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Tacoma Narrows Bridge Collapse

see also the following links about the Tacoma Narrows Bridge Collapse:
Movie formats available- Approx. 8 seconds long

In this lab we look at a particular mechanical system that ended up in destructive oscillations. Your job will be to examine the information and try to determine the cause of the bridge collapse.

Be mindful of the fact that this is not the only bridge to ever collapse. Some have done so after years of neglect in their maintenance, others wre not structurally capable of withstanding the loads applied to them. This one in particular, however, was newly built. It was designed to expand and contract and move about to some degree.

  1. Observe the video describing the bridge and its demise.

    2.  
  2. As part of your laboratory report, comment on the video, your feelings, observations, thoughts, as if you were there at the time.

    3.  
  3. You have a model in the laboratory which may not exactly represent the bridge. It is a model (recall just what models are and do) of the cross section of the road, not the length of the roadway. It does in some manner represent some aspects of a suspension bridge. The Golden Gate Bridge is a suspension bridge as well. Assume the model is a rigid platform supported by two springs of equal spring constant. (Your knowledge of sensitivity to initial conditions might suggest that this is hardly possible, but assume it is "close enough for gov't work." See below:

    4.  

     

    We write down the equations for motion of the bridge using Newton's Second Law:
     
    M ay = - k (y1 + y2) (1)
    I a = - k L/2 (y2 - y1) (2)

    We make the small angle approximation:
     
    q = ( y2 - y1 ) / L (3)

    Your instructor will explain how the resultant differential (this is nonlinear) equations are solved. The result is sinusoidal where:
     
    y1 = A1 sin ( w1t ) and y2 = A2 sin ( w2t ) (4)

    There are two cases to sonsider. If the bridge goes up and down so that both springs move together; secondly, if the springs move in opposite directions, causing a rotation about the center of the span.

    1. Case 1: Here both sides of the bridge move in unison. Both edges have the same frequency of oscillation:
      2. w12 = w22 = w2 = 2 k / m (5)
    2. Case 2: Torsional Mode. The frequencies are:
      3. w12 = w22 = w2 = k L2 / ( 2 mR2) (6)
    The mass per unit length of the bridge was 4.3 X 103 kg/m and the width, L was 12 metres. The radius of gyration was 4.8 m and the spring constant k was 1.5 X 103 N/m for the real bridge. Measure these values for your bridge span (model) and determine the modes of vibration. In a real bridge there are certainly many many frequencies, a distribution. It can be shown that the energy per unit time that is accepted by a mode of vibration is given by:
     
    P = 1 / [ (w - wo )2 + ( Dw / 2 )2 ] (7)

     
  4. Comment on your bridge model and compare to the real bridge. How well do they compare? Is this a realistic model? Why or why not? Describe what happened so the reader understands. Make it a cool writeup so "mom" knows what happened. After all, the reader determines your grade.
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