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Unformatted text preview: 3- 68Problems and Solutions Section 3.8 (3.53 through 3.56) 3.53Show that a critically damped system is BIBO stable. Solution: For a critically damped system h t!"( )=1mt!"( )e!#nt!"()Let f(t) be bounded by the finite constant M. Using the inequality for integrals and Equation (3.96) yields: x t( )!f(")t#h t$"( )d"=Mt#1mt$"( )e$%nt$"()d"The function h(t– τ) decays exponentially and hence is bounded by some constant times 1/t, say M1/t. This is just a statement the exponential decays faster then “one over t” does. Thus the above expression becomes; x(t)<MM1tt!d"=MM1This is bounded, so a critically damped system is BIBO stable. 3- 693.54Show that an overdamped system is BIBO stable. Solution: For an overdamped system, h t!"( )=12m#n$2!1e!$#nt!"()e#n$2!1%&’()*t!"()!e!#n$2!1%&’()*t!"()%&’()*Let f(t) be bounded by M, From equation (3.96), x t( )!Mt"h t#$( )d$x t( )!Mt"12m%n&2#1e#&%nt#$()e%n&2#1’()*+,t#$()#e#%n&2#1’()*+,t#$()’()*+,d$x t( )!...
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This note was uploaded on 03/31/2009 for the course MECHANICAL MAE351 taught by Professor J.g.lee during the Spring '09 term at Korea Advanced Institute of Science and Technology.
- Spring '09