Page 171 - Robotics Designing the Mechanisms for Automated Machinery

4.5 Damping of Harmful Vibrations 159

FIGURE 4.55 Model of a dynamic damper.

Equation (4.47) proves what was stated earlier, i.e., that when co a = (o the numera-
tor of this expression, and obviously also the amplitude a lt equals 0.
Let us consider a model of free vibration. The model shown in Figure 4.55 is free to
oscillate when no excitation force is applied, i.e., P = 0. For this case the equations for
the mass movements are

The solutions are in the form

To find the natural frequencies we must solve the following biquadratic equation:

The amplitudes are related as follows:

From the latter expression we obtain the condition for the minimum value of 02 in the
form

Obviously, if the models discussed here represent rotational vibrations, the mass
characteristic must be replaced by the moments of inertia and the springs must be
described by their angular stiffness. Rotational vibrations have very important effects
on indexing tables (see Chapter 7), which require some time to come to a complete
rest after every step. An example is shown in Figure 4.56 of a pneumatically driven
indexing table. In case a) the table, which has moment of inertia/! +J 2 when stopped,
comes to rest as illustrated by the acceleration recording shown below. This process