Beam stuck in a wall and free on one end with force
Name: rodney b
Date: Around 1995
In differential equations today, we discussed the equation of motion for a
beam that was stuck in the wall on one end but let free to move on the other
end. My professor was unable to explain where that equation came from so I
was wondering if someone on this bbs here could. I would really appreciate
it. If the explanation is too long or involved a book reference would be
great too. Also I was wondering, if instead of using a general equation for
describing the motion, if we used only a single force acting on it in the
middle would that simplify the derivation? And also, would the greatest
force upward be exerted to say a diver on a diving board when he is at the
tip of the beam or in the middle. Is the answer to that question really all
that obvious or is the equation governing the motion of that beam much more
difficult if you vary the position of the downward force (in the diver
example his position of n then board (g))?
Well, you have quite a few questions there.
The equation comes from continuum mechanics, which is the theory of how
solid (in this case - it also does liquids) bodies behave mechanically.
Ordinary mechanics treats how point particles (or at least infinitely rigid
bodies) behave - for a point particle in 3 dimensions that gives 3 variables
for dynamics to act on, for a rigid body the number of variables increases
to at only 6. But for a real solid material, which can bend and stretch
(called strain effects) under imposed forces (called stresses) you really
have an infinite number of such variables (well actually something on the
order of Avogadro's number) which is why a continuum description and
differential equations are necessary.
I am not too familiar with the literature, but try looking for "continuum
mechanics" books for a derivation. This is also important for civil
engineering, but I am not sure if they usually derive it in their texts or
not - you could check a civil engineering textbook though.
Limiting the imposed forces does not simply things because the problem is
not the complexity of the forces, but the complexity of the body (with an
infinite number of degrees of freedom) responding to the forces.
Click here to return to the Mathematics Archives
Update: June 2012