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Fourth order differential equation

  1. Oct 24, 2013 #1
    1. The problem statement, all variables and given/known data
    I have this physics mathematical problem : (see link in comment)


    EI(∂4u)/(∂x4) =f ..........................(1)


    The boundary conditions are: ∂2u/∂x2 =0 and EI ∂3u/∂x3 =±F

    where E is Young’s modulus, I is second moment of area, f is force per unit length applied to the beam and F is the force applied to the edges and the ± applies the the left and right edges.

    The upper load placed on the chip is modelled as two point forces while the loads exerted by the pins of the chip are modelled as (localised) Hookean springs as shown in figure [1 ](in the link). Thus the equations become:

    EI ∂4u/∂x4 =−F[δ(x−B)+δ(x+B)]−k2 u(0)δ(x) .......................................(2)

    with boundary conditions :

    EI∂3u/∂x3 =−k1 u(−A), x=−A .....................(3)

    EI∂3u/∂x3 =k1u(A), x=A .......................(4)

    2u/∂x2 =0, x=±A .........................(5)


    Show that calculating the displacements can be reduced to the problem of solving a set of linear equations.

    could you help please



    2. Relevant equations

    for the full question see link:
    4shared.com/file/rdVHwO6j/strains_in_silicon_chip.html

    3. The attempt at a solution

    Trying to answer:

    x axis:
    −A __(I)____−B __(II)___0 __(III)_______B _____(IV)____A

    •In area I , II , III and IV equation (2) become ZERO: EI∂ 4 u/ ∂x 4 =0 solving this equation we get a linear equation system:

    uI =a1 x 3 +a2 x 2 +a3 x+a4
    uII =b1 x 3 +b2 x 2 +b3 x+b4
    uIII =c 1 x 3 +c2 x 2 +c3 x+c4
    uIV =d1 x 3 +d2 x 2 +d3 x+d4
    which has 16 unknowns (a 's,b 's,c 's and d 's)

    using the doundary conditions and integration equation (1) aroud -B and B we could find 12 of the unknowns.

    After finding them what shuold I do.

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



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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