The Collapse of Tacoma Narrows Bridge: The Ordinary Differential Equations Surrounding the Bridge Known as Galloping Gertie

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The Collapse of Tacoma Narrows Bridge The
Ordinary Differential Equations Surrounding the
Bridge Known as Galloping Gertie

• Math 6700 Presentation
• Qualitative Ordinary Differential Equations
• Laura Lowe and Laura Singletary
• December 15, 2009

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The Tacoma Narrows Bridge
Construction on the Tacoma Narrows Bridge began
in November 1938 and was completed on July 1,
1940. The structure of the bridge was
characterized by lightness, grace, and
flexibility. The completion of the bridge was
heralded as a triumph of man's ingenuity and
perseverance
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Galloping Gerties Gallop
Video
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Galloping Gertie
During construction, the roadbed flexed or
displayed vertical oscillations This raised
questions about the bridges stability. A light
wind of 4 mph could cause these
oscillations Additional engineers were contracted
to fix these oscillations People came from
hundreds of miles away to experience driving on
Galloping Gertie
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Galloping Gertie
On the fateful morning, the center span
experienced vertical oscillations of 3 to 5 ft
under wind speeds of 42 mph The Bridge was
closed by 1000 AM on November 7, 1940
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Galloping Gertie
The motion changed to a two-wave torsional motion
causing the roadbed to tilt as much as 45
degrees This motion continued for 30 minutes
before a panel from the center span broke off
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Galloping Gerties Gallop
By 1100 am, the center span of the Tacoma
Narrows Bridge had fallen into Puget Sound
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The Collapse

Reporters, engineers, and passersby witnessed
the collapse at 1100 AM The only fatality was a
dog abandoned in a car on the bridge Eventually
called the Pearl Harbor of Engineering
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Eyewitness Report by Leonard Coatsworth
"Just as I drove past the towers, the bridge
began to sway violently from side to side. Before
I realized it, the tilt became so violent that I
lost control of the car... I jammed on the brakes
and got out, only to be thrown onto my face
against the curb. "Around me I could hear
concrete cracking. I started to get my dog Tubby,
but was thrown again before I could reach the
car. The car itself began to slide from side to
side of the roadway. "On hands and knees most of
the time, I crawled 500 yards or more to the
towers... My breath was coming in gasps my knees
were raw and bleeding, my hands bruised and
swollen from gripping the concrete curb... Toward
the last, I risked rising to my feet and running
a few yards at a time... Safely back at the toll
plaza, I saw the bridge in its final collapse and
saw my car plunge into the Narrows."
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"A few minutes later I saw a side girder bulge
out on the Gig Harbor side, due to a failure, but
though the bridge was buckling up at an angle of
45 degrees the concrete didn't break up. Even
then, I thought the bridge would be able to fight
it out. Looking toward the Gig Harbor end, I saw
the suspenders -- vertical steel cables -- snap
off and a whole section of the bridge caved in.
The main cable over that part of the bridge,
freed of its weight, tightened like a bow string,
flinging suspenders into the air like so many
fish lines. I realized the rest of the main span
of the bridge was going so I started for the
Tacoma end."
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Purpose
Our project plans to explore the events
surrounding this catastrophe We will explore a
complex model proposed by Dr. P. J. McKenna that
takes into account the potential and kinetic
energy involved with the oscillations of the
Bridge.

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Mathematical Models

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Mathematical Models

Vertical Kinetic Energy Torsional Kinetic
Energy Total Kinetic Energy
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Mathematical Models

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Mathematical Models
Vertical Potential Energy Torsional Potential
Energy Total Potential Energy

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Mathematical Models

Lagrangian Equations
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Mathematical Models
After a bunch of math

is the forcing term are the dampening terms
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Mathematical Models
Assuming the cables never lose tension, let
and

Then simplify and we get
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Mathematical Models

• The mass of the center span was approximately
  2500 kg/ft, and 12m wide.
• The bridge deflected approximately 0.5m per
  100kg/ft.

Hookes Law F -kx So k 1000
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Mathematical Models
Let Then, since the bridge oscillated at a
frequency of 12-14 cycles per minute, And let

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Mathematical Models
Put it all together and we get

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Mathematical Models
System of ODEs

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Mathematical Models
In 1940, the technology was not available to
solve a non-linear system. Instead, the engineers
of the time used a linearized ODE believing that
would give them the information they needed. In
the following slides we will compare these two
ODEs.

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Linear vs. Non-linear

Linear model with initial conditions
and (large push)
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Linear vs. Non-Linear

Non-linear model with initial conditions
and (large push)
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Linear vs. Non-linear

Linear model with initial conditions
and (large push)
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Linear vs. Non-linear

Non-linear model with initial conditions
and (large push)
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Linear vs. Non-linear

Non-linear model with initial conditions
and (large push)
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Linear vs. Non-linear

Non-linear model with initial conditions
and (large push)
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Unforced Model
The unforced, non-linear model with
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Unforced Phase Portrait

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Conclusions
Here we focused on the torsional motion of the
bridge with specific emphasis on the forcing
term. We assumed the cables did not lose tension
and that the torsion was symmetric and
independent from the vertical motion.

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Areas of Further Research

Broughton Bridge Angers Bridge Millennium Bridge
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References

Hobbs, R. S. (2006). Catastrophe to triumph
Bridges of the Tacoma Narrows. Pullman, WA
Washington State University Press. McKenna, P.
J. (1999). Large torsional oscillations in
suspension bridges revisited Fixing an old
approximation. The American Mathematical Monthly,
106, 1-18.