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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=[tex]\int[/tex]0.5*[tex]\rho[/tex]A(w(x,t)^2)dx from 0 to L

Stating U: U=L/6EI *[(-[tex]\int[/tex]q(x,t)xdx+M(t))^2 +M(t)(-[tex]\int[/tex]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=[tex]\overline{W}[/tex]=L/6EI*[M[tex]_{A}[/tex]M[tex]_{B}[/tex]+M[tex]_{A}[/tex]^2+M[tex]_{B}[/tex]^] where M[tex]_{A}[/tex] and M[tex]_{B}[/tex] 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 [tex]^{}_{S_t}[/tex][tex]\int[/tex]F*w(L,t)dS

Does my elastic energy come out right?

Thanks for helping!

Most grateful for all comments.

/J

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# How to formulate the Lagrangian for a cantileverbeam

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