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Topic: Tacoma Bridge (Read 1587 times) |
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Larissa Preedy
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The Tacoma narrows bridge may be approximated by a simple beam with area moment of inertia I, bulk elastic coefficient E and span L. If the mass per unit length is M, use dimensional arguments to find an expression for the order of magnitude of the natural frequency of oscillation of the center span.
now, this much i know
for the units of E
v=sqrt(E/d)
m/s = sqrt (E/(Kg/M^3))
m^2/s^2 *(Kg/m^3) = E
E = kg / (M*s^2)
now i'm stuck as to how to actually find the order of magnitude... by that, i mean something like
A^2 B^-0.25
can anyone help me ? 
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Larissa_Preedy
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Re: Tacoma Bridge
« Reply #1 on: Jun 7th, 2005, 11:51pm » |
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anyone at alL?
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Barukh
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Re: Tacoma Bridge
« Reply #2 on: Jun 8th, 2005, 1:40am » |
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Is this (and also others) your homework exercise?
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Larissa_Preedy
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Re: Tacoma Bridge
« Reply #3 on: Jun 8th, 2005, 2:03am » |
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No, it isn't homework exercises, it is past exam papers questions, and i'm trying to revise. The revision program will only tell you if the answer is correct or not, and not show working until you get it correct once
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Icarus
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Re: Tacoma Bridge
« Reply #4 on: Jun 8th, 2005, 7:06pm » |
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I have looked at this, but am not sure what exactly the problem is talking about. Moments of Inertia I deal with all the time, but I am not sure what is meant by "area moment of inertia". "Elastic Coefficient" I have found used to describe more than one thing. However, they generally all have units of (force)/(area), as you have indicated. However, I am not sure what it means when you put "bulk" in front.
If I try to come up with an answer assuming that one or both of the adjectives "area" and "bulk" do not change the units, then I suffer from an embarassment of riches. There are multiple simple ways of getting units of 1/sec by operating on the given quantities. So simple dimension analysis alone is not sufficient.
I suspect, though, that both adjectives change the units of the described quantities, and if I knew the intended units, I would no longer find multiple simple means to combine them to form the required result.
In particular, I find it peculiar that we are given 3 units with mass: E, I, M. Any two can be combined to cancel out mass - but using the third in any way re-introduces mass into the expression.
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| « Last Edit: Jun 8th, 2005, 7:24pm by Icarus » |
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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SWF
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Re: Tacoma Bridge
« Reply #5 on: Jun 8th, 2005, 10:23pm » |
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This illustrates some of the limitations Icarus mentioned in the guitar string question. You are given 4 parameters:
E = kg/(m-s^2)
I = m^4
L = m
d = kg/m
There are an unlimited number of ways to combine those and be left with only units of time. Also, order of magnitude is not a good description. The frequency can vary by more than a factor of 10 depending on boundary conditions and the mode of vibration. The frequency can be written in terms of tensile modulus, bulk modulus, or shear modulus, all of which have the same units but different values.
Still, there is a parameter that is proportional to the frequency of a beam in a bending mode of vibration (as I recall, the Tacoma Bridge did not vibrate in that mode). I think E is the tensile modulus not the bulk modulus since that is the standard notation. The resistance to a beam to bending is E*I, and if you assume the answer contains E*I, the rest can be found:
Frequency is proportional to: sqrt(E*I/d/L^4) and the constant of proportionality denpends on mode of vibration and boundary conditions. For example, for a simply supported beam at its first fundamental mode, the constant is 9.87.
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Larissa_Preedy
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Re: Tacoma Bridge
« Reply #6 on: Jun 10th, 2005, 12:09pm » |
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i got hte answer guys, dw bout it
i can post it up if ur intersted though
thanks for all the helP 
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