109
C H A P T E R 7
Kinematics of Rigid Bodies
7.1 INTRODUCTION
In this chapter the kinematics of rigid bodies are presented. Section 7.1 is concerned with in-
troduction to various important definitions and concepts in the kinematics of rigid bodies. e
instantaneous center of rotation in plane motion is briefly introduced in Section 7.2. More de-
tails presentation and application in motion analysis may be found in textbooks on kinematics
and dynamics of machinery, for example in [1–3]. Position vector in a rotating frame of refer-
ence is presented in Section 7.3. Rate of change of a vector with respect to a rotating frame of
reference is dealt with in Section 7.4. e 3D motion of a point in a rigid body with respect to a
rotating frame is considered in Section 7.5 in which three representative examples are included
to illustrate the steps in the solution.
In a system of particles or rigid bodies, arguably the most fundamental concept is the
degrees-of-freedom (dof) in spaces. By the dof of a system or rigid body one means the minimum
number of independent coordinates that are required to completely describe the position of the
system. It is independent of the coordinate system adopted in a particular situation. For example,
if the system in a Cartesian coordinate system has 6 dof then it has the same number of dof in
a spherical coordinate system. For a rigid body in 3D space, there are 3 translational dof, and 3
rotational dof, as shown in Figure 7.1. For a rigid body in a 2D space there are 2 translational
dof and 1 rotational dof.
A rigid body may, in general, experiences either one or more of the various types of motion
to be included in the following.
(a) Translation is the motion in which any straight line inside the body maintains the same
direction, as shown in Figure 7.2a. Clearly, all particles in the rigid body move in parallel
paths. However, if these paths are curved lines such motion is called curvilinear translation,
as shown in Figure 7.2b.
(b) Rotation about a fixed axis is the motion in which the particles of the rigid body move in
parallel planes along the circles centered on the same fixed axis, which is called the axis of
rotation, as indicated in Figure 7.3.
(c) General plane motion is one which is neither a rotation nor a translation, as shown in Fig-
ure 7.4.
110 7. KINEMATICS OF RIGID BODIES
Yaw
Sway
Pitch
Roll
Surge
Heave
Z
Y
X
Figure 7.1: Ship floating in a sea.
A
1
B
1
A
2
B
2
A
1
B
1
A
2
B
2
Rigid body at
final position
Rigid body at
star
ting position
Rigid body at
final position
Rigid body at
starting position
(a) Rectiliniar Translation (b) Curviliniar Translation
Figure 7.2: (a) Rectilinear translation and (b) curvilinear translation.
(d) Motion about a fixed point is one that can better be understood through the Euler’s theo-
rem [4] which states that if a rigid body has one point fixed then any motion of that body
can be reduced to a simple angular displacement about a single axis through the point. A
classical example of this type of motion is the top on a rough floor, as shown in Figure 7.5.
It may be appropriate to note that if the component rotations in Euler’s theorem are finite
it is important that the order in which they are applied is kept. For infinitesimally small
rotations the order is not important. In other words, for infinitesimally small rotations
one can represent these rotations as vectors. On the hand, if these rotations are finite they
cannot be represented as vectors. However, they can be represented as the so-called pseudo-
vectors [5]. e latter quantities have important engineering applications and implications
in more advanced dynamics analysis which is beyond the scope of the present book and
therefore, will not be considered here.
7.1. INTRODUCTION 111
Axis of Rotation
A
B
Figure 7.3: Rotation about a fixed axis.
A
1
A
1
A
2
B
2
B
1
A
2
(a) Sliding Rod (b) Rolling Wheel
Figure 7.4: General plane motion: (a) sliding rod in a mechanism and (b) rolling wheel.
(e) General motion is the type which does not belong to any of those presented in the foregoing.
According to Charles’ theorem [4] the most general displacement (motion) of a rigid body
may be reduced to that of a translation, followed by a rotation.
Before leaving this section, the rotation of a rigid body about a fixed axis, and equations
defining the rotation of a rigid body about a fixed axis should be introduced for completeness.
First, consider the rotation of the rigid body about a fixed axis, as shown in Figure 7.6 in
which the fixed frame of reference OXYZ is centered at the origin O, P is a point of the body
and
*
r is its position vector w.r.t. frame OXYZ, and B is called the projection of P on the fixed
axis AA
0
. Note that the distance BP is constant such that when the rigid body rotates point P
describes a circle whose radius is equal to r sin ', where ' is the angle between the position
vector
*
r and AA
0
. Note further that the position of P and the entire body is completely defined
112 7. KINEMATICS OF RIGID BODIES
Top in Motion
Rough Floor
Figure 7.5: Top on a rough floor.
z
x
y
r
P
O
A
'
A
B
θ
φ
A
B
P
O
A'
θ
φ
j
⃑
i
⃑
r
⃑
⃑
k
⃑
ω⃑ = θ
̇
k
⃑
(a) (b)
Figure 7.6: (a) Rotation of rigid body about a fixed axis and (b) velocity vector.
by the angle which is known as the angular coordinate of the body. e angle is expressed in
radians and the right-hand screw rule is applied here.
With reference to the definition for a particle in Chapter 2, one has the velocity of the
point P as
*
v D
d
*
r
dt
whose magnitude v D
ds
dt
D r sin '
d
dt
. It should be noted that the angle
depends on the position of P. However, the rate of change
d
dt
D
P
is itself independent of P .
With reference to Figure 7.6 or recall from vector analysis that the magnitude of the velocity
given above is precisely the vector cross-product of two vectors,
*
! and
*
r : at is,
*
v D
d
*
r
dt
D
*
!
*
r ;
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