Please note:
All locations specified in this document refer to the location in the re-submitted paper and
not the original. Highlighted changes were the changes that were asked to be made.
Reasons for not making changes as specified can be seen at the end of this document.
General
The formatting for all equations with time intervals was changes. i.e.:
  ....
nt m
  ....
qt w
changed to:
....
....
  
nt m
qt w
Other miscellaneous formatting/spacing changes were made
Page 1 - Equation 1.1
The 6 that was originally in the equation was replaced by l since the width is what the
6 represents.
Page 1 – Paragraph 3 (after Equation 1.1)
Added:
…the first derivative taken with respect to time, and a double dot represents the
second derivative taken with respect to time. In Equation 1.1, the first term,
 , represents the wind resistance of the bridge and the second term,
lk
cos sin  , represents the cable resistance.
m
Page 1 – Paragraph 5 (after Equation 1.4)
Removed:
…the forcing term, and t represents time.
Page 2 – Paragraph -1
Changed:
…a large torsional push were used
to
…a large torsional displacement were used
Page 2 – Paragraph 1 (after Equation 2.1)
Added:
…we get approximately 1,000 N/m. It is also known that the width of half the
bridge ( l ) was 6 meters, and it…With all of these constants in place,
Equation 1.1 becomes [Equation 2.2 added and existing 2.2 changed to 2.3]
Page 2 – Paragraph 4
6 in formula was changed to l to be consistent with the changed Equation 1.1
Page 2 – Paragraph 5
Changed:
…on the bridge having different periods on both sides
to
…on the bridge having different periods on each side
Page 3 – Paragraph 3
Changed:
…were acting on either side
to
…were acting on each side
Page 3 – Paragraph 3
Added:
In this Figure, the peaks of the maximum amplitude occur at a later time
than they did in the standard response, and the amplitude in this response
fluctuates a little more after a time of about 500.
Page 4 – Paragraph -1
Added:
…a significant change in the response of the bridge when compared to the
standard response.
Page 4 – Paragraph 1
Changed:
As shown in Figure 3.2-2, there is a more gradual variance of rotation and the
rotation never completely levels out
to
As shown in Figure 3.2-2, there is a more gradual fluctuation in the amplitude,
and it never becomes constant.
Page 5 – Paragraph 3
Changed:
In this case, the rotation of the bridge deck levels out…
to
In this case, the amplitude of the bridge deck’s rotation levels out…
Page 5 – Paragraph 3
Changed:
After this time, the rotation begins to vary and remains in this oscillating
rotation.
to
After this time, the amplitude of the rotation begins to vary and continues
undulating over the shown time.
Page 5 – Paragraph 4
Changed:
Figure 3.2-4 shows the torsional oscillations in the Tacoma Narrows Bridge
going back down to zero.
to
Figure 3.2-4 shows the amplitude of rotation in the Tacoma Narrows Bridge
decreasing towards zero.
Page 6 – Paragraph 1
Corrected Equation number to reflect change in actual number (2.2 changed to 2.3)
Page 6 – Paragraph 3
Added:
In other words, we will show the bridge response where the amplitude of
rotation does not approach zero since we know that the rotation of the
Tacoma Narrows Bridge did not stop.
Page 7 – Paragraph 1
Added:
…than about 61.4, the amplitude of the bridge rotation would…
Page 7 – Paragraph 1
Changed:
For this reason
to
Since the rotation of the Tacoma Narrows Bridge did not stop
Page 7 – Paragraph 1
Changed:
For this reason, Figure 3.2.6 shows…additional force of 61.4, and from this
Figure…
to
Figure 3.2.6 shows…additional force of 61.4, which is the time directly prior to
the amplitude going to zero. From this Figure…
Page 7 – Paragraph 1
Added:
…to about 300, the amplitude of rotation varies more…
Page 7 – Paragraph 2
Added:
…did not stop even as the time interval for the force approached infinity.
Initially the amplitude of rotation varies a…
Page 7 – Paragraph 3
Changed:
…time of 90, the rotation in this case does not stop.
to
…time of 90, the amplitude of rotation in this case does not approach zero as
the time interval for the force approaches infinity.
