Order of rotations due to torque in 3DOF in simulations

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• zonexo
In summary, the author is asking how to rotate a symmetrical rigid body so that it has the same orientation at each time step, given that the body experiences torque in the global x, y, and z axes. The author suggests using a rotation matrix, but notes that this may not be the only orientation the body will have after experiencing torque. If the body also experiences forces at its center of gravity, this may complicate the rotation matrix further.
zonexo
TL;DR Summary
What is the orientation of a body rotating in 3 axes after t=1sec, using very small rotations to each time step
Hi,

I am running a computational fluid dynamics (CFD) simulation. Supposed I have a symmetrical rigid body in space experiencing torque in the global x,y,z axes. It is stationary at t = 0. I also constrain it to only allow rotations in 3DOFs, and no translation.

It will rotate and I need to know its orientation after t = 1s.

Each time step is 1e-5s. And I can get the torque at each time step. So 1st I need to obtain the angular accelerations (alpha) in the global x,y,z axes. I understand that torque folllows: torque_x/y/z = I_xx/yy/zz*alpha_x/y/z

So I can get alpha_x/y/z. Then I can then get the auglar vel and the angle rotated.

But how should I rotate the body to its new orientation at each time step? If I'm using rotational matrix R_x, R_y, R_z (or even quaterions), should the total rotation matrix by R_x*R_y*R_z or R_z*R_y*R_x? Because I thought matrices are not commutative, so would the body get a different orientation if I use different combinations? But there should only be a single correction orientation after the body experienced torque combination in the 3 axes.

At each time step (1e-5s), the angles rotated are small, so does it mean the rotation matrics can be commutative and so it doesn't matter?

Supposed at each time step, the angles at each axis are rotated 0.0001deg, what will be the final orientation of the body at t = 1s? Will it be the same as either using R_x*R_y*R_z or R_z*R_y*R_x and rotating by 10deg (since 0.0001*10000 time steps = 10deg)?

Lastly, if I also include forces acting at the body's CG, would it make any difference to the above rotation formulation (besides the body translating to a new position)?

Hope someone can clarify my doubts. Thanks!

zonexo said:
1st I need to obtain the angular accelerations (alpha) in the global x,y,z axes. I understand that torque folllows: torque_x/y/z = I_xx/yy/zz*alpha_x/y/z
Careful here: I_xx/yy/zz is usually given in the local system of the body, where it is constant. Make sure you are not mixing global and local systems in this equation.

zonexo said:
Then I can then get the auglar vel and the angle rotated.

But how should I rotate the body to its new orientation at each time step?
The angular velocity is a vector, which is along the instantaneous rotation axis. So you need something to convert from axis-angle to matrix or to quaternion:
https://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToMatrix/index.htm
https://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htm

berkeman

What is the significance of the order of rotations in 3DOF simulations?

The order of rotations is crucial in 3DOF (three degrees of freedom) simulations because it determines the final orientation of an object. Different orders can lead to different results due to the non-commutative nature of rotational transformations. This means that rotating an object around the X-axis followed by the Y-axis will yield a different orientation than rotating it around the Y-axis followed by the X-axis.

How do Euler angles affect the order of rotations?

Euler angles represent rotations about the principal axes (usually X, Y, and Z) and are applied in a specific sequence. The order of these rotations (e.g., XYZ, ZYX) affects the final orientation of the object. Euler angles are commonly used in simulations, but they can lead to issues like gimbal lock, where two rotational axes align and cause a loss of one degree of freedom.

What is gimbal lock, and how does it relate to the order of rotations?

Gimbal lock occurs when two of the three rotational axes in a 3DOF system become aligned, causing a loss of one degree of freedom and making it impossible to rotate around one axis. This problem is directly related to the order of rotations because certain sequences of rotations can lead to this alignment. To avoid gimbal lock, alternative representations like quaternions are often used.

Why are quaternions preferred over Euler angles in some simulations?

Quaternions are preferred over Euler angles in some simulations because they do not suffer from gimbal lock and provide a more stable and efficient way to represent rotations. Quaternions allow for smooth interpolation between rotations (slerp) and are computationally more efficient for concatenating multiple rotations.

How can I determine the correct order of rotations for my simulation?

Determining the correct order of rotations depends on the specific requirements and constraints of your simulation. You should consider the initial and final orientations, the potential for gimbal lock, and the ease of implementation. Testing different orders and analyzing their effects on the simulation can help you choose the most appropriate sequence. Additionally, consulting the documentation and best practices for the simulation software you are using can provide valuable guidance.

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