One more thing: I noticed that one of your proposed equations
mixes (beta=(w^2*m*L^4/EI)^0.25) a float with non-floats.
Try not to do that --- especially if you apply a function similar
to trashpositivefactors:
(%i87) product((x-k),k,1,23)$
(%i88) allroots(expand(float(%)));
<junk> ,x=0.42824723478225118*%i+13.437402218704808, <junk>
Yikes! look at the imaginary part.
Barton