Page 8 – Paragraph 3
Added:
…greater than 57.3, the amplitude of rotation of the bridge…
Page 8 – Paragraph 3
Changed:
…large variance in the rotation limits until it levels…
to
…large variance in the amplitude of rotation until it levels…
Page 8 – Paragraph 4
Changed:
…time of 90 to about 208.7, the limits of the rotation in…
to
…time of 90 to about 208.7, the amplitude of the rotation in…
Page 8 – Paragraph 4
Added:
…function is added back into the model, and the amplitude of rotation of the
bridge can…
Page 9 – Paragraph 1
Added:
…than a time of 100 (end time of 700), the amplitude of rotation of the bridge…
Page 9 – Paragraph 1
Changed:
…to the end time of 700, the limits of rotation in the…
to
…to the end time of 700, the amplitude of rotation in the…
Page 9 – Paragraph 2
Changed:
In summary, we found that the oscillations in all three of these cases stopped if
the block…
to
In summary, we found that the amplitude of rotation in all three of these cases
approached zero if the block…
Page 9 – Paragraph 2
Added:
Once again, since we know the rotation of the bridge did not stop, we have
shown the cases directly prior to those that the amplitude of rotation went to
zero.
Page 10 – Paragraph 3
Changed:
The first case that will be looked at is how an…
to
The first case that will be investigated is how an…
Page 10 – Paragraph 3
Changed:
…can be seen below, the limits of the rotation in…
to
…can be seen below, the amplitude of the rotation in…
Page 10 – Paragraph 4
Added:
…deck is very chaotic, and the amplitude of rotation eventually…
Page 11 – Paragraph 2
Changed:
…observed simply shifted the oscillations of the bridge, and made the
oscillations slightly more random.
To
…observed simply shifted the amplitude of rotation of the bridge, and made the
amplitude slightly more random.
Page 11 – Paragraph 2
Added:
…a slight jump in the amplitude of rotation of the bridge…
Page 11 – Paragraph 2
Changed:
…would be introduced, a downward shift in the oscillations would occur…
to
…would be introduced, a downward shift in the amplitude would occur…
Page 12 – Paragraph 4
Changed:
…large variance in the limits of the rotation and…
to
…large variance in the amplitude of the rotation and…
Page 12 – Paragraph 6
Changed:
…velocity before the cables began picking up the force.
to
…velocity before the cables began picking up the weight.
Page 13 – Paragraph -1
Changed:
…Figure 3.4-1, but with the limits of rotation increasing…
to
…Figure 3.4-1, but with the amplitude of rotation increasing…
Page 13 – Paragraph -1
Changed:
…simply increases the limits of rotation quicker.
to
…simply increases the amplitude of rotation more quickly.
Page 13 – Paragraph 1
Changed:
…the deck flips over halfway at a time of about 1,180, and then flipped the rest
of the way over at a time of about 1,220. After the deck flips over completely,
the limits of rotation decrease significantly.
to
…the deck flips over once at a time of about 1,180, and then flipped over again
at a time of about 1,220. After the deck flips over a second time, the amplitude
of rotation decreases significantly.
Page 15 – Paragraph -1
Changed:
This can be mathematically represented with the following equation
to
This can be mathematically represented with the following function
Page 16 – Paragraph 1
Changed:
…the only variable readily available is the mass of the bridge.
to
…the only variable known without further investigation is the mass of the
bridge.
Page 16 – Paragraph 2
Added:
We will approximate the acceleration to be constant since we are only
concerned with obtaining a general idea of what the forces were in the cables.
Page 16 – Paragraph 2
Removed:
By integrating Equation 5.2, we obtain [Equation 5.4] where t represents
time.
Page 16 – Paragraph 2
Added:
In order to find this, we first need to integrate Equation 5.2. [Derivation of
Equation 5.4].
Page 16 – Paragraph 3
Changed:
In this equation, …
to
In Equation 5.4, …
Page 16 – Paragraph 3
Added:
…the time ( t ) it takes…
Page 16 – Paragraph 4
Added:
…velocity of the bridge deck ( v o1 ) would be the same…
Page 16 – Paragraph 4
Added:
…where the cable begins resisting ( v f 2 ).
Page 16 – Paragraph 4
Added:
[Derivation of Equation 5.5]
Page 17 – Paragraph 1
Added:
Other variables in Equation 5.5 are x f and x o which represents the final and
initial positions of the bridge deck respectively.
Page 18 – References
Reference 11 was corrected
Comments:
“p.14, just before equations
Refer to what follows as ‘the following system of equations,’ rather than ‘the
following equation.’”
Reason:
This was not a system of equations it was simply a very large piecewise function. I tried
to make this clearer by reformatting all of the piecewise functions in the paper, and
changing “equation” to “function”.