General Motion of a Free Rigid Body

General motion of a free rigid body is the most general case of rigid-body kinematics, with no fixed point or fixed axis. It can be represented as translation of an arbitrarily selected reference point combined with spherical motion of the body relative to that point.

Decomposition of general motion

Body position is specified by the position of a selected reference point $A$ and the orientation of the body relative to the fixed coordinate system. The selected point affects the translational part of the description, but the instantaneous angular velocity $\vec\omega$ of the body does not depend on that choice.

Velocity of an arbitrary point

In general three-dimensional rigid-body motion, the velocity of point $B$ relative to an arbitrarily selected reference point $A$ is:

$$\vec v_B=\vec v_A+\vec\omega\times\vec r_{B/A}.$$

The first term describes translation with the reference point, while the second describes the instantaneous rotational motion of $B$ relative to that point.

This relation is the three-dimensional generalization of the velocity relation for points of a plane rigid body.

Acceleration of an arbitrary point

For two points $A$ and $B$ of the same rigid body:

$$\vec a_B=\vec a_A+\vec\varepsilon\times\vec r_{B/A}+\vec\omega\times(\vec\omega\times\vec r_{B/A}).$$

The second term is associated with angular acceleration and the third with instantaneous angular velocity.

Choosing the reference point

The reference point is chosen for convenience, often as the center of mass, a point with known motion, or a geometrically significant point of a mechanism. The physical motion of the body does not change with the choice.

Relation to special motions

If $\vec\omega=0$, general motion reduces to translation. If the selected point is fixed, it becomes spherical motion. Plane motion is a special case in which point trajectories lie in parallel planes and $\vec\omega$ has a fixed direction perpendicular to those planes.

Instantaneous screw motion

In the general spatial case, the rigid-body velocity field can be interpreted as an instantaneous screw motion: rotation about an instantaneous axis combined with translation along that axis. This generalizes the concept of an instantaneous rotation axis.

Example

If reference point $A$ has velocity $\vec v_A$ and the position of point $B$ relative to it is known, $B$’s velocity is obtained by adding $\vec v_A$ to $\vec\omega\times\vec r_{B/A}$. Even when $A$ is instantaneously at rest, other points may have nonzero velocities because of the rotational term.

Common mistakes

• treating general motion as merely translation of the center of mass;
• omitting the rotational term $\vec\omega\times\vec r_{B/A}$;
• assuming $\vec\omega$ depends on the selected reference point;
• using planar scalar direction rules instead of full three-dimensional vector analysis;
• confusing the instantaneous screw representation with the trajectory of a particular material point.