Re: resonances etc.

[John Denker <jsd@MONMOUTH.COM>]

At 05:26 PM 12/7/00 -0700, SSHS KPHOX wrote:
> 1. If one tuning fork causes another to begin vibrating, is that
> resonance....I think so.

Certainly it is safe to say that each tuning fork separately has a sharp
resonance.

The transfer of energy from one high-Q system to another is certainly a
consequence of resonance, but it is not _per se_ the definition of resonance.

Just saying we have a resonant (sub)system doesn't tell the whole story
about what the system does.  High-Q systems can do lots of interesting
things, depending on circumstances.

For fun see
  http://unisci.com/stories/20003/0911003.htm


> 2. If I sing near such a tuning fork and part of my melifluous voice (;-0)
>  causes the fork to vibrate can I cite resonance?

The fork will vibrate to some extent whether you sing on-resonance or
off-resonance.  The fact that it vibrates with a much higher amplitude
(Other Things Being Equal) when you are on-resonance is an excellent
demonstration of high-Q resonance.

> 3. If a steady wind stimulates wire cables to vibrate in an assortment of
> modes, is it possible for one of those frequencies to stimulate
> sympathetic vibrations in an neighboring oscillator?

If two wires (not necessarily near neighbors) have resonances that overlap,
then there will be energy transfer between them.  In fact any analysis of
each wire separately will be misleading;  it would be wiser to consider
them jointly, i.e. to analyze the normal modes of the two-wire system.

Coupled-oscillator problems show up all the time in interesting physics
problems.  There's all sorts of wonderful terminology (mode splitting,
avoided crossings, ...).  I can't immediately come up with an elementary
reference for this.

> 4. What mechanism started and supported  the oscillations of that bridge?

First, in all generality, one must be clear about the concepts of
 -- equilibrium
 -- stability, and
 -- damping.

That is:
 -- In equilibrium, forces are in balance.
 -- Stability says that when the position is different from the
equilibrium position, there is a force that pushes things back to equilibrium.
 -- Damping is a force that is proportional to velocity.  It is the R term
in an RLC oscillator.

For details see
  http://www.monmouth.com/~jsd/how/htm/equilib.html

A resonator becomes an oscillator when there is too much stability and not
enough damping.

=============

In this particular case we have a "flutter" problem.

Let's analyze a fluttering leaf, or better yet, a fluttering
airfoil.  Treat it as a mass-on-a-spring problem.  The mass part is
obvious.  Let's ignore any mechanical stiffness;  that is, we consider an
airfoil that is free to move around a hinge.  The only restoring force is
due to the airfoil being hit by the wind.  As the wind velocity increases,
you get more and more stability, i.e. more and more "spring" constant.

Now we ask about the damping.  The usual case is to have positive
damping.  That means that the damping term is a force proportional to
velocity in such a way that the product of force times velocity is
negative;  it sucks energy out of the system.  Any oscillatory motion that
starts by accident will die away exponentially.  In contrast, the exciting
case is where you have zero or even negative damping.  Any oscillation that
starts by accident will last forever, or grow exponentially.

It is quite possible to design an airfoil-on-a-hinge system such that the
aerodynamic force depends on velocity in such a way as to produce negative
damping.  This is particularly easy to arrange if you have a bluff body
that is almost broadside to the wind.  In this regime the angle of attack
is ultra high (around 90 degrees) so your usual intuition about the low-AoA
behavior of airfoils does not apply.

Finally, suppose we have a bridge that is built with some mechanical
damping (but not a whole lot).  That makes a constant positive
contribution, independent of wind speed.  In addition, there will be a
variable negative contribution, depending on wind speed.  When the wind
speed gets large enough, the total damping goes to zero, then
negative.  The Q goes to infinity and beyond.  The resonator becomes an
oscillator.

The "start" of the oscillation is some random fluctuation.  The oscillation
then grows exponentially.

Remark:  The bridge presumably had a fairly high-Q resonance even in
no-wind conditions.  A high-Q bridge or flagpole is vulnerable to
high-amplitude excitation by mischievous adolescents.  But that is far from
being the whole story of this bridge.  The interesting part of the story is
to explain why the damping decreased to zero or beyond when the wind came up.