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Unformatted text preview: Impulse Functions In this section : Forcing functions that model impulsive actions  external forces of very short duration (and usually of very large amplitude). The idealized impulsive forcing function is the Dirac delta function * (or the unit impulse function ), denotes δ ( t ). It is defined by the two properties δ ( t ) = 0, if t ≠ 0, and ∫ ∞ ∞ = 1 ) ( dt t δ . That is, it is a force of zero duration that is only non¡zero at the exact moment t = 0, and has strength (total impulse) of 1 unit. Translation of ( ) The impulse can be located at arbitrary time, rather than just at t = 0. For an impulse at t = c , we just have: δ ( t c ) = 0, if t ≠ c , and ∫ ∞ ∞ = 1 ) ( dt c t δ . * It was introduced by, and is named after the British physicist Paul A. M. Dirac (1902 – 1984), co¡winner of the Nobel Prize in Physics in 1933. A pioneer of quantum mechanics, Dirac is perhaps best known (besides for the delta function) for formulating the Dirac equation, which predicted the existence of antimatter. Laplace transforms of Dirac delta functions L { δ ( t )} = 1, L { δ ( t c )} = e cs , c ≥ 0. Here is an important and interesting property of the Dirac delta function: If f ( t ) is any continuous function, then ∫ ∞ ∞ = ) ( ) ( ) ( c f dt t f c t δ Therefore, for c ≥ 0, L { δ ( t c )} = ∫ ∫ ∞ ∞ ∞ = = ) ( ) ( cs st st e dt e c t dt e c t δ δ . Note : Since the integrand in the above integrals is zero everywhere except at the single point t = c ≥ 0, the contribution from the interval (∞, 0) to the definite integral is, therefore, zero. Hence the two definite integrals above will have the same value despite their different lower limits of integration....
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This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Penn State.
 Spring '08
 CHEZHONGYUAN
 Differential Equations, Equations, Partial Differential Equations

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