Rigid Body Motion

Question

I was studying Rotational Dynamics and came across a statement as follows:

Consider a motion of a rigid body,then there always exist a reference frame A such that there always exist a fixed line in that frame (by fixed line, it means the position vector of all the points,w.r.t frame A,lying on the line remains unchanged with time) for which the motion of the body is pure rotation about it.

The exact statement written in the book was not what I wrote above, actually it's my version of the statement,that is what I understood after reading that statement in the book.

I want to know whether my version of the statement is correct or not?

And if it is,then can you please provide me with some proof of the existence of the frame A.

Answer 1

Frame A must have an instantaneous velocity parallel to the rotation axis for the above to be true.

As measured by an inertial observer the general instantaneous motion of a rigid body, whose center of mass at some time frame is located at $\boldsymbol{r}_C$ and translates with $\boldsymbol{v}_C$, as well as rotates about the center of mass with $\boldsymbol{\omega}$, is that of a of the rotation $\boldsymbol{\omega}$ about a reference frame A with origin at $\boldsymbol{r}_A$ and velocity parallel to the rotation $\boldsymbol{v}_A = h\,\boldsymbol{\omega}$. The scalar factor $h$ is called the pitch and it has units of length.

As seen by the reference frame A the body is only rotating by $\boldsymbol{\omega}$, but the frame isn't fixed in space.

The location $\boldsymbol{r}_A$ and pitch $h$ can be used to reconstruct the motion of the center of mass

$$ \boldsymbol{v}_C =\underbrace{ h \boldsymbol{\omega} }_{\boldsymbol{v}_A} + \boldsymbol{\omega} \times \underbrace{ (\boldsymbol{r}_C - \boldsymbol{r}_A) }_{\boldsymbol{c}} \tag{1}$$

The frame A parameters $h$ and $\boldsymbol{r}_A$ are found from the motion of the center of mass $\boldsymbol{v}_C$ and the rotation $\boldsymbol{\omega}$ using (2) and (3)

$$ \boldsymbol{\omega} \cdot \boldsymbol{v}_C = h\,(\boldsymbol{\omega}\cdot\boldsymbol{\omega}) $$ $$ \Rightarrow \;\; h = \frac{\boldsymbol{\omega}\cdot \boldsymbol{v}_C}{\| \boldsymbol{\omega} \|^2} \tag{2} $$

$$\require{cancel} \boldsymbol{\omega} \times \boldsymbol{v}_C = \boldsymbol{\omega} \times ( \boldsymbol{\omega} \times \boldsymbol{c}) = \boldsymbol{\omega} ( \cancel{\boldsymbol{\omega} \cdot \boldsymbol{c}}) - \boldsymbol{c} ( \boldsymbol{\omega} \cdot \boldsymbol{\omega}) $$ $$ \Rightarrow \;\; \boldsymbol{r}_A = \boldsymbol{r}_C + \frac{ \boldsymbol{\omega} \times \boldsymbol{v}_C}{ \| \boldsymbol{\omega} \|^2} \tag{3} $$

_{The condition that $\boldsymbol{\omega} \cdot \boldsymbol{c} =0$ means that $\boldsymbol{r}_A$ is the point on the rotation axis closest to the center of mass C.}

On in the special cases where $\boldsymbol{v}_C=0$ or $\boldsymbol{v}_C \parallel \boldsymbol{\omega}$ the motion of frame A is zero.

To summarize, the general motion of a rigid body is that of an instantaneous screw with rotation about an axis and a parallel translation along the axis.