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Unformatted text preview: Response to nonperiodic inputs—Ch. 4 “Unit impulse”: ∫ ∞ ∞ = ≠ = 1 ) ( , ) ( dt a t a t a t δ δ for : definition al Mathematic Conceptually: (limit process) e (t a) t a 1/ e Area remains one as pulse narrows ∫ ∞ ∞ = ) ( ) ( ) ( a f dt a t t f δ : property Sampling duration short magnitude, high : on idealizati force Impulsive" " Impulse response ) ( ) ( t t F kx x c x m δ = = + + momentum in change For know we , For = = = + ∫ ∫ + + ) ( ) ( , . 1 ) ( ) ( x m x m dt x m x m dt t t δ δ : at applied impulse unit to response SDOF = t law. 2nd s Newton' in forces of summation the represents EOM : impulse the of duration the over term each in happens what Consider Impulsemomentum: change in momentum determined by impulses of F(t), fD, and fS. An ideal impulse (infinite force → infinite acceleration) changes velocity instantaneously (discontinuous jump). But velocity stays finite , so displacement is continuous. After t=0+, F(t)=0, so impulse response = free vibration with . / 1 ) ( , 1 ) ( ) ( . , . ) ( ) ( m x x m x m t kx dt kx kx cx cx dt x c x c ave = = + + = ∆ = = = + + + ∫ ∫ + + so gives EOM For , For . ) ( , / 1 ) ( = = + + x m x Impulse response: The impulse response is important for SDOF response to “impulsive” forces, and for finding response to arbitrary inputs by using convolution. function. step unit the is for for where for for < = = < = , , 1 ) ( ), ( sin 1 , , sin 1 ) ( t t t u t u t e m t t t e m t g d t d d t d n n ϖ ϖ ϖ ϖ ζϖ ζϖ Step response Unit step function at t=a: where & is a dummy integration variable. For the EOM the particular solution is xp(t) = 1/k; homogeneous solution satisfies the initial conditions ), ( a t u kx x c x m = + + ∫ ∞ = < = t d a a t a t a t u τ τ δ ) ( , , 1 ) ( for for { } { } ) ( ) ( ) ( , ) ( ) ( ) ( = + = = + = = = = = a t h p a t a t h p a t t x t x t x t x t x t x t u(ta) a 1 ) ( ) ( sin ) ( cos 1 1 ) ( ) ( a t u a t a t e k t s d d n d a t n  + = ϖ ϖ ζϖ ϖ ζϖ Step response: ( 29  = + = )...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas.
 Spring '10
 Staff

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