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Coupled differential equations

  1. Oct 30, 2011 #1
    Hi,
    I have to modelize the buckling of a column and I've come up with this system:
    [tex] N'(x) + N(x) \theta ' (x) \theta (x) - Q \theta ' (x) + f = 0 [/tex]
    [tex] Q'(x) + N(x) \theta ' (x) + Q \theta ' (x) \theta (x) = 0 [/tex]

    with f a constant

    The coefficients (thetas) are not constants.
    I've written it as X' = A X + f But I dont know how to diagonalize A since coefficients are not constants.

    Thank You for helping me.
     
  2. jcsd
  3. Oct 30, 2011 #2

    HallsofIvy

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    Actually, it's worse than that. Not only are the coefficients variable, but the second equation includes [itex]\theta \theta'[/itex] so the equations are non-linear and matrix methods cannot be used at all.
     
  4. Oct 30, 2011 #3

    lurflurf

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  5. Oct 30, 2011 #4
  6. Oct 30, 2011 #5

    lurflurf

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    θ is a known function right?
     
  7. Oct 30, 2011 #6
  8. Oct 30, 2011 #7

    lurflurf

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    How complicated is θ?
     
  9. Oct 30, 2011 #8
    Well, i dont know, im a bit lost.
    Actually, theta is my slope angle which is very small.
     
  10. Oct 30, 2011 #9
    May I ask why you just don't solve it numerically? I mean really, he didn't say nothing about analytic solution else I'd keep my mouth shut. You know, numerical methods are perfectly fine for the real-world.

    So if I may be the practical voice in here: work it first numerically even if you need an analytic solution just to get a handle on it, then do it analytically if you have to.
     
  11. Oct 30, 2011 #10
    First I need to solve it analytically. Maybe equations are not good. Maybe I should get simplier equations, i dont know ...
     
  12. Oct 30, 2011 #11
    No you don't even if you have to solve it analytically. You know if you're gonna drive your truck in the dark without lights, it's a good idea to first walk the path with a flash light to see if there are any holes and stuf.

    I'm tellin' you the right way to approach this: solve it first numerically even if you have to just dream-up initial conditions. Get a handle on it, then attempt to solve it analytically.

    Also, I do not believe this is a non-linear system if [itex]\theta(x)[/itex] is known. You effectively have:

    [tex]\frac{dN}{dx}=-Ngh+Qh-f[/tex]

    [tex]\frac{dQ}{dx}=-Nh+Qhg[/tex]

    where g=g(x) and h=h(x) are known functions. Well there you go: I'm dreamin' up g and h, N(0) and Q(0) too, bingo-bango: Numeric solution in hand.
     
    Last edited: Oct 30, 2011
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