## finding local axis of an element

Linear spaces and subspaces, linear transformations, bases, etc.

### finding local axis of an element

Hey All,

This is my first post, and it would be very helpful if someone can help me with this.

Basically, what i'm doing is, lets say i have vector - [1,2,3]

I want to find the 'local axis' of the vector.

In effect, i know in the global axis - the base vectors are

x= [1,0,0]
y= [0,1,0]
z= [0,0,1]

Now i want to know what the 'base vectors' for the vector above [1,2,3] such that the x' axis is [1,2,3]. Then what should y' and z' vectors are?

x'=[1,2,3]
y'=?
z'=?

I hope i've explained my question well, if not, please let me know.

Thanks for any help

-Pete
sublim21

Posts: 2
Joined: Mon Dec 02, 2013 3:50 pm

### Re: finding local axis of an element

Are you maybe talking about basis vectors, where {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is the "standard" basis for R^3, and you're wanting to find a linear transform T to another basis, where T(1, 0, 0) = (1, 2, 3)?
anonmeans

Posts: 44
Joined: Sat Jan 24, 2009 7:18 pm

### Re: finding local axis of an element

exactly!

I apologize about not being clearer in my description, but what i'm trying to do is

find the transformation matrix in 3 dimensions such that the 'x axis' goes from [1,0,0] to [a1,a2,a3]

The reason i'm trying to do this is, I'm working on the direct stiffness method for structural analysis. This method works as follows:

F = K u

Where F = vector of forces acting upon the element
K = The 'stiffness matrix' of the element
u = vector representing 'Degrees of freedom'

There is a standard way of calculating K but, the K matrix is designed for an element whose 'x axis' is oriented down the element.

To transfer an elements stiffness matrix K from the local axis to a global axis, do the following:

F = ( T^-1 K T ) u

where T is a transformation matrix, that transfers an elements 'local axis' to a global axis.

In the book i've been reading, they suggest building the transformation matrix by finding the 'direction cosine matrix', which requires knowing the direction of the local axis.

So, long story short, I have one 'global axis' that is defined by the standard basis or
Global Axis Basis = [ 1 0 0
0 1 0
0 0 1 ]

I know that the element is oriented by the following vector
[v1, v2, v3]

And my question is, how do i find the transformation matrix of the element that takes the elements 'local axis' to a global axis? Knowing that the x axis in the local axis is oriented [v1,v2,v3] and it transformed from [1,0,0]

I hope that explains my question better.

Thanks,

-Pete
sublim21

Posts: 2
Joined: Mon Dec 02, 2013 3:50 pm