Re: Tacoma Narrows resonator

["John S. Denker" <jsd@MONMOUTH.COM>]

At 01:11 PM 12/8/00 -0500, Joseph Bellina wrote:
> Maybe I missing lots here, but I don't really see the distinction
> between the soldiers marching on the bridge, and the bow on the violin
> string.

I think there is a useful distinction.  The two phenomena are not 100%
different, but neither are they 100% the same.

> The only difference is that in the violin case, the magnitude
> of the stimulation is probably large, and so many modes of vibration on
> excited, and a few selected, whereas in the soldier case, the
> stimulation is weak so only the one mode of vibration is excited.

Even if that were true, it would not be the "only" distinction;  see below.

But it is not true anyway.  Soldiers can certainly choose to excite more
than one mode.  A violin can (within limits) be played loud or soft, and
the tone changes relatively little;  the mode selection depends more
directly on other things.

On Fri, 8 Dec 2000, Michael Edmiston wrote:

> > Type-1. (A) I hang a spring pendulum from an electromagnetic driving coil
> > (such as Pasco-scientific SF-9324 or WA-9753) and I drive the coil with an
> > oscillator and find that a certain driving frequency causes build-up of
> huge
> > oscillations in the spring pendulum.  (B) I sing a short-duration steady
> > pitch into my piano with the damper pedal depressed and upon stopping
> > singing note that a particular string(s) has been excited.

Those examples are well chosen.  They have in common that the resonant
object was driven with an external signal of definite frequency.  It of
course responds at a frequency equal to the drive frequency (NOT at its
resonant frequency, except when there is a coincidence, and such
coincidences happen only on a set of measure zero).

> > Type-2.  (C) I strike a tuning fork with a rubber mallet and note that the
> > fork vibrates at particular frequencies.

This is another good example, clearly different from the type-1 examples,
because the excitation has no definite frequency.  After short-lived
transients die out, the response is at the resonant frequency.

>  (D) I bow a violin string and note
> > the string vibrates with particular frequencies.  (E) I blow across the top
> > of a soda-pop bottle and note that I produce particular frequencies of
> > sound.

These are similar to the previous type-2 example, in that the excitation
has no intrinsic frequency.

The new element here is a sustained interaction between the resonator and
the excitation.  The resonating degree of freedom is able to extract energy
from the excitation because of a nonlinear interaction;  it is able to
couple more strongly when it is moving in the right direction (with the
excitation) and to partially decouple when it is moving in the wrong
direction (against the excitation).

If one wanted to convert example (D) into a type-1 example, one would need
to put "teeth" on the violin bow, so that it would excite the string at the
tooth-passage frequency.  This would be very unlike a real violin, where
the output frequency is independent of the speed of the bow (over a wide
range to an excellent approximation).

There is every reason to believe the Tacoma Narrows bridge disaster was
type-2 not type-1, i.e. that there was no relevant intrinsic frequency in
the excitation.

At 09:33 AM 12/8/00 -0800, Leigh Palmer wrote:
> The vortex shedding is not driving the motion. The frequency of
> vortex shedding is determined by the frequency of the torsional
> oscillation of the bridge; it is not inherent in the wind.

I agree.

> This is not the phenomenon usually referred to as (von Karmann)
> vortex shedding.

I agree.

One could imagine a type-1 explanation of Galloping Gertie, involving von
Karman vortex shedding that "just happened" to coincide with the resonant
frequency of the bridge.  But there is no reason to believe that's what
happened, and good reason to believe that's not what happened.  According to
  http://aerodyn.org/Unsteady/unsteady.html
         The largest Reynolds number at which the von
         Karman vortex street is observed is Re=400
and the bridge, in a 40-knot wind, must have had a Reynolds number in the
millions.