I.2
Fundamentals
I.2.1
Degrees of Freedom and Motion
The
number of degrees of freedom
(DOF) of a system is equal to the number of independent
parameters (measurements) that are needed to uniquely define its position in space at any
instant of time. The number of DOF is defined with respect to a reference frame.
Figure I.2.1 shows a rigid body (RB) lying in a plane. The rigid body is assumed to
be incapable of deformation and the distance between two particles on the rigid body is
constant at any time. If this rigid body always remains in the plane, three parameters (three
DOF) are required to completely define its position: two linear coordinates (
x
,
y
) to define
the position of any one point on the rigid body, and one angular coordinate
θ
to define the
angle of the body with respect to the axes. The minimum number of measurements needed
to define its position are shown in the figure as
x
,
y
, and
θ
. A rigid body in a plane then has
three degrees of freedom. Note that the particular parameters chosen to define its position
are not unique. Any alternative set of three parameters could be used. There is an infinity
of sets of parameters possible, but in this case there must always be three parameters per
set, such as two lengths and an angle, to define the position because a rigid body in plane
motion has three DOF.
Sixparametersareneededtodefinethepositionofafreerigidbodyinathreedimensional
(3D) space. One possible set of parameters which could be used are three lengths, (
x
,
y
,
z
),
plus three angles (
θ
x
,
θ
y
,
θ
z
). Any free rigid body in 3D space has six degrees of freedom.
A rigid body free to move in a reference frame will, in the general case, have complex
motion, which is simultaneously a combination of rotation and translation. For simplicity,
only the twodimensional (2D) or planar case will be presented. For planar motion the
following terms will be defined, Figure I.2.2:
•
Pure rotation in which the body possesses one point (center of rotation) which has no
motion with respect to a “fixed” reference frame [Fig. I.2.2(a)]. All other points on the
body describe arcs about that center.
•
Pure translation in which all points on the body describe parallel paths [Fig. I.2.2(b)].
51
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rigid body
(RB)
x
y
θ
X
Y
Z
FIGURE I.2.1
Rigid body in planar motion with three DOF: translation along the x axis, translation
along the y axis, and rotation,
θ
, about the z.
•
Complex motion that exhibits a simultaneous combination of rotation and translation
[Fig. I.2.2(c)]. With general plane motion, points on the body will travel nonparallel
paths, and there will be, at every instant, a center of rotation, which will continuously
change location.
Translation and rotation represent independent motions of the body. Each can exist without
the other. For a 2D coordinate system, as shown in Figure I.2.1, the
x
and
y
terms represent
the translation components of motion, and the
θ
term represents the rotation component.
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 Summer '08
 Gabale,A
 Kinematic Chain, kinematic chains, CEPHA Imaging Pvt, Machine Components Design

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