Cartesian torque to Spherical Coordinates

[Olly613]

I'm writing a function for Matlab and I'm trying to figure out how to apply a torque matrix in cartesian coordinates to an object in spherical coordinates.

The short story is this:

For interest's sake, a friend and I have written a function with creates a tree which random branch orientations. These branches, though later converted to Cartesian for plotting, originate as spherical vectors. What we are attempting to do is have a "wind" push the branches and cause them to deflect (but not stretch). To do so we need to define a delta_theta and delta_phi for our angles (we have it programmed such that phi is relative to the z-axis and theta to the x-axis, thought I should mention that because I know some conventions suggest the opposite). We figure to find the displacement (simplistically) our model ought to find the change in either angle based on the following static case:

SUM(Moments)= 0 =Torque-resisting moment=T-k*dAngle
therefore: dAngle=Torque/k

Where we take k as an equivilent spring constant for a cantilever.

Granted this model isn't perfect, but it ought to produce a reasonable estimate for the deflection.

So, we have created a force vector (x y z) to apply to the branch, but are uncertain as to how we can produce the the spherical torques about the two angles given these values.

Any ideas?

Thank a tonne in advance (metric of course).

 

[Stephen Tashi]

Just a thought, If phi is the angle of the branch projected on tthe xy plane you could use the component of the (x,y,z) force projected on the xy plane and compute the torque it exerts acting on the end of a line segment that is the projection of the branch on the xy plane. The branch is in some vertical plane perpendicular to the xy plane. You could use projections on that plane to compute the torque about theta in a similar way.

 

[Olly613]

Do you reckon it would work that 'simply'? I'm not doubting it haha, I've just been trying to think of it physically, but I can't seem to form a solid visual picture the translation very easily. I'm probably over complicating the issue, but given our torque matrix being representative of the torques about each of the Cartesian axes, would it just be a matter of applying the appropriate projections to get the torques about each spherical axis? If so, great. It got busy all of a sudden going into exams, but thanks for the reply and I'll give it a shot!

 

[Stephen Tashi]

I think (for visual purposes) that it would work that simply for small deflections. The real situation is that as force is applied the angle changes, so the component of the force acting on each "lever" changes. I think you could solve for the equilibrium angle even in that case.

 

[blue_raver22]

Dear writer,

Thank you for sharing your project and question. It sounds like you are working on an interesting and complex problem. To apply a torque matrix in Cartesian coordinates to an object in spherical coordinates, you will need to first convert the torque matrix from Cartesian to spherical coordinates. This can be done using transformation matrices.

Once you have the torque matrix in spherical coordinates, you can use the equations you mentioned to calculate the change in angles (dTheta and dPhi) based on the torque applied. These angles can then be used to update the orientation of your branches.

In terms of finding the displacement, it is important to note that the torque applied will not only cause a change in angles but also a change in position. Therefore, you may need to consider the effect of both the torque and the force vector in your calculations.

I would also suggest considering the mass and moment of inertia of your branches in your model, as these can also affect the deflection.

Overall, it seems like you have a good approach in using statics principles to estimate the deflection of your branches. However, it is important to keep in mind that this is a simplified model and may not account for all the complexities of the real-world scenario.

I hope this information helps and wish you the best of luck with your project.

Sincerely,