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# note6 - MEEN 364 Notes from Session on 304 Fermier Hall...

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MEEN 364 Notes from Session on October 13, 2004 304 Fermier Hall, 6:00~7:00 pm 1) Problem 2.31 of the textbook “Feedback control of dynamic systems”, 4 th Edition, by Franklin et al. (a) At equilibrium: 0 ) ( 0 = + = = + = v u g i dt dv v i dt di ( 1 ) v i = also 0 ) ( = i v u g 0 ) 4 ) )(( 1 ) )(( ( = v v u v u v u Since, 1 = u 0 ) 2 2 ( ) 3 )( )( 1 ( 2 = + = v v v v v v v 3 1 , 0 ± = v From (1): 3 1 , 0 0 0 ± = = v i (b) Writing the deviation equation about the equilibrium: () v v i v i v i i v i i v i v i v i + + + + + = = = = = = = 0 0 0 0 0 0 & 0 0 0 0 1 1 00 11 i i v v u u ii vv uu v i gu v i v u u ig uv i v iv = = = = = = == ∆ =−+ + −+ ∂∂ +− + − & ( 2 ) we get: v i v i i + = + = 0 & u v i v i u v + = + + + = 3 3 ) 3 ( ) 1 ( ) 3 ( 0 & u v i v i dt d + = 3 0 3 1 1 1 ( 3 ) (c) In general the linearized form will be: 1

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MEEN 364 Notes from Session on October 13, 2004 304 Fermier Hall, 6:00~7:00 pm u u g v i v g v i dt d u u v v i i u u v v i i + = = = = = = = 0 0 0 0 0 0 0 1 1 1 Since 1 0 = u ) 2 2 ( ) , 1 ( ) , ( 2 + = = v v v v g v u g 3 2 5 ) 2 2 ( ) 2 2 ( 2 m = + + + = v v v v v g when 3 1 ± = v Also, note 3 2 5 ± = = v g u g u v i v i dt d ± + = 3 2 5 0 3 2 5 1 1 1 m (4) Equation (4) represents the linearized differential equations of motion. 2
MEEN 364 Notes from Session on October 13, 2004 304 Fermier Hall, 6:00~7:00 pm 2) The robot shown the following figure has the differential equations of motion given. Symbols m 1 , m 2 , I 1 , I 2 , l 1 and g are constant parameters, representing the characteristics of the rigid body links. Quantities θ 1 and d 2 are the coordinate variables and are functions of time. The inputs are τ 1 and 2 . Linearize the two equations about the operating point 0 1 = , , and Give your results in the state space form.

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note6 - MEEN 364 Notes from Session on 304 Fermier Hall...

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