SolSec3_8 - 3 68 Problems and Solutions Section 3.8(3.53...

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3- 68 Problems and Solutions Section 3.8 (3.53 through 3.56) 3.53 Show that a critically damped system is BIBO stable. Solution: For a critically damped system h t ! " ( ) = 1 m t ! " ( ) e ! # n t ! " ( ) Let f ( t ) be bounded by the finite constant M . Using the inequality for integrals and Equation (3.96) yields: x t ( ) ! f ( " ) 0 t # h t $ " ( ) d " = M 0 t # 1 m t $ " ( ) e $ % n t $ " ( ) d " The function h ( t τ ) decays exponentially and hence is bounded by some constant times 1/ t , say M 1 / t . This is just a statement the exponential decays faster then “one over t does. Thus the above expression becomes; x ( t ) < M M 1 t 0 t ! d " = MM 1 This is bounded, so a critically damped system is BIBO stable.
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3- 69 3.54 Show that an overdamped system is BIBO stable. Solution: For an overdamped system, h t ! " ( ) = 1 2 m # n $ 2 ! 1 e ! $# n t ! " ( ) e # n $ 2 ! 1 % & ( ) * t ! " ( ) ! e ! # n $ 2 ! 1 % & ( ) * t ! " ( ) % & ( ) * Let f ( t ) be bounded by M , From equation (3.96), x t ( ) ! M 0 t " h t # $ ( ) d $ x t ( ) ! M 0 t " 1 2 m % n & 2 # 1 e # &% n t # $ ( ) e % n & 2 # 1 ( ) * + , t # $ ( ) # e # % n & 2 # 1 ( ) * + , t # $ ( ) ( ) * + , d $ x t ( ) ! M 2 m " n # 2 $ 1 $ 1 " n # 2 $ 1 $ #" % & ( ) * * 1 $ e " n # 2 $ 1 $ #" n % & ( ) * t % & ( ) * + , - - $ $ 1 " n # 2 $ 1 + #" n % & ( ) * * 1 $ e " n # 2 $ 1 $ #" n % & ( ) * t % & ( ) * . / 0 0 Since ! n " 2 # 1 # "! n < 0, then 1 ! e " n # 2 ! 1 ! #" n $ % & ( ) t is bounded. Also, since - ! n " 2 # 1 # "! n < 0, then 1 ! e " n # 2 ! 1 ! #" n $ % & ( ) t is bounded. Therefore, an overdamped system is BIBO stable.
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3- 70 3.55 Is the solution of 2 !! x + 18 x = 4cos2 t + cos t Lagrange stable? Solution: Given 2 !! x + 18 x = 4cos2 t + cos t ! n = k m = 3 The total solution will be x t ( ) = x h t ( ) + x P 1 t ( ) + x P 2 t ( ) From Eq. (1.3): x h t ( ) = A sin
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