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modeling16

# modeling16 - Modeling QUESTION 16 x Linearize the...

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Modeling Q UESTION 16 Linearize the differential equation ( 29 ( 29 ( 29 ( 29 3 2 cos 0 x t x t x t x t + + - = && & . The two states are ( 29 ( 29 1 x t x t = (1) ( 29 ( 29 2 x t x t = & (2) The two first order nonlinear differential equations are ( 29 ( 29 ( 29 1 1 2 f t x t x t = = & (3) ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 1 1 3 2 cos f t x t x t x t x t = = - - + & (4) At equilibrium, the derivatives are equal to zero and equation (3) becomes 2 2 0 0 x x = = (5) At equilibrium equation (4) becomes ( 29 ( 29 2 1 1 1 1 0 3 2 cos 2 cos 0 x x x x x = - - + - = (6) Using graphical techniques or the bisection method to solve equation (6), 1 0.4502 x = . The first incremental state and its first derivative with respect to time are ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 ˆ ˆ x t x t x x t x t = - = & & (7) The second incremental state and its first derivative with respect to time are ( 29 ( 29 ( 29 ( 29 2

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modeling16 - Modeling QUESTION 16 x Linearize the...

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