Solving a 3D force-moment balance using two force application points

[daniel7campbell]

Hi,

I'm attempting to solve a 6x6 system of equations, but am returning an uninvertible matrix.
The system has an arbitrary center point and two load application points defined by [x1,y1,z1] and [x2,y2,z2].
The forces and moments at the center point are [Fx Fy Fz] and [Mx My Mz] respectively.
Using two forces at the load application points,([Fx1 Fy1 Fz1] and [Fx2 Fy2 Fz2]) to zero out the force/moment balance, there are 6 unknowns and 6 variables, yielding:

A=
[-y1 x1 0 -y2 x2 0
z1 0 -x1 z2 0 -x2
0 -z1 y1 0 -z2 y2
1 0 0 1 0 0
0 1 0 0 1 0
0 0 1 0 0 1]
x=
[Fx1;Fy1;Fz1;Fx2;Fy2;Fz2]
B=
[Mz;My;Mx;Fx;Fy;Fz]

Using Ax=b, this matrix cannot be inverted and I am therefore confused.

Any thoughts are appreciated.

Thanks

 

[rb1957]

in hindsight (20/20 !) the problem is difficult to solve. two points make a line (duh !), in this case parallel with the y axis. that means the only My you'll get is Fz*x ('cause Fx = 0). if the ponts were parallel to the x- or z- axis you'd have a better chance to balance the applied loads (eg parallel to z- axis, Mz = Fy*x ... x = 8000/1000 = 8".

adding a third point will fix this, but also make the problem redundant, but that is easy enough to solve (make intelligent decisions about how the load will share, solve the over-defined set of equations with matrix math)

 

[daniel7campbell]

I have not disappeared from this thread, I was away for the weekend.
zekeman: The forces/moments at 0,0 are defined above. They have nothing to do with the forces at points 1 and 2, those forces simply have to add up to equate the forces already present at 0,0. This problem yields a 6dof system with 6 vars, yet a singular matrix in the general case (my original question).

As rb1957 states, the two y-coordinates of the points do create a line parallel to the y-axis, and, using a third point, I have created the over defined system and solved. The problem there is that the two-point system cannot be solved even if you change the two points to any arbitrary location.

Using a third point is not ideal, but doable at least.

 

[rb1957]

i don't know that you can say that two points won't work ... if you had two points at x = 8" (approx) you could satisfy Fy and Mz together, then Fx and a Fx couple for Mz and Fz and an Fz couple for Mz and you should have two loads that'll apply the forces and moments you want.

but we started this discussion as trying to find a pair of forces that'll apply some generalised load (that is the sum of lots of contributing forces). i'm thinking how you could use this information, and the best i can come up with is you've got some structure attached to the rest of the world somehow and instead of applying lots of small forces to set how the rest of the world reacts you want to apply just two ? (and then, of course, you're not looking at the structure, 'cause the applied loads aren't right).

 

[zekeman]

Daniel,
You answered:
"zekeman: The forces/moments at 0,0 are defined above. They have nothing to do with the forces at points 1 and 2, those forces simply have to add up to equate the forces already present at 0,0."

Nothing to do with..??

Is the statement of your problem
Given a vector force on a body M+F at 0,0,0
Find pure force vectors at 2 points (Fx1,Fy1, Fz1) and (Fx2, Fy2, Fz2) that puts the body in static equilibrium?

If not, then I have been fast asleep.

If so, then we have concluded ad nauseum that you cant solve it.
for the reasons stated, so why can't we let this thread die peacefully.

I'm still not convinced that you even need it for your project.

I think, however, as an academic exercise it was useful and while I won't waste any more time on the thesis, I will subsequently submit a non uniqueness "proof".

 

[zekeman]

Start with a new problem.
Same as the OP problem, but Mx=My=Mz=0
i.e.
Is it possible to balance the forces when only a pure force vector is at 0,0,0??

Not unless the force vector at 0,0,0 lies in the plane formed by the 3 points, (0,0,0);(x1,y1,z1);(x2,y2,z2).

Now add the moment vector at 0,0,0. The problem doesn't get easier.

Will continue later.

 

[zekeman]

Continuing....
Don't need the previous exercise..

Rotate the whole system about 0,0,0 making the the line between the 2 points parallel to the new z axis.

Now it is clear that a correcting couple cannot be accomplished for My as RB1957 showed
This proves that, in general, a 2 point solution is not possible.

 

[daniel7campbell]

Apologies for the confusing reply.
And yes, I now understand the problem better. I will add a third point and solve for my application.
Thanks to both of you