5
Kinematics
of
a rigid
body
in plane motion
5.1 Introduction
A
rigid body is defined as being a system of
particles in which the distance between
any
two
particles is !ked in magnitude. The number of
coordinates required to determine the position
and orientation of a body in plane motion is
three: the system is said to have three degrees of
freedom.
Figure 5.3
Figure 5.4
time tl and t2 then the average angular speed is
Figure 5.1
is shown in Fig. 5*1* The position
Of
Some
representative point such as
A
and the angle
which a line
AB
makes with the xaxis are three
possible coordinates.
5.2 Types of motion
As
(t2t1)+0, (024)+0,
and the angular
The simplest type of motion is that of rectilinear
translation, in which
8
remains constant and
A
moves in a straight line (Fig. 5.2). It follows that
all particles move in lines parallel to the path of
A;
thus the velocity and acceleration of all points
are identical.
One way
Of
describing the position
Of
the body
angle (e2

el
).
If this change takes place between
62
 81
t2

tl


waverage
speed is defined as
A0
de
w
=
limAt+o
=

At
dt
(5.1)
The angular velocity vector is defined as having
a magnitude equal to the angular speed and a
direction perpendicular to the plane of rotation,
the positive sense being given by the usual
righthand screw rule. In the present case,
o
=
wk
(5.2)
It should be noted that infinitessimal rotations
This is still true if
A
is describing a curved path,
and
angular velocity are vector quantities,
whereas finite angular displacement is not.
A
very important point to note is that the
angular speed is not affected by the translation,
therefore we do not have to specify any point in
the plane about which rotation is supposed to be
taking place.
Figure 5.2
since if
8
=
constant all paths are identical in
shape but displaced from each other. This motion
is called curvilinear translation (Fig. 5.3).
If the angle
0
changes during translation, then
this motion is described as general plane motion
(Fig.
5.4).
In Fig. 5.4 the body has rotated by an
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View Full Document5.4 Velocity diagrams
55
5.3 Relative motion between two points
on a rigid body
The definition of the vector product of two
vectors has been already introduced in Chapter
4
in connection with the moment of a vector; the
same definition is useful in expressing the relative
velocity between two points on a rigid body due
to rotation.
to the graphical solution of plane mechanisms are
described in the following sections.
5.4 Velocity diagrams
One very simple yet common mechanism is the
fourbar chain, shown in Fig. 5.6. It is seen that if
the motion of
AB
is given then the motion of the
rest of the mechanism may be determined.
Figure 5.6
This problem can be solved analytically, but the
solution is surprisingly lengthy and is best left to a
computer to solve if a large number of positions
of the niechanism are being examined. However,
a simple solution may be found by using vector
diagrams; this also has an advantage of giving
considerable insight into the behaviour
of
mechanisms.
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 Fall '08
 LGKraige

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