# How to formulate the Lagrangian for a cantileverbeam

1. Feb 12, 2009

### Jompa

Hello smarties!
Anyone who can help me out here?

The problem statement is in the attachment, problem.jpg

My attempt:
Lagrangian: L=T-U where T kin. energy and U elastic potential.

Stating T: T=$$\int$$0.5*$$\rho$$A(w(x,t)^2)dx from 0 to L

Stating U: U=L/6EI *[(-$$\int$$q(x,t)xdx+M(t))^2 +M(t)(-$$\int$$q(x,t)xdx+M(t)) +M(t)^2]- what?
(the integrals from 0 to L, sorry couldn't get the hang of how to get the limits to look nice with latex)

The get the expression for W (the expression before "what") comes from W=$$\overline{W}$$=L/6EI*[M$$_{A}$$M$$_{B}$$+M$$_{A}$$^2+M$$_{B}$$^] where M$$_{A}$$ and M$$_{B}$$ are the moments at the ends of the beam. They are determined with help of moment equilibrium.

How should the "what" term look like? Should it look someting like this $$^{}_{S_t}$$$$\int$$F*w(L,t)dS
Does my elastic energy come out right?

Thanks for helping!
/J
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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