From: "Lawrence R. Yeagley" 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! +----------------------------+--------------------------------------------+ | | This poster was brought to you, in part, by| | | a grant from: | | Larry Yeagley | | | |YEAGLEY'S T'SHIRT & LINEAR ALGEBRA EMPORIUM | | lyeagley@gte.net | | | | Come on in !! We're dealing !! | | | Free eigenvalue to last 10 customers ! | +----------------------------+--------------------------------------------+ ============================================================================== Matches for: Anywhere=(tacoma) [1] 98k:73047 Taj\v cov�, Gabriela Mathematical models of suspension bridges. Appl. Math. 42 (1997), no. 6, 451--480. (Reviewer: Takashi Narazaki) 73K12 (35Q72 73D35 73H10 73K05) [2] 95k:34052 Fonda, Alessandro; Schneider, Zdenek; Zanolin, Fabio Periodic oscillations for a nonlinear suspension bridge model. Oscillations in nonlinear systems: applications and numerical aspects. J. Comput. Appl. Math. 52 (1994), no. 1-3, 113--140. (Reviewer: Tong Ren Ding) 34C15 (34B15 34C25 73D25 73K12) [3] 93f:34044 Kwong, Man Kam On the unboundedness of the number of solutions of a Dirichlet problem. Nonlinear Anal. 18 (1992), no. 10, 993--1001. (Reviewer: Eric Benoit) 34B15 [4] 39 #7859 Pittel\cprime , B. G.; Jakubovi\v c, V. A. A mathematical analysis of the stability of suspension bridges based on the example of the Tacoma bridge. (Russian) Vestnik Leningrad. Univ. 24 1969 no. 1 80--91. (Reviewer: I. Aganovi\'c) 73.35 � Copyright American Mathematical Society 1998