RIGID BODY
KINEMATICS
Instantaneous Center of
Zero Velocity
(Ani Dönme Merkezi)
In relative velocity analysis, we determined the
velocity of a point on a rigid body in plane
motion by adding the relative velocity due to
rotation about a convenient reference point to
the velocity of the reference point.
We are now going to solve the problem by
choosing a unique reference point which
momentarily has zero velocity.
Let’s assume that the body in the figure is in plane
motion. For a certain instant, the velocities of all
particles in the body, will be equal to the velocities
as if the body is rotating to an axis perpendicular to
the plane of motion. This axis is called the
instantaneous axis of zero velocity and the
intersection of this axis with the plane of motion is
known as the instantaneous center of zero velocity
(point C). For this certain instant,
the velocity of point C is
zero.This approach provides
us with a valuable means for
visualizing and analyzing
velocities in plane motion.
Locating the Instantaneous Center
The existence of the instantaneous center can be easily
shown. For the body in the figure, let’s assume that the
directions of the absolute velocities of any two points A
and B on the body are known and are not parallel. If there
is a point about which A has absolute circular motion at
the
 instant considered, this point must lie on the normal to
v A through A.



v A vB
 
rA rB
Similar reasoning applies to B, and the intersection of
the two perpendiculars will give point C, the
instantaneous center of zero velocity. Point C may lie on
or off the body. If it lies off the body, it may be
visualized as lying on an imaginary extension of the body.
The instantaneous center of zero velocity need not be
a fixed point in the body or a fixed point in the
plane.



v A vB
 
rA rB
If the magnitude of the velocity of one of the points is
known, for example vA, the angular velocity  of the body
and the linear velocity of every point in the body may be
obtained. In this case, the angular velocity  of the body
will be
vA

rA
which, of course, is also the angular velocity of every line in
the body.



v A vB
 
rA rB
The velocity of B will be vB = rB = (rB/rA)vA. Once the
instantaneous center is located, the direction of the
instantaneous velocity of every point in the body is readily
found since it must be perpendicular to the radial line
joining the point to C.



v A vB
 
rA rB
If the velocities of two points in a body having plane motion
are parallel, and the line joining the points is perpendicular to
the direction of the velocities, the instantaneous center is
located by direct proportion. If the velocities of points A
and B are parallel and equal in magnitude then the body will
be in translation and the instantaneous center of zero
velocity will approach infinity.
A
B

C in ∞

vA
vA  vB
vB
Motion of the Instantaneous Center
As the body changes its position, the instantaneous center
C also changes its position in space and on the body. The
locus of the instantaneous centers in space is known as the
space centrode, and the locus
of the positions of the
instantaneous centers on the
body is known as the body
centrode. At the instant
considered, the two curves
are tangent at the position of
point C. It can be shown that
the body-centrode curve rolls
on the space-centrode curve
during the motion of the body
as indicated in the figure.
Motion of the Instantaneous Center
Although the velocity of the instantaneous center is zero,
its acceleration may not be equal to zero. Thus, this point
may not be used as an instantaneous center of zero
acceleration.
If two or more bodies are connected by pins, the
instantaneous center (IC) will be determined separately
for each body. In a rotating disk the IC will be the point
of contact of the disk with the surface.
IC for AB
vB
IC for BC
BC
AB
vA
instantaneous
center
Absolute IC
Point O absolute IC
Point O absolute IC
Points O1 and O2 absolute ICs
If the instantaneous center of velocity is fixed
for a certain motion of the body, it can be named
as “absolute IC”.
Relative IC
Point C relative IC
Relative IC in infinity
Point P relative IC
For the position shown, rod AB
translates, AB=0
If the instantaneous center of velocity changes
position for a certain motion of the body, it can
be named as “relative IC”.