Page 144 - Dynamics of Mechanical Systems

0593_C05_fm Page 125 Monday, May 6, 2002 2:15 PM

5

Planar Motion of Rigid Bodies — Methods of Analysis

5.1 Introduction
In this chapter, we consider planar motion — an important special case of the kinematics
of rigid bodies. Planar motion characterizes the movement of the vast majority of machine
elements and mechanisms. When a body has planar motion, the description of that motion
is greatly simplified. Special methods of analysis can be used that provide insight not
usually obtained in three-dimensional analyses. We begin our study with a general dis-
cussion of coordinates, constraints, and degrees of freedom. We then consider the planar
motion of a body and the special methods of analysis that are applicable.

5.2 Coordinates, Constraints, Degrees of Freedom
In our discussion, we will use the term coordinate to refer to a parameter locating a particle
or to a parameter defining the orientation of a body. In this sense, a coordinate is similar
to the measurement used in elementary mathematics to locate a point or to orientate a
line. We can bridge the difference between mathematical and physical objects by simply
identifying particles with points and bodies with line segments.
Coordinates are not unique. For example, a point P in a plane is commonly located
either by Cartesian coordinates or polar coordinates as shown in Figure 5.2.1.
In Cartesian coordinates, P is located by distances (x, y) to coordinate axes. In polar
coordinates, P is located by the distance r to the origin (or pole) and by the inclination θ
of the line connecting P with the pole. In both coordinate systems, two independent
parameters are needed to locate P.
If P is free to move in the plane, the values of the coordinates will change as P moves.
Because these changes can occur independently for each coordinate, P is said to have two
degrees of freedom. If, however, P is restricted in its movement so that it must remain on,
say, a curve C, then P is said to be constrained. The coordinates of P are then no longer
independent.
Suppose, for example, that P is required to move on a circle C, centered at O with radius
a, as in Figure 5.2.2. The coordinates of P are then restricted by a constraint equation. Using
Cartesian coordinates, the constraint equation is:

x + y = a 2 (5.2.1)
2
2

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