An investigation of resonance

Note that I have chosen to do this all in Mathematica. This doesn't matter. The important thing here is the math, not the computer code.

For the following, let's consider the system
    

pure resonance

For pure resonance, take .  For convenience we'll also choose and   Thus the complementary homogeneous solution is
    
If we force at the natural frequency we will see resonance.  For we'll see beats:

If is away from , we won't see either:

practical resonance

Now let's put the damping back in.  In this case we have a transient and a long-term steady state solution:

The amplitude of this steady state solution is about

-0.106001, as shown in the next graph:

Now, what happens if we change the frequency of the forcing?  Let's decrease it some and see what happens:

Hmm!  Now the amplitude is about

0.188144--a bit bigger!  So we've changed the amplitude of the response just by changing the frequency of the forcing, without changing the amplitude of the forcing!  We could plot the amplitude as a function of the forcing frequency to see what's happening--the two points we have so far give us

Let's add a bunch more points to this graph to see what's happening:

The maximum looks to be between 2.5 and 2.75:

So practical resonance occurs when .