Re: Midterm Question - Sort of

[Leigh Palmer <palmer@SFU.CA>]

> > On a related topic: It always bothers me when the Tacoma
> > Narrows Bridge collapse is cited as an example of resonance.
> > It's not. The wind was steady, more or less, and there was
> > no significant component at the bridge's resonant frequency.
> > It is a case of nearly pure DC excitation.
> Huh?  Isn't this precisely the reason that it *is* an example of
> resonance?  Because of a high Q, the extremely small driving force at
> the resonant frequency was able to drive a large amplitude oscillation.

No, I don't believe that is relevant. The component of the wind at
the relevant frequency and within the bandwidth necessary probably
did not have nearly enough energy to drive the bridge to the
amplitude observed. I know that is not a very quantitative statement,
but it certainly resonates with my physical intuition. Besides,
there is the issue of phase coherence. In order to drive a resonance
the excitation must be within the proper 180 degree phase window.
The coherence time required to maintain this phase relationship, if
the observations can be believed, is of the order of an hour. That
even more stringently constrains the requisite bandwidth, and no one
would buy that, would they? No, the mechanism must have its own
positive feedback mechanism to keep the force in the proper phase.

It seems from your question that you have indeed constructed an
explanation of the event which employs the proper meaning of
resonance. That is exactly my point; if a chemistry PhD-to-be is
exposed to this film as an example of resonance *he comes away with
a misconception about the event*. Introductory physics teachers owe
their students more than that!

> If the wind had been purely DC, wouldn't the bridge have just bent to a
> new equilibrium position?  What, in more detail, do you mean by "pure DC
> excitation?"

Von Karmann (I think) showed the hydrodynamic mechanism at work here.
Turbulence cannot be avoided above a critical Reynolds number. It is
the turbulent mechanism which extracts the energy from the otherwise
steady wind. Introducing a system of small size into a much larger
laminar flow (such as placing a bridge in the wind through a river
valley) induces periodic turbulence. This is most dramatically seen
by "decorating" the wind with smoke which, downstream from the
obstacle, will be seen to form a spatially periodic "vortex street".

Leigh