Rigid body kinematic motion and rotation

[fog37]

Hello everyone,
A rigid body is a system whose points, pairwise, always keep a constant mutual distance. Let's say the body is in a certain configuration $C_0$ at time $t_0$ (which means that each point has a specific velocity and position relative to a fixed lab reference frame) and the body is later found in a different configuration $C_1$ at time $t_1$. Motion is essentially a temporal sequence of configurations.

I read that it is geometrically possible for the body to pass from one configuration to the next one with a translation of a point + rotation about that point and the point itself is arbitrary (the point can be on the rigid body or even outside the body as long as it is part of the body reference frame). That means there are infinite combinations (translation+rotation) that can be carries out to get to the same configurations and each combination is different and depends on the different and arbitrary point. However, I think that when the rigid body is moving, it is passing between different configurations in a unique way, correct? How does it help us to know that there are infinite possible, viable transformations (rotation+translation or vice versa) about arbitrary points if that is not how the rigid body realistically behave when it moves? Depending on the body and initial conditions, a body tumbling in the air must pass between configurations in a very specific way.

Can anyone clarify these concepts? I know a rigid body motion is known once the location and velocity of an arbitrary point $Q$ and the angular velocity $\omega$ are known: $$v_P(t) = v_Q(t) +\omega \times (P-Q)$$
The rotational axis has the direction of the angular velocity vector $\omega$ and passes from $Q$. However, any point other than $D\neq Q$ could be chosen and $\omega$ would not change but the axis of rotation would be passing from $D$ in that case:
$$v_P(t) = v_D(t) +\omega \times (P-D)$$

 

[jbriggs444]

How does it help us to know that there are infinite possible, viable transformations (rotation+translation or vice versa) about arbitrary points if that is not how the rigid body realistically behave when it moves?

There is only one motion. There are many ways of describing that same motion.

It is important to know that so that we do not simply reject a description that someone else proposes just because it differs from the description that we had in mind.

 

[A.T.]

... the point itself is arbitrary

So is the orientation of your coordinate system axes. If you change them, then you also get different numbers for your transformation. Does that also bother you?

 

[fog37]

No orientation and type of coordinate system does not bother me. I see

There is only one motion. There are many ways of describing that same motion.

It is important to know that so that we do not simply reject a description that someone else proposes just because it differs from the description that we had in mind.

I agree there is ONE specific motion and so it seems important to determine how the object truly evolves in going from one configuration to the next. Motion is just a sequence of snapshots of configurations...I get that we can go from one state to the next state in many arbitrary ways with rotations about arbitrary points but if I study the motion of a specific object (say a boomerang flying in the air), it would be important to know the exact sequence, instant by instant, of those configurations and not just that there are many ways to go between two configurations to separated by an interval of time...

 

[A.T.]

agree there is ONE specific motion and so it seems important to determine how the object truly evolves in going from one configuration to the next.

It seems your are talking about two different issues here:

- The total transformation from A to B (unique, but description coordinate dependent)
- The transformation interpolation from A to B (infinite possibilities).

It seems you confuse infinite possible descriptions of a unique transformation, with infinite possible transformation interpolations.

 

[fog37]

Sorry for being unclear. Let me try to explain what I mean with the picture below:

The T-shaped body is initially in configuration $C_1$ and later passes to configuration $C_2$ through motion. To achieve this, we can translate point $A$ to its new position in configuration $C_2$ and make a rotation about $A'$ of a angle $\theta$ counterclockwise. Or, equivalently, we can translate point $B'$ to its new position in configuration $C_2$ and rotate the figure about it CCW by an angle also equal to $\theta$. Any other point other than $A$ or $B$ would work too...

But during its motion, the t-shape body must have gone from $C_1$ to $C_2$ through a very specific motion regardless of the fact that there are also infinite different translations+rotations about arbitrary points...

 

[A.T.]

Sorry for being unclear. Let me try to explain what I mean with the picture below:
View attachment 237145

The T-shaped body is initially in configuration $C_1$ and later passes to configuration $C_2$ through motion. To achieve this, we can translate point $A$ to its new position in configuration $C_2$ and make a rotation about $A'$ of a angle $\theta$ counterclockwise. Or, equivalently, we can translate point $B'$ to its new position in configuration $C_2$ and rotate the figure about it CCW by an angle also equal to $\theta$. Any other point other than $A$ or $B$ would work too...

These are different ways to describe the same total transformation C1 -> C2 mathematically. They don't deal with what actually happens in between C1 and C2, they just describe the net difference between the two.

But during its motion, ...

That's a completely different issue, independent of the description chosen above. The physical object could have gotten from C1 to C2 in an infinite number of ways, which are actually distinct.

It still seems you confuse the mathematical notion of rotation/translation, as an operation that takes points and gives transformed points back, with the physical act of rotating/translating through all the states between C1 and C2.

 

[fog37]

Thank you A.T. I guess I do confuse the two topics. My topic of interest right now is kinematics and trying to understand the motion of a rigid body that spins and tumbles in the air. Many believe that the object rotates about an imaginary axis passing through the center of mass but that is not the case (it is just a convenient mathematical choice). So I am trying to understand how this hypothetical object is "rotating" (what axis, etc.)
Is rotation just a relative concept like many others in mechanics? For instance, from the fixed reference frame, can we say that the tumbling object rotate and translate? Rotate about what point? In the case of planar motion and a disk rotating while translating in the air, if we position ourselves in the moving center of mass frame of reference, there is only one point not moving (the center of mass) and everything rotates. If our frame of reference is the body frame of reference, the body is completely at rest and everything else rotate...Anyway, you see where I going with this. I am just wondering if from the fixed reference frame perspective, rotation can be described in different ways, about different rotation axes, for an object tumbling in the air...

