Rotation of rigid body - wobbling

[Nick R]

When a rigid body experiences a rotation about an axis other than one of its "principal axis of rotation", it experiences a "wobble"

I have been trying to understand why this is so (intuitively).

Here is what I have come up with - please tell me if I have misconceptions or misunderstandings, or if I'm just full of crap :(

I have come up with this qualitative explanation - the particles that comprise the rigid body are subject to two constraints - they must rotate about $$\omega$$, but at the same time they must stay fixed relative to each other. The latter constraint introduces the wobble when the rotation is not about a "principal axis of rotation".

Is that a correct way of looking at it?

A more mathematical explanation I have come up with is (tell me if any of these concepts are flawed... hopefully this makes some sense without a drawing) -

Rigid body experiences a rotation about $$\omega$$, where $$\omega$$ is not parallel to a principal axis of rotation.

Viewing the system from a coordinate system that is fixed with respect to the "inertial tensor" I of the rigid body,
$$\omega$$ actually appears to rotate about a principle axis

(imagine instead rotating the coordinate system about $$\omega$$ - $$\omega$$ effectivly rotates about a principal axis(s)).

Therefore, because by definition

L = I $$\bullet$$ $$\omega$$

coordinate system is fixed with respect to inertial mass tensor I
so dI/dt = 0

but $$\omega$$ is not fixed in the coordinate system so
d$$\omega$$/dt $$\neq$$ 0

therefore, dL/dt = torque $$\neq$$ 0

And this torque is responsible for the wobble.

 

[eljose79]

First of all, congratulations on trying to understand this concept intuitively! It shows a great curiosity and desire to truly understand the physics behind it. Your qualitative explanation is on the right track, but there are a few misconceptions that I would like to clarify.

Firstly, when a rigid body experiences a rotation about an axis other than a principal axis, it doesn't necessarily experience a "wobble." The term "wobble" usually refers to a small, unsteady motion of an object. In this case, the rigid body is still rotating smoothly, but the direction of the rotation is not aligned with any of its principal axes.

To understand why this happens, let's start by defining what we mean by "principal axis of rotation." A principal axis of rotation is an axis about which an object can rotate without experiencing any torque. In other words, if an object is rotating about a principal axis, it will continue to rotate at a constant angular velocity without any external forces acting on it.

Now, let's consider a rigid body rotating about an axis that is not a principal axis. In this case, the particles that make up the rigid body are still moving in circular paths, but the direction of these paths is not aligned with the rotation axis. This means that there is a component of the velocity of each particle that is not in the direction of the rotation axis, and this component is changing over time. This change in velocity creates a net torque on the object, causing it to rotate about a different axis. This is what leads to the "wobble" that you mentioned.

Your mathematical explanation is on the right track, but there are a few things to clarify. Firstly, the term "inertial mass tensor" is not commonly used in this context. The term "moment of inertia tensor" is more commonly used to refer to the tensor that describes how mass is distributed in an object and how it affects its rotational motion. Secondly, the equation L = I * ω is only valid for rotations about a principal axis. For rotations about any other axis, this equation becomes L = I * ω + ω x I * ω, where ω x I * ω is the "cross product" of ω and I * ω. This term represents the additional torque that is created when the rotation axis is not aligned with a principal axis.

Overall, your understanding of this concept is on the right track, but I would recommend doing some more

 

[astrokreng]

Your qualitative explanation is correct. When a rigid body rotates about an axis that is not one of its principal axes, the particles that make up the body are no longer rotating about their own individual axes of rotation. Instead, they are rotating about a common axis that is not aligned with any of the principal axes. This creates a wobbling motion because the particles are trying to maintain their relative positions while also rotating about this new axis.

Your mathematical explanation is also correct. When the coordinate system is fixed with respect to the inertial tensor, the angular momentum is a constant since the moment of inertia is not changing. However, because the rotation is not about a principal axis, the angular velocity is not fixed in this coordinate system. This results in a non-zero torque and causes the wobbling motion.

In summary, both explanations provide valid insights into why a wobbling motion occurs when a rigid body experiences a rotation about an axis other than one of its principal axes. It is important to note that this wobbling motion can have practical implications in various fields such as aerospace engineering, where it must be accounted for in the design and control of rotating bodies.