# Balancing a 3d weight vs lift system

1. Oct 30, 2012

### LordSimpson

Hi I'm trying to balance any 3d system with various fixed weights by selecting the correct lift forces for points of lift an example system would be the following shape ( view from above , think a plane missing the right wing) where X is a point of weight and L is a lift point (also has weight)

[X1][X2][L1][X4][L2]
[X3]​
[L3]​

to be clear
X’s are fixed weights
L’s can have variable lift forces
L3 needs to counteract the rotation from its own weight and the weight of X3
L1 is mostly central along the main body so will can counter X1,2,4 and L2's weight
L2 may not be needed

so something like L3 = 2(cancelling its and X3 weight / moment), L1 = 5 (cancelling the main body’s weight) L2 = 0 else it would flip

This is just an arbitrary shape, I need to be able to solve it for any number of lifts / weights in any shape / positions.

I’m sure I learnt how to solve this sort of problem during my degree however I guess I can’t find the correct set of words to put into the all-knowing google!

Any help would be appreciated!

2. Oct 30, 2012

### Staff: Mentor

You can get 6 conditions for balance:

- the sum of all forces has to be 0 (otherwise the object would accelerate). This gives 3 equations, one for each dimension
- the sum of all torques has to be 0 (otherwise the object would rotate). Again, you get one equation per dimension. It does not matter which point you use as center of your coordinate system, as the conditions above make sure that you get the same result everywhere.

Any combination of lift forces which satisfies those conditions is fine. If all forces are vertical, you can reduce the system to 3 equations (one for the horizontal position and two for rotations).

3. Nov 3, 2012

### LordSimpson

Thanks :D, Realised the goggle phrase i was missing was 'Linear Equations'

I've now got most of the equations worked out but currently cant get it to work quite right hopefully someone will see where I've messed up. Currently I have a problem where when I'm attempting to move the object in the X axis it will rotate around the Y axis as-well.

I'm Achieving the X axis movement by rotating the lift objects about the Z axis causing the force to change form completely down (0.0,-1,0.0) to slightly to the left or right (0.7,-0.7,0.0).

For the Translational equations I have the following
Where F is a vector for the force direction.

F1.y + F2.y ... +Fn.y = Weight
F1.x +..... = Xaxis movement (left right)
F1.z +..... = Zaxis movement (forward back)

then the rotational equations

Where T = the torque calculated by r cross F, where r is a vector from the centre of mass to the location of F (Centre Of Mass - Location of F)
giving me

T1.x + T2.x ...+ Tn.x = Current Torque about X axis (pitch)
T1.y + T2.y ...+ Tn.y = Current Torque about Y axis (yaw)
T1.z + T2.z ...+ Tn.z = Current Torque about Z axis (roll)

I'm calculating the Current Torque by finding the amount of torque necessary to move the objects Up direction vector to straight up (0,1,0).

Any ideas where I may have gone wrong?

I have a feeling its the torque calculation as the this torque calculation doesn't include any Y axis rotation, I assume i have to calculate the torque required using the froward vector (and right ? ) however im unsure how i should combine them to get the total torque required for each rotational axis.

Last edited: Nov 3, 2012
4. Nov 3, 2012

### Staff: Mentor

I don't see your problem at the moment.

If you cannot avoid this... well, then you'll need more points of support to get a pure movement.

Torque around the Y axis will give a rotation around the Y axis.
Which "forward vector"?

5. Nov 4, 2012

### LordSimpson

Ok the extra supports helped, however I'm definitely missing something as the simulation I've built is less than completely stable.

Ok hopefully this will make the torque bit a little clearer. Currently I'm calculating the torque required to rotate the object from one vector direction to another. This means I cant get a full picture of the torque from one calculation, For example, If I calculate the torque required to rotate from the objects up direction to straight up (to make the object level) I will only get information about the X and Z axis as there is no way of telling what the Y axis rotation is. A vector cant tell you the rotation about its self, but it can tell you the rotation about its two orthogonal counterparts (forward and right).

So I'm unsure how to get the complete picture of the torque.