The Failure of the Tacoma Narrows Bridge

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The Failure of the Tacoma Narrows Bridge
By Thomas Fein Advisor Dr. Jyoti Champanerkar
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Brief History of the Bridge

• Built in Washington State
• Construction started in November of 1938.
• Completed on July 1, 1940.
• Third Largest Suspension Bridge of its time.
• Total Length1524 m (5000ft)
• Length of Center Span854 m (2800 ft)
• Width 11.9 m (39 ft, two lanes)
• Cost 6.4M (90M in Todays Dollars)

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Behavior of the Bridge on the Day of Collapse

• Amplitude of Oscillations were about 5 feet
• Frequency of oscillations were 38 cycles per
  minute
• Vertical Oscillation of approximately 12 to14
  cycles per second
• Instantaneous change from vertical to torsional
  motion

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Classic Explanation for CollapseResonance

• What is Resonace-The increase in amplitude of
  oscillation of an electric or mechanical system
  exposed to a periodic force whose frequency is
  equal or very close to the natural undamped
  frequency of the system.
• Resonance fails to account to change from
  vertical to torsional motion
• Also fails to account for vertical oscillation of
  bridge due to a single gust of wind

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Outline

• Derive a mechanical model of the Bridge
• Choose Physical Constants for the differential
  equations based on Historical Data
• Compare the linear and nonlinear models
• Explain the instantaneous change from vertical to
  torsional oscillation by analyzing nonlinear model

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Modeling the Bridge
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Energy Equations from Bridge Model

• Potential Energy-energy associated with position
• Kinetic Energy-energy of motion
• Potential Energy ky2/2, where kspring constant
  and y is the extended distance of the spring
• Kinetic Energy(1/6)mL2(? ) 2, where a rod of
  mass m and length 2L rotates about its center of
  gravity (cg) with angular velocity ?

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Energy Equations Continued

• Let ?angle of the rod from horizontal and
  ydownward distance of the cg of the from the
  unloaded state (dashed lines)
• Let y be the positive part of y
• Gravitational Potential Energy is mgy
• Extension of One Spring is (y-l sin ?) and the
  other is (y l sin ?)
• Total PE(k/2) (y-l sin ?) (y l sin
  ?)2-mgy
• Total KEm y2 /2 (1/6)mL2(? ) 2

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Model Continued
(1/3)ml2? (kl) cos ? (y- L sin ?) - (y L
sin ?) y -k (y- L sin ?) - (y L sin ?)
mg Solving for ? and y adding a small
viscous damping term d? to the first equation
and dy to the second. Also, we are adding a
forcing term, f(t) to the second equation. ? -
d? (3k/m L) cos ? (y- L sin ?) - (y L sin
?) f(t) y - dy -(k/m) (y- L sin ?)
(y L sin ?) g Assuming cables never lose
tension ? - d? (6k/m)cos ? sin ? f(t) y
- dy - (2k/m)y g
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Numerical Experiment Comparison of the Linear
and Non-linear Models

• The nonlinear model ( for torsion) is
• ? (-.01)? (6k/m)cos ? sin ? f(t)
• Choosing Physical Constants
• Letting K1000 and m2500, f(t) ?sin(µt) and
  .01 for the viscous damping terms. µ will be
  taken from 1.2 to 1.6.
• ? (-.01)? (2.4)cos ? sin ? ? sin(µt)
• For small ?, we let sin ? ? and cos ?1. Thus,
  the linearized version of the equation is
• ? (- .01)? (2.4)? ? sin(µt)

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• Large torsional Push- ICs ? (0) 1.2 and ? (0)
  0 with µ1.2 and ?0.06
• The linear model (Top) has settled down to an
  oscillation of 3 degrees.
• The nonlinear model (Bottom) has settled into
  large amplitude periodic oscillation of about one
  radian.

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• µ1.2 and ?.06
• Small initial conditions ,?(0)1.2 and ?(0)0.
  Similar to linear model from previous graph (Top)
• Large torsional push combined with the same small
  torsional forcing term results in a large
  amplitude oscillation. (Bottom)
• ConclusionNonlinear Model can have different
  responses to same forcing term depending on
  whether or not there is a single push

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• Increase µ 1.3 but decrease ? to 0.02 produces
  an oscillation of 1 degrees in the linear model.
  Here the linear model settles down to near
  equilibrium (Bottom)
• Nonlinear model,-with its large initial push-
  produces a large torsional oscillation.
• The period is 4.83 which is somewhat above that
  observed when the bridge started its torsional
  oscillation.

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• µ is increased to 1.4-same results.
• Period is reduced to 4.5 which is much closer to
  the reported 4.3.
• Similar result if µ is increased to 1.5.
• Results vanish until integral multiples of µ are
  reached.

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• Historical Observation- vertical motions that
  came before the torsional ones had a frequency of
  about 40 per minute( µ near 4).
• When the initial conditions are large or small
  combined with a forcing term of ? sin (4t) the
  results in either a small linear response or a
  large nonlinear response respectively.

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• Compare the transient behavior for the nonlinear
  (correct) model and the linear model.
• Different initial conditions are used and the
  forcing term, when used, takes the form ?
  sin(1.2t).
• Figure Below- Large Initial Push and No Forcing
  Term. There is very little difference as both
  models settle back to equilibrium.

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• Now, a forcing term is introduced, ?.05 but with
  initial conditions at equilibrium.
• As seen in Figure below, the results for both
  models are again similar.

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• In Figure Below both effects are combined.
• As predicted by the Principle of Superposition,
  the linear system (top) dies out.
• The nonlinear model goes into large torsional
  oscillation, eventually settling down to a large
  periodic motion.

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• The sensitivity of the nonlinear system to the
  amplitude of the forcing term is exhibited in the
  Figure below.
• If ? is reduced from 0.05 (top) to 0.04 (bottom)
  the motion is reduced to near-equilibrium.

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The Transition to Torsional Motion

• Investigate initial value problem for nonlinear
  system if cables lose tension briefly
• Start with large vertical push y(0)26 and tiny
  torsional forcing
• As seen in figure below, the cables lose tension
  for about 2 periods but torsional motions are
  around zero.

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Vertical and Torsional Responses to large push
(y(0)26) in vertical direction
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The Transition to Torsional Motion

• Repeat Experiment but change y(0)31.
• At first almost no torsional oscillations then it
  instantaneously approaches 1 radian
• With no torsional forcing term, the motion is
  damped and settles down to near equilibrium

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Vertical and Torsional Responses to large push
(y(0)31) in vertical direction
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• Repeat Experiment with tiny displacement ? (0)
  .001 and no torsional forcing term.
• Results Below- Again, no torsional oscillation
  for a while, then and instantaneous jump to large
  torsional oscillation up to around 1 radian
• With no torsional forcing term, oscillation dies
  down due to damping.

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• Now, we combine a small periodic forcing term and
  a large vertical push.
• Results- A quick transition to large torsional
  oscillation as the vertical oscillation is damped
  away.

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Conclusion

• Linearizing for small oscillations was a good
  approximation but for large oscillations it
  should be categorized as an error.
• The error of treating cables, in both their
  loaded and unloaded state, as springs that obey
  Hookes law was corrected.
• Most important conclusion-Within frequencies
  corresponding to those of the bridge, the final
  oscillations produced by the nonlinear model can
  be both very large or very small.

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• Thank You For Your Time.
• Questions?