How to Modelize Column Buckling with Coupled Differential Equations?

[tdcaupv]

Hi,
I have to modelize the buckling of a column and I've come up with this system:
$$N'(x) + N(x) \theta ' (x) \theta (x) - Q \theta ' (x) + f = 0$$
$$Q'(x) + N(x) \theta ' (x) + Q \theta ' (x) \theta (x) = 0$$

with f a constant

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

Thank You for helping me.

 

[HallsofIvy]

Actually, it's worse than that. Not only are the coefficients variable, but the second equation includes $\theta \theta'$ so the equations are non-linear and matrix methods cannot be used at all.

 

[lurflurf]

How complicated is θ, you could use the method of peano baker among others.

generalizing separation technique

Note on the Peano-Baker Method of solving Linear Differential Equations by
Arch Milne[/QUOTE]

http://arxiv.org/pdf/1011.1775v1

 

[tdcaupv]

edit

 

[lurflurf]

θ is a known function right?

 

[tdcaupv]

Yes.

 

[lurflurf]

How complicated is θ?

 

[tdcaupv]

Well, i don't know, I am a bit lost.
Actually, theta is my slope angle which is very small.

 

[jackmell]

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.

 

[tdcaupv]

First I need to solve it analytically. Maybe equations are not good. Maybe I should get simplier equations, i don't know ...

 

[jackmell]

No you don't even if you have to solve it analytically. You know if you're going to 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 $\theta(x)$ is known. You effectively have:

$$\frac{dN}{dx}=-Ngh+Qh-f$$

$$\frac{dQ}{dx}=-Nh+Qhg$$

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.