What Is the Instantaneous Center of Rotation and How Is It Chosen?

[fog37]

Hello,

A force couple is compose of two forces of equal magnitude, opposite direction and parallel lines of actions separated by a distance $d$. The moment due to a force couple is called a pure moment because its value does not depend on the point about which the moment is computed. The force couple can be moved anywhere and its value and physical effect on the body will remain the same.

A force couple only causes a rotation of the rigid body (not a translation). About which point $P$ does the body rotation happen? Is the center of mass? If so, why? Is there a mathematical derivation? In the past, I realized that the concept of rotation is somewhat arbitrary in the sense that we can see an object rotate about different points (for ex., a free rigid body rotating/spinning in the air does not objectively rotate about the CM; it is just a mathematical convenience).

Thank you,
Fog37

 

[A.T.]

A force couple only causes a rotation of the rigid body (not a translation).

No translation of the center of mass, assuming it was initially at rest.

About which point $P$ does the body rotation happen? Is the center of mass? If so, why?

Since the linear impulse remains zero, the center of mass doesn't move.

 

[fog37]

Thank you A.T. !

I understand: when only a pure couple acts on a rigid body at rest, the $CM$ does not move and the body rotates exactly about the $CM$.

To expand, let's consider a rigid body initially at rest and a single force $F$ applied to point $P$ on the body whose line of action does not pass through the $CM$. In virtue of some important theorem, we can "move" the force $F$ so that its point of application becomes the $CM$ as long as we also introduce a force couple producing a pure moment $M$.

Result: the body translates and rotates about the $CM$ simultaneously. However, if we moved the force $F$ to a point $Q \neq CM$, the magnitude of the added pure moment will be different. Will the body rotation still occur about the $CM$ or not?

 

[vanhees71]

To describe a rigid body (of course only in non-relativistic physics ;-)) you need 6 degrees of freedom: The location of one body-fixed reference point (usually but not necessarily inside the body) in some given (inertial frame of reference) and the relative orientation of a body-fixed reference frame with its origin in the reference point. This describes the 3 translational degrees of freedom (location of the body-fixed reference point) and the 3 degrees of freedom of the intrinsic rotation around the reference point (i.e., the relative orientation of the body-fixed Cartesian basis to the space-fixed Cartesian basis in terms of, e.g., Euler angles). In principle the choice of the body-fixed reference point and the body-fixed Cartesian basis is arbitrary. The most convenient body-fixed Cartesian basis is given by the principle axes of the intertia tensor of the body around the body-fixed reference point.

In many cicumstances, however the description becomes simpler: If the body is freely falling in the constant gravitational field of the Earth, it's most convenient to introduce the center-of-mass point as the body-fixed reference point and the body-fixed Cartesian basis as the principle axis of the tensor of inertia around this center-mass reference point. Then it turns out that the center of mass is just behaving like a freely falling point mass and the intrinsic rotation around this reference point is described by Euler's equations of a free top.

 

[A.T.]

...the $CM$ does not move and the body rotates exactly about the $CM$...

...the body translates and rotates about the $CM$ simultaneously...

You have to decide what "rotates about X" means to you. In the first case, the CM remains stationary relative to the body and relative to the inertial frame. In the second case (non-zero net force) there might be no such constant point, just instantaneous ones.

If you are talking about non-stationary rotation centers, then there are many decompositions into rotation and translation possible.

 

[fog37]

To describe a rigid body (of course only in non-relativistic physics ;-)) you need 6 degrees of freedom: The location of one body-fixed reference point (usually but not necessarily inside the body) in some given (inertial frame of reference) and the relative orientation of a body-fixed reference frame with its origin in the reference point. This describes the 3 translational degrees of freedom (location of the body-fixed reference point) and the 3 degrees of freedom of the intrinsic rotation around the reference point (i.e., the relative orientation of the body-fixed Cartesian basis to the space-fixed Cartesian basis in terms of, e.g., Euler angles). In principle the choice of the body-fixed reference point and the body-fixed Cartesian basis is arbitrary. The most convenient body-fixed Cartesian basis is given by the principle axes of the intertia tensor of the body around the body-fixed reference point.

