note5 - MEEN 364 Notes from Session on October 6, 2004 304...

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MEEN 364 Notes from Session on October 6, 2004 304 Fermier Hall, 6:00~7:00 pm 1 ) Consider the electro-mechanical system shown below. The input to the system is the voltage, ‘e(t)’ and the output of the system is the displacement of the block of mass, ‘m’. The voltage drop across the core and the force developed by the core are proportional to the current, ‘i 2 (t)’ and are given by the relations, ) ( ) ( 2 t i k t e b b = and ) ( ) ( 2 t i k t F t = , where k b and k t are constants. a) Derive the governing differential equations of motion for this system. b) Represent the derived equations in state-space form. k b i 1 (t) i 2 (t) R F(t) e(t) c e b (t) C o r e LOOP 1 LOOP 2 L m Writing the voltage balance equation for LOOP 1, we get ). ( ) ( ) ( 1 t e t V t Ri c = + ( 1 ) Similarly writing the voltage balance equation for LOOP 2, we get . 0 ) ( ) ( ) ( , 0 ) ( ) ( ) ( 2 2 2 = + = + t V t i k dt t di L t V t e dt t di L c b c b ( 2 ) Free body diagram of the block of mass, ‘m’ ) ( t kx ) ( t x b & ) ( t F Writing the Newton’s second law of motion, we get ), ( ) ( ) ( ) ( t F t kx t x b t x m = + + & & & ). ( ) ( ) ( ) ( 2 t i k t kx t x b t x m t = + + & & & ( 3 ) 1
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MEEN 364 Notes from Session on October 6, 2004 304 Fermier Hall, 6:00~7:00 pm Equations (1), (2) and (3) represent the governing differential equations of motion. State-space Representation Let the states of the system be chosen as ). ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( 4 . 3 2 1 2 t x t x t x t x t x t V t x t i c = = = = Substituting the above relations in the differential equations we get () ). ( ) ( ) ( ) ( ), ( ) ( , ) ( ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( 1 ) ( ), ( 1 ) ( ) ( 4 3 1 4 . 4 3 . 1 2 2 2 1 2 . 2 1 1 . t x m b t x m k t x m k t x t x t x t x R t x R t e C t i R t V R t e C t i t i C t x t x L t x L k t x t c b = = = = = + = Therefore representing the above equations in matrix form, we get ). ( 0 0 1 0 ) ( ) ( ) ( ) ( 0 1 0 0 0 0 0 1 1 0 0 1 ) ( ) ( ) ( ) ( 4 3 2 1 4 .
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note5 - MEEN 364 Notes from Session on October 6, 2004 304...

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