How to Solve a Space Truss Statics Problem with Ball and Socket Supports?

[LucasSG]

Homework Statement

A telescope mirror housing is supported by 6 bars in form of a space truss with geometry defined by the figure below. The total mass is 3000[kg] with center of mass being point G. The distance between:
z axis and points A, B and C is 1 [m]
z axis and points D, E and F is 2.5[m]

Using the exact same coordinate system the problem gives, the points coordinates are:
G=(0,0,1)
A=(-1/2, √3/2, 0)
B=(-1/2, -√3/2, 0)
C=(1,0,0)
D=(-5/2, 0, -4)
E=(5/4, (-5√3)/4, -4)
F=(5/4, (5√3)/4, -4)

The force vector P is equal to:
P=(30*cos(20)cos(60), -30cos(20)sin(60), -30sin(20))[kN] (using g=10m/s²)(Ball and socket supports)

Homework Equations

∑Fx=0 (1)
∑Fy=0 (2)
∑Fz=0 (3)
∑Mx=0 (4)
∑My=0 (5)
∑Mz=0 (6)

The Attempt at a Solution

First thing i thought was to consider A, B and C "ball and socket supports" detaching them from the bars. So i would have reactions on x, y and z directions in each of these three points. Then i would consider the entire part between G and the points A, B and C to be a rigid body. The forces that act on each point would be (using the same coordinate system the problem gives me):
Fa=(Fax, Fay, Faz) [kN]
Fb=(Fbx, Fby, Fbz) [kN]
Fc=(Fcx, Fcy, Fcz) [kN]
P=(30*cos(20)cos(60), -30cos(20)sin(60), -30sin(20))[kN]
Then i used ∑M=0 on point A, so i got:
∑Ma= AB x Fb +Ac x Fc + AG x P = 0
= (-√3Fbz, 0, √3Fbx) + (-0.86Fcz, -1.5Fcz, 0.86Fcx + 1.5Fcy) + (15√3*(cos(20)+sin(20)), 15cos(20), 15sin(20)) And that's where I'm stuck. Using equations 1-6 now i got 6 equations and 9 unknowns, i guess now i need to find symmetry between the forces on those 3 points, but i can't seem to find any new information about those forces that can help me get the other 3 equations i need.

I suppose finding the sum of the moments on points B or C won't give me any new information.

Am i trying to solve this problem the right way or there's a better way of finding the reactions on A, B and C to analyze the truss? If so, what can i do next? I'm really stuck right now...

 

[AlephZero]

Each ball and socket is connected to only two links, so you don't have three independent component of each reaction force. The reaction force can only lie in the plane containing the two links.

Probably the best choice of unknowns are the tensions in the 6 links. That way, you have 6 equations in 6 unknowns.

(But solving a problem with the geometry as messy as this by hand seems rather pointless, compared with setting up a simple finite element model...)