- #1
smolloy
- 10
- 0
Hi all,
I'm trying to calculate the rocking frequency of a half-cylindrical arch. That is, a half-cylinder, that has had a smaller half-cylinder "bitten" out of it. If placed with the curved surface on the floor, it can be made to rock from side to side (sort of like an inverted pendulum).
In calculating this, I have come up with the following differential equation:
[tex]\theta'' = -g\frac{3}{\pi}R\theta[/tex]
Where [tex]\theta[/tex] is the angle of the arch from its nominal position, g is the gravitational constant, and R is the radius of the arch.
The problem is that I interpret this as meaning that the frequency is proportional to the square root of the radius, when observation tells me that it should be an inverse relationship (i.e. large radius leads to low frequency).
Have I interpreted the differential equation incorrectly, or have I made an error somewhere in my derivation?
I'm trying to calculate the rocking frequency of a half-cylindrical arch. That is, a half-cylinder, that has had a smaller half-cylinder "bitten" out of it. If placed with the curved surface on the floor, it can be made to rock from side to side (sort of like an inverted pendulum).
In calculating this, I have come up with the following differential equation:
[tex]\theta'' = -g\frac{3}{\pi}R\theta[/tex]
Where [tex]\theta[/tex] is the angle of the arch from its nominal position, g is the gravitational constant, and R is the radius of the arch.
The problem is that I interpret this as meaning that the frequency is proportional to the square root of the radius, when observation tells me that it should be an inverse relationship (i.e. large radius leads to low frequency).
Have I interpreted the differential equation incorrectly, or have I made an error somewhere in my derivation?
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