Traslate in matlab equations of rotational displacement

Hi everyone. I would like to ask for your help to 'translate' these three equations for seismic motion in Matlab.

These are the equations I want to implement and solve in Matlab to get the values of omega.

I translated it like this and I don't know if this is ok.

equationX = omega_x == 1/2 * (diff(uz, y) - diff(uy, z));
equationY = omega_y == 1/2 * (diff(ux, z) - diff(uz, x));
equationZ = omega_z == 1/2 * (diff(uy, x) - diff(ux, y));

X_solution = solve(equationX, ux);
Y_solution = solve(equationY, uy);
Z_solution = solve(equationZ, uz);

CLARIFY WHAT THE VARIABLES ARE

When we record an earthquake with a seismic station,

we get the values of the displacements (actually oscillations) in the 3 spatial directions: ux uy and uz.

Now I want to calculate the rotational movement around the axes: ωx, ωy and ωz.

Answer 1

[William Rose]

@Ilary,

The equations you have given are for a rigid body. If you are comfortable regarding the earth as a rigid body within a zone of interest*, then you need to measure the x,y,z displacements at three non-collinear points in order to compute (or estimate) the rotation. You can understand this by considering a cube. If I know the x,y,z displacement of one corner of the cube during a movement, I will not be able to determine if the cube rotated, or translated, or did a little of both. But if I know the x,y,z displacements of three corners of the cube, then I will be able to determine how much translation and rotation occurred. If I know the x,y,z displacements of four or more corners, then I can use the "extra" information to estimate whether the cube is in fact moving as a rigid body, or not.

A mathematical way of understanding the requirement for displacements at three points is that if I only have at one point, I will not be able to compute and the other partials. I need at three non-collinear points in order to estimate the different partials.

* In seismology, one often does NOT regard the earth as a rigid body.

[William Rose]

@Ilary,

You said in your initial post

"we get the values of the displacements (actually oscillations) in the 3 spatial directions: ux uy and uz"

Please note that ux, uy, uz, in the equations for , are the linear velocities, not the linear displacements.

[Ilary]

Thank you both!