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

In summary, the problem involves a telescope mirror housing supported by a space truss with 6 bars. The total mass is 3000 kg and the center of mass is at point G. The distance between the z axis and points A, B, and C is 1 m, while the distance between the z axis and points D, E, and F is 2.5 m. The problem also provides coordinates for the points and a force vector P. The equations used for solving the problem are also listed. The solution involves considering the ball and socket supports at points A, B, and C and using the equations to find the reactions at these points. However, since each ball and socket is only connected to two links, the
  • #1
LucasSG
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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]
76f3a2e2aca3612136545f2b2f4eeb84.png


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)
26f43ff721f9354d1e8a04b91c07d196.png


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...
 
Last edited:
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  • #2
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...)
 

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

1. What is a space truss statics problem?

A space truss statics problem is a type of engineering problem that involves analyzing the forces and stresses acting on a 3-dimensional truss structure. This type of problem is commonly encountered in the design and construction of large structures such as bridges, towers, and spacecraft.

2. What are the main principles of space truss statics?

The main principles of space truss statics are the laws of equilibrium, which state that the sum of all forces acting on a body must equal zero and the sum of all moments acting on a body must also equal zero. These principles are used to analyze the stability and strength of a space truss structure.

3. How do you solve a space truss statics problem?

To solve a space truss statics problem, you must first identify the external forces acting on the structure, such as gravity, wind, or applied loads. Then, using the principles of equilibrium, you can calculate the internal forces and stresses within the truss members. This can be done using methods such as the method of joints or the method of sections.

4. What are the common assumptions made in space truss statics problems?

Some common assumptions made in space truss statics problems include assuming that all members are perfectly rigid, that all joints are pinned and can only support axial forces, and that the structure is in a state of static equilibrium. These assumptions may not always reflect real-world conditions but are necessary simplifications for solving the problem.

5. What are some real-world applications of space truss statics?

Space truss statics problems have many applications in engineering, including the design of bridges, towers, and cranes. They are also used in the aerospace industry for designing and analyzing spacecraft structures. In addition, these principles can be applied to other types of structures, such as buildings and offshore platforms.

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