Page 79 - Robotics Designing the Mechanisms for Automated Machinery

68 Dynamic Analysis of Drives

In a more realistic approach, we must consider a resisting force acting on the mass
m during its motion, as shown in Figure 3.4b). Since the nature of the force can vary,
so can its analytic description. For example, if it is caused by dry friction, the force may
be described analytically in the form

This graphic interpretation of Equation (3.19) is given in Figure 3.7. The movement of
mass m can be described by

which can be replaced by a system of equations in the form

2
Here // = \F\/m and co = c/m.
2
Substituting k = fj./co , in Equations (3.21), we obtain

It is convenient to transform these equations multiplying them by 2 x and integrating
them into the following form:

The value R is an integration constant which must be defined for every change of sgnx.
This form of interpretation permits us to express the behavior of the mass in the
terms of the phase plane which is shown in Figure 3.8. The oscillating movement of
the mass ceases at the moment when R n =£ 2kco. In our case, the spring moves the mass
from a point x = x 0 through a distance L to a point x = x^ In accordance with the diagram
given in Figure 3.8, the value R equals cox 0 - cok. This enables us to rewrite the first of
the two Equations (3.23) in the following way:

FIGURE 3.7 Force developed by dry friction versus
speed of the body.