pendulum shown:- l inear tors ional stiffnesses l- length of each rod P - external (conservative) load m mass of each rod 1 2 k ,k ! 1 ! 2 l l k 2 k 1 P g O A y x 2 Inverted Double Pendulum equation of motion The equations of motion can be determined by us ing Lagranges equations: Here, are the generalized coordinates , T and V are, respectivel y, the kinetic and potential Energies for the system, are the generalized forces due to non-conservative effects. nc i i i i d T T V ( ) Q , i 1,2,3, ...... dt q q q " " " # $ % % " " " ! i q nc i Q 3 Inverted Double Pendulum equations of motion For the double pendulum: kinetic energy: 2 2 2 2 1 2 G1 G1 1 G2 G2 2 2 2 1 1 G2 A G2/ A A 1 OA OA 1 1 A 1 1 1 G2/ A 2 G2/ A G2/ A 2 2 G2/ A 2 2 2 T T T [ (mv I ) (mv I )] /2 T (ml/3) /2; v v v ; v k r r l (cos i s in j ); v l ( s in i cos j ); v k r ; r l (cos i s in j )/2; v l ( s in i cos j )/2 % $ % $ ! $ $ ! & % ! % $ % ! % ! $ ! % ! # ! $ ! % ! % ! $ ! % ! # ! $ ! ! ! ! ! ! ! ! 2 2 2 2 2 2 2 1 2 1 2 2 1 ; T (ml/12) /2 ml[ /4 cos( )/2] /2 & % ! $ ! $ ! $ ! ! ! # ! ! ! ! ! ! 4 Inverted Double Pendulum equations of motion potential energy: The work done by the external force P in a v irtual displacement f rom straight vertical pos ition is: 1 2 1 2 2 1 2 1 1 2 2 1 V V V mglcos /2 mg[lcos lcos /2] [k k ( ) ] /2 % $ % ! $ ! $ ! $ ! $ ! # ! 1 1 2 2 1 1 1 2 2 2 1 1 2 2 1 1 2 2 B B B W P i r ; r l (cos i s in j ) l (cos i s in j ) r l ( s in i cos j ) l ( s in i cos j ) W P[ ls in ls in ] The are: Q Pls in ; Q Pls in ; general ized forces ( % # )( % ! $ ! $ ! $ ! ( % (! # ! $ ! $ (! # ! $ ! & ( % (! ! $ (! ! & % ! % ! 5 Inverted Double Pendulum equations of motion: Equation for ! 1: 2 1 1 2 2 1 1 2 1 2 2 1 2 2 1 2 1 1 1 1 2 1 2 2 1 2 2 1 2 2 1 2 2 1 1 2 1 2 2 d T ( ) ml[ /3 cos( )/4 dt ( )s in( )/4] T ml s in( )/4 V 3mgls in /2 (k k ) k ml[4 /3 cos( )/4 s in( )/4] (k k ) k 3mgls in " % ! $ ! $ ! ! # ! "! $ ! ! # ! ! # ! " % ! ! ! # ! "! " % # ! $ $ ! # ! "! ! $ ! ! # ! # ! ! # ! & $ $ ! # ! # !! !! !! ! ! ! ! ! ! !! !! ! 1 1 /2 Pls in ! % ! 6 Inverted Double Pendulum equations of motion: Equation for ! 2: 2 2 1 2 1 2 1 1 2 2 1 2 2 1 2 1 2 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 2 1 2 2 2 2 d T ( ) ml[ /3 cos( )/4 dt ( )s in( )/4] T ml s in( )/4 V mgls in /2 k k ml[ /3 cos( )/4 s in( )/4] k k mgls in /2 Pls in " % ! $ ! ! # ! "! $ ! ! # ! ! # ! " % # ! ! ! # ! "! " % # ! # ! $ ! "! ! $ ! ! # ! $ ! ! # ! & # ! $ ! # ! % ! !! !! ! ! ! ! ! ! !! !! ! 7 Inverted Double Pendulum equations of motion: W e now cons ider a s impl if ied vers ion with k 1 = k 2 =k Let Then, the equations are: Equation for !...
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set2 - 1 2. Some analytical models of nonlinear physical...

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