The basic idea is like this: suppose some system exhibits
all the requisite properties for *SHM*, namely a linear restoring
"force"
and an inertial factor .
Then *if once set in motion* it will oscillate forever
at its "resonant frequency"
,
unless of course there is a "damping force"
to dissipate the energy stored in
the oscillation. As long as the damping is weak
[
], any oscillations will
persist for many periods. Now suppose the system is initially
at rest, in equilibrium, ho hum.
What does it take to "get it going?"

The *hard* way is to give it a great whack to start it
off with lots of kinetic energy, or a great tug to stretch
the "spring" out until it has lots of potential energy,
and then let nature take its course. The *easy* way
is to give a tiny push to start up a small oscillation,
then wait exactly one full period and give another tiny push
to increase the amplitude a little, and so on. This works
because *the frequency ** is independent of the
amplitude **q*_{0}. So if we "drive" the system
*at its natural "resonant" frequency* ,
no matter how small the individual "pushes" are,
we will slowly build up
an *arbitrarily large oscillation*.^{13.11}

Such resonances often have dramatic results.
A vivid example is the famous movie of the collapse of
the Tacoma Narrows bridge, which had a torsional [twisting]
resonance^{13.12}
that was excited by a steady breeze blowing past the bridge.
The engineer in charge anticipated all the other
more familiar resonances [of which there are many]
and incorporated devices specifically designed
to safely damp their oscillations, but forgot this one.
As a result, the bridge developed huge twisting oscillations
[mistakes like this are usually painfully obvious
when it is too late to correct them] and tore itself apart.

A less spectacular example is the trick of getting yourself
going on a playground swing by leaning back and forth
with arms and legs in synchrony with the natural frequency
of oscillation of the swing [a sort of pendulum].
If your kinesthetic memory is good enough you may recall
that it is important to have the "driving" push exactly
radians [a quarter cycle] "out of phase"
with your velocity - *i.e.* you *pull* when you reach
the *motionless* position at the top of your swing,
if you want to achieve the maximum result.
This has an elegant mathematical explanation, but
I promised . . . .

Jess H. Brewer - Last modified: Sun Nov 15 13:52:16 PST 2015