Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.

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A very brief look at Driven Harmonic Motion

The simplest case is a simple harmonic oscillator,
which responds to a simple force:

So its equation of motion can be written with just two terms:

The solution to this differential equation is a sine or cosine
function which oscillates with
a natural frequency

If we add a resistance which depends on velocity,

the equation of motion grows a bit:

(I'm going to get rid of the vector signs at this point, for simplicity)

Now, let's add something new:
a driving force with magnitude **F**_{d}
(the subscribe **d** stands for "driving")
which varies sinusoidally with angular frequency **omega**_{d}.
That means that Newton's Second Law for the object becomes

Or, if we write this as a differential equation for the
position of the object as a function of time, **x(t)**, we have

The general solution to this equation is .... messy.
It consists of several terms.
Some of those terms, however, include negative exponential
factors of time, and so decay as time progresses.
If we ask "what is happening after some long time passes?"
we find that the **steady state**
can be described by a simple formula:

It should be no surprise that the object will oscillate with
a frequency that matches the driving force;
after all, the driving force is, well, DRIVING the object.

What is more interesting is the SIZE of the oscillations.
Their amplitude, written above as simply **A**, actually
depends on a combination of the driving frequency
and the **natural frequency omega**_{o} of the object itself.

Note that the amplitude of motion depends crucially
on how far the driving frequency
is from the natural frequency.

####
Resonance

Resonance occurs when
the driving frequency **omega**_{d} is equal
to the natural frequency **omega**_{o} of
a system.
Every little forward push applied by the driving force
happens as the object is moving forward,
and every little backward pull applied by the
driving force occurs as the object is moving backward.
That means that the WORK done by the driving force
on the object is always positive

and so the driving force transfers energy to the object
very efficiently.

You all have experience with driving forces and
resonance, even if you don't realize it.

Image taken from The Coffee Ring blog
Engineers who design buildings and bridges must
figure out AHEAD OF TIME the resonant frequencies
of their structures, and make sure that
these frequencies are far from any which might
be excited naturally by
earthquakes, traffic, wind, columns of marching soldiers,
rock bands, and other primal forces.

Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.