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Mohammad I. Kilani
Dynamics and Vibrations
Kinematics of Rigid Bodies
What is a Rigid Body?
A body is considered to be a rigid body when the
distance between any two points on it remain
unchanged as the body moves.
A result of the above definition is that the angle
between any two lines on the body does not change
as the body moves. This is because the angle between
any two lines can be seen as an angle in a triangle
containing the two lines. Since the lengths of the
sides triangle do not change, the angles do not change
either.
What is a Rigid Body?
Note that a rigid body contains an infinite number of
particles. However, the above observations on rigid body
motion allows determining the position of any point on the
body if the position of one reference point and the angular
orientation of one reference line on it are known.
For example, the figure shows the location of all points on
the body if reference point A is located at the origin and
reference line AB is aligned with the x- axis (orientation = 0
degrees)
ABO
A
B
O
What is a Rigid Body?
In the previous example, three scalar quantities were
needed to fully define the location of a body in the
plane. Those were the x- and y- coordinates of point
A and the angular orientation of line AB.
The degrees of freedom (DOF) for a system is the
number of independent coordinates that need to be
specified to completely describe the configuration of
the system in space. A rigid body in plane motion has
DOF = 3.
ABO
A
B
O
What is a Rigid Body?
The independent coordinates for determining the location of a
rigid body is not unique. For example, one could choose the x-
and y- coordinates of point B and the x- coordinates of point A.
Another choice could be the x- coordinates of point A, the y-
coordinates of point B, and the orientation of line AB.
There is an infinite number of possible choices for the
coordinates needed to determine the location of the body.
However, regardless of the choice, there number is always the
same, and once they are given, the position of any point on the
body can be determined.
ABO
A
B
O
Analytical Determination of the location of points
Knowing the location of the reference point A, and the
orientation of the reference line AB, the location of a point C on
a body can be determined from the relative position equation
below:
In using the above equation, note that the angle γ and the
magnitude of the vector rc/a
does not change as the body
moves.
ACAC rrr
ABO
A
B
O
rA
rC/A
γ
Example:Determination of the location of a point on a rectangle
The rectangle shown is 3 m long and 1 m wide. Determine the
location of point C if point A is located at point (1,1) and the line
AB is oriented at 90 degrees from the x- axis
)4,0(
)3,1()1,1(
)43.108sin10,43.108cos10()1,1(
43.1081043.189010
3
1tan9013
90
122
C
C
C
AC
AC
ACAC
ACAC
r
r
r
r
r
rr
rrr
O
AB
C
A
BC
rA
rC/A γ
O
Example:Determination of the location of a point on a rectangle
The lengths of the sides AB and AC right angle triangle are 2 m and 1.5 m, respectively.
Determine the coordinates of the apex A if point B is located on the y- axis and point C is
on the x-axis, and the line BC makes 45 degrees with the horizontal
77.12
5.2
77.12
5.2
)2
5.2,2
5.2(),0()0,(
)2
5.2,2
5.2(
455.2
455.12 22
B
C
BC
BCBC
BC
BC
BC
BCBC
y
x
yx
rrr
r
r
r
rrr
O
C
O
A B
C
A
B
C
98.1,28.0
)98.1,28.0()77.1,0(),(
)98.1,28.0(
87.812
2
5.1tan452 1
yx
yx
rrr
r
r
r
rrr
AA
BABA
BA
BA
BA
BABA
Example:Determination of the location of a point on a rectangle
The lengths of the sides AB and AC right angle triangle are 2 m and 1.5 m, respectively.
Determine the coordinates of the apex A if point B is located on the x- axis and point C is on
the y-axis, and the line AB makes 45 degrees with the horizontal
O
AC
O
A B
C
A
B
C
77.12
5.2
77.12
5.2
)2
5.2,2
5.2()0,(),0(
)2
5.2,2
5.2(
225.2
2255.12 22
B
C
BC
BCBC
BC
BC
BC
BCBC
x
y
xy
rrr
r
r
r
rrr
98.1,49.1
)98.1,28.0()0,77.1(),(
)98.1,28.0(
87.2612
2
5.1tan2252 1
AA
AA
BABA
BA
BA
BA
BABA
yx
yx
rrr
r
r
r
rrr
TYPES OF MOTION
Three Dimensional Motion
A rigid body free to move within a
reference frame will, in the general
case, have a simultaneous combination
of rotation and translation.
