PROFESSOR: Hi everyone.
So today, I'd like to talk about frequency response.
And specifically, we're going to take
a look at a couple different differential equations.
And we're asked to graph the amplitude response
for each equation.
And you'll note that these equations a, b and c have
varying amounts of damping.
So for part a we're asked to plot the amplitude response
for x dot dot plus 4x equals F_0 cosine omega*t.
For part b it's the same equation,
however we have an x dot term added.
For part c, again we've increase the damping.
So now we have 6x dot.
And then lastly for part d, we'd like
to discuss the resonance for each system.
So I'll let you work out this problem,
and I'll be back in a moment.
Hi everyone.
Welcome back.
So for part a, we're asked to graph the amplitude response
to the differential equation x dot dot plus 4x equals F_0
cosine omega*t.
And from a previous recitation, we already
wrote down the particular response
to this differential equation.
So I'm just going to write down the particular response,
which has the form F_0 4 minus omega squared cosine omega*t.
Now the amplitude response is defined as a ratio,
and specifically it's the ratio of the output amplitude
of a differential equation to the input amplitude
of a differential equation.
So in the case at hand, we have the output
is a sinusoidal function whose amplitude is F_0 divided
by 4 minus omega squared.
So it's the output divided by the input.
These are both amplitudes.
And in our case, we have F_0 divided
by 4 minus omega squared.
This is the output amplitude.
And the input amplitude is just F_0.
So we see when we compute this ratio the F_0's divide out.
And at the end of the day, we're left
with 4 minus omega squared.
So I'm going to draw the amplitude response now.
So we have omega.
And we see that when omega is equal to 2,
there's an asymptote.
When omega is equal to 0, we have 1/4.
And specifically, we have this tent-like function.
So this is the amplitude response.
So notice how when we drive the system with frequency two,
the amplitude response goes to infinity.
As a result, we call this frequency
the resonant frequency.
So this concludes part a.
For part b, we have a differential equation
with damping now.
And to compute the particular solution,
we follow the standard procedure of first complexifying
the right-hand side and then using the exponential response
formula.
So I'm just going to write down the particular solution.
If we follow these steps, we find that it's the real part
of the right-hand side complexified,
which is F_0 e to the i*omega*t divided by the characteristic
polynomial evaluated at i*omega.
And in this case, the characteristic polynomial p
of s is s squared plus s plus 4.
p of i*omega is then 4 minus omega squared plus i*omega.
And when we put the pieces together,
we end up with a particular solution,
which looks like the real part of 1 over 4 minus omega squared
plus i*omega, F_0 upstairs, e to the i*omega*t.
So we're asked to compute the amplitude response graph.
And if we take a look at this, we
see that the denominator here is really just a complex number.
So we can convert it into the form of r e to the i*phi.
Now, the amplitude response is defined
as the ratio of the output divided by the input.
And so the output amplitude is going
to be the magnitude of this complex number.
So as a result, the amplitude response is just the magnitude
of 1 over the characteristic polynomial evaluated
at i*omega.
This is also sometimes referred to as the complex gain.
Moreover, this term right here contains
two pieces of information.
Not only does it contain the amplitude response,
but it also contains the phase information.
When we take the absolute value, we're
throwing out the phase information,
and we're just remembering the amplitude response.
So what is this amplitude response look
like for this case?
Well, we have 1 over and it's the magnitude of this complex
number, which is 4 minus omega squared plus i*omega.
So I just take the real part, square it,
add it to the imaginary part squared,
and square root the whole quantity.
Now there's a question of how to graph this.
And we see that first off, the square root's
an increasing function.
And we see that we're 1 over an increasing function.
So there's a trick, which is to just
look first at sketching this piece which
is under the radical sign.
And if you look at trying to maximize this function--
so finding the critical points-- we'd see that in this case,
we have one maximum, to 4 minus omega squared plus omega
squared.
And this is at when omega equals the square root of 7/2.
Sorry.
This is a minimum.
So when I go to sketch this now, we have omega,
we have the amplitude response.
Now, I'm going to draw in 2 from our previous diagram.
Now, the square root of 7/2 is just below 2,
so square root of 7/2.
So we end up with a maximum at 7/2,
and then a decay to infinity.
And again, this is going to be 1/4 when omega is 0.
So this is the peak amplitude response.
So note that in this case, by adding damping, what we've done
is we no longer have an asymptote at omega equals 2.
But we now have a finite amplitude,
which occurs at omega equals the square root of 7/2.
So I'm just going to clean up the board,
and I'll be back with part c in a second.
For part a, we have an amplitude response diagram, which
looks like a tent function.
And at 2, omega equals 2, we have a resonance.
So the amplitude response diverges.
Just like to point out, I made a small error before.
I forgot to include absolute values on the denominator here.
The amplitude response function, it's
always a positive quantity.
We always throw out any phase information
and leave that for the phase in the description of the response
of the linear system.
So the amplitude response is always positive.
For part b, we added dampening to the system.
And we see that there's actually a peak point which
is at the square root of 7/2.
And the amplitude response is bounded at this point,
but it achieves a maximum.
And then again it decays to infinity.
So I'd like now to take a look at part c.
And in part c, we have the differential equation x dot dot
plus 6x dot plus 4x equals F_0 cosine omega*t.
And again, the amplitude response is going to equal 1
over the absolute value of p of i*omega.
And in this case, p of i*omega is going to be 1 over-- Well,
we still have the 4 minus omega squared term.
Instead of x dot, we now have 6x dot, which gives us 6i*omega.
And then again, we want to take the absolute value
of this complex number.
And when we take the absolute value,
we just get the sum of the real parts
squared plus the sum of the imaginary parts squared,
which in this case is going to be 36 omega squared,
the whole quantity squared rooted,
and then we have 1 over this value.
So now if we'd like to plot this function,
we can still do the same trick and try to maximize or find
the critical points of the denominator under the radical.
And if we did this, in this case we
would find that the only critical point
is when omega is equal to 0.
Secondly, if we look at omega going to infinity,
we see that the denominator goes to infinity.
So this whole quantity must go to 0.
So if I were to go back here to the amplitude response
for part c, again, when omega is equal to 0
it's going to start off at 1/4.
I've just argued that it goes to 0 as omega goes to infinity.
And since there are no critical points,
we must smoothly paste the function between the two.
And in fact, it's always decreasing.
So the amplitude response, in this case,
is just a decreasing function.
So this concludes part c.
And now I'll take a look at part d.
Discuss the resonance for each system.
So in part a, we had no damping.
And we saw that there was a resonance at omega equals 2.
And the resonance manifested itself
in the amplitude response graph with a divergent asymptote
at omega is equal to 2.
So as you drive the system close to omega equals 2,
the amplitude of the system starts to diverge.
In case two we introduced damping into the system.
So we still have a very large amplitude response at omega
equals the square root of 7/2, however it's
no longer infinite.
And then lastly, when we increased damping even further
so we had the 6x dot term, the presence of a peak disappeared.
And in fact, the amplitude response
just monotonically decayed from 1/2 to infinity.
So just constantly decreased to 0.
So I'd just like to conclude there,
and I'll see you next time.