Ch3-1 - Chile Mine Rescue Operation Chapter 3 General...

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Chile Mine Rescue Operation
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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 1:20:36 00 11 0 ( ) ( ) ( ) cos mx t cx t kx t F t + + = cos( ) + sin( ) harmonic forcing functions mx cx kx c Y t kY t
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Review: Linear Superposition Principle 12 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 11 2 22 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  3 10:37:07 00 02
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3.1 Impulse Response Function F ( t ) t ˆ 2 F + - Formal Definition of Impulse Excitation: F ( t ) 0 t  t  ˆ F 2 t t  t   0 t  t   The parameter is a “small” positive number (compared to a time scale). t is the time instance at which the pulse is applied. 4 t ( ) ( ) ( ) ( ) mx t cx t kx t F t is a “small” positive number. How small?
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Impulse = ( ) ( ) ( ) ( ) ( ) N s ˆ ˆ 2 2 F t dt F t I F t dt F t dt F F t t     F ( t ) ˆ F 2 + - Area under pulse Concept of Impulse and Momentum The impulse Imparted to an Object is Equal to the Change in its Momentum 5 10:37:10 00 05 t m k x(t) c () Ft
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( ) 0, F t t tt    Impulse and the Dirac Delta function F ( t ) t Equal impulses Dirac Delta function ˆ For 1, as tends to 0, the impulse function is the Dirac Delta (t) F ˆ () F t dt F t   6 10:37:12 00 07 ( ) ( - ) ( ) oo F t t t dt F t
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This note was uploaded on 11/01/2010 for the course ME 455 taught by Professor Centinkaya during the Fall '10 term at Clarkson University .

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Ch3-1 - Chile Mine Rescue Operation Chapter 3 General...

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