In three-dimensional space, there may
be rotation about any axis and
translation that can be resolved into
components along three axes.
Plane Motion
In a plane, or two-dimensional space, rigid body
motion becomes a combination of simultaneous
rotation about one axis (perpendicular to the
plane) and also translation resolved into
components along two axes in the plane.
Planar motion of a body occurs when all the
particles of a rigid body move along paths which
are equidistant from a fixed plane
Translation
All points on the body describe
parallel (curvilinear or rectilinear)
paths.
A reference line drawn on a body in
translation changes its linear
position but does not change its
angular orientation.Rectilinear Translation
Curvilinear Translation
Fixed Axis Rotation
The body rotates about one axis that has no motion with
respect to the “stationary” frame of reference. All other
points on the body describe arcs about that axis. A
reference line drawn on the body through the axis
changes only its angular orientation.
When a rigid body rotates about a fixed axis, all the
particles of the body, except those which lie on the axis of
rotation, move along circular paths
General Plane Motion
When a body is subjected to
general plane motion, it
undergoes a combination of
translation and rotation, The
translation occurs within a
reference plane, and the rotation
occurs about an axis
perpendicular to the reference
plane.
DEGREES OF FREEDOM (DOF) OR MOBILITY
Definition of the DOF
The number of degrees of freedom (DOF)
that a system possesses is equal to the
number of independent parameters
(measurements) that are needed to
uniquely define its position in space at any
instant of time.
Note that DOF is defined with respect to a
selected frame of reference.
xA
yA
xB
YB
θB
DOF of a Rigid Body in a 2D Plane
If we constrain the pencil to always remain in the plane of the
paper, three parameters are required to completely define its
position on the paper, two linear coordinates (x, y) to define
the position of any one point on the pencil and one angular
coordinate (θ) to define the angle of the pencil with respect to
the axes.
The minimum number of measurements needed to define its
position is shown in the figure as x, y, and θ. This system o has
three DOF.
DOF of a Rigid Body in a 2D Plane
Note that the particular parameters chosen to define the
position of the pencil are not unique. A number of alternate
set of three parameters could be used.
There is an infinity of sets of parameters possible, but in this
case there must be three parameters per set, such as two
lengths and an angle, to define the system’s position because
a rigid body in plane motion always has three DOF.
DOF of a Rigid Body in 3D Space
If the pencil is allowed to move in a three-dimensional space,
six parameters will be needed to define its position. A
possible set of parameters that could be used is three
coordinates of a selected point, (x, y, z), plus three angles (θ,
φ, ρ).
Any rigid body in a three-dimensional space has six degrees of
freedom. Note that a rigid body is defined as a body that is
incapable of deformation. The distance between any two
points on a rigid body does not change as the body moves.
ϕ
θρ
DOF of Mechanisms
Rotation about a Fixed Axis
Rotation about a Fixed Axis
Since a point is without dimension, it cannot have angular
motion. Only lines or bodies undergo angular motion. For
example, consider the body shown and the angular motion of
a radial line r located within the shaded plane.
At the instant shown, the angular position of r is defined by
the angle θ, measured from a fixed reference line to r
Rotation about a Fixed Axis
Rotation about a Fixed Axis
PPP
PPP
PrPP
PPrPPP
PPPPrPP
OPP
OPOP
rra
rra
erera
errerra
rererererv
vv
vvv
2
2
2
/
/
2
Relative Motion Analysis: Velocity
ABAB
ABAB
/
/
vvv
rrr
Relative Motion Analysis: Acceleration
ABABAB
ABABAB
rABABAB
ABAB
ABAB
ABAB
/2
/
//
//
/
/
/
rrαaa
rωωrαaa
aaaa
aaa
vvv
rrr
Problem 16-2
Problem 16-2
Problem 16-6
Problem 16-6
Problem 16-9
Problem 16-9
Problem 16-18
Problem 16-18
Problem 16-29
Problem 16-29
Problem 16-29
Problem 16-29
Problem 16-42
Problem 16-42
Problem 16-45
Problem 16-45