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PROFESSOR: Hi everyone.

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So today, I'd like to talk
about frequency response.

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And specifically,
we're going to take

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a look at a couple different
differential equations.

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And we're asked to graph
the amplitude response

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for each equation.

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And you'll note that these
equations a, b and c have

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varying amounts of damping.

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So for part a we're asked to
plot the amplitude response

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for x dot dot plus 4x
equals F_0 cosine omega*t.

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For part b it's
the same equation,

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however we have an
x dot term added.

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For part c, again we've
increase the damping.

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So now we have 6x dot.

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And then lastly for
part d, we'd like

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to discuss the resonance
for each system.

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So I'll let you work
out this problem,

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and I'll be back in a moment.

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Hi everyone.

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Welcome back.

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So for part a, we're asked to
graph the amplitude response

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to the differential equation
x dot dot plus 4x equals F_0

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cosine omega*t.

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And from a previous
recitation, we already

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wrote down the
particular response

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to this differential equation.

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So I'm just going to write
down the particular response,

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which has the form F_0 4 minus
omega squared cosine omega*t.

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Now the amplitude response
is defined as a ratio,

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and specifically it's the
ratio of the output amplitude

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of a differential equation
to the input amplitude

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of a differential equation.

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So in the case at hand,
we have the output

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is a sinusoidal function
whose amplitude is F_0 divided

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by 4 minus omega squared.

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So it's the output
divided by the input.

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These are both amplitudes.

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And in our case,
we have F_0 divided

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by 4 minus omega squared.

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This is the output amplitude.

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And the input
amplitude is just F_0.

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So we see when we compute this
ratio the F_0's divide out.

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And at the end of
the day, we're left

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with 4 minus omega squared.

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So I'm going to draw the
amplitude response now.

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So we have omega.

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And we see that when
omega is equal to 2,

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there's an asymptote.

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When omega is equal
to 0, we have 1/4.

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And specifically, we have
this tent-like function.

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So this is the
amplitude response.

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So notice how when we drive
the system with frequency two,

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the amplitude response
goes to infinity.

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As a result, we
call this frequency

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the resonant frequency.

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So this concludes part a.

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For part b, we have a
differential equation

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with damping now.

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And to compute the
particular solution,

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we follow the standard
procedure of first complexifying

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the right-hand side and then
using the exponential response

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formula.

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So I'm just going to write
down the particular solution.

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If we follow these steps, we
find that it's the real part

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of the right-hand
side complexified,

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which is F_0 e to the i*omega*t
divided by the characteristic

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polynomial evaluated at i*omega.

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And in this case, the
characteristic polynomial p

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of s is s squared plus s plus 4.

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p of i*omega is then 4 minus
omega squared plus i*omega.

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And when we put the
pieces together,

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we end up with a
particular solution,

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which looks like the real part
of 1 over 4 minus omega squared

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plus i*omega, F_0 upstairs,
e to the i*omega*t.

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So we're asked to compute
the amplitude response graph.

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And if we take a
look at this, we

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see that the denominator here
is really just a complex number.

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So we can convert it into
the form of r e to the i*phi.

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Now, the amplitude
response is defined

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as the ratio of the output
divided by the input.

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And so the output
amplitude is going

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to be the magnitude of
this complex number.

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So as a result, the amplitude
response is just the magnitude

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of 1 over the characteristic
polynomial evaluated

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at i*omega.

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This is also sometimes referred
to as the complex gain.

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Moreover, this term
right here contains

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two pieces of information.

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Not only does it contain
the amplitude response,

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but it also contains
the phase information.

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When we take the
absolute value, we're

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throwing out the
phase information,

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and we're just remembering
the amplitude response.

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So what is this
amplitude response look

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like for this case?

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Well, we have 1 over and it's
the magnitude of this complex

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number, which is 4 minus
omega squared plus i*omega.

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So I just take the
real part, square it,

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add it to the
imaginary part squared,

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and square root
the whole quantity.

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Now there's a question
of how to graph this.

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And we see that first
off, the square root's

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an increasing function.

