Transformation matrix using Rotation Matrix producing identity

Good Afternoon All,

I am trying to find a transformation matrix from a local coordinate system to a global system using Euler angles (goal is to transform the local stiffness to global stiffness). The matrix is simply the rotational matrix, i.e. euler to DCM (rotation in xyz). According to all the published articles in regards to the mount transformation its DCM*K*DCM'. Easy enough to code but my logic may be faulty in this aspect....isn't DCM*DCM' the identity matrix? I am sure I am missing something here because the same thing has been published in multiple articles (<http://www.mecheng.osu.edu/adl/files/adl/conference_papers/C07-8.pdf)>. Everything works wonderful when I have zero degrees for the Euler angles but when I start applying the rotational (transformation) matrix the DCM cancel one another out. Anyone have any suggestions on how to resolve this issue? Should I just use DCM*K?

Here is my code:

K=[100 100 400]; Angles=[30,0,0];
%K_mi Mount Stiffness Matrix Caclulation
%Note: K_mi is the static input stiffness by user

%Obtaining Initial Individual Stiffness Values (Initial Input)
Kx=K(1);Ky=K(2); Kz=K(3);

%Setting up initial matrix
K_mi=zeros(3,3);
K_mi(1,1)=Kz;
K_mi(2,2)=Ky;
K_mi(3,3)=Kx;

%R_gmi: Rotational Transformation Matrix of K_mi

%Obtaining the Initial Orientation Value of Mounts
Theta=Angles(1); Rho=Angles(2); Phi=Angles(3);
Theta=Theta*pi/180; Rho=Rho*pi/180; Phi=Phi*pi/180;

%Rotational Matrix using Euler Angles

% Mathetmatical Representation XYZ Rotation
%[ cy*cz, sz*cx+sy*sx*cz, sz*sx-sy*cx*cz]
%[-cy*sz, cz*cx-sy*sx*sz, cz*sx+sy*cx*sz]
%[ sy,         -cy*sx,          cy*cx]

Rot_gmi=zeros(3,3);
Rot_gmi(1,1)=cos(Rho)*cos(Phi);
Rot_gmi(1,2)=cos(Theta)*sin(Phi)+sin(Theta)*sin(Rho)*cos(Phi);
Rot_gmi(1,3)=sin(Theta)*sin(Phi)-cos(Theta)*sin(Rho)*cos(Phi);
Rot_gmi(2,1)=-cos(Rho)*sin(Phi);
Rot_gmi(2,2)=cos(Theta)*cos(Phi)-sin(Theta)*sin(Rho)*sin(Phi);
Rot_gmi(2,3)=sin(Theta)*cos(Phi)+cos(Theta)*sin(Rho)*sin(Phi);
Rot_gmi(3,1)=sin(Rho);
Rot_gmi(3,2)=-sin(Theta)*cos(Rho);
Rot_gmi(3,3)=cos(Theta)*cos(Rho);

Rot_gmi;

%K_gmi: Local Stiffness Matrix transformed to Global coordinates
K_gmi=Rot_gmi*K_mi*Rot_gmi';

Answer 1

[Matt J]

Everything works wonderful when I have zero degrees for the Euler angles but when I start applying the rotational (transformation) matrix the DCM cancel one another out.

Only because of the fortuitous combination of data that you've chosen. Try using

K=[200 300 400];

and you will see that you get K_gmi~=K_mi

>> K_gmi, K_mi

K_gmi =

    400.0000         0         0
           0  275.0000  -43.3013
           0  -43.3013  225.0000

K_mi =

     400     0     0
       0   300     0
       0     0   200

[Melissa]

Hm you are right. But if two spring values are the same K_gmi=K_mi. And I feel like that is wrong. Thanks for answering by the way.

[Matt J]

I should mention that I have no background in the physics of your problem. Nor, I would assume, do most people on this board. However, your equations are behaving as the math predicts they should.

But if two spring values are the same K_gmi=K_mi

No, not always. Try this combination of input data

    K=[100 100 400]; Angles=[30,20,10];

and you will get

    K_mi =

       400     0     0
         0   100     0
         0     0   100

    K_gmi =

      356.9188  -45.3017   94.9533
      -45.3017  107.9879  -16.7428
       94.9533  -16.7428  135.0933

It is true that if all three of the spring values are the same, then you will get K_gmi=K_mi independently of the rotation angles. I don't know if that's what you expect physically, but it is the expected mathematical behavior of the equations you've presented. In that case, K_mi is just a multiple of the identity matrix c*I and so

 K_gmi = DCM*c*I*DCM.'
       = c*(DCM*DCM.')
       = c*I
       = K_mi

If your rotation is purely about the z-axis, then something similar happens when Kx=Ky, which is what you were observing before.

[Iain]

I think I figured it out:

Reactive force = K * Displacement

If we have a matrix "K" defined in axis "A", and the displacement measured in axis "B", with a set of euler angles or a DCM (DCM, the way I'm using it transfers B to A):

 K * DCM * Displacement = reactive force in axis A (not wanted)
 DCM' * K * DCM * Displacement = reactive force in axis B (wanted)

 Thats why its got both.