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1. What exactly is this? Is it like a geometric constraint? This equation doesn't seem to depend explicitly on external loading or beam boundary conditions.

2. Can I derive this starting from the Euler-Bernoulli beam equation?

I am dealing with a beam with non-uniform cross-section and stiffness. The beam is cantilevered. It also has a point load at the tip. I tried to start with:

[itex]EI(x)\frac{∂^4w}{∂x^4}=-μ\frac{∂^2w}{∂t^2}+Fδ(x-L)[/itex]

My goal is to get bending moment as a function of x and t. I thought about integrating with respect to x, because maybe that would get me a "moment" term Fx. But then, integrating the left side seems really ugly, and I am also confused because μ is mass per length? So you get mass per length times acceleration times x, which definitely is not moment? And

I am hopelessly confused... I just want a way to make sure that I am not misusing the bending moment equation, because it seems weird to me that it would hold for ever single case (for example, shouldn't it be different if the beam had an axial load?).