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And we see that we're 1
over an increasing function.

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So there's a trick,
which is to just

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look first at sketching
this piece which

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is under the radical sign.

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And if you look at trying
to maximize this function--

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so finding the critical points--
we'd see that in this case,

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we have one maximum, to 4
minus omega squared plus omega

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squared.

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And this is at when omega
equals the square root of 7/2.

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Sorry.

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This is a minimum.

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So when I go to sketch
this now, we have omega,

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we have the amplitude response.

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Now, I'm going to draw in 2
from our previous diagram.

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Now, the square root
of 7/2 is just below 2,

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so square root of 7/2.

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So we end up with
a maximum at 7/2,

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and then a decay to infinity.

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And again, this is going
to be 1/4 when omega is 0.

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So this is the peak
amplitude response.

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So note that in this case, by
adding damping, what we've done

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is we no longer have an
asymptote at omega equals 2.

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But we now have a
finite amplitude,

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which occurs at omega equals
the square root of 7/2.

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So I'm just going to
clean up the board,

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and I'll be back with
part c in a second.

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For part a, we have an amplitude
response diagram, which

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looks like a tent function.

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And at 2, omega equals
2, we have a resonance.

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So the amplitude
response diverges.

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Just like to point out, I
made a small error before.

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I forgot to include absolute
values on the denominator here.

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The amplitude response
function, it's

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always a positive quantity.

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We always throw out
any phase information

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and leave that for the phase in
the description of the response

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of the linear system.

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So the amplitude response
is always positive.

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For part b, we added
dampening to the system.

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And we see that there's
actually a peak point which

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is at the square root of 7/2.

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And the amplitude response
is bounded at this point,

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but it achieves a maximum.

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And then again it
decays to infinity.

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So I'd like now to
take a look at part c.

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And in part c, we have the
differential equation x dot dot

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plus 6x dot plus 4x
equals F_0 cosine omega*t.

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And again, the amplitude
response is going to equal 1

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over the absolute
value of p of i*omega.

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And in this case, p of i*omega
is going to be 1 over-- Well,

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we still have the 4
minus omega squared term.

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Instead of x dot, we now have
6x dot, which gives us 6i*omega.

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And then again, we want
to take the absolute value

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of this complex number.

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And when we take
the absolute value,

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we just get the sum
of the real parts

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squared plus the sum of the
imaginary parts squared,

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which in this case is going
to be 36 omega squared,

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the whole quantity
squared rooted,

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and then we have
1 over this value.

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So now if we'd like
to plot this function,

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we can still do the same trick
and try to maximize or find

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the critical points of the
denominator under the radical.

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And if we did this,
in this case we

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would find that the
only critical point

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is when omega is equal to 0.

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Secondly, if we look at
omega going to infinity,

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we see that the denominator
goes to infinity.

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So this whole
quantity must go to 0.

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So if I were to go back here
to the amplitude response

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for part c, again, when
omega is equal to 0

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it's going to start off at 1/4.

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I've just argued that it goes
to 0 as omega goes to infinity.

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And since there are
no critical points,

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we must smoothly paste the
function between the two.

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And in fact, it's
always decreasing.

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So the amplitude
response, in this case,

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is just a decreasing function.

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So this concludes part c.

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And now I'll take
a look at part d.

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Discuss the resonance
for each system.

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So in part a, we had no damping.

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And we saw that there was a
resonance at omega equals 2.

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And the resonance
manifested itself

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in the amplitude response graph
with a divergent asymptote

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at omega is equal to 2.

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So as you drive the system
close to omega equals 2,

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the amplitude of the
system starts to diverge.

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In case two we introduced
damping into the system.

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So we still have a very large
amplitude response at omega

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equals the square root
of 7/2, however it's

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no longer infinite.

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And then lastly, when we
increased damping even further

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so we had the 6x dot term, the
presence of a peak disappeared.

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And in fact, the
amplitude response

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just monotonically decayed
from 1/2 to infinity.

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So just constantly
decreased to 0.

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So I'd just like
to conclude there,

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and I'll see you next time.