Big example, superposition, and Frequency Response

[1] Example.

PLEASE KNOW the solution to the homogeneous harmonic oscillator

are sinusoids of circular frequency omega !

Here, .

In the real example I drive it: .

The complex equation is .

If it weren't for the t we could try to apply ERF:
,

, though, so it doesn't apply; we do have
the

resonance

response formula , which gives

so .

But there is a t there. We should then use "Variation of Parameters":

Look for solutions of the form

for u an Unknown
function .

so

Reduction of order :

Use undetermined coefficients :

so

The general solution is then
the homogeneous solution.

[3] Superposition: putting special cases together .

Suppose a bank is giving I percent per year interest:

Suppose that I open TWO bank accounts and proceed to save at rates

(t)

and in them.

Is this any different than opening ONE bank account and saving at the

rate

Say the solutions with savings rates and
are and
.

Is a solution with savings rate
?

since differentiation respects sums (and multiplying by I
does too).

In general if and

then

In fact this is true for nonconstant coefficient linear equations too.

It is the essence of linearity, and it's the most general form of the

superposition principle.

It lets you break up the input signal into constituent parts, solve for

them separately, and then put the results back together. This is why it isn't

so

bad that we spent all that time studying very special input signals.

One example is when : then
is a solution to the

homogeneous

equation, and we find again that adding such a function to a solution of

gives another solution.

Our work has shown a general result:

Theorem: If q(t) is any linear combination of products of polynomials

and exponential functions, then all solutions to
are

again

linear combinations of products of polynomials and exponential

functions.

Here we mean *complex* linear combinations and *complex* exponentials,

so for example is a possible signal

or solution.

[4] Frequency response

Polar form of a complex number :

Frequency response is about the amplitude and phase lag of
a sinusoidal

(steady state) response of a system to a sinusoidal signal of some

frequency.

It is based on the following method of finding a sinusoidal system

response in "polar" (amplitude/phase lag) form:

Example:

Now write in polar
form . Do the denominator first :

Lesson: if

then

Suppose now that I let the input frequency be anything:

So the amplitude of the sinusoidal response is

This takes value 1 at omega = 0 , and when omega is large
it

falls off like . In this case, it reaches a
modest

"near resonance" peak at omega = 1 .

The phase lag is

There's no particular advantage in writing out a more explicit formula

for this.

Good luck!