Component of angular velocity in rigid body motion -- which is right?

[mcheung4]

Let w = (w1, w2, w3) wrt to the body frame of a rigid body, where the body frame is right-handed orthonormal. I have gathered 2 definitions of w from different sources and I am confused at how they connect to one another. One is that the RB rotates about w through its CoM at rate abs(w), the other is that each component of w represents the rate at which the RB rotates about that particular basis axis. Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis? Thanks!

 

[Andrew Mason]

Let w = (w1, w2, w3) wrt to the body frame of a rigid body, where the body frame is right-handed orthonormal. I have gathered 2 definitions of w from different sources and I am confused at how they connect to one another. One is that the RB rotates about w through its CoM at rate abs(w), the other is that each component of w represents the rate at which the RB rotates about that particular basis axis. Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis? Thanks!

Yes, if there is a simple finite rotation. That is Euler's rotation theorem. See:http://vmm.math.uci.edu/PalaisPapers/EulerFPT.pdf

Where there is continual rotation about more than one axis, one axis of rotation will rotate about another axis, which means the axis will precess. Warning: that analysis of rotational motion can be rather complicated and difficult.

AM

 

[Stephen Tashi]

Does this mean we can somehow add the 3 rotations (which are about different axes) and get an equivalent rotation about some other (single) axis?

You can add them as vectors. If you want to "add" them by doing 3 separate rotations, one about each axis, this is a different matter. A velocity is a rate of rotation, not a finite rotation. You'd have to define what it means to "apply each angular velocity component separately about its corresponding axis". I think you can make this definition meaningful if you think in terms of integrating angular velocity to get net displacement. As I recall, it has to do with taking the exponential of matrices.

Finite rotations don't commute. So if you say "perform each rotation separately about its correspondin axis" this doesn't define a unique answer. The order in which you perform them can give different results - i.e. a specific point on the rotating object can end up in different positions depending on the order of rotations. So if you treated the velocity numbers as finite rotations and performed the component rotations, you would not, in general, end up with a rotation equal to their vector sum.

 

[Khashishi]

At any instant in time, you can view the combined rotation as a rotation about a single axis, but unless the object is spherically symmetric (or has an inertial tensor with just a single eigenvalue), the rotation rate and axis will change over time.

 

[PJC01]

I would like to clarify the two definitions of w that you have mentioned. The first definition, where the rigid body (RB) rotates about w through its center of mass (CoM) at a rate of abs(w), refers to the magnitude of the angular velocity vector. This means that the RB is rotating at a constant rate around an axis represented by the vector w, and the magnitude of this rotation is given by abs(w).

The second definition, where each component of w represents the rate at which the RB rotates about a particular basis axis, refers to the direction of the angular velocity vector. In a right-handed orthonormal body frame, the basis axes are represented by the unit vectors i, j, and k along the x, y, and z directions respectively. So, if w1 represents the rotation rate around the x-axis, w2 around the y-axis, and w3 around the z-axis, then the RB is rotating in all three dimensions simultaneously.

To answer your question, yes, we can add the three rotations (w1, w2, and w3) to get an equivalent rotation about a single axis. This is known as the Euler's rotation theorem, which states that any rotation in three-dimensional space can be represented by a single rotation around a fixed axis.

In summary, the component of angular velocity in rigid body motion is a vector quantity that has both magnitude and direction. The magnitude represents the rate of rotation, while the direction represents the axis around which the rotation is occurring. By adding the three components, we can determine the equivalent rotation around a single axis. I hope this clarifies your confusion.