Differential Equations Solutions 124

Differential Equations Solutions 124 - 134 Chapter 22...

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134 Chapter 22. Solutions: Case Study: Robot Control: Swinging Like a Pendulum so the solution to the diferential equation is y ( t )= α 1 · 1 λ 1 ¸ e λ 1 t + α 2 · 1 λ 2 ¸ e λ 2 t , where α 1 and α 2 are constants determined by two additional conditions. IF the discriminant satis±es c 2 / (4 m 2 ± 2 ) g/± > 0, then the solution decays; otherwise it can have an oscillatory component in addition to a decaying one. CHALLENGE 22.2. We note that v (0 , 0) = 0, and it is easy to see that v> 0 For all other values oF its arguments. We diferentiate: d d t v ( y ( t )) = g sin θ ( t ) ± d θ ( t ) d t + d θ ( t ) d t d 2 θ ( t ) d t 2 = g sin θ ( t ) ± d θ ( t ) d t d θ ( t ) d t 1 ( c d θ ( t ) d t + mg sin( θ ( t ))) = c µ d θ ( t ) d t 2 0 . ThereFore, we can now conclude that the point θ =0 , d θ/ d t = 0 is stable For both the damped ( c> 0) and undamped ( c = 0) cases. ²or the undamped case, d v ( y ( t )) / d t is identically zero, and we cannot conclude that we have asymptotic stability. ²or the damped case, we note that the set de±ned by d v ( y ( t )) / d t = 0 contains all points ( θ, d d t = 0), and the only invariant set is the one containing the single point (0
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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