ALGEBRA PROBLEM - VECTORS

[Lyster]

This may not be the correct place to post this but please feel free to direct me to a suitable resource.
I've been given dimensions to some bolt locations in one axis system (lets call it axes 1) and I have only some of those bolt locations in a different axis system (lets call that axes 2).
So basically I want to calculate the transformation matrix between axes 1 & 2 but I don't know the origin of axes 1.
What I thought is that since I have the bolt locations in axes 1 which are basically vectors in axes 1 and I have a similar situation for only some of them in axes 2 then there must be a way to figure out the origin of axes 1 in terms of axes 2 and the transformation matrix between the two from the info I have, I'm just not sure how. This will allow me to get the rest of the bolt locations in axes 2 which I need. Note that the (bolt) vectors may not be orthogonal.

So if anyone can help me with this or knows a place I might find the answer I'd much appreciate it.

Thanks,

Wade.

 

[rb1957]

if you've got locations in a co-ord system, then you know where the origin is !

what you need is points in both systems ... without them you cannot co-ordinate the two systems.

 

[rb1957]

reading your post again ! ... don't think you can backtrack, 'cause you don't know the angular relationship between the two x (or y or z) axes ... you can go from the origin of sytem 1 to the location defined in system 1, and continue to the location defined in system 2 (presumably you can see the two locations and using sytem 1 axes create a vector between them) but then the origin of system 2 (at the other end of the co-ords) can't be placed in system 1 co-cords). maybe if you had three pairs you could ... maybe two would be enough, but i think three is the number needed ...

 

[GregLocock]

It depends a bit on colinearity and so on. You ultimately need to find 6 unknowns for LCS1, that is, location of the origin, and rotations of the axes, compared with LCS2.

I think you'll need at least three and probably 4 pairs for the general case where there is no particular alignment of the axes. No three points may be colinear.

The topic is generally called transformation of cartesian coordinates.

It is much easier if you know for instance that the axes are parallel





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[spshone]

Not sure if understood fully, and making a few assumptions... Does it matter where the origin is? Can you not define a local system in the "axis 1" structure? If the bolts are in the same place on the original and new structure (but have been translated and rotated), then transformation in local "bolt" coordinates is always the same for each bolt?

Now, I assume the bolt is oriented normal to the surface it bolts "into" (maybe I'm wrong) but either that or at a fixed angle to some reference plane. So define the local z-axis on the shaft line of your bolt so the vector is in the positive k-direction (for example). Now the XY plane is your surface. The orientation of X and Y only matter if rotation about the z axis makes a difference (e.g. a cylindrical rod is such that no matter how you rotate the XY plane about Z, where z is the on the shaft line, it changes nothing NB... unless you have an force at certain point! then it matters!). If it does matter, define the x or y axis to be perpendicular or parallel to some structural feature common to both cases, thus creating a XZ or YZ plane to rotate.

You know where the surface is on your new target structure ( I assume) so you know where your target XY surface is. You also know the coordinates of where your bolt goes on the structure. So what is missing is the rotations of the XY plane about the y-axis and about the x axis (and z if you need to , see above comments). You should be able to get the rotation angles from the fact that you know the orientation of your XY plane, given by the bolt vector being normal (or at a fixed angle) to your target surface, i.e. your z direction.

Maybe I'm wrong, a long day with FE has been had, but coordinate systems to me are relative - using structural details like the orientation of a bolt hole and surface plane is no different to some reference point or origin. However, if you are putting something like this into a built-up model, whatever system you create above must be transformed into whatever global coordinates you are using. Again, the global system is arbitrarily chosen. You can look at it as if to say, if the local coordinate system is equivalent to the global system, the transformation is unity.

I hope maybe this helps,

Best regards,
S.