Page 78 - Robot Builders Source Book - Gordon McComb

3.1 Mechanically Driven Bodies 67

FIGURE 3.5 Characteristic of the spring:
force versus deformation.

The force P always acts in the direction opposite to x.
Thus, the movement of the mass m is described by the following equation based
on the Dalamber principle:

This differential homogeneous equation has a simple solution:

where the unknown parameters A, B, and co must be determined. Substituting Expres-
sion (3.15) into Equation (3.14), we obtain

and

The parameter co is known as the natural frequency of the system. To find the unknown
parameters A and B, we have to use the initial conditions of the system. Say, at the
moment t = 0 the deformation of the spring x = x 0 and x = 0. We then substitute these
data into expression (3.15) and obtain directly A = x 0 and B = 0. Thus the complete solu-
tion of Equation (3.14) is

Expression (3.17) is interpreted graphically in Figure 3.6.

To find the time needed to move the mass from the point JC Q to any other point x l
located at a distance L from JC G, we rewrite Expression (3.17) in the following way:

FIGURE 3.6 Displacement versus time for a
TEAM LRN body driven by a spring.