In many cicumstances, however the description becomes simpler: If the body is freely falling in the constant gravitational field of the Earth, it's most convenient to introduce the center-of-mass point as the body-fixed reference point and the body-fixed Cartesian basis as the principle axis of the tensor of inertia around this center-mass reference point. Then it turns out that the center of mass is just behaving like a freely falling point mass and the intrinsic rotation around this reference point is described by Euler's equations of a free top.

Thank you vanhees71. I am making progress.

So there are two frames of reference: the first frame $OXYZ$, the lab frame, is inertial. The rigid body is inside that frame. The second frame $O'X'Y'Z'$ is a body-frame that moves with the body. The three axes of $O'X'Y'Z'$ change orientation (i.e. change the three relative angles) w.r.t. the axes of $OXY'$ in time. Is that what rotation means, i.e. the time change of those three angles between the axes of the two reference frames?

If that is what rotation means, does it mean that rotation happens about the point $O'$ belonging to the body frame $O'X'Y'Z'$? But the point $O'$ is arbitrary so rotation becomes arbitrary...Is that what you intend and what A.T. intends when describing rotation ABOUT X?

I am still missing the meaning of the instantaneous center of rotations...What if we stick with the frame $O'X'Y'Z'$? Wouldn't the rotation always be described relative to the point $O'$?

 

[fog37]

If you are talking about non-stationary rotation centers, then there are many decompositions into rotation and translation possible.

I guess the rotation center is not stationary because the body frame moves? Or does the rotation center varies as the body moves inside the inertial frame $OXYZ$?

 

[vanhees71]

Thank you vanhees71. I am making progress.

So there are two frames of reference: the first frame $OXYZ$, the lab frame, is inertial. The rigid body is inside that frame. The second frame $O'X'Y'Z'$ is a body-frame that moves with the body. The three axes of $O'X'Y'Z'$ change orientation (i.e. change the three relative angles) w.r.t. the axes of $OXY'$ in time. Is that what rotation means, i.e. the time change of those three angles between the axes of the two reference frames?

If that is what rotation means, does it mean that rotation happens about the point $O'$ belonging to the body frame $O'X'Y'Z'$? But the point $O'$ is arbitrary so rotation becomes arbitrary...Is that what you intend and what A.T. intends when describing rotation ABOUT X?

I am still missing the meaning of the instantaneous center of rotations...What if we stick with the frame $O'X'Y'Z'$? Wouldn't the rotation always be described relative to the point $O'$?

It's pretty much right now. Note that in general the momentaneous rotation axis changes too, i.e., the relation between the body-fixed and the space-fixed (inertial) basis is something like
$$\vec{e}_j'=\vec{e}_k R_{kj} (\theta,\psi,\phi),$$
where $R_{kj} \in \mathrm{SO}(3)$ and $\theta$, $\psi$, $\phi$ are, e.g., Euler angles.

https://en.wikipedia.org/wiki/Euler_angles

 

[fog37]

Thanks vanhees71 and sorry for beating a seemingly dead horse. I see how the 3x3 matrix $R$ works with its time dependent components capturing the idea that the body-fixed Cartesian triad changes orientation w.r.t. to the fixed lab Cartesian triad of unit vectors.

a) For a rigid body with a fixed and "imposed" axis of rotation, the points on the said rotational axis are all at rest (relative to the lab reference frame) while all other body points rotate (follow circular paths) around the rotational axis itself. To note that ALL points share the same rotational velocity $\omega$ except for the points sitting on the fixed rotational axis which are not moving.

b) In the case of a free rigid body rotating in the air, all points truly have the same same rotational velocity $\omega$.

So around which rotational axis is the body rotating (my dilemma)? How would you answer this question?

