1. ## Simultaneous equations/Matrices...

I've just started a PhD on wind turbine blade testing. The blades are mounted on a hub and resonated with an reciprocating mass and I am modelling this. I am looking to create a Finite Element model of the blade modelled as a beam. I have mass/unit length, dimensional and stiffness data for the blade at different stations along it's length. What I am looking to do is model the blade as a rectangular box section that steps down in size along the blade. So on to the maths bit...

The blade is modelled as a hollow rectangular tube, b wide, d deep with a recess in the middle that is b1 wide and d1 deep. Knowing this we can formulate 3 equations for the stiffness in the width direction, stiffness in the thickness direction and mass per unit length. The density is assumed constant along the length and so is the young's modulus (don't worry about this, it's an engineering term) the width b is given in the blade data i recieve, and I want to use it as it is useful for it's aerodynamic properties so I want to retain it. The other dimensions can be adjusted to suit.

Ixx=b.d^3/12-b1.d1^3/12 -Stiffness in thickness direction
Iyy=b^3.d/12-b1^3.d1/12 -Stiffness in width direction
q=density*(b.d-b1.d1)

There is also the requirement that b>b1 and d>d1. So we have 3 equations and 3 unknowns which should be solvable.

I've tried solving this with simultaneous equations by rearranging the third equation for d then substituting it into the second equation, which can then be rearranged for d1 and substituted into the first equation. But I'm getting unstuck rearranging that for b1! Is there a way of doing this with matrices? I need to remind myself of how to do them anyway as I suspect they will be very useful. Sorry if this doesn't seem university level to people but it's definitely harder than anything I did at A-Level! I'm not after having this solved for me (wouldn't hurt though...) but I need to be pointed in the right direction. Any help will be much appreciated!

Thanks,

Pete

2. So, let us clarify this a little:

First let us distinguish between pre-given constants and un-knowns by a more suggestive notation: let $d= x, b_1=y,d_1= z$

Denote the constants:
$c =$ density
$b =$ width blade.

So if I understand correctly we must solve the following system for $x,y,z$:

$12[Ixx]=bx^3-yz^3$
$12[Iyy]=b^3x-y^3z$
$\frac{1}{c}[q] = bx-yz$

contitions: $y < b, z < x$
Where I assume [Ixx],[Iyy],[q] are given values as well.

3. All correct, I'm just about to write a macro to do it by trial and error but I'd much prefer a more elegant solution!