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How to formulate the Lagrangian for a cantileverbeam

  1. Feb 12, 2009 #1
    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=[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.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution

    Attached Files:

  2. jcsd
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