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        riddles >> putnam exam (pure math) >> Tacoma Bridge
        
(Message started by: Larissa Preedy on Jun 7^th, 2005, 6:43am)

Title: Tacoma Bridge
Post by Larissa Preedy on Jun 7^th, 2005, 6:43am

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 ? :(

Title: Re: Tacoma Bridge
Post by Larissa_Preedy on Jun 7^th, 2005, 11:51pm

???

anyone at alL? :(

Title: Re: Tacoma Bridge
Post by Barukh on Jun 8^th, 2005, 1:40am

Is this (and also others) your homework exercise?

Title: Re: Tacoma Bridge
Post by Larissa_Preedy on Jun 8^th, 2005, 2:03am

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

Title: Re: Tacoma Bridge
Post by Icarus on Jun 8^th, 2005, 7:06pm

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.

Title: Re: Tacoma Bridge
Post by SWF on Jun 8^th, 2005, 10:23pm

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.

Title: Re: Tacoma Bridge
Post by Larissa_Preedy on Jun 10^th, 2005, 12:09pm

i got hte answer guys, dw bout it

i can post it up if ur intersted though

thanks for all the helP :P