Instant centre of rotation

The instant centre of rotation, also called instantaneous velocity center, or also instantaneous centre or instant centre, is the point fixed to a body undergoing planar movement that has zero velocity at a particular instant of time. At this instant, the velocity vectors of the trajectories of other points in the body generate a circular field around this point which is identical to what is generated by a pure rotation.

Planar movement of a body is often described using a plane figure moving in a two-dimensional plane. The instant centre is the point in the moving plane around which all other points are rotating at a specific instant of time.

The continuous movement of a plane has an instant centre for every value of the time parameter. This generates a curve called the moving centrode. The points in the fixed plane corresponding to these instant centres form the fixed centrode.

The generalization of this concept to 3-dimensional space is that of a twist around a screw. The screw has an axis which is a line in 3D space (not necessarily through the origin, and the screw also has a finite pitch (a fixed translation along its axis corresponding to a rotation about the screw axis.

The instant centre can be considered the limiting case of the pole of a planar displacement.

The planar displacement of a body from position 1 to position 2 is defined by the combination of a planar rotation and planar translation. For any planar displacement there is a point in the moving body that is in the same place before and after the displacement. This point is the pole of the planar displacement, and the displacement can be viewed as a rotation around this pole.

Construction for the pole of a planar displacement: First, select two points A and B in the moving body and locate the corresponding points in the two positions; see the illustration. Construct the perpendicular bisectors to the two segments A¹A² and B¹B². The intersection P of these two bisectors is the pole of the planar displacement. Notice that A¹ and A² lie on a circle around P. This is true for the corresponding positions of every point in the body.

If the two positions of a body are separated by an instant of time in a planar movement, then the pole of a displacement becomes the instant centre. In this case, the segments constructed between the instantaneous positions of the points A and B become the velocity vectors V_A and V_B. The lines perpendicular to these velocity vectors intersect in the instant centre.

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