I know that we can identify, at every instant $t$, an axis called instantaneous axis of rotation whose points have the "minimum" velocity (smallest magnitude I guess) such that all other body points not on the axis perform rotation, i.e. follow circular paths, around the instantaneous axis itself. The instantaneous axis can change (or not) from instant to instant. In 2D cases, the instantaneous axis is perpendicular to the plane and its points have zero instantaneous velocity (see disk rolling without slipping).

My confusion: there is surely one instantaneous axis of rotation at each instant of time. But we can also (can we?) say that the body is rotating about other axes (even if they are not the instantaneous), like the axis passing from the CM...So there may be a single unique instantaneous axis...but are we able to talk about rotation around other arbitrary axes?

 

[jbriggs444]

I know that we can identify, at every instant t, an axis called instantaneous axis of rotation

We can only do this by assuming a particular frame of reference. Choose a different frame of reference, get a different instantaneous center of rotation.

That should be a large red flag suggesting that the "axis of rotation" is not a physically meaningful concept.

 

[fog37]

Thanks jbriggs444. Let me think about this.

I guess I am keeping the discussion focused on only two reference frames: the lab, inertial fixed frame (an observer looking at the rigid body rotating in the air) and the body-fixed frame which is glued somewhere on the moving rigid body. The inertial observer on the fixed lab frame sees the body rotate while an observer sitting on the fixed body frame does not see the body rotate (body looks at rest) while the rest of the world rotates instead.

I think the instantaneous axis pertains to the rotation as viewed by the inertial observer who sees the body as rotating...

 

[jbriggs444]

I think the instantaneous axis pertains to the rotation as viewed by the inertial observer who sees the body as rotating...

Which inertial observer? There are lots of inertial frames to choose from.

 

[A.T.]

I think the instantaneous axis pertains to the rotation as viewed by the inertial observer who sees the body as rotating...

The body fixed frame also sees the world rotate around some axis.

 

[fog37]

Hello, the inertial reference frame is planet Earth in my case: imagine an observer at rest watching the rigid body freely spinning/rotating in the air. This observer, located at the origin $O$ of the fame $OXYZ$ states that the body is rotating, i.e. it perceives a change in orientation and I am wondering what rotational axis he/she thinks it is rotating about...

I found this on the internet last night ()
The combined effects of translation of the centre of mass and rotation about an axis through the centre of mass are equivalent to a pure rotation with the same angular speed about an axis passing through a point of zero velocity. Such an axis is called the instantaneous axis of rotation.

In the 2D case of a rolling disk (internet example), the same physical situation, i.e. a wheel rolling, is described in two different ways: a) translation+rotation about the CM and b) pure rotation about the instantaneous axis of rotation are called equivalent. I guess both perspectives are from the same inertial reference frame, an Earth bound observer? What could be a third possible and equivalent description from an Earth bound observer?

 

[jbriggs444]

In the 2D case of a rolling disk, translation+rotation about the CM and pure rotation about the instantaneous axis of rotation are called equivalent. I guess both perspectives are from the same inertial reference frame,the Earth bound observer?

No.

Two different inertial reference frames. One in which the center of mass is not moving and, accordingly, is the instantaneous center of rotation. One in which the rim at the contact patch is not moving and, accordingly, is the instantaneous center of rotation.

 

[fog37]

Mmm...ok...so you are saying that the first description (translation of CM+rotation about CM) pertains to an inertial reference frame centered on the CM. This inertial frame moves with the disk but does not rotate with it. So this inertial frame is indeed a body-frame of reference in the sense that is move with body but it is not a body-fixed frame since it does not rotate rigidly as the body rotates, correct? This frame is different from the body fixed frame $O'X'Y'Z'$ which changes orientation w.r.t. the lab frame.

I guess I am confused about the translation part of the CM. Clearly, if we sit on the CM and place our inertial frame origin on the CM, the CM looks at rest. But what about the CM translation? That translation is perceived from the standpoint of an earthbound observer. Of course, the observer sitting on the CM sees the world translating...

