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Ch3-1_to_3-8_revised_Nov03_2010

Ch3-1_to_3-8_revised_Nov03_2010 - Chile Mine Rescue...

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Chile Mine Rescue Operation 1 11/3/2010 9:41:46 AM
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Chapter 3: General Forced Response In the topics covered so far, all of the driving forces have been harmonic excitation, i.e., sine or cosine excitations Linear combination of harmonic excitation has been also used. Here we examine the response to any form of excitation such as: ± Impulse ± Sums of harmonic functions: sines and cosines ± Any (integrable) function 2 0 ( ) ( ) ( ) cos m x t c x t k x t F t Z ± ± + + = cos( ) + sin( ) harmonic forcing functions m x c x k x c Y t k Y t Z Z Z
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Review: Linear Superposition Principle 1 2 1 1 2 2 If ( ) and (t) are solutions of a linear homogeneous equation of motion, then ( ) ( ) (t) is also a solution to the same equation of motions. x t x x t a x t a x ± This principle allows us to break up complicated forces into sums of simpler forces, compute the response and add to get the total solution 2 1 1 2 2 2 2 1 2 1 2 If ( ) is the particular solution of ( ) and ( ) is for ( ) ( ) ( ) ( ) solves ( ) ( ) n n n x t x x f t x t x x f t x t ax t bx t x x a f t b f t Z Z Z ± ± ± ± ± 3
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3.1 Impulse Response Function F ( t ) W Ö 2 F H W + H W H Formal Definition of Impulse Excitation: F ( t ) 0 t ² W ³ H Ö F 2 H W ³ H ² t ² W ± H 0 t ! W ± H ­ ® ° ¯ ° The parameter H LV D ³VPDOO´ SRVLWLYH QXPEHU ±FRPSDUHG WR D WLPH VFDOH²µ W is the time instance at which the pulse is applied. 4 t ( ) ( ) ( ) ( ) m x t c x t k x t F t ± ± H LV D ³VPDOO´ SRVLWLYH QXPEHUµ How small?
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Impulse = ( ) ( ) ( ) ( ) ( ) N s Ö Ö 2 2 F t d t F t I F t d t F t d t F F W H W H W H H H ± f ³ ³f ' ³ ³ ³ F ( t ) W Ö F 2 H W + H W H Area under pulse Concept of Impulse and Momentum The impulse Imparted to an Object is Equal to the Change in its Momentum 5 t m k x(t) c ( ) F t
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( ) 0, F t t W W ³ z Impulse and the Dirac Delta function F ( t ) t Equal impulses Dirac Delta function Ö For 1, as tends to 0, the impulse function is the Dirac Delta (t) F H G Ö ( ) F t d t F W f ³f ³ ³ 6 ( ) ( - ) ( ) o o F t t t d t F t G f ³f ³
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impulse=momentum change 0 0 [ ( ) ( )] F t m v m v t v t ± ³ ' ' ³ 0 0 Ö Ö F F t F m v v m m ' The effect of an impulse on a spring-mass-damper is related to its change in momentum. Thus, the response to impulse with zero IC is equal to the free response with IC: x 0 =0 and v 0 = F ' t / m Just after impulse Just before impulse 7 Impulse and Momentum Area under pulse 0
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Free Response 0 0 The governing equation with I.C.: 0 (0) (0) m x c x k x x x x v ± ± - 0 0 0 In underdamped case ( 1), the solution becomes: ( ) sin cos n t n d d d v x x t e t x t ]Z ] ]Z Z Z Z ² ª º § · ± ± « » ¨ ¸ © ¹ ¬ ¼ m k x(t) c
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Impulse Response (Underdamped) unit impulse response function Ö Ö ( ) sin (response to ) Ö ( ) ( ), where ( ) sin n n t d d t d d F e x t t F m e x t F h t h t t m ]Z ]Z Z Z Z Z ³ ³ 0 10 20 30 40 -1 -0.5 0 0.5 1 Time h(t) m k x(t) c F Ö 9 - 0 0 0 ( ) sin cos for 1 n t n d d d v x x t e t x t ]Z ]Z Z Z ] Z ª º § · ± ± ² « » ¨ ¸ © ¹ ¬ ¼ 0 0 Ö Ö Use and 0 to obtain the impulse response.
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