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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.:
EMBED Equation.3
changed to:
EMBED Equation.3
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, EMBED Equation.3 , represents the wind resistance of the bridge and the second term, EMBED Equation.3 , represents the cable resistance.
Page 1 Paragraph 5 (after Equation 1.4)
Removed:
the forcing term, and EMBED Equation.3 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 ( EMBED Equation.3 ) was 6 meters, and itWith 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 decks 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 showsadditional force of 61.4, and from this Figure
to
Figure 3.2.6 showsadditional 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 EMBED Equation.3 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 ( EMBED Equation.3 ) it takes
Page 16 Paragraph 4
Added:
velocity of the bridge deck ( EMBED Equation.3 ) would be the same
Page 16 Paragraph 4
Added:
where the cable begins resisting ( EMBED Equation.3 ).
Page 16 Paragraph 4
Added:
[Derivation of Equation 5.5]
Page 17 Paragraph 1
Added:
Other variables in Equation 5.5 are EMBED Equation.3 and EMBED Equation.3 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.
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