From: "Lawrence R. Yeagley" <GTE/lyeagley@gte.net>
Newsgroups: sci.math.num-analysis
Subject: Re: Collapse of  Tacoma Narrows Bridge
Date: 6 Dec 1997 06:20:32 GMT

Apropos Galloping Gertie:

Most of the following comments on the failure of the Tacoma Narrows
Bridge in November, 1940 are based on repeated viewings of the 7 minute
video tape from APT (Association of Physics Teachers??).

1. There were two modes of vibration vertical (or galloping) in low wind
speeds (down to circa 3 or 4 mph) and twisting or torsional (seen, I
believe, only on the day of    the collapse when wind speed was 38-40
mph). Even in low winds the "galloping" motion was fairly large.
However, once the bridge transistioned to a torsional mode, the
amplitude was eye-popping. It makes for a truly impressive show.

2. Since there were two modes of oscillation, there probably were two
mechanisms for driving these modes. My guess would be that shedding of
von Karman vortices explains   only one mode. Guessing some more, I
would say that vortex shedding drove the    "galloping" mode. The best
explanation I've heard for the torsional mode is that of    a wing whose
moment (due to lift) twists the wing up until it stalls and the
moment    decreases significantly. At which point the elasticity of the
wing snaps it back and   the cycle begins again. However, it might be
that "stalling" is just another way of shedding a vortex.

3. The mechanism was probably not resonance - certainly not linear
resonance. The    reason for saying this is that in both modes of
oscillation the maximum amplitude appears constant. In linear resonance
the amplitude would grow without limit. Most
likely the two modes were limit cycles of a non-linear motion. If you
wish to call that nonlinear resonance that's up to you but I think that
is deceptive.

4. A few years ago one of the SIAM journals carried an article on the
Tacoma Narrows Bridge by two mathematicians (whose names escape me at
the moment). They went to       considerable lengths to build lots of
structural nonlinearities into their model       and worked through lots
of flashy nonlinear math but in the end I think it was         pointless
because they ended up with a nonlinear structural dynamics problem
in        which the driving force (the wind) was an independent (i.e.
known) variable rather than a dependent (i.e. unknown) variable which
would be determined by solving a fluids problem with moving boundaries.
They completely missed the point that the problem was one of
aeroelasticity (in which structural and fluid motions are coupled)
rather than just a structural dynamics problem in which the driving
force is given or known. The latter type of problem is much more
difficult to solve.

When I have some time this weekend I will dig out some references and
post them next week.

One final note: Many thanks to Lynn Killingbeck for the URL. It was
excellent!
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==============================================================================
   Matches for: Anywhere=(tacoma)
   
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   24 1969 no. 1 80--91. (Reviewer: I. Aganovi\'c) 73.35
   
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