Page 96 - Dynamics of Mechanical Systems

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4

Kinematics of a Rigid Body

4.1 Introduction
The majority of machine elements can be modeled as rigid bodies. The same is true for
mechanical systems in general. Therefore, in this chapter we will review the kinematics
of rigid bodies. Although the motion of many machines and mechanical systems is
restricted to a plane, we will initially develop a general (three-dimensional) analysis.
Application with planar motion will then be simple, direct, and well founded. In the course
of our discussions we will also develop several useful expressions regarding the relative
motion of particles and rigid bodies.

4.2 Orientation of Rigid Bodies
Consider a rigid body B moving in a reference frame R as in Figure 4.2.1. Let n , n , n 3
1
2
and N , N , N be mutually perpendicular dextral unit vector sets in B and R as shown.
1
2
3
Then, the orientation of B in R can be defined in terms of the relative inclinations of the
unit vector sets. To explore and develop this, recall from Section 2.11 that a transformation
matrix S between the unit vector sets has the elements:
⋅
S = Nn (4.2.1)
ij i j
Recall that S has the convenient property of being orthogonal. That is, the inverse and
transpose are equal:

S = S T (4.2.2)
−1
Recall further that the inclinations of the n relative to the N can be described in terms of
j
i
three angles α, β, and γ, defined as follows: Let the n be mutually aligned with the N . i
j
Then, three successive dextral rotations of B about n , n , and n through the angles α, β,
2
3
1
and γ bring the n (and, hence, B itself) into a general inclination relative to the N (and,
j
i
hence, relative to R). In this procedure, the transformation matrix S may be expressed as
(see Eqs. (2.11.18) and (2.11.19)):

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