The other 2nd description (IC is the contact point) derives from the Earth bound, lab-reference frame instead.

 

[jbriggs444]

I guess I am confused about the translation part of the CM. Clearly, if we sit on the CM and place our inertial frame origin on the CM, the CM looks at rest. But what about the CM translation? That translation is perceived from the standpoint of an earthbound observer. Of course, the observer sitting on the CM sees the world translating...

If you adopt a reference frame, adopt the reference frame. The earthbound observer is irrelevant. In the frame of reference where the CM is at rest, the CM is at rest. It is not translating.

 

[fog37]

Thank you for the patience.

So, in the spirit of what you said, from the CM reference frame (observer sitting on the CM) the physical motion of the rolling disk would simply be described as a pure rotation (without a translation). But the full motion is generally described as a translation of CM+rotation about CM.My conclusion was that both different descriptions, a) translation of CM+rotation about CM and b) pure rotation about instantaneous contact point, are from the fixed lab reference frame...

 

[vanhees71]

My confusion: there is surely one instantaneous axis of rotation at each instant of time. But we can also (can we?) say that the body is rotating about other axes (even if they are not the instantaneous), like the axis passing from the CM...So there may be a single unique instantaneous axis...but are we able to talk about rotation around other arbitrary axes?

Sure, you can. That's what I tried to say above: You need all 6 degrees of freedom of a (non-relativistic ;-)) rigid body: three translational degrees of freedom, i.e., the location of one arbitrarily chosen point always at rest with respect to the body (often the center of mass is a good choice, but in principle it's arbitrary which point you choose) and three rotational degrees of freedom describing the orientation of the body-fixed cartesian basis relative to the space-fixed cartesian basis, given by the three degrees of a rotation. Usually the Euler angles are convenient descriptions for this, because they are holonomous coordinates.

You can of course describe any rotation by giving a unit vector in the direction of the rotation axis (defined by the right-hand rule) and a rotation angle around this axis. These are non-holonomous coordinates but maybe a bit more intuitive: It's giving the momentary rotation axis and the angle of the rotation around it. If you consider an infinitesimal change during an infinitesimal time interval the corresponding infinitesimal rotation of the body-fixed coordinate axes relative to the space-fixed coordinate axes is described by an (axial) vector, the angular velocity. Of course this depends on the reference point chosen, i.e., if you change your body-fixed reference point also the angular velocity of spin changes. Of course this is just a change of description. The actual motion of the body itself is independent of it.

 

[fog37]

Hello vanhees71 and thank you. Let's see if we can bring this to closure :)

This is a key statement for me: "...three translational degrees of freedom, i.e., the location of one arbitrarily chosen point always at rest with respect to the body (often the center of mass is a good choice, but in principle it's arbitrary which point you choose) and three rotational degrees of freedom describing the orientation of the body-fixed cartesian basis relative to the space-fixed cartesian basis, given by the three degrees of a rotation..."

I am paraphrasing: we can pick any arbitrary point $P$ that is on the body (or not exactly of the body but part of the body extension in space). This point moves at a velocity $v_P$. After that, we need the angular velocity vector $\omega(t)$. But I thought that the vector$\omega$ was common to all points on the body which seems to be in contrast with your other statement "...if you change your body-fixed reference point also the angular velocity of spin changes..." which I probably I did not correctly understand.

So where the rotation axis is located is "arbitrary" and depends on the choice of the reference point $P$.

You also mention that the $CM$ (let's call it point A) is a popular point to use as the instantaneous center of rotation (ICR) but the contact point of the wheel which is momentarily at rest (let's call it point B) is another possible and very popular choice. What advantage does point B offer compared to choosing the $CM$ as rotation center where the rotation axis passes from?

In 2D, we can find a point on the body whose velocity is instantaneously zero. In 3D instead, if the body is moving and rotating, no point has zero velocity (relative to the space fixed frame). The ICR can be chosen as the point with the smallest velocity, like point B in the 2D situation..?

Thanks!