SolSec3.8 - 3- 49 Problems and Solutions Section 3.8 (3.46...

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3- 49 Problems and Solutions Section 3.8 (3.46 through 3.49) 3.46 Show that a critically damped system is BIBO stable. Solution: For a critically damped system ht m te n t () =− −− ττ ωτ 1 Let f ( t ) be bounded by the finite constant M . Using the inequality for integrals and Equation (3.96) yields: xt f d M m d tt t n ≤− ∫∫ τ 00 1 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; M M t dM M t <= 1 0 1 This is bounded, so a critically damped system is BIBO stable.
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3- 50 3.47 Show that an overdamped system is BIBO stable. Solution: For an overdamped system, ht m ee e n t tt n nn () = −− τ ωζ ζω 1 21 2 11 22 Let f ( t ) be bounded by M , From Eq. (3.96), xt M ht d M m e d t t n t n ≤− () ≤ 0 0 2 1 ττ M m e e t t
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SolSec3.8 - 3- 49 Problems and Solutions Section 3.8 (3.46...

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