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

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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
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2 Inverted Double Pendulum –equation of motion The equations of motion can be determined by us ing Lagrange’s 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
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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 & % ! $ ! $ ! $ ! ! ! # ! ! ! ! ! !
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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 ( % # ) ( % ! $ ! $ ! $ ! ( % (! # ! $ ! $ (! # ! $ ! & ( % (! ! $ (! ! & % ! % !
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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 ! % !
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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 " % ! $ ! ! # ! "! $ ! ! # ! ! # ! " % # ! ! ! # ! "! " % # ! # ! $ ! "! ! $ ! ! # ! $ ! ! # ! & # ! $ ! # ! % ! !! !! ! ! ! ! ! ! !! !! !
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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 ! 1: Equation for ! 2: 2 2 1 2 1 1 2 1 1 2 2 [ /3 cos( )/ 4 s in( )/ 4] k k (P M)s in ! $ ! ! # ! $ ! ! # ! # ! $ ! % $ ! !! !! ! 2 1 2 2 1 2 2 1 1 2 1 [4 /3 cos( )/ 4 s in( )/ 4] 2k k (P 3M)s in ! $ ! ! # ! # ! ! # ! $ ! # ! % $ ! !! !! ! 2 2 2 k k /ml, P Pl /ml, M mgl / 2ml * % %
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8 Discrete dynamical systems… … . 2. Inverted Double Pendulum with Follower Force : Cons ider the same system as in last example, except that the force P changes direction depending on the orientation of the body on which it acts. The force P now acts at an angle + to the rod AB and always maintains this direction relative to the rod regardless of the pos ition in space of the system during its osc illations. ! 1 ! 2 l l k 2 k 1 P g O A y x + B
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9 Inverted Double Pendulum with Follower… .– equations of motion Note that the onl y change is in the effect of the external force P. The potential and kinetic energy express ions remain the same. So, 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 % $ % !
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