Ch3-1_to_3-8_revised_Nov10_2010

Ch3-1_to_3-8_revised_Nov10_2010 - Chile Mine Rescue...

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Unformatted text preview: Chile Mine Rescue Operation 1 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 ( ) ( ) ( ) 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 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 ¡ ¡ ¡ ¡ ¡ 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 3.1 Impulse Response Function F ( t ) W Ö 2 F H W + H W- H Formal Definition of Impulse Excitation: F ( t ) t ¡ W ¢H Ö F 2 H W ¢H ¡ t ¡ W £ H t ! W £ H ­ ® ° ¯ ° The parameter H LV D ³VPDOO´ SRVLWLYH QXPEHU ¡FRPSDUHG WR D WLPH VFDOH¢£ 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 QXPEHU£ How small? 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 ¢ ¢ f ' £ 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 ( ) , F t t W W ¡ 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 ¡f ¡ ³ 6 ( ) ( - ) ( ) o o F t t t d t F t G f ¡f ³ impulse=momentum change [ ( ) ( ) ] F t m v m v t v t ¡ ¢ ' ' ¢ 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....
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Ch3-1_to_3-8_revised_Nov10_2010 - Chile Mine Rescue...

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