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18.03 Class 29
, April 16, 2010
Laplace Transform IV:
The pole diagram
1. Another example
2. tshift rule
3. Poles
4. What the pole diagram of
F(s)
says about
f(t)
[1]
We saw that if
p(D)w = delta(t) ,
i.e.
a_n w^(n) + .
.. + a_1 w' + a_0 w = delta(t)
with rest initial conditions, then
p(s) W(s) = 1
,
or
W(s) = 1/p(s) ,
or
w(t) = L^{1}(1/p(s))
w(t) = weight function
W(s) = transfer function
This is another RULE.
Other input signals?
x" + 4x = cos(2t)
rest initial conditions
(No delta functions in this signal, so we know that
x(0+) = 0
and x'(0+) = 0,
but that will come out automatically.)
X = W(s) LT[cos(2t)] = 2s/(s^2+4)^2
From our latest addition to the table, we find that
x
= u(t) 1/2 t sin(t)
(Resonance!)
GENERAL FACT:
p(D) x = f(t)
with rest initial conditions
has Laplace transformed equation
p(s) X(s) = F(s)
with solution
X(s) = W(s) F(s)
In the sdomain, the system response is obtained by multiplying by the
transfer function!
[2] Another rule:
Let
a > 0 and define
f_a(t) = 0
if
t < a
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View Full Document= f(ta)
if
t > a
(If
f_s(t)
contains
c delta(t) ,
I also want
f_a(t)
to contain
c delta(ta) .
So for example
delta_a(t) = delta(ta) .)
If you know
F(s) , what is the LT of
f_a(t) ?
integral_{0}^infty f_a(t) e^{st} dt = integral_{a}^\infty f(ta) e^{st} dt
Substitution:
u = ta ,
du = dt ,
... = integral_{0}^\infty f(u) e^{s(u+a)} du
= e^{as} integral_{0}^infty f(u) e^{su} du
= e^{as} F(s)
So we have the
tshift rule:
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 Spring '09
 vogan

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