Ch 6'5: Impulse Functions

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Ch 6.5 Impulse Functions

• For example, a mechanical system subject to a
  force g(t) of large magnitude that acts over a
  short time interval about t0.

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1/ Unit impulse function

• In a mechanical system, where g(t) is a force,
  the total impulse of this force is measured by
  the integral
• Now if g(t) has the form
• then
• In particular, if c 1/(2?), then I(?) 1
  (independent of ? ).

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• Suppose the forcing function g(t) has the form (g
  is called d?)
• And so I(?) 1.
• Now we are interested d?(t) acting over
• shorter and shorter time intervals
• (i.e., ? ? 0, c ? inf.), but I(?) 1 !
• So d?(t) gets taller and narrower
• as ? ? 0. In this case we have

for t ? 0
anyway
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• The unit impulse function is called ? and is
  defined to have the properties
• ? is usually called the Dirac delta function.
• In general, for a unit impulse at an arbitrary
  point t0,

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2/ Laplace Transform of ?

• The Laplace Transform of ? is
• For Laplace Transform of ? at t0 0, take limit
  as follows

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• Proof the Laplace Transform of ? is defined by
  
• and thus

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3/ Product of Continuous Functions and ?

• The product of the delta function and a
  continuous function f is
• Analytical proof

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Example 1 Initial Value Problem (1 of 3)

• Consider the solution to the initial value
  problem
• Then
• Letting Y(s) Ly,
• Substituting in the initial conditions, we obtain
• or

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• We have
• The partial fraction expansion of H(s) yields
  (see 6.4 ex1)
• and hence

and
Plot of the Solution
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• t lt 5 No initial conditions and no external
  excitation there is no response on (0, 5).
• t 5 The impulse at t 5 produces a decaying
  oscillation that persists indefinitely.

Plot of the Solution