 

[jbriggs444]

Is rotation just a relative concept like many others in mechanics? For instance, from the fixed reference frame, can we say that the tumbling object rotate and translate? Rotate about what point?

Any point. For any point you choose, the infinitesimal motion can be described in terms of a translation rate and a rotation rate about the chosen point. But you already know this, so I am having a hard time understanding the question.

 

[fog37]

Thanks jbriggs444. I guess I am looking for some validation since I am still second guessing myself a little bit. I got so used to thinking that there is an absolute axis of rotation about which an object must be rotating that it is hard to break that mold. For example, considering the 2D motion of a disk rotating while translating in the air, the center of mass follows a smooth parabolic trajectory while all other points on the disk have more intricate cycloid-looking paths. So no particle is at rest at any moment in time $t$ or follows nice a circular path/trajectory around a fixed axis of rotation from the perspective of the fixed terrestrial frame. Does that mean that rotation is not happening from the fixed frame of reference?
I read about the instantaneous center of rotation though, which is a point that changes over time. From which reference frame is the object's rotation in terms of the instantaneous rotation axis described from?

 

[jbriggs444]

From which reference frame is the object's rotation in terms of the instantaneous rotation axis described from?

If we restrict ourselves to two dimensions then for any rotating object and for any frame, there will be an instantaneous center of rotation. A common example is a tire rolling on the road.

If we adopt the car frame, the center is at the axle. If we adopt the road frame, the center is at the contact patch. If we adopt the frame of a jet fighter screaming overhead, overtaking the car from behind, the center may be a hundred feet in the air.

In each of these frames the center of rotation is different, the rate at which the tire is translating is different and the rotation rate of the tire is the same. All are equivalent ways of describing the motion. None are more absolute than the rest. None have any claim to be the one true description.

Some descriptions may make it easier to analyze a particular situation than others. We end up using

 

[fog37]

Cool. Thank you.

Would it be correct to say that, even if each situation you described has a different axis of rotation (constant in time, changing in time, etc.) the angular velocity $\omega \neq 0$ in each scenario and that $\omega$ is the same for all the situations?

Would it be correct to think that, every time we don't have pure translational motion, i.e. when all points move with the same velocity at any time $t$, it automatically means that different points have different velocity and this automatically implies that there is some rotation involved?

Just to wrap this topic up, purely from a kinematic standpoint and from the same fixed reference frame, why again is it useful to know that we can geometrically and potentially move the body from one configuration to the next one (finite displacement) using an infinite number of transformations where the rotation occurs about an arbitrary point? I still feel like the instantaneous displacement and how that is achieved is the most relevant aspect...

 

[A.T.]

why again is it useful to know that we can geometrically and potentially move the body from one configuration to the next one (finite displacement) using an infinite number of transformations where the rotation occurs about an arbitrary point?

It is useful to know what can be freely chosen:
- to be able to chose what is convenient for you
- to avoid misinterpreting different choices as different situations

 

[vanhees71]

It is useful to know what can be freely chosen:
- to be able to chose what is convenient for you
- to avoid misinterpreting different choices as different situations

To make it a bit more concrete, let me briefly discuss two cases.

(a) a rigid body allowed to freely rotate around its fixed center of mass ("free top")

Here you chose the body-fixed frame such that its origin is the center of mass and the Cartesian basis vectors along the principles axes of the tensor of inertia. If the top is symmetric (i.e., 2 of the eigenvalues of the tensor of inertia are equal), then you should choose the $3'$-axis of the body-fixed frame in direction of the principal axis, for which the eigenvalue is the one different from the other two. This is because one of the usual Euler angles is a rotation around the body-fixed 3' axis, and due to the symmetry of a symmetric top, this variable becomes cyclic and the corresponding canonical momentum conserved.

The lab frame (i.e., the frame at rest relative to Earth which is to a good enough approximation an inertial frame) then should have the origin also in the center of mass of the top, and if the usual Euler angles are later used to describe the dynamics, it's convenient to orient its Cartesian basis such that the 3-axis points into the direction of the conserved angular momentum of the spinning top. That's because one of the Euler angles is a rotation around the lab-frame 3-axis, and thus it's nice to put the 3-axis in direction of the angular momentum. When treated in Lagrangian mechanics then the corresponding Euler angle becomes cyclic and the canonical momentum conserved (which then of course is nothing than the modulus of the angular-momentum vector).

(b) a rigid body allowed to rotate freely along an arbitrary of its points

Then the body-fixed frame's origin is chosen at the fixed point and the basis vectors along the principle axes of the tensor of inertia. For the symmetric top the qualification discussed for the free top above also hold. The lab frame's 3-axis should now be oriented in direction of the gravitational acceleration $\vec{g}$. Since the problem is symmetric wrt. rotations around this axis, the corresponding lab-frame angular momentum component is conserved, which again is the canonical momentum of the corresponding Euler angle for the rotation around the lab-fixed 3-axis.

Note: The tensor of inertia of course depends on the chosen origin of the body-fixed system. So it's a different tensor for the cases (a) and (b)!